Coupling of the Peridynamic Theory and Finite Element Method

Abstract

The PD theory provides deformation, as well as damage initiation and growth, without resorting to external criteria since material failure is invoked in the material response. However, it is computationally more demanding compared to the finite element method. Furthermore, the finite element method is very effective for modeling problems without damage. Hence, it is desirable to couple the PD theory and FEM to take advantage of their salient features if the regions of potential failure sites are identified prior to the analysis. Then, the regions in which failure is expected can be modeled by using the PD theory and the rest can be analyzed by using FEM.

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The PD theory provides deformation, as well as damage initiation and growth, without resorting to external criteria since material failure is invoked in the material response. However, it is computationally more demanding compared to the finite element method. Furthermore, the finite element method is very effective for modeling problems without damage. Hence, it is desirable to couple the PD theory and FEM to take advantage of their salient features if the regions of potential failure sites are identified prior to the analysis. Then, the regions in which failure is expected can be modeled by using the PD theory and the rest can be analyzed by using FEM.

A simple coupling approach is submodeling, demonstrated by Oterkus et al. (2012) and Agwai et al. (2012); it involves FEM for global analysis and PD theory for submodeling in order to perform failure prediction. The primary assumption in submodeling is that the structural details of the submodel do not significantly affect the global model. Also, the boundaries of the submodel should be sufficiently far away from local features so that St. Venant’s principle holds for a valid submodeling analysis. The solution obtained from the global model along the boundary of the domain of interest is applied as displacement boundary conditions on the submodel. The global model should be refined enough to enable accurate calculation of the displacement on the boundary of the submodeling region. Also, different time discretizations of the displacement boundary condition should be considered because the time-dependent nature of boundary conditions may affect the results in submodeling.

Another straightforward way of coupling was suggested by Macek and Silling (2007) where the PD interactions are represented by using traditional truss elements and an embedded element technique for the overlap region. Lall et al. (2010) also utilized this approach to study shock and vibration reliability of electronics.

Recently, Liu and Hong (2012) introduced interface elements between FEM and PD regions. A finite number of peridynamic points are embedded inside the interface element to transfer information between PD and FEM regions. The peridynamic forces exerted on these embedded material points are distributed as nodal forces to the interface element based on two particular schemes. In the first scheme, coupling forces are distributed to all nodes of the interface element. However, in the second scheme, the coupling forces are only distributed to nodes that are located on the interface plane between FEM and PD regions. The displacement of the embedded material points are not computed through a PD equation of motion. Instead, they are determined by utilizing the nodal displacements of interface elements and their shape functions.

Also, Lubineau et al. (2012) coupled local and nonlocal theories by introducing a transition (morphing) strategy. The definition of the morphing functions relies on the energy equivalence, and the transition region affects only constitutive parameters. The influence of local and nonlocal theories is captured by defining a function that automatically converges to full local and nonlocal formulations along their respective boundaries. In a recent study, Seleson et al. (2013) proposed a force-based blended model that coupled PD theory and classical elasticity by using nonlocal weights composed of integrals of blending functions. They also generalized this approach to couple peridynamics and higher-order gradient models of any order.

In addition to these techniques, Kilic and Madenci (2010) introduced a direct coupling of FEM and PD theory using an overlap region, shown in Fig. 11.1a, in which equations of both PD and FEM are solved simultaneously. The PD region is discretized with material points and the finite element region with traditional elements (Fig. 11.1b). Both the PD and FE equations are satisfied in the overlap region. Furthermore, the displacement and velocity fields are determined using finite element equations in the overlap region. These fields are then utilized to compute the body force densities using the PD theory. Finally, these body force densities serve as external forces for finite elements in the overlap region.

Fig. 11.1

Schematic for coupling of the finite element method and peridynamics: (a) finite element (FEA) and peridynamic regions; (b) discretization

11.1 Direct Coupling

The direct coupling of PD theory and FEM presented herein concerns steady-state or quasi-static solutions. However, the PD equation of motion, Eq. 7.1, includes dynamic terms that need to be eliminated. Thus, the adaptive dynamic relaxation method, described in Chap. 7, is utilized to obtain a steady-state solution. The damping coefficient is changed adaptively in each time step. The dynamic relaxation method is based on the fact that the static solution is the steady-state part of the transient response of the solution.

In order to achieve direct coupling, the discrete PD equation of motion, Eq. 7.1, is rewritten as

$$ \left\{ {\begin{array}{*{20}{c}} {\mathbf{{\underline{\ddot{U}}}}_p^n} \\ {\mathbf{{\underline{{\underline{\ddot{U}}}}}}_p^n} \\ \end{array}} \right\}+{c^n}\left\{ {\begin{array}{*{20}{c}} {\mathbf{{\underline{\dot{U}}}}_p^n} \\ {\mathbf{{\underline{{\underline{\dot{U}}}}}}_p^n} \\ \end{array}} \right\}=\left[ {\begin{array}{*{20}{c}} {{{{\underline{\mathbf{D}}}}^{-1 }}} & 0 \\ 0 & {{{{\mathbf{{\underline{{\underline{D}}}}}}}^{-1 }}} \\ \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\underline{\mathbf{F}}_p^n} \\ {\mathbf{{\underline{{\underline{F}}}}}_p^n} \\ \end{array}} \right\}, $$

(11.1)

in which U is a vector that contains displacements at the PD material points and the vector F is the summation of internal and external forces. The subscript p denotes the variables associated with the PD region, and single and double underscores denote the variables located outside and inside the overlap region, respectively. The parameter ${c^n}$ represents the damping coefficient at the nth time increment. The coefficients of the fictitious diagonal density matrix, $\mathbf{D}$, are determined through Greschgorin’s theorem (Underwood 1983). A detailed description of these parameters is presented in Chap. 7.

In order to achieve coupling of FEM with the PD theory, the direct assembly of finite element equations without constructing the global stiffness matrix is utilized so that the FE equations can be expressed as

$$ \left\{ {\begin{array}{*{20}{c}} {\mathbf{{\underline{\ddot{U}}}}_f^n} \\ {\mathbf{{\underline{{\underline{\ddot{U}}}}}}_f^n} \\ \end{array}} \right\}+{c^n}\left\{ {\begin{array}{*{20}{c}} {\mathbf{{\underline{\dot{U}}}}_f^n} \\ {\mathbf{{\underline{{\underline{\dot{U}}}}}}_f^n} \\ \end{array}} \right\}=\left[ {\begin{array}{*{20}{c}} {{{{\underline{\mathbf{M}}}}^{-1 }}} & 0 \\ 0 & {{{{\mathbf{{\underline{{\underline{M}}}}}}}^{-1 }}} \\ \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\underline{\mathbf{F}}_f^n} \\ {\mathbf{{\underline{{\underline{F}}}}}_f^n} \\ \end{array}} \right\}, $$

(11.2)

in which subscript f denotes the variables associated with the finite element region, and M is the diagonal mass matrix. The components of the mass matrix can be approximated as

$$ \mathbf{M}=\mathbf{I}\tilde{\mathbf{m}}, $$

(11.3)

in which $\mathbf{I}$ is the identity matrix. The mass vector, $\tilde{\mathbf{m}}$, is constructed as

$$ \tilde{\mathbf{m}} =\mathop{\mathbf{A}}\limits_e{{\hat{\mathbf{m}}}^{(e) }}, $$

(11.4)

where $\mathbf{A}$ is the assembly operator and the operations are strictly performed as additions (Belytschko 1983). The components of vector ${{\hat{\mathbf{m}}}^{(e) }}$ can be written as

$$ \hat{m}_i^{(e) }=\sum\limits_{j=1}^8 {\left| {k_{ij}^{(e) }} \right|}, $$

(11.5)

in which $k_{ij}^{(e) }$ indicates the components of the element stiffness matrix given by Zienkiewicz (1977).

The force vector ${{\mathbf{F}}^n}$ at the nth time increment can be expressed as

$$ {{\mathbf{F}}^n}={{\mathbf{f}}^{ext }}({{\boldsymbol{ t}}^n})-{{\mathbf{f}}^{{\operatorname{int}}}}({{\mathbf{u}}^n}), $$

(11.6)

where t is time and ${{\mathbf{f}}^{ext }}$ is the vector of external forces.

The internal forces resulting from the deformation of the elements can be assembled into a global array of internal forces by using the convention of Belytschko (1983) as

$$ {{\mathbf{f}}^{{\operatorname{int}}}}=\mathop{\mathbf{A}}\limits_e{{\mathbf{f}}^{(e) }}, $$

(11.7)

where ${{\mathbf{f}}^{(e) }}$ is the element force vector.

The element force vector is expressed as

$$ {{\mathbf{f}}^{(e) }}={{\mathbf{k}}^{(e) }}{{\mathbf{u}}^{(e) }}, $$

(11.8)

in which ${{\mathbf{k}}^{(e) }}$ is the element stiffness matrix described by Zienkiewicz (1977) and ${{\mathbf{u}}^{(e) }}$ is the vector representing the nodal displacements of the eth element.

The vector ${{\mathbf{u}}_p}$ representing displacements of a PD material point located inside the eth element can be obtained from

$$ {{\mathbf{u}}_p}=\sum\limits_{i=1}^8 {{N_i}\mathbf{u}_i^{(e) }}, $$

(11.9)

where ${N_i}$ are the shape functions given by Zienkiewicz (1977). The vector $\mathbf{u}_i^{(e) }$ is the ith nodal displacements of the eth element and is extracted from the global solution vector, ${{\mathbf{{\underline{{\underline{U}}}}}}_f}$, denoting nodal FE displacements. Determination of the vector ${{\mathbf{u}}_p}$ leads to the computation of vector ${{\mathbf{{\underline{{\underline{U}}}}}}_p}$. The force density vector $\mathbf{{\underline{{\underline{F}}}}}_p^n$ can then be computed by utilizing the force density vector ${{\mathbf{F}}_p}$ associated with the PD material point ${{\mathbf{x}}_p}$ inside the eth element (subdomain) as given by Eq. 7.1

$$ \begin{array}{lll} {{\mathbf{F}}_p} =\mathbf{b}({{\mathbf{x}}_p},\boldsymbol{ t})+\sum\limits_{e=1}^N {\sum\limits_{j=1}^{{{N_e}}} {{w_{(j) }}\left[ {\mathbf{t}\left( {\mathbf{u}\left( {{{\mathbf{x}}_{(j) }},\boldsymbol{ t}} \right)-\mathbf{u}\left( {{{\mathbf{x}}_p},\boldsymbol{ t}} \right),{{\mathbf{x}}_{(j) }}-{{\mathbf{x}}_p},\boldsymbol{ t}} \right)} \right.}} \hfill \\ \quad-\left. {\mathbf{t}\left( {\mathbf{u}\left( {{{\mathbf{x}}_p},\boldsymbol{ t}} \right)-\mathbf{u}\left( {{{\mathbf{x}}_{(j) }},\boldsymbol{ t}} \right),{{\mathbf{x}}_p}-{{\mathbf{x}}_{(j) }},\boldsymbol{ t}} \right)} \right]{V_{(j) }}, \end{array} $$

(11.10)

where $N$ is the number of elements within the horizon and ${N_e}$ is the number of collocation points in the ${e^{th }}$ element. The position vector ${{\mathbf{x}}_{(j) }}$ represents the location of the ${j^{th }}$ collocation (integration) point.

Since, for a quasi-static problem, the equilibrium should be satisfied for all material points, i.e., ${{\mathbf{F}}_p}=\mathbf{0}$, the body load exerted on material point ${{\mathbf{x}}_p}$ can then be calculated as

$$ \begin{array}{lll} \mathbf{b}({{\mathbf{x}}_p},\boldsymbol{ t}) =-\sum\limits_{e=1}^N {\sum\limits_{j=1}^{{{N_e}}} {{w_{(j) }}\left[ {\mathbf{t}\left( {\mathbf{u}\left( {{{\mathbf{x}}_{(j) }},\boldsymbol{ t}} \right)-\mathbf{u}\left( {{{\mathbf{x}}_p},\boldsymbol{ t}} \right),{{\mathbf{x}}_{(j) }}-{{\mathbf{x}}_p},\boldsymbol{ t}} \right)} \right.}} \hfill \cr \quad-\left. {\mathbf{t}\left( {\mathbf{u}\left( {{{\mathbf{x}}_p},\boldsymbol{ t}} \right)-\mathbf{u}\left( {{{\mathbf{x}}_{(j) }},\boldsymbol{ t}} \right),{{\mathbf{x}}_p}-{{\mathbf{x}}_{(j) }},\boldsymbol{ t}} \right)} \right]{V_{(j) }}. \end{array} $$

(11.11)

The total body load associated with the element where the material point ${{\mathbf{x}}_p}$ is located can be computed as

$$ \mathbf{g}^{(e) }=\sum\limits_{j=1}^{{{N_e}}} {\mathbf{b}({{\mathbf{x}}_p},\boldsymbol{ t})}. $$

(11.12)

Furthermore, the calculated total body load can be lumped into the finite element nodes as

$$ \mathbf{f}_I^{(e) }=\int\limits_{{{V_e}}} {d{V_e}{N_I}\rho\mathbf{g}^{(e) }}, $$

(11.13)

in which $\rho$ is the mass density of the eth element and I indicates the Ith node of the eth element. Hence, $\mathbf{f}_I^{(e) }$ indicates the external force acting on the Ith node. The body force density is only known at the PD material points, which serve as integration points for the eth element in Eq. 11.13. Furthermore, $\mathbf{{\underline{{\underline{F}}}}}_f^n$ is constructed by assembling the nodal forces given by Eq. 11.13.

Finally, the coupled system of equations can be expressed as

$$ {{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\ddot{U}}}}}^n}+{c^n}{{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}}^n}={{\underset{\scriptscriptstyle\thicksim}{\mathbf{M}}}^{-1 }}{{\underset{\scriptscriptstyle\thicksim}{\mathbf{F}}}^n}, $$

(11.14)

in which $\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}$ and $\mathbf{{\underset{\scriptscriptstyle\thicksim}{\ddot{U}}}}$ are the first and second time derivatives of the displacements, respectively, and can be expressed as

$$ {{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}}^n}={{\left\{ {\begin{array}{*{20}{c}} {\mathbf{{\underset{\scriptscriptstyle-}{\dot{U}}}}_p^n} & {\mathbf{{\underset{\scriptscriptstyle-}{\dot{U}}}}_f^n} & {\mathbf{{\underset{\scriptscriptstyle-}{{\underset{\scriptscriptstyle-}{\dot{U}}}}}}_f^n} \\ \end{array}} \right\}}^T}, $$

(11.15a)

$$ {{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\ddot{U}}}}}^n}={{\left\{ {\begin{array}{*{20}{c}} {\mathbf{{\underset{\scriptscriptstyle-}{\ddot{U}}}}_p^n} & {\mathbf{{\underset{\scriptscriptstyle-}{\ddot{U}}}}_f^n} & {\mathbf{{\underset{\scriptscriptstyle-}{{\underset{\scriptscriptstyle-}{\ddot{U}}}}}}_f^n} \\ \end{array}} \right\}}^T}. $$

(11.15b)

The matrix $\underset{\scriptscriptstyle\thicksim}{\mathbf{M}}$ can be written as

$$ \underset{\scriptscriptstyle\thicksim}{\mathbf{M}}=\left[ {\begin{array}{*{20}{c}} {\underset{\scriptscriptstyle-}{\mathbf{D}}} & 0 & 0 \\ 0 & {\underset{\scriptscriptstyle-}{\mathbf{M}}} & 0 \\ 0 & 0 & {\mathbf{{\underset{\scriptscriptstyle-}{{\underset{\scriptscriptstyle-}{M}}}}}} \\ \end{array}} \right]. $$

(11.16)

The vector $\underset{\scriptscriptstyle\thicksim}{\mathbf{F}}$ is given as

$$ {{\underset{\scriptscriptstyle\thicksim}{\mathbf{F}}}^n}={{\left\{ {\begin{array}{*{20}{c}} {\underset{\scriptscriptstyle-}{\mathbf{F}}_p^n} & {\underset{\scriptscriptstyle-}{\mathbf{F}}_f^n} & {\mathbf{{\underset{\scriptscriptstyle-}{{\underset{\scriptscriptstyle-}{F}}}}}_f^n} \\ \end{array}} \right\}}^T}. $$

(11.17)

As suggested by Underwood (1983), the damping coefficient ${c^n}$ can be determined as

$$ {c^n}=2\sqrt{{{{{\left( {{{{\left( {{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{U}}}}^n}} \right)}}^T}{\,^1}{{\mathbf{K}}^n}{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{U}}}}^n}} \right)}} \left/ {{\left( {{{{\left( {{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{U}}}}^n}} \right)}}^T}\,{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{U}}}}^n}} \right)}} \right.}}}, $$

(11.18)

in which ${^1}{{\mathbf{K}}^n}$ is the diagonal “local” stiffness matrix expressed as (Underwood 1983)

$$ ^1K_{ii}^n=-{{{\left( {{{{{\underset{\scriptscriptstyle\thicksim}{F}}_i^n}} \left/ {{{{{{\underset{\scriptscriptstyle\thicksim}{m}}}}_{ii }}}} \right.}-{{{{\underset{\scriptscriptstyle\thicksim}{F}}_i^{n-1 }}} \left/ {{{{{{\underset{\scriptscriptstyle\thicksim}{m}}}}_{ii }}}} \right.}} \right)}} \left/ {{\,\dot{U}_i^{{n-{1 \left/ {2} \right.}}}}} \right.}. $$

(11.19)

The time integration is performed by utilizing the central-difference explicit integration, with a time step size of unity, as

$$ {{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}}^{{n+{1 \left/ {2} \right.}}}}=\frac{{\left( {2-{c^n}} \right){{{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}}}^{{n-{1 \left/ {2} \right.}}}}+2{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{M}}}}^{-1 }}{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{F}}}}^n}}}{{\left( {2+{c^n}} \right)}}, $$

(11.20a)

$$ {{\underset{\scriptscriptstyle\thicksim}{\mathbf{U}}}^{n+1 }}={{\underset{\scriptscriptstyle\thicksim}{\mathbf{U}}}^n} + {{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}}^{{n+{1 \left/ {2} \right.}}}}. $$

(11.20b)

However, the integration algorithm given by Eq. 11.20a, 11.20b cannot be used to start the integration due to an unknown velocity field at ${t^{{-{1 \left/ {2} \right.}}}}$, but integration can be started by assuming that ${{\underset{\scriptscriptstyle\thicksim}{\mathbf{U}}}^0}\ne 0$ and ${{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}}^0}=0$, which yields

$$ {{\mathbf{{\underset{\scriptscriptstyle\thicksim}{\dot{U}}}}}^{{{1 \left/ {2} \right.}}}}={{{{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{M}}}}^{-1 }}{{{\underset{\scriptscriptstyle\thicksim}{\mathbf{F}}}}^{{{1 \left/ {2} \right.}}}}}} \left/ {2} \right.}. $$

(11.21)

Finally, the steps in coupling FEM with PD can be summarized as:

• 1. Utilize displacement and velocity fields which are known at time steps i, where $i\leq n.$

• 2. Compute displacement of collocation points within the overlap region using nodal displacements within the overlap region.

• 3. Compute force densities associated with collocation points within the overlap region.

• 4. Apply force densities as body force to finite elements within the overlap region.

• 5. Integrate to find displacements and velocities at time step $(n+1)$.

• 6. Repeat previous steps to reach desired number of time steps.

11.2 Validation of Direct Coupling

The validity of the direct coupling approach is demonstrated by considering a bar and a plate with a hole, both of which are under tension. In the case of a bar, there exists only one overlap region between PD and FEM solution domains. In the case of a plate, the region of the hole where failure is expected to occur is modeled with the PD theory, and the regions far away from the hole are modeled with FEM, resulting in two overlap regions.

11.2.1 Bar Subjected to Tensile Loading

The isotropic bar under tension at both ends is divided into two regions for modeling with FEM and PD theory, as illustrated in Fig. 11.2.

Fig. 11.2

Dimensions of the bar

Geometric Parameters

• Length of the beam: $L=10\ \mathrm{ in}.$ (FEM region, ${L_f}=5\ \mathrm{ in}.;$ PD region, ${L_p}=5\ \mathrm{ in}.$)

• Cross-sectional area: $A=h\times h=0.16\ \mathrm{ i}{{\mathrm{ n}}^2}$

Material Properties

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

• Poisson’s ratio: $\nu =0.25$

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

Boundary Conditions

• Free of displacement constraints

Applied Loading

• Uniaxial tensile force: $F = 1600\ \mathrm{ lbs}.$

PD Discretization Parameters

• Total number of material points in the x-direction: $200$

• Total number of material points in the y-direction: $8$

• Total number of material points in the z-direction: $8$

• Spacing between material points: $\Delta =0.05\ \mathrm{ in}.$

• Incremental volume of material points: $\Delta V=125\times {10^{-6 }}\ \mathrm{ i}{{\mathrm{ n}}^3}$

• Boundary layer volume: $\Delta {V_{\Delta}}=1\times 8\times 8\times 125\times {10^{-6 }}\ \mathrm{ i}{{\mathrm{ n}}^3}=8\times {10^{-3 }}\ \mathrm{ i}{{\mathrm{ n}}^3}$

• Applied body force density: ${b_x}={F \left/ {{\Delta {V_{\Delta}}}} \right.}=2\times {10^5}\ \mathrm{ lb}/\mathrm{ i}{{\mathrm{ n}}^3}$

• Overlap region: ${L_b}=0.125\ \mathrm{ in}.$

• Horizon: $\delta =3\Delta$

• Adaptive Dynamic Relaxation: ON

• Incremental time step size: $\Delta t=1\ \mathrm{ s}$

In addition to the coupled approach, the entire bar is also modeled by using either the PD theory or the FEM. The FE model was constructed using SOLID45 brick elements of ANSYS, a commercially available program. The uniaxial tension is applied as surface tractions at the end surfaces of the bar. A comparison of the displacements from the coupled approach with those of the PD theory and the FEM is shown in Fig. 11.3. There is an approximately 5 % difference among the models using only the PD theory and FEM. The contour plot of horizontal displacements from the coupled approach is shown in Fig. 11.4.

Fig. 11.3

Comparison of horizontal displacements of the bar

Fig. 11.4

Horizontal displacement contour plot of the bar

11.2.2 Plate with a Hole Subjected to Tensile Loading

The isotropic plate with a hole under tension at both ends is divided into three different regions for coupled modeling of the FEM and PD theory, as illustrated in Fig. 11.5. The extent of the PD and FEM regions is defined by ${L_p}$ and ${L_f}$, respectively.

Fig. 11.5

Dimensions of the plate with a circular cutout

Geometric Parameters

• Length of the plate: $a=9\ \mathrm{ in}.$ (${L_f}=2.5\ \mathrm{ in}.$ and ${L_p}=4\ \mathrm{ in}.$)

• Width of the plate: $b=3\ \mathrm{ in}.$

• Thickness of plate, $h=0.2\ \mathrm{ in}.$

• Hole radius: $r=0.5\ \mathrm{ in}.$

Material Properties

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

• Poisson’s ratio: $\nu =0.25$

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

Boundary Conditions

• Free of displacement constraints

Applied Loading

• Uniaxial tensile force: $F = 6000\ \mathrm{ lbs}.$

PD Discretization Parameters

• Total number of material points in the x-direction: 180

• Total number of material points in the y-direction: $60$

• Total number of material points in the z-direction: $4$

• Spacing between material points: $\Delta =0.05\ \mathrm{ in}.$

• Incremental volume of material points: $\Delta V=125\times {10^{-6 }}\ \mathrm{ i}{{\mathrm{ n}}^3}$

• Overlap region: ${L_b}=0.125\ \mathrm{ in}.$

• Boundary layer volume: $\Delta {V_{\Delta}}=1\times 4\times 60\times 125\times {10^{-6 }}\ \mathrm{ i}{{\mathrm{ n}}^3}=0.03\ \mathrm{ i}{{\mathrm{ n}}^3}$

• Applied body force density: ${b_x}={F \left/ {{\Delta {V_{\Delta}}}} \right.}=2\times {10^5}\ \mathrm{ lb}/\mathrm{ i}{{\mathrm{ n}}^3}$

• Horizon: $\delta =3\Delta$

• Adaptive Dynamic Relaxation: ON

• Incremental time step size: $\Delta t=1\ \mathrm{ s}$

The three-dimensional model is constructed by discretizing the domain, as shown in Fig. 11.6. The validity of the coupled approach is established by comparing the steady-state displacements from the PD theory and FEM using ANSYS, a commercially available program. Both the PD and FE models are constructed by utilizing the same discretization as that of coupled model shown in Fig. 11.6. The FE model was constructed using the SOLID45 brick elements of ANSYS. Figure 11.7 shows the horizontal displacements along the bottom line of the plate. The comparison of horizontal displacements indicates a close agreement among the coupled analysis, peridynamic theory, and finite element method.

Fig. 11.6

Three dimensional discretization of the plate for coupled analysis

Fig. 11.7

Horizontal displacements along the lower edge of plate