Cohesive Zone Models—Theory, Numerics and Usage in High-Temperature Applications to Describe Cracking and Delamination

Abstract

This treatise deals with Cohesive Zone Models which were developed around 1960 through Barenblatt and Dugdale. At first, we present an overview about these models and the numerical treatment of these models in the sense of the Finite Element Method. Further on, a rate-dependent Cohesive Zone Model is presented and tested through a simulation of a Four-Point-Bend-Test with a metal compound. The required material parameters are determined through numerical optimisation by using a neural network which is explained, as well.

Appendix

Arranging of displacement, position and temperature vector for a 2D cohesive element

$$\begin{aligned} \begin{aligned} {{\mathbf {\mathsf{{u}}}}}^{\mathrm {e}}&=\begin{bmatrix} u_{1x}&u_{1y}&u_{2x}&\ldots&u_{ny} \end{bmatrix}^{\top }\, ,\quad n={NN}\\ {{\mathbf {\mathsf{{x}}}}}^{\mathrm {e}}&=\begin{bmatrix} x_{1x}&x_{1y}&x_{2x}&\ldots&x_{ny} \end{bmatrix}^{\top }\, ,\quad n={NN}\\ {{\varvec{\uptheta }}}^{\mathrm {e}}&=\begin{bmatrix} \theta _{1}&\theta _{2}&\ldots&\theta _{n} \end{bmatrix}^{\top }\, ,\quad n={NN}\end{aligned} \end{aligned}$$

Arranging of total ${DOF}$ and residual vector for a 2D cohesive element

$$\begin{aligned} \begin{aligned} {{\mathbf {\mathsf{{p}}}}}^{\mathrm {e}}&=\begin{bmatrix} {{\mathbf {\mathsf{{u}}}}}^{\mathrm {e}} \\[1ex] {{\varvec{\uptheta }}}^{\mathrm {e}} \end{bmatrix} \\ {{\mathbf {\mathsf{{r}}}}}^{\mathrm {e}}_{\mathrm {i}}&=\begin{bmatrix} {{\mathbf {\mathsf{{r}}}}}^{\mathrm {e}}_{\mathrm {i,u}} \\[1ex] {{\mathbf {\mathsf{{r}}}}}^{\mathrm {e}}_{\mathrm {i,}\uptheta } \end{bmatrix} \end{aligned} \end{aligned}$$

Shape functions for a 2D cohesive element

$$\begin{aligned} \begin{aligned} N_{1}&=\frac{1}{2}\left( 1-\xi _{1}\right) \\ N_{2}&=\frac{1}{2}\left( 1+\xi _{1}\right) \end{aligned} \end{aligned}$$

Arranging of the displacement and temperature shape function matrix

$$\begin{aligned} \begin{aligned} {{\mathbf {\mathsf{{N}}}}}_{\mathrm {u}}&=\begin{bmatrix} N_{1}&0&N_{2}&0 \\ 0&N_{1}&0&N_{2} \end{bmatrix}\\ {{\mathbf {\mathsf{{N}}}}}_{\uptheta }&=\begin{bmatrix} N_{1}&N_{2} \end{bmatrix}\\ \end{aligned} \end{aligned}$$

Arranging of the displacement and temperature mean value matrix

$$\begin{aligned} \begin{aligned} {{\mathbf {\mathsf{{M}}}}}_{\mathrm {u}}&=\begin{bmatrix} 1&0&0&0&0&0&1&0 \\ 0&1&0&0&0&0&0&1 \\ 0&0&1&0&1&0&0&0 \\ 0&0&0&1&0&1&0&0 \end{bmatrix}\\ {{\mathbf {\mathsf{{M}}}}}_{\uptheta }&=\begin{bmatrix} 1&0&0&1 \\ 0&1&1&0 \end{bmatrix} \end{aligned} \end{aligned}$$

Arranging of the displacement and temperature separation relation matrix

$$\begin{aligned} \begin{aligned} {{\mathbf {\mathsf{{L}}}}}_{\mathrm {u}}&=\begin{bmatrix} -1&0&0&0&\quad 0&\quad 0&\quad 1&\quad 0 \\ 0&-1&0&0&\quad 0&\quad 0&\quad 0&\quad 1 \\ 0&0&-1&0&\quad 1&\quad 0&\quad 0&\quad 0 \\ 0&0&0&-1&\quad 0&\quad 1&\quad 0&\quad 0 \end{bmatrix} \\ {{\mathbf {\mathsf{{L}}}}}_{\uptheta }&=\begin{bmatrix} -1&0&\quad 0&\quad 1 \\ 0&-1&\quad 1&\quad 0 \end{bmatrix} \end{aligned} \end{aligned}$$

Arranging of the elasticity matrix

$$\begin{aligned} {{\mathbf {\mathsf{{C}}}}}_{\mathrm {TSL}}=\begin{bmatrix} C_{\mathrm {n}}&0 \\ 0&C_{\mathrm {t}} \end{bmatrix} \end{aligned}$$