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...