Solutions for a viscoelastic axisymmetric plane problem involving time-dependent boundary regions under mixed boundary condition

The stress and displacement fields for a viscoelastic axisymmetric plane problem involving time-dependent boundary regions under mixed boundary condition are presented in this paper. The viscoelastic fields are determined by solving two unknown functions; one is governed by a resulting second kind Volterra integral equation from the stress condition at the outer radius of an annular region, which is dependent on the other by the displacement condition at the inner radius. The integral equation has an analytical solution for the case of an infinite region with large enough outer radius while it can be solved using a numerical integral method for the case of a finite region. Numerical examples for the Boltzmann viscoelastic model are given, and the responses of the displacement and stresses are presented in detail to illustrate the effects of velocity of change in the inner radius and the magnitude of the outer radius. Meanwhile, the effect of the aspect ratio (void concentration parameter) on the viscoelastic fields is presented. When the ratio of the outer radius to the inner radius is large enough, the responses can be evaluated by directly adopting the resulting analytical solution to avoid a complex numerical procedure. The resulting solutions and computational results are helpful to a better understanding of mechanical behaviors for (large) excavation or finite void growth in viscoelastic media.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

Bifurcation of cavitation solutions for incompressible transversely isotropic hyper-elastic materials

In this paper, the bifurcation problem of void formation and growth in a solid circular cylinder, composed of an incompressible, transversely isotropic hyper-elastic material, under a uniform radial tensile boundary dead load and an axial stretch is examined. At first, the deformation of the cylinder, containing an undetermined parameter-the void radius, is given by using the condition of incompressibility of the material. Then the exact analytic formulas to determine the critical load and the bifurcation values for the parameter are obtained by solving the differential equation for the deformation function. Thus, an analytic solution for bifurcation problems in incompressible anisotropic hyper-elastic materials is obtained. The solution depends on the degree of anisotropy of the material. It shows that the bifurcation may occur locally to the right or to the left, depending on the degree of anisotropy, and the condition for the bifurcation to the right or to the left is discussed. The stress distributions subsequent to the cavitation are given and the jumping and concentration of stresses are discussed. The stability of solutions is discussed through comparison of the associated potential energies. The bifurcation to the left is a `snap cavitation'. The growth of a pre-existing void in the cylinder is also observed. The results for a similar problem in three dimensions were obtained by Polignone and Horgan.     

饱和多孔介质一维瞬态波动问题的解析分析

采用基于混合物理论的多孔介质模型,提出了饱和多孔介质一维动力响应的初边值问题。利用拉氏变换和卷积定理,分别得到了边界自由排水时在任意应力边界条件和任意位移边界条件下瞬态波动过程的解析表达。几种典型的数值算例同时给出了两类边界条件下瞬态波动过程中多孔固体的位移场、应力场和孔隙流体的速度场、压力场。结果表明,饱和多孔介质的波动过程是多孔固体和孔隙流体中以同一速度传播的两种波动的耦合过程,时效特性分析也揭示了饱和多孔介质固有的表观粘弹性性质。     

State space approach to one-dimensional thermal shock problem for a semi-infinite piezoelectric rod

The theory of generalized thermoelasticity, based on the theory of Lord and Shulman with one relaxation time, is used to solve a boundary value problem of one-dimensional semi-infinite piezoelectric rod with its left boundary subjected to a sudden heat. The governing partial differential equations are solved in the Laplace transform domain by the state space approach of the modern control theory. Approximate small-time analytical solutions to stress, displacement and temperature are obtained by means of the Laplace transform and inverse transform. It is found that there are two discontinuous points in both stress and temperature solutions. Numerical calculation for stress, displacement and temperature is carried out and displayed graphically.     

饱和地基上弹性圆板的固结位移

 采用Laplace变换及Hankel变换研究弹性圆板在饱和地基上的固结位移．在Laplace变换域上建立了求解弹性圆板固结位移的第二类Fredholm积分方程，由数值反变换获得时域解，给出了对工程实践有参考价值的计算结果．计算结果表明，弹性板的固结位移随饱和土的排水泊松比的减少而增加，而弹性板的初始位移随饱和土的不排水泊松比的减少而增加．     

压电层合纤维在动载荷作用下的应力和电势的瞬态响应

利用压电层合纤维层间的连续性条件和外表面的应力边界条件,通过有限Hankel积分变换和Laplace变换求解压电层合纤维的力电耦合的动力学方程,给出了压电层合纤维在动载荷作用下的应力和电势瞬态响应的解析解。从解析表达式和算例,可以得到在径向动载荷下层合压电纤维的应力和电势的瞬态响应历程和实心处动态集中效应的规律。     

Antiplane electro‐mechanical field of a piezoelectric finite wedge under a pair of concentrated forces and free charges

Abstract

This paper presents general antiplane electro‐mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as traction free and insulated. Using finite Mellin transform methods, the stress and electrical displacement in all fields of the piezoelectric finite wedge are derived analytically. Singularity orders and intensity factors of stress and electrical displacement can be obtained too. After being reduced to a problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.     

Analytical solutions describing the consolidation of a multi-layered soil under circular loading

Analytical solutions describing the consolidation of a multi-layered soil under circular loading are presented. From the governing equations of saturated poroelastic soil in a cylindrical coordinate system, the eighth-order state-space equation of consolidation is obtained by eliminating the variation of time t using the Laplace transform together with the technique of Fourier expansions with respect to the coordinate θ and the Hankel transform with respect to coordinate r. The solution of the eighth-order state-space equation is derived directly by using the Laplace transform and its inversion of the z-domain. Based on the continuity between layers and the boundary conditions, the transfer-matrix method is utilized to derive the solutions for the consolidation of a multi-layered soil under circular loading in the transformed domain. By the inversion of the Laplace transform and the Hankel transform, the analytical solutions in the physical domain are obtained. A numerical analysis based on the solutions is carried out by a corresponding program.     

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.     

反平面冲击荷载作用下圆柱形洞室的动力响应

对无限弹性土体内圆柱形洞室在反平面冲击荷载的作用下的瞬态响应进行了探讨。用残余变量的方法结合拉普拉斯变换及其逆变换,在频域内的得到土体位移和应力一般表达式,并采用拉普拉斯逆变换的数值方法,给出了问题的数值解。在时域内分析了无限弹性土体内圆柱形孔洞在沿轴向方向的冲击荷载作用下的土体动力响应的变化规律,并将计算结果与静力情况下的作了比较：反平面荷载作用后,波向外发散传播,并沿半径方向衰减,衰减速度较静力情况下的衰减要慢,且距离波源越远二者差别越大;波到达后,该点土体的应力和位移均发生与外荷载相同的三角形变化：先增大,随后减小最后保持为零不变。     

离散多层圆筒在热冲击载荷下的弹性动力响应

离散多层圆筒由薄内筒和倾角错绕的钢带层组成,具有制造简便、成本低等优点。预测筒体在热冲击载荷下的热应力对强度设计和安全操作具有重要的应用价值。该文首次研究了离散多层圆筒在热冲击载荷作用下的热弹性动态响应。将内筒和钢带层的径向位移分别分解为满足给定应力边界条件的准静态解和满足初始条件的动态解,准静态解通过齐次线性方法确定,热弹性动态解通过有限Hankel积分变换和Laplace变换确定。根据内外层界面处位移连续条件,得到层间压力关于时间的第二类Volterra积分方程,利用Hermit二次三项式插值方法可求得该层间应力。最后将离散多层圆筒的热弹性动力响应与单层厚壁圆筒的响应进行了比较,并分析了钢带缠绕倾角和材料参数对热弹性动力响应的影响。     

INFLUENCE OF SECONDARY VOID DAMAGE IN THE MATRIX MATERIAL AROUND VOIDS

Abstract— The mechanism of ductile damage caused by secondary void damage in the matrix around primary voids is studied by large strain, finite element analysis. A cylinder embedding an initially spherical void, a plane stress cell with a circular void and plane strain cell with a cylindrical or a flat void are analysed under different loading conditions. Secondary voids of smaller scale size nucleate in the strain hardening matrix, according to the requirements of some stress/strain criteria. Their growth and coalescence, handled by the empty element technique, demonstrate distinct mechanisms of damage as circumstances change. The macroscopic stress-strain curves are decomposed and illustrated in the form of the deviatoric and the volumetric parts. Concerning the stress response and the void growth prediction, comparisons are made between the present numerical results and those of previous authors. It is shown that loading condition, void growth history and void shape effect incorporated with the interaction between two generations of voids should be accounted for besides the void volume fraction.     

The symplectic approach for two-dimensional thermo-viscoelastic analysis

This paper presents an exact symplectic approach for two dimensional isotropic viscoelastic solids subjected to external force and temperature boundary conditions. With the use of the state space method and the Laplace transform, all general solutions of the governing equations are obtained analytically. By applying the inverse integral transform, the time domain adjoint symplectic relationships between the general solutions are established. Therefore, the problems of the particular solution and the boundary conditions can be analysed either in the Laplace domain or directly in the time domain. As its applications, the boundary condition problems are discussed in the numerical calculations. The results show that, due to the displacement constraints and the temperature influence, local effects are distinct near the boundary, and the effects decay rapidly with the distance from the boundary.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

Elastic deformation of the rotating functionally graded annular disk with rigid casing

An accurate solution for a rotating functionally graded annular disk is presented. Material properties of the present annular disk are assumed to be graded in the radial direction according to a simple exponential-law distribution. The inner surface of the disk is pure metal whereas the outer surface of the disk is pure ceramic. The boundary condition of rigid casing is considered herein, that is the vanishing of the radial displacement at the outer surface. The boundary condition at the inner surface of the disk is taken to be vanishing either radial displacement or radial stress. Analytical solutions for the elastic deformation of the rotating functionally graded annular disks subjected to these boundary conditions are obtained. Numerical results for radial displacement, circumferential and radial stresses are presented. Comparisons between the different rotating homogeneous and functionally graded annular disks are made at the same angular velocity. The results show that distributions of stresses and displacement through the radial direction of the rotating annular disk vary with different parameters.     

The influence of Maxwell stresses on the fracture mechanics of piezoelectric materials

The influence of Maxwell stresses on the generalized 2D fracture mechanics problem of piezoelectric materials under combined mechanical and electric loads at infinity is studied. The electrically semi-permeable crack boundary condition is adopted in this paper. Based on the Stroh’s formalism, explicit and closed-form solutions of electric displacement inside the crack, stress and electric intensity factors are obtained. Numerical results are also given to discuss the effects of Maxwell stresses on the stress and electric displacement intensity factors when the interior of the crack and the surrounding space at infinity are filled with different dielectric medium. It is found that the stress intensity factor increases rapidly with increasing value of the applied electric displacement load for the case of the dielectric constant of the surrounding at infinity is smaller than that inside the crack. The electric displacement intensity factor always increases as the applied electric loads or the applied mechanical loads increase.     

考虑中间主应力和剪胀特性的深埋圆巷弹塑性应力位移解

平面应变条件下的深埋圆形巷道问题一般忽略中间主应力的影响,但这会与塑性区围岩的实际情况产生较大差异。为了充分考虑中间主应力对深埋圆形巷道的影响,基于平面应变假设与非关联流动法则将Mohr-Coulomb准则精确匹配为Drucker-Prager准则,在此基础上推导了考虑剪胀特性的理想弹塑性材料在塑性阶段的中间主应力表达式,中间主应力与剪胀角密切相关;根据所得的中间主应力表达式结合非关联流动法则,不引入任何假设,得出深埋圆巷塑性区由于剪胀角而发生体变的关系式;进一步推导了考虑中间主应力和剪胀特性的深埋圆形巷道塑性区应力位移解析式,其中径向应力、切向应力及塑性区半径的表达式与卡斯特奈(Kastner)解完全一致,但卡斯特奈(Kastner)解无法得出中间主应力,而新的位移解析式则与以往的文献完全不同;经与以往文献的位移理论解比较分析知,新的位移解答更加合理。因此考虑中间主应力和剪胀特性的解答为深埋圆形巷道的计算与设计提供一定的理论基础。     

Inclusion problem of a two-dimensional finite domain: The shape effect of matrix

With the advance in composite mechanics and micromechanics, there are increasing demands for analytical solutions of inclusion problems in a bounded domain. To echo this need, this study is focused on establishing explicit expressions of elastic fields for a 2D elastic domain containing a circular inclusion at center. Unlike the configuration in the classical Eshelby formulation, the elastic domain in this study is bounded and has shapes other than a circle. To circumvent the mathematical difficulty in solving Green’s function in a finite domain, an approach powered by complex potential method, which has been successfully employed to formulate the elastic fields for inclusion problems where matrix is unbounded or bounded by a circle, is extended to finite domains displaying complicated shapes, particularly, a Pascal’s limaçon and a curved square (an approximation of perfect square) in this study. In order to take advantage of the mathematical simplicity inherent in expressing a circular geometry, conformal mapping is used to transform the complex geometry of the finite domain of interest to a unit circle. The governing complex potentials, which capture the discontinuity on the inclusion–matrix interface due to the uniform eigenstrain within the inclusion, are formulated with the aid of Cauchy integral and then explicitly identified by satisfying the prescribed boundary conditions. In this study, the displacement fields for finite domains bounded by a Pascal’s limaçon and a curved square are obtained based on Dirichlet (displacement) boundary conditions imposed by the far field strain. In addition to asymptotical behaviors, firm agreement is also achieved when the analytical solutions based on complex potentials are compared with the FEM results. Furthermore, inverse of the conformal mapping is discussed here in order to get the explicit expression for elastic fields.     

Time history kernel functions for square lattice

We propose an exact non-reflecting boundary condition for simulations of a square lattice in finite atomic sub-domain. The boundary condition is based on the series expansion of the single source kernel functions. The time history kernel Laplace transform is first derived in a closed form, where the Fourier transform inversion is performed analytically. An efficient numerical procedure is also developed for the inverse Laplace transform. Differences in the finite boundary condition and the infinite boundary approximation elucidate the cause of the long-lasting difficulty of corner effects. Numerical tests verify accuracy of the proposed approach.