A Coupled Peridynamic and Finite Element Approach in ANSYS Framework for Fatigue Life Prediction Based on the Kinetic Theory of Fracture

Abstract

This study presents a coupled peridynamics (PD) and finite element (FE) approach to simulate the process of failure due to cyclic loading based on the kinetic theory of fracture (KTF). It specifically leads to the prediction of number of load cycles to crack initiation and its propagation path. The PD representation of the equilibrium equations and the stress–strain relations are derived based on the PD least square minimization (PD LSM) method. The PD governing equations are constructed by using MATRIX27 element in the ANSYS framework and solved by employing an implicit method. The PD interactions are considered in the region of potential failure sites; otherwise, traditional finite elements are employed in the discretization of the domain. The coupling between the MATRIX27 elements and traditional finite elements are achieved through the coupled degrees of freedom (DOF) command available in the ANSYS framework. The verification of the coupled PD-FE approach is demonstrated by comparison against the FE prediction of displacement fields in a plate with and without a hole under tension. Its validity for predicting crack growth is established by simulating compact tension experiments under cyclic loading. The predictions capture the number of cycles to failure as well as the crack propagation paths.

Appendix

Based on the PD LSM introduced by Madenci et al. [38], the first and second order derivatives of the function, $f\left(\mathrm{x}\right)$ with ${\mathrm{x}}^{T}=\left\{{x}_{1},{x}_{2}\right\}$ can be obtained by solving for the unknown vectors, ${\mathbf{D}}_{1}\left(f\left(\mathrm{x}\right)\right)$ and ${\mathbf{D}}_{2}\left(f\left(\mathrm{x}\right)\right)$ in the system of equations of the form

$$\mathbf{A}\mathbf{D}=\mathbf{R}$$

(58)

with

$$\mathbf{A}=\left[\begin{array}{c}{\mathbf{A}}_{11} {\mathbf{A}}_{12}\\ {\mathbf{A}}_{21} {\mathbf{A}}_{22}\end{array}\right];\mathbf{D}=\left\{\begin{array}{c}{\mathbf{D}}_{1}\left({{f}}\left({\varvec{x}}\right)\right)\\ {\mathbf{D}}_{2}\left({{f}}\left({\varvec{x}}\right)\right)\end{array}\right\};\mathbf{R}=\left\{\begin{array}{c}{\mathbf{R}}_{1}\\ {\mathbf{R}}_{2}\end{array}\right\}$$

(59)

in which the unknown vectors are defined as

$${\mathbf{D}}_{1}={\left\{{{{D}}}_{1} {{{D}}}_{2}\right\}}^{{{T}}}$$

(60)

$${\mathbf{D}}_{2}={\left\{{{{{D}}}_{1}^{2} {{{D}}}_{2}^{2} {{D}}}_{1} {{{D}}}_{2}\right\}}^{{{T}}}$$

(61)

with

$${{{D}}}_{{{i}}}^{{{n}}}=\frac{{\partial }^{{{n}}}}{{\partial {{x}}}_{{{i}}}^{{{n}}}}$$

(62)

In reference to Fig. 4, the discrete form of the known vectors, ${\mathbf{R}}_{1}$ and ${\mathbf{R}}_{2}$ can be expressed as

$$\mathbf{R}_1 (\mathbf{x}_{k}) = \sum^{N_{(k)}}_{j=1}w_{(k)(j)} \left\{\begin{array}{c}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}\\ {{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\end{array}\right\} (f \; (\mathrm {x}_{(k)}) - f \; (\mathrm{x}_k)) \; V_{j}$$

(63)

and

$$\mathrm{R}_2 (\mathrm{x}_k)= \sum^{N_k}_{j=1} w_{(k)(j)}\left\{ \begin{array} {c}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}\\ {{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ \end{array}\right\} (f \; (\mathrm{x}_{(j)}) - f \; (\mathrm{x}_{(k)})) \; V_{(j)}$$

(64)

The weight function can be specified as ${w}_{\left(k\right)\left(j\right)}={e}^{-{\left(2{\xi }_{\left(k\right)\left(j\right)}/\delta \right)}^{2}}$. The known submatrices can be expressed as

$${\mathbf{A}}_{11}\left({\mathbf{x}}_{\left({{k}}\right)}\right)=\sum_{{{j}}=1}^{{{{N}}}_{{\left({{k}}\right)}}}{{{w}}}_{\left({{k}}\right)\left({{j}}\right)}\left[\begin{array}{c}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2} {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)} {{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}\end{array}\right]\\$$

(65)

$${\mathbf{A}}_{12}\left({\mathbf{x}}_{\left({{k}}\right)}\right)=\sum_{{{j}}=1}^{{{{N}}}_{{\left({{k}}\right)}}}{{{w}}}_{\left({{k}}\right)\left({{j}}\right)}\left[\begin{array}{c}\frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{3}}{2} \frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}}{2} {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ \frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}}{2} \frac{{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{3}}{2} {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}\end{array}\right]{{{V}}}_{\left({{j}}\right)}$$

(66)

$${\mathbf{A}}_{21}\left({\mathbf{x}}_{\left({{k}}\right)}\right)=\sum_{{{j}}=1}^{{{{N}}}_{{\left({{k}}\right)}}}{{{w}}}_{\left({{k}}\right)\left({{j}}\right)}\left[\begin{array}{c}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{3} {{{\xi}}}_{1\left(k\right)\left({{j}}\right)}^{2}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2} {{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{3}\\ {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)} {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}\end{array}\right]{{{V}}}_{\left({{j}}\right)}$$

(67)

$${\mathbf{A}}_{22}\left({\mathbf{x}}_{\left({{k}}\right)}\right)=\sum_{{{j}}=1}^{{{{N}}}_{{\left({{k}}\right)}}}{{{w}}}_{\left({{k}}\right)\left({{j}}\right)}\left[\begin{array}{c}\frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{4}}{2} \frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}}{2} {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{3}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ \frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}}{2} \frac{{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{4}}{2} {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{3}\\ \frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{3}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}}{2} \frac{{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{3}}{2} {{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}\end{array}\right]{{{V}}}_{\left({{j}}\right)}$$

(68)

The solution to the system of equations leads to

$${\mathbf{D}}_{1}\left({{f}}\left({\mathbf{x}}_{\left({{k}}\right)}\right)\right)=\sum_{{{j}}=1}^{{{{N}}}_{{\left({{k}}\right)}}}{{{w}}}_{\left({{k}}\right)\left({{j}}\right)}{\mathbf{g}}_{1}\left({{{\xi}}}_{\left({{k}}\right)\left({{j}}\right)}\right)\left({{f}}\left({\mathbf{x}}_{\left({{j}}\right)}\right)-{{f}}\left({\mathbf{x}}_{\left({{k}}\right)}\right)\right){{{V}}}_{\left({{j}}\right)}$$

(69)

and

$${\mathbf{D}}_{2}\left({{f}}\left({\mathbf{x}}_{\left({{k}}\right)}\right)\right)=\sum_{{{j}}=1}^{{{{N}}}_{{\left({{k}}\right)}}}{{{w}}}_{\left({{k}}\right)\left({{j}}\right)}{\mathbf{g}}_{2}\left({{{\xi}}}_{\left({{k}}\right)\left({{j}}\right)}\right)\left({{f}}\left({\mathbf{x}}_{\left({{j}}\right)}\right)-{{f}}\left({\mathbf{x}}_{\left({{k}}\right)}\right)\right){{{V}}}_{\left({{j}}\right)}$$

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where

$${\mathbf{g}}_{1}\left({{{\xi}}}_{\left({{k}}\right)\left({{j}}\right)}\right)=\left\{\begin{array}{c}{{{a}}}_{11}^{11}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{12}^{11}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{11}^{12}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}+{{{a}}}_{12}^{12}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}{+{{a}}}_{13}^{12}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ {{{a}}}_{21}^{11}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{22}^{11}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{21}^{12}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}+{{{a}}}_{22}^{12}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}{+{{a}}}_{23}^{12}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\end{array}\right\}$$

(71)

and

$${\mathbf{g}}_{2}\left({{{\xi}}}_{\left({{k}}\right)\left({{j}}\right)}\right)=\left\{\begin{array}{c}{{{a}}}_{11}^{21}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{12}^{21}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{11}^{22}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}+{{{a}}}_{12}^{22}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}{+{{a}}}_{13}^{22}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ {{{a}}}_{21}^{21}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{22}^{21}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{21}^{22}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}+{{{a}}}_{22}^{22}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}{+{{a}}}_{23}^{22}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\\ {{{a}}}_{31}^{21}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{32}^{21}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}+{{{a}}}_{31}^{22}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}^{2}+{{{a}}}_{32}^{22}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}^{2}{+{{a}}}_{33}^{22}{{{\xi}}}_{1\left({{k}}\right)\left({{j}}\right)}{{{\xi}}}_{2\left({{k}}\right)\left({{j}}\right)}\end{array}\right\}$$

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in which the coefficients are given by

$${\mathbf{A}}^{-1}=\left[\begin{array}{c}{{{a}}}_{11}^{11} {{{a}}}_{12}^{11} {{{a}}}_{11}^{12} {{{a}}}_{12}^{12} {{{a}}}_{13}^{12}\\ {{{a}}}_{21}^{11} {{{a}}}_{22}^{11} {{{a}}}_{21}^{12} {{{a}}}_{22}^{12} {{{a}}}_{23}^{12}\\ {{{a}}}_{11}^{21} {{{a}}}_{12}^{21} {{{a}}}_{11}^{22} {{{a}}}_{12}^{22} {{{a}}}_{13}^{22}\\ {{{a}}}_{21}^{21} {{{a}}}_{22}^{21} {{{a}}}_{21}^{22} {{{a}}}_{22}^{22} {{{a}}}_{23}^{22}\\ {{{a}}}_{31}^{21} {{{a}}}_{32}^{21} {{{a}}}_{31}^{22} {{{a}}}_{32}^{22} {{{a}}}_{33}^{22}\end{array}\right]$$

(73)

The matrix, $\mathbf{G}\left(\boldsymbol\xi \right)$ for 2D analysis is defined in terms of the components of the vector, ${\mathbf{g}}_{2}\left(\boldsymbol\xi \right)$ as

$$\mathbf{G}\left({\boldsymbol{\xi}}\right)=\left[\begin{array}{cc}{\mathbf{g}}_{2\left(1\right)}& {\mathbf{g}}_{2\left(3\right)}\\ {\mathbf{g}}_{2\left(3\right)}& {\mathbf{g}}_{2\left(2\right)}\end{array}\right]$$

(74)