Sea Ice in Civil Engineering Applications

Abstract

This chapter is devoted to the behaviour of sea ice on civil engineering length and time scales. Several problems of the interaction between a coherent sea ice cover and an engineering structure are analysed. First, the problem of elastic response of floating ice during its short-time interaction (measured in seconds) with a rigid vertical structure is analysed, with the aim to evaluate horizontal forces that are exerted by ice on the structure during an elastic buckling failure of a floating ice plate under compressive and bending loadings. Next, ice–structure interaction events lasting for hours and days are investigated, in which the deformations of ice are dominated by its creep. Thus, the mechanism of creep buckling of a floating ice plate is analysed, with the purpose to estimate the magnitudes of forces acting on the structure until the time at which the flexural failure of the ice cover occurs. This s followed by the analysis of a dynamic impact of floating ice on a rigid structure, during which the floating ice behaves in a typically brittle manner.

In civil engineering applications, in which an interaction between a structure and a floating ice cover occurs, the main interest is in the evaluation of forces which the ice exerts on the structure, with the particular importance of their maximum magnitudes, since these determine design loads for a given engineering object. As already discussed in Chap. 3, a variety of deformation mechanisms can be observed in ice, depending on stress, strain and strain-rate to which the material is subjected. Typically, at the beginning of an ice–structure interaction process, elastic strains develop in ice, but these are small in magnitude compared to other modes of deformation. When the ice is in good contact with the structure (is frozen to its walls), so that the forces (induced by winds and water currents) which drive the ice change slowly in time, and when the sea waves are small, then the material deforms in a continuous way, by creep. If, however, the stresses in ice are large, and/or the sea wave action breaks the ice cover, then the ice–structure interaction has a dynamic character, and the ice behaves in a typical brittle manner.

In what follows in this chapter, the three above-mentioned types of the ice mechanical behaviour, that is, the elastic, creep and brittle responses of the material, will be analysed. Hence, several problems of the interaction between a coherent floating ice cover and an engineering structure are discussed. First, in Sect. 4.1, the problem of purely elastic response of ice during its short-time (measured in seconds) interaction with a rigid vertical structure is investigated, with the aim to evaluate the magnitudes of the maximum horizontal forces exerted on the structure; these forces are assumed to be those which lead to an elastic buckling failure of an ice plate under compressive and transverse loadings. Then, in Sect. 4.2, quasi-static ice–structure interaction events lasting for hours and days are investigated, in which the deformations of ice are dominated by its creep. Hence, rheological models describing the creep of sea ice are first discussed, and these models are used to analyse the mechanism of creep buckling of a floating ice sheet, and then the interaction problems involving rigid structures of rectangular and cylindrical planar cross-sections are considered. Further, on the basis of the results of numerical simulations, the effects of different sea ice rheologies on the predicted magnitudes of the ice–structure interaction forces are examined. Finally, Sect. 4.3 is devoted to the problem of a dynamic impact of an ice floe on a structure, during which the ice behaves in a brittle manner.

4.1 Elastic Interaction of Ice with a Rigid Wall

The purely elastic behaviour of ice is rarely observed in floating ice, which is due to the fact that at typical stress levels in sea ice (1–$5\,{\mathrm {MPa}}$), creep strains overtake elastic ones within a period of seconds after the application of loading forces. Despite this, the problem of evaluating the forces in ice during the very short period of its elastic response is of practical importance, since these forces may reach the values which exceed the magnitudes of forces in ice at later, creep or fracture, modes of deformation. Hence, the elastic response forces need to be taken into account when determining design loads on an engineering structure.

It is a common assumption (Sanderson 1988) that the maximum values of the elastic response forces exerted on a structure are bounded by the magnitude of a force which is required to cause an elastic buckling of the part of an ice cover which directly interacts with the structure. Usually, the floating ice sheet that interacts with a structure is supposed to have, in the horizontal plane, a shape of a truncated wedge of a finite or semi-infinite length. Such a geometry reflects the conditions frequently occurring in the field, when radial cracks propagating from the vertical edges of the structure develop, bounding thus the domain of the ice cover which effectively interacts with the structure.

When analysing the behaviour of coherent floating ice, it is usually assumed that the ice cover can be treated as a continuous plate. The problem of the elastic buckling of a wedge-shaped plate on an elastic foundation (the underlying water) has been investigated in a number of papers, for instance, by Kerr (1978) , Nevel (1980) and Sanderson (1988) , in which approximate estimates for the buckling forces, derived analytically, are given. Some relevant analytical results can be also found in the work by Kerr and Palmer (1972) , and experimental data on the elastic buckling of ice have been reported by Sodhi et al. (1983) . The results presented here have been obtained by the author Staroszczyk (2002) by applying a finite-element method (FEM). These results are compared with those predicted by approximate solutions proposed by Kerr (1978) , and it will be shown that the approximations of the latter author lead to a significant overestimation of the buckling forces that a wedge-shaped elastic plate can sustain. Moreover, some inconsistency in the analytical results by Kerr (1978) has been discovered. Therefore, by fitting to the FEM results obtained, new approximate formulae enabling simple, but reliable, calculations of the buckling forces in floating wedge-shaped elastic plates have been proposed for the use by engineers.

4.1.1 Interaction Problem Formulation

The behaviour of a continuous ice cover floating on the free surface of water is considered. The ice is assumed to be driven by horizontal drag forces due to the wind and water currents action. As the ice cover deforms, not only in the horizontal direction but also transversely, it undergoes vertical loading resulting from the reaction of the underlying water. In this analysis, the ice cover is treated as an elastic plate that floats on a liquid foundation, and is subject to the combined action of the in-plane as well transverse (out-of-plane) forces. The definitions of the internal forces (axial and shear forces and bending moments) and external loadings, together with the adopted frame of rectangular coordinates, are shown in Fig. 4.1a. The plate of ice is assumed to be of uniform thickness, denoted by h, and to be in perfect contact with the underlying water (that is, there are no air pockets between the ice and the water), see Fig. 4.1b. Due to the possible variation of the ice porosity and temperature with depth, the mechanical properties (such as the Young modulus and the ice viscosity) may change across the ice cover (usually, the ice is weaker near its base). The consequence of this is the plate inhomogeneity along the vertical direction, which implies that, in general, the neutral and middle surfaces in the plate do not coincide.

Fig. 4.1

a Adopted coordinate system and definitions of internal forces acting on a plate element, b and c vertical cross-sections of the plate

The vertical z-axis, directed downwards, is chosen in such a way that $z=0$ at the top surface of the plate, and $z=h$ at its bottom surface. The plate transverse displacement along the z-axis is denoted by w. The ice is assumed to be in contact with an engineering structure and to exert forces on it. For simplicity, the structure is modelled as a rigid body which interacts with the floating ice along vertical walls. The prime objective is to evaluate the forces which the wind-driven ice exerts on the structure. In particular, the magnitude of the horizontal compressive force in ice under which the ice plate buckles in an elastic manner will be determined, as this buckling force defines the maximum horizontal load exerted on the structure.

In order to solve the problem, the classical Kirchhoff–Love theory of thin plates (Timoshenko and Woinowsky-Krieger 1959) is applied, which is based on the assumptions that (1) the plate thickness is small compared to its characteristic lengths, (2) the plate deflections are small, that is, not exceeding its thickness, (3) the effects due to shear stresses are neglected, so (4) the plate cross-sections which are normal to the middle plane prior to bending remain plane and normal to the middle surface in the deformed state, and (5) the normal stresses in the direction transverse to the plate surfaces are disregarded.

In the horizontal plane Oxy, the internal loads acting in the plate are the axial forces $N_x$ and $N_y$ (considered positive in tension) and the shear forces $N_{xy}=N_{yx}$, all measured per unit length. Apart from them, there are tangential forces (tractions) acting over the top and bottom surfaces of the plate, caused by the wind stress and water drag. The two components of these forces, defined per unit area of the middle plane of the plate, are denoted by $\tau _x$ and $\tau _y$ (see Fig. 4.1c). The equilibrium balances of the forces along the x and y axes, ignoring inertia forces due to small horizontal velocities of ice, are expressed by

$$\begin{aligned} \frac{\partial N_x}{\partial x}+\frac{\partial N_{xy}}{\partial y}+\tau _x=0, \qquad \frac{\partial N_{xy}}{\partial x}+\frac{\partial N_{y}}{\partial y}+\tau _y=0. \end{aligned}$$

(4.1)

Along the z-axis, the plate is subject to the vertical shear forces, $Q_x$ and $Q_y$, and the transverse distributed load q. In a deformed state, also the forces $N_x$, $N_y$ and $N_{xy}$, all acting in directions tangential to the deflection surface w(x, y), have relevant vertical components. Hence, neglecting the own weight of ice, the projection of all forces on the z-axis direction gives

$$\begin{aligned} \begin{aligned}&\frac{\partial Q_x}{\partial x}+\frac{\partial Q_y}{\partial y}+q+ N_x\frac{\partial ^{2}w}{\partial x^{2}}+\frac{\partial N_x}{\partial x}\frac{\partial w}{\partial x}+ N_y\frac{\partial ^{2}w}{\partial y^{2}}+\frac{\partial N_y}{\partial y}\frac{\partial w}{\partial y}\,+ \\&+\, 2N_{xy}\frac{\partial ^2 w}{\partial x \partial y}+\frac{\partial N_{xy}}{\partial x}\frac{\partial w}{\partial y}+\frac{\partial N_{xy}}{\partial y}\frac{\partial w}{\partial x}=0. \end{aligned} \end{aligned}$$

(4.2)

Considering the equilibrium of moments acting on an infinitesimal plate element with respect to the y and x axes, we find that

$$\begin{aligned} \frac{\partial M_x}{\partial x}-\frac{\partial M_{xy}}{\partial y}-Q_x=0, \qquad \frac{\partial M_y}{\partial y}-\frac{\partial M_{xy}}{\partial x}-Q_y=0, \end{aligned}$$

(4.3)

where $M_x$ and $M_y$ are the bending moments, and $M_{xy}=M_{yx}$ are the twisting moments, all defined per unit length. The only transverse load that is exerted on the plate comes from the reaction of the underlying water, when the plate is either lifted or depressed from its floating equilibrium state. It is assumed that the reaction of the water is purely elastic and is proportional to the plate deflection w (thus, the liquid base can be regarded as the Winkler–Zimmerman-type foundation). Accordingly,

$$\begin{aligned} q=-\varrho _w gw, \end{aligned}$$

(4.4)

where $\varrho _w$ is the density of water and g is the acceleration due to gravity.

It is useful to eliminate the vertical shearing forces $Q_x$ and $Q_y$ from the equilibrium balances (4.2) and (4.3), thus reducing the number of equations to be solved. Accordingly, by inserting in (4.2) the expressions for $Q_x$ and $Q_y$ defined by (4.3), and then using relations (4.1) and (4.4) in the resulting equation, one arrives at the equilibrium equation of the form:

$$\begin{aligned} \begin{aligned}&\frac{\partial ^{2}M_x}{\partial x^{2}}-2\frac{\partial ^2 M_{xy}}{\partial x \partial y}+\frac{\partial ^{2}M_y}{\partial y^{2}}+ N_x\frac{\partial ^{2}w}{\partial x^{2}}+2N_{xy}\frac{\partial ^2 w}{\partial x \partial y}+ N_y\frac{\partial ^{2}w}{\partial y^{2}}\,+ \\&-\,\varrho _w gw-\tau _x\frac{\partial w}{\partial x}-\tau _y\frac{\partial w}{\partial y}=0. \end{aligned} \end{aligned}$$

(4.5)

This equation involves the bending moments and in-plane loads, the plate deflection w and its spatial derivatives, and the driving forces $\tau _x$ and $\tau _y$.

The internal forces in Eq. (4.5) can be expressed in terms of the stresses $\sigma _{xx}$, $\sigma _{yy}$ and $\sigma _{xy}$ which act in the transverse cross-sections of the plate. Hence, the in-plane axial and shear forces are given by

$$\begin{aligned} N_x=\int \limits _0^h\!\sigma _{xx}\mathrm{d}z, \quad N_y=\int \limits _0^h\!\sigma _{yy}\mathrm{d}z, \quad N_{xy}=\int \limits _0^h\!\sigma _{xy}\mathrm{d}z, \end{aligned}$$

(4.6)

and the bending and twisting moments are defined by

$$\begin{aligned} \begin{aligned} M_x&=\int \limits _0^h\!\sigma _{xx} (z-z_0) \mathrm{d}z, \quad M_y=\int \limits _0^h\!\sigma _{yy} (z-z_0) \mathrm{d}z, \\ M_{xy}&=-\int \limits _0^h\!\sigma _{xy} (z-z_0) \mathrm{d}z. \end{aligned} \end{aligned}$$

(4.7)

In relations (4.7), $z_0$ is the position of the neutral surface in the plate in its undeformed state.

To proceed further, one needs specific constitutive laws describing the material response of ice; that is, the equations which relate stresses to strains (in the case of the elastic response), strain-rates (in the case of the creep response), or both (in the case of the viscoelastic response of ice or more complex material responses).

The equilibrium equation (4.5) describes the two-dimensional behaviour of a plate floating on the water surface. In order to solve this equation, even for simple two-dimensional geometries, one has to resort to one of discrete methods, since no analytical solutions are available. As already mentioned, in typical ice–structure interaction phenomena, due to the propagation of cracks in the ice cover, the domain of ice which actively interacts with an engineering object has, to a good approximation, the horizontal shape of a truncated wedge, with the ice which is outside the wedge playing only a passive role. The geometry of the problem is illustrated in Fig. 4.2a, showing the planar view of the wedge-shaped plate being in contact with a flat, vertical wall at $x=0$.

Fig. 4.2

Geometry of a wedge-shaped plate of floating ice interacting with a rigid structure of width $b_0$: a plane view, b plate cross-section, c definition of internal forces

Before solving the elastic plate buckling problem, the latter is simplified by assuming that the plate wedge, defined by the wall width $b_0$ and the angle $\alpha \ge 0$, is symmetric about the x-axis, and extends to infinity. Further, it is also assumed that the forces induced by the wind and water drag action are symmetric about the x-axis as well, so that the plate is pushed towards the structure along the negative direction of the x-axis. Due to the above assumptions, another simplification is introduced, by supposing that the plate deflection w and all the loads and forces acting on the plate are functions of only one horizontal coordinate, x. Thus, $w=w(x)$, $q=q(x)$, $M_x=M_x(x)$, etc., which implies that there is no bending of the plate in the lateral y-direction; that is, the plate is bent cylindrically in the vertical plane Oxz. Accordingly, the problem is in fact reduced to that of a beam of thickness h and variable width b(x) floating on water, and subjected to bending and axial compression (however, the dependence of the plate flexural rigidity on the Poisson ratio, in a form typical of plates, will be retained to account for the constraints in the lateral y-axis direction). In spite of the considerable simplifications, it is believed that the results obtained, at least for small wedge angles $\alpha$, will not differ significantly from those that can be obtained by solving a fully two-dimensional plate problem, and therefore they will prove useful for engineering practice.

Since the problem is treated as a one-dimensional, we omit henceforth the subscripts in the notations of relevant internal forces, as shown in Fig. 4.2c. The equilibrium equation (4.5) now becomes

$$\begin{aligned} \frac{\mathrm{d}^{2}M}{\mathrm{d}x^{2}}+N\,\frac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}=\varrho _w gw + \tau \frac{\mathrm{d}w}{\mathrm{d}x}, \end{aligned}$$

(4.8)

where $\tau =\tau _x$. The plane cross-section assumption implies a linear variation of axial strains $\epsilon _{xx}(x,z)$ and $\epsilon _{yy}(x,z)$ with z, with zero strains at the neutral surface. Supposing that the axial strain $\epsilon _{xx}(x,z)$ across the plate is proportional to the curvature $\kappa _x$ of the middle surface of the deformed plate, one can express $\epsilon _{xx}$ as

$$\begin{aligned} \epsilon _{xx}=\kappa _x (z-z_0). \end{aligned}$$

(4.9)

By Hooke’s law, the axial strains are related to axial stresses by

$$\begin{aligned} \epsilon _{xx}=\frac{1}{E}\,(\sigma _{xx}-\nu \sigma _{yy}), \quad \epsilon _{yy}=\frac{1}{E}\,(\sigma _{yy}-\nu \sigma _{xx}), \end{aligned}$$

(4.10)

in which the Young modulus is a function of z. To ensure the plate continuity in the lateral y-direction, the constrain $\epsilon _{yy}=0$ is introduced. Due to this constraint, relations (4.10) supply the following expression for the axial stress $\sigma _{xx}$ due to bending:

$$\begin{aligned} \sigma _{xx}=\frac{E(z)}{1-\nu ^2}\;\kappa _x(z-z_0), \end{aligned}$$

(4.11)

which shows that the axial stress distribution across the plate is not linear in z because of E being a function of z. Using the latter expression for $\sigma _{xx}$ in the bending moment definition (4.7), on obtains the relation

$$\begin{aligned} M=\kappa _x D_p\,, \end{aligned}$$

(4.12)

in which $D_p$ is the flexural rigidity of the plate defined by

$$\begin{aligned} D_p=\frac{1}{1-\nu ^2}\int \limits _0^h (z-z_0)^2\,E(z)\mathrm{d}z. \end{aligned}$$

(4.13)

The position of the neutral surface, $z_0$, can be determined from the condition that the resultant axial force N due to bending, obtained by integrating over $0 \le z \le h$ the stress $\sigma _{xx}$ defined by (4.11), is zero. This yields

$$\begin{aligned} \int \limits _0^h (z-z_0)\,E(z) \mathrm{d}z = 0. \end{aligned}$$

(4.14)

For practical purposes, it can be assumed that Young’s modulus varies linearly with the depth of ice. Let $E=E_0$ at the upper surface of ice ($z=0$), and $E=\beta E_0$ at the ice base ($z=h$), with $0 \le \beta \le 1$. Then, for the linear variation of E(z) between the two limit values, relation (4.14) gives

$$\begin{aligned} z_0=h\,\frac{1+2\beta }{3(1+\beta )}\,. \end{aligned}$$

(4.15)

With $z_0$ defined by (4.15), the plate flexural rigidity (4.13) becomes

$$\begin{aligned} D_p=\frac{E_0 h^3}{12(1-\nu ^2)} \left[ \frac{1+4\beta +\beta ^2}{3(1+\beta )}\right] . \end{aligned}$$

(4.16)

Obviously, for $\beta =1$ (E is uniform across the plate thickness), (4.15) yields $z_0=h/2$, and the plate rigidity (4.16) is equal to $D=E_0 h^3/[12(1-\nu ^2)]$. On the other hand, for the limit case of $\beta =0$, the neutral surface position is $z_0=h/3$, and the plate flexural rigidity $D_p$ reduces to 1/3 of that for the homogeneous ice plate.

Now, let us return to the plate equilibrium equation (4.8). Due to the small deflection assumption, implying $(dw/dx)^2 \ll 1$, the plate middle surface curvature (considered positive if the deformed plate is convex downward) can be approximated by $\kappa _x=-d^2w/dx^2$. By substituting the latter expression for $\kappa _x$ in the bending moment definition (4.12), then using the resulting expression for M in (4.8), and finally by multiplying both sides of the ensuing equation by the plate width b(x), one arrives at the following fourth-order differential equation for the plate deflection w(x):

$$\begin{aligned} D_p\,b(x)\,\frac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}+P\,\frac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}+\varrho _w gb(x)w=0, \quad 0< x < \infty , \end{aligned}$$

(4.17)

where the varying width of the wedged-shaped plate is defined by

$$\begin{aligned} b(x)=b_0+2x\tan \alpha . \end{aligned}$$

(4.18)

In Eq. (4.17), $P=-Nb$ is the total compressive load acting on the whole cross-section of width b of the plate wedge (see Fig. 4.2). The load P is assumed to be independent of x in the region adjacent to the rigid structure located at $x=0$; that is, in the region in which, for $\alpha > 0$, elastic buckling of the plate is expected to occur. Moreover, in derivation of (4.17), the horizontal traction $\tau$ has been neglected. Such a simplification seems to be permitted, since a typical magnitude of the buckling force P is much larger than the resultant tangential load due to wind and water drag stresses applied over a relatively small area of the ice cover in the immediate vicinity of the structure.

The differential equation (4.17) must be supplemented by boundary conditions at the ice–structure contact area at $x=0$. Two types of these conditions are considered. The first type describes the case of a simply-supported edge of the plate, with zero deflection w and bending moment M, whereas the other type corresponds to the case of a rigidly-supported (clamped) end, with zero deflection and zero slope at $x=0$. According to Sanderson (1988) , the first case of the simply-supported plate at $x=0$ is more realistic in practice, since the perfect contact between floating ice and a structure is rarely observed in the field. These two types of boundary conditions are expressed, respectively, by the relations

$$\begin{aligned}&\text {simply-supported end:} \quad w(0)=0, \quad \frac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}(0)=0, \end{aligned}$$

(4.19)

$$\begin{aligned}&\text {rigidly-supported end:} \quad w(0)=0, \quad \frac{\mathrm{d}w}{\mathrm{d}x}(0)=0. \end{aligned}$$

(4.20)

Besides the boundary conditions (4.19) and (4.20), the regularity condition of the plate deflection being bounded at $x \rightarrow \infty$ has to be satisfied.

4.1.2 Finite-Element Solution of the Problem

The fourth-order differential equation (4.17), supplemented by the boundary conditions, either (4.19) or (4.20), describes an eigenvalue problem from which the buckling force P can be calculated. Because of the presence of the variable coefficient b(x) in (4.17), no exact closed-form analytical solution is available for the general case of a wedge-shaped plate defined by $\alpha > 0$. However, in a particular case of $\alpha =0$, corresponding to the case a plate of uniform width $b(x)=b_0$, Eq. (4.17) simplifies to that with constant coefficients. For such an equation, an analytical solution can be obtained in a straightforward manner, and has the form

$$\begin{aligned} P_0=2b_0\sqrt{\varrho _w g D_p}\,, \end{aligned}$$

(4.21)

which is valid for both simply-supported and clamped boundary conditions at $x=0$ (Kerr 1978) . The corresponding buckling mode, described by $w(x)=\sin (k_0 x)$, yields the buckling half-wave length $L_0$ given by

$$\begin{aligned} L_0=\pi /k_0\,, \quad \text {with} \quad k_0^4={\frac{\varrho _w g}{D_p}}\,. \end{aligned}$$

(4.22)

An attempt to construct a semi-analytical solution of the problem defined by Eqs. (4.17), (4.19) and (4.20) for the case of $\alpha > 0$ was made by Kerr (1978) . However, it has proved that the approximate relations proposed in the latter paper are erroneous (Staroszczyk 2002) , since these relations are inconsistent with the solution (4.21) for $\alpha =0$, and they significantly overestimate the magnitudes of buckling forces for the case of a rigidly-supported plate; this will be demonstrated in Sect. 4.1.4.

In order to solve the eigenvalue problem defined by Eqs. (4.17), (4.19) and (4.20), one can apply a discrete method, for instance, a finite-difference method. It turns out, however, that then the discretization of the problem leads to the necessity of solving a generalized eigenvalue problem for non-symmetric matrices, which is much more difficult to solve numerically than a problem involving symmetric matrices. For this reason, a finite-element method (FEM) is applied, in which case all the matrices resulting from the discretization of the problem are symmetric, which significantly simplifies numerical calculations.

A weighted residual, or Galerkin, method (Zienkiewicz et al. 2005) is employed, in which the problem equation is satisfied in an integral mean sense. Following this method, the plate of variable width b is discretized along the x-axis by introducing one-dimensional finite elements. To ensure the continuity of both the plate deflection curve and its slope between elements, at each discrete nodal point two parameters are used to describe the plate deformations, namely w and $\mathrm{d}w/\mathrm{d}x$. Assuming that a given finite element is defined by two nodes i and j, located at $x_i$ and $x_j$, respectively, we approximate the continuous function w(x) within the element ij by means of four interpolation (shape) functions $\varPhi _r(x)$ ($r=1,\ldots ,4$) as follows:

$$\begin{aligned} w(x)=w_i \varPhi _1 + \theta _i \varPhi _2 + w_j \varPhi _3 +\theta _j \varPhi _4\,, \end{aligned}$$

(4.23)

where $\theta _i=(\mathrm{d}w/\mathrm{d}x)_i$ and $\theta _j=(\mathrm{d}w/\mathrm{d}x)_j$ are the nodal values of the plate slope. Let introduce a dimensionless local coordinate $\xi$ defined by

$$\begin{aligned} \xi =\frac{x-x_c}{a}, \quad x_c=\frac{x_i+x_j}{2}, \quad -1 \le \xi \le 1, \end{aligned}$$

(4.24)

where 2a is the length of the element ij. Then, the adopted shape functions are given by

$$\begin{aligned} \begin{aligned} \varPhi _1&=\frac{1}{4}\,(\xi -1)^2(2+\xi ), \\ \varPhi _3&=\frac{1}{4}\,(\xi +1)^2(2-\xi ), \end{aligned} \quad \begin{aligned} \varPhi _2&=\frac{a}{4}\,(\xi -1)^2(\xi +1), \\ \varPhi _4&=\frac{a}{4}\,(\xi +1)^2(\xi -1). \end{aligned} \end{aligned}$$

(4.25)

By multiplying Eq. (4.17), in turn, by a set of weighting functions, which in the Galerkin method are identical with the interpolation functions $\varPhi _r$, and then integrating the resulting relations over the plate length $x \ge 0$ and applying in the process Green’s theorem (Zienkiewicz et al. 2005) to reduce by one the order of differentiation, one obtains a system of 2N linear algebraic equations, with N being the number of discrete nodes. This system of equations can be expressed in matrix form as

$$\begin{aligned} \left( \varvec{K}+P\varvec{B}+\varvec{C}\right) \varvec{w}=\varvec{O}, \end{aligned}$$

(4.26)

where the vector $\varvec{w}=(w_1,\theta _1,\ldots ,w_i,\theta _i,w_j,\theta _j,\ldots ,w_N,\theta _N)^T$ contains the values of the plate deflections and slopes at all nodal points of the discrete system. The plate stiffness matrix $\varvec{K}$ and the matrices $\varvec{B}$ and $\varvec{C}$ are aggregated from respective element matrices $\varvec{K}^e$, $\varvec{B}^e$ and $\varvec{C}^e$ in a way typical of the finite-element method (Zienkiewicz et al. 2005) . The element matrices, each of the dimension $4 \times 4$, have the entries which for the element ij are defined by the following integrals:

$$\begin{aligned} \begin{aligned} K_{rs}^e&= D_p\int \limits _{x_i}^{x_j}\!b(x)\,\frac{\mathrm{d}^{2} \varPhi _r}{\mathrm{d}x^{2}}\frac{\mathrm{d}^{2} \varPhi _s}{\mathrm{d}x^{2}}\mathrm{d}x, \quad B_{rs}^e=\int \limits _{x_i}^{x_j}\! \varPhi _r\frac{\mathrm{d}^{2} \varPhi _s}{\mathrm{d}x^{2}}\mathrm{d}x, \\ C_{rs}^e&=\varrho _w g\int \limits _{x_i}^{x_j}\!b(x)\, \varPhi _r \varPhi _s\mathrm{d}x, \quad (r,s=1,\ldots ,4; \;\; i,j=1,\ldots ,N), \end{aligned} \end{aligned}$$

(4.27)

in which the shape functions involved are given by (4.25).

Equation (4.26) defines a generalized eigenvalue problem from which the value of the buckling force P, the lowest eigenvalue of the problem, can be calculated, together with the associated eigenvector $\varvec{w}$. To accomplish this, the matrix $\varvec{B}$ is first decomposed into a product of the lower and upper triangular matrices, and then, by matrix inversions and multiplications, the general eigenvalue problem is reduced to a standard problem for a real and symmetric matrix. The latter problem can be solved by using standard numerical tools.

4.1.3 Numerical Simulations

The finite-element model presented in Sect. 4.1.2 was applied to simulate the interaction od a wedge-shaped ice cover with a rigid wall. The model included 200 finite elements of the uniform length for plates thinner than $h=0.2\,{\mathrm {m}}$, and 100 elements otherwise, and the length of each element was assumed to be equal to 3h. Thus, the length of the plate adopted to approximate the behaviour of a semi-infinite ice cover was equal to either 600h or 300h. The material constants were taken to be those pertinent to isotropic granular T1 ice at temperature $-5\,^\circ {\mathrm {C}}$. Hence, on account of relations (3.2) and (3.7), the Young modulus was $E=8.99\,{\mathrm {GPa}}$, and the Poisson ratio was $\nu =0.308$. The water density was assumed to be $\varrho _w=10^3\,{\mathrm {kg\,m}}^{-3}$, and $g=9.81\,{\mathrm {m\,s}}^{-2}$.

The results of simulations illustrating the dependence of the elastic buckling load P on the ice cover thickness h and the wedge angle $\alpha$, for a rigid structure of the width $b_0=10\,{\mathrm {m}}$, are plotted in Fig. 4.3. The solid lines in the figure show the results obtained for a simply-supported edge of the plate at $x=0$, and the dashed lines correspond to the case of a rigidly-supported (clamped) edge. The labels by the curves indicate the ice cover thickness in metres. The values of the buckling forces are normalized by the magnitude of the load $P_0$ defining the buckling force of a parallel-sided plate of width $b_0$ and the respective thickness h, see Eq. (4.21); accordingly, the ratios $P/P_0$ are plotted in the graph.

Fig. 4.3

Dependence of the normalized buckling load $P/P_0$ on the wedge angle $\alpha$ and the ice plate thickness h for simply-supported (solid lines) and rigidly-supported (dashed lines) edge conditions at $x=0$, for a structure width $b_0=10\,{\mathrm {m}}$

Figure 4.4 presents the average pressures, defined by $P/(b_0 h)$, which are exerted on a structure by the floating ice cover. Again, the rigid wall is assumed to be $10\,{\mathrm {m}}$ long. The dependence of contact pressures on the plate geometry described by the angle $\alpha$ for different ice thicknesses h is illustrated for a plate which is simply-supported at its edge at $x=0$. The horizontal dashed-dotted line in the figure indicates a pressure level at which ice fails by crushing. The value of the latter limit pressure, corresponding to the compressive strength of ice (see p. 51) was adopted as $5\,{\mathrm {MPa}}$. Above this value, elastic buckling is unlikely to occur in the ice cover since the ice strength is exceeded earlier than the critical magnitude of compressive load P needed to buckle the ice is reached. The plots in Fig. 4.4 show that, for the range of most realistic wedge angles $30^\circ \lesssim \alpha \lesssim 45^\circ$, elastic buckling can occur only for the ice which is thinner than about $0.2\,{\mathrm {m}}$. In such a case, the average contact stresses at the structure wall are smaller than the ice compressive strength of $5\,{\mathrm {MPa}}$. For thicker ice, in turn, the stresses in ice at its buckling exceed the ice compressive strength. Hence, such thick ice crushes in a brittle manner, and the maximum contact pressures on the wall are equal to $5\,{\mathrm {MPa}}$. When the plate is clamped at the wall, rather than simply-supported, then the limit ice thicknesses h above which no elastic buckling of ice can occur decreases to about $0.12\,{\mathrm {m}}$.

Fig. 4.4

Dependence of the average contact pressure on the wedge angle $\alpha$ and the ice sheet thickness h for simply-supported edge conditions at $x=0$ and a structure width $b_0=10\,{\mathrm {m}}$. The dashed-dotted line indicates the compressive strength of ice

Fig. 4.5

Variation of the average contact pressure with angle $\alpha$ and the parameter $\beta$ for the plate $h=0.2\,{\mathrm {m}}$ thick and $b_0=10\,{\mathrm {m}}$ wide at $x=0$, with a simply-supported edge. The dashed-dotted line indicates the compressive strength of ice

Corresponding to the previous diagram is Fig. 4.5, showing the effect of weakening of ice with its increasing depth on the average contact pressures sustained by a structure. The plots, presenting the predictions for the ice cover of thickness $h=0.2\,{\mathrm {m}}$ and the structure width $b_0=10\,{\mathrm {m}}$, illustrate the dependence of the average contact pressure on the parameter $\beta$, the latter describing a linear variation of Young’s modulus E(z) from its maximum value $E_0$ to $\beta E_0$ between the top and the bottom surfaces of ice, respectively.

Fig. 4.6

Buckling deflection modes for various wedge angles $\alpha$ (given in $\deg$), for the plate $h=0.2\,{\mathrm {m}}$ thick and $b_0=10\,{\mathrm {m}}$ wide at $x=0$, with a simply-supported edge

Figure 4.6 illustrates the shapes of buckling deflection modes for an ice plate of thickness $h=0.2\,{\mathrm {m}}$ and width $b_0=10\,{\mathrm {m}}$, with simply-supported edge conditions at $x=0$. The curves depict the plate deflections w for different wedge angles $\alpha$, expressed in $\deg$. For a plate of uniform width, defined by $\alpha =0$, the fundamental buckling mode is described by the function $w(x)=\sin (k_0 x)$, with $k_0$ given by relation (4.22). For the adopted material constants, the latter parameter determines the buckling half-wave length $L_0=\pi /k_0$ equal to $16.0\,{\mathrm {m}}$. We note that the buckling mode length obtained by solving numerically the eigenvalue problem (4.26) agrees well with the length determined analytically. It can be also observed that for wedge angles $\alpha > 0$, even as small as $5^\circ$, the deflection of the ice plate in buckling attenuates rapidly with the distance x from the structure, showing that elastic buckling of the ice cover can take place only in a small region adjacent to the structure wall at $x=0$. It also confirms that the length of the wedge-shaped plate adopted in the discrete model, equal to minimum 300h (i.e. $60\,{\mathrm {m}}$ for $h=0.2\,{\mathrm {m}}$) seems to be adequate, as the deflections w at the truncated edge of the buckled plate are negligibly small for $\alpha \ge 5^\circ$.

4.1.4 Approximate Analytical Solution

As already noted earlier in this section, Kerr (1978) attempted to construct semi-analytical formulae that enable simple estimations of elastic buckling forces in wedge-shaped floating ice plates for the general case of $\alpha > 0$. The relations derived by Kerr are expressed by

$$\begin{aligned} \begin{aligned}&\text {simply-supported end:}&\quad P&=c_1 k_0 D_p\,(k_0 b_0+2\tan \alpha ),\\&\text {rigidly-supported end:}&\quad P&=c_2 k_0 D_p\,(2k_0 b_0+2\tan \alpha ), \end{aligned} \end{aligned}$$

(4.28)

where $k_0$ is defined in (4.22), and $c_1=5.3$ and $c_2=8$ are constants. It can be easily proved, however, that for the angle $\alpha =0$ (the case of a parallel-sided plate) relations (4.28) give $P/P_0=c_1/4=1.325$ for a simply-supported plate, and $P/P_0=c_2/2=4$ for a rigidly-supported plate, instead of unity in either case. Therefore, the above formulae by Kerr (1978) are apparently erroneous, with a particularly large error occurring in the case of a plate clamped at its edge at $x=0$. For this reason, the finite-element results presented in the previous Sect. 4.1.3 have been used to construct alternative approximate relations to be used to estimate plate buckling forces with accuracy levels that are satisfactory for an engineer. Such an approximate relation, common for both simply and rigidly-supported plate ends, can be expressed in the following dimensionless form (Staroszczyk 2002) :

$$\begin{aligned} \frac{P}{P_0}=1+\left( \frac{h}{h^*}\right) ^{r_1} \left( \frac{b_0}{b_0^*}\right) ^{-r_2} \left( r_3\,\alpha +r_4\,\alpha ^2+r_5\,\alpha ^3\right) , \end{aligned}$$

(4.29)

in which $P_0$ is defined by (4.21), and $r_i$ ($i=1,\ldots ,5$) are coefficients. In Eq. (4.29), two characteristic scales are introduced: $h^*=0.1\,{\mathrm {m}}$ for the plate thickness, and $b_0^*=10\,{\mathrm {m}}$ for the width of a structure interacting with floating ice.

The coefficients $r_i$ have been determined by correlating relation (4.29) with the finite-element results by using the method of least-squares. The correlations have been carried out for ice thicknesses h ranging from 0.05 to $0.5\,{\mathrm {m}}$ and structure widths $b_0$ ranging from 5 to $50\,{\mathrm {m}}$, separately for the two types of boundary conditions at the ice–structure interface, defined by (4.19) and (4.20). The best results have been achieved with the two sets of the coefficients $r_i$ listed in Table 4.1.

Table 4.1 Coefficients $r_i$ for two types of boundary conditions at the edge $x=0$

Fig. 4.7

Comparison of the finite-element results (solid lines) with the estimates given by Eq. (4.29) (dashed lines) for a floating ice sheet $h=0.2\,{\mathrm {m}}$ thick and $b_0=10\,{\mathrm {m}}$ wide, for simply-supported and rigidly-supported edge conditions. Dotted lines show the results predicted by Eq. (4.28) proposed by Kerr (1978)

Figure 4.7 compares the predictions of the finite-element method (solid lines) with the approximations (4.28) (dotted lines) and (4.29) (dashed lines), on an example of the plate $h=0.2\,{\mathrm {m}}$ thick and $b_0=10\,{\mathrm {m}}$ wide. It is immediately seen that, for the whole range of wedge angles $\alpha \le 50^\circ$, the results obtained by Kerr (1978) , and subsequently repeated by Sanderson (1988) , significantly overestimate elastic buckling forces for a plate which is clamped at $x=0$; in the case of a simply-supported plate edge the FEM results and Kerr’s estimates differ by about 20–30%. On the other hand, a good agreement between the results given by the proposed analytical approximation (4.29) and the finite-element results, for both simply-supported and clamped plate boundary conditions, is observed. For the adopted parameters ($h=0.2\,{\mathrm {m}}$ and $b_0=10\,{\mathrm {m}}$), the maximum relative discrepancies between the FEM results and those determined by (4.29) are equal to 2.8% for the simply-supported plate and 4.0% for the clamped plate.

For other combinations of h and $b_0$ than that illustrated in Fig. 4.7 (the ranges $5\,{\mathrm {m}}\le b_0\le 50\,{\mathrm {m}}$, $0.05\,{\mathrm {m}}\le h\le 0.5\,{\mathrm {m}}$, and $0^\circ \le \alpha \le 50^\circ$ have been explored), the maximum relative discrepancies between the FEM and approximate analytical predictions (4.29) are of similar order. As the results presented by Staroszczyk (2002) demonstrate, in the case of narrow plates ($b_0=5\,{\mathrm {m}}$), these discrepancies do not exceed $6.6\%$ for simply-supported plates, and 9.5% for clamped plates. For much wider plates ($b_0=50\,{\mathrm {m}}$), in turn, the maximum relative differences are smaller, and equal to 3.8% and 5.7% for simply-supported and clamped plates, respectively. This shows that the proposed approximate formula (4.29) provides the predictions which can be regarded as sufficiently accurate for civil engineering applications.

4.2 Interaction of Creeping Ice with a Structure

As has already been pointed out on a few occasions in this book, the creep behaviour of sea ice is a dominant mechanism of its deformation in a wide range of stress, strain and strain-rate regimes. This section starts with the discussion of rheological models describing the creep behaviour of ice. Then, creep buckling of a floating ice plate is investigated, and key features of this mechanism are analysed in the context of the elastic buckling phenomenon considered in Sect. 4.1. Further in this section, based on the equations formulated in Sect. 4.1.1, a two-dimensional finite-element model is presented for an ice–structure interaction problem, in which the creep rheology is implemented. This model is applied to calculate the forces exerted by sea ice on a rigid structure of a rectangular planar cross-section; the results are shown in Sect. 4.2.4. Finally, in Sect. 4.2.5, the interaction of sea ice with a cylindrical structure is investigated. Hence, the ice equilibrium equations are formulated, and solved numerically, in cylindrical polar coordinates, and the results obtained for different rheological models for ice are compared to examine their effect on the predicted forces acting on the structure walls.

4.2.1 Rheological Models for Sea Ice

Various forms of the constitutive laws describing the rheology of creeping ice have been formulated, tested in numerical models and verified by field observations. Essentially, these laws include, among others, non-linearly viscous, viscous-plastic and elastic-viscous-plastic rheologies, all constructed with the purpose to describe as well as possible the complex mechanical response of sea ice to loading. It appears that two major classes of the rheological models are of the greatest significance to the sea ice modelling; these are: non-linearly viscous fluid models, which are formally and computationally simpler, but less realistic, and more realistic, and more complex, non-linearly viscous-plastic models, which describe distinct material responses below and above a critical level of a strain-rate invariant. The second class of models also comprises constitutive formulations which describe the failure of ice by other mechanisms than plasticity, for instance by brittle fracture.

The non-linearly viscous fluid models are usually based on a Reiner-Rivlin constitutive equation. The first model of this kind was introduced to the sea ice applications by Smith (1983) , and was followed by Overland and Pease (1988) and a few other authors, including Gray and Morland (1994) , Schulkes et al. (1998) and Morland and Staroszczyk (1998) .

The viscous-plastic rheological models originate from the work by Hibler (1979) . The main idea of these models is to use an elliptic plastic yield curve which restricts permissible stress states in the floating ice pack. For strain-rates below a critical value the stresses in ice are determined by a viscous flow relation, while for those above the critical value the stresses lie on a yield curve. To eliminate some drawbacks of the original formulation, various modifications of the model were subsequently proposed (Ip et al. 1991; Hibler and Ip 1995; Tremblay and Mysak 1997; Hibler 2001) , including one in which non-physical elasticity was introduced (Hunke and Dukowicz 1997) in order to improve the numerical performance of the model. A simplified form of the viscous-plastic rheological model is the cavitating fluid model proposed by Flato and Hibler (1992) . Comparisons between the predictions of the above two classes of constitutive laws are made by Schulkes et al. (1998) , who have applied four different ice rheologies to simulate the behaviour of an ice cover.

In what immediately follows, we focus on the viscous-fluid and viscous-plastic rheologies, since these rheologies will be implemented in numerical models for sea ice, the predictions of which will be discussed in the further part of this chapter, and also later in Chap. 5.

Viscous Fluid Model

Commonly, viscous fluid (VF) rheological models are constructed as forms of a general, frame-indifferent Reiner-Rivlin constitutive law (Chadwick 1999) , which expresses the Cauchy stress tensor $\varvec{\sigma }$ in terms of the strain-rate tensor and its invariants. When applied to the sea ice pack, which is assumed to be stress-free in diverging flow (that is, when adjacent ice floes move away from each other), the Reiner-Rivlin law is expressed as

$$\begin{aligned} \varvec{\sigma }=\left[ \phi _1(\eta ,\gamma )\varvec{I}+\phi _2(\eta ,\gamma )\varvec{D}\right] H(-\eta ). \end{aligned}$$

(4.30)

In the above equation, $\varvec{I}$ is the unit tensor, and $\varvec{D}$ denotes the two-dimensional strain-rate tensor, the components of which are defined in terms of the components $v_i$ of the ice velocity vector $\varvec{v}$ by

$$\begin{aligned} D_{ij}=\frac{1}{2}\left( \frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right) \quad (i,j=1,2). \end{aligned}$$

(4.31)

In (4.30), $\eta$ and $\gamma$ are two invariants of $\varvec{D}$, defined for the two-dimensional deformation by

$$\begin{aligned} \eta ={\text {tr}}\varvec{D}, \quad \gamma ^2=\tfrac{1}{2}{\text {tr}}(\hat{\varvec{D}}^2), \end{aligned}$$

(4.32)

where ${\text {tr}}(\cdot )$ denotes the trace of a tensor, and $\hat{\varvec{D}}$ is the deviatoric strain-rate tensor given by

$$\begin{aligned} \hat{\varvec{D}}=\varvec{D}-\tfrac{1}{2}\eta \varvec{I}. \end{aligned}$$

(4.33)

In strain-rate components, the two invariants, the dilatation-rate $\eta$ and the shear-rate invariant $\gamma$, are expressed by

$$\begin{aligned} \eta =D_{11}+D_{22}, \quad \gamma ^2=D_{12}^2+\tfrac{1}{4}\left( D_{11}-D_{22}\right) ^2. \end{aligned}$$

(4.34)

The invariant $\eta$ is positive in diverging flow, and negative in converging flow.

The function $H(\cdot )$, entering Eq. (4.30), denotes the Heaviside unit step function, which is zero for the negative values of its argument, and is unity for the positive values of the argument; hence,

$$\begin{aligned} H(-\eta )= {\left\{ \begin{array}{ll} 1 &{} \text {for} \quad \eta < 0, \quad \text {i.e. in converging flow}, \\ 0 &{} \text {for} \quad \eta > 0, \quad \text {i.e. in diverging flow}. \end{array}\right. } \end{aligned}$$

(4.35)

In view of the above definition, the constitutive equation (4.30) gives $\varvec{\sigma }=\varvec{0}$ for the diverging flow ($\eta > 0)$.

The two response functions $\phi _1$ and $\phi _2$, appearing in the general flow law (4.30), describe the material behaviour of the medium under applied deformation-rates. In order to express the viscous fluid constitutive law in a more conventional way, that is, in terms of viscosities, let first introduce the deviatoric stress $\varvec{S}$, defined by

$$\begin{aligned} \varvec{S}=\varvec{\sigma }+p\varvec{I}, \quad p=-\tfrac{1}{2}{\text {tr}}\varvec{\sigma }, \end{aligned}$$

(4.36)

where p denotes a mean pressure in ice. Then, by taking the spherical and deviatoric parts of both sides of Eq. (4.30), one can determine the bulk and shear responses of the material, the former response associated with the action of pressure, and the latter with the action of the deviatoric stress (Morland and Staroszczyk 1998; Staroszczyk 2005) . By introducing standard definitions of the bulk and shear viscosities, $\zeta$ and $\mu$ respectively, the viscous behaviour of ice can be described by

$$\begin{aligned} p=-\zeta \eta , \quad \varvec{S}=2\mu \hat{\varvec{D}}, \end{aligned}$$

(4.37)

and the material response functions can then be expressed as

$$\begin{aligned} \phi _1=(\zeta -\mu )\eta , \quad \phi _2=2\mu . \end{aligned}$$

(4.38)

With the above relations, the viscous flow law (4.30) becomes

$$\begin{aligned} \varvec{\sigma }=\left[ (\zeta -\mu )\eta \varvec{I}+2\mu \varvec{D}\right] H(-\eta ). \end{aligned}$$

(4.39)

Note that this is a constitutive law for an ice pack, that is, a material consisting of a large number of floes and, therefore, having little ability to sustain tensile stresses on the scale of a pack. The latter feature is accounted for by the inclusion of the Heaviside function term in (4.39). When only a single, continuous, floe is considered, then this term should be omitted in the law.

Since the viscosities $\mu$ and $\zeta$ are the functions of the current deformation-rate invariants $\eta$ and $\gamma$, Eq. (4.39) represents, in general, a non-linear constitutive law. In a particular case of $\zeta =\mu$, relation (4.39) simplifies to the form

$$\begin{aligned} \varvec{\sigma }=2\mu \varvec{D}H(-\eta ), \end{aligned}$$

(4.40)

with a single viscosity measure. While this is not a realistic rheological model for the sea ice pack, it may have some value for numerical testing (Schulkes et al. 1998; Staroszczyk 2003) .

An example of the rheological model based on the general Reiner-Rivlin viscous fluid law (4.30) is the model proposed by Overland and Pease (1988) , sometimes referred to as the OP-rheology. In this constitutive model, the isotropic stress in ice (described by the response function $\phi _1$) is assumed to depend on the ice thickness h, whereas the deviatoric stress (defined by the function $\phi _2$) depends also on the shear-rate invariant $\gamma$ (that is, there is no dependence on the dilatation-rate $\eta$). This specific form of the rheological law was investigated by Schulkes et al. (1998) , who used it in finite-element simulations of a large sea ice pack behaviour under the action of wind. Another, more complex, form of the non-linearly viscous fluid rheological model was developed by Morland and Staroszczyk (1998) . In their model, possible stress states in sea ice are assumed to lie within an envelope in the principal stress plane, which makes it a little similar to viscous-plastic rheological models considered further in this section.

Viscous-Plastic Model

In the viscous-plastic rheological models, often referred to as the VP-models, introduced to the sea ice dynamics by Hibler (1979) , it is assumed that floating ice has zero tensile strength, and when it is subject to compressive stresses, which occur during converging flow of the ice pack, the ice can behave in two ways, depending on the current rate of its deformation. Below a certain critical level of strain-rate, the ice behaves as viscous fluid, whereas above that critical strain-rate it deforms by plastic yield. The limit stress state in the ice in its plastic flow is defined by a yield curve, the shape of which prescribes admissible stresses in the material. When viewed on the principal stress plane, the stress states on the yield curve occur during plastic flow, while those inside the yield curve occur during viscous flow.

Various shapes of the yield envelopes have been proposed and used in simulations so far; some of them, plotted in principal stress axes $(\sigma _1, \sigma _2)$, with positive values denoting tension, are presented in Fig. 4.8. Usually, an elliptic curve has been adopted (Hibler 1979; Ip et al. 1991) , though other shapes, including tear-drop curves (Rothrock 1975; Morland and Staroszczyk 1998) have also been tried. For comparison, the straight lines, representing the Coulomb-Mohr rheology widely applied for granular media, are also plotted in the figure.

Fig. 4.8

Comparison of different yield curves for viscous-plastic rheology: H—elliptic curve by Hibler (1979) , R—tear-drop curve by Rothrock (1975) , MS—teardrop curve by Morland and Staroszczyk (1998) , CM—Coulomb–Mohr straight lines. P is the compressive strength of ice

In the original formulation of the viscous-plastic model (Hibler 1979) zero-tensile strength of ice was postulated. It turned out, however, that such an assumption gives rise to some problems with numerical stability during simulations, since an arbitrarily small change in the divergence rate through zero results in a large change in the creep response of the medium. For this reason, to remove this source of numerical instability, the original viscous-plastic model has been modified in such a way that a small tensile strength of ice is allowed in diverging flow regime (Staroszczyk 2006) . Hence, two strength parameters: $P_1 > 0$ for compression and $P_2 > 0$ for tension, with $P_1 \gg P_2$, are used in the modified viscous-plastic model to describe the properties of sea ice. In this manner, without introducing any non-physical diffusive terms in the flow equations (which is a common practice in numerical modelling of sea ice), the stability of a numerical method is significantly improved. The adopted elliptic yield curve, plotted in the two-dimensional principal stress space, is illustrated in Fig. 4.9 (the solid line). The tensile stress states are those in the first quadrant (near point B) of the ($\sigma _1,\sigma _2$) plane.

Fig. 4.9

Elliptic yield curve (solid line) with the ice compressive strength $P_1$ (stress point A) and the small tensile strength $P_2$ (point B), and a smaller ellipse (dashed line) describing stress states in viscous flow

The elliptic yield curve presented in Fig. 4.9 is specified by the equation

$$\begin{aligned} F(\sigma _1,\sigma _2)=(\sigma _1+\sigma _2+P_1-P_2)^2+e^2(\sigma _1-\sigma _2)^2-(P_1+P_2)^2=0, \end{aligned}$$

(4.41)

where $\sigma _1$ and $\sigma _2$ are the principal stress components, and $e \ge 1$ defines the ellipse eccentricity (the ratio of the major to the minor axis lengths of the ellipse). In physical terms, e defines the ratio of the maximum shear yield stress in the material to the maximum mean pressure: the larger value of the parameter e, the smaller is the shear resistance of ice, with $e \rightarrow \infty$ describing the cavitating fluid rheology.

Following Hibler (1977) , it is assumed that sea ice during its yield (when the stress lies on the yield curve) obeys a normal flow rule, implying that the principal strain-rate vector is normal to the yield curve $F(\sigma _1,\sigma _2)$. Hence, an associated flow law is applied, expressed in the form:

$$\begin{aligned} D_{ij}=\left. \lambda \,\frac{\partial F(\sigma _{ij})}{\partial \sigma _{ij}}\;\right| _{\,F=0} \,, \quad i,j=1,2, \quad \lambda > 0, \end{aligned}$$

(4.42)

where $\lambda$ is a function of strain-rate. It can be shown (Staroszczyk 2006) that the latter function is given by the relation

$$\begin{aligned} \lambda =\frac{ \Delta }{4(P_1+P_2)}\,, \end{aligned}$$

(4.43)

with

$$\begin{aligned} \Delta ^2=\eta ^2+4\gamma ^2/e^2, \quad \Delta \ge 0. \end{aligned}$$

(4.44)

Since $\Delta$ is a function of two invariants $\eta$ and $\gamma$, already defined by formulae (4.32) or (4.34), it itself is an invariant of the strain-rate tensor $\varvec{D}$ (recall that the dimensionless rheological parameter e is a constant).

Substitution of the definition (4.43) for $\lambda$ into the plastic flow rule (4.42), with $F(\sigma _1,\sigma _2)$ defined by the formula (4.41), gives expressions for the strain-rates in terms of the stresses. Inversion of the latter expressions prescribes the stresses $\sigma _{ij}$ in terms of the strain-rates $D_{ij}$. These relations, when set in the tensor form, give the following frame-indifferent flow law:

$$\begin{aligned} \varvec{\sigma }=2\mu \varvec{D}+\left[ (\zeta -\mu )\eta -\frac{1}{2}(P_1-P_2)\right] \varvec{I}, \end{aligned}$$

(4.45)

where the parameters $\zeta$ and $\mu$ are defined by

$$\begin{aligned} \zeta =\frac{P_1+P_2}{2\Delta }\,, \quad \mu =\frac{\zeta }{e^2}=\frac{P_1+P_2}{2\Delta e^2}\,. \end{aligned}$$

(4.46)

Comparison of the viscous-plastic flow law (4.45) with the viscous fluid flow relation (4.39) shows that the parameters $\zeta$ and $\mu$ can be identified as the bulk and shear viscosities of ice, respectively, both being now functions of the ice strength parameters $P_1$ and $P_2$ and the strain-rate invariant $\Delta$.

The law (4.45), in conjunction with the viscosity definitions (4.46), describes the behaviour of ice during plastic yield. The latter is assumed to occur when the strain-rate invariant $\Delta$ reaches a certain critical value, denoted by $\Delta _c$; that is, plastic deformations take place when $\Delta \ge \Delta _c$. Below the critical value, when $\Delta < \Delta _c$, ice is supposed to undergo viscous deformations, with constant (that is, independent of the current strain-rates) viscosities $\zeta$ and $\mu$. Following Hibler (1979) , the magnitudes of the latter two parameters are set to be equal to the viscosities at the onset of plastic yield. Hence, on account of (4.46), they are defined by

$$\begin{aligned} \zeta _m=\frac{P_1+P_2}{2\Delta _c}\,, \quad \mu _m=\frac{P_1+P_2}{2\Delta _c e^2}\,. \end{aligned}$$

(4.47)

The above parameters $\zeta _m$ and $\mu _m$ can be considered as the upper bounds on the viscosities of sea ice. The critical level $\Delta _c$ of the strain-rate invariant can be inferred from in situ observations of the sea ice behaviour. A typical value is $\Delta _c={\,2\times 10^{-9}\,}\,{\mathrm {s}}^{-1}$ (Hibler 1979; Schulkes et al. 1998) .

When sea ice pack deforms in the viscous regime, which occurs for $\Delta < \Delta _c$, then, following the idea of Ip et al. (1991) , the ice strength parameters $P_1$ and $P_2$ must be scaled down by a factor $\Delta / \Delta _c$ as follows:

$$\begin{aligned} P_1 \rightarrow \frac{ \Delta }{ \Delta _c}\,P_1, \quad P_2 \rightarrow \frac{ \Delta }{ \Delta _c}\,P_2, \quad \Delta < \Delta _c\,, \end{aligned}$$

(4.48)

to avoid a physically unsound response, in which isotropic non-zero stress arises in the ice pack in the absence of any deformation-rate. Therefore, two distinct relations must be used to describe two distinct deformation regimes, plastic and viscous, depending on the current strain-rate magnitude relative to its critical level:

$$\begin{aligned} \varvec{\sigma }= {\left\{ \begin{array}{ll} 2\mu \varvec{D}+\left[ (\zeta -\mu )\eta -\frac{1}{2}(P_1-P_2)\right] \varvec{I}\quad &{} \text {for} \quad \Delta \ge \Delta _c\,,\\ 2\mu _m\varvec{D}+ \left[ (\zeta _m-\mu _m)\eta -\frac{1}{2}\frac{ \Delta }{ \Delta _c}(P_1-P_2)\right] \varvec{I}\quad &{} \text {for} \quad \Delta < \Delta _c\,. \end{array}\right. } \end{aligned}$$

(4.49)

It can be shown that in the viscous flow, when $\Delta < \Delta _c$, the stresses predicted by the second equation (4.49) lie on an ellipse, the centre of which approaches the stress origin, and the major and minor axes decrease monotonically to zero, as $\Delta \rightarrow 0$. One such an ellipse is plotted in Fig. 4.9 (see the dashed line).

The constitutive model represented by Eq. (4.49) has four free parameters: $P_1$ and $P_2$ defining, respectively, compressive and tensile strength of ice, $\Delta _c$ prescribing the critical strain-rate at which plastic yield starts, and e defining, through (4.46) and (4.47), the ratio of the shear to bulk viscosities of ice.

4.2.2 Creep Buckling of Floating Ice

In Sect. 4.1 the mechanism of elastic buckling of a floating ice plate is considered. It has been shown that this failure mechanism is possible to occur only in relatively thin ice sheets, of thicknesses usually not exceeding 0.3–$0.4\,{\mathrm {m}}$, depending on the geometry of the plate and the type of boundary conditions at the ice–structure contact zone (see Fig. 4.4 on p. 71). There is, however, a vast field evidence (Sanderson 1988) that under certain conditions, in particular at very low horizontal velocities of the floating ice cover, ice sheets significantly thicker than $0.5\,{\mathrm {m}}$, and sometimes even more than $1.0\,{\mathrm {m}}$, are also susceptible to the out-of-plane buckling. Typically, during late Arctic spring, when ice becomes softer and undergoes thermal expansion, buckles form in floating ice sheets over the periods of up to several days, until tensile cracks develop at the surface of ice, leading to its gradual failure. Similar buckle features occur when ice is pushed against a vertical structure at very low loading levels.

Certainly, the reason for such a behaviour of floating ice is its creep, which is substantial comparing with other materials encountered in civil engineering. At typical stress levels of $1\,{\mathrm {MPa}}$, the time required for creep strains to exceed elastic strains in ice is about one minute (Mellor 1980; Sanderson 1988) . This clearly indicates that not only elastic, but also, and first of all, creep (viscous) effects in ice must be taken into account to properly determine realistic contact forces between floating ice and an engineering structure. That is, the maximum forces occurring in ice during its creep buckling must be found, as these are the forces which can be exerted by ice on a structure. Relatively little research has been devoted to this topic so far (Sjölind 1985; Sanderson 1988) , and this is limited to the problems of plates of a uniform width.

Here an extension of the analyses carried out in the two latter papers is presented, in which a floating plate having in the horizontal plane a shape of a truncated wedge is considered. This is the same geometry as that adopted in Sect. 4.1 for analysing the mechanism of the elastic buckling of ice (see Fig. 4.2 on p. 64). Accordingly, the same Eqs. (4.1) to (4.7) from Sect. 4.1 are used to describe the equilibrium balances of forces acting on the plate in the horizontal and vertical planes. The significant difference is the constitutive description of the material: instead of the equations of elasticity (4.10), the viscous flow law (4.39) is applied to express stresses in ice in terms of deformation-rates. An important factor now is temperature, since the viscosity of ice is strongly temperature-dependent, making the material mechanically inhomogeneous across the plate thickness.

The equation of equilibrium of the ice sheet floating on water, under the combined action of axial compression and bending, can be derived by the method analogous to that described Sect. 4.1 for the case of the elastic response of ice. A simplified form of this equation for a wedge-shaped plate is obtained by treating the latter as a beam of its width varying with x. For the viscous behaviour of ice, the equilibrium equation has the form (Staroszczyk and Hedzielski 2004) :

$$\begin{aligned} Rb(x)\,\frac{\partial ^{4}\dot{w}}{\partial x^{4}}+P\,\frac{\partial ^{2}w}{\partial x^{2}}+\varrho _w gb(x)w=0, \quad 0< x < \infty , \end{aligned}$$

(4.50)

where $P=-Nb$ is the compressive force acting along the x-axis, and b(x) is the varying plate width in the lateral y-direction. The quantity R describes the flexural viscous behaviour of the ice plate and is defined by

$$\begin{aligned} R=\int \limits _{0}^{h}\zeta _a(z)\,z\,(z-z_0)\mathrm{d}z, \end{aligned}$$

(4.51)

where $\zeta _a=\mu +\zeta$ denotes the axial viscosity of ice, with $z_0$ denoting the position of a neutral plane in the vertical cross-section of the plate. Comparison of Eq. (4.17) on p. 66 with (4.50) shows that they differ only in the first term, in which, instead of the elastic plate rigidity $D_p$, the parameter R is used, and instead of the plate deflection w its time-derivative $\dot{w}$ now appears.

The temperature-dependence of the viscous properties of sea ice is described by the three Eqs. (3.18) to (3.20) on p. 40, defining a dimensionless factor a(T) (Smith and Morland 1981) that is used to scale the viscosities with the ice temperature. For instance, the change of ice temperature from $-1$ to $-5\,^\circ {\mathrm {C}}$ increases the ice viscosity by a factor of about 3.5. Since temperature variations of such a magnitude are quite usual in sea ice due to diurnal (24 h) cycles of heating and cooling, this clearly indicates how substantially the creep properties of ice can change over relatively short time scales. The solution of the heat conduction equation

$$\begin{aligned} \frac{\partial T}{\partial t}=k\,\frac{\partial ^{2}T}{\partial z^{2}}\,, \end{aligned}$$

(4.52)

where $k=1.15\times 10^{-6}\,{\mathrm {m}}^2\,{\mathrm {s}}^{-1}$ is the thermal diffusivity coefficient for ice, shows that a free surface temperature perturbation during 24-h temperature cycles is attenuated by a factor of 10 at a depth of ice of about $0.4\,{\mathrm {m}}$ (more rapid temperature variations decay faster). Hence, it may be assumed that, typically, the daily temperature changes affect only the upper layer of thick sea ice.

Before proceeding further, it is useful to realize an essential difference between the mechanisms of elastic and creep buckling. Elastic buckling occurs instantly after a critical load has been reached, and is followed by unstable failure of ice. In contrast, creep buckling is a rather slow time-dependent process which occurs at any load level, and leads to the failure of ice only if sufficiently large strain-rates (and hence stresses reaching the flexural strength of ice) develop in the medium.

Analytical Results for a Uniform-Width Plate

In general, the solution of the ice plate equilibrium equation (4.50) for the plate deflection w(x, t), with the variable coefficient b(x), is possible only by an approximate method. An analytic solution is possible only in a particular case of a plate of uniform width $b(x)=b_0$, when (4.50) simplifies to the equation with constant coefficients

$$\begin{aligned} Rb_0\,\frac{\partial ^{4}\dot{w}}{\partial x^{4}}+P\,\frac{\partial ^{2}w}{\partial x^{2}}+\varrho _w gb_0 w=0. \end{aligned}$$

(4.53)

Unlike elastic buckling, creep buckling requires an initial perturbation in the plate deflection w; this perturbation will subsequently evolve under applied loading. However, not any initial buckle w(x, 0) will grow with time under a given load level P. In order to prove this, let re-write Eq. (4.53) in the form

$$\begin{aligned} Rb_0\,\frac{\partial ^{4}\dot{w}}{\partial x^{4}}=q(x,t), \end{aligned}$$

(4.54)

where

$$\begin{aligned} q(x,t)=-\left( P\,\frac{\partial ^{2}w}{\partial x^{2}}+\varrho _w gb_0 w\right) . \end{aligned}$$

(4.55)

The expression q(x, t) in (4.54) can be treated as a transverse load depending on the axial force P and the current plate deflection w(x, t). The existing deflection will grow with time only if $q(x,t) > 0$, and, reversely, it will decay with time if $q(x,t)<0$; a stationary state, with w not evolving, occurs for $q(x,t)=0$. Assuming that (4.55) can be solved by the method of separation of variables, and adopting the boundary conditions $w(0,t)=0$ and $\partial ^2 w/\partial x^2 (0,t)=0$ (a simply-supported plate at its edge $x=0$), a general solution for the plate deflection w can be expressed in the form

$$\begin{aligned} w(x,t)=A(t)\sin (\pi x/L), \end{aligned}$$

(4.56)

where A(t) is a time-dependent buckle amplitude, and L is an arbitrary half-wavelength of a buckle. By substituting (4.56) into (4.55), a critical length of a buckle half-wave, denoted by $L_c$, can be determined as:

$$\begin{aligned} L_c=\pi \sqrt{\frac{P}{\varrho _w gb_0}}\,. \end{aligned}$$

(4.57)

This critical length $L_c$ determines the longest buckling half-wave, the amplitude of which can increase with time. Any existing buckles of lengths $L > L_c$ will decrease with time, as long as P is not increasing.

The length of a buckle (for $L < L_c\,$) affects the rate of growth of its current amplitude. It is supposed here that the amplitudes A(t) of creeping buckles increase in an exponential manner, that is

$$\begin{aligned} A(t)=w_0 \exp (t/\tau ), \end{aligned}$$

(4.58)

where $w_0$ is an initial small deflection amplitude of a given buckle, and $\tau$ is a time constant. On inserting relation (4.58) into (4.56), and then substituting the resulting expression for w(x, t) into the differential equation (4.53), the following relation is obtained:

$$\begin{aligned} \frac{1}{\tau }=\frac{1}{R}\left[ \frac{P}{b_0}\Big (\frac{L}{\pi }\Big )^2 -\varrho _w g\Big (\frac{L}{\pi }\Big )^4\right] , \end{aligned}$$

(4.59)

which describes the growth-rate parameter $\tau$ in terms of the buckle length L and the axial load P. From among all possible perturbations of different lengths L, the fastest growing is the one for which $\tau$ attains the minimum value. By differentiating (4.59) with respect to L and setting it to zero, one can find that $\tau$ is minimized for the buckle half-wavelength $L_0$ given by

$$\begin{aligned} L_0=\pi \sqrt{\frac{P}{2\varrho _w gb_0}}\,. \end{aligned}$$

(4.60)

The corresponding growth-rate parameter $\tau$, obtained by substituting (4.60) into (4.59), is expressed by

$$\begin{aligned} \tau _0=\frac{4R\varrho _w gb_0^2}{P^2}\,. \end{aligned}$$

(4.61)

Like the critical buckle length $L_c$, the length $L_0$ of the fastest-growing buckle depends on the load magnitude P, but does not depend on the viscous properties of the ice plate described by the parameter R. As the creep deformation of the plate develops from its initial state with small perturbations of various lengths L, the buckle of the half-wavelength $L_0$, with the largest growth-rate defined by $\tau _0$, gradually becomes the dominant buckling mode.

By comparing expressions (4.57) and (4.60) one can note that, independently of the loading level P, the critical and dominant buckle half-wavelengths remain always at a constant ratio given by

$$\begin{aligned} \frac{L_c}{L_0}=\sqrt{2}\,. \end{aligned}$$

(4.62)

Numerical Results for a Wedge-Shaped Plate

The fourth-order in space, and first-order in time, partial differential equation (4.50), which describes the creep behaviour of a wedge-shaped floating ice sheet in response to the compressive load P, has been solved numerically by applying a finite-element method. Essentially, the same discretization scheme is used as that employed in Sect. 4.1.2 for solving the problem of elastic buckling of a wedge-shaped plate. Hence, linear finite elements are used for the discretization of the plate along the x-axis, with the approximation method described by Eq. (4.23), and the element shape functions given by (4.25). One significant difference is that now, instead of the stiffness matrix $\varvec{K}$ with components depending on the elastic flexural rigidity $D_p$, as defined by Eq. (4.27), a similar in structure damping matrix is formed, with its components depending on the parameter R involving the ice viscosity. The details are omitted here; they will be addressed in the next Sect. 4.2.4, devoted to the finite-element solution of a fully two-dimensional problem of creep behaviour of a floating ice sheet.

In numerical simulations, 400 finite elements of the same length, equal to $1.5\,h$, were used. Thus, the behaviour of a semi-infinite plate was approximated by the plate of the finite length of $600\,h$. The axial viscosity of ice at the melting point was adopted to be $\zeta _a=1\times 10^{11}\,{\mathrm {kg\,m}}^{-1}\,{\mathrm {s}}^{-1}$. The results presented below were obtained for ice temperature equal to $-2\,^\circ {\mathrm {C}}$ at the top surface, and $0\,^\circ {\mathrm {C}}$ at the bottom surface of the plate, with the ice viscosities adjusted accordingly across the plate depth to account for the temperature dependence of creep properties of ice. The elastic constants, Young’s modulus E and Poisson’s ratio $\nu$, were equal to $9.0\,{\mathrm {GPa}}$ and 0.31, respectively (they were needed to evaluate the magnitude of the elastic buckling force, $P_e$, for the plate). The flexural strength of ice was assumed to be $\sigma _f=0.2\,{\mathrm {MPa}}$, corresponding to the ice of about 10% porosity.

The initial perturbed small deflection of the plate was adopted as a sum of twenty harmonic components, given by

$$\begin{aligned} w_0(x)=\sum _{i=1}^{20}\pm w_0^{(i)}\sin \left( \frac{i\pi x}{L}\right) , \end{aligned}$$

(4.63)

where the signs (±) were selected at random, and all the component amplitudes $w_0^{(i)}$ were equal and such that the maximum initial deflection was $w_0=0.001\,{\mathrm {m}}$. L, defining the length of the longest initial perturbation, was chosen to be three times the length $L_0$ of the dominant buckle half-wavelength for a plate of uniform width. In this way, the initial deflection $w_0(x)$ includes two components which are longer than the critical half-wavelength $L_c$ determined by (4.57). In the simulations, the value of the compressive load P exerted on the floating plate was normalized by the magnitude of the force $P_e$ causing elastic buckling of the plate; the latter force was calculated by using the method presented in Sect. 4.1.

Fig. 4.10

Reprinted from Staroszczyk and Hedzielski (2004), Fig. 2, with permission of the Institute of Fundamental Technological Research of the Polish Academy of Sciences

Evolution of the deflection w(x, t) of a uniform-width plate of thickness $h=0.2\,{\mathrm {m}}$ under the axial load $P\!{/}\!P_e=0.1$: a for $t\le 7.5\,{\mathrm {h}}$, b for $t\ge 7.5\,{\mathrm {h}}$. The solid circles in (b) show the results of the analytic solution for the critical time $t=t_c=10.55\,{\mathrm {h}}$.

Figure 4.10 illustrates the time variation of the deflection w(x, t) of the plate of a unit width and the thickness $h=0.2\,{\mathrm {m}}$, subjected to the compressive axial force $P=0.1\,P_e$. The plots show how the plate vertical displacements, plotted at the intervals of $1.25\,{\mathrm {h}}$ (hours), gradually evolve from the initial, random distribution of small perturbations, into a regular pattern which, with increasing time, is more and more dominated by the buckling mode of the length $L_0$ defined by (4.60). The evolution of the plate deflection w(x, t) from its initial state, prescribed by (4.63), is followed up to the time $t_c$, called the critical time, at which the tensile stress at any point in the plate exceeds the value of the ice flexural strength $\sigma _f$ and the plate begins to fail due to the propagation of tensile cracks. The deflection of the plate at the critical time $t=t_c=10.55\,{\mathrm {h}}$ is plotted in Fig. 4.10b by the solid line. We note that the maximum deflections $w(x,t_c)$ at the onset of the plate failure are equal to about h / 2. For comparison, the results of the analytic solution, indicated by the solid circles, are also presented in Fig. 4.10b to demonstrate the accuracy of the finite-element solution.

Figure 4.11 illustrates the effect of the in-plane axial load magnitude $P/P_e$ on the ice displacement at the failure times $t_c$. The results, obtained for the plate of a unit width and the thickness $h=0.2\,{\mathrm {m}}$, show that while the maximum plate deflections $w(x,t_c)$ decrease by a factor of about two with a fourfold increase in the load level, the values of the critical time at which the ice cover starts to fail change with the normalized load very substantially, decreasing by a factor of about 18 for the same, fourfold increase in loading.

Fig. 4.11

Reprinted from Staroszczyk and Hedzielski (2004), Fig. 3, with permission of the Institute of Fundamental Technological Research of the Polish Academy of Sciences

Deflection of a uniform-width plate at the critical time $t=t_c$ (expressed in hours) as a function of the normalized load $P/P_e$, for the ice thickness $h=0.2\,{\mathrm {m}}$.

Fig. 4.12

Reprinted from Staroszczyk and Hedzielski (2004), Fig. 5, with permission of the Institute of Fundamental Technological Research of the Polish Academy of Sciences

Variation of the critical time $t_c$ (given in hours) with the angle $\alpha$ and the normalized axial load $P\!{/}\!P_e$, for the ice thickness $h=0.2\,{\mathrm {m}}$ and the structure width $b_0=10\,{\mathrm {m}}$.

The values of the critical time $t_c$ required to fail a floating ice sheet due to its creep deformation started from initial, small-amplitude imperfections, are plotted in Figs. 4.12 and 4.13 as functions of the angle $\alpha$ defining the in-plane geometry of the truncated wedge (see Fig. 4.2 on p. 64). Fig. 4.12 illustrates, for the structure width $b_0=10\,{\mathrm {m}}$ and the ice cover thickness $h=0.2\,{\mathrm {m}}$ kept constant, the dependence of the critical time $t_c$ on the normalized axial load $P/P_e$ (the corresponding plate deflections for selected ratios $P/P_e$ and $\alpha =0$ are shown in Fig. 4.12).

Figure 4.13 displays, at the constant load $P/P_e=0.1$, the variation of $t_c$ for different plate thicknesses h. One can note that for thinner ice plates the values of the critical time initially slightly increase with the increasing angle $\alpha$, while for thicker ice the values of $t_c$ decrease monotonically with $\alpha$.

Fig. 4.13

Reprinted from Staroszczyk and Hedzielski (2004), Fig. 6. Copyright by the Institute of Fundamental Technological Research of the Polish Academy of Sciences

Variation of the critical time $t_c$ (given in hours) with the angle $\alpha$ and the ice thickness h, for the normalized load $P/P_e=0.1$ and the structure width $b_0=10\,{\mathrm {m}}$.

More results obtained by the discrete method described above and illustrating the mechanism of ice creep buckling can be found in the paper by Staroszczyk and Hedzielski (2004) .

4.2.3 Ice Plate Failure Due to Its Thermal Expansion

In the preceding part of this section it was tacitly assumed that the in-plane axial forces N (and hence P) which caused the creep buckling of floating ice were generated by stresses arising on the ice surface due to the action of wind. Further, it was assumed that the temperature of ice did not change during the process of ice creep; that is, the properties of ice (in the first place its viscosity) which depend on temperature remained constant in time. Now we proceed to the problem in which the forces driving the creep buckling of floating ice are of a different origin—they are caused by the phenomenon of thermal expansion of ice due to its heating at the free surface of the ice cover. In order for such buckling force to develop in ice, the latter must be somehow constrained in the lateral (horizontal) directions. It is assumed here, that the lateral deformation of ice is prohibited by vertical, rigid walls representing elements of an engineering structure. Certainly, of the main interest are then the magnitudes of the forces exerted by ice on the constraining walls, and the evolution of these forces as the ice creep deformation progresses until the instant of the ice failure due to its flexural fracture. The evolution of the thermally-induced forces within the ice plate is not only due to the rise in temperature at the ice top surface, but also due to the vertical heat transfer through the ice cover from its top to the bottom. The latter process results in the variation, in time and space, of the elastic and viscous properties of the material; therefore, the plate of ice cannot be treated as a homogeneous, since its mechanical properties vary with depth.

Thermal Creep Plate Buckling Problem Formulation

The problem under consideration is sketched in Fig. 4.14. As previously, the floating ice cover is idealized by a plate of uniform thickness h. The lateral span of the plate is denoted by L, and is equal to the distance between the two constraining vertical walls at the ends of the plate. It is assumed that the top surface of ice is subjected to the action of varying in time temperature T(t), with the ice at the base ($z=h$) being at the melting point temperature $T_m$ at all time, and $T < T_m$ throughout the ice plate. It is also assumed that at the initial time $t=0$ the plate is stress-free; that is, it is in equilibrium under an initial distribution of temperature in the plate. For simplicity, a plane-strain problem is analysed, so that the ice plate can be treated as a beam of uniform width, with its elastic flexural rigidity adjusted accordingly to account for the zero deformations in the direction normal to the plane Oxz.

Fig. 4.14

Floating ice plate of thickness h and span L constrained by vertical rigid walls at its ends

All the equations describing the ice plate buckling problem are essentially those presented earlier in this section. Again, the viscous behaviour of ice is assumed to obey the Reiner-Rivlin-type constitutive law (4.39) on p. 77 (with the Heaviside function term $H(-\eta )$ omitted, since a single, continuous ice floe, not an ice pack, is now considered). Then, the equilibrium equation (4.53), for a plate of unit width $b_0$, becomes

$$\begin{aligned} R\,\frac{\partial ^{4}\dot{w}}{\partial x^{4}}+P\,\frac{\partial ^{2}w}{\partial x^{2}}+\varrho _w gw=0, \end{aligned}$$

(4.64)

where the parameter R, defining the viscous flexural ‘rigidity’ of the plate, is given by Eq. (4.51) which involves the axial viscosity $\zeta _a=\mu +\zeta$ being strongly sensitive to temperature. The above differential equation for the plate deflection curve w(x, t) is solved with the boundary conditions at the plate edges at $x=0$ and $x=L$ representing the case of a simply-supported plate, as being regarded (Sanderson 1988) as the most realistic conditions encountered in the field. Thus,

$$\begin{aligned} x=0: \;\;w=0, \;\;\frac{\partial ^{2}w}{\partial x^{2}}=0; \quad x=L: \;\;w=0, \;\;\frac{\partial ^{2}w}{\partial x^{2}}=0. \end{aligned}$$

(4.65)

The initial condition for the function w(x, t) is prescribed by assuming that the plate deflection $w_0(x)$ at $t=0$ consists of a number of small-amplitude, harmonic in x perturbations:

$$\begin{aligned} w_0(x)=\sum _{k=1}^{m} \pm A_k^0\sin \left( \frac{k\pi x}{L}\right) , \end{aligned}$$

(4.66)

where the m component initial amplitudes $A_k^0$ and their signs (±) are selected at random. The number of components m is such that the shortest buckle half-wavelength is of order $1\,{\mathrm {m}}$.

Besides the plate deflection evolution Eq. (4.64), also a heat conduction equation (4.52) on p. 83 which governs the evolution of the temperature field T(z, t) in ice must be solved. The solution of that equation provides current vertical distributions of ice temperature, which in turn determine the current vertical distributions of elastic and viscous parameters of ice. The boundary conditions for the temperature field are prescribed at the top, $z=0$, and the bottom, $z=h$, surfaces of the ice plate. At $z=0$ a time-varying temperature distribution T(0, t) is adopted to represent the ice heating conditions, whereas at $z=h$ it is assumed that the temperature of ice is constant and equal to the melting point, that is $T(h,t)=T_m$. Further, it is assumed that at the time $t=0$ the temperature along the ice depth varies linearly between the initial top surface temperature $T_0(0)$ and the bottom surface temperature $T_m$.

In contrast to the problem considered in Sect. 4.2.2 in which the axial force P was treated as independent in time during the ice creep buckling process, now this force is time-dependent, since the elastic response of the constrained ice to the changing temperature field evolves as the ice heating progresses. The axial force P(t) is defined by

$$\begin{aligned} P(t) = \alpha \int \limits _0^h\! \Delta T(z,t)\,\frac{E(z,T)}{1-\nu ^2(z,T)} \mathrm{d}z, \end{aligned}$$

(4.67)

where $\alpha ={\,5.2\times 10^{-5}\,}{\mathrm {K}}^{-1}$ is the thermal expansion coefficient for ice. The quantity $\Delta T(z,t)$ describes the difference between the current local temperature T(z, t) and the initial local temperature $T_0(z)$ at the stable (stress-free) state of the plate. The force P(t) given by (4.67) cannot exceed the force $P_0$ causing the instantaneous elastic buckling of the plate, which is given by Kerr (1978)

$$\begin{aligned} P_0=2\sqrt{\varrho _w g D_p}\,, \end{aligned}$$

(4.68)

where the temperature-dependent plate elastic flexural rigidity $D_p$ is

$$\begin{aligned} D_p=\int \limits _0^h\!(z-z_0)^2 \frac{E(z,T)}{1-\nu ^2(z,T)}\,\mathrm{d}z. \end{aligned}$$

(4.69)

In order to describe the temperature-dependence of the elastic constants, the Young modulus E and the Poisson ratio $\nu$, the relations given in Chap. 3 are used. Hence, relation (3.2) on p. 34 is adopted to describe the function E(T) for granular T1 ice, and relation (3.7) is employed for the function $\nu (T)$. The temperature-dependence of the ice viscous properties, $\mu$ and $\zeta$, is, in turn, described by relations (3.18) to (3.20) on p. 40 due to Morland (1993 , 2001) ; the same temperature scaling is applied to both shear and bulk viscosities. It is assumed here that the ice is porous. The degradation of the elastic properties with increasing ice porosity is described by formulae (3.9) on p. 36 due to Hutter (1983) . Because of the lack of relevant data, it is supposed that the ice viscosity magnitudes $\mu$ and $\zeta$ are reduced for porous ice in the same manner as E; that is, by applying (3.9) with $\mu$ and $\zeta$ replacing E in the formulae. Finally, the weakening effect of ice porosity on the ice flexural strength $\sigma _f$ is taken into account. For this purpose, relation (3.33) on p. 48 proposed by Timco and O’Brien (1994) is applied.

Results of Numerical Simulations

The differential equation (4.64) describing the evolution of the plate deflection surface w(x, t) can be solved by either the analytical method for a uniform-width plate, or by a more general finite-element method for a wedge-shaped plate; both methods are described in the preceding Sect. 4.2.2. An enhanced analytical method is discussed by Staroszczyk (2018) . In the calculations, the following material parameters were adopted: Young’s modulus $E\,(T=0{\,{^\circ }\mathrm{C}})=8.93\,{\mathrm {GPa}}$, Poisson’s ratio $\nu =0.308$, ice porosity $\phi _b=0.05$, ice viscosities $\mu ={\,1\times 10^{11}\,}\,{\mathrm {{Pa}\cdot {s}}}$ and $\zeta ={\,2\times 10^{11}\,}\,{\mathrm {{Pa}\cdot {s}}}$, the ice flexural strength $\sigma _f=0.47\,{\mathrm {MPa}}$ (at $\phi _b=0.05$) and the ice compressive strength $\sigma _c=5\,{\mathrm {MPa}}$. The sea water density was assumed to be $\varrho _w=1020\,{\mathrm {kg\,m}}^{-3}$ and its freezing temperature was $T_m=-1{\,{^\circ }\mathrm{C}}$ (a compromise value between $T_m=0{\,{^\circ }\mathrm{C}}$ for freshwater and $T_m=-1.9{\,{^\circ }\mathrm{C}}$ for ocean water of salinity $35\,{\mathrm {ppt}}$). The ice thickness h and the plate length L were adopted of magnitudes typical of civil engineering applications.

The numerical simulations were run for a series of idealized sinusoidal ice surface temperature scenarios depicted in Fig. 4.15. The ice temperature T at the start of calculations was assumed to be equal to $T_0$, the maximum temperature reached during the daytime heating was $T_{max}$ (supposed to be below the ice melting temperature $T_m$), and the minimum temperature at night was $T_{min}$. The results presented below were obtained for $T_0 \ge -5{\,{^\circ }\mathrm{C}}$ and $\Delta T_{max}=T_{max}-T_0 \le 4{\,{^\circ }\mathrm{C}}$. For such moderate daily temperature increases, the maximum thermally-induced axial elastic force P determined from (4.67) was equal to around 1 / 3 of the magnitude of the elastic buckling force $P_0$ calculated from (4.68).

Fig. 4.15

Time variation of temperature T(t) (time is given in hours) at the ice top surface $z=0$. $T_0$ is the temperature at $t=0$ (the start of simulations), $T_{max}$ is the maximum temperature during the daytime, and $T_{min}$ is the minimum temperature at night

The temperature distribution along the ice plate depth was calculated by solving the heat conduction equation (4.52) by a finite-difference method, with the time integration performed by applying a Crank-Nicolson scheme. An example evolution of temperature depth profiles calculated by solving (4.52) for a plate of thickness $h=0.3\,{\mathrm {m}}$ is illustrated in Fig. 4.16. These profiles were obtained by adopting in the temperature scenario shown in Fig. 4.15 the values $T_0=-5{\,{^\circ }\mathrm{C}}$ and $\Delta T_{max}=4{\,{^\circ }\mathrm{C}}$, and for the day (heating period) lasting for 9 h. The solid lines in the figure illustrate the ice warming phase (increasing temperature period), and the dashed lines illustrate the cooling phase (decreasing temperature period), with the same colours corresponding to the same ice temperature at the upper surface of ice.

Fig. 4.16

Reprinted with permission from Staroszczyk (2018), Fig. 3. Copyright 2018 by Elsevier

Time variation of depth profiles of ice temperature T(z) across a floating plate of thickness $h=0.3\,{\mathrm {m}}$, for an initial temperature at the top surface of the plate $T_0=-5{\,{^\circ }\mathrm{C}}$ and a daily temperature increase $\Delta T_{max}=4{\,{^\circ }\mathrm{C}}$.

The initial perturbed ice plate deflection curve, see Eq. (4.66), which was adopted for the plate of length $L=20\,{\mathrm {m}}$ and thickness $h=0.2\,{\mathrm {m}}$ is shown in Fig. 4.17. This curve was obtained by assuming $m=20$ harmonic components, with the randomly selected component amplitudes scaled in such a way that the maximum initial plate deflection was equal to 1 / 100 of the ice thickness.

Fig. 4.17

Initial perturbed plate deflection curve $w_0(x)$ as a superposition of 20 harmonic components $A_0^{(k)}\,\sin (k\pi x/L$) with random amplitudes, for the plate of thickness $h=0.2\,{\mathrm {m}}$ and length $L=20\,{\mathrm {m}}$

Fig. 4.18

Reprinted with permission from Staroszczyk (2018), Fig. 7. Copyright 2018 by Elsevier

Plate deflection curves w(x) at corresponding failure times $t=t_f$ for different temperature amplitudes $\Delta T_{max}$, for the plate of thickness $h=0.2\,{\mathrm {m}}$ and span $L=20\,{\mathrm {m}}$.

The plots in Fig. 4.18 show the shapes of the plate deflection curves at the failure time for different maximum temperature amplitudes $\Delta T_{max}$. The presented results illustrate the creep response of the plate of length $L=20\,{\mathrm {m}}$ and thickness $h=0.2\,{\mathrm {m}}$, for $\Delta T_{max}=3$, 4 and $5{\,{^\circ }\mathrm{C}}$; smaller values of $\Delta T_{max}$ do not generate sufficiently large stresses needed to fail the ice sheet. One can see that the creep behaviour of the plate, and in particular the length and the shape of the buckles, is similar for all the cases plotted in the figure, with the maximum plate deflections increasing monotonically with increasing temperature amplitudes. The corresponding values of the failure time ranged from about $4.5\,{\mathrm {h}}$ for $\Delta T_{max}=3{\,{^\circ }\mathrm{C}}$ to about $3\,{\mathrm {h}}$ for $\Delta T_{max}=5{\,{^\circ }\mathrm{C}}$, see also the next figure.

Fig. 4.19

Reprinted with permission from Staroszczyk (2018), Fig. 9. Copyright 2018 by Elsevier

Evolution of the horizontal force P(t) exerted by ice on the rigid walls for different values of the maximum temperature increase $\Delta T_{max}$, for the plate of thickness $h=0.2\,{\mathrm {m}}$ and span $L=20\,{\mathrm {m}}$.

The curves displayed in Fig. 4.19 show the dependence of the ice–wall reaction force P(t) on the temperature amplitude $\Delta T_{max}$, for the ice of thickness $h=0.2\,{\mathrm {m}}$. The lines illustrating the evolution of P(t) break at the corresponding ice failure time instants, except the cases $\Delta T_{max}=1$ and $2{\,{^\circ }\mathrm{C}}$ when the ice does not fracture during the heating period $0< t < 9\,{\mathrm {h}}$. As can be expected, an increase in the ice temperature-rate results in the ice failure occurring at earlier time. It can be noted that prior to the failure time the reaction force P increases in a smooth monotonic manner.

Finally, Fig. 4.20 illustrates the dependence of the maximum reaction forces P on the ice plate thickness h and the temperature increase amplitude $\Delta T_{max}$, calculated on the assumption that no plate flexural failure occurs. The results plotted in Fig. 4.20 represent the maximum magnitudes of the compressive forces P which, at given h and $\Delta T_{max}$, can theoretically develop in an ice sheet due to its heating. For safety reasons, the presented maximum forces should be adopted by an engineer as design loads on a structure that can be subjected to the action of floating ice. It is seen that the forces displayed in the figure are approximately proportional to the maximum temperature increase $\Delta T_{max}$. Their increase with the plate thickness is most pronounced for $h \lesssim 0.5\,{\mathrm {m}}$, whereas for thicker ice the thermally-induced axial forces caused by daily temperature variations do not significantly exceed those for $h \sim 0.5\,{\mathrm {m}}$.

Fig. 4.20

Reprinted with permission from Staroszczyk (2018), Fig. 10. Copyright 2018 by Elsevier

Variation of the horizontal compressive force P with the plate thickness h for different temperature increase amplitudes $\Delta T_{max}$, on the assumption that the plate does not fail.

The results of calculations presented above show that a moderate temperature increase by a few degrees Celsius during a period of a few hours can lead to the fracture of a floating ice plate of thickness up to about $0.5\,{\mathrm {m}}$. The thermally-induced forces exerted by floating ice on the walls constraining its lateral deformation can be of the magnitudes of several hundred kN per unit width of the ice plate. For instance, for the ice of thickness $0.5\,{\mathrm {m}}$ and a daily peak temperature increase by $5{\,{^\circ }\mathrm{C}}$, the maximum ice–structure reaction force predicted in the numerical simulations was equal to about $350\,{\mathrm {kN/m}}$. More results illustrating the mechanism of floating ice creep buckling due to the phenomenon of thermal expansion of ice can be found in the paper by Staroszczyk (2018) .

4.2.4 Plane Ice–Structure Interaction Problem

In Sect. 4.2.2, the creep behaviour of a floating ice sheet is analysed under the assumption that the ice sheet has a wedge-shaped geometry in the Oxy plane and the problem is symmetric with respect to the x-axis. This, in fact, simplifies the problem to a one-dimensional in the horizontal plane. Now a fully two-dimensional problem is considered, without introducing any simplifications regarding the planar geometry of an ice sheet.

Governing Equations

The equations of the floating ice plate equilibrium in the horizontal and vertical planes, together with the definitions of internal forces (in-plane axial and shear forces and bending and twisting moments) in terms of stresses have already been formulated in Sect. 4.1.1, see relations (4.1) to (4.7) (also refer to Fig. 4.1 on p. 61). In the case of the creep deformations of ice, the stresses are expressed in terms of strain-rates, see the constitutive relations discussed in Sect. 4.2.1. Here we focus on solving the plane problem in which the creep of ice is described by the viscous fluid rheological model, expressing the stresses in terms of strain-rates by the flow law (4.39) on p. 77. We note, however, that the methodology of constructing a solution for the viscous-plastic rheology, due to the formal similarities between the laws (4.39) and (4.49) on p. 81, is not much different from that described below for the viscous fluid rheology.

When expressed in components in the Cartesian coordinate plane Oxy, the viscous fluid flow law (4.39) takes the form:

$$\begin{aligned} \sigma _{ij}=\left[ (\zeta -\mu )D_{kk}\delta _{ij}+2\mu D_{ij}\right] H(-\eta ) \quad (i,j,k=1,2), \end{aligned}$$

(4.70)

where the summation convention applies for a repeated suffix. Recall that in (4.70) $\zeta$ and $\mu$ denote the bulk and shear viscosities, respectively, and $D_{ij}$ are the components of the two-dimensional strain-rate tensor $\varvec{D}$, see definition (4.31). The subscripts i and j stand for either x and y, with the equivalence $x_1=x$ and $x_2=y$.

Deformations in the ice cover can be expressed as a sum of the deformations in the neutral plane of the plate caused by the forces $N_{ij}$, and these can be regarded as functions of the horizontal coordinates x and y alone, and the deformations due to bending and twisting of the plate, which are functions of the depth z as well. Accordingly, the in-plane strain-rates are determined by using the horizontal velocity components $v_x(x,y)$ and $v_y(x,y)$ in (4.31), while the strain-rates due to bending and twisting of the plate are given in terms of the curvatures and twist of the deflection surface w(x, y) as follows

$$\begin{aligned} \begin{aligned} D_{xx}&= \dot{\kappa }_x (z-z_0)=-\frac{\partial ^{2}\dot{w}}{\partial x^{2}} (z-z_0), \\ D_{yy}&= \dot{\kappa }_y (z-z_0)=-\frac{\partial ^{2}\dot{w}}{\partial y^{2}} (z-z_0), \\ D_{xy}&=-\dot{\kappa }_{xy} (z-z_0)=-\frac{\partial ^2 \dot{w}}{\partial x \partial y} (z-z_0). \end{aligned} \end{aligned}$$

(4.71)

In these equations, $\kappa _x$ and $\kappa _y$ are the curvatures of the deflection surface along the x and y axes, respectively, $\kappa _{xy}$ is the twist with respect to the x and y axes, and $z_0$ is the position of the neutral plane in the undeformed state. With the strain-rates given by (4.31) and (4.71), and the stresses determined through the constitutive law (4.70), the in-plane axial and shear forces (4.6) become

$$\begin{aligned} \begin{aligned} N_x&= (H_1+H_2)\,\frac{\partial v_x}{\partial x}+(H_1-H_2)\,\frac{\partial v_y}{\partial y}, \\ N_y&= (H_1-H_2)\,\frac{\partial v_x}{\partial x}+(H_1+H_2)\,\frac{\partial v_y}{\partial y}, \\ N_{xy}&= H_2\left( \frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x} \right) , \end{aligned} \end{aligned}$$

(4.72)

and the bending and and twisting moments (4.7) are given by

$$\begin{aligned} \begin{aligned} M_x&= -\left[ (R_1+R_2)\frac{\partial ^{2}\dot{w}}{\partial x^{2}}+(R_1-R_2)\frac{\partial ^{2}\dot{w}}{\partial y^{2}} \right] ,\\ M_y&= -\left[ (R_1-R_2)\frac{\partial ^{2}\dot{w}}{\partial x^{2}}+(R_1+R_2)\frac{\partial ^{2}\dot{w}}{\partial y^{2}} \right] ,\\ M_{xy}&= 2R_2\,\frac{\partial ^2 \dot{w}}{\partial x \partial y}\,. \end{aligned} \end{aligned}$$

(4.73)

In the above expressions for the internal forces, the parameters defining the plate viscous properties are given by

$$\begin{aligned} H_1=\int \limits _0^h\!\zeta \mathrm{d}z, \quad H_2=\int \limits _0^h\!\mu \mathrm{d}z, \quad R_1=\int \limits _0^h\!\zeta z(z-z_0) \mathrm{d}z, \quad R_2=\int \limits _0^h\!\mu z(z-z_0) \mathrm{d}z. \end{aligned}$$

(4.74)

For the sake of brevity of the notations, it is tacitly assumed henceforth in this chapter that the Heaviside unit step factor $H(-\eta )$ is included in the viscosity terms $\zeta$ and $\mu$.

By substituting now the definitions (4.73) for the internal moments into the equilibrium relation (4.5), we obtain the following differential equation

$$\begin{aligned} \begin{aligned} R\left( \frac{\partial ^{4}\dot{w}}{\partial x^{4}}+2\frac{\partial ^4 \dot{w}}{\partial x^2 \partial y^2}+\frac{\partial ^{4}\dot{w}}{\partial y^{4}}\right) =&\, N_x\frac{\partial ^{2}w}{\partial x^{2}}+2N_{xy}\frac{\partial ^2 w}{\partial x \partial y}+N_y\frac{\partial ^{2}w}{\partial y^{2}}\,+ \\&-{}\varrho _w gw-\tau _x\frac{\partial w}{\partial x}-\tau _y\frac{\partial w}{\partial y}\,, \end{aligned} \end{aligned}$$

(4.75)

with the definition

$$\begin{aligned} R=R_1+R_2=\int \limits _0^h\!\zeta _a z(z-z_0) \mathrm{d}z, \end{aligned}$$

(4.76)

where $\zeta _a=\mu +\zeta$ is the axial viscosity. Equation (4.75) describes the time and space variation of the plate deflection w in terms of the in-plane forces $N_{ij}$ and the external driving forces $\tau _i$ $(i,j=1,2)$. The forces $N_{ij}$, the functions of the plate horizontal velocities $v_x$ and $v_y$ as given by (4.72), can be determined independently of (4.75) by solving the equations of the in-plane equilibrium (4.1).

The driving forces $\tau _i$ are due to wind and water drag on the top and bottom surfaces of ice. In this work, formulae that are quadratic in velocities are adopted to describe these drag forces in terms of the ice, wind and water velocities. Hence, the surface tractions are expressed in the following forms (Sanderson 1988) :

$$\begin{aligned} \varvec{\tau }_a=C_a\varrho _a(\varvec{u}_a-\varvec{v})|\varvec{u}_a-\varvec{v}|, \quad \varvec{\tau }_w=C_w\varrho _w(\varvec{u}_w-\varvec{v})|\varvec{u}_w-\varvec{v}|, \end{aligned}$$

(4.77)

where $\varrho _a$ and $\varrho _w$ are, respectively, the air and water densities, and $\varvec{u}_a$ and $\varvec{u}_w$ are, respectively, the wind and ocean current velocity vectors. The parameters $C_a$ and $C_w$ in relations (4.77) denote dimensionless wind stress and water drag coefficients . On the basis of the data presented in the literature (Sanderson 1988; Kara et al. 2007; Lu et al. 2011) , the values $C_a={\,2\times 10^{-3}\,}$ and $C_w={\,4\times 10^{-3}\,}$ have been adopted for numerical simulations.

Typical boundary conditions, with which the three Eqs. (4.75) and (4.1) are solved in sea ice applications, are those of a simply-supported plate edge at the contact region with the structure, with zero horizontal velocities in the direction normal to the ice-structure interface (so-called free-slip conditions). These conditions are expressed by

$$\begin{aligned} w=0, \quad M_n=0, \quad \varvec{v}\cdot \varvec{n}=0, \end{aligned}$$

(4.78)

where n is the direction normal to the edge of the plate, defined by the outward unit vector $\varvec{n}$, and $M_n$ is the bending moment acting on the plate section normal to $\varvec{n}$.

Finite-Element Formulation

The system of three differential equations for the ice plate deflection w and the horizontal velocities $v_x$ and $v_y$, given by (4.75) and (4.1) with (4.72), is solved approximately by applying the finite-element method. The weighted residual, or Galerkin, version of the method is employed. The plate is discretized in the horizontal plane Oxy by using a mesh of triangular elements, with the unknown variables defined at the corner nodes. At each discrete node, apart from the two horizontal velocities, $v_x$ and $v_y$, and the plate vertical displacement, w, also the two plate slopes, $\partial w/\partial x$ and $\partial w/\partial y$, are treated as unknown variables. Such an approach is typical of the plate theory and is applied in order to ensure the continuity of the plate deflection surface along the sides of the adjacent elements (Zienkiewicz and Taylor 2005) . Thus, there are five discrete parameters to be calculated at each node, so altogether there are 15 degrees of freedom per each triangular element. The continuous functions w, $v_x$ and $v_y$ are approximated by the following representations:

$$\begin{aligned} \begin{aligned} w(x,y,t)&= \Phi _j^w(x,y) w_{j}(t), \quad (j=1,\ldots ,9), \\ v_i(x,y,t)&= \Phi _j^v(x,y) v_{ij}(t), \quad (i=1,2;\; j=1,2,3), \end{aligned} \end{aligned}$$

(4.79)

where $w_j$ and $v_{ij}$ are the unknown nodal parameters, the displacements and the velocities respectively, with the former including both the plate deflections and the plate slopes. $\Phi _j^w$ and $\Phi _j^v$ are shape functions, which are different for the displacement and the velocity fields. While the velocity field is interpolated by simple linear shape functions, for the plate deflection approximation fourth-order polynomials in both x and y are used, following the formulation due to Specht (1988) .

By applying a typical finite-element procedure for so-called weak formulations of the problem equation, in which the latter are multiplied by weighting functions (which in the Galerkin method are identical to the shape functions $\Phi _j^w$ and $\Phi _j^v$). Integration of the resulting relations (Staroszczyk 2003) reduces the problem defined by Eqs. (4.75), (4.1) and (4.72) to the solution of a system of first-order differential equations given in a matrix form by

$$\begin{aligned} \varvec{C}\varvec{\dot{w}}+\varvec{K}\varvec{w}=\varvec{f}, \end{aligned}$$

(4.80)

where the vector $\varvec{w}$ includes the values of the plate deflections $w_j$, the plate slopes $(\partial w/\partial x)_j$ and $(\partial w/\partial y)_j$, and the velocities $v_{xj}$ and $v_{yj}$ at all nodal points j of the discrete system. We note that the matrix $\varvec{K}$ depends on the horizontal velocities, so $\varvec{K}=\varvec{K}(\varvec{w})$, which means that the system of Eqs. (4.80) is non-linear in $\varvec{w}$. The matrices $\varvec{C}$, $\varvec{K}$ and the forcing vector $\varvec{f}$ are aggregated from the respective element matrices and vectors in a way characteristic of the finite-element method. The element matrices, $\varvec{C}^e$ and $\varvec{K}^e$, each of size $15 \times 15$, are, in turn, composed of 9 submatrices of dimension $5 \times 5$ each. The non-zero entries in these component submatrices are given for the matrix $\varvec{C}$ by

$$\begin{aligned}&c_{rs}^{mn}=\int \limits _{A}\left[ (R_1+R_2)\left( \frac{\partial ^{2}\Phi _i^w}{\partial x^{2}}\,\frac{\partial ^{2}\Phi _j^w}{\partial x^{2}} +\frac{\partial ^{2}\Phi _i^w}{\partial y^{2}}\,\frac{\partial ^{2}\Phi _j^w}{\partial y^{2}}\right) +\right. \nonumber \\&\left. +\,4R_2\,\frac{\partial ^2 \Phi _i^w}{\partial x \partial y}\,\frac{\partial ^2 \Phi _j^w}{\partial x \partial y} +(R_1-R_2)\left( \frac{\partial ^{2}\Phi _i^w}{\partial x^{2}}\,\frac{\partial ^{2}\Phi _j^w}{\partial y^{2}}+ \frac{\partial ^{2}\Phi _i^w}{\partial y^{2}}\,\frac{\partial ^{2}\Phi _j^w}{\partial x^{2}}\right) \right] \mathrm{d}A, \end{aligned}$$

(4.81)

and for the matrix $\varvec{K}$ they are

$$\begin{aligned} \begin{aligned} k_{rs}^{mn}&= \int \limits _{A}\Phi _i^w\left( \!\varrho _w g\Phi _j^w +\tau _x\frac{\partial \Phi _j^w}{\partial x}+\tau _y\frac{\partial \Phi _j^w}{\partial y}+ \right. \\&\left. {\;} -N_x\frac{\partial ^{2}\Phi _j^w}{\partial x^{2}}-2N_{xy}\frac{\partial ^2 \Phi _j^w}{\partial x \partial y} -N_y\frac{\partial ^{2}\Phi _j^w}{\partial y^{2}}\!\right) \mathrm{d}A, \\ k_{rs}^{44}&= \int \limits _{A}\left[ (H_1+H_2)\frac{\partial \Phi _r^v}{\partial x}\frac{\partial \Phi _s^v}{\partial x} +H_2\,\frac{\partial \Phi _r^v}{\partial y}\frac{\partial \Phi _s^v}{\partial y}\right] \mathrm{d}A, \\ k_{rs}^{45}&= \int \limits _{A}\left[ (H_1-H_2)\frac{\partial \Phi _r^v}{\partial x}\frac{\partial \Phi _s^v}{\partial y} +H_2\,\frac{\partial \Phi _r^v}{\partial y}\frac{\partial \Phi _s^v}{\partial x}\right] \mathrm{d}A, \\ k_{rs}^{54}&= \int \limits _{A}\left[ H_2\,\frac{\partial \Phi _r^v}{\partial x}\frac{\partial \Phi _s^v}{\partial y} +(H_1-H_2)\frac{\partial \Phi _r^v}{\partial y}\frac{\partial \Phi _s^v}{\partial x}\right] \mathrm{d}A, \\ k_{rs}^{55}&= \int \limits _{A}\left[ H_2\,\frac{\partial \Phi _r^v}{\partial x}\frac{\partial \Phi _s^v}{\partial x} +(H_1+H_2)\frac{\partial \Phi _r^v}{\partial y}\frac{\partial \Phi _s^v}{\partial y}\right] \mathrm{d}A. \end{aligned} \end{aligned}$$

(4.82)

The indices in (4.81) and (4.82) are

$$\begin{aligned} r,s,m,n=1,2,3, \quad i=3(r-1)+m, \quad j=3(s-1)+n, \end{aligned}$$

(4.83)

and A denotes the plane domain of integration. The components of the forcing vector $\varvec{f}$ are given by

$$\begin{aligned} \begin{aligned} f_r^m&= \oint \limits _ \Gamma \!\Phi _i^w Q \mathrm{d}{\varGamma }, \\ f_r^4&= \int \limits _{A}\Phi _r^v\tau _x \mathrm{d}A +\oint \limits _ \Gamma \!\Phi _r^v T_x \mathrm{d}{\varGamma }, \\ f_r^5&= \int \limits _{A}\Phi _r^v\tau _y \mathrm{d}A +\oint \limits _ \Gamma \!\Phi _r^v T_y \mathrm{d}{\varGamma }, \end{aligned} \end{aligned}$$

(4.84)

where ${\varGamma }$ denotes the boundary of the domain A. In the first of the above equations, Q is the vertical shear force acting on the boundary ${\varGamma }$, and $T_x$ and $T_y$ are, respectively, the x- and y-components of the in-plane traction vector $\varvec{T}$ acting on ${\varGamma }$, and are defined by

$$\begin{aligned} T_x=N_x n_x+N_{xy} n_y, \quad T_y=N_{xy} n_x+N_y n_y, \end{aligned}$$

(4.85)

with $n_x$ and $n_y$ being the components of the outward unit vector $\varvec{n}$ normal to the boundary ${\varGamma }$.

Fig. 4.21

A rectangular rigid structure of horizontal dimensions $a \times b$ interacting with sea ice driven by wind blowing at the angle $\alpha$ to the x-axis

The system of equations (4.80) is integrated in time by applying an implicit weighted residual $\theta$-method (Zienkiewicz et al. 2005) . Application of this method gives the relationship that connects the solution vectors $\varvec{w}_n$ and $\varvec{w}_{n+1}$ at two consecutive time levels, $t_n$ and $t_{n+1}$:

$$\begin{aligned} \left( \varvec{C}+\theta \Delta t \varvec{K}\right) \varvec{w}_{n+1}= \left[ \varvec{C}-(1-\theta ) \Delta t \varvec{K}\right] \varvec{w}_n+ \Delta t \bar{\varvec{f}}, \end{aligned}$$

(4.86)

where $\Delta t=t_{n+1}-t_n$ is the time-step length. The vector $\bar{\varvec{f}}$ is the time-averaged forcing vector which, assuming a linear variation of $\varvec{f}$ from $t_n$ to $t_{n+1}$, is defined by

$$\begin{aligned} \bar{\varvec{f}}=(1-\theta )\varvec{f}_n+\theta \varvec{f}_{n+1}. \end{aligned}$$

(4.87)

In numerical calculations, the value of $\theta =0.6$ has been adopted, for which the method is unconditionally stable, and which guarantees that the time-discretization error is nearly of the order $( \Delta t)^2$.

Ice–Structure Interaction Simulations

Before applying the above-described finite-element model to simulate a plane ice–structure interaction event, the discrete model was tested on a one-dimensional problem, for which a closed-form analytical solution is available, as described in Sect. 4.2.2. Hence, the model was run for a uniform-width plate undergoing creep buckling under the action of a in-plane compressive horizontal force, with the initial plate deflection consisting of a number of small harmonic perturbation of various lengths and random amplitudes, see Eq. (4.63) on p. 86. It turned out (Staroszczyk 2003) that the finite-element predictions were in a very good agreement with the analytical results (the maximum relative error in the plate deflections given by the two methods was less than 3%).

After the successful verification of the accuracy of the discrete model in the one-dimensional configuration, a two-dimensional problem sketched in Fig. 4.21 was solved, in which the behaviour of a coherent floating ice cover interacting with a rigid structure, the horizontal cross-section of which has the shape of a rectangle of dimensions defined by a and b. The ice cover was assumed to be driven towards the structure by air drag forces caused by a wind blowing in the direction defined by the angle $\alpha$ shown in the figure.

The simulations were carried out for a rigid structure situated at the centre of a rectangular in shape coherent ice field of the size $1\,{\mathrm {km}} \times 1\,{\mathrm {km}}$ and the ice thickness $h=0.5\,{\mathrm {m}}$. At the ice–structure interface, the plate was assumed to be simply-supported, and the free-slip boundary conditions, prescribed, by (4.78), were adopted. The wind had a speed $u_a=30\,{\mathrm {m\,s}}^{-1}$, and its direction has been varied within the range $0< \alpha < 90^\circ$ in order to investigate how this affects the total loading exerted by the ice on the structure. Three particular cases of the structures of different shapes were considered, in which the width of the structure b was kept constant and equal to $10\,{\mathrm {m}}$, and the length a was varied and equal to 20, 30 and $40\,{\mathrm {m}}$, respectively. The results of numerical calculations, conducted with the mesh consisting of 4000 triangular finite elements and 10,400 degrees of freedom, are presented in Fig. 4.22. The results shown in the figure have been obtained for the ice viscosities $\zeta =\mu ={\,1.0\times 10^{9}\,}\,{\mathrm {kg\,m}}^{-1}{\mathrm {s}}^{-1}$. The temperature at the top surface of the ice was assumed to be equal to $-2\,^\circ {\mathrm {C}}$, and that at the bottom surface to be $0\,^\circ {\mathrm {C}}$.

Fig. 4.22

Total horizontal forces F, together with their components $F_x$ and $F_y$, exerted on the structure by the ice cover as a function of the wind direction angle $\alpha$. Shown are the results for three rectangles of the same width $b=10\,{\mathrm {m}}$ and the lengths $20\,{\mathrm {m}}$ (solid lines), $30\,{\mathrm {m}}$ (dashed lines) and $40\,{\mathrm {m}}$ (dashed-dotted lines)

The plots illustrate the dependence of the magnitude of the total horizontal force F exerted by the ice on the structure on the wind direction angle $\alpha$. Also shown are the components of the total force along the x and y axes, $F_x$ and $F_y$ respectively. The results obtained for the rectangle $20\,{\mathrm {m}} \times 10\,{\mathrm {m}}$ are indicated by the solid lines, those for the rectangle $30\,{\mathrm {m}} \times 10\,{\mathrm {m}}$ are given by the dashed lines, and those for the longest rectangle $40\,{\mathrm {m}} \times 10\,{\mathrm {m}}$ are shown by the dashed-dotted lines. One can see in the figure that the geometry of the structure cross-section has a relatively small effect on the total force F sustained by the object during its interaction with creeping ice. Further, a rather small influence of the wind direction on the total force F is also noted. For the structure for which $a/b=4$, the maximum and minimum forces, for $\alpha =90^\circ$ and $\alpha =0^\circ$ respectively, differ by about 20%, while for the structure, for which $a/b=2$, the corresponding relative difference is about 10%. The results of a similar character, that is showing a relatively small effect of the wind angle $\alpha$ on the total force acting on a structure, have been also obtained for other thicknesses of the ice cover.

Fig. 4.23

Plate deflections w along the x-axis at the critical times t (given in hours) for the wind direction angle $\alpha =180^\circ$. Shown are the results for the plates of thickness $h=0.2\,{\mathrm {m}}$ and $h=0.5\,{\mathrm {m}}$ (solid lines); for the thicker plate also the deflections and corresponding critical times for colder ice (dashed line) and the ice with larger shear and bulk viscosities (dashed-dotted line) are presented

Figure 4.23 illustrates the ice plate deflections in the vicinity of the structure vertical wall. The results plotted in the figure have been obtained for the structure dimensions $a=20\,{\mathrm {m}}$ and $b=10\,{\mathrm {m}}$, and for the wind blowing along the negative direction of the x-axis (that is for the angle $\alpha =180^\circ$, see Fig. 4.21). Plotted are the plate deflection curves along the positive x-axis at the critical times at which the process of flexural failure of ice starts (the origin of the x-axis is on the structure vertical wall; that is, it is shifted to the right by a / 2 compared to Fig. 4.21). The two solid lines in the figure illustrate the plate deflections for two different thicknesses of the ice: $h=0.2\,{\mathrm {m}}$ and $h=0.5\,{\mathrm {m}}$. It can be immediately noticed that the plate failure times for these two plates differ quite considerably: $t=0.07\,{\mathrm {h}}$ for the thinner ice and $t=1.10\,{\mathrm {h}}$ for the thicker ice. The dashed line displays, for $h=0.5\,{\mathrm {m}}$, the plate deflection in the case of the top surface of the ice having the temperature $-4\,^\circ {\mathrm {C}}$ (compared to $-2\,^\circ {\mathrm {C}}$ for the ice represented by the respective solid line in the figure). Finally, the dashed-dotted line shows, for $h=0.5\,{\mathrm {m}}$, the plate deflection for the ice of the viscosities $\zeta$ and $\mu$ increased by 30% with respect to the reference case plotted by the solid line; such a difference in viscosities occurs between isotropic ice and transversely isotropic columnar ice. It is seen that both the temperature and the type of ice anisotropy have quite a pronounced effect on the strength of ice, significantly increasing the values of the failure time (by about $30\%$ in the presented example). On the other hand, the maximum plate deflections do not change much with the change of temperature and the type of ice.

4.2.5 Interaction of Ice with Cylindrical Structures

In the previous Sect. 4.2.4 the plane problem of the interaction of floating ice with a rectangular in shape structure is discussed. Here a similar problem is considered, in which creeping ice interacts with a vertically-walled circular cylinder, see Fig. 4.24. Since the problem involves a single circular structure, cylindrical polar coordinates $r,\theta ,z$ $(0\le \theta <2\pi )$ are adopted, with the vertical z-axis coinciding with the axis of the rotational symmetry of the cylinder. As before, it is assumed that in the immediate vicinity of the structure the floating ice cover has a constant thickness, h. The z-axis, directed downwards, is chosen in such a way that $z=0$ corresponds to the top surface of the ice sheet, and $z=h$ to its bottom. A circular cylinder, of radius $R_0$, is treated as a fixed rigid body that interacts with the ice sheet along its vertical walls at $r=R_0$. The purpose is to evaluate the values of the horizontal forces which the floating ice exerts on the structure during an interaction event.

Fig. 4.24

Geometry of the problem and cylindrical polar coordinates: a plane view, b ice sheet cross-section

Similar problems, of a circular cylinder interacting with sea ice, were previously investigated by Wang and Ralston (1983) and Sjölind (1985) . In the first of these papers, the ice was treated as an elastic-plastic material, while in the second a viscoelastic rheology was adopted to describe the ice deformation. In this work, the forces acting on the structure are determined by adopting either the non-linearly viscous fluid rheology, or the viscous-plastic rheology, both discussed in Sect. 4.2.1.

Governing Equations

The definitions of internal forces acting on an infinitesimal plate element, with their components expressed in the adopted polar coordinates, are given in Fig. 4.25. Basically, all the equations describing the equilibrium of forces acting on a plate element in the horizontal and vertical planes are derived in a way analogous to that presented in Sect. 4.1.1. Hence, the balances of the forces acting in the horizontal plane $Or\theta$, involving the axial forces $N_r$ and $N_\theta$ and the shear forces $N_{r\theta }=N_{\theta r}$, are expressed by

$$\begin{aligned} \begin{aligned}&\frac{\partial (rN_r)}{\partial r}-N_\theta +\frac{\partial N_{\theta r}}{\partial \theta }+rq_r =0, \\&\frac{1}{r}\,\frac{\partial (r^2N_{r\theta })}{\partial r}+\frac{\partial N_\theta }{\partial \theta }+rq_\theta =0, \end{aligned} \end{aligned}$$

(4.88)

where $q_r$ and $q_\theta$ denote the components of the external forces acting in the horizontal direction, which arise due to the wind stress and water current drag.

Fig. 4.25

Definitions of internal forces acting on a plate element: a axial and shear forces, b bending and twisting moments

Along the z-direction, a plate element is subject to the vertical shear forces $Q_r$ and $Q_\theta$, and also to the transverse distributed load $q_z$ coming from the underlying water. Since in our problem the in-plane forces $N_r$, $N_\theta$ and $N_{r\theta }$, all acting in the directions tangential to the deflection surface $w(r,\theta )$, can have magnitudes considerably larger than those of the vertical shear forces $Q_r$ and $Q_\theta$, we include the z-components of the former in the equilibrium balance. Accordingly, the projection of all forces on the vertical direction, with the own weight of ice neglected, gives

$$\begin{aligned} \begin{aligned}&\frac{\partial (rQ_r)}{\partial r}+\frac{\partial Q_\theta }{\partial \theta }+\frac{\partial }{\partial r}\left( rN_r\frac{\partial w}{\partial r}\right) + \frac{1}{r}\,\frac{\partial }{\partial \theta }\left( N_\theta \frac{\partial w}{\partial \theta }\right) \,+ \\&{}+\frac{\partial }{\partial r}\left( N_{r\theta }\frac{\partial w}{\partial \theta }\right) +\frac{\partial }{\partial \theta }\left( N_{\theta r}\frac{\partial w}{\partial r}\right) +rq_z =0. \end{aligned} \end{aligned}$$

(4.89)

The above relations, apart from the internal forces, also involve the plate deflection spatial derivatives. The equilibrium of all moments (see Fig. 4.25b) acting on an infinitesimal plate element with respect to the radial (r) and circumferential ($\theta$) directions yields the expressions

$$\begin{aligned} \begin{aligned}&\frac{\partial (rM_r)}{\partial r}-\frac{\partial M_{\theta r}}{\partial \theta }-rQ_r =0, \\&\frac{\partial M_\theta }{\partial \theta }-\frac{\partial (rM_{r\theta })}{\partial r}-rQ_\theta =0, \end{aligned} \end{aligned}$$

(4.90)

where $M_r$ and $M_\theta$ are the bending moments, and $M_{r\theta }=M_{\theta r}$ are the twisting moments, all per unit width of the plate. Elimination of the shear forces $Q_r$ and $Q_\theta$ from (4.89) by means of relations (4.90) gives the equilibrium equation

$$\begin{aligned} \begin{aligned}&\frac{\partial ^{2}(rM_r)}{\partial r^{2}}+\frac{1}{r}\,\frac{\partial ^{2}M_\theta }{\partial \theta ^{2}}-\frac{\partial ^2 M_{r\theta }}{\partial r \partial \theta }-\frac{1}{r}\,\frac{\partial ^2 (rM_{r\theta })}{\partial r \partial \theta }\,+ \\&{}+rN_r\frac{\partial ^{2}w}{\partial r^{2}}+N_\theta \left( \frac{1}{r}\,\frac{\partial ^{2}w}{\partial \theta ^{2}}+\frac{\partial w}{\partial r}\right) +2rN_{r\theta }\frac{\partial ^2 }{\partial r \partial \theta }\left( \frac{w}{r}\right) \,+ \\&{}-rq_r\frac{\partial w}{\partial r}-q_\theta \frac{\partial w}{\partial \theta }+rq_z =0. \end{aligned} \end{aligned}$$

(4.91)

The transverse distributed load $q_z$, resulting from the response of the underlying water, is assumed to be proportional to the plate deflection w:

$$\begin{aligned} q_z=-\varrho _w g w. \end{aligned}$$

(4.92)

The internal forces in the vertical plate cross-sections are determined in terms of the axial, $\sigma _{rr}$ and $\sigma _{\theta \theta }$, and shear, $\sigma _{r\theta }$, stresses by the integrals:

$$\begin{aligned} N_r=\int \limits _0^h\!{\sigma _{rr}}\mathrm{d}z, \quad N_\theta =\int \limits _0^h\!{\sigma _{\theta \theta }}\mathrm{d}z, \quad N_{r\theta }=\int \limits _0^h\!{\sigma _{r\theta }}\mathrm{d}z, \end{aligned}$$

(4.93)

and

$$\begin{aligned} M_r=\int \limits _0^h\!{\sigma _{rr}z}\mathrm{d}z, \quad M_\theta =\int \limits _0^h\!{\sigma _{\theta \theta }z}\mathrm{d}z, \quad M_{r\theta }=-\int \limits _0^h\!{\sigma _{r\theta }z}\mathrm{d}z. \end{aligned}$$

(4.94)

The stresses in (4.93) and (4.94), in the case of creeping behaviour of ice, are functions of the strain-rates and their invariants, as prescribed by the viscous fluid and viscous-plastic flow laws, (4.39) and (4.49) respectively. The components of the strain-rate tensor $\varvec{D}$ due to the motion of ice in the horizontal plane, when expressed in polar coordinates, are defined by

$$\begin{aligned} D_{rr}=\frac{\partial v_r}{\partial r}\,, \quad D_{\theta \theta }=\frac{1}{r}\,\left( v_r+\frac{\partial v_\theta }{\partial \theta }\right) , \quad D_{r\theta }=\frac{1}{2}\left[ \frac{1}{r}\,\frac{\partial v_r}{\partial \theta }+ r\frac{\partial }{\partial r}\left( \frac{v_\theta }{r}\right) \right] , \end{aligned}$$

(4.95)

and their invariants $\eta$ and $\gamma$ are given by

$$\begin{aligned} \eta =D_{rr}+D_{\theta \theta }, \quad \gamma ^2=D_{r\theta }^2+\frac{1}{4}\left( D_{rr}-D_{\theta \theta }\right) ^2. \end{aligned}$$

(4.96)

The strain-rates, developing in ice due to the bending and twisting of the plate, vary across its depth and are defined in terms of the time rates of the plate curvatures, $\kappa _r$ and $\kappa _\theta$, and the twist, $\kappa _{r\theta }$, of the deflection surface $w(r,\theta )$. Hence,

$$\begin{aligned} D_{rr}=\dot{\kappa }_r\,(z-z_0), \quad D_{\theta \theta }=\dot{\kappa }_\theta \,(z-z_0), \quad D_{r\theta }=-\dot{\kappa }_{r\theta }\,(z-z_0), \end{aligned}$$

(4.97)

where $z_0$ denotes the position of the neutral plane in the undeformed state. In terms of the plate deflection function $w(r,\theta )$, the curvatures and the twist are given by

$$\begin{aligned} \kappa _r=-\frac{\partial ^{2}w}{\partial r^{2}}\,, \quad \kappa _\theta =-\frac{1}{r}\,\frac{\partial w}{\partial r}-\frac{1}{r^2}\,\frac{\partial ^{2}w}{\partial \theta ^{2}}\,, \quad \kappa _{r\theta }=\frac{\partial }{\partial r}\left( \frac{1}{r}\,\frac{\partial w}{\partial \theta }\right) . \end{aligned}$$

(4.98)

So far in this chapter, when analysing the creep buckling of ice and the plane ice–structure interactions, the viscosities $\mu$ and $\zeta$ were assumed constant in the viscous fluid flow law (4.39). In the latter law, the Heaviside step function factor $H(-\eta )$ appears, the role of which is to ensure that no axial stresses develop in ice during its diverging flow (when $\eta < 0$), in order to model zero tensile strength of ice. Such an abrupt cut-off to zero stress during a change from converging to diverging flow, however, gives rise to instabilities in two-dimensional ice flow numerical models (Schulkes et al. 1998) . For this reason, Morland and Staroszczyk (1998) proposed a replacement of the abrupt cut-off by a smooth transition to zero stress over a dilatation-rate range equal to approximately one-tenth to one-hundredth of the maximum convergence-rate typically appearing in sea ice flow problems, which significantly improved the stability of numerical algorithms. Hence, a scaling factor $\bar{H}(\eta )$ (Staroszczyk 2005) defined by

$$\begin{aligned} \bar{H}(\eta )= {\left\{ \begin{array}{ll} \quad 1 &{} \text {if}\quad \eta < 0, \\ \quad \exp [-(\eta /\eta _c)^2] &{} \text {if}\quad \eta \ge 0, \end{array}\right. } \end{aligned}$$

(4.99)

is adopted to reduce ice viscosities in a narrow range of divergence-rates, which is unity at $\eta =0$, tends to zero as $\eta \rightarrow \infty$, and has zero derivatives at $\eta =0$. The free parameter $\eta _c > 0$ is a divergence-rate magnitude around which significant changes in viscosities occur. Accordingly, with the function $\bar{H}$, the flow law (4.39) is modified to take, in components, the form

$$\begin{aligned} \sigma _{ij}=\left[ (\zeta -\mu )D_{kk}\delta _{ij}+2\mu D_{ij}\right] \bar{H}(\eta ) \quad (i,j,k=1,2), \end{aligned}$$

(4.100)

with the indices i, j, k denoting either r or $\theta$.

The stress tensor components given by the flow law (4.100), with the strain-rate tensor components expressed by (4.95), after their insertion into the definitions (4.93), yield the in-plane axial and shear forces in the forms

$$\begin{aligned} \begin{aligned} N_r&= (H_1+H_2)\frac{\partial v_r}{\partial r}+(H_1-H_2)\,\frac{1}{r}\,\left( v_r+\frac{\partial v_\theta }{\partial \theta }\right) ,\\ N_\theta&= (H_1-H_2)\frac{\partial v_r}{\partial r}+(H_1+H_2)\,\frac{1}{r}\,\left( v_r+\frac{\partial v_\theta }{\partial \theta }\right) ,\\ N_{r\theta }&= H_2\left[ \frac{\partial v_\theta }{\partial r}+\frac{1}{r}\,\left( \frac{\partial v_r}{\partial \theta }-v_\theta \right) \right] . \end{aligned} \end{aligned}$$

(4.101)

Similarly, the moment definitions (4.94), when combined with the relations (4.100), (4.97) and (4.98), express the bending and twisting moments as

$$\begin{aligned} \begin{aligned} M_r&= -\left[ (R_1+R_2)\,\frac{\partial ^{2}\dot{w}}{\partial r^{2}} +(R_1-R_2)\,\frac{1}{r}\,\left( \frac{\partial \dot{w}}{\partial r}+\frac{1}{r}\,\frac{\partial ^{2}\dot{w}}{\partial \theta ^{2}}\right) \right] , \\ M_\theta&= -\left[ (R_1-R_2)\,\frac{\partial ^{2}\dot{w}}{\partial r^{2}} +(R_1+R_2)\,\frac{1}{r}\,\left( \frac{\partial \dot{w}}{\partial r}+\frac{1}{r}\,\frac{\partial ^{2}\dot{w}}{\partial \theta ^{2}}\right) \right] , \\ M_{r\theta }&={} 2R_2\,\frac{\partial ^2}{\partial r\partial \theta }\left( \frac{\dot{w}}{r}\right) . \end{aligned} \end{aligned}$$

(4.102)

The parameters $H_1$, $H_2$, $R_1$ and $R_2$ in Eq. (4.102) describe the viscous properties of the ice plate and are defined by relations (4.74) on p. 97, with the bulk and shear viscosities $\zeta$ and $\mu$ now replaced by $\zeta \bar{H}(\eta )$ and $\mu \bar{H}(\eta )$, accordingly.

Substitution of the moment expressions (4.102) into the equilibrium relation (4.91), with the distributed load $q_z$ given by (4.92), yields the differential equation for the plate deflection function w in the form

$$\begin{aligned} \begin{aligned} R\,\nabla ^2\nabla ^2\dot{w}=&\,N_r\,\frac{\partial ^{2}w}{\partial r^{2}}+N_\theta \,\frac{1}{r}\,\left( \frac{\partial w}{\partial r} +\frac{1}{r}\,\frac{\partial ^{2}w}{\partial \theta ^{2}}\right) +2N_{\theta r}\frac{\partial ^2}{\partial r\partial \theta }\left( \frac{w}{r}\right) \,+ \\&{}-q_r\frac{\partial w}{\partial r}-\frac{q_\theta }{r}\,\frac{\partial w}{\partial \theta }-\varrho _w g w, \end{aligned} \end{aligned}$$

(4.103)

where

$$\begin{aligned} \nabla ^2=\frac{\partial ^{2}}{\partial r^{2}}+\frac{1}{r}\,\frac{\partial }{\partial r}+\frac{1}{r^2}\,\frac{\partial ^{2}}{\partial \theta ^{2}} \end{aligned}$$

(4.104)

is the Laplace operator expressed in polar coordinates, and $R=R_1+R_2$, as defined by (4.76).

The above Eq. (4.101) for the in-plane forces, and (4.103) for the plate deflection evolution, have been derived for the ice creep behaviour described by the viscous fluid flow law (4.39), in its slightly modified version given by (4.100) with (4.99). When the viscous-plastic rheological model (4.49) is used instead to describe the creep of ice, then the ensuing equations are similar, which is due to the formal similarities between the flow laws (4.39) and (4.49). Both laws include the same two viscosity parameters, $\mu$ and $\zeta$ (though their physical meanings are different in the two laws), and additionally two ice strength parameters $P_1$ and $P_2$ enter the viscous-plastic flow relation (4.49). This makes the resulting equations more elaborate compared to (4.101) and (4.103), but the formal structure of the equations is retained.

Numerical Simulations

The system of two partial differential equations (4.88) with (4.93) for the unknown ice horizontal velocity components $v_r$ and $v_\theta$, and Eq. (4.103) for the unknown plate deflection w, was solved numerically by applying a finite-difference method in order to simulate the creep behaviour of a coherent ice cover interacting with a cylindrical structure (Staroszczyk 2005, 2006) . The simulations were carried out for two sea ice rheological models: the non-linearly viscous fluid flow law (4.39) with its modification given by (4.100), and the viscous-plastic flow law described by (4.49). Owing to the symmetry of the problem with respect to the wind direction which was assumed to blow along the coordinate line $\theta =0$, only the region $0\le \theta \le \pi$ has been considered in the numerical model. In the radial direction, the ice domain was assumed to extend from the cylinder wall at $r=R_0$ to the free edge of the ice cover at $r=R_{max}$. The adopted computational mesh had 300 discrete nodes in the radial direction and 61 nodes in the circumferential direction, uniformly distributed along both r and $\theta$ ranges, so that there were 18,300 nodes in all, with 54,900 unknown values of the ice velocities and the plate deflections to be calculated.

At the ice–structure contact surface either no-slip (full bonding) or free-slip boundary conditions were assumed for the ice horizontal deformation, and the simply supported conditions for the ice plate bending. For a no-slip boundary these conditions are expressed by

$$\begin{aligned} r=R_0: \quad \varvec{v}=\varvec{0}, \quad w=0, \quad M_r=0, \end{aligned}$$

(4.105)

and for a free-slip boundary by

$$\begin{aligned} r=R_0: \quad \varvec{v}\cdot \varvec{n}=0, \quad N_{r\theta }=0, \quad w=0, \quad M_r=0, \end{aligned}$$

(4.106)

where $\varvec{n}$ denotes the unit vector normal to the cylinder wall. The ice at the outer edge $r=R_{max}$ was assumed to be stress-free, that is,

$$\begin{aligned} r=R_{max}: \quad N_r=0, \quad N_{r\theta }=0. \end{aligned}$$

(4.107)

Regarding the initial conditions, it was assumed that at the start of simulations the floating ice was undeformed and stress-free. The results presented below correspond to the ice flow stages when the magnitudes of the forces sustained by a structure attain their maximum values.

The simulations were carried out for a cylinder of the radius $R_0=10\,{\mathrm {m}}$, situated at the centre of a circular ice field extending to $R_{max}=500\,{\mathrm {m}}$, with the thickness of the ice cover equal to $h=0.2\,{\mathrm {m}}$. The ice was assumed to be driven onto the structure by a wind of a constant velocity, blowing along the coordinate line $\theta =0$ in the negative direction of r. The dimensionless wind and drag coefficients appearing in (4.77) on p. 98 were adopted of the values $C_a={\,2\times 10^{-3}\,}$ and $C_w={\,4\times 10^{-3}\,}$, and the air and water densities were $\varrho _a=1.3\,{\mathrm {kg\,m}}^{-3}$ and $\varrho _w={\,1.02\times 10^{3}\,}\,{\mathrm {kg\,m}}^{-3}$.

As first, the results obtained for the ice treated as a viscous fluid are presented. They have been obtained for the wind velocity $u_a=30\,{\mathrm {m\,s}}^{-1}$ (such a wind generates a tangential stress $\tau _a \approx 2.3\,{\mathrm {Pa}}$ on the ice surface). The viscous fluid rheological model involves three material parameters: two viscosities $\mu$ and $\zeta$, and the critical strain-rate parameter $\eta _c$ which describes the tensile ice strength reduction rate at the beginning of diverging flow. The adopted values of the viscosities were $\mu ={\,1.0\times 10^{9}\,}\,{\mathrm {kg\,m}}^{-1}\,{\mathrm {s}}^{-1}$ and $\zeta ={\,2.0\times 10^{9}\,}\,{\mathrm {kg\,m}}^{-1}\,{\mathrm {s}}^{-1}$, which can be regarded as typical viscosity magnitudes for floating ice. Since the maximum horizontal strain-rates occurring in the problem considered are of magnitudes equal to about ${\,5\times 10^{-5}\,}\,{\mathrm {s}}^{-1}$, the parameter $\eta _c$ has been adopted from within a range embracing the latter value. Thus, $\eta _c$ has been chosen to vary from ${\,5\times 10^{-6}\,}$ to ${\,1\times 10^{-4}\,}\,{\mathrm {s}}^{-1}$ to explore the effect of $\eta _c$ on the magnitudes of the total contact forces exerted by ice on the cylinder.

Fig. 4.26

Distribution of the radial forces $N_r$ along the cylinder wall for no-slip boundary conditions and different magnitudes of the rheological parameter $\eta _c$

The plots in Figs. 4.26 and 4.27 present the distribution of the forces exerted by the ice cover on the cylinder wall in the case of no-slip boundary conditions defined by (4.105). Illustrated is the dependence of the loads on the wall on the magnitude of the rheological parameter $\eta _c$; for comparisons, the contact forces generated in the case of a linearly viscous response of ice are also plotted. Figure 4.26 shows the variation of the radial force $N_r$ with the angle $\theta$. It can be noted that the effect of the ice viscosity reduction in diverging flow occurring at the leeward side of the cylinder ($90^\circ < \theta \le 180^\circ$) is hardly observed on the opposite, windward part of the wall ($0^\circ \le \theta \le 90^\circ$), where the forces are practically insensitive to the value of $\eta _c$. In stark contrast, the radial forces on the leeward side decrease dramatically with decreasing $\eta _c$. The results in the plots suggest that for realistic modelling of the floating ice creep behaviour (small tensile strength of ice compared to its compressive strength), the magnitudes of the critical dilatation-rate $\eta _c$ should be chosen of the order ${\,1\times 10^{-5}\,}\,{\mathrm {s}}^{-1}$.

Fig. 4.27

Distribution of the shear forces $N_{r\theta }$ along the cylinder wall for no-slip boundary conditions and different magnitudes of the rheological parameter $\eta _c$

A similar pattern is seen in Fig. 4.27, illustrating the variation of the contact shear forces $N_{r\theta }$ with the angle $\theta$ and the value of the rheological parameter $\eta _c$. Again, the tangential forces exerted on the walls on the windward side of the cylinder are roughly independent of $\eta _c$, while those on the leeward side rapidly approach zero values with $\eta _c$ approaching the value ${\,1\times 10^{-5}\,}\,{\mathrm {s}}^{-1}$. Comparing this figure with the previous one, the change in the loading coming from the ice cover, for small values of $\eta _c$, seems even more dramatic.

Figure 4.28 displays the distribution of the normal forces $N_r$ on the cylinder wall in the case of free-slip boundary conditions (4.106), when the tangential forces $N_{r\theta }$ are zero by definition. Due to $N_{r\theta }\equiv 0$, the whole loading from the ice is passed on the cylinder walls through the normal contact forces. For this reason, the normal forces at $\theta =0$ (the wind direction) are by about 40 per cent larger than those in the case of no-slip conditions at the interface, see Fig. 4.26. Otherwise, qualitatively very similar features are observed in the plots for the free-slip and no-slip boundary conditions at the walls, with practically unchanged contact loading on the windward side, and a significant reduction of the $N_r$ forces on the leeward side of the cylinder for the critical dilatation-rates $\eta _c \sim {\,1\times 10^{-5}\,}\,{\mathrm {s}}^{-1}$.

Fig. 4.28

Distribution of the radial forces $N_r$ along the cylinder wall for free-slip boundary conditions and different magnitudes of the rheological parameter $\eta _c$

Of a particular interest to civil engineers are the magnitudes of total forces sustained by a cylindrical structure during its interaction with floating ice. These magnitudes, obtained by integrating the radial and shear forces $N_r$ and $N_{r\theta }$ along the whole perimeter of the cylinder, are listed in Table 4.2. Compared are the results for different cylinder diameters $R_0$, for different values of the critical dilatation-rate $\eta _c$, and for the two types (no-slip and free-slip) of the boundary conditions.

Table 4.2 Values of the total horizontal force F exerted on a cylindrical structure by the sea ice cover as a function of the rheological parameter $\eta _c$, for different cylinder diameters $R_0$ and boundary conditions on the wall ($\eta _c\rightarrow +\infty$ corresponds to the linearly viscous fluid solution). Results for the wind velocity of $30\,{\mathrm {m\,s}}^{-1}$

The viscous-plastic behaviour of sea ice predicted by the constitutive law (4.49) is determined by the values of the four constitutive parameters: $P_1$, $P_2$, e and $\Delta _c$, from among which the first and the last, the ice compressive strength $P_1$ and the critical strain-rate invariant $\Delta _c$, are most important in terms of quantitative results. Regarding the compressive strength of ice, $P_1$, there is no clarity in the literature as to its most proper magnitude. In the original formulation of the viscous-plastic model, Hibler (1979) used the value ${\,5\times 10^{3}\,}\,{\mathrm {Pa}}$ for large-scale Arctic ice simulations, and the latter value was subsequently used by him and co-authors in a number of papers (Ip et al. 1991; Hibler and Ip 1995) . Flato and Hibler (1992) , in turn, applied a larger value, ${\,2.75\times 10^{4}\,}\,{\mathrm {Pa}}$, also for describing the large-scale behaviour of ice. In our simulations, a value of $P_1={\,5\times 10^{4}\,}\,{\mathrm {Pa}}$ which is slightly larger than the latter one was adopted, in belief that the strength of ice increases with decreasing spatial scales encountered in civil engineering applications, in accordance with the empirical data discussed in Sect. 3.4. Regarding the magnitude of the critical strain-rate invariant, a value of $\Delta _c={\,2\times 10^{-5}\,}\,{\mathrm {s}}^{-1}$ was used in the simulations. The latter value was adopted on the basis of the results presented above for the viscous fluid rheology, showing that the most realistic predictions were obtained for $\eta _c \sim {\,1\times 10^{-5}\,}\,{\mathrm {s}}^{-1}$, and also assuming that the strain-rate invariants $\eta$ and $\gamma$ are of comparable magnitudes in the relation (4.44) defining $\Delta _c$. The rheological model parameter e, defining the shape of the yield curve, and hence the magnitude of the shear viscosity relative to the bulk viscosity, was commonly assumed (Hibler 1979) as 2 (implying $\mu /\zeta =1/4$). In our simulations the range $1 \le e \le 3$ was explored. Finally, the remaining constitutive model parameter, $P_2$, used in the flow law (4.49) to define the tensile strength of ice, was adopted as a small fraction of the compressive strength $P_1$. Accordingly, a value $P_2={\,1\times 10^{3}\,}\,{\mathrm {Pa}}$, that is, $P_2=P_1/50$ was used in the simulations. Recall that the small parameter $P_2 > 0$ was introduced in (4.49) to avoid numerical instabilities encountered in earlier viscous-plastic rheological models. Computational tests showed that this parameter (as long as it is small) has a very limited effect on the magnitudes of forces sustained by an engineering object, since most of the loading on structure walls comes from the ice that is under compression on the windward side of the structure.

Fig. 4.29

Distribution of the radial forces $N_r$ along the cylinder wall for no-slip boundary conditions and different values of the rheological parameter e

The results plotted in Figs. 4.29 and 4.30 illustrate the distributions of the forces exerted by the ice on the structure walls in the case of no-slip boundary conditions (4.105). Shown is the dependence of the ice–structure contact forces on the rheological parameter e; that is, the effect of the ratio of the shear to bulk viscosities of ice is presented. The range of e varying from 1 to 3 corresponds to the viscosity ratios $\mu /\zeta$ (or $\mu _m/\zeta _m$) decreasing from 1 (for $e=1$) to $1/9\sim 0.111$ (for $e=3$), with the bulk viscosity $\zeta$ held constant for a given value of the strain-rate invariant $\Delta$, as prescribed by relations (4.46) on p. 80. Figure 4.29 illustrates the variation of the radial force $N_r$ with the polar angle $\theta$. One can observe that the effect of the shear viscosity $\mu$ on the magnitude of $N_r$, for the no-slip boundary, is moderate, especially on the windward side of the structure. A little surprising is the prediction that most of the cylinder walls (for the no-slip conditions) is under the action of compressive contact forces.

The distribution of the shear forces $N_{r\theta }$ on the cylinder wall is shown in Fig. 4.30. It is seen that the shear forces exerted by the floating ice vary smoothly with the angle $\theta$, with maximum values occurring at the angle $\theta \sim 60^\circ$. Thus, the magnitudes of $N_{r\theta }$ on the windward side of the structure are larger than those on the leeward side, though the differences are not considerable, especially for smaller values of the shear viscosity (larger values of the parameter e).

Fig. 4.30

Distribution of the shear forces $N_{r\theta }$ along the cylinder wall for no-slip boundary conditions and different values of the rheological parameter e

Fig. 4.31

Distribution of the radial forces $N_r$ along the cylinder wall for free-slip boundary conditions and different values of the rheological parameter e

Figure 4.31 illustrates the variation of the normal contact forces $N_r$ with the angle $\theta$ and the rheological parameter e in the case of a free-slip boundary (4.106), when the tangential forces $N_{r\theta }$ are identically zero. Comparing this figure with the analogous plots in Fig. 4.29 for the no-slip boundary conditions, one can note qualitatively distinct distributions of the radial forces along the cylinder walls. While in the no-slip case the forces $N_r$ vary in a monotonic manner over the entire range of the angles $\theta$, some rapid changes in the magnitudes of $N_r$ are predicted by the viscous-plastic rheological model in the case of the free-slip boundary. These changes occur within the range of the angles $60^\circ \lesssim \theta \lesssim 90^\circ$, where the normal forces switch from compressive to tensile ones, with the maximum tensile forces occurring at $\theta \sim 90^\circ$. It turns out that the above dramatic changes in the contact forces within the range $60^\circ \lesssim \theta \lesssim 90^\circ$ are associated with the change in the creep behaviour of ice, which is in viscous flow for the latter range of $\theta$, in contrast to the rest of the wall, where it is in plastic yield (Staroszczyk 2006) . As concerns the case of the no-slip boundary conditions illustrated in Fig. 4.29, the ice is in plastic flow for all e and $\theta$, except for the case of $e=1$ (the solid line) when, for $\theta \lesssim 30^\circ$, viscous deformation of ice takes place.

More results, regarding the ice horizontal deformation-rates and transverse plate deflection variations in time and space in the vicinity of a cylindrical structure interacting with sea ice, can be found in the papers by Staroszczyk (2005, 2006) .

4.3 Ice Floe Impact on an Engineering Structure

In the previous part of this section, ice–structure interaction problems are considered in which, due to the stress, deformation or deformation-rate levels involved, the behaviour of ice can be sufficiently well approximated by that of a continuous slab of ice floating on the surface of water. Hence, the ice can be treated as an either elastic or creeping material which deforms in a continuous (ductile) manner, and it has been assumed that the ice remains in perfect contact with an engineering structure walls throughout an interaction event.

A different situation arises when strains, strain-rates and stresses in ice reach the magnitudes at which cracks start to develop in the material, giving rise to the brittle fracture of ice. Typically, sea ice undergoes a transition from ductile to brittle behaviour when stresses exceed the fracture strength of the material (that is, about $5\,{\mathrm {MPa}}$ in compression and about $1\,{\mathrm {MPa}}$ in tension), or strains exceed the value of about 0.01, or strain-rates reach the level of about $10^{-4}$ to $10^{-3}\,{\mathrm {s}}^{-1}$ (Hawkes and Mellor 1972; Sanderson 1988; Schulson and Duval 2009) . During this creep-to-brittle transition phase (see Sect. 3.4), the loads exerted by floating ice on a structure attain their maximum or near-maximum values, and further increase in the ice deformation and its rate, usually associated with the fast-progressing process of crack formation and their subsequent growth, does not increase the forces in ice. Typically, the brittle failure of ice takes place, at a given time instant, only at a number of relatively small regions of the ice–structure interface, therefore the total loads exerted by ice on the object show a highly irregular variation in time, with characteristic sharp spikes appearing at irregular time intervals.

In order to investigate the main features of the mechanism of brittle failure of floating ice during its interaction with an engineering object, a problem is considered in which an ice floe impacts dynamically on a rigid cylindrical structure. It is believed that, in spite of a number of simplifying assumptions adopted in the course of the analysis, the results obtained will realistically describe the complex nature of the dynamic ice–structure interaction phenomenon, and will be in reasonable agreement with the behaviour of ice observed in Arctic seas (Sanderson 1988; Jordaan 2001) .

As already noted in Sect. 3.4, the modelling of the ice fracture mechanism is difficult, and requires the knowledge of advanced methods of mechanics (Ashby and Hallam 1986; Sjölind 1987; Nixon 1996; Pralong et al. 2006) . The derived theoretical solutions seem to be by far too complicated to be effectively implemented into realistic, engineering applications. Therefore, a simple approach (Staroszczyk 2007) , extending the method suggested by Sanderson (1988) , is applied to construct a model that enables the estimation of forces exerted by brittle-failing ice on a vertically-walled rigid structure. In this approach, the mechanism of ice fracture is described, essentially, by only three physical parameters: (1) axial compressive fracture stress, (2) an associated axial strain at which the fracture occurs, and (3) ice clearing axial stress, which is a stress occurring in already fractured blocks of ice. There is no doubt that many interesting small-scale processes occurring during the fracture of ice are disregarded in this way. However, such small-scale effects are deemed unimportant for the purpose of this analysis, the objective of which is to evaluate total net forces sustained by the structure during its dynamic impact by a large ice floe.

An ice floe that hits the rigid structure is treated as a compact plate of uniform thickness. The interaction between the moving floe and the structure vertical wall is assumed to occur, at any time instant, at a number of small zones, with local fracture events taking place non-simultaneously at different points of the ice-structure contact interface. The local failure of ice at each small zone is supposed to occur independently of the other zones, and all these independent local fracture events are treated as separate random processes. The total force sustained by the structure is then determined as a statistical sum of individual loads occurring at all the contact zones. By simulating numerically a large number of separate floe–structure collision events, with randomly varied input parameters, the probability distributions of the total contact force magnitudes during an ice–structure impact phenomenon are calculated. In particular, the probability distributions illustrating the dependence of peak interaction forces on the floe thickness, its size and its initial velocity are presented. These distributions can be used by an engineer to perform a risk assessment analysis for an off-shore structure during the stage of its designing.

4.3.1 Fracture of Ice at a Structure Wall

The problem under consideration is sketched in Fig. 4.32. An ice floe, being initially at some distance from a rigid structure, is driven by wind and/or water current drag forces towards the object at a horizontal free-drift velocity $V_0\,$. After arriving at the structure at time $t_0$ and establishing first contact with its walls, the impacting ice floe starts to break at points at which the magnitudes of local contact stresses exceed the brittle fracture strength of ice. As the ice undergoes crushing, its chunks pile up or sink near the structure walls, and the initial kinetic energy of the floe is gradually dissipated. The floe velocity, v(t), with t measuring the time elapsed from the instant $t_0$, steadily decreases, until the floating ice slab eventually comes to rest.

Fig. 4.32

Definition of the ice floe impact problem: a planar and b cross-sectional views

In real field conditions, an ice floe leading edge is commonly irregular in shape. Thus, the contact between the floe and the structure wall is unlikely to take place over the entire possible interface between the ice and the object. Instead, as illustrated in Fig. 4.33a, the ice interacts with the structure walls at a number of locations of small areas, and these contact locations change all the time as the ice floe advances. Once the ice–structure interaction has been initiated and the ice starts to fail, broken blocks of ice of various size and shape are formed in a chaotic manner at each local interaction zone. These local ice fracture events occur non-simultaneously, as different ice fragments fail at different times at different places at the interface surface. At any one small contact zone, the ice fragments are supposed to arrive and fail one by one: as one fragment fails and the debris is cleared by the process of ice piling up or sinking, another ice fragment arrives immediately to start its interaction with the wall and to fail after some time, etc.

Fig. 4.33

a Imperfect contact between an ice floe and a structure wall, b problem idealization, c detailed view and definitions

In order to model such a complex interaction phenomenon as depicted above, a method that refines the approach originally proposed by Ashby and Hallam (1986) and subsequently followed by Sanderson (1988) is applied here. In this method, the floe is treated as a collection of regular in shape and independent cells, as shown in Fig. 4.33b, c. Each discrete cell is assumed to have the same size, and to be a square of dimensions $b \times b$ in the horizontal plane. As observed by Sanderson (1988) , the characteristic dimension b of the fractured ice blocks is similar to the ice thickness h; therefore, it is assumed in this analysis that $b \sim h$. As a particular cell starts to interact with the structure (see Fig. 4.33c), it is supposed that it fails (that is, the axial stress component normal to the contact surface reaches the fracture strength of ice) when the whole discrete element is advanced by a distance $\Delta$; the latter parameter represents a critical displacement at which the crushing of ice occurs.

Fig. 4.34

Contact stress history for a single discrete zone

The history of loading experienced by a discrete zone (of width b) at the contact interface, as successive discrete ice blocks arrive and fail there, is idealized in a manner illustrated in Fig. 4.34, adapted from Sanderson (1988). When a given ice block comes to the rigid wall and then moves by a distance $\Delta$, the contact stress is assumed to grow linearly from zero to its peak value, equal to the ice fracture strength, $\sigma _f$. Next, immediately after the failure of ice, the contact stress falls sharply to a much lower level, $\sigma _c$, which is a stress in ice caused by forces that are needed to clear the debris formed during the failure (that is, to move the fractured ice fragments up or down, since the debris cannot be cleared by pushing it aside, in the direction lateral to the impact direction). This clearing stress is supposed to remain constant until the time when the next ice block arrives at the wall and starts to fail, rising the stress gradually to the $\sigma _f$ level again, etc. The failure and clearing stresses, $\sigma _f$ and $\sigma _c$, are assumed to have different magnitudes for different ice cells, in order to reflect both a stochastic character of the fracture mechanism and an associated statistical scatter in available empirical data. The respective mean values of $\sigma _f$ and $\sigma _c$ are illustrated in the figure by the two horizontal dashed lines. For simplicity, the variations of $\sigma _f$ and $\sigma _c$ about their mean values are supposed to follow the normal Gauss distribution (though it is possible that the Weibull distribution might be more appropriate). Moreover, the distance between consecutive failure stress peaks is not uniform, but also varies in a stochastic manner; in such a way the randomness of an individual ice block size is accounted for. It is assumed in the model that the average distance separating two successive failure stress peaks is equal to b—the average size of a fractured block. Further, it is supposed that there is an equal probability of a stress peak to lie anywhere within a given stretch of length b (that is, a uniform probability distribution function is used for this purpose).

The model for the mechanism of the ice–structure brittle interaction developed on the basis of the above-described ideas involves three main parameters: the stress magnitudes $\sigma _f$ and $\sigma _c$ (their mean values and statistical variation) and the critical displacement $\Delta$ at which an ice block of length b fails. The latter parameter will be expressed by means of a critical axial strain, $\epsilon _f$, being the strain at which brittle fracture occurs.

Of the above three parameters, the failure stress level $\sigma _f$, equal to the brittle fracture strength of ice, is the most significant. As already noted in Sect. 3.4, the ice fracture mechanism exhibits a pronounced scale-dependence, illustrated by the pressure–area curve plotted in Fig. 3.11 on p. 44. The weakening of ice strength with increasing contact area can be roughly approximated by a functional relationship expressed by

$$\begin{aligned} p \propto A^{-\beta }, \quad \beta > 0, \end{aligned}$$

(4.108)

where p is the pressure in ice at failure, A denotes the contact area, and the symbol ‘$\propto$’ means ‘proportional to’. There is some discussion in the literature, concerning the most appropriate value of the parameter $\beta$ in the above pressure–area relationship. Generally, it is accepted that $\beta$ takes a value from the range 1 / 4 to 1 / 2, with the lower limit value appropriate for smaller scales ($A \lesssim 0.1\,{\mathrm {m}}^2$), and the upper limit value relevant for larger scales ($A \gtrsim 10^3\,{\mathrm {m}}^2$). In the model proposed here, the value $\beta =1/4$ is adopted, which has been derived theoretically by Palmer and Sanderson (1991) and Xu et al. (2004) on the basis of a fractal analysis of the size distribution of fragmented sea ice. Hence, the ice size effect on the failure pressure, or the compressive fracture strength $\sigma _f$, is expressed in the following, normalized, form as

$$\begin{aligned} \sigma _f=\sigma _f^*\,\left( {\frac{A}{A_0}}\right) ^{-1/4}, \end{aligned}$$

(4.109)

where $A_0$ is a reference contact area, assumed here to be equal to $1\,{\mathrm {m}}^2$, and $\sigma _f^*$ is a normalized ice failure strength (that is, that corresponding to $A_0$). Xu et al. (2004) recommend a value of $\sigma _f^*=1.66\,{\mathrm {MPa}}$, as the one providing the best fit to empirical data. The latter value is particularly suitable for the case of contact areas of the order of $1\,{\mathrm {m}}^2$; that is, those occurring in typical engineering problems.

The ice failure strength defined by Eq. (4.109) represents its mean value. The experimental data for sea ice, however, show a significant statistical scatter. Sanderson (1988) carried out some detailed statistical calculations for the Arctic sea ice and found that the variation coefficient (the ratio of the standard deviation to the mean value) of the data for first-year ice is as high as about $45\%$, and for multi-year ice it is about $65\%$. As in our ice–structure interaction problem we are concerned mostly with first-year ice, the variation coefficient equal to $50\%$ has been chosen to describe the scatter in possible values of $\sigma _f^*$. Moreover, on the basis of statistical analysis, Sanderson (1988) observed that the probability distribution of experimental data for $\sigma _f^*$ is approximately normal, and, therefore, such a type of distribution will be used in the numerical simulations presented further in this section.

While the failure strength of ice, $\sigma _f$, can be relatively easily determined by small-scale indentation tests or large-scale observations, the magnitude of the ice clearing stress, $\sigma _c$, the second free parameter in the proposed model, is more difficult to identify, since, to the author’s knowledge, no in situ measurements of this quantity have been conducted yet during real ice floe impact events. For this reason, several authors have made attempts to estimate the ice clearing stress levels indirectly, by theoretical arguments. For instance, Sanderson (1988) analysed the work done against gravity which is required to rise or sink ice fragments after their failure, and compared it to the work done by the forces acting in impacting ice. On this basis it has been inferred that the typical clearing stress is appreciably smaller than the failure stress, and the value of $\sigma _c=0.05\,{\mathrm {MPa}}$ has been proposed for multi-year ice. It seems, however, that for thinner, first-year ice considered here the clearing stresses are even smaller. Therefore, the value of $0.02\,{\mathrm {MPa}}$ has been adopted as a mean magnitude of $\sigma _c$, together with a $50\%$ variation coefficient accounting for its possible scatter, which is the same value of the variation coefficient as that assumed for the failure stress scatter.

Finally, the third free parameter in the proposed ice floe impact model is the critical strain $\epsilon _f$, developing during the process of brittle crushing of ice, and determining the critical axial displacement $\Delta$ (see Fig. 4.34) through the relation $\Delta =\epsilon _f b$. There are some experimental data regarding the strain magnitudes at which ice fractures, but these are limited to small-scale laboratory tests on fresh-water ice samples (Schulson and Gratz 1999; Iliescu and Schulson 2002) , and thus have little relevance to large-scale field conditions. Therefore, as in the above case of $\sigma _c$, the value of the critical strain $\epsilon _f$ has been inferred by theoretical considerations (Sanderson 1988) , and the values ranging from $\epsilon _f=0.02$ to $\epsilon _f=0.05$ have been obtained for multi-year ice. In the present analysis the lower value is adopted, that is $\epsilon _f=0.02$, in belief that pre-failure strains that develop in young ice are smaller than those occurring in thick, multi-year ice.

4.3.2 Numerical Method

The ice floe impact model, based on the assumptions and simplifications discussed above, has been applied to simulate dynamic ice–structure interaction events. In these simulations, the direction of the floe movement is defined by an axis x, with the origin $x=0$ on the structure wall and its increasing coordinate measuring the ice penetration distance, shown in Fig. 4.34 (see also Fig. 4.35 on p. 123). Further, it is assumed that the first contact between the floe and the structure takes place at time $t=0$ and the floe velocity is then $v=V_0$.

The computations proceed in the following steps (Staroszczyk 2007) :

• 1. Given the initial geometry of an ice floe, its in-plane dimensions and a mean thickness h, the ice sheet is discretized in the way shown in Fig. 4.33, by choosing the ice cell size b of a magnitude close to h. Also the floe mass, m, and its initial kinetic energy are evaluated.

• 2. For each discrete contact zone at the ice–structure interface, a separate stochastic realization of loading, as illustrated in Fig. 4.34, is prescribed, with the values of the failure and clearing stress as well as the distance between consecutive failure stress peaks randomized about their mean values. For this purpose, standard random number generators for the uniform and normal probability distributions are used.

• 3. At each calculation step, for $t>0$, the floe is advanced by a small increment $\delta x$, chosen to be a fraction of the critical displacement $\Delta =\epsilon _f b$. For the current value of x, a local contact stress, determined from the respective realization of loading, is calculated for each discrete zone, and all these local stresses, multiplied by the respective local contact areas, are summed up to yield a total impact force, F, at current x and t.

• 4. Assuming that during a given displacement step k ($k=1,2,3,\ldots$) the interaction force, $F_k$, is constant, the total work done by this force over the distance $\delta x$ is determined as $F_k \delta x$. Equating that work with the amount of the total kinetic energy of the floe lost due to the decrease in its velocity from the value of $v_{k-1}$ to $v_k$, the current velocity $v_k$ can be evaluated from the relation

  $$\begin{aligned} v_k^2=v_{k-1}^2-\frac{2F_k}{m}\,\delta x, \quad k=1,2,3,\ldots , \quad v_0=v(t=0)=V_0. \end{aligned}$$

  (4.110)

• 5. Assuming a linear variation of the floe velocity at each step, the time that elapsed during the advance of the floe at the k-th step, denoted by $(\delta t)_k$, is calculated from the formula

  $$\begin{aligned} (\delta t)_k=\frac{2\,\delta x}{v_{k-1}+v_k}\,. \end{aligned}$$

  (4.111)

  All the time increments, added up over all preceding displacement steps, determine the current value of time t elapsed since the beginning of the interaction event.

The procedure outlined above yields time histories of the total interaction force F, the floe velocity v, and the penetration distance x for one particular stochastic realization of an impact event. For each such a realization, a magnitude of the maximum force F occurring during an entire interaction event is found, and then, by simulating a large number of random realizations, probability distributions for the peak ice impact loads F are calculated.

Fig. 4.35

Cylindrical structure of radius $r_0$ in contact a with a floe of radius $R_0$, b with a straight-edged floe. The hatched areas show the regions of ice penetrated by the structure

4.3.3 Simulations of Forces Exerted on a Structure

The proposed model has been used to simulate a series of dynamic impact events of the geometries shown in Fig. 4.35. The basic configuration investigated in the simulations is that depicted in Fig. 4.35a, showing a circular cylindrical structure of radius $r_0$ interacting with a circular ice floe of radius $R_0$ and an average thickness h. In order to examine the effect of the shape of the impacting floe edge on the forces sustained by the structure, also the configuration presented in Fig. 4.35b has been considered, in which the projection of the leading edge on the horizontal plane is a straight line normal to the direction of the floe movement.

The numerical computations have been carried out for a vertically-walled circular cylinder of radius $r_0=10\,{\mathrm {m}}$. As discussed earlier in Sect. 4.3.1, the three basic parameters of the model have the values: $\sigma _f^*=1.66\,{\mathrm {MPa}}$, $\sigma _c=0.02\,{\mathrm {MPa}}$ and $\epsilon _f=0.02$. The ice density (needed to determine the total mass of the floe) was taken as $900\,{\mathrm {kg\,m}}^{-3}$. All probability distributions presented below have been obtained by running the model repeatedly for 10,000 times.

Fig. 4.36

a Typical history of the total force $F_x$, b the floe velocity and c the ice penetration distance during an impact event, for the floe radius $R_0=100\,{\mathrm {m}}$ and thickness $h=0.5\,{\mathrm {m}}$, and its initial velocity $V_0=0.5\,{\mathrm {m\,s}}^{-1}$

Typical time histories of (a) the total force, $F_x(t)$, acting on the structure in the direction of the floe advance, (b) the floe velocity, v(t), and (c) the ice penetration distance, x(t), are shown in Fig. 4.36. These results have been obtained for a floe of radius $100\,{\mathrm {m}}$ and thickness $0.5\,{\mathrm {m}}$, moving towards the structure at the velocity of $0.5\,{\mathrm {m\,s}}^{-1}$. It can be seen that, for the particular realization illustrated, the impact event lasts nearly 25 s, the ice floe moves a distance of about $6.8\,{\mathrm {m}}$ before it comes to rest, and the peak forces exerted on the structure have magnitudes close to $1.6\,{\mathrm {MN}}$. As anticipated, the time-variation of the impact forces is very irregular. On the contrary, the floe velocity v and its position x vary in a relatively smooth way. A characteristic feature is a gradual increase of peak forces as the collision progresses—this is because the total ice–cylinder contact area increases with the interaction time, so that the number of local zones at which ice fails increases.

The results plotted in Fig. 4.36 illustrate a single impact event, which due to the intrinsic randomness of the process may not be representative for the dynamic phenomenon under consideration. More general information can be obtained by a statistical analysis of a series (in our case 10,000) of randomized runs. It turns outs that, for the same input data as above, the mean values of the peak load x- and y-components (see Fig. 4.35) are $\bar{F}_x=1.58\,{\mathrm {MN}}$ and $\bar{F}_y=0.59\,{\mathrm {MN}}$, with respective standard deviations 0.18 and $0.08\,{\mathrm {MN}}$. Similar ratios of the $F_y$ to $F_x$ components have been obtained for other combinations of the floe parameters (Staroszczyk 2007) . This means that the average lateral peak force equals nearly 2 / 5 of the average longitudinal component, indicating thus that the ice–structure interaction loads are far from symmetric with respect to the x-axis direction (the direction of the floe advancement).

Fig. 4.37

Exceedance probability distributions of peak impact loads exerted on a structure for different floe thicknesses h ($R_0=100\,{\mathrm {m}}$, $V_0=0.5\,{\mathrm {m\,s}}^{-1}$). Compared are the results for circular (lines) and straight-edged (symbols) floes

The following three diagrams show density probability distributions of the peak ice–structure interaction forces. These figures illustrate the influence of the floe velocity and its thickness and planar size on the magnitude of the total longitudinal contact force $F_x$ and the statistics of its occurrence. In the plots, for each value of the load $F_x$ obtained from the simulations, the probability that this particular value will be exceeded is shown. Fig. 4.37 demonstrates the effect of the floe thickness h on the exceedance probabilities of the total impact loads, for the floe of radius $R_0=100\,{\mathrm {m}}$ and its initial velocity $V_0=0.5\,{\mathrm {m\,s}}^{-1}$. It is seen that the influence of the floe edge geometry on the load probability distributions is negligibly small—the maximum relative discrepancies are of order 1%. On the other hand, the influence of the ice thickness on the impact load magnitudes is, obviously, significant. However, the total loads sustained by the cylinder are not roughly proportional to the ice floe thickness, as could be expected at first sight, which is due to the scale-effects (the thinner ice has larger fracture strength $\sigma _f^*$ than the thicker one).

The plots in Fig. 4.38 display the exceedance probability curves for the peak loads $F_x$ as a function of the initial velocity of ice, $V_0$, with the ice floe thickness and radius kept constant. Hence, the probability distributions for $F_x$ are plotted for the values of $V_0$ ranging from 0.2 to $0.5\,{\mathrm {m\,s}}^{-1}$. It can be noted that, despite an increase in the floe kinetic energy by a factor of 6.25 when the velocity changes from 0.2 to $0.5\,{\mathrm {m\,s}}^{-1}$, the magnitudes of the impact forces, at the same exceedance probability, increase only by a factor of about 1.25 to 1.4.

Fig. 4.38

Exceedance probability distributions of peak impact loads exerted on a structure for different initial floe velocities $V_0$ ($h=0.5\,{\mathrm {m}}$, $R_0=100\,{\mathrm {m}}$)

In Fig. 4.39 the floe size effect on the exceedance probabilities of the peak impact forces is illustrated, by presenting the results of simulations carried out for circular floes of radii $R_0$ varying between 50 and $200\,{\mathrm {m}}$. In a way, the character of the exceedance curves resembles that in the previous diagram. Although the total kinetic energy of the moving floe increases significantly with increasing floe radius (by a factor of 16 between the smallest and the largest floes considered), the corresponding peak load magnitudes vary merely by a factor of about 1.5 at the same exceedance probability. This demonstrates once again the complexity of the interaction mechanism in which extensive brittle fracture of ice takes place.

Fig. 4.39

Exceedance probability distributions of peak impact loads exerted on a structure for different floe radii $R_0$ ($h=0.5\,{\mathrm {m}}$, $V_0=0.5\,{\mathrm {m\,s}}^{-1}$)

There is no doubt that the above-presented model considerably simplifies the real phenomenon of the ice floe impact on an engineering structure. However, its main purpose has been to provide an engineer with the estimations of the magnitudes of the total forces exerted by ice on a structure in order to carry out a risk assessment analysis, without a detailed consideration of the local mechanisms occurring in the immediate vicinity of the structure walls and the processes taking place in the ice itself. Such a more insightful analysis of the brittle behaviour of ice impacting a structure has become possible in the past decade with the fast development of the discrete-element method (DEM) and its application to sea ice problems. Some examples of the application of this still relatively new discrete method to the ice–structure interaction problems can be found in the papers by Polojärvi and Tuhkuri (2009) and Polojärvi et al. (2015) . In recent papers by Herman (2016, 2017) , the DEM has been employed for the numerical analysis of the problem of surface wave-induced breaking of floating ice. It seems that an extension of the latter model by accounting for the ice–structure and wave–structure interaction mechanisms would make possible a very realistic dynamic analysis of the coupled wave–ice–structure system (though, certainly, such an analysis would involve high computational costs typical of the discrete-element method applications).