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.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10999-014-9272-z/MediaObjects/10999_2014_9272_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10999-014-9272-z/MediaObjects/10999_2014_9272_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10999-014-9272-z/MediaObjects/10999_2014_9272_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10999-014-9272-z/MediaObjects/10999_2014_9272_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10999-014-9272-z/MediaObjects/10999_2014_9272_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10999-014-9272-z/MediaObjects/10999_2014_9272_Fig6_HTML.gif)
Similar content being viewed by others
References
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)
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)
Adams, B.L., Kalidindi, S.R., Fullwood, D.T.: Microstructure Sensitive Design for Performance Optimization. Butterworth-Heinemann, Waltham, MA (2013)
Ashby, M.F.: Materials Selection in Mechanical Design, 4th edn. Butterworth-Heinemann, Waltham, MA (2010)
Böhlke, T.: Texture simulation based on tensorial Fourier coefficients. Comput. Struct. 84(17–18), 1086–1094 (2006)
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)
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)
Bunge, H.J.: Zur Darstellung allgemeiner Texturen. Z. Metallkd. 56, 872–874 (1965)
Bunge, H.J.: Texture analysis in materials science: mathematical methods. Butterworth, London (1982)
Fedorov, F.I.: Theory of Elastic Waves in Crystals. Plenum Press, New York (1968)
Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elasticity 43(2), 81–108 (1996)
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)
Gel'fand, I.M., Minlos, R., Shapiro, Z.: Representations of the Rotation and Lorentz Groups and Their Applications. Pergamon Press, Oxford (1963)
Grimsditch, M., Zouboulis, E.S., Polian, A.: Elastic constants of boron nitride. J. Appl. Phys. 76(2), 832 (1994)
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)
Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10, 343–352 (1962)
Hill, R.: The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. A 65, 349–354 (1952)
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)
Kocks, U., Tomé, C., Wenk, H.: Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties. Cambridge University Press, Cambridge (2000)
Kröner, E.: Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids 25(3), 137–155 (1977)
Li, D., Garmestani, H., Adams, B.L.: A texture evolution model in cubic-orthotropic polycrystalline system. Int. J. Plasticity 21(8), 1591–1617 (2005)
Li, D., Garmestani, H., Ahzi, S.: Processing path optimization to achieve desired texture in polycrystalline materials. Acta Mater. 55(2), 647–654 (2007)
Mehrabadi, M.M., Cowin, S.C.: Eigentensors of linear anisotropic elastic materials. Quart. J. Mech. Appl. Math. 43(1), 15–41 (1990)
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)
Paufler, P., Schulze, G.: Physikalische Grundlagen mechanischer Festkörpereigenschaften. Vieweg, Berlin (1978)
Reuss, A.: Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech. 9, 49–58 (1929)
Simmons, G., Wang, H.: Single Crystal Elastic Constants and Calculated Aggregate Properties. A Handbook. The MIT Press, Cambridge (1971)
Sundararaghavan, V., Zabaras, N.: Linear analysis of texture-property relationships using process-based representations of Rodrigues space. Acta Mater. 55(5), 1573–1587 (2007)
Sundararaghavan, V., Zabaras, N.: A statistical learning approach for the design of polycrystalline materials. Stat. Anal. Data Min. 1(5), 306–321 (2009)
Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New-York (2002)
Voigt, W.: Lehrbuch der Kristallphysik: (mit Ausschluss der Kristalloptik). Teubner Leipzig, Berlin (1910)
Author information
Authors and Affiliations
Corresponding author
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
The transformed regions are denoted by \(R'_{1..6}\)
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.
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
is equal to
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
which together with \(R'_1\) is the affine transformation of the alternative parametrization (28). This completes the proof.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10999-014-9272-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10999-014-9272-z