# On Elastic Symmetry Identification for Polycrystalline Materials

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Elasticity Class Identification

**Lemma 1.**

**Proof.**

**Problem 1.**

**Problem 2.**

**Problem 3.**

## 4. Elasticity of Polycrystalline Aggregates

## 5. Changes in Elastic Symmetry of Polycrystals during Inelastic Deformation

#### 5.1. Simple Shear

#### 5.2. Quasi-Axial Tension

#### 5.3. Quasi-Axial Upsetting

## 6. Discussion and Conclusions

**Problem 4.**

**Problem 5.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Acharjee, S.; Zabaras, N. A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to optimal control of microstructure-sensitive properties. Acta Mater.
**2003**, 51, 5627–5646. [Google Scholar] [CrossRef] - 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
**2001**, 49, 1639–1663. [Google Scholar] [CrossRef] - Clement, A. Prediction of deformation texture using a physical principle of conservatiol. Mater. Sci. Eng.
**1982**, 55, 203–210. [Google Scholar] [CrossRef] - Ganapathysubramanian, S.; Zabaras, N. Design across length scales: A reduced-order model of polycrystal plasticity for the control of microstructure-sensitive material properties. Comput. Methods Appl. Mech. Eng.
**2004**, 193, 5017–5034. [Google Scholar] [CrossRef] - Ganapathysubramanian, S.; Zabaras, N. Modeling the thermoelastic-viscoplastic response of polycrystals using a continuum representation over the orientation space. Int. J. Plast.
**2005**, 21, 119–144. [Google Scholar] [CrossRef] - Kumar, A.; Dawson, P.R. Computational modeling of f.c.c. deformation textures over Rodrigues’ space. Acta Mater.
**2000**, 48, 2719–2736. [Google Scholar] [CrossRef] - Kuramae, H.; Sakamoto, H.; Morimoto, H.; Nakamachi, E. Process metallurgy design for high-formability aluminum alloy sheet metal generation by using two-scale FEM. Procedia Eng.
**2011**, 10, 2250–2255. [Google Scholar] [CrossRef] - McDowell, D.L.; Olson, G.B. Concurrent design of hierarchical materials and structures. Sci. Model. Simul. SMNS
**2008**, 15, 207–240. [Google Scholar] [CrossRef] - Nakamachi, E.; Kuramae, H.; Sakamoto, H.; Morimoto, H. Process metallurgy design of aluminum alloy sheet rolling by using two-scale finite element analysis and optimization algorithm. Int. J. Mech. Sci.
**2010**, 52, 146–157. [Google Scholar] [CrossRef] - Proust, G.; Kalidindi, S.R. Procedures for construction of anisotropic elastic-plastic property closures for face-centered cubic polycrystals using first-order bounding relations. J. Mech. Phys. Solids
**2006**, 54, 1744–1762. [Google Scholar] [CrossRef] - Sundararaghavan, V.; Zabaras, N. On the synergy between texture classification and deformation process sequence selection for the control of texture-dependent properties. Acta Mater.
**2005**, 53, 1015–1027. [Google Scholar] [CrossRef] - Sundararaghavan, V.; Zabaras, N. Classification and reconstruction of three-dimensional microstructures using support vector machines. Comput. Mater. Sci.
**2005**, 32, 223–239. [Google Scholar] [CrossRef] - Sundararaghavan, V.; Zabaras, N. Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization. Int. J. Plast.
**2006**, 22, 1799–1824. [Google Scholar] [CrossRef] - Sundararaghavan, V.; Zabaras, N. A statistical learning approach for the design of polycrystalline materials. Stat. Anal. Data Min.
**2009**, 1, 306–321. [Google Scholar] [CrossRef] - Busso, E.P.; Matériaux, C.; Paristech, M. Multiscale Approaches: From the Nanomechanics to the Micromechanics. In Computational and Experimental Mechanics of Advanced Materials; Springer: Vienna, Austria, 2010; pp. 141–165. [Google Scholar]
- Luscher, D.J.; McDowell, D.L. An extended multiscale principle of virtual velocities approach for evolving microstructure. Procedia Eng.
**2009**, 1, 117–121. [Google Scholar] [CrossRef] - Luscher, D.J.; McDowell, D.L.; Bronkhorst, C.A. A second gradient theoretical framework for hierarchical multiscale modeling of materials. Int. J. Plast.
**2010**, 26, 1248–1275. [Google Scholar] [CrossRef] - Trusov, P.V.; Shveykin, A.I. Multilevel crystal plasticity models of single- and polycrystals. Direct models. Phys. Mesomech.
**2013**, 16, 99–124. [Google Scholar] [CrossRef] - Trusov, P.V.; Shveykin, A.I. Multilevel crystal plasticity models of single- and polycrystals. Statistical Models. Phys. Mesomech.
**2013**, 16, 23–33. [Google Scholar] [CrossRef] - Bunge, H.J. Texture Analysis in Materials Science. Mathematical Methods; Butterworths: London, UK, 1982; ISBN 978-0-408-10642-9. [Google Scholar]
- Kalidindi, S.R.; Houskamp, J.R.; Lyons, M.; Adams, B.L. Microstructure sensitive design of an orthotropic plate subjected to tensile load. Int. J. Plast.
**2004**, 20, 1561–1575. [Google Scholar] [CrossRef] - Kalidindi, S.R.; Houskamp, J.; Proust, G.; Duvvuru, H. Microstructure sensitive design with first order homogenization theories and finite element codes. Mater. Sci. Forum
**2005**, 495–497, 23–30. [Google Scholar] [CrossRef] - Sundararaghavan, V.; Zabaras, N. A dynamic material library for the representation of single-phase polyhedral microstructures. Acta Mater.
**2004**, 52, 4111–4119. [Google Scholar] [CrossRef] - Kumar, A.; Dawson, P.R. Modeling crystallographic texture evolution with finite elements over neo-Eulerian orientation spaces. Comput. Methods Appl. Mech. Eng.
**1998**, 153, 259–302. [Google Scholar] [CrossRef] - Becker, R.; Panchanadeeswaran, S. Crystal rotations represented as rodrigues vectors. Textures Microstruct.
**1989**, 10, 167–194. [Google Scholar] [CrossRef] - Morawiec, A.; Field, D.P. Rodrigues parameterization for orientation and misorientation distributions. Philos. Mag. A
**1996**, 73, 1113–1130. [Google Scholar] [CrossRef] - Truesdell, C.A. A First Course in Rational Continuum Mechanics; Academic Press: New York, NY, USA, 1991. [Google Scholar]
- Cowin, S.C.; Mehrabadi, M.M. On the identification of material symmetry for anisotropic elastic materials. Q. J. Mech. Appl. Math.
**1987**, 40, 451–476. [Google Scholar] [CrossRef] - Gurevich, G.B. Foundations of the Theory of Algebraic Invariants; Noordhoff: Groningen, The Netherlands, 1964. [Google Scholar]
- Rychlewski, J. On Hooke’s law. J. Appl. Math. Mech.
**1984**, 48, 303–314. [Google Scholar] [CrossRef] - Ostrosablin, N.I. On invariants of a fourth-rank tensor of elasticity moduli. Sib. Zh. Ind. Mat.
**1998**, 1, 155–163. [Google Scholar] - Spencer, A.J.M. Isotropic Polynomial Invariants and Tensor Functions. In Applications of Tensor Functions in Solid Mechanics; Boehler, J.P., Ed.; Springer: Vienna, Austria, 1987; pp. 141–169. [Google Scholar]
- Zhilin, P.A. The modified theory of the tensor symmetry and tensor invariants. Izv. Vyss. Uchebn. Zaved. Sev.-Kavkaz. Reg. Estestv. Nauki
**2003**, 1, 176–195. [Google Scholar] - Bos, L.; Slawinski, M.A. 2-Norm Effective Isotropic Hookean Solids. J. Elast.
**2015**, 120, 1–22. [Google Scholar] [CrossRef] - Gazis, D.C.; Tadjbakhsh, I.; Toupin, R.A. The elasticity tensor of a given symmetry nearest to an anisotropic elastic tensor. Acta Cryst.
**1963**, 16, 917–922. [Google Scholar] [CrossRef] - Moakher, M.; Norris, A.N. The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry. J. Elast.
**2006**, 85, 215–263. [Google Scholar] [CrossRef] - Norris, A.N. The isotropic material closest to a given anisotropic material. Mater. Struct.
**2006**, 1, 223–238. [Google Scholar] [CrossRef] - Arts, R.J.; Helbig, K.; Rasolofosaon, P.N.J. General anisotropic elastic tensor in rocks: Approximation, invariants, and particular directions. In SEG Technical Program Expanded Abstracts 1991; Society of Exploration Geophysicists: Tulsa, OK, USA, 1991; pp. 1534–1537. [Google Scholar] [CrossRef]
- Danek, T.; Kochetov, M.; Slawinski, M.A. Uncertainty analysis of effective elasticity tensors using quaternion-based global optimization and Monte-Carlo method. Q. J. Mech. Appl. Math.
**2013**, 66, 253–272. [Google Scholar] [CrossRef] - Danek, T.; Slawinski, A. On effective transversely isotropic elasticity tensors based on Frobenius and L
_{2}operator norms. Dolomit. Res. Notes Approx.**2014**, 7, 1–6. [Google Scholar] [CrossRef] - Danek, T.; Slawinski, M.A. On choosing effective elasticity tensors using a monte-carlo method. Acta Geophys.
**2015**, 63, 45–61. [Google Scholar] [CrossRef] - Diner, Ç.; Kochetov, M.; Slawinski, M. Identifying symmetry classes of elasticity tensors using monoclinic distance function. J. Elast.
**2011**, 102, 175–190. [Google Scholar] [CrossRef] - Kochetov, M.; Slawinski, M.A. On obtaining effective transversely isotropic elasticity tensors. J. Elast.
**2009**, 94, 1–13. [Google Scholar] [CrossRef] - Ostapovich, K.V.; Trusov, P.V. On elastic anisotropy: Symmetry identification. Mekhanika Kompositsionnykh Mater. I Konstr.
**2016**, 22, 69–84. [Google Scholar] - Sevostianov, I.; Kachanov, M. On approximate symmetries of the elastic properties and elliptic orthotropy. Int. J. Eng. Sci.
**2008**, 46, 211–223. [Google Scholar] [CrossRef] - Hayes, M. A simple statical approach to the measurement of the elastic constants in anisotropic media. J. Mater. Sci.
**1969**, 4, 10–14. [Google Scholar] [CrossRef] - Norris, A.N. On the acoustic determination of the elastic moduli of anisotropic solids and acoustic conditions for the existence of symmetry planes. Q. J. Mech. Appl. Math.
**1989**, 42, 413–426. [Google Scholar] [CrossRef] - Tsvelodub, I.Y. Determining the elastic characteristics of homogeneous anisotropic bodies. J. Appl. Mech. Tech. Phys.
**1994**, 35, 455–458. [Google Scholar] [CrossRef] - Khristich, D.V. Criterion of experimental identification of isotropic and cubic materials. Izv. Tul. Gos. Univ. Est. Nauki
**2012**, 1, 110–118. [Google Scholar] - Khristich, D.V. Criterion of experimental identification of rhombic, monoclinic and triclinic materials. Izv. Tul. Gos. Univ. Est. Nauk.
**2013**, 1, 166–178. [Google Scholar] - Khristich, D.V. On the problem of material main anisotropy axes identification. Izv. Tul. Gos. Univ. Est. Nauki
**2014**, 1, 203–213. [Google Scholar] - Sokolova, M.Y.; Khristich, D.V. Program of experiments to determine the type of initial elastic anisotropy of material. J. Appl. Mech. Tech. Phys.
**2015**, 56, 913–919. [Google Scholar] [CrossRef] - Astapov, Y.V.; Khristich, D.V. Numerical modeling of experiments by detecting of initial anisotropy type of elastic materials. Comput. Contin. Mech.
**2015**, 8, 386–396. [Google Scholar] [CrossRef] - Shveykin, A.I.; Trusov, P.V. Correlation between geometrically nonlinear elasto-visco-plastic constitutive relations formulated in terms of the actual and unloaded configurations for crystallites. Phys. Mesomech.
**2016**, 19, 48–57. [Google Scholar] - Hazewinkel, M.; Gubareni, N.; Kirichenko, V.V. Algebras, Rings and Modules; Kluwer: Dordrecht, The Netherlands, 2004; Volume 1. [Google Scholar]
- Trusov, P.V.; Dudar’, O.I.; Keller, I.E. Tensor Algebra and Analysis; Perm State Technical University: Perm, Russia, 1998. [Google Scholar]
- Curnier, A. Computational Methods in Solid Mechanics; Kluwer: Dordrecht, The Netherlands, 1994. [Google Scholar]
- Bertram, A. Elasticity and Plasticity of Large Deformations; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 2012; ISBN 978-3-642-24614-2. [Google Scholar]
- Olshevsky, V. Structured Matrices in Mathematics, Computer Science, and Engineering I; American Mathematical Society: Washington, DC, USA, 2001; ISBN 0821819216. [Google Scholar]
- Trenogin, V.A. Functional Analysis; Nauka: Moscow, Russia, 1980. [Google Scholar]
- Love, A. A Treatise on the Mathematical Theory of Elasticity; Dover: New York, NY, USA, 1944. [Google Scholar]
- Green, A.E.; Adkins, J.E. Large Elastic Deformations and Non-Linear Continuum Mechanics; Oxford Clarenden Press: Oxford, UK, 1960. [Google Scholar]
- Bóna, A.; Bucataru, I.; Slawinski, M.A. Material symmetries of elasticity tensors. Q. J. Mech. Appl. Math.
**2004**, 57, 583–598. [Google Scholar] [CrossRef] - Forte, S.; Vianello, M. Symmetry classes for elasticity tensors. J. Elast.
**1996**, 43, 81–108. [Google Scholar] [CrossRef] - Minkevich, L.M. Presentation of elasticity and compliance tensors via eigentensors. Issues Dyn. Mech. Syst. Vib. Eff.
**1973**, 1, 107–110. [Google Scholar] - Ostrosablin, N.I. On the structure of the elasticity moduli tensor. Elastic eigenstates. Dyn. Contin. Media
**1984**, 1, 113–125. [Google Scholar] - Sutcliffe, S. Spectral Decomposition of the Elasticity Tensor. J. Appl. Mech.
**1992**, 59, 762. [Google Scholar] [CrossRef] - Weyl, H. The Classical Groups: Their Invariants and Representations, 2nd ed.; Princeton University Press: Princeton, NJ, USA, 1997. [Google Scholar]
- Sirotin, Y.I.; Shaskolskaya, M.P. Fundamentals of Crystal Physics; Mir Publishers: Moscow, Russia, 1982. [Google Scholar]
- Voigt, W. Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik); Teubner Verlag: Leipzig, Germany, 1928. [Google Scholar]
- Clerc, M.; Kennedy, J. The particle swarm—Explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evol. Comput.
**2002**, 6, 58–73. [Google Scholar] [CrossRef] - Bunge, H.J.; Kiewel, R.; Reinert, T.; Fritsche, L. Elastic properties of polycrystals—Influence of texture and stereology. J. Mech. Phys. Solids
**2000**, 48, 29–66. [Google Scholar] [CrossRef] - Borovkov, A.A. Probability Theory; Springer: London, UK, 2013. [Google Scholar]
- Kuksa, L.V.; Arzamaskova, L.M.; Sergeev, A.V. Vectorial models of cubic, hexagonal, trigonal crystals and elasticity scale effect of composites based on them. Izv. Volgogr. Gos. Tekh. Univ.
**2005**, 1, 85–90. [Google Scholar] - Kuksa, L.V.; Arzamaskova, L.M. Comparative studies on scale effect of physical and mechanical properties of single-phase and two-phase polycrystalline materials. Izv. Volgogr. Gos. Tekh. Univ.
**2009**, 11, 127–133. [Google Scholar] - Shermergor, T.D. Theory of Elasticity of Micro-Inhomogeneous Media; Nauka: Moscow, Russia, 1977. [Google Scholar]
- Trusov, P.V.; Shveykin, A.I.; Yanz, A.Y. Motion decomposition, frame-independent derivatives and constitutive relations at large displacement gradients from the viewpoint of multilevel modeling. Phys. Mesomech.
**2016**, 19, 47–65. [Google Scholar] - Böhlke, T.; Bertram, A. Isotropic orientation distributions of cubic crystals. J. Mech. Phys. Solids
**2001**, 49, 2459–2470. [Google Scholar] [CrossRef] - Bertram, A.; Böhlke, T.; Gaffke, N.; Heiligers, B.; Offinger, R. On the generation of discrete isotropic orientation distributions for linear elastic cubic crystals. J. Elast.
**2000**, 58, 233–248. [Google Scholar] [CrossRef] - Paroni, R.; Man, C.-S. Constitutive equations of elastic polycrystalline materials. Arch. Ration. Mech. Anal.
**1999**, 150, 153–177. [Google Scholar] [CrossRef]

**Figure 1.**Symmetry classes of elasticity tensors. Numbers of independent components and cardinalities of the symmetry groups are specified in the parentheses ($\mathfrak{c}$ for continuum). The arrows indicate the directions of inclusion between the corresponding groups [63].

**Figure 2.**Dimensionless MEs in the isotropic class with respect to the elasticity tensors of random realizations of polycrystalline copper aggregates with uniformly distributed orientations versus the number of crystallites: (

**a**) $\Vert {{\u043f}^{\prime}}^{-1}\Vert {\overline{ME}}_{I}^{\left(iso\right)}$; (

**b**) $\Vert {{\u043f}^{\prime}}^{-1}\Vert {\overline{ME}}_{II}^{\left(iso\right)}$.

**Figure 3.**Regressions for mean dimensionless MEs in the isotropic class as functions of the number of crystallites with respect to the elasticity tensors of random polycrystalline aggregates with uniformly distributed orientations: (

**a**) $\Vert {{\u043f}^{\prime}}^{-1}\Vert {\overline{ME}}_{I}^{\left(iso\right)}$; (

**b**) $\Vert {{\u043f}^{\prime}}^{-1}\Vert {\overline{ME}}_{I}^{\left(iso\right)}$.

**Figure 4.**Changes in the DPFs (projecting along ${l}_{3}$) of the polycrystalline copper aggregate during the simple shear test: (

**a**) <111>; (

**b**) <110>; (

**c**) <100>.

**Figure 5.**Changes in the elastic symmetry properties of the polycrystalline copper aggregate during the simple shear test: (

**a**) RMEs in the isotropic class; (

**b**) ${\overline{RME}}_{I}^{\left(s\right)}$.

**Figure 6.**Changes in the ${\Vert \cdot \Vert}_{{\mathbb{E}}_{3}^{4}}$-optimal approximation orientation parameters and the corresponding values of ${\overline{RME}}_{II}^{\left(s\right)}$ for the elasticity tensor of the polycrystalline copper aggregate during the simple shear test.

**Figure 7.**Changes in the minimal non-zero Eigen-value of the elasticity tensor of the polycrystalline copper aggregate during the simple shear test.

**Figure 8.**Changes in the DPFs (projecting along ${l}_{3}$) of the polycrystalline copper aggregate during the quasi-axial tension test: (

**a**) <111>; (

**b**) <110>; (

**c**) <100>.

**Figure 9.**Changes in the elastic symmetry properties of the polycrystalline copper aggregate during the quasi-axial tension test: (

**a**) RMEs in the isotropic class; (

**b**) ${\overline{RME}}_{I}^{\left(s\right)}$.

**Figure 10.**Changes in the ${\Vert \cdot \Vert}_{{\mathbb{E}}_{3}^{4}}$-optimal approximation orientation parameters and the corresponding values of ${\overline{RME}}_{II}^{\left(s\right)}$ for the elasticity tensor of the polycrystalline copper aggregate during the quasi-axial tension test.

**Figure 11.**Changes in the minimal non-zero Eigen-value of the elasticity tensor of the polycrystalline copper aggregate during the quasi-axial tension test.

**Figure 12.**Changes in the DPFs (projecting along ${l}_{3}$) of the polycrystalline copper aggregate during the quasi-axial upsetting test: (

**a**) <100>; (

**b**) <110>; (

**c**) <111>.

**Figure 13.**Changes in the elastic symmetry properties of the polycrystalline copper aggregate during the quasi-axial upsetting test: (

**a**) RMEs in the isotropic class; (

**b**) ${\overline{RME}}_{I}^{\left(s\right)}$.

**Figure 14.**Changes in the ${\Vert \cdot \Vert}_{{\mathbb{E}}_{3}^{4}}$-optimal approximation orientation parameters and the corresponding values of ${\overline{RME}}_{II}^{\left(s\right)}$ for the elasticity tensor of the polycrystalline copper aggregate during the quasi-axial upsetting test.

**Figure 15.**Changes in the minimal non-zero Eigen-value of the elasticity tensor of the polycrystalline copper aggregate during the quasi-axial upsetting test.

Symmetry Class ($\mathit{s}$) | Elastic Modulus Index ($\mathsf{\alpha}$) | Non-Zero Components of ${K}_{\mathsf{\alpha}}^{\left(\mathit{s}\right)}$ in Canonical Basis | |
---|---|---|---|

Indexes | Value | ||

Isotropic | 1122 | 1111, 1122, 1133,2211, 2222, 2233,3311, 3322, 3333 | 1 |

1212 | 1212, 1221, 1313, 1331,2112, 2121, 2323, 2332,3113, 3131, 3223, 3232 | 1 | |

1111, 2222, 3333 | 2 | ||

Transversely Isotropic | 1111 | 1212, 1221, 2112, 2121 | 0.5 |

1111, 2222 | 1 | ||

1122 | 1212, 1221, 2112, 2121 | –0.5 | |

1122, 2211 | 1 | ||

1133 | 1133, 2233, 3311, 3322 | 1 | |

2323 | 1212, 1221, 1313, 1331,2112, 2121, 3113, 3131 | 1 | |

3333 | 3333 | 1 | |

Cubic | 1111 | 1111, 2222, 3333 | 1 |

1122 | 1122, 1133, 2211, 2233, 3311, 3322 | 1 | |

1212 | 1212, 1221, 1313, 1331,2112, 2121, 2323, 2332,3113, 3131, 3223, 3232 | 1 | |

Orthotropic | 1111 | 1111 | 1 |

1122 | 1122, 2211 | 1 | |

1133 | 1133, 3311 | 1 | |

1212 | 1212, 1221, 2112, 2121 | 1 | |

1313 | 1313, 1331, 3113, 3131 | 1 | |

2222 | 2222 | 1 | |

2233 | 2233, 3322 | 1 | |

2323 | 2323, 2332, 3223, 3232 | 1 | |

3333 | 3333 | 1 |

Notation | Expression | Minimized Objective |
---|---|---|

${\overline{ME}}_{I}^{\left(s\right)}$ | ${\Vert {\Psi}^{\left(s\right)}\Vert}_{{\mathbb{E}}_{3}^{4}}$ | ${\Vert {\Psi}^{\left(s\right)}\Vert}_{{\mathbb{E}}_{3}^{4}}$ |

${\overline{ME}}_{II}^{\left(s\right)}$ | ${\Vert {\Psi}^{\left(s\right)}\Vert}_{\text{\hspace{1em}}}$ | ${\Vert {\Psi}^{\left(s\right)}\Vert}_{{\mathbb{E}}_{3}^{4}}$ |

${\overline{ME}}_{III}^{\left(s\right)}$ | ${\Vert {\Psi}^{\left(s\right)}\Vert}_{\text{\hspace{1em}}}$ | ${\Vert {\Psi}^{\left(s\right)}\Vert}_{\text{\hspace{1em}}}$ |

**Table 3.**Regression parameters for mean dimensionless MEs in the isotropic class as functions of the number of crystallites with respect to the elasticity tensors of random polycrystalline aggregates with uniformly distributed orientations.

Crystal | Elastic Moduli, GPa | $\Vert {{\mathsf{\u043f}}^{\prime}}^{-1}\Vert {\overline{\mathit{M}\mathit{E}}}_{\mathit{I}\text{\hspace{0.17em}}1}^{\left(\mathit{i}\mathit{s}\mathit{o}\right)}$ | ${\mathsf{\lambda}}_{\mathit{I}}$ | $\Vert {{\mathsf{\u043f}}^{\prime}}^{-1}\Vert {\overline{\mathit{M}\mathit{E}}}_{\mathit{I}\mathit{I}\text{\hspace{0.17em}}1}^{\left(\mathit{i}\mathit{s}\mathit{o}\right)}$ | ${\mathsf{\lambda}}_{\mathit{I}\mathit{I}}$ | ||||
---|---|---|---|---|---|---|---|---|---|

${\text{\u043f}}_{1111}$ | ${\text{\u043f}}_{1122}$ | ${\text{\u043f}}_{2323}$ | ${\text{\u043f}}_{1133}$ | ${\text{\u043f}}_{3333}$ | |||||

Transversely Isotropic Elasticity | |||||||||

Mg | 59.7 | 26.2 | 16.4 | 21.7 | 61.7 | 0.22 | 0.50 | 0.19 | 0.55 |

$\mathsf{\alpha}$-Ti | 162.4 | 92.0 | 46.7 | 69.0 | 180.7 | 0.52 | 0.51 | 0.36 | 0.54 |

Zn | 161.0 | 34.2 | 38.3 | 50.1 | 61.0 | 3.08 | 0.51 | 2.36 | 0.54 |

Cubic Elasticity | |||||||||

Al | 108.4 | 62.3 | 28.5 | - | - | 0.26 | 0.51 | 0.14 | 0.48 |

Cu | 168.4 | 121.4 | 75.4 | - | - | 2.42 | 0.51 | 1.36 | 0.48 |

$\mathsf{\alpha}$-Fe | 287.0 | 141.0 | 116.0 | - | - | 0.65 | 0.50 | 0.35 | 0.47 |

**Table 4.**Crystallite mutual orientations in the polycrystalline copper aggregates with the most ${\Vert \cdot \Vert}_{{\mathbb{E}}_{3}^{4}}$- isotropic elastic properties.

$\mathit{M}$ | Optimal Mutual Orientations | DPFs (Projecting along ${\mathbf{l}}_{3}$) | ${\overline{\mathit{M}\mathit{E}}}_{\mathit{I}}^{\left(\mathit{i}\mathit{s}\mathit{o}\right)}$, GPa | ${\overline{\mathit{R}\mathit{M}\mathit{E}}}_{\mathit{I}}^{\left(\mathit{i}\mathit{s}\mathit{o}\right)}$ | ||
---|---|---|---|---|---|---|

<111> | <110> | <100> | ||||

2 | 57.897 | 0.710 | ||||

3 | 18.951 | 0.192 | ||||

4 ^{1} | 0 | 0 |

^{1}The general case is considered in [78].

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Trusov, P.V.; Ostapovich, K.V.
On Elastic Symmetry Identification for Polycrystalline Materials. *Symmetry* **2017**, *9*, 240.
https://doi.org/10.3390/sym9100240

**AMA Style**

Trusov PV, Ostapovich KV.
On Elastic Symmetry Identification for Polycrystalline Materials. *Symmetry*. 2017; 9(10):240.
https://doi.org/10.3390/sym9100240

**Chicago/Turabian Style**

Trusov, Peter V., and Kirill V. Ostapovich.
2017. "On Elastic Symmetry Identification for Polycrystalline Materials" *Symmetry* 9, no. 10: 240.
https://doi.org/10.3390/sym9100240