Approximate description of the inelastic deformation of an isotropic material with allowance for the stress mode

Abstract

The elastoplastic deformation of an isotropic material is described using constitutive equations and allowing for the stress mode. The equations include two nonlinear functions that relate the first and second invariants of the stress and linear-strain tensors to the stress mode angle. It is proposed to use a linear rather than nonlinear relationship between the first invariants of the tensors. This simplification is validated by comparing calculated and experimental strains under loading with constant and variable stress mode angle Keywords: isotropic material, elastoplastic deformation, constitutive equations, stress mode angle Introduction. Constitutive equations describing the elastoplastic deformation of isotropic materials along small-curvature paths and allowing for the stress mode were proposed in [10,11]. These equations relate the stress components and the linear strain components and can be used both at small and large strains. It was assumed that stress deviators and differentials of plastic strains are coaxial. The equations contain two nonlinear functions that depend on a stress mode parameter. For this parameter, the stress mode angle was used in [3]. One function relates the mean stress and strain and the stress mode angle, while the other function relates the intensities of tangential stresses and shear strains and the stress mode angle. These functions are determined from tests on tubular specimens under tension and internal pressure increased proportionally, i.e., at several constant stress mode angles. When these functions are assumed independent of the stress mode angle and determined from uniaxial-tension tests, the above equations transform into the standard equations of the theory of deformation along small-curvature paths [2, 4, 5, 13], which are widely used to solve boundary-value problems [6-9, etc.].The assumptions underlying the constitutive equations were validated in [10][11][12] against the data of tests on tubular specimens subject to tension and internal pressure. The specimens were made of Kh18N10T steel and preliminarily annealed.An approximate method to calculate the above functions from test data for stress mode angles w p p s = 0 6 3 , / , / (base functions) was proposed in [1]. For intermediate values of w s , linear interpolation was used. The base functions and the algorithm developed in [1] were used to analyze several deformation processes for tubular specimens for different stress mode angles. The calculated strains were in satisfactory agreement with experimental data.In support of [10][11][12] and in contrast to [1], the present paper uses a more simple approximate approach to describe the inelastic deformation of isotropic materials with allowance for the stress mode. This approach assumes that the relationship between the mean stress and the mean strain is independent of the stress mode, while the relationship between the intensities of tangential stresses and shear strains is dependent on the stress mode. The approach will be tested against specific examples.1. Co...

Method of successive approximations for solving boundary-value problems of plasticity with allowance for the stress mode

“…The difference increases with the strains. The values obtained by the equations [1] differ from the exact values at the 15th step (end of active loading) by 25% for the meridional strains and by 14% for the circumferential strains. This difference is due to the replacement of the nonlinear function (1.16) by a linear one, i.e., only function (1.13) depends on the stress mode.…”

“…If we use the simplified constitutive equations [1], then F K Table. 2. It can be seen that these values are less than the exact ones.…”

“…The same problem has been solved using the simplified constitutive equations [1] that include one nonlinear function dependent on the stress mode. The maximum values of the meridional and circumferential strains are less than their values obtained by the constitutive equations with two nonlinear functions by 25% and 14%, respectively.…”

Describing the elastoplastic deformation of layered shells under axisymmetric loading with allowance for the third invariant of the stress deviator

“…In [2], a method of solving the axisymmetric problem of the theory of thin shells is outlined and numerical results obtained with the same constitutive equations are presented. These equations relating the components of the stress tensors and the linear strain components (to be called strains, for short) were experimentally validated in [8,9,[12][13][14]. It was assumed that the strains can be represented as the sum of elastic and plastic components and that the stress deviators and the deviators of plastic strain differential are coaxial.…”

“…describing deformation along paths of small curvature [5], which in the case of active loading coincide with the equations of incremental plasticity [3, 10, etc.] associated with the von Mises yield criterion (Prandtl-Reuss equations).Simplified constitutive equations [12][13][14] were considered in [9], where the former nonlinear function was replaced by a linear relation between the mean stress and mean strain and the possibility of such a replacement was justified. The simplified constitutive equations [9] were used in [11] to solve a spatial problem of plasticity for a body of revolution based on a method of successive approximations.…”