On the Symmetry Properties of Various Relations for the Transversely Isotropic Material

Abstract

The paper considers nonlinear, tensor-linear, and linear relationships linking the stresses and strains in the transversely isotropic materials. Based on their structural symmetry properties, the transversely isotropic materials are divided into five classes: D_∞, D_∞h, C_∞, C_∞ν, C_∞. Relationships based on the tensor function theory are obtained to describe the nonlinear relationship between stresses and strains in the transversely isotropic materials exhibiting the plastic properties. The paper shows that two types of the tensor functions are sufficient to describe plastic properties of the transversely isotropic materials of five classes. At this point, the symmetry principle is used, which states that the material structure symmetry elements should be contained in the symmetry group of the D_∞h or C_∞h transversely isotropic material. For the C_∞h symmetry, two possible function representations are available: polynomial and non-polynomial. The simplified versions of the nonlinear relationship between strains and stresses are the tensor-linear (quasilinear) relations. For the transversely isotropic materials, the relations are derived from the nonlinear functions of a general form. Linear and elastic-linear relations are special cases of the polynomial tensor-linear relations. The paper shows that compliance matrices of the transversely isotropic materials have the same form. The elastic-linear relations of five classes of the transversely isotropic materials belong to the same D_∞h symmetry class. The paper indicated that a symmetry group of plastic properties of each transversely isotropic material is equal to or higher than the structure symmetry group; and symmetry group of the material elastic properties is equal to or higher than the plastic properties group

Please cite this article in English as:

Tsvetkov S.V. On the symmetry properties of various relations for the transversely isotropic material. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 2025, no. 4 (155), pp. 102--117 (in Russ.). EDN: HKLJBO

References

[1] Vasin R.A. On experimental verification of basic hypothesis and models of plasticity. Vestnik Nizhegorodskogo universiteta im. N.I. Lobachevskogo, 2011, no. 4, pp. 1415--1417 (in Russ.). EDN: TBGNWX

[2] Drukker D. Plastichnost, techenie i razrushenie [Plasticity, yielding and fracture]. V kn.: Neuprugie svoystva kompozitsionnykh materialov [In: Inelastic behavior of composite materials]. Moscow, Mir Publ., 1978, pp. 9--32 (in Russ.).

[3] Sarbaev B.S. Constitutive equations for high temperature composite materials based on endochronic theory thermoplasticity. Problemy mashinostroeniya i nadezhnosti mashin, 2019, no. 7, pp. 97--104 (in Russ.). DOI: https://doi.org/10.1134/S0235711919070113

[4] Treshchev A.A., Gvozdev A.E., Yushchenko N.S., et al. Nonlinear mathematical model of relation of second-rank tensors for composite materials. Chebyshevskiy sbornik, 2022, vol. 23, no. 3, pp. 224--237 (in Russ.). DOI: https://doi.org/10.22405/2226-8383-2022-23-3-224-237

[5] Rabotnov Yu.N. Polzuchest elementov konstruktsiy [Creep of structure elements]. Moscow, Nauka Publ., 1966.

[6] Pobedrya B.E. Theory of plasticity of anisotropic materials. Prikladnye zadachi prochnosti i plastichnosti, 1984, pp. 110--115 (in Russ.).

[7] Hamermesh M. Group theory and its application to physical problems. New York, Addison-Wesley, 1989.

[8] Chernykh K.F. Vvedenie v anizotropnuyu uprugost [Introduction to anisotropic elasticity]. Moscow, Nauka Publ., 1988.

[9] Curie P. O simmetrii v fizicheskikh yavleniyakh: simmetriya elektricheskogo i magnitnogo poley [On symmetry in physical phenomena: symmetry of electric and magnetic fields]. V kn.: Izbrannye Trudy [In: Selected works]. Moscow, Nauka Publ., 1966, pp. 95--113 (in Russ.).

[10] Spencer A.J.M. Teoriya invariantov [Theory of invariants]. Moscow, Mir Publ., 1974.

[11] Lokhin V.V., Sedov L.I. Nonlinear tensor functions of several tensor arguments. Prikladnaya matematika i mekhanika, 1963, vol. 27, no. 3, pp. 393--417 (in Russ.).

[12] Boehler J.P. A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. ZAMM, 1979, vol. 59, no. 4, pp. 157--167. DOI: https://doi.org/10.1002/zamm.19790590403

[13] Tsvetkov S.V. Non-linear constitutive equations for transversely isotropic materials belonging to the C_∞ and C_∞h symmetry groups. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2019, no. 3 (84), pp. 46--59 (in Russ.).DOI: http://dx.doi.org/10.18698/1812-3368-2019-3-46-59

[14] Zheng Q.S. On transversely isotropic, orthotropic and relative isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Part I: Two dimensional orthotropic and relative isotropic functions and three dimensional relative isotropic functions. Int. J. Eng. Sc., 1993, vol. 31, no. 10, pp. 1399--1409. DOI: https://doi.org/10.1016/0020-7225(93)90005-F

[15] Xiao H. On anisotropic functions of vectors and second order tensors --- all subgroups of the transverse isotropy group C_∞h. Arch. Mech., 1998, vol. 50, no. 2, pp. 281--319. DOI: https://doi.org/10.24423/aom.1468

[16] Zheng Q.S. Theory of representations for tensor functions --- a unified invariant approach to constitutive equations. Appl. Mech. Rev., 1994, vol. 47, no. 11, pp. 545--587. DOI: https://doi.org/10.1115/1.3111066

[17] Tsvetkov S.V. Strength criteria of transversally-isotropic materials of different classes of structure symmetry. Herald of the Bauman Moscow State Technical University, Series Mechanical Engineering, 2009, no. 1 (74), pp. 86--99 (in Russ.). EDN: KDYHDH

[18] Boehler J.P., Sawczuk A. On yielding of oriented solids. Acta Mech., 1977, vol. 27, no. 1, pp. 185--204. DOI: https://doi.org/10.1007/BF01180085

[19] Pobedrya B.E. Theory of plasticity of transversely isotropic materials. Mekhanika tverdogo tela, 1990, no. 3, pp. 96--101 (in Russ.).

[20] Tsvetkov S.V. Elastic and plastic anisotropy. Inzhenernyy zhurnal: nauka i innovatsii [Engineering Journal: Science and Innovation], 2012, no. 8 (in Russ.). EDN: QZPOOZ