Linear Response in Finite Elasticity

This paper presents objective finite elastic equations that reproduce the linear stress-strain behaviour observed in uniaxial tension tests with some materials. More generally, the new equations reduce to linear elastic equations when the deformation is a pure stretch. Further, the response of the proposed equations for the case of isotropic materials in uniaxial tension-compression and simple shear is examined. It is also shown how the new equations can be considered as linear approximations of general equations of finite elasticity.     

关于线性非局部弹性理论的应力边界条件

应力边界条件的提法是线性非局部弹性理论尚未解决的一个理论问题。文中针对这一问题进行了研究，所导出的应力边界条件包含了物体微观结构的长程相互作用，这个结果不仅解释了在裂纹混合边界值问题中非线性局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以自然地得到裂纹尖端的分子内聚力模型。     

基于非局部弹性理论的纳米板横向振动

本文利用非局部弹性理论研究了单层石墨烯的纳米板的横向自由振动响应．通过迭代法推导了非局部应力表达,进一步通过哈密顿原理推导了纳米板的控制方程,应用纳维解法得到四边简支纳米板振动固有频率的数值解,并将本文研究结果与已有文献结果进行对比,进一步讨论了小尺寸效应,以及纳米板的三维尺寸和半波数对振动频率的影响．结果表明：非局部效应的存在使得纳米板的等效刚度和固有频率降低;半波数的增加则使得纳米板的固有频率提高．相关分析结果对基于二维纳米材料的新设备的设计和优化具有重要意义．     

Weak Local Minimizers in Finite Elasticity

This paper deals with necessary conditions and sufficient conditions for a weak local minimum of the energy of a hyperelastic body. We consider anisotropic bodies of arbitrary shape, subject to prescribed displacements on a given portion of the boundary. As an example, we consider the uniaxial stretching of a cylinder, in the two cases of compressible and incompressible material. In both cases we find that there is a continuous path across the natural state, made of local energy minimizers. For the Blatz-Ko compressible material and for the Mooney-Rivlin incompressible material, explicit estimates of the minimizing path are given and compared with those available in the literature. Dedicated to the memory of Victor J. Mizel.     

Azimuthal Shear in Compressible Finite Elasticity

In an earlier paper, the broadest classes of compressible isotropic strain energies that support irrotational universal deformations were identified and the problems of cylindrical and spherical inflation or compaction were solved in closed form for all of these strain energies. Similar closed form solutions of the problem of azimuthal shear are presented here.      

Linear Constitutive Relations in Isotropic Finite Elasticity

For simple shearing and simple extension deformations of a homogeneous and isotropic elastic body, it is shown that a linear relation between the second Piola-Kirchhoff stress tensor and the Green-St. Venant strain tensor does not predict a physically reasonable response of the body. This constitutive relation implies that the slope of the curve between an appropriate component of the first Piola-Kirchhoff stress tensor and a deformation measure is an increasing functions of the deformation measure. This revised version was published online in August 2006 with corrections to the Cover Date.     

Linear Equations of Motion in Finite Elasticity

In this note we show that it may be possible and useful to construct valid strain-energy functions that lead directly to linear equilibrium equations for problems in isotropic homogeneous unconstrained nonlinear elasticity. While it is possible to make some general progress the final outcome will depend on the geometry and kinematics of the problem under consideration. Specific examples are given to show how exact solutions, via the linear equations of motion, can be found to non-trivial problems for physically meaningful constitutive models.      

Finite Third-order Gradient Elasticity and Thermoelasticity

A constitutive format for the third-order gradient elasticity is suggested. It includes both isotropic and anisotropic non-linear behavior under finite deformations. Appropriate invariant stress and strain variables are introduced, which allow for reduced forms of the elastic energy law that identically fulfill the objectivity requirement. After working out the transformation behavior under a change of the reference placement, the symmetry transformations for third-order materials can be introduced. After the mechanical third-order theory, an extension to thermoelasticity is given, and necessary and sufficient conditions are derived from the Clausius-Duhem inequality.     

On the Uniqueness of Energy Minimizers in Finite Elasticity

The uniqueness of absolute minimizers of the energy of a compressible, hyperelastic body subject to a variety of dead-load boundary conditions in two and three dimensions is herein considered. Hypotheses under which a given solution of the corresponding equilibrium equations is the unique absolute minimizer of the energy are obtained. The hypotheses involve uniform polyconvexity and pointwise bounds on derivatives of the stored-energy density when evaluated on the given equilibrium solution. In particular, an elementary proof of the uniqueness result of Fritz John (Commun. Pure Appl. Math. 25:617–634, 1972) is obtained for uniformly polyconvex stored-energy densities.     

非局部弹性理论概述及在当代材料背景下的一些进展

局部作用原理在发展经典连续介质力学的本构关系中起着重要的作用，由此导出的简单物质理论得到了广泛的应用．然而，随着科技的发展，各种具有微结构的新材料不断涌现，理论和实验表明，非局部理论可以更好地刻画这些材料的宏观力学行为．本文简要介绍了一些传统的非局部弹性理论，包括Eringen 理论、Kunin 理论、Mindlin 理论；阐述了针对复合材料发展的，具有时间-空间非局部特征的Willis 方程、最新的时间-空间耦合非局部弹性动力学理论以及近场动力学理论．时间-空间非局部理论反映了复合材料宏观性能固有的非局部特征，而具有空间非局部特征的近场动力学理论便于处理具有不连续性的问题．最后，本文讨论了非局部理论的发展中值得关注的一些问题.     

On the Commutability of Homogenization and Linearization in Finite Elasticity

We consider a family of non-convex integral functionals

$\frac{1}{h^2}\int_\Omega W(x/\varepsilon,{\rm Id}+h\nabla g(x))\,\,{\rm d}x,\quad g\in W^{1,p}({\Omega};{\mathbb R}^n)$

where W is a Carathéodory function periodic in its first variable, and non-degenerate in its second. We prove under suitable conditions that the Γ-limits corresponding to linearization (h → 0) and homogenization (\({\varepsilon\rightarrow 0}\)) commute, provided W is minimal at the identity and admits a quadratic Taylor expansion at the identity. Moreover, we show that the homogenized integrand, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the second variation of W.

     

Necessary Conditions at the Boundary for Minimizers in Incompressible Finite Elasticity

New necessary conditions for energy-minimizing states of an incompressible, elastic body are found. These must hold at boundary points at which dead-load traction data is prescribed.     

On the Size of RVE in Finite Elasticity of Random Composites

This paper presents a quantitative study of the size of representative volume element (RVE) of random matrix-inclusion composites based on a scale-dependent homogenization method. In particular, mesoscale bounds defined under essential or natural boundary conditions are computed for several nonlinear elastic, planar composites, in which the matrix and inclusions differ not only in their material parameters but also in their strain energy function representations. Various combinations of matrix and inclusion phases described by either neo-Hookean or Ogden function are examined, and these are compared to those of linear elastic types.     

Invariance of the Equilibrium Equations of Finite Elasticity for Compressible Materials

For homogeneous, isotropic, compressible nonlinearly elastic materials, a wide class of strain-energy density functions are obtained that leave the equations of equilibrium invariant under simple scaling transformations of the material and spatial coordinates. These strain-energy densities are homogeneous functions of the principal stretches. Several illustrative examples of particular strain-energies are provided. For axisymmetric problems, the invariance discussed here ensures that the equations of equilibrium can be solved by quadratures and thus often leads to analytic solutions in parametric or closed-form. Mathematics Subject Classifications (2000) 74B20, 74G55.     

多层横观各向同性圆柱壳的三维弹性理论解

本文从三维弹性理论出发,用特征函数法研究多层横观各向同性圆柱壳的轴对称问题.把位移和应力分量的齐次解表达成特征函数展开式,并把特解部分用Fourier级数表示.以多层圆柱壳的内、外柱面作为齐次边界,同时考虑层间的连续条件,推导出问题的特征方程并用Muller法求解.文中运用传递矩阵技术处理多层问题,并用边界型最小二乘配点法处理端部边界条件.作为实例,对双层圆柱壳作了数值计算.     

A Study of Two-Dimensional Plane Elasticity Finite Elements for Optimal Design

ABSTRACT

Design of structures using a variety of two-dimensional finite elements is considered in this paper. An efficient technique of computing first-order derivatives of pointwise stress constraints for simple and higher-order two-dimensional (membrane) finite elements is presented. Computational aspects of design sensitivity vector calculation, using a semi-analytical method versus traditional methods are presented. Implementation of a fully-stressed design approach to find a suitable initial estimate leads to increased computational efficiency. These aspects of the design procedure are illustrated through analysis of numerical examples. Experience indicates that a suitable mix of low and higher-order elements yields the most efficient and accurate design model.     

有限位移弹性理论变分泛函的代入构成法

提出了变分泛函的一种新的数学构成法—代入法.将此法应用于弹性有限位移理论中,除了得到前人得到的变分泛函外,还得到了一对新泛函—狭义泛函.并通过狭义泛函,给出应力应变关系是目前所得到的各种变分泛函的Euler方程的结论.     

非线性弹性及全量弹塑性有限元分析中的一致性切线模量

本文导出了非线性弹性及全量弹塑性有限元分析中的一致性切线模量,从而可以保持牛顿迭代法固有的平方收敛速度.指出了某些文献中关于切线模量的不正确表述,并以数值算例验证了本文方法的正确有效.     

Non-local criteria for the borehole problem: Gradient Elasticity versus Finite Fracture Mechanics

Two nonlocal approaches are applied to the borehole geometry, herein simply modelled as a circular hole in an infinite elastic medium, subjected to remote biaxial loading and/or internal pressure. The former approach lies within the framework of Gradient Elasticity (GE). Its characteristic is nonlocal in the elastic material behaviour and local in the failure criterion, hence simply related to the stress concentration factor. The latter approach is the Finite Fracture Mechanics (FFM), a well-consolidated model within the framework of brittle fracture. Its characteristic is local in the elastic material behaviour and non-local in the fracture criterion, since crack onset occurs when two (stress and energy) conditions in front of the stress concentration point are simultaneously met. Although the two approaches have a completely different origin, they present some similarities, both involving a characteristic length. Notably, they lead to almost identical critical load predictions as far as the two internal lengths are properly related. A comparison with experimental data available in the literature is also provided.