A gradient crystal plasticity theory for large deformations with a discontinuous accumulated plastic slip

Abstract

The implementation of novel material models in the microscale gives a deeper understanding of inner and intercrystalline effects of crystalline materials. For future works, this allows more precise predictions of macroscale models. Here, we present a finite gradient crystal plasticity theory which preserves the single crystal slip kinematics. However, the model is restricted to one gradient-stress, associated with the gradient of the accumulated plastic slip, in order to account for long range dislocation interactions in a physically simplified, numerically efficient approach. In order to model the interaction of dislocations with and their transfer through grain boundaries, a grain boundary yield condition is introduced. The grain boundary flow rule is evaluated at sharp interfaces using discontinuous trial functions in the finite element implementation, thereby allowing for a discontinuous distribution of the accumulated plastic slip. Simulations of crystal aggregates are performed under different loading conditions which reproduce well the size dependence of the yield strength. An analytical solution for a one-dimensional single slip supports the numerical results.

Appendix

1.1 Derivation of field equations

The application of the chain rule and Gauss’ theorem on the contributions of the virtual mechanical energy balance (cf. Eq. 13), yields

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and for the micromorphic contribution

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1.2 Transformation of internal power contributions

The power of the internal forces is formulated with respect to the current placement (cf. Eq. 11). A transformation of the individual contributions to the reference placement is performed.

The transformation of the contribution of the Cauchy stress $\varvec{\sigma }$ yields

$$\begin{aligned} \begin{aligned} \int _{{\mathcal B}} \varvec{\sigma }\cdot \varvec{ L}{\,\mathrm d}v&= \int _{{\mathcal B}_0} \varvec{\sigma }\cdot \left( \dot{\varvec{ F}}{\varvec{ F}}^\mathrm{-1}\right) J {\,\mathrm d}V \\&= \int _{{\mathcal B}_0} \underbrace{\left( J {\varvec{ F}}^\mathrm{-1}\varvec{\sigma }{\varvec{ F}}^\mathsf{-T}\right) }_{\varvec{ S}} \cdot \underbrace{\left( {\varvec{ F}}^\mathsf{T}\dot{\varvec{ F}}\right) }_{\dot{\varvec{ E}}} {\,\mathrm d}V \\&= \int _{{\mathcal B}_0} \varvec{ S}\cdot \dot{\varvec{ E}} {\,\mathrm d}V. \end{aligned} \end{aligned}$$

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The contribution of the scalar micromorphic stress $\pi$ is transformed

$$\begin{aligned} \begin{aligned} \int _{{\mathcal B}} \pi {\,\mathrm d}v = \int _{{\mathcal B}_0} \underbrace{J \pi }_{\pi _{\mathrm {L}}} {\,\mathrm d}V = \int _{{\mathcal B}_0} \pi _{\mathrm {L}} {\,\mathrm d}V, \end{aligned} \end{aligned}$$

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as well as the contribution of the vector valued micromorphic stress $\varvec{\xi }$

$$\begin{aligned} \begin{aligned}&\int _{{\mathcal B}} \varvec{\xi }\cdot \mathrm{grad}\left( \dot{\breve{\gamma }}_{\mathrm {ac}} \right) {\,\mathrm d}v = \int _{{\mathcal B}_0} J \varvec{\xi }\cdot \mathrm{grad}\left( \dot{\breve{\gamma }}_{\mathrm {ac}} \right) {\,\mathrm d}V \\&\qquad = \int _{{\mathcal B}_0} J \xi _i \displaystyle \frac{\displaystyle \partial \dot{\breve{\gamma }}_{\mathrm {ac}}}{\displaystyle \partial x_i} {\,\mathrm d}V = \int _{{\mathcal B}_0} J \xi _i \displaystyle \frac{\displaystyle \partial \dot{\breve{\gamma }}_{\mathrm {ac}}}{\displaystyle \partial X_j} \displaystyle \frac{\displaystyle \partial X_j}{\displaystyle \partial x_i} {\,\mathrm d}V \\&\qquad = \int _{{\mathcal B}_0} \underbrace{\left( J {\varvec{ F}}^\mathrm{-1}\varvec{\xi }\right) }_{\varvec{\xi }_{\mathrm {L}}} \cdot \mathrm{Grad}\left( \dot{\breve{\gamma }}_{\mathrm {ac}} \right) {\,\mathrm d}V \\&\qquad = \int _{{\mathcal B}_0} \varvec{\xi }_{\mathrm {L}} \cdot \mathrm{Grad}\left( \dot{\breve{\gamma }}_{\mathrm {ac}} \right) {\,\mathrm d}V. \end{aligned} \end{aligned}$$

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Finally, the transformation of the internal forces on the grain boundaries, meaning the grain boundary micromorphic tractions ${\Xi }_{\mathrm {j}}$ and  ${\Xi }_{\mathrm {m}}$, yields

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1.3 Details on the global Newton algorithm

We build the scalar product of the balance of linear momentum (cf. Eq. 15) with a trial function $\delta \underline{w}^{u} = \underline{\underline{N}}^{u} ~\hat{\underline{w}}$ and integrate over an arbitrary volume ${\mathcal B}$

$$\begin{aligned} \begin{aligned} 0&= \int _{{\mathcal B}} \mathrm{div }\left( \underline{\sigma } \right) \cdot \delta \underline{w}^{u} {\,\mathrm d}v, \\ \underline{r}^{u}&= \int _{{\mathcal B}} {\underline{\underline{B}}^{u}}^\mathsf{T} \underline{\sigma } {\,\mathrm d}v - \int _{{\mathcal B}_t} {\underline{\underline{N}}^{u}}^\mathsf{T} \underline{\bar{t}} {\,\mathrm d}a. \\ \end{aligned} \end{aligned}$$

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Hereby, the matrix of shape function gradients $\underline{\underline{B}}$ was introduced. Furthermore, we use the micromorphic field equation, the grain boundary condition and the grain boundary yield condition (cf. Eqs. 15 and 37), and the trial function $\delta w^{\gamma } = \underline{N}^{\gamma } ~\hat{\underline{w}}$

$$\begin{aligned}&0 = \int _{{\mathcal B}} \left( \beta - \breve{p} - K_\mathrm {g} \mathrm{div }\left( \mathrm{grad}\left( \breve{\gamma }_{\mathrm {ac}} \right) \right) \right) \delta w^{\gamma } {\,\mathrm d}v, \nonumber \\&r^{\gamma } = \int _{{\mathcal B}} {\underline{N}^{\gamma }}^\mathsf{T} \left( \beta - \breve{p} \right) {\,\mathrm d}v + \int _{{\mathcal B}} {\underline{B}^{\gamma }}^\mathsf{T} K_{\mathrm {g}} \left( \underline{B}^{\gamma } \underline{\hat{\gamma }} + \underline{B}^{\delta } \underline{\hat{\delta }}\right) {\,\mathrm d}v \nonumber \\&\quad = - \int _{\partial {\mathcal B}_\mathrm {\Xi }} {\underline{N}^{\gamma }}^\mathsf{T} \bar{\mathrm {\Xi }} {\,\mathrm d}a + \int _{{\Gamma }_\mathrm {act}} \left( \tilde{{\Xi }}_{\mathrm {m}}^{\mathrm {e}} + \tilde{{\Xi }}_{\mathrm {m}}^{\mathrm {c}} \right) {\,\mathrm d}a, \end{aligned}$$

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where ${\Gamma }_\mathrm {act}$ denotes the plastically active part of the grain boundaries. By using Eqs. (15) and (37) with the trial function $\delta w^{\delta } = \underline{N}^{\delta } ~\hat{\underline{w}}$, the shape function enrichment results in an additional equation

$$\begin{aligned} 0&= \int _{{\mathcal B}} \left( \beta - \breve{p} - K_\mathrm {g} \mathrm{div }\left( \mathrm{grad}\left( \breve{\gamma }_{\mathrm {ac}} \right) \right) \right) \delta w^{\delta } {\,\mathrm d}v, \nonumber \\ r^{\delta }&= \int _{{\mathcal B}} {\underline{N}^{\delta }}^\mathsf{T} \left( \beta - \breve{p} \right) {\,\mathrm d}v + \int _{{\mathcal B}} {\underline{B}^{\delta }}^\mathsf{T} K_{\mathrm {g}} \left( \underline{B}^{\gamma } \underline{\hat{\gamma }} + \underline{B}^{\delta } \underline{\hat{\delta }}\right) {\,\mathrm d}v \nonumber \\&= - \int _{\partial {\mathcal B}_{\Xi }} {\underline{N}^{\delta }}^\mathsf{T} \bar{{\Xi }} {\,\mathrm d}a + \int _{{\Gamma }_\mathrm {act}} \left( \tilde{{\Xi }}_{\mathrm {j}}^{\mathrm {e}} + \tilde{{\Xi }}_{\mathrm {j}}^{\mathrm {c}} \right) {\,\mathrm d}a. \end{aligned}$$

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The total derivatives of Eqs. (81)–(83) read in matrix-vector notation

$$\begin{aligned} \begin{pmatrix} {\,\mathrm d}\underline{r}^{u} \\ {\,\mathrm d}r^{\gamma } \\ {\,\mathrm d}r^{\delta } \end{pmatrix} = \begin{pmatrix} \displaystyle \frac{\displaystyle \partial \underline{r}^{u}}{\displaystyle \partial \underline{\hat{u}}} &{}\quad \displaystyle \frac{\displaystyle \partial \underline{r}^{u}}{\displaystyle \partial \underline{\hat{u}}} &{}\quad \displaystyle \frac{\displaystyle \partial \underline{r}^{u}}{\displaystyle \partial \underline{\hat{u}}}\\ \displaystyle \frac{\displaystyle \partial {r^{\gamma }}}{\displaystyle \partial \underline{\hat{\gamma }}} &{}\quad \displaystyle \frac{\displaystyle \partial {r^{\gamma }}}{\displaystyle \partial \underline{\hat{\gamma }}} &{}\quad \displaystyle \frac{\displaystyle \partial {r^{\gamma }}}{\displaystyle \partial \underline{\hat{\gamma }}} \\ \displaystyle \frac{\displaystyle \partial {r^{\delta }}}{\displaystyle \partial \underline{\hat{\delta }}} &{}\quad \displaystyle \frac{\displaystyle \partial {r^{\delta }}}{\displaystyle \partial \underline{\hat{\delta }}} &{}\quad \displaystyle \frac{\displaystyle \partial {r^{\delta }}}{\displaystyle \partial \underline{\hat{\delta }}}\\ \end{pmatrix} \begin{pmatrix} {\,\mathrm d}\underline{\hat{u}} \\ {\,\mathrm d}\underline{\hat{\gamma }} \\ {\,\mathrm d}\underline{\hat{\delta }} \end{pmatrix}. \end{aligned}$$

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The entries of the global algorithmic tangent are

$$\begin{aligned} \displaystyle \frac{\displaystyle \partial \underline{r}^{u}}{\displaystyle \partial \underline{\hat{u}}}&= \int _{{\mathcal B}} {\underline{\underline{B}}^{u}}^\mathsf{T} \displaystyle \frac{\displaystyle \partial \underline{\sigma }}{\displaystyle \partial \underline{E}} \underline{\underline{B}}^{u} {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial \underline{r}^{u}}{\displaystyle \partial \underline{\hat{\gamma }}}&= \int _{{\mathcal B}} {\underline{\underline{B}}^{u}}^\mathsf{T} \displaystyle \frac{\displaystyle \partial \underline{\sigma }}{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} \underline{N}^{\gamma } {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial \underline{r}^{u}}{\displaystyle \partial \underline{\hat{\delta }}}&= \int _{{\mathcal B}} {\underline{\underline{B}}^{u}}^\mathsf{T} \displaystyle \frac{\displaystyle \partial \underline{\sigma }}{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} \underline{N}^{\delta } {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial {r^{\gamma }}}{\displaystyle \partial \underline{\hat{u}}}&= \int _{{\mathcal B}} {\underline{N}^{\gamma }}^\mathsf{T} \left( - \displaystyle \frac{\displaystyle \partial \breve{p}}{\displaystyle \partial \underline{E}} \right) \underline{\underline{B}}^{u} {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial {r^{\gamma }}}{\displaystyle \partial \underline{\hat{\gamma }}}&= \int _{{\mathcal B}} {\underline{N}^{\gamma }}^\mathsf{T} \underline{N}^{\gamma } \left( \displaystyle \frac{\displaystyle \partial \beta }{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} -\displaystyle \frac{\displaystyle \partial \breve{p}}{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} \right) {\,\mathrm d}v\nonumber \\&\quad +\,\int _{{\mathcal B}} {\underline{B}^{\gamma }}^\mathsf{T} \underline{B}^{\gamma } K_\mathrm {g} {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial {r^{\gamma }}}{\displaystyle \partial \underline{\hat{\delta }}}&= \int _{{\mathcal B}} {\underline{N}^{\gamma }}^\mathsf{T} \underline{N}^{\delta } \left( \displaystyle \frac{\displaystyle \partial \beta }{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} -\displaystyle \frac{\displaystyle \partial \breve{p}}{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} \right) {\,\mathrm d}v\nonumber \\&\quad +\,\int _{{\mathcal B}} {\underline{B}^{\gamma }}^\mathsf{T} \underline{B}^{\delta } K_\mathrm {g} {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial {r^{\delta }}}{\displaystyle \partial \underline{\hat{u}}}&= \int _{{\mathcal B}} {\underline{N}^{\delta }}^\mathsf{T} \left( - \displaystyle \frac{\displaystyle \partial \breve{p}}{\displaystyle \partial \underline{E}} \right) \underline{\underline{B}}^{u} {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial {r^{\delta }}}{\displaystyle \partial \underline{\hat{\gamma }}}&= \int _{{\mathcal B}} {\underline{N}^{\delta }}^\mathsf{T} \underline{N}^{\gamma } \left( \displaystyle \frac{\displaystyle \partial \beta }{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} -\displaystyle \frac{\displaystyle \partial \breve{p}}{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} \right) {\,\mathrm d}v\nonumber \\&\quad +\,\int _{{\mathcal B}} {\underline{B}^{\delta }}^\mathsf{T} \underline{B}^{\gamma } K_\mathrm {g} {\,\mathrm d}v, \nonumber \\ \displaystyle \frac{\displaystyle \partial {r^{\delta }}}{\displaystyle \partial \underline{\hat{\delta }}}&= \int _{{\mathcal B}} {\underline{N}^{\delta }}^\mathsf{T} \underline{N}^{\delta } \left( \displaystyle \frac{\displaystyle \partial \beta }{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} -\displaystyle \frac{\displaystyle \partial \breve{p}}{\displaystyle \partial \breve{\gamma }_{\mathrm {ac}}} \right) {\,\mathrm d}v\nonumber \\&\quad +\,\int _{{\mathcal B}} {\underline{B}^{\delta }}^\mathsf{T} \underline{B}^{\delta } K_\mathrm {g} {\,\mathrm d}v. \end{aligned}$$

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Fig. 13

Convergence study for a varying mesh discretization of the crystalline structure. A good compromise of the computational time and the accuracy of the results is given for a finite element mesh with about 6 $\times$ $10^4$ DOF

1.4 Details on the convergence

In Fig. 13 the results of the convergence study are shown. The loading case as well as the material parameters were chosen in the manner of the comparison to the analytical solution with a grain rotation angle of $\varphi _{x}=45\,\mathrm{^{\circ }}$. Hereby, the mesh discretization was varied by using 400, 600, 800, 1000 and 1200 Elements.