Analysis of partly wrinkled membranes by the method of dynamic relaxation

A version of the method of dynamic relaxation is developed to analyze equilibrium configurations of partly wrinkled membranes. In this method equilibria are regarded as long time limits of a damped dynamical problem. The membrane theory considered is based on the concept of a relaxed strain energy function that automatically incorporates the effects of wrinkling. For neo-Hookean materials, existence theorems of nonlinear elasticity are used to show that the relaxed potential energy possesses minimizers in a certain function space. Moreover, solutions of the equilibrium equations furnish global minima of the energy, for certain classes of boundary data. Such deformations are automatically stable according to the minimum-energy criterion. This result motivates the search for solutions of the equilibrium equations, although the existence theory does not guarantee that energy minimizers possess the degree of regularity required by these equations. Several examples of two-and three-dimensional deformations are presented.     

Finite deformations of compressible hyperelastic tubes subjected to circumferential shear

Finite deformations of homogeneous, isotropic, hyperelastic and compressible tubes subjected to circumferential shearing forces are studied using the theory of finite elasticity. The coupled, non-linear differential equations governing the problem are solved numerically to obtain the angle of twist and radial displacement. The qualitative difference of the present problem from the corresponding problem with incompressible material being the presence of radial deformations. the significance of this coupling is studied for various material constants. Limiting cases of infinitesimal deformations and incompressible material are compared with the findings of this work.     

Large deformations of a protoelastic sheet

We consider the deformation of a plane sheet of variable thickness composed of a protoelastic, simple, isotropic and homogeneous material. The deformations examined here are symmetric with respect to the middle plane of the sheet. With the assumption that the thickness of the sheet is small compared to its other dimensions we derive two-dimensional equations of motion and stress-strain relations. The equations obtained are considerably simpler than the exact equations of the general, three-dimensional, theory of protoelasticity.     

Nonpolar macrodeformation fields in micropolar media

Conditions are stated under which the field equations for homogeneous, isotropic, linearly elastic, micropolar media have solutions, containing a macrodeformation field which satisfies Navier's equations of nonpolar elasticity. Such solutions are found to have a microrotation equal to the local macrorotation of the medium.     

Anisotropic separable free energy functions for elastic and non-elastic solids

In incompressible isotropic elasticity, the Valanis and Landel strain energy function has certain attractive features from both the mathematical and physical view points. This separable form of strain energy has been widely and successfully used in predicting isotropic elastic deformations. We prove that the Valanis–Landel hypothesis is part of a general form of the isotropic strain energy function. The Valanis–Landel form is extended to take anisotropy into account and used to construct constitutive equations for anisotropic problems including stress-softening Mullins materials. The anisotropic separable forms are expressed in terms of spectral invariants that have clear physical meanings. The elegance and attractive features of the extended form are demonstrated, and its simplicity in analysing anisotropic and stress-softening materials is expressed. The extended anisotropic separable form is able to predict, and compares well with, numerous experimental data available in the literature for different types of materials, such as soft tissues, magneto-sensitive materials and (stress-softening) Mullins materials. The simplicity in handling some constitutive inequalities is demonstrated. The work here sets an alternative direction in formulating anisotropic solids in the sense that it does not explicitly use the standard classical invariants (or their variants) in the governing equations.     

Plane deformations of shear-resisting membranes formed of elastic cords

The governing equations are formulated and some exact solutions are obtained for plane deformation of membranes formed of two families of elastic cords. The cords are assumed to be continuously distributed and every cord of one family is joined to each cord of the other family at their point of intersection. The membranes are incapable of withstanding in-plane compression but they exhibit shear resistance and a general nonlinear stretch-tension relation. The solutions include deformations with constant tensions (or stretches), deformations with straight cords and a half-universal state of tensions which satisfies the governing equations for any stretch-tension relations but a particular shear-deformation relation.     

On the decay of solutions for the heat conduction with two temperatures

This paper is devoted to the study of the asymptotic behavior of the solutions of the system of equations that models the heat conduction with two temperatures. That is, we consider a mixture of isotropic and homogeneous rigid solids. We analyze the static problem in a semi-infinite cylinder where every material point has two temperatures with nonlinear boundary conditions on the lateral side. A Phragmén–Lindelöf alternative for the solutions is obtained by means of energy arguments. Estimates for the decay and growth of the solutions are presented. We also prove that the only solution vanishing in the exterior of a bounded set is the null solution for a particular subfamily of problems. Cone-like domains are considered in the last section, and we obtain decay estimates for the solutions when the total energy is bounded.     

两边固支条形薄板的非线性磁弹性效应分析

在给出载流薄板的磁弹性非线性运动方程,电动力学方程的基础上,通过变量代换将描述载流薄板的磁弹性状态方程整理成含有10个基本未知函数的标准柯西(Cauchy)型。并通过差分法及准线性化方法,将标准柯西型的非线性偏微分方程组,变换成为能够用离散正交法编程求解的准线性微分方程组。通过具体算例,得到了两边固支载流条形薄板的磁弹性应力与变形的数值解。变换电磁参量讨论了载流条形薄板的应力及变形的变化规律,通过实例说明了通过变化电磁参量可实现对板的变形控制。     

Some basic half-plane problems of the cohesive elasticity theory with surface energy

Summary We outline a procedure for obtaining the relevant influence functions for the cohesive plane strain half-plane with surface energy under any distribution of normal and tangential loads on its bounding surface, in terms of solutions of classical elasticity. This is achieved by modifying the classical isotropic elasticity theory, to account for the existence of higher order terms in the constitutive equations and additional boundary conditions. We also deal with the half-plane problem under concentrated edge forces and under a uniform distribution of shearing tractions, both of which involve load-induced concentrations of stress (or strain), and it is illustrated how the proposed cohesive elasticity theory can remove the strain singularities.     

An algorithm for solving the plane elastoplastic problem,based on use of the method of complex potentials and the method of conformal representations

The physically nonlinear problem leads to modified Beltrami-Mitchell relationships, which are distinguished from similar ones in the theory of elasticity by the presence of additional terms on the right side. These equations are solved by selecting stress functions in the form of the sum of general and partial solutions. The boundary conditions are varied using the partial solution. The method of complex potentials is used to determine the general solution. Physical nonlinearity is satisfied by the iteration procedure of the method of additional deformations.Translated from Problemy Prochnosti, No. 3, pp. 40–42, March, 1991.     

Finite deformation plasticity and viscoplasticity laws exhibiting nonlinear hardening rules

 This paper deals with plasticity and viscoplasticity laws exhibiting nonlinear kinematic hardening as well as nonlinear isotropic hardening rules. In Tsakmakis (1996a, b) a constitutive theory has been formulated within the framework of finite deformations, which is based on the concept of so-called dual variables and associated time derivatives. Within two families of dual variables, two different formulations have been proposed for kinematic hardening, referred to as Models 1 and 2. In particular, rigid plastic deformations without isotropic hardening have been considered. In the present paper, the constitutive theory of Tsakmakis (1996a, b) is appropriately extended to take into account isotropic hardening as well as elastic deformations. Care is taken that the evolution equations governing the hardening response fulfill the intrinsic dissipation inequality in every admissible process. For the case of small elastic strains combined with a simplification concerning kinematic hardening, to be explained in the paper, an efficient, implicit time-integration algorithm is presented. The algorithm is developed with a view to implementation in the ABAQUS Finite Element code. Also, explicit formulas for the consistent tangent modulus are derived. Received 22 September 1999     

The MFS–MPS for two-dimensional steady-state thermoelasticity problems

We consider the numerical approximation of the boundary and internal thermoelastic fields in the case of two-dimensional isotropic linear thermoelastic solids by combining the method of fundamental solutions (MFS) with the method of particular solutions (MPS). A particular solution of the non-homogeneous equations of equilibrium associated with a planar isotropic linear thermoelastic material is derived from the MFS approximation of the boundary value problem for the heat conduction equation. Moreover, such a particular solution enables one to easily develop analytical solutions corresponding to any two-dimensional domain occupied by an isotropic linear thermoelastic solid. The accuracy and convergence of the proposed MFS–MPS procedure are validated by considering three numerical examples.     

Finite axisymmetric deformations of elastic tubes: An approximate method

Finite axisymmetric deformation of a hollow circular cylinder with a finite length, composed of a neo-Hookean material, is studied. The inner surface of the tube is subjected to both normal and tangential tractions, while the outer surface is free of tractions. The cylinder will undergo both radial and axial deformations. An asymptotic-expansion method is used to determine the stress and shape of the deformed tube. The deformed radial and axial coordinates, the stress tensor and the surface tractions are expanded into a power series of an appropriate thickness parameter. A hierarchy of equilibrium equations, boundary conditions and constitutive equation are derived following the usual procedure. The theories corresponding to the lowest two order members in this hierarchy are studied in detail. It is shown that the zeroth-order theory corresponds to the membrane theory. The shape of the deformed tube, up to the second-order in the thickness parameter, is determined in terms of the zeroth-order radial and axial deformations. The zeroth-order radial and axial deformations are governed by a coupled pair of nonlinear ordinary differential equations, both of which are of second order. For illustrative purposes the present approach is then applied to a simple representative problem: simultaneous extension and inflation of a cylindrical elastic tube. Finally, the solutions corresponding to the zeroth and first-order approximations of the present theory and the exact solutions obtained from finite elasticity theory are compared for the above-mentioned problem.     

热过屈曲梁振动的解析解

该文导出了面内热载荷作用下, 梁在其过屈曲构形附近微幅振动的解析解。首先基于经典梁理论, 推导了控制轴向和横向变形的基本方程。然后, 将2 个非线性方程化为一个关于横向挠度的四阶非线性积分-微分方程。假设梁的振幅以及由此引起的附加应变为无限小, 另设其响应为谐振, 则该非线性积分-微分方程将化为两组耦合的微分方程:一组控制非线性静态响应;另一组就是叠加于梁屈曲构形之上的线性振动方程。直接求解这些问题, 可以得到梁热过屈曲构形以及固有频率的解析解, 这些解是外加热载荷的函数。该文得到的精确解可以用于验证或改进各类近似理论和数值方法。     

非线性声波方程的几种解法对比

在医学诊疗领域及微、介观损伤的无损检测行业中,经常需要对介质的材料非线性系数进行表征,以得到局部区域更加精细的力学性能变化.文章在简述各向同性固体和理想流体介质中的非线性声波方程的基础上,证实了它们具有相同的形式,这表明它们的解也应具有相同的形式和性质.介绍了求解非线性声波方程的五种方法,包括有限差分、有限元、摄动法、...     

Analytical solutions for an infinite transversely isotropic functionally graded sectorial plate subjected to a concentrated force or couple at the tip

This paper considers the elastic responses of an infinite sectorial plate made of transversely isotropic functionally graded material (FGM), which is subjected to a concentrated force or couple at the tip. There is no load acting on the upper and lower surfaces, and the elastic coefficients can vary arbitrarily through the plate thickness. No constraint is required on the symmetry of the plate in the thickness direction. Based on the displacement assumption for the bending of an FGM plate and by using the complex variable method, this paper presents the general solutions to the basic equations governing transversely isotropic FGM plates, which are expressed in terms of four analytical functions (or complex potentials). The boundary conditions are a combination of those from the plane elasticity and those from the classical plate theory. For a particular boundary value problem, such as the ones considered here for a sectorial plate, with the specific conditions for determining solutions, the four analytic functions can be assumed in appropriate forms, which contain only some unknown constants. Once these constants are determined from the specific conditions, the complete solutions are readily derived too. Among the solutions presented here, the solutions for the infinite FGM sectorial plate under a concentrated couple are absolutely new to the literature, and they are also applicable to isotropic FGM sectorial plates. The solutions degenerate into the ones for a homogeneous sectorial plate, which coincide with the available solutions from the plane elasticity theory. There are three-dimensional correction terms in the mid-plane displacements.     

Some results on finite amplitude elastic waves propagating in rotating media

Summary. Two questions related to elastic motions are raised and addressed. First: in which theoretical framework can the equations of motion be written for an elastic half-space put into uniform rotation? It is seen that nonlinear finite elasticity provides such a framework for incompressible solids. Second: how can finite amplitude exact solutions be generated? It is seen that for some finite amplitude transverse waves in rotating incompressible elastic solids with general shear response the solutions are obtained by reduction of the equations of motion to a system of ordinary differential equations equivalent to the system governing the central motion problem of classical mechanics. In the special case of circularly-polarized harmonic progressive waves, the dispersion equation is solved in closed form for a variety of shear responses, including nonlinear models for rubberlike and soft biological tissues. A fruitful analogy with the motion of a nonlinear string is pointed out.     

Buckling of a functionally graded coating with an embedded crack bonded to a homogeneous substrate

In an attempt to simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a functionally graded coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an isotropic stress-strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to give the critical buckling strain and the corresponding crack opening displacement shapes. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions and coating thicknesses.     

两类非线性拟抛物方程解的有限时间爆破和长时间行为

本文研究两类非线性拟抛物方程的初边值问题,它们包括了GBBM方程,Sobolev—Galpern方程,多维粘性扩散方程及半线性拟抛物方程作为特殊情形。对这两类方程我们分别采用积分估计方法和特征函数法证明了,当方程的非线性项满足某些条件时,问题的解按时间t的指数形式衰减为零,而当方程的非线性项满足另外某些条件时,问题的解在有限时间内爆破。本文从实质上改进和推广了已有结果。     

Universal relations for nonlinear electroelastic solids

Summary Electro-sensitive elastomers are materials that can support large elastic deformations under the influence of an electric field. There has been growing interest recently in their applications as so-called ``smart materials'. This paper is devoted to the derivation of universal relations in the context of the nonlinear theory of electroelasticity that underpins such applications. Universal relations are equations relating the components of the stress, the electric variables and the deformation that are independent of the constitutive law for a family of materials. For the general constitutive equations of an isotropic electroelastic material derived from a free energy function and for some special cases of these equations, we obtain universal relations, the word ``universal' being relative to the considered class or subclass of constitutive laws. These universal relations are then applied to some controllable states (homogeneous and non-homogeneous) in order to highlight some examples that may be useful from the point of view of experimental characterization of the material properties. Additionally, we examine the (non-controllable) problem of helical shear of a circular cylindrical tube in the presence of a radial electric field, and we find that a nonlinear universal relation that has been obtained previously for an elastic material also holds when the electric field is applied.