An incremental elastic–plastic triaxiality dependent fatigue model

Abstract

A stress-state dependent cyclic cohesive model, which accounts for accumulation of plasticity, during both tensile as well as compressive deformations, and incorporates accumulation of irreversible damage due to macroscopic plasticity as well as microstructural mechanisms, is formulated. The model is implemented in mode-I plane strain fatigue crack growth simulations. The model is validated by reproducing the effect of retardation in crack growth rates after different combinations of tensile and compressive overloads. We show accurate description of the elastic–plastic behaviour of the process zone is vital, in particular for negative stress ratios subsequent to a tensile over-load, as considerable plasticity occurs in compression at the crack-tip, significantly reducing retardation effects.

Appendices

Summary of the traction–separation behaviour.

Table 1 Summary of traction–separation law

Consistent tangent operator

The incremental form of the stress-state dependent traction separation behaviour for the cohesive elements is shown in Eq. 3. The associated tangent modulus, K, is then defined as

$$\begin{aligned}&K = \frac{\partial T_n}{\partial {\hat{\delta }}_{n}} \end{aligned}$$

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$$\begin{aligned}&\frac{\partial T_n}{\partial {\hat{\delta }}_{n}} = (1-D) {\bar{E}} \left( 1-\frac{\partial {\hat{\delta }}_{n}^p}{\partial {\hat{\delta }}_{n}} \right) - \frac{\partial D}{\partial {\hat{\delta }}_{n}} {\bar{E}} \left( {\hat{\delta }}_{n}-{\hat{\delta }}_{n}^p \right) \end{aligned}$$

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and is implemented into FEM as per (Segurado and LLorca 2004). From Eqs. 8 and 11, $\frac{\partial {\hat{\delta }}_{n}^p}{\partial {\hat{\delta }}_{n}}$ and $\frac{\partial D}{\partial {\hat{\delta }}_{n}}$ are determined to be of the form

$$\begin{aligned} \begin{aligned} \frac{\partial {\hat{\delta }}_{n}^p}{\partial {\hat{\delta }}_{n}}&= \frac{\partial {\hat{\delta }}_{n}^p}{\partial \lambda } \frac{\partial \lambda }{\partial {\hat{\delta }}_{n}}\\&= \frac{3}{2}\;\frac{\left( 2-\alpha -\beta \right) }{3\sqrt{\frac{(\alpha -1)^2+(\beta -1)^2+(\alpha -\beta )^2}{2}}} \mathrm{{sign}}(T_n)\; \frac{\partial \lambda }{\partial {\hat{\delta }}_{n}} \\ \frac{\partial D}{\partial {\hat{\delta }}_{n}}&= \frac{\partial D}{\partial \lambda } \frac{\partial \lambda }{\partial {\hat{\delta }}_{n}} =\frac{0.02}{\epsilon _{y0}+\lambda _{s}}\\&\left( \frac{\lambda -\lambda _{s}}{\epsilon _{y0}+\lambda _{s}} \right) ^3e^{-0.005\left( \frac{\lambda -\lambda _{s}}{\epsilon _{y0}+\lambda _{s}} \right) ^4} H (\lambda -\lambda _{s})\;\frac{\partial \lambda }{\partial {\hat{\delta }}_{n}}. \end{aligned} \end{aligned}$$

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where $\frac{\partial \lambda }{\partial {\hat{\delta }}_{n}}$ is found by enforcing consistency condition to the triaxiality dependent one dimensional yield surface formulated, earlier in Eq. 15, so that during yielding traction remains on the yield surface. From consistency condition of the damage coupled yield surface, ${\dot{\varPhi }}$ = 0, $\frac{\partial \lambda }{\partial {\hat{\delta }}_{n}}$ is found and takes the form

$$\begin{aligned} \begin{aligned}&\frac{\partial \lambda }{\partial {\hat{\delta }}_{n}} =\\&\frac{(1-D) {\bar{E}} \; \mathrm {sign}(T_n)}{ (1-D)\left( {\bar{E}}\; \frac{\partial {\hat{\delta }}_{n}^p}{\partial \lambda }\; \mathrm {sign}(T_n) + \frac{\partial {\bar{\sigma }}_y}{\partial \lambda } \right) + \frac{\partial D}{\partial \lambda } \left( {\bar{E}}\;({\hat{\delta }}_{n}-{\hat{\delta }}_{n}^p)\; \mathrm {sign}(T_n) - {\bar{\sigma }}_y \right) } \end{aligned} \end{aligned}$$

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