自旋张量的绝对表示及其在有限变形理论中的应用

基于对一类线性张量方程的一般解法,导出了任一对称张量所对应的自旋张量的绝对表示。该结果可以很自然地用于研究左和右伸长张量的自旋并研讨在连续介质力学中常见到的各种转动率张量间的关系。一个重要的公式,即Hill意义下广义应变的共轭应力和Cauchy应力之间的关系,从功共轭原理建立了起来。尤其是详细讨论了对数应变的时间变率及相应的共轭应力。无疑,上述结果对有限变形条件下本构理论的研究是颇为重要的。     

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.     

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).     

A frame-indifferent model for a thermo-elastic material beyond the three-dimensional Eulerian and Lagrangian descriptions

The covariance principle of differential geometry within a four-dimensional (4D) space-time ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space-time and the use of 4D tensors, operations and operators. This enables to separate covariance (i.e. frame-indifference) and material objectivity (i.e. material-indifference). We propose here a method to build a constitutive relation for thermo-elastic materials using such a 4D formalism. A 4D generalization of the classical variational approach is assumed leading to a model for a general thermo-elastic material. The isotropy of the relation can be ensured by the use of the invariants of variables, which offers new possibilities for the construction of constitutive relations. It is then possible to build a general frame-indifferent but not necessarily material-indifferent constitutive relation. It encompasses both the 3D Eulerian and Lagrangian thermo-elastic isotropic relations for finite transformations.     

A theory of thermo-visco-elastic-plastic materials: thermomechanical coupling in simple shear

The constitutive relations of a theory of thermo-visco-elastic-plastic continuum have been formulated in Lagrangian form. The Lagrangian strains, strain rates, temperature, temperature rate and temperature gradients are considered as the independent constitutive variables. Three internal state variables (plastic strain tensor, back strain tensor and a scalar hardening parameter) are also incorporated. The axioms of objectivity and equipresence are followed. The Clausius–Duhem inequality is taken as the second law of thermodynamics. Several special theories are deduced based on material symmetries and/or conventionally adopted assumptions. The applications to the formation of shear bands and dynamic crack propagation are discussed.     

Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains

The familiar small strain thermodynamic 3D theory of isotropic pseudoelasticity proposed by Raniecki and Lexcellent is generalized to account for geometrical effects. The Mandel concept of mobile isoclinic, natural reference configurations is used in order to accomplish multiplicative decomposition of total deformation gradient into elastic and phase transformation (p.t.) parts, and resulting from it the additive decomposition of Eulerian strain rate tensor. The hypoelastic rate relations of elasticity involving elastic strain rate are derived consistent with hyperelastic relations resulting from free energy potential. It is shown that use of Jaumann corotational rate of stress tensor in rate constitutive equations formulation proves to be convenient. The formal equation for p.t. strain rate , describing p.t. deformation effects is proposed, based on experimental evidence. Phase transformation kinetics relations are presented in objective form. The field, coupled problem of thermomechanics is specified in rate weak form (rate principle of virtual work, and rate principle of heat transport). It is shown how information on the material behavior and motion inseparably enters the rate virtual work principle through the familiar bridging equation involving Eulerian rate of nominal stress tensor.

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Strictly conjugate stress and strain tensors

a subclass of strictly conjugate tensors, namely, the tensors that satisfy the requirement for transformation by the same law upon rigid motion of the neighborhood of a material particle, is separated into the class of work-conjugate stress and strain tensors. The advantage of the use of strictly conjugate stress and strain tensors in formulating the variational principles for bodies from a hyperelastic material is shown. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 149–154, May–June, 2000.     

A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept

Starting from the issue of what is the correct form for a Legendre transformation of the strain energy in terms of Eulerian and two-point tensor variables we introduce a new two-point deformation tensor, namely H=(F−F^−T)/2, as a possible deformation measure involving points in two distinct configurations. The Lie derivative of H is work conjugate to the first Piola–Kirchhoff stress tensor P. The deformation measure H leads to straightforward manipulations within a two-point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation. The manipulations are analogous to those used for the Lagrangian and Eulerian frameworks. It is also shown that the Legendre transformation in terms of two-point tensors and spatial tensors require Lie derivatives. As an illustrative example we propose a simple Saint Venant–Kirchhoff type of a strain-energy function in terms of H. The constitutive model leads to physically meaningful results also for the large compressive strain domain, which is not the case for the classical Saint Venant–Kirchhoff material.     

Strain Rates and Material Spins

The material time rate of Lagrangean strain measures, objective corotational rates of Eulerian strain measures and their defining spin tensors are investigated from a general point of view. First, a direct and rigorous method is used to derive a simple formula for the gradient of the tensor-valued function defining a general class of strain measures. By means of this formula and the chain rule as well as Sylvester's formula for eigenprojections, explicit basis-free expressions for the material time rate of an arbitrary Lagrangean strain measure can be derived in terms of the right Cauchy–Green tensor and the material time rate of any chosen Lagrangean strain measure, e.g. Hencky's logarithmic strain measure. These results provide a new derivation of Carlson–Hoger's general gradient formula for an arbitrary generalized strain measure and supply a new, rigorous proof for Carlson–Hoger's conjecture concerning the n-dimensional case. Next, by virtue of the aforementioned gradient formula, a general fact for objective corotational rates and their defining spin tensors is disclosed: Let Ω = ϒ ( B, D, W) be any spin tensor that is continuous with respect to B, where B, D and B are the left Cauchy–Green tensor, the stretching tensor and the vorticity tensor. Then the corotational rate of an Eulerian strain measure defined by Ω is objective iff Ω = W + υ ( B, D), where Υ is isotropic. By means of this fact and certain necessary or reasonable requirements, it is further found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. A general basis- free expression for all such spin tensors and accordingly a general basis-free expression for a general class of objective corotational rates of an arbitrary Eulerian strain measure are established in terms of the left Cauchy–Green tensor B and the stretching tensor B as well as the introduced antisymmetric function. By choosing several particular forms of the latter, all commonly-known spin tensors and corresponding corotational rates are shown to be incorporated into these general formulas in a natural way. In particular, with the aid of these general formulae it is proved that an objective corotational rate of the Eulerian logarithmic strain measure ln V is identical with the stretching tensor D and moreover that in all possible strain tensor measures only ln V enjoys this property. This revised version was published online in August 2006 with corrections to the Cover Date.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

An investigation on microscopic and macroscopic stability phenomena of composite solids with periodic microstructure

An analysis of the effects of microscopic instabilities on the homogenized response of heterogeneous solids with periodic microstructure and incrementally linear constitutive law is here carried out. In order to investigate the possibility to obtain a conservative prediction of microscopic primary instability in terms of homogenized properties, novel macroscopic constitutive stability measures are introduced, corresponding to the positive definiteness of the homogenized moduli tensors relative to a class of conjugate stress–strain pairs.Numerical simulations, addressed to hyperelastic microstructural models representing cellular solids and reinforced composites, are worked out through the implementation of an innovative one-way coupled finite element formulation able to determine sequentially the principal equilibrium solution, the incremental equilibrium solutions providing homogenized moduli and the stability eigenvalue problem solution, for a given monotonic macrostrain path. Both uniaxial and equibiaxial loading conditions are considered.The exact microscopic stability region in the macrostrain space, obtained by taking into account microstructural details, is compared with the macroscopic stability regions determined by means of the introduced macroscopic constitutive measures. These results highlight how the conservativeness of the adopted macroscopic constitutive stability measure with respect to microscopic primary instability, strictly depends on the type of loading condition (tensile or compressive) and the kind of microstructure.     

The finite element method of viscoelastic large deformation plane problem with Kirchhoff stress tensors-Green strain tensors constitutive relation

Based upon the updated Lagrangian approach, the principle of virtual work denoted by the updated Kirchhoff stress increment tensors and the updated Green strain increment tensors and the integral constitutive relation expressed by Kirchhoff stress tensors and Green strain tensors are used and the viscoelastic large deformation incremental variational equation is derived. By means of the 8-nodes isoparametric finite element the program of two-dimensional problem is written. Good agreement is found among the results obtained from this paper and other literatures.     

Hyperelastic dielectrics with saturated polarization2

Variational and invariance principles of modern continuum mechanics are used to establish the field equations, boundary conditions and constitutive relations of a non-linear hyperelastic dielectric with constant magnitude ‘saturated’ polarization. Euclidean invariance places restrictions on the Lagrangian and implies the basic conservation laws. The principles of objectivity and material symmetry restrict the form of the constitutive equations. Four equivalent forms of the free energy functional are listed and for one of these forms the minimal isotropy integrity basis. consisting of eleven invariants, is constructed. The positive definiteness of the energy functional is used to derive various inequalities for the material constants of isotropic dielectrics.     

Extremum principles for certain hyperelastic materials

A hyperelastic material is here said to be of class H_m if the elastic potential is a homogeneous function of order m + 1 in the components of the Lagrangian displacement gradient. It is shown that a single solution to a boundary value problem generates an infinite family of solutions to a family of related boundary value problems. Assuming that a solution to a boundary value problem exists, it is shown that it is unique provided that the material is stable in the sense of Hill in a deleted neighbourhood of the stress-free state. A minimum theorem concerning the strain energy and the virtual work of the prescribed forces is established for the equilibrium configurations, and a maximum theorem concerning the virtual work of the prescribed surface displacements and the complementary stress energy is established for compatible stress fields. As an application, upper and lower bounds are found for the torsional stiffness of a cylindrical bar of square cross section under infinitesimal deformation.     

Finite Anticlastic Bending of Hyperelastic Solids and Beams

This paper deals with the equilibrium problem in nonlinear elasticity of hyperelastic solids under anticlastic bending. A three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains three unknown parameters. A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. An Eulerian analysis is then accomplished to evaluating stretches and stresses in the deformed configuration. Cauchy stress distributions are investigated and it is shown how, for wide ranges of constitutive parameters, the obtained solution is quite accurate. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.