Energy-coupled stress and strain measures are defined in Euler coordinates. They are used to analyze the relationship between the first invariants of the stress and strain tensors for linearity and to determine strains at which the plastic component of the first strain invariant can be neglected. It is established that this relationship remains linear within an engineering plastic-strain tolerance of 0.2% irrespective of the value of strain intensity, which depends on the type of material and its stress state Keywords: solid body, Eulerian and Lagrangian coordinates, stress and strain tensors, first invariants, linear relationshipIntroduction. The modern theories of plasticity with strain hardening [11][12][13][14][15], which refer to Bridgman's study [2], postulated the generalized Hooke's law in both elastic and plastic strain ranges; i.e., it is supposed that the volume of a solid does not change during elastoplastic deformation. In this case, the first invariant of the stress tensor is used as hydrostatic pressure and the first invariant of the strain tensor as volume strain; i.e., all modern theories of plasticity assume that the first invariants of the stress and strain tensors are in a linear relationship. However, the first strain invariant can define volume strain only approximately.In this connection, we will discuss the values of the strain components at which the plastic component of the first strain invariant can be neglected, which, in fact, is done in each modern theory of plasticity. We will use the principles of solid mechanics based on Cauchy's continuum hypothesis. Unlike Lagrange's approach, this hypothesis suggests using the method of sections to determine the stresses at an arbitrary point of a body on an area element somehow oriented in space and going through this point. According to this method, a solid subjected to specified external loads is partitioned by this plane, and one of the parts is rejected. The equilibrium condition for the remaining part is used to determine the principal vector and the principal moment exerted by the rejected part onto the remaining part in the specified section going through the specified point. Dividing the main vector and principal moment by the area of the section and letting it tend to zero at the point, we obtain that the limit of the ratio of the principal moment to this area tends to zero and the limit of the ratio of the principal vector to this area tends to the value of the stress vector acting on this plane at the specified point [3]. Projecting the vector onto the axes of an orthogonal coordinate system fixed at one point to the solid before deformation and having constant directions during deformation (Eulerian coordinate system), we obtain the stress components in a plane passing through the point of interest.1. In a deformed solid, we select an elementary rectangular parallelepiped with sizes dx i in the Eulerian coordinate system x i , apply the stresses obtained above to each of its sides, and set up differential equilibrium equations in t...

Equations relating the components of the stress and strain tensors (constitutive equations) are formulated in terms of Euler coordinates. The equations describe the finite elastoplastic deformation of an isotropic body along paths of small curvature. It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator. The relationships between the first and second invariants of the stress and strain tensors in the case of complex elastoplastic deformation of the body's elements are determined from base tests on tubular specimens loaded along rectilinear paths for several values of the stress mode angle. Methods for specification of these relationships are proposed. The assumptions adopted to derive the constitutive equations are validated experimentally Keywords: finite strains, isotropic body, stress, constitutive equations, small-curvature pathsIn formulating the constitutive equations (relating the components of the stress and strain tensors) to describe the elastoplastic deformation of elements of a solid body, it is usually assumed (see [9-13, etc.]) that the relationship between the invariants of these tensors is independent of the stress mode and is determined from uniaxial tension/torsion tests on specimens made of an isotropic material. This assumption is in good agreement with experimental data obtained at small strains (6-10%)[6] and in poor agreement with experimental data obtained at strains of more than 10% [4,7]. In this connection, here we derive constitutive equations describing the elastoplastic deformation of elements of an isotropic solid along small-curvature paths on the assumption that the relationship between the invariants of the stress and strain tensors depends on the stress mode.1. Constitutive Equations. To derive constitutive equations, we will use the stress components σ ij on the faces of an elementary rectangular parallelepiped cut out in a deformed solid and described in a spatial system of coordinates x i (Euler coordinates) referred to the dimensions of this parallelepiped prior to deformation. These components constitute a second-rank tensor (as the coordinate system turns, they are transformed by quadratic formulas for the direction cosines of one coordinate system in the other) [3] and satisfy the following linear differential equations of equilibrium for the parallelepiped in the Cartesian coordinate system:(1.1)Here and later on, K i are the projections of the body force onto the axes of the spatial coordinate system; the summation is over the repeating indices in monomial expressions within the limits specified in parentheses and is not over the indices in angular brackets.The strain components ε ij in the spatial coordinate system x i are defined by the following formulas [2, 7, 8]:

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: