On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities

The problems of singularity formation and hydrostatic stress created by an inhomogeneity with eigenstrain in an incompressible isotropic hyperelastic material are considered. For both a spherical ball and a cylindrical bar with a radially symmetric distribution of finite possibly anisotropic eigenstrains, we show that the anisotropy of these eigenstrains at the center (the center of the sphere or the axis of the cylinder) controls the stress singularity. If they are equal at the center no stress singularity develops but if they are not equal then stress always develops a logarithmic singularity. In both cases, the energy density and strains are everywhere finite. As a related problem, we consider annular inclusions for which the eigenstrains vanish in a core around the center. We show that even for an anisotropic distribution of eigenstrains, the stress inside the core is always hydrostatic. We show how these general results are connected to recent claims on similar problems in the limit of small eigenstrains.     

On Structured Surfaces with Defects: Geometry,Strain Incompatibility,Stress Field,and Natural Shapes

In this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic. Similarly, the stress in a cylindrical inclusion contained in an incompressible orthotropic cylindrical bar is uniform hydrostatic if the radial and circumferential eigenstrains are equal and the axial stretch is equal to a value determined by the axial eigenstrain. We also prove that for a compressible isotropic spherical ball and a cylindrical bar containing a spherical and a cylindrical inclusion, respectively, with uniform eigenstrains the stress in the inclusion is uniform (and hydrostatic for the spherical inclusion) if the radial and circumferential eigenstrains are equal. For compressible transversely isotropic and orthotropic solids, we show that the stress field in an inclusion with uniform eigenstrain is not uniform, in general. Nevertheless, in some special cases the material can be designed in order to maintain a uniform stress field in the inclusion. As particular examples to investigate such special cases, we consider compressible Mooney-Rivlin and Blatz-Ko reinforced models and find analytical expressions for the stress field in the inclusion.     

On Non-Symmetric Deformations of an Incompressible Nonlinearly Elastic Isotropic Sphere

A class of non-symmetric deformations of a neo-Hookean incompressible nonlinearly elastic sphere are investigated. It is found via the semi-inverse method that, to satisfy the governing three-dimensional equations of equilibrium and the incompressibility constraint, only three special cases of the class of deformation fields are possible. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation describing inflation and stretching. The implications of these results for cavitation phenomena are also discussed. In the course of this work, we also present explicitly the spherical polar coordinate form of the equilibrium equations for the nominal stress tensor for a general hyperelastic solid. These are more complicated than their counterparts for Cauchy stresses due to the mixed bases (both reference and deformed) associated with the nominal (or Piola-Kirchhoff) stress tensor, but more useful in considering general deformation fields. This revised version was published online in August 2006 with corrections to the Cover Date.     

Wedge indentation into elastic-plastic single crystals. 2: Simulations for face-centered cubic crystals

The asymptotic stress and deformation fields associated with the contact point singularity of a nearly-flat wedge indenter impinging on a specially-oriented single face-centered cubic crystal are derived analytically in a companion paper. In order to investigate the extent of the asymptotic fields, the indentation process is simulated numerically using single crystal plasticity. The quasistatically translating asymptotic fields consist of four angular elastic sectors separated by plastically deforming sector boundaries, as predicted in the companion paper. The asymptotic stress distributions are in accord with the analytical predictions. In addition, simulations are performed for a wedge indenter with a 90° included angle in order to investigate the consequences of finite deformation and finite lattice rotation. Several salient features of the deformation field for the nearly-flat indenter persist in the deformation field for the 90° wedge indenter. The existence of the salient features is validated by comparison to experimental measurements of the lower bound on geometrically necessary dislocation (GND) densities.     

Vibrations of neo-Hookean elastic wedge

This paper considers small amplitude vibrations superimposed upon large planar deformations of an infinite wedge composed of a neo-Hookean elastic material. It is shown herein that even though the static deformation of the entire wedge and the vibrations of the wedge faces are planar, out-of-plane vibrational modes must necessarily be excited in the wedge interior even to first order in an asymptotic expansion of the motion with small parameter being the amplitude of the vibration applied to the wedge faces. In addition, it is demonstrated that this result is fundamentally due to the non-linearity of the problem by demonstrating that the corresponding problem for an incompressible, isotropic, homogeneous linear elastic wedge does not exhibit the same behavior.     

Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two neo-Hookean materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the underformed configuration. Such a configuration is the only stable solution for sufficiently large loads. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right orto the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems.Since this paper was written, the authors have carried out further analysis of the class of problems of concern here [11]. In particular the stress distribution in the composite neo-Hookean sphere has been described in [11].Paper presented at the 17th International Congress of Theoretical and Applied Mechanics, Grenoble, France, August 1988.     

Problems on elastic equilibrium of a wedge with cracks on the axis of symmetry

We use the Wiener-Hopf method to obtain exact solutions of plane deformation problems for an elastic wedge whose lateral sides are stress free and which has rectilinear cracks on its axis of symmetry. In problem 1, a finite crack issues from the wedge apex edge; in problem 2, a half-infinite crack originates at a certain distance from the wedge apex edge; and in problem 3, the wedge contains an internal finite crack.     

Finite deformations of fibre-reinforced vascular prosthesis

We present an mechanical study of a vascular prosthesis subjected to finite radial expansion, torsion and circular shearing. The material strain-energy function is expressed in the form of the fibre-reinforced neo-Hookean. The governing equations are solved analytically and the results present effects of the combined deformation on the stress distributions.     

Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress

A new method is developed to determine the dominant asymptotic stress and deformation fields near the tip of a Mode-I traction free plane stress crack. The analysis is based on the fully nonlinear equilibrium theory of incompressible hyperelastic solids. We show that the dominant singularity of the near tip stress field is governed by the asymptotic solution of a linear second order ordinary differential equation. Our method is applicable to any hyperelastic material with a smooth work function that depends only on the trace of the Cauchy-Green tensor and is particularly useful for materials that exhibit severe strain hardening. We apply this method to study two types of soft materials: generalized neo-Hookean solids and a solid that hardens exponentially. For the generalized neo-Hookean solids, our method is able to resolve a difficulty in the previous work by Geubelle and Knauss (1994a). Our theoretical results are compared with finite element simulations.     

三维结构安定分析的直接算法

基于线性随动强化理论和Von. Mises屈服准则,对蒙板结构直接安定分析法进行了扩展,建立了结构的三维安定直接分析法。根据投射原理,推导出结构发生塑性安定的存在条件,便于调整控制加载步长和载荷历程。采用逐次增量加载方式,确定出背应力的偏移范围,克服了原始直接分析法不能获得安定极限的缺陷,并得到安定极限条件下结构中残余应力与应变的分布状况。该数值方法将弹塑性问题分解为弹性问题和特征应变决定的残余问题,节约计算时间,提高计算效率,将该算法应用于相关算例,并与有关数值结果相比较,验证了该算法的有效性。     

Characterization of random microstructural stresses and fracture estimation

In this paper, the analytical approach to quantitative characterization of random microstructural residual stress field in brittle elastic materials is presented. The analysis consists of two parts. First, we expound the basic features of random microstructural stresses and show how the eigenstrain approach (of “classical” micromechanics) can be extended to the situations where randomness of the initial eigenstrains has to be taken into account. The second part of the paper deals with the effects of random microstresses on crack growth phenomena. The stress intensity induced by random dilatant transformation eigenstrains and by thermally-induced random microstresses are treated in detail, including numerical and graphical illustrations of specific crack problems.     

Cavitation in finite elasticity with surface energy effects

Cavity formation in incompressible as well as compressible isotropic hyperelastic materials under spherically symmetric loading is examined by accounting for the effect of surface energy. Equilibrium solutions describing cavity formation in an initially intact sphere are obtained explicitly for incompressible as well as slightly compressible neo-Hookean solids. The cavitating response is shown to depend on the asymptotic value of surface energy at unbounded cavity surface stretch. The energetically favorable equilibrium is identified for an incompressible neo-Hookean sphere in the case of prescribed dead-load traction, and for a slightly compressible neo-Hookean sphere in the case of prescribed surface displacement as well as prescribed dead-load traction. In the presence of surface energy effects, it becomes possible that the energetically favorable equilibrium jumps from an intact state to a cavitated state with a finite cavity radius, as the prescribed loading parameter passes a critical level. Such discontinuous cavitation characteristics are found to be highly dependent on the relative magnitude of the surface energy to the bulk strain energy.     

循环接触下安定状态问题的研究

基于线性随动强化理论,运用算子分离技术,研究将弹塑性问题转换为弹性问题和残余问题的分析方法,且针对循环载荷接触安定状态,建立了计算机分析程序,该研究能够分析计算弹塑性接触载荷在安定状态下的应力、残余累积应变及残余应力,分析计算了不同载荷的安定状态,并探讨其残余应力场的分析方法。     

Asymptotic Axially Symmetric Deformations for Perfectly Elastic Neo-Hookean and Mooney Materials

For axially symmetric deformations of the perfectly elastic neo-Hookean and Mooney materials, formal series solutions are determined in terms of expansions in appropriate powers of 1/R, where R is the cylindrical polar coordinate for the material coordinates. Remarkably, for both the neo-Hookean and Mooney materials, the first three terms of such expansions can be completely determined analytically in terms of elementary integrals. From the incompressibility condition and the equilibrium equations, the six unknown deformation functions, appearing in the first three terms can be reduced to five formal integrations involving in total seven arbitrary constants A, B, C, D, E, H and k ², and a further five integration constants, making a total of 12 integration constants for the deformation field. The solutions obtained for the neo-Hookean material are applied to the problem of the axial compression of a cylindrical rubber tube which has bonded metal end-plates. The solution so determined is approximate in two senses; namely as an approximate solution of the governing equations and for which the stress free and displacement boundary conditions are satisfied in an average manner only. The resulting load-deflection relation is shown graphically. The solution so determined, although approximate, attempts to solve a problem not previously tackled in the literature.      

Inverse Eigenstrain Analysis of the Effect of Non-uniform Sample Shape on the Residual Stress Due to Shot Peening

This paper deals with the analysis of elastic strain and eigenstrain in non-uniformly shaped shot-peened 17-4PH stainless steel samples. Based on residual strain measurements by synchrotron X-ray diffraction, the finite element (FE) models are established for the inverse problem of eigenstrain analysis in slice conical sample. The eigenstrains obtained in the slice are then implemented into the FE model of the solid conical sample. It is found that the dependence of elastic strain distributions on the peening intensity and sample shape/thickness could be elucidated via the understanding of underlying permanent strain, or eigenstrain. The effect of the peening process is therefore best described in terms of the induced eigenstrain. The proposed framework is useful for the predictive modelling of residual stresses in non-uniformly shaped shot-peened materials, in that it allows efficient reconstruction of complete residual stress states. In addition, it provides an excellent basis for developing predictive tools for in service performance and design optimisation.     

On Non-Symmetric Deformations of Neo-Hookean Solids

Deformations possible (i.e., those satisfying the governing three-dimensional equations of equilibrium and the incompressibility constraint) within a class of non-symmetric deformations for a neo-Hookean nonlinearly elastic body were determined in [1], where it was found that only three special cases of the class of deformation fields considered could be solutions. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation contained within a family of universal deformations. In this paper, the results reported in [1] are shown to hold for a substantially broadened deformation field. This revised version was published online in July 2006 with corrections to the Cover Date.     

Constitutive branching analysis of cylindrical bodies under in-plane equibiaxial dead-load tractions

Finite homogeneous deformations of hyperelastic cylindrical bodies subjected to in-plane equibiaxial dead-load tractions are analyzed. Four basic equilibrium problems are formulated considering incompressible and compressible isotropic bodies under plane stress and plane deformation condition. Depending on the form of the stored energy function, these plane problems, in addition to the obvious symmetric solutions, may admit asymmetric solutions. In other words, the body may assume an equilibrium configuration characterized by two unequal in-plane principal stretches corresponding to equal external forces. In this paper, a mathematical condition, in terms of the principal invariants, governing the global development of the asymmetric deformation branches is obtained and examined in detail with regard to different choices of the stored energy function. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. With reference to neo-Hookean, Mooney-Rivlin and Ogden-Ball materials, a broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. Finally, using the energy criterion, a number of considerations are put forward about the stability of the computed solutions.     

考虑塑性和刚度不匹配的焊接残余应力估算

现有残余应力计算方法未能考虑材料塑性变形和焊接接头刚度不匹配的影响,使得焊接残余应力计算结果和实际残余应力存在较大偏差.在2219-T87铝合金钨极氩弧焊焊接头残余应力测试基础上,提出一种基于非线性有限元和材料弹性模量分区的残余应力—释放应变曲线的残余应力计算方法,研究了材料塑性变形和接头刚度不匹配对焊接残余应力计算的影响.结果表明,焊接接头中非均质材料塑性不匹配可以引起对于残余应力计算的较大误差;材料塑性变形对残余应力的影响大于接头刚度不匹配对残余应力的影响.所提出方法修正了传统方法在焊接接头的残余应力计算中由于未考虑接头非均质材料塑性不匹配而引起的误差.     

A simplified viscoelastic constitutive equation applied to a dynamic finite deformation problem

A single integral type of constitutive equation for finite viscoelastic deformation is proposed. A special case, which is a viscoelastic generalization of the constitutive equation for a neo-Hookean elastic solid, is used to consider the finite deformation problem of shock wave propagation resulting from the sudden application of compressive force at the end of a semi-infinite viscoelastic bar. An approximate method is used to determine the shock front path and shock strength when the viscoelastic dissipative effect is small.