Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

Abstract

Constitutive equations relating the components of the stress tensor in a Eulerian coordinate system and the linear components of the finite-strain tensor are derived. These stress and strain measures are energy-consistent. It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator and that the first invariants of the stress and strain tensors are in a nonlinear relationship. In the case of combined elastoplastic deformation of elements of the body, this relationship, as well as the relationship between the second invariants of the stress and strain deviators, is determined from fundamental tests on a tubular specimen subjected to proportional loading at several values of stress mode angle (the third invariant of the stress deviator). Methods to individualize these relationships are proposed. The initial assumptions are experimentally validated. The constitutive equations derived underlie an algorithm for solving boundary-value problems Keywords: isotropic body, stress tensor, linear components of finite-strain tensor, Eulerian coordinates, elastoplastic deformation, path of small curvature, stress mode angle, constitutive equations Introduction. In deriving the constitutive equations describing the elastoplastic deformation of elements of an isotropic solid along paths of small curvature, we [9] accounted for the stress mode and made and experimentally validated certain assumptions. In these equations, the stress components are defined in a Eulerian coordinate system fixed at one point to the body and remaining unchanged during its deformation. The strain components defined in the same coordinate system are considered finite, i.e., the strain tensor has linear and nonlinear components. As shown in [10], the sum of products of these stress components and the variations of the finite-strain components is not the work done by forces during displacement variation-this work is equal to the sum of products of these stress components and the variations of not the total finite-strain components, but their linear parts only. We will examine the effect of the energy inconsistency of the stress and strain measures [2,9] on the reliability of the assumptions made to derive the constitutive equations [9]. To this end, we will validate the assumptions on the basis of fundamental tests for the energy-consistent stress and strain measures, taking the stress mode of the material into account.1. Constitutive Equations. To establish the relationship between the stress components and the linear finite-strain components in a Eulerian coordinate system, we will represent these tensors s ij and e ij as sums of deviatoric (s ij and e ij ) and spherical (a s 0 and e 0 ) components:

Constitutive equations of elastoplastic isotropic materials that allow for the stress mode

“…These functions should be determined from tests on tubular specimens proportionally (at different constant stress mode angles) loaded by tension and internal pressure. If the functions are assumed independent of the stress mode angle and determined in tension tests, the equations transform into the equations of deformation along small-curvature paths [6].To validate the above assumptions, test data for tubular specimens subject to tension and internal pressure were used in [13][14][15]. The specimens were made of Kh18N10T steel and preliminarily annealed.…”

“…The radial strain needed to calculate the functions and not measured in these tests was determined approximately. In [13][14][15], however, the strains were not determined based on the proposed constitutive equations and these functions and were not compared with experimental data.In contrast to [13][14][15], the present paper proposes a different approximate approach to determine the radial strain and finds new values of base functions. We use these functions to analyze deformation processes from given loading processes at different stress mode angles.…”

“…To validate the above assumptions, test data for tubular specimens subject to tension and internal pressure were used in [13][14][15]. The specimens were made of Kh18N10T steel and preliminarily annealed.…”

“…The radial strain needed to calculate the functions and not measured in these tests was determined approximately. In [13][14][15], however, the strains were not determined based on the proposed constitutive equations and these functions and were not compared with experimental data.…”

“…In contrast to [13][14][15], the present paper proposes a different approximate approach to determine the radial strain and finds new values of base functions. We use these functions to analyze deformation processes from given loading processes at different stress mode angles.…”