A study of elasticity and plasticity equations under arbitrary displacements and strains

Equations of geometrically nonlinear theory of elasticity with finite displacements and strains are analyzed. The equations are composed using three versions of physical relations and applied to solve the problem of tension-compression of a straight bar. It is shown that the use of the classical relations between the components of the stress tensor and the Cauchy-Green strain tensor in the problem of compression of the bar results in the appearance of “spurious” static loss of stability such that the bar axis remains straight if the stresses are referred to unit areas before the deformation (conditional stresses). However, in the problem of tension, the classical relations do not permit one to describe the phenomenon of static instability (neck formation as the plastic instability occurs). These drawbacks disappear if one uses the third version of the physical equations, composed as relations between the true stresses referred to unit areas of the deformed faces on which they act and the true elongations and shears. The relations of the third version are most correct; they permit one to pass to self-consistent equations of elasticity and plasticity under small strains and finite displacements, and they should be recommended for practical use. As an example, such relations are composed for the flow theory.     

Non parabolic band transport in semiconductors: closure of the moment equations

The problem of the closure of the moment equations of the semiconductor Boltzmann equation is studied in the framework of the Kane dispersion relation (therefore avoiding the limitations of the parabolic band approximation). By using the maximum entropy ansatz for the closure one obtains, in the limit of small anisotropy, explicit constitutive relations for the stress tensor and the flux of energy flux tensor. The results obtained are in remarkable agreement with those arising from Monte Carlo simulations. Received October 27, 98     

New Statement of the elasticity problem for a layer

The problem on the equilibrium of an inhomogeneous anisotropic elastic layer is considered. The classical statement of the problem in displacements consists of three partial differential equations with variable coefficients for the three displacements and of three boundary conditions posed at each point of the boundary surface. Sometimes, instead of the statement in displacements, it is convenient to use the classical statement of the problem in stresses [1] or the new statement of the problem in stresses proposed by B. E. Pobedrya [2]. In the case of the problem in stresses, it is necessary to find six components of the stress tensor, which are functions of three coordinates. The choice of the statement of the problem depends on the researcher and, of course, on the specific problem. The fact that there are several statements of the problem makes for a wider choice of the method for solving the problem. In the present paper, for a layer with plane boundary surfaces, we propose a new statement of the problem, which, in contrast to the other two statements indicated above, can be called a mixed statement. The problem for a layer in the new statement consists of a system of three partial differential equations for the three components of the displacement vector of the midplane points. The system is coupled with three integro-differential equations for the three longitudinal components of the stress tensor. Thus, in the new statement, just as in the other statements in stresses, one should find six functions. In the new statement, three of these functions (the displacements of the midplane points) are functions of two coordinates, and the other three functions (the longitudinal components of the stress tensor) are functions of three coordinates. It is shown that all equations in the new statement are the Euler equations for the Reissner functional with additional constraints. After the problem is solved in the new statement, three components of the displacement vector and three transverse components of the stress tensor are determined at each point of the layer. The new statement of the problem can be used to construct various engineering theories of plates made of composite materials.     

The invariant representation of spins with applications in the theory of finite deformations

Based on the general solution given to a kind of linear tensor equations, the spin of a symmetric tensor is derived in an invariant form. The result is applied to find the spins of the left and the right stretch tensors and the relation among different rotation rate tensors has been discussed. According to work conjugacy, the relations between Cauchy stress and the stresses conjugate to Hill's generalized strains are obtained. Particularly, the logarithmic strain, its time rate and the conjugate stress have been discussed in detail. These results are important in modeling the constitutive relations for finite deformations in continuum mechanics. The project is supported by the National Natural Science Foundation of China and the Chinese Academy of Sciences (No. 87-52).     

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.      

A generalized continuum theory for granular materials

A generalized continuum theory for granular media is formulated by allowing for the possibility of rotation of granules. The basic balance laws are presented and based on thermodynamical consideration a set of constitutive equations are derived. The theory naturally gives rise to the generation of antisymmetric stress tensor and existence of couple stresses. The basic equations of motion are derived and it is shown that the theory contains Mohr-Coulomb criterion of limiting equilibrium as a special case. The problem of coupled porosity and microrotational wave propagation is investigated and the rectilinear shear flow of granular materials is discussed.     

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     

Momentum equation for straight electrically charged jet

A one-dimensional momentum conservation equation for a straight jet driven by an electrical field is developed. It is presented in terms of a stress component, which can be applied to any constitutive relation of fluids. The only assumption is that the fluid is incompressible. The results indicate that both the axial and radial constitutive relations are required to close the governing equations of the straight charged jet. However, when the trace of the extra stress tensor is zero, only the axial constitutive relation is required. It is also found that the second normal stress difference for the charged jet is always zero. The comparison with other developed momentum equations is made.     

n the fourth order tensor valued function of the stress in return map algorithm

针对各向同性材料,基于一组相互正交的基张量,建立了一套有 效的相关运算方法. 基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则 使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴. 根据张量函数表示 定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组 基所表示. 推演结果表明：这些张量之间的运算,表现为对应系数矩阵之间的简单 关系. 其中,四阶张量求逆归结为对应的3\times3系数矩阵求逆,它对二阶张量的变换 则表现为该矩阵对3times 1列阵的变换. 最后,对这些变换关系应用于返回映 射算法的迭代格式进行了相关讨论.     

Three-dimensional constitutive relations of ideal plasticity and the flow on the coulomb-tresca prism edge

In the present paper, we consider basic relations of the mathematical theory of plasticity for the spatial state corresponding to the edge of the Coulomb-Tresca prism, which follow from the generalized associated flow law restricting the plastic flow freedom for the above states to the minimal possible extent. We found that the spatial relations of the theory of plasticity, formulated by A. Yu. Ishlinsky in 1946, can be derived from the above version of the theory of flow. We show that the A. Yu. Ishlinsky constitutive relations for states on the Coulomb-Tresca prism edge express the commutativity of the stress tensor and the tensor of plastic strain increments. We obtained one explicit form of the constitutive relation relating the stress tensor to the plastic strain increments for the stressed states corresponding to the Coulomb-Tresca prism edge.     

Basis free relations for the conjugate stresses of the strains based on the right stretch tensor

In this paper, general relations between two different stress tensors T^f and T^g, respectively conjugate to strain measure tensors f(U) and g(U) are found. The strain class f(U) is based on the right stretch tensor U which includes the Seth–Hill strain tensors. The method is based on the definition of energy conjugacy and Hill’s principal axis method. The relations are derived for the cases of distinct as well as coalescent principal stretches. As a special case, conjugate stresses of the Seth–Hill strain measures are then more investigated in their general form. The relations are first obtained in the principal axes of the tensor U. Then they are used to obtain basis free tensorial equations between different conjugate stresses. These basis free equations between two conjugate stresses are obtained through the comparison of the relations between their components in the principal axes, with a possible tensor expansion relation between the stresses with unknown coefficients, the unknown coefficients to be obtained. In this regard, some relations are also obtained for T⁽⁰⁾ which is the stress conjugate to the logarithmic strain tensor lnU.     

Dynamics of deformation of an elastic medium with initial stresses

The constitutive equations of motion of an elastic medium with given initial stresses are formulated in the form of a hyperbolic system of first order differential equations. Equations describing the propagation of small perturbations in a prestressed isotropic medium with an arbitrary dependence of the elastic strain energy on the strain tensor are derived, and equations for the quadratic dependence of elastic strain energy on the strain tensor are given.     

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.     

Anisotropic damage mechanics for viscoelastic ice

We present a formulation of continuum damage in glacier ice that incorporates the induced anisotropy of the damage effects but restricts these formally to orthotropy. Damage is modeled by a symmetric second rank tensor that structurally plays the role of an internal variable. It may be interpreted as a texture measure that quantifies the effective specific areas over which internal stresses can be transmitted. The evolution equation for the damage tensor is motivated in the reference configuration and pushed forward to the present configuration. A spatially objective constitutive form of the evolution equation for the damage tensor is obtained. The rheology of the damaged ice presumes no volume conservation. Its constitutive relations are derived from the free enthalpy and a dissipation potential, and extends the classical isotropic power law by elastic and damage tensor dependent terms. All constitutive relations are in conformity with the second law of thermodynamics.PACS 83.60.Df, 62.20.Mk     

A comparative study of kinematic hardening rules at finite deformations

Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong-Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong-Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).     

Solution of a Three-Dimensional Elastic Problem for a Layer

A closed system of differential equations for stresses and boundary and integral equilibrium and compatibility conditions for the components of the stress tensor are derived in solving a three-dimensional elastic problem for an unbounded layer. These equations are proposed to integrate directly.     

Perturbations of flows of incompressible nonlinearly viscous and viscoplastic fluids caused by variations in material functions

Material functions are necessary element of the constitutive relations determining any model of continuum. These functions can be defined as a collection of objects from which the operator of constitutive relations can be reconstructed completely. The material functions are found in test experiments and show the differences between a given medium and other media in the framework of the same model [1]. The “test experiment theory” is an important part of modern experimental mechanics.Just as in any experiment, from determining the viscosity coefficient by using the rotational viscosimeters to constructing the yield surface by using machines combined loading, the material functions are determined with an unavoidable error. For example, experimenters know that, in experiments with arbitrary accuracy, the moduli of elasticity can only be measured with an unimprovable tolerance of about 7%. Starting already from [2], the investigators’ attention has been repeatedly drawn to the fact that it is necessary to take into account this tolerance in determining the material constants, functions, and functionals in problems of mechanics and especially in analyzing the stability of deformation processes. Mathematically, this means that problems of stability under perturbations of the initial data, external constantly acting forces, domain boundaries, etc. should be supplemented with the assumption that the material functions have unknown perturbations of a certain class [3].The variations of material functions in the framework of the linearized stability theory were considered in [2, 4, 5]. In what follows, we study isotropic tensor functions in the most general case of scalar and tensor nonlinearity. These functions are assigned the meaning of constitutive relations between the stress and strain rate tensors in continuum. These constitutive relations contain scalar material functions of invariants on which, as follows from the above, some variations proportional to a small physical parameter α can be imposed. These variations imply perturbations of the tensor function itself. The components of such perturbations linear and quadratic in α are determined. In each of the approximations, we write out a closed system of equations consisting of the equations of motion (linear in the variables of the respective approximation) and the incompressibility condition.We analyze tensor-linear functions with arbitrary scalar rheology inmore detail. Materials with such constitutive relations include non-Newtonian viscous fluids and viscoplastic materials. Viscoplastic materials are characterized by the existence of rigidity zones, where the stress intensity is less than the yield strength. We derive equations for the boundaries of the rigidity zones in the perturbed motion, in particular, for the case in which the unperturbed medium is a viscous Newtonian fluid. Throughout the paper, index-free notation is used.     

The Rayleigh and Stokes problems with an incompressible non-Newtonian fluid

Summary The Rayleigh problem or impulsive motion of a flat plate has been solved using a perturbation scheme when the surrounding fluid is representable by the constitutive equations of Oldroyd or Coleman and Noll. The shear stress and normal stress at the wall were expressed analytically for this unsteady motion. Further, an exact solution of the equations was found for a special case of the constitutive equations.The motion of the fluid above a harmonically oscillating plate or the Stokes problem has been determined for a special non-Newtonian fluid. The penetration of the shear wave into the fluid, the energy dissipation, the velocity profiles and the shear and normal stresses at the wall were expressed and compared to an equivalent Newtonian fluid.Some of the features of these non-Newtonian fluids were examined in simple shearing flows, and techniques to calculate some of the material constants discussed.     

Implicit equations for thermoelastic bodies

In this paper we generalize the recent implicit models that have been put into place to describe the elastic response of bodies when thermal effects come into play. The implicit constitutive relations for thermoelastic response presented here provide a very natural way to overcome a serious problem associated with the celebrated model due to Fourier, namely infinite speed of the propagation of temperature. We also study some boundary value problems within the context of the implicit equations that we have developed. We carry out a linearization based on the classical assumption that the displacement gradient is small and obtain constitutive relations that allow the linearized strain to be a non-linear function of the stress and temperature.     

Nonlocal continuum crystal plasticity with internal residual stresses

We derive a three-dimensional constitutive theory accounting for length-scale dependent internal residual stresses in crystalline materials that develop due to a non-homogeneous spatial distribution of the excess dislocation (edge and screw) density. The second-order internal stress tensor is derived using the Beltrami stress function tensor φ that is related to the Nye dislocation density tensor. The formulation is derived explicitly in a three-dimensional continuum setting for elastically isotropic materials. The internal stresses appear as additional resolved shear stresses in the crystallographic visco-plastic constitutive law for individual slip systems. Using this formulation, we investigate two boundary value problems involving single crystals under symmetric double slip. In the first problem, the response of a geometrically imperfect specimen subjected to monotonic and cyclic loading is investigated. The internal stresses affect the overall strengthening and hardening under monotonic loading, which is mediated by the severity of initial imperfections. Such imperfections are common in miniaturized specimens in the form of tapered surfaces, fillets, fabrication induced damage, etc., which may produce strong gradients in an otherwise nominally homogeneous loading condition. Under cyclic loading the asymmetry in the tensile and compressive strengths due to this internal stress is also strongly influenced by the degree of imperfection. In the second example, we consider simple shear of a single crystalline lamella from a layered specimen. The lamella exhibits strengthening with decreasing thickness and increasing lattice incompatibility with shearing direction. However, as the thickness to internal length-scale ratio becomes small the strengthening saturates due to the saturation of the internal stress.Finally, we present the extension of this approach for crystalline materials exhibiting elastic anisotropy, which essentially depends on the appropriate Green function within φ.