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

Babeshko, M. E.; Shevchenko, Yu. N.; Tormakhov, N. N.

doi:10.1007/s10778-010-0258-8

Cited by 9 publications

(31 citation statements)

References 10 publications

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“…Unlike [1], we will approximately calculate the radial strain from an expression derived from (1. In using formulas (1.6) and (2.1), it is assumed that the mean plastic strain is zero (e 0 0 (p) = ).…”

Section: Calculation Of Base Functionsmentioning

confidence: 99%

“…In using formulas (1.6) and (2.1), it is assumed that the mean plastic strain is zero (e 0 0 (p) = ). As shown in [1,11], e 0 2 10 2 (p) × = after the tension of a tubular specimen is completed [11], i.e., e 0 0 (p) = in this process. Let us evaluate the disagreement between the values of strain e e jj rr = predicted by formula (2.1) and measured in tests.…”

Section: Calculation Of Base Functionsmentioning

confidence: 99%

“…Figure 1 shows the strain e e jj rr = versus the mean stress s 0 . Here the solid line corresponds to formula (2.1), the dashed line to the approach [1], and the triangles to experiment. As is seen, values e e jj rr = calculated by the approach [1] Dashed curve 2 in Fig.…”

Section: Calculation Of Base Functionsmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…This angle is related to the Lode parameter m s in a simple manner [3]: To individualize the function F 2 (1.11), we will use, as in [1], data of base tests on tubular specimens proportionally loaded by a tensile axial force and internal pressure at several constant values of w s as well as the assumption…”

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

2010

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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...

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Section: Calculation Of Base Functionsmentioning

confidence: 99%

Section: Calculation Of Base Functionsmentioning

confidence: 99%

Section: Calculation Of Base Functionsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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

2010

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Constitutive equations of thermoplasticity including the third invariant

Shevchenko

Tormakhov

2010

Int Appl Mech

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Constitutive equations describing the nonisothermal deformation of elements of a body along paths of small curvature are formulated taking into account the stress mode. The equations include two scalar functions, one relating the first invariants of the tensors and the other relating the second invariants of the stress and strain deviators. Both scalar functions are nonlinear, dependent on temperature and stress mode, and determined in tests on tubular specimens. The plastic incompressibility condition is validated in uniaxial-tension tests on tubular and solid specimens. The proposed equations are used to design a loading process that differs from the base ones and proceeds at high temperature. The calculated results are compared with experimental data Introduction. The constitutive equations describing the deformation of elements of a body along paths of small curvature at room temperature and taking into account the stress mode were formulated in [15,16]. These equations are based on the assumption that the total strain can be additively decomposed into elastic and plastic components and that the deviators of stresses and plastic-strain increments are coaxial. The constitutive equations contain two scalar functions, one relating the first invariants of the tensors and the other relating the second invariants of the stress and strain deviators. The scalar functions are determined in tests on tubular specimens subject to axial force and internal pressure. Both scalar functions are nonlinear and dependent on the stress mode. The assumptions that underlay these equations are validated experimentally. A possible algorithm of using the equations to solve boundary-value problems is outlined. Approximate constitutive equations in which the first scalar function is independent of the stress mode and the first invariants of the stress and strain tensors are in linear relationship are presented in [13].In the tests on tubular specimens, the radial strains are not measured. In [13,15,16], two different approximate techniques are used to calculate these strains, and these experimental data are used to design processes of deformation along small-curvature paths.In the present paper, we will determine the radial strains more accurately by using the plastic incompressibility condition for finite strains. It will be assumed that an element of the body is at high temperature, and the scalar functions appearing in the constitutive equations will be determined at different high temperatures. The components of the strain tensor will be calculated from the components of the stress tensor and compared with the experimental data.1. Constitutive Equations. Let us derive the constitutive equations that relate the stress and strain components in an initially isotropic material nonisothermally loaded along paths of small curvature and take into account the stress mode. The material is considered initially isotropic if its properties will be similar in different directions, the stress mode and the first invariant of the stress tensor bein...

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

Babeshko

Shevchenko

2010

Int Appl Mech

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A method for solving boundary-value problems of plasticity with allowance for the stress mode is developed. To describe the elastoplastic deformation of an isotropic material, use is made of constitutive equations that include two nonlinear functions dependent on the stress mode and determined experimentally. The elastoplastic state of a thin cylindrical shell under axisymmetric loading is calculated as a numerical example. The numerical results demonstrate good convergence of the method. The effect of the stress mode on the strain distribution in a cylindrical shell is assessed Keywords: boundary-value problems of plasticity, stress mode, third stress invariant, thin cylindrical shell Introduction. Constitutive equations describing the elastoplastic deformation of isotropic materials along paths of small curvature were proposed and experimentally validated in [7,[10][11][12]. These equations relate the components of the stress tensors and linear strains and assume that the strains have elastic and plastic components and the stress deviators and plastic strain differentials are coaxial. The equations include two nonlinear functions determined experimentally, one relating the mean stress and mean strain to the stress mode parameter, while the other relating the shear-stress and shear-strain intensities to the same parameter. This parameter is taken to be the stress mode angle [2]. If we replace the former nonlinear function by a linear relation between the first invariants of the stress and strain tensors and assume that the latter function is independent of the stress mode and determined from simple-tension tests, then the constitutive equations go over into those describing deformation along paths of small curvature [4], which in the case of active loading coincide with the equations of incremental plasticity [2, 8, etc.] associated with the von Mises yield criterion.To linearize the constitutive equations [10-12], the method of additional stresses was used in [10], where the general process of successive approximations for solving boundary-value static problems was outlined. Simplified constitutive equations [10-12] were considered in [1], 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 were used in [9] to solve a spatial problem of plasticity for a body of revolution based on the method of successive approximations. Unlike [9] and further to [10], the present paper details the algorithm of successive approximations based on the simplified constitutive equations that include the two nonlinear functions mentioned above and allow for unloading. The algorithm will be tested by solving a problem for a thin shell under axisymmetric loading. The method will be analyzed for practical convergence. Problem Formulation and Governing Equations.Consider an isotropic body (its mechanical properties are similar in all directions, the stress mode parameter being constan...

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

Cited by 9 publications

References 10 publications

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

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

Constitutive equations of thermoplasticity including the third invariant

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

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