Disorder is good for you: the influence of local disorder on strain localization and ductility of strain softening materials

Abstract

We formulate a generic concept model for the deformation of a locally disordered, macroscopically homogeneous material which undergoes irreversible strain softening during plastic deformation. We investigate the influence of the degree of microstructural heterogeneity and disorder on strain localization (formation of a macroscopic shear band) in such materials. It is shown that increased microstructural heterogeneity delays strain localization and leads to an increase of the plastic regime in the macroscopic stress–strain curves. The evolving strain localization patterns are characterized.

Appendix

In our evaluation of internal stresses we use that the evaluation of stresses in any plane strain deformation problem can be mapped onto the evaluation of the stress field of a 2D dislocation arrangement consisting of straight parallel edge dislocations with line direction perpendicular to the considered plane. In particular, the shear stress field $\sigma _{xy}({\varvec{r}}) =: \tau ({\varvec{r}})$ generated by a two dimensional distribution of dislocations with Burgers vector parallel to the x axis, line direction along the z axis, and Burgers vector density $\alpha$ can be expressed as

$$\begin{aligned} \tau \left( {{\varvec{r}}} \right) = \int {{\tau _{{\text {ind}}}}\left( {{{{\varvec{r}}}} - {{{\varvec{r}}}}'} \right) {\alpha }\left( {{{{\varvec{r}}}}'} \right) {d^2}r'} \end{aligned}$$

(8)

where $\tau _\mathrm{ind}$ is the shear stress field created by a dislocation of unit Burgers vector length,

$$\begin{aligned} {\tau _{{\text {ind}}}}\left( {{{\varvec{r}}}} \right) = \frac{{\mu }}{{2\pi \left( {1 - \nu } \right) }}\frac{{x\left( {{x^2} - {y^2}} \right) }}{{{{\left( {{x^2} + {y^2}} \right) }^2}}}. \end{aligned}$$

(9)

Due to the relation $\alpha = - {\partial _x}{\gamma \left( {{{\varvec{r}}}} \right) }$ (Groma et al. 2003), the shear stress field can then be expressed in terms of the shear strain field as

$$\begin{aligned} \tau \left( {{\varvec{r}}} \right) = - \int {{\tau _{{\text {ind}}}}\left( {{{{\varvec{r}}}} - {{{\varvec{r}}}}'} \right) {\partial _x {\gamma \left( {{{{\varvec{r}}}}'} \right) }}{d^2}r'} . \end{aligned}$$

(10)

By a partial integration one obtains

$$\begin{aligned} \tau \left( {{\varvec{r}}} \right) = \int {\gamma \left( {{{{\varvec{r}}}}'} \right) {\partial _x}{\tau _{{\text {ind}}}}\left( {{{{\varvec{r}}}} - {{{\varvec{r}}}}'} \right) {d^2}r'} . \end{aligned}$$

(11)

Here ${\partial _x}{\tau _{{\text {ind}}}}\left( {{{{\varvec{r}}}}} \right) = {G^E}$ is the Green’s function which allows to calculate the shear stress by convolution with the strain.

We now need to adapt the above reasoning to a discrete lattice system with periodic boundary conditions. In doing so we aim at a correct representation on large scales, and at a correct representation of the symmetries of the Green’s function, but not at a faithful representation of the shearing process inside a single grid cell (which cannot be represented anyway in a lattice based simulation). Thus we calculate the stress and strain fields generated by an elementary slip event in a cell as follows (Fig. 8 illustrates the calculation.) The cell under deformation is cut along the x and y direction. The upper part is moved by a distance b along the x direction and the right side is moved by b along the y direction, according to the sign of the shear stress acting on the cell. Then, the four parts are glued back together. Next, an elastic deformation is applied which transforms the cell back to its original shape so it fits its original place in the sample. The cell is placed back to its original position and the sample is elastically relaxed. The average plastic strain generated by this process in the cell is $\varDelta \gamma ^\mathrm{pl} = 2b/d$.

Fig. 8

In the elementary slip event, a cell is cut into four pieces which are displaced according to the acting shear stress and then glued back together. This cell is inserted back into the original lattice and forced elastically to fit, generating an internal stress field

The process is formally equivalent to adding four edge dislocations with the respective virtual Burgers vectors $b{{\varvec{e}}_{x}}$, $b{{\varvec{e}}_{y}}$, $-b{{\varvec{e}}_{x}}$, $-b{{\varvec{e}}_{y}}$ at the centerpoints of the right, top, left and bottom sides of the cell. Accordingly, the stress field can be evaluated as the superposition of the stress fields of these four dislocations. Periodic boundary conditions are implemented by adding to the stress fields of the four dislocations those of their periodic images which form an infinite lattice of period L (for details of the method used for evaluating the lattice sum, see Bako et al. 2006). We evaluate stresses at the cell centerpoints, hence, the stress field induced by a elementary slip event $\varDelta {\gamma ^{{\text {pl}}}}$ at the centerpoint of the active cell is obtained from summation of the stress fields of the four edge dislocations as $G_{0,0}^E\varDelta {\gamma ^{{\text {pl}}}} = - 2\mu \varDelta {\gamma ^{{\text {pl}}}}/\left[ {\pi \left( {1 - \nu } \right) } \right]$ where $\mu$ is the shear modulus and $\nu$ is Poisson’s ratio. The result is shown in Fig. 9.

Fig. 9

The stress field of a unit slip event located in the origin in the units of $\left| {G_{0,0}^E} \right|$, assuming periodic boundary conditions with $L = 32$. Note the symmetry in the x and y directions and the logarithmic color scale. In the upper right corner we show a magnification for $k \in \left[ {0,2} \right] , l\in \left[ {0,2} \right]$

We emphasize that the use of dislocations to evaluate internal stresses is, in our present context, a mere computational device which allows us to treat the periodic boundary conditions in a simple and efficient manner, but which does not necessarily reflect the physical processes which govern the elementary slip event $\varDelta {\gamma ^{{\text {pl}}}}$ (e.g., in the context of amorphous materials, this could be a shear transformation which does not involve any dislocations).