Engineering Transactions, 66, 4, pp. 443–459, 2018

Material Spin and Finite-Strain Hypo-Elasticity for Two-Dimensional Orthotropic Media

Ivan Yurievitch ZUBKO
Perm National Research Polytechnic University
Russian Federation

A constitutive material spin tensor in the case of purely elastic finite-strain deformation is introduced for a two-dimensional orthotropic media using the minimizing principle applied to obtain the reloaded configuration of the material volume. This material spin explains the rotation of the orthonormal vector frame which coincides with the material symmetry axes in the initial configuration of the material volume and uniquely corresponds to a set of these axes in the current configuration although it does not coincide with the latter. The given definition is followed by the exact expression which includes the deformation gradient tensor, unit vectors of the initial material anisotropy axes and their axial parameters. This definition allows obtaining a new variant of decomposing any elastic finite-strain motion onto rigid and deformational parts and introducing the material corotational rate. The latter is used for the formulation of the anisotropic rate-type elastic law in the current configuration based on the strain measure which does not belong to the Seth-Hill family. For isotropic as well as for tetragonal media, the introduced material rotation tensor coincides with the rotation tensor from the polar decomposition of a deformation gradient.
Keywords: material spin; material corotational rate; elastic anisotropy; finite strains; hypo-elasticity; material strain tensor
Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


Kratochvil J., Finite-strain theory of crystalline elastic-inelastic materials, Journal of Applied Physics, 42: 1104–1108, 1971.

Mandel J., Classic plasticity and viscoplasticity [in French: Plasticité classique et viscoplasticité], Courses and Lectures, No. 97, ICMS, Udine, Italy, Springer, Vienna–New York, 1971.

Dafalias, Y.F., Corotational rates for kinematic hardening at large plastic deformation, Journal of Applied Mechanics, 50: 561–565, 1983.

Dafalias Y.F., Plastic spin: necessity or redundancy?, International Journal of Plasticity, 14: 909–931, 1998.

Van der Giessen E., Micromechanical and thermodynamic aspects of the plastic spin, International Journal of Plasticity, 7: 365–386, 1991.

Xiao H., Bruhns O.T., Meyers A., Logarithmic strain, logarithmic spin and logarithmic rate, Acta Mechanica, 124: 89–105, 1997.

Xiao H., Bruhns O.T., Meyers A., Hypo-elasticity model based upon the logarithmic stress rate, Journal of Elasticity, 47: 51–68, 1997.

dell'Isola F., Della Corte A., Giorgio I., Scerrato D., Pantographic 2D sheets: discussion of some numerical investigations and potential applications, International Journal of Non-Linear Mechanics, 80: 200–208, 2016.

Zubko I.Yu., Computation of elastic moduli of graphene monolayer in nonsymmetric formulation using energy-based approach, Physical Mesomechanics, 19(1): 93–106, 2016.

Palmov V., Corotational variables in nonlinear mechanics of solids, Proceedings of SPIE 3687 – Society of Photo-Optical Instrumentation Engineers (SPIE), International Workshop on Nondestructive Testing and Computer Simulations in Science and Engineering, SPIE Russia, pp. 391–394, 1999,

Hill R., On constitutive inequalities for simple materials – I, Journal of the Mechanics and Physics of Solids, 16(4): 229–242, 1968.

Hill R., Aspects of invariance in solid mechanics, Advances in Applied Mechanics, 18: 1–75, 1979.

Xiao H., Bruhns O.T., Meyers A., Objective corotational rates and unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures, Archives of Mechanics, 50: 1015–1045, 1998.

Korobeynikov S.N., Objective tensor rates and applications in formulation of hyperelastic relations, Journal of Elasticity, 93: 105–140, 2008.

Zhilin P.A., Altenbach H., Ivanova E.A., Krivtsov A.M., Material strain tensor, [in:] Mechanics of Generalized Continua, Altenbach H., Forest S., Krivtsov A.M. (Eds.), Berlin, Springer, pp. 321–331, 2013.

Truesdell C., Hypo-elasticity, Journal of Rational Mechanics and Analysis, 4: 83–133, 1955.

Truesdell C., Remarks on hypo-elasticity, Journal of Research of the National Bureau of Standards, 67B(3), 141–143, 1963.

Simo J.C., Pister K.S., Remarks on rate constitutive equations for finite deformation problems: computational implications, Computer Methods in Applied Mechanics and Engineering, 46: 201–215, 1984.

Kojic M., Bathe K.J., Studies of finite element procedures – stress solution of a closed elastic strain path with stretching and shearing using the updated Lagrangian Jaumann formulation, Computers and Structures, 26: 175–179, 1987.

Lehmann Th., Anisotropic plastic deformations [in German: Anisotrope plastische Formänderungen], Romanian Journal of Technical Science – Applied Mechanics, 17: 1077–1086, 1972.

Nagtegaal J.C., De Jong J.E., Some aspects of non-isotropic work-hardening in finite strain plasticity, Proceedings of Workshop on Plasticity of Metals at Finite Strain: Theory, Experiment and Computation, Lee E.H., Mallett R.L. (Eds.), Stanford, CA: Stanford University Press, pp. 65–102, 1982.

Dienes J.K., On the analysis of rotation and stress rate in deforming bodies, Acta Mechanica, 32: 217–232, 1979.

Xiao H., Bruhns O.T., Meyers A., Objective stress rates, path-dependence properties and non-integrability problems, Acta Mechanica, 176: 135–151, 2005.

Rychlewski J., On Hooke’s law [in Russian], Prikl. Mat. Mekh., 48: 420–435, 1984.

Rychlewski J., Unconventional approach to linear elasticity, Archives of Mechanics, 47(2): 149–171, 1995.

Kowalczyk-Gajewska K., Ostrowska-Maciejewska J., Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material, Engineering Transactions, 57(3–4): 145–183, 2009.

Blinowski A., Ostrowska-Maciejewska J., On the elastic orthotropy, Archives of Mechanics, 48(1), 129–141, 1996.

Blinowski A., Ostrowska-Maciejewska J., Rychlewski J., Two-dimensional Hooke’s tensors- isotropic decomposition, effective symmetry criteria, Archives of Mechanics, 48(2): 325–345, 1996.

Doyle T.C., Ericksen J.L., Nonlinear elasticity, Advances in Applied Mechanics, 4: 53–115, 1956.

Lurie A.I., Non-linear theory of elasticity, North Holland, Amsterdam, 1990.

Seth B.R., Generalized strain measure with applications to physical problems, Math. Res. Centre Technical Summary Report #248, University of Wisconsin, Madison, USA, 1961.

DOI: 10.24423/EngTrans.799.20181203