Microstructure in plasticity without nonconvexity

Abstract

A simplified one dimensional rate dependent model for the evolution of plastic distortion is obtained from a three dimensional mechanically rigorous model of mesoscale field dislocation mechanics. Computational solutions of variants of this minimal model are investigated to explore the ingredients necessary for the development of microstructure. In contrast to prevalent notions, it is shown that microstructure can be obtained even in the absence of non-monotone equations of state. In this model, incorporation of wave propagative dislocation transport is vital for the modeling of spatial patterning. One variant gives an impression of producing stochastic behavior, despite being a completely deterministic model. The computations focus primarily on demanding macroscopic limit situations, where a convergence study reveals that the model-variant including non-monotone equations of state cannot serve as effective equations in the macroscopic limit; the variant without non-monotone ingredients, in all likelihood, can.

Appendix 1

1.1 Simplified one-dimensional form of conservation law of Burgers vector content

The conservation law for Burgers vector content in mesoscale field dislocation mechanics is given by

$$\begin{aligned} \dot{{\varvec{\alpha }} }=-curl\left( {{\varvec{\alpha }} \times {\varvec{V}}+{\varvec{L}}^{p}} \right) \end{aligned}$$

(41)

where ${\varvec{\alpha }}$ is the mesoscale Nye tensor, ${\varvec{V}}$ is the mesoscale averaged dislocation velocity vector, and ${\varvec{L}}^{p}$ is the plastic strain rate produced by unresolved, at the mesoscale, and hence ‘statistical’ dislocations [4]. Here,

$$\begin{aligned} -curl~{\varvec{U}}^{p}={\varvec{\alpha }}. \end{aligned}$$

(42)

Consider a tensor field of the form

$$\begin{aligned} {\varvec{A}}=A_{12} {\varvec{e}}_1 \otimes {\varvec{e}}_2 \end{aligned}$$

(43)

and we assume that all fields vary in only the $x_3$ direction. Then the only non-zero component of

$$\begin{aligned} \left( {curl~{\varvec{A}}} \right) _{ri} =e_{ijk} A_{rk,j} \end{aligned}$$

(44)

is

$$\begin{aligned} \left( {curlA} \right) _{11} =e_{132} A_{12,3} =-A_{12,3}. \end{aligned}$$

(45)

We assume the ansatz

$$\begin{aligned}&{\varvec{U}}^{p}=U_{12}^p {\varvec{e}}_1 \otimes {\varvec{e}}_2 \nonumber \\&{\varvec{L}}^{p}=L_{12}^p {\varvec{e}}_1 \otimes {\varvec{e}}_2 \nonumber \\&{\varvec{\alpha }} =\alpha _{11} {\varvec{e}}_1 \otimes {\varvec{e}}_1 \nonumber \\&{\varvec{V}}=V_3 {\varvec{e}}_3 . \end{aligned}$$

(46)

Then

$$\begin{aligned}&{\varvec{\alpha }} \times {\varvec{V}}=-\alpha _{11} V_3 {\varvec{e}}_1 \otimes {\varvec{e}}_2\end{aligned}$$

(47)

$$\begin{aligned}&-\left[ {curl\left( {{\varvec{\alpha }} \times {\varvec{V}}} \right) } \right] =-\left[ {-\left( {-\alpha _{11} V_3 } \right) _{,3} } \right] {\varvec{e}}_1 \otimes {\varvec{e}}_1\end{aligned}$$

(48)

$$\begin{aligned}&-curl~{\varvec{L}}^{p}=-\left[ {-L_{12,3}^p } \right] {\varvec{e}}_1 \otimes {\varvec{e}}_1\end{aligned}$$

(49)

$$\begin{aligned}&{\varvec{\alpha }} =-curl~{\varvec{U}}^{p}=-\left[ {-U_{12,3}^p } \right] {\varvec{e}}_1 \otimes {\varvec{e}}_1 . \end{aligned}$$

(50)

Denoting $U_{12}^p =\varphi$, $x_3 =x, \;x_1 =y,\;x_2 =z$, $V_3=V$, and $L_{12}^p =L^{p}$ the conservation law reduces to the equation

$$\begin{aligned} \left( {\varphi _x } \right) _t =-\left( {\varphi _x V-L^{p}} \right) _x. \end{aligned}$$

(51)