Dynamic crack problems in three-dimensional transversely isotropic solids

In this paper, a general boundary element approach for three-dimensional dynamic crack problems in transversely isotropic bodies is presented for the first time. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The procedure is based on the subdomain technique, the displacement integral representation for elastodynamic problems and the expressions of the time-harmonic point load fundamental solution for transversely isotropic media. Numerical results corresponding to cracks under the effects of impinging waves are presented. The accuracy of the present approach for the analysis of dynamic fracture mechanics problems in transversely isotropic solids is shown by comparison of the obtained results with existing solutions.     

Three-dimensional fracture analysis in transversely isotropic solids

In this paper a boundary element formulation for three-dimensional crack problems in transversely isotropic bodies is presented. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The point load fundamental solution for transversely isotropic media is implemented. Numerical solutions to several three-dimensional crack problems are obtained. The accuracy and robustness of the present approach for the analysis of fracture mechanics problems in transversely isotropic bodies are shown by comparison of some of the results obtained with existing analytical solutions. The approach is shown to be a simple and useful tool for the evaluation of stress intensity factors in transversely isotropic media.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

Boundary elements for half‐space problems via fundamental solutions: A three‐dimensional analysis

An efficient solution technique is proposed for the three‐dimensional boundary element modelling of half‐space problems. The proposed technique uses alternative fundamental solutions of the half‐space (Mindlin's solutions for isotropic case) and full‐space (Kelvin's solutions) problems. Three‐dimensional infinite boundary elements are frequently employed when the stresses at the internal points are required to be evaluated. In contrast to the published works, the strongly singular line integrals are avoided in the proposed solution technique, while the discretization of infinite elements is independent of the finite boundary elements. This algorithm also leads to a better numerical accuracy while the computational time is reduced. Illustrative numerical examples for typical isotropic and transversely isotropichalf‐space problems demonstrate the potential applications of the proposed formulations. Incidentally, the results of the illustrative examples also provide a parametric study for the imperfect contact problem. Copyright © 2001 John Wiley & Sons, Ltd.     

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.     

Elastostatic Green’s functions for an arbitrary internal load in a transversely isotropic bi-material full-space

The problem of a full-space which is composed of two half-spaces with different transversely isotropic materials with an internal load at an arbitrary distance from the interface is considered. By virtue of Hu-Nowacki-Lekhnitskii potentials, the equations of equilibrium are uncoupled and solved with the aid of Hankel transform and Fourier decompositions. With the use of the transformed displacement- and stress-potential relations, all responses of the bi-material medium are derived in the form of line integrals. By appropriate limit processes, the solution can be shown to encompass the cases of (i) a homogeneous transversely isotropic full-space, and (ii) a homogeneous transversely isotropic half-space under arbitrary surface load. As the integrals for the displacement- and stress-Green’s functions, for an arbitrary point load can be evaluated explicitly, illustrative results are presented for the fundamental solution under different material anisotropy and relative moduli of the half-spaces and compared with existing solutions.     

Analysis of elliptical cracks perpendicular to the interface of two joined transversely isotropic solids

This paper presents a boundary element analysis of elliptical cracks in two joined transversely isotropic solids. The boundary element method is developed by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of stress intensity factors (SIFs) are obtained by using crack opening displacements. The results of the proposed method compare well with the existing exact solutions for an elliptical crack parallel to the isotropic plane of a transversely isotropic solid of infinite extent. Elliptical cracks perpendicular to the interface of transversely isotropic bi-material solids of either infinite extent or occupying a cubic region are further examined in detail. The crack surfaces are subject to the uniform normal tractions. The stress intensity factor values of the elliptical cracks of the two types are analyzed and compared. Numerical results have shown that the stress intensity factors are strongly affected by the anisotropy and the combination of the two joined solids.     

Boundary integral equation solution of three‐dimensional elastostatic problems in transversely isotropic solids using closed‐form displacement fundamental solutions

The boundary integral equation method is used for the solution of three‐dimensional elastostatic problems in transversely isotropic solids using closed‐form fundamental solutions. The previously published point force solutions for such solids were modified and are presented in a convenient form, especially suitable for use in the boundary integral equation method. The new presentations are used as a basis for accurate numerical computations of all Green's functions necessary in the BEM process without inaccuracy and redundant computations. The validity of the new presentation is shown through three numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.     

横观各向同性结构边界元分析

本文利用边界单元法分析三维横观各向同性结构,引用三个位势函数并利用叠加原理导出了基本解,并且利用基本解的新型结构形式避免了在分界单元计算中所遇到的所谓“奇异积分”的问题。     

Analyzing interaction between coplanar square cracks using an efficient boundary element‐free method

This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Circular crested waves in anisotropic thermoelastic plates bordered with inviscid liquid

The propagation of circularly crested waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides is investigated in the context of conventional coupled thermoelasticity, Lord-Shulman and Green-Lindsay theories of thermoelasticity. Secular equations for circular homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. The special cases such as short wavelength waves, thin plate waves and leaky Lamb waves of the secular equation are also deduced and discussed. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic circular plate of cobalt material bordered with water. The dispersion curves for symmetric and antisymmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The analytical and numerical results are found to be in close agreement.     

A new analysis of the complex two-dimensional multilayered anisotropic soil in time domain

The dynamic analysis of two-dimensionalmultilayered anisotropic soilwith rigid bedrock is studied. An efficient numerical approach named the modified scaled boundary finite element method (SBFEM) is proposed in the time domain. Based on introducing the continued fraction method and auxiliary variables, the time domain solution is obtained. This solution can be applied to the transversely isotropic medium without any difficulty. For the modified SBFEM, the original scaling center is replaced by a scaling line. These characteristics enable the modified SBFEM to model the horizontal layered medium. Three significant technologies have been introduced in the formula derivation and solving process. First, the dual system is used to derive the displacement equation of the modified SBFEM, which is built on a Hamilton system. According to the principle of virtual work, the displacement equation is transformed to the dynamic stiffness equation. Second, the new continued fraction method for the unbounded domain resting on rigid bedrock is proposed. By introducing auxiliary variables, the displacement equation of motion of an unbounded domain is built. Third, it is an extremely important point that the accurate precise time-integration method is first employed to solve the global equation of motion of the modified SBFEM. This numerical integral method can achieve the machine precision. By using this method in solving the equation of motion of the modified SBFEM, an extremely accurate solution can be achieved. Finally, numerical examples validate the accuracy of the new proposed method, especially for the complex inclined model with anisotropic soil.     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

On wave propagation in anisotropic elastic cylinders at nanoscale: surface elasticity and its effect

Material heterogeneity induced by a surface or interface may be neglected at macroscale since the surface-to-volume ratio is usually small. However, its effect can become significant for structures at nanoscale with a large surface-to-volume ratio. In this paper, we incorporate such surface material heterogeneity into wave propagation analysis of a nanosized transversely isotropic cylinder. This is achieved by using the concept of surface elasticity. Instead of directly using the well-known Gurtin–Murdoch (GM) surface elasticity, we develop here another general framework based on a thin layer model. A novel approach based on state-space formalism is used to derive the approximate governing equations. Three different sources of surface effect can be identified in the first-order surface elasticity, i.e., the elasticity effect, the inertia effect and the thickness effect. It is found that the derived theory becomes identical to the GM surface elasticity if the thickness effect is dropped and when the material is isotropic. The axisymmetric wave propagation in a transversely isotropic cylinder with surface effect is then studied, and an exact solution is presented. Numerical results are finally given to show that the surface effect will play a very pronounced role in wave propagation in cylinders at nanoscale.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Analytical layer-element approach for wave propagation of transversely isotropic pavement

Asphalt pavements have been recognised as transversely isotropic multi-layered structures. In this paper, an analytical layer-element approach is utlised to solve the wave propagation of transversely isotropic multi-layered pavement structures under the falling weight deflectometer impact load. After the application of Fourier–Hankel transform, the Navier's equation for transversely isotropic layer by impulsive force are solved analytically. The global stiffness matrix equation of multilayered structures is further obtained by assembling the interrelated layer-elements, and the actual solution is achieved by numerical inversion of the Fourier–Hankel transform after the solution in the transformed domain is obtained. The layer-element of a single layer and the global stiffness matrix only contain negative exponential functions, which leads to a considerable improvement in computation ef?ciency and stability. Numerical examples are presented to demonstrate the accuracy of this method and to inversitgate the influence of the properties of transversely isotropic elastic materials on the load-displacement responses.     

The general solution of three-dimensional problems in magnetoelectroelastic media

The general solution of three-dimensional problems in transversely isotropic magnetoelectroelastic media is obtained through five newly introduced potential functions. The displacements, electric potential, magnetic potential, stresses, electric displacements and magnetic inductions can all be expressed concisely in terms of the five potential functions, all of which are harmonic. The derived general solution is then applied to find the fundamental solution for a generalized dislocation and also to derive Green's functions for a half-space magnetoelectroelastic solid.     

Thermoelastic Lamb waves in a transversely isotropic plate bordered with layers of inviscid liquid

Analysis for the propagation of thermoelastic waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides, is investigated in the context of coupled theory of thermoelasticity. Secular equations for homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion of wave propagation and occurs along an in-plane axis of symmetry. The special cases, such as short wavelength waves and thin plate waves of the secular equations are also discussed. The secular equations for leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic plate of zinc material bordered with water. The dispersion curves for symmetric and anti-symmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement.     

Three‐dimensional fundamental solution for transversely isotropic piezothermoelastic material

Fundamental solutions play an important role in electroelastic analyses and numerical methods of piezoelectric material. However, most works available on this topic are on the case of identical temperature. We use the compact mono‐harmonic general solutions of transversely isotropic piezothermoelastic material to construct the three‐dimensional fundamental solution of a steady point heat source in an infinite piezothermoelastic material by four newly introduced mono‐harmonic functions. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for cadmium selenide are given graphically by contours. Copyright © 2008 John Wiley & Sons, Ltd.     

关于横观各向同性体在其各向同性面内弹性问题的一个常数

研究了在平面应力和平面应变情况下, 横观各向同性材料在其各向同性面内的应力2应变关系以及用位移表达的平衡方程可以被表示成与各向同性材料完全相同的形式。这种等同关系是通过引入一个与横观各向同性材料的泊松比有关的常数得到的。该常数的引入消除了已发表的文献中求解横观各向同性材料平面问题时出现的矛盾。这个常数的引入也便于正确计算单向纤维增强复合材料的横向切变模量。