On non-linear Beltrami-Mitchell equations

Beltrami-Mitchell equations for non-linear elasticity theory are derived using the first Piola-Kirchhoff stress and the deformation gradient tensors as field variables so as to yield linear equilibrium and compatibility equations, respectively. In the derivation it is assumed that a strain energy density and, correspondingly, a complementary strain energy density exist, and satisfy the axiom of objectivity. Substitution for the deformation gradient in the compatibility equations yields non-linear differential equations in terms of the first Piola-Kirchhoff stress tensor which may be regarded as the Beltrami-Mitchell equations of non-linear elasticity. The equations are also derived for “semi-linear” isotropic elastic materials and the theory is illustrated by three simple examples.     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

The Thermoviscoelastoplastic Axisymmetric Stress–Strain State of Laminated Flexible Orthotropic Shells

A technique is proposed to determine the thermoviscoelastoplastic axisymmetric stress–strain state of laminated shells made of isotropic and orthotropic materials. The paper deals with processes of shell loading such that both instantaneous elastoplastic and creep strains occur in isotropic materials and elastic and creep strains in orthotropic materials. The technique is developed within the framework of the Kirchhoff–Love hypotheses for a stack of layers with the use of the equations of the geometrically nonlinear theory of shells in a quadratic approximation. The deformation of isotropic materials is described by the equations of the theory of deformation along slightly curved trajectories, while the deformation of orthotropic materials is described by Hooke's law with additional terms allowing for creep. A numerical example is given     

A finite-deformation theory of plasticity

Based on a multiplicative decomposition of local deformation into elastic and plastic deformations general constitutive equations of elastic-plastic materials are proposed. Two alternative approaches are discussed: one in which the elastic deformation is used as an independent variable, and the other in which the stress is one of the independent variables. The appropriate material symmetries are defined, and it is shown that the plastic spin is absent in the theory of isotropic materials. Analysis of a simple extension is given as an example.     

Incremental elastic motions superimposed on a finite deformation in the presence of an electromagnetic field

The equations describing the interaction of an electromagnetic sensitive elastic solid with electric and magnetic fields under finite deformations are summarized, both for time-independent deformations and, in the non-relativistic approximation, time-dependent motions. The equations are given in both Eulerian and Lagrangian form, and the latter are then used to derive the equations governing incremental motions and electromagnetic fields superimposed on a configuration with a known static finite deformation and time-independent electromagnetic field. As a first application the equations are specialized to the quasimagnetostatic approximation and in this context the general equations governing time-harmonic plane-wave disturbances of an initial static configuration are derived. For a prototype model of an incompressible isotropic magnetoelastic solid a specific formula for the acoustic shear wave speed is obtained, which allows results for different relative orientations of the underlying magnetic field and the direction of wave propagation to be compared. The general equations are then used to examine two-dimensional motions, and further expressions for the wave speed are obtained for a general incompressible isotropic magnetoelastic solid.     

ON THE FREE VIBRATION OF A SUBMERGED FGM HOLLOW SPHERE

The free vibration of a functionally graded material hollow sphere submerged in a compressible fluid medium is exactly analyzed. The sphere is assumed to be spherically isotropic with material constants being inhomogeneous along the radial direction. By employing a separation technique as well as the spherical harmonics expansion method, the governing equations are simplified to an uncoupled second-order ordinary differential equation, and a coupled system of two such equations. Solutions to these equations are given when the elastic constants and the mass density are power functions of the radial coordinate. Numerical examples are finally given to show the effect of the material gradient on the natural frequencies. The project is supported by the National Natural Sciences Foundation of China(No. 19872060).     

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.     

On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

The paper compares two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic materials with different elastic moduli in tension and compression. The approaches use different nonlinear constitutive equations describing the elastic deformation of the transversally isotropic bimodulus materials. Both approaches employ the theory of deformation along paths of small curvature and the Kirchhoff-Love hypothesis for the whole laminate to describe the deformation of the isotropic materials. The problem is solved by the method of successive approximations. The solutions of specific problems obtained by the two approaches are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 59–69, June 2008.     

A crystal plasticity model that incorporates stresses and strains due to slip gradients

This work is concerned with incorporating the kinematic and stress effects of excess dislocations in a constitutive model for the elastoplastic behavior of crystalline materials. The foundation of the model is a three term multiplicative decomposition of the deformation gradient in which the two classical terms of plastic and elastic deformation are included along with an additional term for long range strain due to the collective effects of excess dislocations. The long range strain is obtained from an assumed density of Volterra edge dislocations and is directly related to gradients in slip. A new material parameter emerges which is the size the region about a continuum point that contributes to long range strains.Using Hookean elasticity, the stress at a point is linearly related to the sum of the elastic plus the long range strain fields. However, the driving force for slip is postulated to be due only to the elastic stress so that the long range stress is a back stress in the constitutive relationship for plastic deformation. A consistent balance of the total deformation rate with the three proposed mechanisms of deformation leads to a set of differential equations that can be solved for the elastic stress, rotation and pressure which then implicitly defines the material state and equilibrium stress. Results from the simulation of a tapered tensile specimen demonstrate that the constitutive model exhibits isotropic and kinematic type hardening effects as well as changes in the pattern of plastic deformation and necking when compared to a material without slip gradient effects.     

On thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic solids with application to biomechanics

In this paper, we are interested in developing thermodynamically consistent constitutive equations for fiber-reinforced nonlinearly viscoelastic bodies, in particular for transversely isotropic nonlinearly viscoelastic solids, in isothermal processes. It follows from results in the theory of algebraic invariants that constitutive equations for such materials can be expressed in terms of functions of 18 independent invariants associated with deformation and fiber orientation: 10 of them are isotropic invariants and 8 of them are associated with the deformation and the orientation of the fiber. Among the 8 anisotropic invariants just 6 are related to the viscoelastic response. The terms in the Cauchy stress tensor associated to these 6 invariants are analyzed with respect to thermodynamical consistency, and we obtain restrictions for the corresponding constitutive coefficients. This framework is applied to viscoelastic potentials within the context of biomaterials.     

The mechanics of pyramids

The elastostatic fields and free-vibration characteristics are studied for a class of continuous solids in the shape of homogeneous or layered pentahedral pyramids. The pyramids possess an arbitrary number of linear elastic layers containing dissimilar elastic constants. The layers are assumed to be perfectly bonded in the out-of-plane direction of the pyramid. The change in elastic stiffness across each interface requires a model that allows for a jump in displacement gradient at these location. A discrete-layer representation is used that combines the Ritz method with polynomial in-plane approximations with one-dimensional Lagrangian interpolation polynomials in the thickness direction. The free-vibration characteristics are examined for a variety of isotropic and anisotropic materials and representative bulk stiffness estimates are given for homogeneous pyramids under static deformation.     

Statistical continuum theory for inelastic behavior of a two-phase medium

A statistical continuum mechanics formulation is presented to predict the inelastic behavior of a medium consisting of two isotropic phases. The phase distribution and morphology are represented by a two-point probability function. The isotropic behavior of the single phase medium is represented by a power law relationship between the strain rate and the resolved local shear stress. It is assumed that the elastic contribution to deformation is negligible. A Green’s function solution to the equations of stress equilibrium is used to obtain the constitutive law for the heterogeneous medium. This relationship links the local velocity gradient to the macroscopic velocity gradient and local viscoplastic modulus. The statistical continuum theory is introduced into the localization relation to obtain a closed form solution. Using a Taylor series expansion an approximate solution is obtained and compared to the Taylor’s upper-bound for the inelastic effective modulus. The model is applied for the two classical cases of spherical and unidirectional discontinuous fiber-reinforced two-phase media with varying size and orientation.     

均布荷载作用下功能梯度悬臂梁弯曲问题的解析解

采用弹性力学半逆解法,假设所有材料常数沿梁厚度方向按同一函数规律变化,求得了功能梯度悬臂梁在均布载荷作用下的解析解.该解退化到各向同性均匀弹性情况时与已有的理论解相一致.对弹性模量按指数函数梯度变化的算例进行了分析.所得到的解对任意梯度函数均成立,可作为数值解以及简化理论的检验依据.     

Asymptotic Analysis of Finite Deformation in a Nonlinear Transversely Isotropic Incompressible Hyperelastic Half-space Subjected to a Tensile Point Load

The Boussinesq problem, that is, determining the deformation in a hyperelastic half-space due to a point force normal to the boundary, is an important problem of engineering, geomechanics, and other fields to which elasticity theory is often applied. While linear solutions produce useful Greens functions, they also predict infinite displacements and other physically inconsistent results nearby and at the point of application of the load where the most critical and interesting material behavior occurs. To illuminate the deformation due to such a load in the region of interest, asymptotic analysis of the nonlinear Boussinesq problem has been considered in the context of isotropic hyperelasticity. Studies considering transversely isotropic materials have also been broadly used in the linear theory, but have not been treated within the nonlinear framework. In this paper we extend the nonlinearly elastic isotropic analysis to transverse isotropy, producing a more general theory which also better encompasses applications involving layered media. The governing equations for nonlinearly elastic, transversely isotropic solids are derived, conservation laws of elastostatics are invoked, asymptotic forms of the deformation solutions are hypothesized, and the differential equations governing deformation near the point load are determined. The analysis also develops sequences of simple tests to determine if a transversely isotropic material can possibly sustain a finite deflection under the point load. The results are applied to a variety of transversely isotropic materials, and the effects of the anisotropy considered is demonstrated by comparison of the resulting deformation with similar asymptotic solutions in the isotropic theory. Mathematics Subject Classifications (2000) 74B20, 74E10, 74G10, 74G15, 74G70.     

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.     

Local and global universal relations for first-gradient materials

Local universal relations are relations between stress and kinematic variables which hold for all materials of a particular class irrespective of specific material parameters. A method is developed for obtaining local universal relations for most first gradient materials. The currently known local universal relations for isotropic elastic materials have been extended to all isotropic first gradient materials under constant step deformation histories and have also been extended to all isotropic first gradient materials undergoing arbitrary time dependent triaxial extensions along fixed material directions. It has been shown that universal relations exist for some anisotropic materials. A set of pseudo-universal relations has been obtained for anisotropic elastic materials which can be used to decouple the material functions. These pseudo-universal relations contain some, but not all, material functions. A global universal relation has been developed for the extension and torsion of an isotropic cylindrical shaft which holds for all incompressible first gradient materials.     

功能梯度材料动态断裂力学的径向积分边界元法

采用径向积分边界元法分析功能梯度材料动态断裂力学问题. 该方法使用与弹性模量无关的弹性静力学开尔文基本解作为问题的基本解,在导出的边界-域积分方程中含有由材料的非均质性和惯性项引起的域积分,通过径向积分法将域积分转化为等效的边界积分,得到只含边界积分的纯边界积分方程;从而建立只需边界离散的无内部网格边界元算法. 采用候博特方法求解关于时间二阶导数的系统离散的常微分方程组. 最后通过数值算例验证本文方法的精度和有效性.      

Short-Term Microdamageability of Fibrous Composites with Transversally Isotropic Fibers and a Microdamaged Binder

The theory of microdamageability of fibrous composites with transversally isotropic fibers and a microdamaged isotropic porous matrix is proposed. Microdamages in the matrix are simulated by pores filled with particles of the destroyed material that resist compression. The criterion of damage in the matrix microvolume is taken in the Schleicher–Nadai form. It accounts for the difference between the ultimate tensile and compressive loads. The ultimate strength is a random function of coordinates with Weibull distribution. The stress–strain state and effective properties of the material are determined from the stochastic equations of the elastic theory for a fibrous composite with porous components. The equations of deformation and microdamage are closed by the equations of porosity balance in the matrix. Nonlinear diagrams of the concurrent processes of deformation of fibrous materials and microdamage of the matrix are plotted. The effect of the physical and geometrical parameters on them is studied     

The Micromechanics of Short-Term Damageability of Fibrolaminar Composites

A theory of microdamageability is constructed for fibrous laminated composites consisting of transversally isotropic fibers and a microdamaged isotropic porous binder. Microdamages in the binder are simulated by pores filled with compression-resisting particles of the destroyed material. Damage in a microvolume of the binder is described by the Schleicher–Nadai strength criterion, which allows for the difference between the ultimate tensile and compressive loads. The ultimate strength is a random function of coordinates with the Weibull distribution. The stress–strain state and effective characteristics of the material are determined by solving the stochastic equations of elastic theory for a fibrous laminated composite with a porous binder. The equations of deformation and microdamageability are closed by the equations of porosity balance in the binder. Nonlinear diagrams of the concurrent processes of deformation of the fibrous laminated material and microdamage of the matrix for various physical and geometrical parameters are constructed