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


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.

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  1. Adams, B.L., Henrie, A., Henrie, B., Lyon, M., Kalidindi, S.R., Garmestani, H.: Microstructure-sensitive design of a compliant beam. J. Mech. Phys. Solids 49(8), 1639–1663 (2001)

    Article  MATH  Google Scholar 

  2. Adams, B.L., Lyon, M., Henrie, B.: Microstructures by design: linear problems in elastic-plastic design. Int. J. Plasticity 20(8–9), 1577–1602 (2004)

    Article  MATH  Google Scholar 

  3. Adams, B.L., Kalidindi, S.R., Fullwood, D.T.: Microstructure Sensitive Design for Performance Optimization. Butterworth-Heinemann, Waltham, MA (2013)

    Google Scholar 

  4. Ashby, M.F.: Materials Selection in Mechanical Design, 4th edn. Butterworth-Heinemann, Waltham, MA (2010)

    Google Scholar 

  5. Böhlke, T.: Texture simulation based on tensorial Fourier coefficients. Comput. Struct. 84(17–18), 1086–1094 (2006)

    Article  Google Scholar 

  6. Böhlke, T., Bertram, A.: The evolution of Hooke’s law due to texture development in FCC polycrystals. Int. J Solids Struct. 38(52), 9437–9459 (2001)

    Article  MATH  Google Scholar 

  7. Böhlke, T., Lobos, M.: Representation of Hashin-Shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with application in materials design. Acta Mater. 67, 324–334 (2014)

    Article  Google Scholar 

  8. Bunge, H.J.: Zur Darstellung allgemeiner Texturen. Z. Metallkd. 56, 872–874 (1965)

    Google Scholar 

  9. Bunge, H.J.: Texture analysis in materials science: mathematical methods. Butterworth, London (1982)

    Google Scholar 

  10. Fedorov, F.I.: Theory of Elastic Waves in Crystals. Plenum Press, New York (1968)

    Book  Google Scholar 

  11. Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elasticity 43(2), 81–108 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fullwood, D.T., Niezgoda, S.R., Adams, B.L., Kalidindi, S.R.: Microstructure sensitive design for performance optimization. Prog. Mater. Sci. 55(6), 477–562 (2010)

    Article  Google Scholar 

  13. Gel'fand, I.M., Minlos, R., Shapiro, Z.: Representations of the Rotation and Lorentz Groups and Their Applications. Pergamon Press, Oxford (1963)

    MATH  Google Scholar 

  14. Grimsditch, M., Zouboulis, E.S., Polian, A.: Elastic constants of boron nitride. J. Appl. Phys. 76(2), 832 (1994)

    Article  Google Scholar 

  15. Guidi, M., Adams, B.L., Onat, E.T.: Tensorial representation of the orientation distribution function in cubic polycrystals. Texture Microstruct. 19(3), 147–167 (1992)

    Article  Google Scholar 

  16. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10, 343–352 (1962)

    Article  MathSciNet  Google Scholar 

  17. Hill, R.: The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. A 65, 349–354 (1952)

    Article  Google Scholar 

  18. Kalidindi, S.R., Knezevic, M., Niezgoda, S., Shaffer, J.: Representation of the orientation distribution function and computation of first-order elastic properties closures using discrete Fourier transforms. Acta Mater. 57(13), 3916–3923 (2009)

    Article  Google Scholar 

  19. Kocks, U., Tomé, C., Wenk, H.: Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  20. Kröner, E.: Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids 25(3), 137–155 (1977)

    Article  MATH  Google Scholar 

  21. Li, D., Garmestani, H., Adams, B.L.: A texture evolution model in cubic-orthotropic polycrystalline system. Int. J. Plasticity 21(8), 1591–1617 (2005)

    Article  MATH  Google Scholar 

  22. Li, D., Garmestani, H., Ahzi, S.: Processing path optimization to achieve desired texture in polycrystalline materials. Acta Mater. 55(2), 647–654 (2007)

    Article  Google Scholar 

  23. Mehrabadi, M.M., Cowin, S.C.: Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math. 43(1), 15–41 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  24. Nadeau, J., Ferrari, M.: On optimal zeroth-order bounds with application to Hashin–Shtrikman bounds and anisotropy parameters. Int. J. Solids Struct. 38(44), 7945–7965 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Paufler, P., Schulze, G.: Physikalische Grundlagen mechanischer Festkörpereigenschaften. Vieweg, Berlin (1978)

    Book  Google Scholar 

  26. Reuss, A.: Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech. 9, 49–58 (1929)

    Article  MATH  Google Scholar 

  27. Simmons, G., Wang, H.: Single Crystal Elastic Constants and Calculated Aggregate Properties. A Handbook. The MIT Press, Cambridge (1971)

    Google Scholar 

  28. Sundararaghavan, V., Zabaras, N.: Linear analysis of texture-property relationships using process-based representations of Rodrigues space. Acta Mater. 55(5), 1573–1587 (2007)

    Article  Google Scholar 

  29. Sundararaghavan, V., Zabaras, N.: A statistical learning approach for the design of polycrystalline materials. Stat. Anal. Data Min. 1(5), 306–321 (2009)

    Article  MathSciNet  Google Scholar 

  30. Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New-York (2002)

    Book  Google Scholar 

  31. Voigt, W.: Lehrbuch der Kristallphysik: (mit Ausschluss der Kristalloptik). Teubner Leipzig, Berlin (1910)

    Google Scholar 

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Correspondence to Thomas Böhlke.



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

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

The transformed regions are denoted by \(R'_{1..6}\)

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

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

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

is equal to

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

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

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

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

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Lobos, M., Böhlke, T. Materials design for the anisotropic linear elastic properties of textured cubic crystal aggregates using zeroth-, first- and second-order bounds. Int J Mech Mater Des 11, 59–78 (2015).

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  • Cubic crystal aggregates
  • Materials design
  • Hashin–Shtrikman bounds
  • Tensorial texture coefficients