Materials design for the anisotropic linear elastic properties of textured cubic crystal aggregates using zeroth-, first- and second-order bounds

Abstract

For polycrystals made of cubic materials like copper, aluminum, iron and other metals and ceramics, the macroscopic elastic behavior can be bounded using minimum energy principles. Böhlke and Lobos (Acta Mater. 67:324–334, 2014) have shown that not only the Voigt and the Reuss bound but also the Hashin–Shtrikman bounds can be represented explicitly depending on the texture in form of the fourth-order texture coefficient. Considering the inequalities due to these bounds, the texture can be enclosed independently of the specific cubic material parameters. This implies domains for the texture parameters. Materials design is defined as the identification of materials and microstructures such that the effective constitutive properties correspond best to a prescribed properties profile. The design space is proposed to be constituted by the material design space and microstructure design space, delivering a total of twelve scalar design variables in the present model for linear elasticity of cubic crystal aggregates. Based on analytical results, materials design is established as an algorithm following Adams et al. (Microstructure Sensitive Design for Performance Optimization, 2013). In the present work, the scheme consists of four steps: (i) material selection, (ii) homogenization scheme, (iii) properties closure, and (iv) microstructure optimization. As an example, Young’s modulus of a polycrystal is designed with respect to four prescribed directions for a macroscopical orthotropic sample symmetry. For the orthotropic texture domain, a mathematically equivalent parametrization is derived in order to facilitate the constrained numerical optimizations.

Appendices

Appendix

Proof of the equivalent parametrization of the orthotropic texture domain

The following proof is based on the geometric properties of the constraint regions (26). The regions are represented in Fig. 2. The resulting texture domain described by the inequalities in (24), reducible to (27), is equivalent to the resulting texture domain described by (28).

In order to facilitate the visibility of the different regions $R_{1..6}$ the following affine transformation is used

$$ \left[\begin{array}{l} V_1 \\ V_2 \\V_3 \end{array}\right] = \frac{1}{3}\sqrt{\frac{2}{15}} \left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] + \sqrt{\frac{5}{6}} \left[\begin{array}{lll} -1/3&1/\sqrt{3}&-1/3 \\ -1/3&-1/\sqrt{3}&-1/3 \\ 2/3&0&-1/3 \end{array}\right] \left[\begin{array}{l} W_1 \\ W_2 \\ W_3 \end{array}\right]. $$

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The transformed regions are denoted by $R'_{1..6}$

$$\begin{aligned}&R'_1: \sqrt{3}(W_1 - \sqrt{3}W_2 + W_3) \le \sqrt{3} \nonumber \\&\quad \quad \wedge \sqrt{3}(W_1 + \sqrt{3}W_2 + W_3) \le \sqrt{3} \ \wedge -2 W_1 + W_3 \le 1 \ , \nonumber \\&R'_2: 6 W_2 \le \sqrt{3} (2 W_1 + 2 W_3+1) \nonumber \\&\quad \quad \wedge 2 \sqrt{3} W_1+6 W_2+\sqrt{3} (2 W_3+1)\ge 0 \ \wedge 4 W_1 \le 2 W_3+1 \ , \nonumber \\&R'_3: \sqrt{W_1^2+W_2^2} \le W_3 \ , \ R'_4: \sqrt{W_1^2+W_2^2} \ge W_3 - 1 \ , \nonumber \\&R'_5: \sqrt{W_1^2+W_2^2} \ge -W_3 \ , \ R'_6: \sqrt{W_1^2+W_2^2} \le -W_3 + 1 \ , \end{aligned}$$

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with $R'_i$ corresponding to $R_i$. The pyramid corners $R'_1$ and $R'_2$ are the transformed cuboid corners $R_1$ and $R_2$. $R'_3$ makes visible that $R_3$ is only the affine transformation of a cone. $R'_4$ and $R'_5$ are the outside regions of the cones $\sqrt{W_1^2+W_2^2} \le W_3 - 1$ starting at $W_3=1$ in positive $W_3$ direction and $\sqrt{W_1^2+W_2^2} \le -W_3$ starting at $W_3=0$ in negative $W_3$ direction, respectively. The last region $R'_6$ represents a cone starting at $W_3=1$ (as the pyramid corner $R'_1$) in negative $W_3$ direction. The original and the affine transformed orthotropic texture domains are illustrated in Fig. 7.

Fig. 7

Original and affine transformed orthotropic texture domain

Having this geometrical interpretation of the different regions, it can be trivially shown using elementary geometry that the regions $R'_2$, $R'_4$, $R'_5$ and $R'_6$ (compare with Fig. 2) are unnecessary for the final intersection of all regions. The final intersection is described by the pyramid corner $R'_1$ and the cone $R'_3$, which are the affine transformations of (27).

The inequality

$$\begin{aligned} W_1^2 + W_2^2 \le W_3^2 \end{aligned}$$

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is equal to

$$\begin{aligned} \sqrt{W_1^2+W_2^2} \le W_3 \quad \vee \quad \sqrt{W_1^2+W_2^2} \le -W_3 \end{aligned}$$

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which is the union of the cones $R'_3$ and the one in opposite direction. It is therefore trivial that $R'_3$ can equivalently be described by

$$\begin{aligned} W_1^2 + W_2^2 \le W_3^2 \quad \wedge \quad W_3 \ge 0 \end{aligned}$$

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which together with $R'_1$ is the affine transformation of the alternative parametrization (28). This completes the proof.