Coupled crystal plasticity and damage model for micro crack propagation in polycrystalline microstructures

Abstract

Micro-crack propagation in polycrystalline materials can strongly depend on the defect size and its ratio to specimen size, and local variation in the microstructural features such as grain orientation, size, etc. While the dependencies are understood heuristically, the use of mechanistic models to capture the effect of various factors influencing micro-crack propagation can enable accurate prediction of fracture properties of polycrystalline materials and their engineering. To this end, a crystal plasticity coupled to damage model for micro-crack propagation on cleavage planes has been developed in this work and is shown to successfully capture the grain orientation dependent growth. In order to identify a suitable integration scheme for the coupled model, a one-dimensional model is developed and a detailed comparative analysis of three different schemes is performed. The analysis shows that the coupled explicit–implicit scheme is the most suitable and is a key finding of this work. Subsequently, a two-scale multi-scale method has been developed to include the interaction between the defect, its surrounding microstructure and the specimen. The two-scale method along with the coupled crystal plasticity-damage model has been applied to perform finite element method based micro-crack growth simulations for a microstructurally short and physically long crack with two different microstructures with random orientation and texture. Such a study comparing microstructural effects on crack growth from pre-existing defects of drastically disparate sizes hasn’t been performed before and is a novelty of this work. The analyses clearly show that though the micro-crack path from long crack is different depending on the orientation distribution, the rates are nearly independent of the local behavior. Moreover, the micro-crack propagation rate from long crack is significantly larger at the initial stages, with the latter showing significant acceleration after a small growth. Overall, the influence of microstructure on the crack growth behavior is stronger for short cracks, which conform with experimental observations and is successfully captured by the proposed model.

Appendix: Numerical integration schemes

1.1 Case 1: Fully implicit numerical integration scheme

The residual equations corresponding to Eqs. 18 to 22 are,

$$\begin{aligned} R_1= & {} \sigma _{t+1}-\sigma _t-E(\Delta \varepsilon _{t+1}-\Delta \varepsilon ^p_{t+1} -\Delta \varepsilon ^d_{t+1}) \end{aligned}$$

(A1)

$$\begin{aligned} R_2= & {} \Delta \varepsilon ^p_{t+1} - {\dot{a}}_0\left| \frac{ \sigma _{t+1} }{{g_{t+1}}}\right| ^{m} {\text {sign}}(\sigma _{t+1}) \Delta {t} \end{aligned}$$

(A2)

$$\begin{aligned} R_3= & {} g_{t+1} - g_t - h \left| \Delta \varepsilon _{t+1}^p\right| \end{aligned}$$

(A3)

$$\begin{aligned} R_4= & {} \Delta \varepsilon ^d_{t+1} - {\dot{o}}_{0}\left| \frac{\left\langle \sigma _{t+1}-\phi _{t+1}^{2}d_{cr}\right\rangle }{\phi _{t+1}^{2}d_{cr}}\right| ^p \Delta {t}\end{aligned}$$

(A4)

$$\begin{aligned} R_5= & {} \phi _{t+1}-\phi _t+M \phi _{t+1} \Delta {\varepsilon }^{d}_{t+1} \end{aligned}$$

(A5)

1.2 Case 2: Decoupled explicit–implicit numerical integration scheme

The residual equations and the Jacobian for the first loop are explained as follow,

$$\begin{aligned} R_{\sigma }=\sigma _{t+1}-\sigma _t-E(\Delta \varepsilon _{t+1} -\Delta \varepsilon ^p_{t+1}-\Delta \varepsilon ^d_{t}) \end{aligned}$$

(A6)

where plastic strain increment is calculated as

$$\begin{aligned} \Delta \varepsilon ^p_{t+1} = {\dot{a}}_0\left| \frac{ \sigma _{t+1}}{{g_{t+1}}}\right| ^{m} {\text {sign}}(\sigma _{t+1}) \Delta {t} \end{aligned}$$

The Jacobian is obtained as,

$$\begin{aligned} J_{\sigma }=\frac{\partial R_{\sigma }}{\partial \sigma _{t+1}} =1+{\dot{a}}_{0} m \left| \frac{ \sigma _{t+1} }{{g_{t+1}}}\right| ^{m-1} \frac{\Delta t}{g_{t+1}} \end{aligned}$$

(A7)

The stress, $\sigma$, is updated following,

$$\begin{aligned} {[}\sigma _{t+1}]^{i+1}=[\sigma _{t+1}]^{i}-J_{\sigma }^{-1}R_\sigma \end{aligned}$$

(A8)

Once $\left| R_{\sigma }\right| <\eta$, a chosen tolerance, the g is updated following

$$\begin{aligned} g_{t+1} = g_t + h \left| \Delta \varepsilon _{t+1}^p\right| \end{aligned}$$

(A9)

In the second loop, the variables $\left( \phi \right)$ and $\left( \Delta \varepsilon ^d\right)$ are updated using a similar approach as above where the residual and the Jacobian are as follows:

$$\begin{aligned} R_{\phi } = \phi _{t+1}-\phi _t+M \phi _{t+1} \Delta {\varepsilon }^{d}_{t+1} \end{aligned}$$

(A10)

where

$$\begin{aligned} \Delta \varepsilon ^d_{t+1} = {\dot{o}}_{0}\left| \frac{\left\langle \sigma _{t+1}-\phi _{t+1}^{2} d_{cr}\right\rangle }{\phi _{t+1}^{2}d_{cr}}\right| ^p \Delta {t} \end{aligned}$$

The Jacobian is obtained as,

$$\begin{aligned} J_{\phi }=1+M\Delta \varepsilon ^d_{t+1}-2 M\phi _{t+1}\, p\, {\dot{o}}_0\,(\frac{q_1+q_2}{q_3})\Delta t\nonumber \\ \end{aligned}$$

(A11)

The damage variable $\phi$ is updated as follows,

$$\begin{aligned} {[}\phi _{t+1}]^{i+1}=[\phi _{t+1}]^{i}-J_{\phi }^{-1}R_\phi \end{aligned}$$

(A12)

1.3 Case 3: Coupled explicit–implicit numerical integration scheme

The residual $R_{\sigma }$ and the Jacobian $J_{\sigma }$ for the first loop are as follows,

$$\begin{aligned} R_{\sigma }=\sigma _{t+1}-\sigma _t-E(\Delta \varepsilon _{t+1}- \Delta \varepsilon ^p_{t+1}-\Delta \varepsilon ^d_{t+1})\nonumber \\ \end{aligned}$$

(A13)

where

$$\begin{aligned} \Delta \varepsilon ^p_{t+1}, \Delta \varepsilon ^d_{t+1} \end{aligned}$$

are as given previous case.

$$\begin{aligned} J_{\sigma }{} & {} =1+E\left( \frac{{\dot{a}}_{0} m}{g_{t+1}} \left| \frac{ \sigma _{t+1} }{{g_{t+1}}}\right| ^{m-1}\right. \nonumber \\{} & {} \quad \left. +\frac{{\dot{o}}_{0} p}{\phi ^2 d_{cr}} \left| \frac{\left\langle \sigma _{t+1}-\phi _{t+1}^{2}d_{cr} \right\rangle }{\phi _{t+1}^{2}d_{cr}}\right| ^{p-1} \right) \Delta {t} \end{aligned}$$

(A14)

and the stress is updated following

$$\begin{aligned} {[}\sigma _{t+1}]^{i+1}=[\sigma _{t+1}]^{i}-J_{\sigma }^{-1}R_\sigma \end{aligned}$$

(A15)

Once $\left| R_{\sigma }\right| <\eta$, a predefined tolerance, the resistance stress g is updated following,

$$\begin{aligned} g_{t+1} = g_t + h \left| \Delta \varepsilon _{t+1}^p\right| \end{aligned}$$

(A16)

and damage variable $\phi$ using

$$\begin{aligned} \phi _{t+1}=\frac{\phi _t}{1+M \Delta {\varepsilon }^{d}_{t+1}} \end{aligned}$$

(A17)