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.     

Modeling cylindrical waves in nonlinear elastic composites

A procedure of deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves in composite materials modeled by a mixture with two elastic constituents is outlined. Nonlinearity is introduced by metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential. It is the quadratic nonlinearity of all governing relations. For a configuration (state) dependent on the radial coordinate and independent of the angular and axial coordinates, quadratically nonlinear wave equations for stresses are derived and a relationship between the components of the stress tensor and partial strain gradient is established. Four combinations of physical and geometrical nonlinearities in systems of wave equations are examined. Nonlinear wave equations are explicitly written for three of the combinations __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 63–72, June 2007.     

各向同性率无关材料本构关系的不变性表示

在内变量理论的框架下,针对各向同性率无关材料,使用张量函数表示理论建立了塑性应变全量及增量本构关系的最一般的张量不变性表示. 它们均由3个完备不可约的基张量组合构成,这3个基张量分别是应力的零次幂、一次幂和二次幂. 因此得出,塑性应变、塑性应变增量与应力三者共主轴. 通过对基张量的正交化,给出了本构关系式在主应力空间中的几何解释. 进一步,全量(或增量)本构关系中3个组合因子被表达为应力、塑性应变(或塑性应变增量)的不变量的函数. 当塑性应变(或塑性应变增量)的3个不变量之间满足一定关系时,所给出的本构关系将退化为经典的形变理论(或塑性势理论).最后,还讨论它与奇异屈服面理论的关系,当满足一定条件时,两者是一致的.      

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     

General invariant representations of the constitutive equations for isotropic nonlinearly elastic materials

This paper develops general invariant representations of the constitutive equations for isotropic nonlinearly elastic materials. Different sets of mutually orthogonal unit tensor bases are constructed from the strain argument tensor by using the representation theorem and corresponding irreducible invariants are defined. Their relations and geometrical interpretations are established in three dimensional principal space. It is shown that the constitutive law linking the stress and strain tensors is revealed to be a simple relationship between two vectors in the principal space. Relative to two different sets of the basis tensors, the constitutive equations are transformed according to the transformation rule of vectors. When a potential function is assumed to exist, the vector associated with the stress tensor is expressed in terms of its gradient with respect to the vector associated with the strain tensor. The Hill’s stability condition is shown to be that the scalar product of the increment of those two vectors must be positive. When potential function exists, it becomes to be that the 3 × 3 constitutive matrix derived from its second order derivative with respect to the vector associated with the strain must be positive definite. By decomposing the second order symmetric tensor space into the direct sum of a coaxial tensor subspace and another one orthogonal to it, the closed form representations for the fourth order tangent operator and its inversion are derived in an extremely simple way.     

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading conditions.     

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.     

正装结构非线性应力应变累加规律研究

基于有限位移理论的正装结构非线性分析理论对解决超大跨度、柔性结构的非线性计算有重大意义.文中探讨了正装结构非线性的分析特点,研究了其应变场与应力场的Kirchhoff应力张量与Lagrange应变张量的适用性,提出了正装结构非线性分析中应力场与应交场的累加规律,导出了拖动坐标法的虚功增量方程,以此对杆系结构非线性分析常用的CR法和UL列式进行了精度比较分析.文中研究成果可为正装结构的非线性分析提供理论指导.     

Stress solutions to the three-dimensional problem of elasticity

New representations of the stress tensor in the linear theory of elasticity and thermoelasticity are proposed. These representations satisfy the equilibrium equations and the strain compatibility equation. The stress tensor is expressed in terms of a harmonic tensor or a harmonic vector. The second boundary-value problem for an elastic half-space and an elastic layer is solved as an example __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 3–35, August 2006.     

The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

A Recasting of Anisotropic Poroelasticity in Matrices of Tensor Components

The equations associated with the theory of anisotropic poroelastic materials undergoing small deformations are recast in a matrix notation where the matrices are composed of proper tensor components. Using this notation the compatibility conditions on the components of the strain tensor are expressed in terms of the stress tensor, the pore pressure and the anisotropic elastic coefficients of the medium.     

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.     

The angle between the stress deviator and the strain-rate deviator in a tensor nonlinear isotropic medium

An expression for the angle between the symmetric stress deviator and the strain-rate deviator in a tensor nonlinear isotropic continuous medium is derived. A dependence of this angle on a certain orientation parameter in the three-dimensional space of principal strain rates is analyzed.     

基于微面有效应力矢量的各向异性屈服准则

基于微面模型，定义损伤变量为微面上有效承载面积的减少. 将Kachanov的一维有效 应力概念推广到三维，提出微面有效应力矢量的概念. 根据微面的有效应力矢量，将无损材 料的宏观应力张量及不变量与微面应力矢量的积分关系拓展到有损材料，得到了有损材料的 宏观有效应力张量及其不变量与宏观名义应力张量、微面面积损伤组构张量之间的关系. 将 无损材料的以应力张量不变量表示的Drucker-Prager准则推广到有损材料，建立了含缺陷 材料的各向异性屈服准则. 对有损材料，宏观有效应力张量与Murakami的有效应力张量具 有相同的形式，各向异性强度准则与Liu等提出的扩展Hill准则有相同的形式，当不考虑 静水应力对屈服的影响时，它与Hill准则具有相同的形式.     

Linear relationship between the first invariants of the stress and strain tensors in theories of plasticity with strain hardening

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 __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 60–72, March 2007.     

A novel internal dissipation inequality by isotropy and its implication for inelastic constitutive characterization

In the present work a novel inelastic deformation caused internal dissipation inequality by isotropy is revealed. This inequality has the most concise form among a variety of internal dissipation inequalities, including the one widely used in constitutive characterization of isotropic finite strain elastoplasticity and viscoelasticiy. Further, the evolution term describing the difference between the rate of deformation tensor and the “principal rate” of the elastic logarithmic strain tensor is set, according to the standard practice by isotropy, to equal a rank-two isotropic tensor function of the corresponding branch stress, with the tensor function having an eigenspace identical to the eigenspace of the branch stress tensor. Through that a general form of evolution equation for the elastic logarithmic strain is formulated and some interesting and important results are derived. Namely, by isotropy the evolution of the elastic logarithmic strain tensor is embodied separately by the evolutions of its eigenvalues and eigenprojections, with the evolution of the eigenprojections driven by the rate of deformation tensor and the evolution of the eigenvalues connected to specific material behavior. It can be proved that by isotropy the evolution term in the present dissipation inequality stands for the essential form of the evolution term in the extensively applied dissipation inequality.     

Application of Sharafutdinov’s Ray Transform in Integrated Photoelasticity

We explain the main concepts centered around Sharafutdinovs ray transform, its kernel, and the extent to which it can be inverted. It is shown how the ray transform emerges naturally in any attempt to reconstruct optical and stress tensors within a photoelastic medium from measurements on the state of polarization of light beams passing through the strained medium. The problem of reconstruction of stress tensors is crucially related to the fact that the ray transform has a nontrivial kernel; the latter is described by a theorem for which we provide a new proof which is simpler and shorter as in Sharafutdinovs original work, as we limit our scope to tensors which are relevant to Photoelasticity. We explain how the kernel of the ray transform is related to the decomposition of tensor fields into longitudinal and transverse components. The merits of the ray transform as a tool for tensor reconstruction are studied by walking through an explicit example of reconstructing the ₃₃-component of the stress tensor in a cylindrical photoelastic specimen. In order to make the paper self-contained we provide a derivation of the basic equations of Integrated Photoelasticity which describe how the presence of stress within a photoelastic medium influences the passage of polarized light through the material. Mathematics Subject Classifications (2000) 53C65, 53C80, 44-02, 44A12.     

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.     

Dyadic method for tensor functions

In this paper, we discuss tensor functions by dyadic representation of tensor. Two different cases of scalar invariants and two different cases of tensor invariants are calculated. It is concluded that there are six independent scale invariants for a symmetrical tensor and an antisymmetrical tensor, and there are twelve invariants for two symmetrical tensors and an antisymmetrical tensor. And we present a new list of tensor invariants for the tensor-valued isotropic function. The project supported by the Special Funds for Major State Basic Research Project “Nonlinear Science” and the National Basic Research Project “The Several Key Problems of Fluid and Aerodynamics”