• К. М. Рудаков The National Technical University of Ukraine «Kyiv Polytechnic Institute», Kyiv, Ukraine
  • О. А. Добронравов The National Technical University of Ukraine «Kyiv Polytechnic Institute», Kyiv, Ukraine



large strains, multiplicate decomposition, thermoelastic, plastic, creep


The authors generalizes an idea of Lee's multiplicative decomposition for the case of simultaneous presence of four types of strains: thermal, elastic, plastic and creep. This decomposition uses group properties of operators of reflection from an abstract algebra.
Using multiplicative decomposition of matrix of Cauchy-Green strain gradient for three times, the matrix is found to be equal to the product of four matrices of gradients separately from each type of strain. This allowed writing Green's-Lagrange's tensors for the different types of strains, as well as exactly additive decomposition of the matrix of the spatial gradient of the strain rate for each type of strain. The matrix of the spatial gradient of strain rate is multiplied on the transpose matrix of the gradient of the elastic strain on the left side and on the normal matrix of the gradient of the elastic strain on the right side for use of the energetically integrated second stress tensor of Piola-Kirhgof. The resulting expressions will be used for an establishment of the equations of thermoelasto-plasticity and creep in the case of large strains by means of the second law of the thermodynamics that is written down in the form of Clausius-Duhem's inequality


Green A.E., Naghdi P.M. A general theory of an elastic-plastic continuum. Arch. Rat. Mech. Analysis, 1965. 18. pp. 251-281.

Vashizu K. Variacionnye metody v teorii uprugosti i plastichnosti [Variational methods in elasticity and plasticity] Moscow: Mir, 1987. 542 p.

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. Comput. Struct., 1987. 26. pp. 175-179.

Eterović A.L., Bathe K-J. A hyperelastic-based large strain elasto–plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. Int. J. Num. Meth. Enging, 1990. 30. pp. 1099-1114.

Weber G., Anand L. Finite deformation constitutive equations and a time integration procedure for isotropic hyperelastic-viscoplastic solids. Comput. Meth. App. Mech. Enging, 1990. 79. pp. 173–202.

Lee E.H. Elastic–plastic deformations at finite strains. J. Appl. Mech. (ASME), 1969. 36. pp. 1–6.

Bathe K-J. Finite Element Procedures. New-York: Prentice Hall, 1996. 1037 p.

Montáns F.J., Bathe K-J. Computational issues in large strain elasto-plasticity: an algorithm for mixed hardening and plastic spin. Int. J. Num. Meth. Enging, 2005. 63. pp. 159-196.

Stojanović R., Djurić S., Vujošević L. On finite thermal Deformations. Arch. Mech. Mech., 1964. 16. pp. 103-108.

Vujošević L., Lubarda V.A. Finite-strain thermoelasticity based on multiplicative decomposition of deformation gradient. Theor. Appl. Mech. Enging, 2002. 28-29. pp. 379-399.

Lubarda V.A. Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics. Appl. Mech. Rev., 2004. 57. no 2. pp. 95-108.

Weber G.G., Boyce M.C. A framework for finite strain thermoelasto-plastic deformation of Solids. D. Hui, T.J. Kosik, eds., Symp. on Viscoplastic Behavior of New Materials, ASME Winter Annual Meeting Proceedings, 1989. 1. pp. 17.

Cleja-Tigoiu S., Soo´s E. Elastoplastic models with relaxed configurations and internal state variables. Appl. Mech. Rev., 1990. 43. pp. 131- 151.

Lion A., Höfer P. On the phenomenological representation of curing phenomena in continuum mechanics. Arch. Mech., 2007. 59. pp. 59-89.