Onset of Cavitation in Compressible,Isotropic, Hyperelastic Solids

In this work, we derive a closed-form criterion for the onset of cavitation in compressible, isotropic, hyperelastic solids subjected to non-symmetric loading conditions. The criterion is based on the solution of a boundary value problem where a hyperelastic solid, which is infinite in extent and contains a single vacuous inhomogeneity, is subjected to uniform displacement boundary conditions. By making use of the “linear-comparison” variational procedure of Lopez-Pamies and Ponte Castañeda (J. Mech. Phys. Solids 54:807–830, 2006), we solve this problem approximately and generate variational estimates for the critical stretches applied on the boundary at which the cavity suddenly starts growing. The accuracy of the proposed analytical result is assessed by comparisons with exact solutions available from the literature for radially symmetric cavitation, as well as with finite element simulations. In addition, applications are presented for a variety of materials of practical and theoretical interest, including the harmonic, Blatz-Ko, and compressible Neo-Hookean materials.     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Distortion of Anisotropic Hyperelastic Solids Under Pure Pressure Loading: Compressibility, Incompressibility and Near-Incompressibility

The purpose of this note is to examine distortion during pure pressure loading for anisotropic hyperelastic solids. We contrast the corresponding issues in compressible and incompressible hyperelasticity, and then use these results to examine nearly incompressible materials. An anisotropic compressible hyperelastic solid will generally exhibit both volume change and distortion under hydrostatic pressure loading. In contrast, an incompressible hyperelastic solid—both isotropic and anisotropic—exhibits no change to its current state of deformation as the hydrostatic pressure is varied. Nearly incompressible hyperelastic materials are compressible, but approach an incompressible response in an appropriate limit. We examine this limiting process in the context of transverse isotropy. The issue arises as to how to implement a nearly incompressible version of a given truly incompressible material model. Here we examine how certain implementations eliminate distortion under pure pressure loading and why alternative implementations do not eliminate the distortion.     

Anticlastic behavior of flat plates

The purpose of this paper is to show that the readings from strain gages can be used effectively to compute small transverse deflections in a rectangular plate and, further, show that the theory developed by Lamb for the rectangular-plate problem agrees with experiment. A numerical procedure is developed, based on the trapezoidal rule, which determines the transverse deflections from the readings of strain gages mounted to the top and bottom surface of a rectangular plate subjected to large longitudinal curvatures. It is shown using the strain-gage technique that experiment agrees with Lamb's theory forb ² /Rt ratios up to 50.     

Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids

Based on the Theory of Porous Media (TPM), a formulation of a fluid-saturated porous solid is presented where both constituents, the solid and the fluid, are assumed to be materially incompressible. Therefore, the so-called point of compaction exists. This deformation state is reached when all pores are closed and any further volume compression is impossible due to the incompressibility constraint of the solid skeleton material. To describe this effect, a new finite elasticity law is developed on the basis of a hyperelastic strain energy function, thus governing the constraint of material incompressibility for the solid material. Furthermore, a power function to describe deformation dependent permeability effects is introduced.After the spatial discretization of the governing field equations within the finite element method, a differential algebraic system in time arises due to the incompressibility constraint of both constituents. For the efficient numerical treatment of the strongly coupled nonlinear solid-fluid problem, a consistent linearization of the weak forms of the governing equations with respect to the unknowns must be used.     

Mixed Formulations of Bending Problems for Homogeneous Elastic Plates and Beams

Mixed formulations of bending problems for homogeneous plates (beams) are proposed, whose essence is that the deformation of a plate (beam) near its fixed boundary is described by the threedimensional elasticity equations, and the remaining part by the conventional equations of plate (beam) bending. At the interface between these regions, the solutions of these equations are joined. The mixed formulation allows one to describe the threedimensional stress state in the neighborhood of the fixed boundaries of plates (beams) and take into account the complex nature of the fixing conditions. Finiteelement implementation is more efficient for the mixed formulations of plate (beam) bending problems than for the wellknown threedimensional formulations.     

考虑轴力作用时梁的弯曲损伤失效分析

对考虑轴力作用时梁的弯曲损伤进行了分析,推导了其损伤分析的基本方程,分析表明,中性轴随损伤的发展而偏移,有向上、向下和保持不动三种情况,其偏移的方向和范围完全由比例加载系数决定,并且在特定的比例加载系数时,中性轴能保持不动,最后获得了梁在弯曲损伤时的中性轴起始偏移点、极限偏移点和极限载荷的计算公式,以及它们的简化计算公式;讨论了损伤极限状态时轴力和弯矩的交互作用曲线;轴力对弯曲损伤的影响较为显著,有轴力作用时应尽可能考虑其影响。     

Anticlastic action of flat sheets in bending

A theoretical analysis is made of the stresses induced in thin elastic strips which are bent (by uniform bending moments) into arcs with substantially constant radii of curvatures. Stresses determined experimentally on aluminum and steel strips agree very well with the theory.     

Finite Amplitude Transverse Waves in Special Incompressible Viscoelastic Solids

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part is a linear combination of A ₁, A ₁ ² and A ₂, where A ₁, A ₂ are the first and second Rivlin–Ericksen tensors. The body is first subject to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that a finite amplitude circularly polarized wave may propagate along either n ⁺ or n ^?, where n ⁺, n ^? are the normals to the planes of the central circular section of the ellipsoid x?B ^?1 x=1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.     

多孔有限大弹性薄板弯曲应力集中问题

应用弹性力学的复变函数理论,采用多保角变换的方法,推出了含有任意多孔有限大弹性薄板弯曲的多复变量应力函数的表达式.在内边界上进行复Fourier级数展开,在外边界采用配点法来确定应力函数的未知系数,从而计算有限大弹性薄板的应力场.本文以外边界为矩形,内边界为任意多椭圆孔的有限薄板为例,编制了相应的计算程序,进行了算例分析.结果表明本方法对处理多孔有限大弹性平面问题是简单且行之有效的.     

梁的动力稳定性分析的有限元方法

提出了对梁进行动力学稳定性分析的有限元方法──给出了单元质量矩阵，抗弯刚度矩阵，几何刚度矩阵及相应的Ｍａｔｈｉｅｕ方程，通过坐标变换消除了方程的动力与静力耦合，然后说明了由这种具有参数激励耦合的多自由度系统的Ｍａｔｈｉｅｕ方程求得系统一般参数共振及组合参数共振的过渡曲线的约束参数方法与多尺度方法。最后作为算例求出了均匀简支梁受简谐轴向力作用时的过渡曲线。     

Longitudinal–Transverse Bending of Layered Beams in a Three-Dimensional Formulation

Three-dimensional equations of the elasticity theory for layered beams are solved by the method of asymptotic splitting without additional hypotheses or restrictions.     

纯弯曲梁的应变损伤失效分析和预测

提出了纯弯曲梁的应变损伤失效分析方法,与Kachanov的材料受载横截面减少定义拉伸损伤变量类似,以梁的弯曲惯性矩阵减少定义弯曲损伤变量。推导了梁的弯曲应变损伤基本方程,其中的材料常数可由Kachanov拉伸损伤模型的材料常数确定。并且提出了便于工程应用的应变失效预测方程。     

梁的循环弯曲蠕变脆断损伤失效分析和预测

提出了梁的循环弯曲蠕变损伤失效分析方法，推导了梁的循环弯曲蠕变损伤基本方程，以梁的弯曲惯性矩减少定义弯曲损伤变量，给出了便于工程应用的失效预测方程。     

TEST AND CHARACTERIZATION FOR THE INCOMPRESSIBLE HYPERELASTIC PROPERTIES OF CONDITIONED RUBBERS UNDER MODERATE FINITE DEFORMATION

In this paper, the automated grid method is applied to test for the mechanical properties of conditioned rubbers under the moderate ?nite deformation (not exceeding 100%). More accurate stress-strain curves of conditioned rubber specimens unde…     

Finite Element Analysis of Dynamically Loaded Homogeneous Viscous Beams

The paper is concerned with impulsively loaded beams in which the material is treated as homogeneous viscous as an approximation of a rigid-viscoplastic constitutive relation. As opposed to the standard displacement method finite element formulation, where interpolation functions describing the velocity field across elements is given, a mixed formulation is used in which nodal velocities and nodal moments are carried as parameters. At each instant the accelerations (by the Tamuzh principle) and the rates of change of moment (by a virtual velocities formulation) are found, and velocities and moments are integrated forward independently. The properties of the mode solution are also introduced, and the forward integration is carried through only for the difference between the mode solution and the actual solution. This leads to a very efficient scheme for the numerical solution of a cantilever beam problem shown as an illustration.     

混凝土梁三点弯曲断裂模式转变的近场动力学模拟

混凝土结构的宏观损伤开裂与其非均质微观结构紧密相关。底部带切口的混凝土梁在进行三点弯曲破坏时,随着切口的位置由梁中向梁边转移,裂纹由从切口处萌生并生长转变为从梁的中部萌生。本文采用半均质化近场动力学（IH-PD）模型和全均质化近场动力学（FH-PD）模型,分别对混凝土梁三点弯断裂问题进行模拟研究。IH-PD模型根据混凝土中骨料体积分数随机生成不同键的组合方式,将微观尺度的非均质性引入模型,无需详细描绘骨料形状和分布即可考虑混凝土非均质性。本文将IH-PD与FH-PD模型得到的断裂模式随切口位置的变化关系,与实验结果对比,分析微观结构对混凝土梁开裂的影响;基于非均质材料特征尺寸与IH-PD模型网格参数的相关性,模拟骨料大小对混凝土梁断裂模式的影响;另外,通过在IH-PD模型中设置预损伤的方式引入随机分布的孔隙,探讨孔隙率对混凝土断裂模式的影响。     

在有限变形条件下损伤粘弹性梁的动力学行为

本文在有限变形条件下，根据损伤粘弹性材料的一种卷积型本构关系和温克列假设，建立了粘弹性基础上损伤粘弹性Timoshenko梁的控制方程。这是一组非线性积分——偏微分方程。为了便于分析，首先利用Galerkin方法对该方程组进行简化，得到一组非线性积分一常微分方程。然后应用非线性动力学中的数值方法，分析了粘弹性地基上损伤粘弹性Timoshenko梁的非线性动力学行为，得到了简化系统的相平面图、Poincare截面和分叉图等。考察了材料参数和载荷参数等对梁的动力学行为的影响。特别，考察了基础和损伤对粘弹性梁的动力学行为的影响。