Constitutive equations of thermoplasticity including the third invariant

Abstract

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...

Experimental validation of the constitutive equations of thermoelastoplasticity including the third invariant of the stress deviator

“…The stresses in elements of the body are calculated from given strains by the method of successive approximations Introduction. The constitutive equations describing the thermoelastoplastic deformation of elements of a body along paths of small curvature and taking into account the third invariant of the stress deviator were formulated in [6]. These equations include two scalar functions, one relating the spherical components of the stress and strain tensors and the other relating the second invariants of the stress and strain deviators.…”

“…These equations were partially validated in [6] against a non-base process of loading along a nearly linear path at high temperature (500°C). The strains were calculated from the stresses measured in the test [6].The present paper deals with the experimental validation of the constitutive equations [6] describing thermoelastoplastic deformation along nonstraight paths of small curvature and taking into account the stress mode. As in [6-8], we will use the stress mode angle w s that characterizes the orientation of the octahedral shear stress relative to the negative direction of the projection of the principal axis of the minimum normal stress onto the octahedral plane.…”

“…These equations were partially validated in [6] against a non-base process of loading along a nearly linear path at high temperature (500°C). The strains were calculated from the stresses measured in the test [6].…”

“…The constitutive equations were experimentally validated against tubular Kh18N10T-steel specimens by the procedure outlined in [6]. The test setup consisted of a TsDMU-30t testing machine, electric heater, and loading device [5].…”

“…Then, this level of axial force being kept constant, the specimen was subjected to internal pressure. The data of this test were processed by the procedure of [6] and are collected in Table 1, where s zz and s jj are the axial and hoop stresses; e zz , e jj , and e rr are the axial, hoop, and radial strains; w s is the stress mode angle, and T is the temperature of the specimen. …”