On stress-state dependent plasticity modeling: Significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule

It has been shown that the plastic response of many materials, including some metallic alloys, depends on the stress state. In this paper, we describe a plasticity model for isotropic materials, which is a function of the hydrostatic stress as well as the second and third invariants of the stress deviator, and present its finite element implementation, including integration of the constitutive equations using the backward Euler method and formulation of the consistent tangent moduli. Special attention is paid for the adoption of the non-associated flow rule. As an application, this model is calibrated and verified for a 5083 aluminum alloy. Furthermore, the Gurson-Tvergaard-Needleman porous plasticity model, which is widely used to simulate the void growth process of ductile fracture, is extended to include the effects of hydrostatic stress and the third invariant of stress deviator on the matrix material.     

J^＊ INTEGRAL OF THE SPECIFIC DEVIATOR STRAIN ENERGY AND ITS APPLICATION

First the deviator strain energy is introduced, then the problem of plane-crack critical growth was discussed, a path independent line integral J^* was defined, furthermore its conservation was proved strictly. As application examples, Mode-Ⅰ stress intensity factors of cracked beams were obtained with present approach. The results are shown to agree well with those available in the open literature.     

Tresca-type plastic materials in the theory of hypoelasticity II. Optical constitutive equations and birefringence in simple shear

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.     

Describing the thermoelastoplastic deformation of compound shells under axisymmetric loading with allowance for the third invariant of the stress deviator

The paper outlines a procedure for the numerical analysis of the thermoelastoplastic stress–strain state of thin compound shells of revolution under axisymmetric nonisothermal loading. The constitutive equations describing the thermoelastoplastic deformation of isotropic materials along paths of small curvature and incorporating the third invariant of the stress deviator are used. A numerical example is presented     

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

The equations describing thermoelastoplastic deformation along nonstraight paths and taking into account the third invariant of the stress deviator are experimentally validated. The equations contain two scalar functions that are specified in base tests on tubular specimens. The test data are tabulated. The values of the scalar functions for strains, temperature, and stress mode are found by using nonlinear interpolation of the data and the temperature similarity of the functions. The stresses in elements of the body are calculated from given strains by the method of successive approximations     

Determining the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator

A technique to determine the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator is developed. The technique is based on the theory of thin shells that takes into account transverse shear and torsional strains. Plastic equations that relate the components of the stress tensor in Eulerian coordinates with the linear components of the finite-strain tensor are used as constitutive equations. The nonlinear scalar functions in the constitutive equations are found from base tests on tubular specimens under proportional loading for different stress modes. The boundary-value problem is solved by numerically integrating a system of ordinary differential equations     

Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

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 __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 43–55, June 2007.     

Constitutive equations for describing the elastoplastic deformation of elements of a body along small-curvature paths in view of the stress mode

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 __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 62–72, April 2006.     

Characteristic cones of the equations in the nonlinear theory of elasticity

The roots of the equation for the characteristic normals for two systems of differential equations in the nonlinear theory of elasticity are investigated. The first model is constructed using a thermodynamic identity. The second is a very simple hypoelastic model (the deviator of the stress-rate tensor is proportional to the deviator of the strain-rate tensor). It is shown that the roots of the equations for the normals to the characteristics for the second model are the same as the first-order terms in the expansion of the roots of the first model with respect to the strain-tensor deviator.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–132, May–June, 1974.The author is grateful to S. K. Godunov for discussions.     

Effects of the stress state on plasticity and ductile failure of an aluminum 5083 alloy

The experimental and numerical work presented in this paper reveals that stress state has strong effects on both the plastic response and the ductile fracture behavior of an aluminum 5083 alloy. As a result, the hydrostatic stress and the third invariant of the stress deviator (which is related to the Lode angle) need to be incorporated in the material modeling. These findings challenge the classical J₂ plasticity theory and provide a blueprint for the establishment of the stress state dependent plasticity and ductile fracture models for aluminum structural reliability assessments. Further investigations are planned to advance, calibrate and validate the new plasticity and ductile fracture models.     

体外预应力梁摩擦效应的非线性分析

构造了简单的体外预应力梁的摩擦单元,摩擦单元位于转向块和体外筋之间的角平分线上,能模拟转向块和体外筋之间的有摩擦或无摩擦滑移。考虑混凝土、钢筋和体外筋应力-应变的非线性关系,采用梁截面弯矩-轴力-曲率的三折线模型,探讨了体外预应力梁的性能。对简支梁和连续梁的不同因素进行计算,包括不同摩擦系数、不同体外筋和钢筋面积、不同偏心距以及对称和非对称荷载形式。计算结果表明,对于简支梁和对称荷载下的连续梁,承载力的摩擦效应可以忽略,最大预应力增量和挠度的摩擦效应不宜忽略,最小预应力增量的摩擦效应明显;对于非对称荷载下的连续梁,承载力、最大和最小预应力增量以及挠度的摩擦效应不可忽略。     

Formulation of stress-strain relation for plastic deformation of mild steel for strain trajectory consisting of two straight branches

A relation between the stress and incremental strain deviators, which are not coaxial, is derived by considering the characteristics in tensor space. On the bases of this relation and of precise experimental results for mild steel, a specific plastic stress-strain relation is formulated for a strain trajectory with a comer in a 3D-vector space corresponding to the strain deviator, as a typical example of plastic behaviour under complex loading. It may be confirmed that the effects of plastic anisotropy and of the third invariant of the strain deviator on the plastic behaviour of mild steel under complex loading are expressed precisely by means of the above formulation.     

New analytical criterion for porous solids with Tresca matrix under axisymmetric loadings

In this paper, a new analytic criterion for porous solids with matrix obeying Tresca yield criterion is derived. The criterion is micromechanically motivated and relies on rigorous upscaling theorems. Analysis is conducted for both tensile and compressive axisymmetric loading scenarios and spherical void geometry. Finite element cell calculations are also performed for various triaxialities. Both the new model and the numerical calculations reveal a very specific coupling between the mean stress and the third invariant of the stress deviator that results in the yield surface being centro-symmetric and void growth being dependent on the third-invariant of the stress deviator. Furthermore, it is verified that the classical Gurson’s criterion is an upper bound of the new criterion with Tresca matrix.     

A unified thermodynamic theory of elasticity and plasticity

A thermodynamics is developed for a unified theory of elasticity and plasticity in infinitesmal strain. The constitutive equations which relate stress and strain deviators are rate type differential equations. When they satisfy a Lipschitz condition, uniqueness for the initial value problem dictates that the stress and strain will be related through elastic relations. Failure of the Lipschitz condition occurs when a von Mises yield condition is achieved: Plastic yield then occurs and the deviator relations turn into the Prandtl-Reuss equations. The plastic yield solution is stable during loading and unstable during unloading. The requirement that the solution followed during unloading be stable dictates entry into an elastic regime. Appropriate thermodynamic functions are constructed. It then appears that stress deviator (not strain deviator) is a viable state variable, and the thermodynamic relations are constructed in terms of a Gibbs function. The energy balance leads to satisfaction of the Clausius-Duhem inequality (and thus the second law of thermodynamics) in an elastic regime because it is shown that in an elastic regime entropy production is caused only by heat flux. During yield, the proper method of differentiating yields entropy production terms in addition to those arising from heat flux. These terms are positive during loading, whence it is concluded that the requirement that a stable solution be followed leads to satisfaction of the Clausius-Duhem inequality during plastic as well as elastic behavior.     

卸荷条件下粉质粘土变形性状研究

通过室内应力控制的三轴试验研究卸荷条件下土体的应力-应变关系和孔隙水压力变化特征,得到变形模量的确定方法。研究表明:在初始偏压应力固结条件下,无论是轴向压力保持不变而侧向压力不断卸荷,还是轴向压力和侧向压力同时卸荷,其应力-应变关系均表现为较好的双曲线函数关系。而且,侧向固结压力愈大,强度值愈大。在卸荷过程中,随剪切应力的增大和剪切应变的发展,土样呈现剪胀趋势并在后期产生较大的负孔隙水压力。不排水条件下由于负孔隙水压力的存在,土样的表观强度略有增大。     

A pressure-smoothing scheme for incompressible flow problems

A pressure-smoothing scheme for Stokes and Navier–Stokes flows of Newtonian fluids and for Stokes flow of Maxwell fluids is described. The stress deviator obtained from the calculated velocity field is substituted into the governing equilibrium equation. The resulting equation is then solved to obtain a new, smoothed pressure by a least square finite element method.     

On the stability of the flow of a viscoelastic fluid down an inclined plane

The paper presents an analysis of laminar flow of a film of viscoelastic fluid flowing under gravity down an infinite inclined plane. It is assumed that the mechanical behavior of the fluid can be represented by a generalized Maxwell model, whose constitutive equation contains a time derivative of the deviator of the stress tensor in the Jaumann sense [1. 2]. The equations of motion of the viscoelastic fluid considered here admit an exact solution for the case of rectilinear laminar flow with a plane free boundary. The stability of this flow with respect to surface waves is investigated by the method of successive approximations described in [3, 4].     

Creep of a heat-resistant aluminum alloy in a complex state of stress

The article gives the results of experimental investigations of the creep of heat-resistant aluminum alloy AK4-1 with constant and variable loads, at a temperature of 175 °C and a duration of the tests equal to 100 h. Based on the experimental data, a verification of the theory of creep is adduced, based on the following hypotheses: 1) the change in the volume is elastic; 2) the deviator of the creep rates is proportional to the deviator of the stresses; 3) the intensities of the stresses, of the creep deformations, and their rates are connected by a relationship which does not depend on the type of the state of stress. It should be pointed out that the results of investigation of creep, under a complex state of stress, in carbon, low-alloy, austenitic steels, copper, and certain light alloys, are given in [1–6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 83–86, March–April, 1971.     

Deformable spherical devices to measure stresses within field soils

The design, calibration and use of two deformable spherical stress transducers are described. They are suitable for detecting principal stresses in deforming media such as soil and have major advantages over many rigid transducers previously used in such situations. One of the transducers is a water-filled rubber ball (WFRB) sensitive to hydrostatic and deviator stresses, the other is a mastic ball which deforms plastically and is sensitive to only deviator stresses. When the two devices are used at similar depths under a surface load, e.g. a wheel, the combined measurements of internal pressure of the WFRB and axial deformation of the mastic ball can be used to derive values for first and third principal stresses (assuming second and third principal stresses are equal). Calibration of the transducers at different temperatures is described.

Field measurements made with the transducer under loaded wheels are compared with predicted values of first principal stresses using equations developed by Söhne. Close correspondence between predicted and measured values was observed, when the existing soil strength conditions were taken into account.

The transducers promise to be useful in the measurement of stresses in field soils.     

J* INTEGRAL OF THE SPECIFIC DEVIATOR STRAIN ENERGY AND ITS APPLICATION

IntroductionItiswell_knowthatJintegraloffracturemechanicswasproposedbyJ.R .RiceinRef.[1 ] .Jintegralisusedfordescribingtheaverageintensityofstressandstrainnearcracktipinelastic_plasticsituation .Underthelinearelasticcondition ,J=G ,whereGistheenergyreleaserate.However,Jintegralisnottheuniqueconservativelawofstaticelasticity[2 ],forinstance,Jk,L ,M ,allarepathindependentconservativeintegral.Y .J.Xie[3]suggestedthemouthwideningenergyrateintegralG ,whichisusedforcalculatingtheMode_Istress…