Nonimpact Problems

Abstract

In Chap. 8, peridynamic solutions of many benchmark problems were presented and compared with the classical theory in the absence of failure prediction. This chapter presents solutions to various problems while considering failure initiation and propagation. When available and suitable, the peridynamic (PD) predictions are compared with the finite element analysis (FEA) solutions.

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In Chap. 8, peridynamic solutions of many benchmark problems were presented and compared with the classical theory in the absence of failure prediction. This chapter presents solutions to various problems while considering failure initiation and propagation. When available and suitable, the peridynamic (PD) predictions are compared with the finite element analysis (FEA) solutions.

An isotropic plate with a hole is slowly stretched along its horizontal boundaries. Its solution is straightforward when failure prediction is not a concern; however, it poses a challenge from a failure analysis point of view. Based on the Linear Elastic Fracture Mechanics (LEFM) concept, the traditional finite elements fail to address crack initiation and growth when there is no pre-existing crack in the structure. This problem demonstrates the capability of peridynamics when addressing crack initiation and its propagation. Next, an isotropic plate with a pre-existing crack is stretched rather fast along its horizontal boundaries. The peridynamic solution to this problem captures the effect of rate of loading (stretching) on the evolution of dynamic crack growth. In order to include the presence of thermal loading, a bimaterial strip with mismatch in thermal expansion coefficients is subjected to a uniform temperature change. Finally, an isotropic plate is subjected to a temperature gradient rather than a uniform temperature change. The peridynamic solutions to these problems are obtained by developing specific FORTRAN programs, which are available on the website http://extras.springer.com.

9.1 Plate with a Circular Cutout Under Quasi-Static Loading

As shown in Fig. 9.1, an isotropic plate with a circular cutout is subjected to a slow rate of stretch along its horizontal edges, representing quasi-static loading. There exist no initial cracks of any form in its domain. The solution is obtained by specifying the geometric parameters, material properties, initial and boundary conditions as well as the peridynamic discretization and time integration parameters as:

Fig. 9.1

Geometry of a plate with a circular cutout under slow stretch and its discretization

Geometric Parameters

• Length of the plate: $L=50\ \mathrm{ mm}$

• Width of the plate: $W=50\ \mathrm{ mm}$

• Thickness of the plate: $h=0.5\ \mathrm{ mm}$

• Diameter of the cutout: $D=10\ \mathrm{ mm}$

Material Properties

• Young’s modulus: $E=192\ \mathrm{ GPa}$

• Poisson’s ratio: $\nu =1/3$

• Mass density: $\rho =8000\ \mathrm{ kg}/{{\mathrm{ m}}^3}$

Boundary Conditions

• ${{\dot{u}}_y}(x,{{{\pm L}} \left/ {2} \right.},t)=\pm 2.7541\times {10^{-7 }}\mathrm{ m}/\mathrm{ s}$

PD Discretization Parameters

• Total number of material points in the x-direction: $100$

• Total number of material points in the y-direction: $100+3+3$

• Total number of material points in the z-direction: $1$

• Spacing between material points: $\Delta =0.0005\ \mathrm{ m}$

• Incremental volume of material points: $\Delta V=1.25\times {10^{-10 }}{{\mathrm{ m}}^3}$

• Volume of fictitious boundary layer: $\Delta {V_{\delta }}=3\times 100\times 1\times \Delta V=3.75\times {10^{-8 }}\ {{\mathrm{ m}}^3}$

• Horizon: $\delta =3.015\ \Delta$

• Critical stretch (failure off): ${s_c}=1$

• Critical stretch (failure on): ${s_c}=0.02$

• Adaptive Dynamic Relaxation: ON

• Time step: $\Delta t=1.0\ \mathrm{ s}$

• Total number of time steps: 1,000

Numerical Results: First, the displacement field due to the applied loading is obtained and compared against finite element predictions in the absence of failure. The variations of horizontal and vertical displacements along the central x-axis and y-axis, respectively, are shown in Fig. 9.2. A close agreement is observed between PD predictions and FEA results with ANSYS, a commercially available code. This indicates that the values of PD parameters such as grid size and horizon, and the volume of the boundary region, provide accurate results. After establishing the values of the PD parameters, failure among the material points is allowed by specifying a critical stretch value of ${s_c}=0.02$, and the damage progression is examined at different time steps. Although there is no pre-existing crack in the plate, failure initiates in the form of a crack at the stress concentration sites. This is clearly an exceptional feature of the PD theory, unlike the other existing techniques that require pre-existing cracks. As shown in Fig. 9.3a, the damage initiates in the stress concentration sites at the end of 650 time steps. At the end of 700 time steps (Fig. 9.3b), the local damage value of some material points exceeds $\varphi =0.38$, resulting in self-similar crack growth. Due to the low value of applied velocity along the boundary, representative of quasi-static loading, the crack continues to propagate toward the external vertical boundaries, as shown in Fig. 9.3c, d.

Fig. 9.2

Variation of horizontal displacement (a) and vertical displacement (b) along the central axes at the end of 1,000 time steps when failure is not allowed

Fig. 9.3

Damage plots for the plate with a circular cutout at the end of (a) 650 time steps, (b) 700 time steps, (c) 800 time steps, and (d) 1,000 time steps

9.2 Plate with a Pre-existing Crack Under Velocity Boundary Conditions

The circular hole is replaced with a pre-existing crack, as shown in Fig. 9.4. Also, its horizontal edges are subjected to a very fast rate of stretch (velocity) in order to observe how the rate of loading affects the evolution of dynamic crack growth. The plate properties and geometry are the same as before except for the thickness. The thickness and boundary conditions, as well as the peridynamic discretization and time integration parameters, are specified as:

Fig. 9.4

Geometry of a plate with a pre-existing crack under velocity boundary conditions and its discretization

Geometric Parameters

• Thickness of the plate: $h=0.0001\ \mathrm{ m}$

• Initial length of the pre-existing crack: $2a=0.01\ \mathrm{ m}$

Boundary Conditions

• Case 1: ${{\dot{u}}_y}(x,{{{\pm L}} \left/ {2} \right.},t)=\pm 20.0\ \mathrm{ m}/\mathrm{ s}$

• Case 2: ${{\dot{u}}_y}(x,{{{\pm L}} \left/ {2} \right.},t)=\pm 50.0\ \mathrm{ m}/\mathrm{ s}$

PD Discretization Parameters

• Total number of material points in the x-direction: $500$

• Total number of material points in the y-direction: $500+3+3$

• Total number of material points in the z-direction: $1$

• Spacing between material points: $\Delta =0.0001\ \mathrm{ m}$

• Incremental volume of material points: $\Delta V=1\times {10^{-12 }}\ {{\mathrm{ m}}^3}$

• Volume of fictitious boundary layer: $\Delta {V_{\delta }}=3\times 100\times 1\times \Delta V=3\times {10^{-10 }}\ {{\mathrm{ m}}^3}$

• Horizon: $\delta =3.015\ \Delta$

• Adaptive Dynamic Relaxation: OFF

• Time step: $\Delta {}t=1.3367\times {10^{-8 }}\ \mathrm{ s}$

• Total number of time steps: 1,250

• Critical stretch (failure off): ${s_c}=1$

• Critical stretch (failure on): ${s_c}=0.04472$

Numerical Results: First, failure is not allowed (the interaction between the material points never ceases), and the crack opening displacement is computed as shown in Fig. 9.5. Unlike the elliptical crack opening displacement of classical continuum mechanics, the PD analysis predicts a cusp-like crack opening displacement near the crack tip. As explained by Silling (2000), the elliptical crack opening displacement is a mathematical requirement of the unbounded stresses (physically impossible) near the crack tip. The PD theory successfully captures a more physically meaningful crack opening shape. When failure is allowed by using a critical stretch value of ${s_c}=0.04472$, a self-similar crack growth is observed at the end of 1,250 time steps, as shown in Fig. 9.6a. This growth is typical for a mode-I type of loading. The position of the crack tip or crack growth is determined based on the local damage value of any material point that exceeds $\varphi =0.38$ along the $x\text{--}\mathrm{ axis}$. The growth of a crack as a function of time is shown in Fig. 9.6b, and the crack growth speed can be evaluated as 1,650 m/s. This crack speed is less than the Rayleigh wave speed of 2,800 m/s, which is considered to be the upper limit of the crack growth speed for a mode-I type of loading (Silling and Askari 2005). If the applied velocity boundary condition is increased from ${V_0}(t)=20\mathrm{ m}/\mathrm{ s}$ to ${V_0}(t)=50\mathrm{ m}/\mathrm{ s}$, the crack growth characteristics change from self-similar to branching, as shown in Fig. 9.7. It is worth noting that the only parameter that was different between the two PD analyses while obtaining Figs. 9.6a and 9.7 is due to the applied velocity boundary condition. All other parameters remain the same. The PD theory captures a very complex phenomenon of crack branching without resorting to any external criteria that triggers branching.

Fig. 9.5

Crack opening displacement near the crack tip at the end of 1,250 time steps when failure is not allowed

Fig. 9.6

(a) Damage indicating self-similar crack growth at the end of 1,250 time steps under the velocity boundary condition of ${V_0}(t)=20\mathrm{ m}/\mathrm{ s}$ and (b) crack growth as a function of time

Fig. 9.7

Damage indicating crack branching at the end of 1,000 time steps under a velocity boundary condition of ${V_0}(t)=50\mathrm{ m}/\mathrm{ s}$

9.3 Bimaterial Strip Subjected to Uniform Temperature Change

A bimaterial strip is subjected to a uniform temperature change, as shown in Fig. 9.8. Both the top and bottom regions have the same length and thickness, but different widths and thermal expansion coefficients. Although the bimaterial strip is free of constraints and the temperature change is uniform, the mismatch between the thermal expansion coefficients causes bending deformation. The interface has the same properties as those of the top plate, and failure is not allowed. The solution is obtained by specifying the geometric parameters, material properties, initial and boundary conditions, as well as the peridynamic discretization and time integration parameters as:

Fig. 9.8

Geometry of a bimaterial strip subjected to uniform temperature change and its discretization

Geometric Parameters

• Length of plate: $L=30\ \mathrm{ mm}$

• Thickness of plate: $h = 0.1\ \mathrm{ mm}$

• Width of bottom plate: ${W_b} = 1\ \mathrm{ mm}$

• Width of top plate: ${W_t} = 3\ \mathrm{ mm}$

Material Properties

• Young’s modulus of bottom plate: ${E_b}=128.0\mathrm{ GPa}$

• Poisson’s ratio of bottom plate: ${\nu_b}=1/3$

• Thermal expansion coefficient of bottom plate: ${\alpha_b}=16.6\times {10^{-6 }}/^{\circ}\mathrm{ C}$

• Young’s modulus of top plate: ${E_t}=5.1\ \mathrm{ GPa}$

• Poisson’s ratio of top plate: ${\nu_t}=1/3$

• Thermal expansion coefficient of top plate: ${\alpha_t}=50\times {10^{-6 }}/^{\circ}\mathrm{ C}$

Applied Loading

• Uniform temperature change: $\Delta T = 50^{\circ}\mathrm{ C}$

PD Discretization Parameters

• Total number of material points in x-direction: $300$

• Total number of material points in z-direction: $1$

• Total number of material points in bottom plate in y-direction: $10$

• Total number of material points in top plate in y-direction: $30$

• Spacing between material points, $\Delta =0.1\ \mathrm{ mm}$

• Horizon: $\delta =3.015{}\ \Delta$

• Adaptive Dynamic Relaxation: ON

• Incremental time step size: $\Delta t=1.0\ \mathrm{ s}$

• Total number of time steps: 20,000

Numerical Results: The PD predictions for displacement components, ${u_x}$ and ${u_y}$, along the interface between two plates are compared with those of FEA simulations. As observed in Fig. 9.9, the results from both approaches agree very well with each other. As expected, the bimaterial strip is curled down, presented in Fig. 9.9b, due to the mismatch between the coefficients of thermal expansion.

Fig. 9.9

Variation of (a) ${u_x}$ displacement component, and (b) ${u_y}$ displacement component along the interface between two materials

9.4 Rectangular Plate Subjected to Temperature Gradient

An isotropic plate is subjected to a nonuniform temperature change, as shown in Fig. 9.10. It is free of any constraints and, by specifying a unit critical stretch value, failure is not allowed. The solution is obtained by specifying the geometric parameters, material properties, initial and boundary conditions, as well as the peridynamic discretization and time integration parameters as:

Fig. 9.10

Geometry of a rectangular plate subjected to a temperature gradient and its discretization

Geometric Parameters

• Length of the plate: $L=10\ \mathrm{ in}.$

• Width of the plate: $W=4\ \mathrm{ in}.$

• Thickness of the plate: $h=0.04\ \mathrm{ in}.$

Material Properties

• Young’s modulus: $E=1\times {10^7}\mathrm{ psi}$

• Poisson’s ratio: $\nu =1/3$

• Mass density: $\rho =0.1\ \mathrm{ lb}/\mathrm{ i}{{\mathrm{ n}}^3}$

• Thermal Expansion Coefficient: $\alpha =24\times {10^{-6 }}/^{\circ}\mathrm{ C}$

Applied Loading

• Temperature change: $\Delta T = 5.0\left( {x+5.0} \right)$

PD Discretization Parameters

• Total number of material points in the x-direction: $250$

• Total number of material points in the y-direction: $100$

• Total number of material points in the z-direction: $1$

• Spacing between material points: $\Delta =0.04\ \mathrm{ in}.$

• Horizon: $\delta =3.015{}\ \Delta$

• Adaptive Dynamic Relaxation: ON

• Time step size: $\Delta t=1.0\ \mathrm{ s}$

• Total number of time steps: 4,000

Numerical Results: The steady-state solution from PD is compared with the FEA predictions. The comparison of ${u_x}$ displacement values along the central x-axis and the ${u_y}$ displacement values along the central y-axis, shown in Fig. 9.11, indicates a close agreement.

Fig. 9.11

(a) ${u_x}$ displacement variation along central x-axis and (b) ${u_y}$ displacement variation