The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Modeling of surface and subsurface crack behavior under contact load in the presence of lubricant

A new mathematical model for lubricated elastic solids weakened by cracks is proposed. Surface and subsurface cracks are taken into account, and the interaction of lubricant with elastic solids within cavities of surface cracks is regarded as the most interesting aspect of the problem. The boundary conditions characterizing the behavior of lubricant within crack cavities such as pressure rise in crack cavities fully filled with lubricant as well as other boundary and additional conditions are derived. The problem is reduced to a system of integro-differential equations with nonlinear boundary conditions in the form of alternating equations and inequalities. A new iterative numerical method is developed for solution of the proposed problem. The method guarantees conservation of lubricant volumes trapped within closed crack cavities and allows for all three functions (normal and tangential displacement jumps and normal stress applied to crack faces) characterizing the problem solution to be determined simultaneously. Examples of numerical results for surface and subsurface cracks are presented and numerical and asymptotic results for small subsurface cracks are compared to each other. The numerical analysis indicates that depending on a surface crack orientation its normal stress intensity factor may be two or more orders of magnitude higher than the one for a similar subsurface one.     

Impact response of a cracked soft ferromagnetic medium

A solution is given for the problem of an infinite soft ferromagnetic solid containing a central crack subjected to normal impact load. The solid is permeated by a uniform magnetostatic field normal to the crack surface. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of integral equations in the Laplace transformed plane. A numerical Laplace inversion technique is used to compute the values of the dynamic stress-intensity factor, and the results are compared with the corresponding elastodynamic values to reveal the influence of magnetic field on the dynamic stress-intensity factor. The dynamic stress intensity factor is found to increase with increasing values of the magnetic field.With 4 Figures     

Boundary element analysis of dynamic coupled thermoelasticity problems

Some possibility of numerical analysis of coupled dynamic problems of linear elastic heat conductors on classical thermoelasticity theory by using the boundary element method is shown in this paper. The boundary integral equation formulation and its numerical implementation of the two-dimensional problem are developed in the manner by the newly derived fundamental solution for the coupled equations of elliptic type in the transformed space and the numerical inversion of Laplace transformation. The boundary element unsteady solutions of the first and second Danilovskaya problems and the Sternberg and Chakravorty problem in the half-space are demonstrated through comparison with the existing solutions.     

Three-dimensional dynamic fracture analysis with the dual reciprocity method in Laplace domain

In this paper, the dual reciprocity boundary element method in the Laplace domain has been developed for the analysis of three-dimensional elastodynamic fracture mechanics mixed-mode problems. The boundary element method is used to calculate the unknowns of transformed boundary displacement and traction and the domain integrals in the elastodynamic equation are transformed into boundary integrals by the use of the dual reciprocity method. The transformed dynamic stress intensity factors are determined by the crack opening displacement (COD) directly in the Laplace domain. By using Durbin's inversion technique, the dynamic stress intensity factors in the time domain are obtained. Several numerical examples are presented to demonstrate the good agreement with existing solutions.     

A fast multipole boundary element method for 2D viscoelastic problems

A fast multipole formulation for 2D linear viscoelastic problems is presented in this paper by incorporating the elastic–viscoelastic correspondence principle. Systems of multipole expansion equations are formed and solved analytically in Laplace transform domain. Three commonly used viscoelastic models are introduced to characterize the time-dependent behavior of the materials. Since the transformed multipole formulations are identical to those for the 2D elastic problems, it is quite easy to implement the 2D viscoelastic fast multipole boundary element method. Besides, all the integrals are evaluated analytically, leading to highly accurate results and fast convergence of the numerical scheme. Several numerical examples, including planar viscoelastic composites with single inclusion or randomly distributed multi-inclusions, as well as the problem of a crack in a pressured viscoelastic plane, are presented. The results are verified by comparison with the developed analytical solutions to illustrate the accuracy and efficiency of the approach.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

无限长条板中弹性与粘弹性界面裂纹尖端场

研究无限长条板中粘弹性-弹性界面Griffith裂纹在 Ⅰ 型突加载荷作用下,裂纹尖端动态应力强度因子的时间响应。利用积分变换方法、Fourier和Laplace变换,分别推导出了弹性和粘弹性问题的控制方程组;引入位错密度函数,并结合边界条件,导出了反映裂纹尖端奇异性的Cauchy型奇异积分方程组,运用Chebyshev正交多项式化奇异积分方程组为代数方程组,用配点法进行求解;最后用Laplace积分变换数值反演方法,将拉氏域内的解反演到时间域内,求得动态应力强度因子的时间响应,并对材料参数的影响进行了分析。结果表明,剪切松弛参量对 Ⅰ 型动应力强度因子的影响小于对 Ⅱ 型的影响,而膨胀松弛参量对 Ⅰ 型动应力强度因子的影响大于对 Ⅱ 型的影响。      

Weight Function for a Crack in a Two-dimensional Orthotropic Medium Under Impact Shear Loading

The paper examines the elastodynamic response of an infinite two-dimensional orthotr- opic medium containing a central crack under impact shear loading. Laplace and Fourier integral transforms are employed to reduce the problem to a pair of dual integral equations in the Laplace transformed plane. These equations are reduced to integral differential equations, which have been solved in the low frequency domain by iterations. To determine time dependence, these equations are inverted to yield the dynamic stress intensity factor (SIF) for shear point force loading that corresponds to the weight function for the crack under shear loading. It is used to derive SIF for polynomial loading.     

Indirect Trefftz method for boundary value problem of Poisson equation

Trefftz method is the boundary-type solution procedure using regular T-complete functions satisfying the governing equation. Until now, it has been mainly applied to numerical analyses of the problems governed with the homogeneous differential equations such as the two- and three-dimensional Laplace problems and the two-dimensional elastic problem without body forces. On the other hand, this paper describes the application of the indirect Trefftz method to the solution of the boundary value problems of the two-dimensional Poisson equation. Since the Poisson equation has an inhomogeneous term, it is generally difficult to determine the T-complete function satisfying the governing equation. In this paper, the inhomogeneous term containing an unknown function is approximated by a polynomial in the Cartesian coordinates to determine the particular solutions related to the inhomogeneous term. Then, the boundary value problem of the Poisson equation is transformed to that of the Laplace equation by using the particular solution. Once the boundary value problem of the Poisson equation is solved according to the ordinary Trefftz formulation, the solution of the boundary value problem of the Poisson equation is estimated from the solution of the Laplace equation and the particular solution. The unknown parameters included in the particular solution are determined by the iterative process. The present scheme is applied to some examples in order to examine the numerical properties.     

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.     

Boundary element analysis of stress intensity factor KI in some two-dimensional dynamic thermoelastic problems

This paper presents a numerical technique for the calculation of stress intensity factor as a function of time for coupled thermoelastic problems. In this task, effect of inertia term considering coupled theory of thermoelasticity is investigated and its importance is shown.A boundary element method using Laplace transform in time-domain is developed for the analysis of fracture mechanic considering dynamic coupled thermoelasticity problems in two-dimensional finite domain. The Laplace transform method is applied to the time-domain and the resulting equations in the transformed field are discretized using boundary element method. Actual physical quantities in time-domain is obtained, using the numerical inversion of the Laplace transform method.The singular behavior of the temperature and stress fields in the vicinity of the crack tip is modeled by quarter-point elements. Thermal dynamic stress intensity factor for mode I is evaluated using J-integral method. By using J-integral method effects of inertia term and other terms such as strain energy on stress intensity factor may be investigated separately and their importance may be shown. The accuracy of the method is investigated through comparison of the results with the available data in literature.     

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.     

Axisymmetric elastodynamic response of a flat annular crack to normal impact waves

The axisymmetric response of a flat annular crack in an infinite medium subjected to normal impact load is investigated in this study. A step stress is applied to the crack surface. The singular solution is equivalent to solutions of the problem of diffraction of normally incident tension wave by a flat annular crack, and the problem of the sudden appearance of a flat annular crack in a uniform tensile stress field. Laplace and Hankel transforms are used to reduce the problem to the solution of a set of triple integral equations in the Laplace transform domain. These equations are solved by using a integral transform technique and the result is expressed in terms of a singular integral equation of the first kind with the kernel which is improved by means of a contour integration on the Riemann surface. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results of the dynamic stress intensity factor are obtained to show the influence of inertia, the ratio of the inner radius to the outer one and Poisson's ratio on the load transmission to the crack tip.     

Impact response of an anti-plane crack in a FGPM bonded to a homogeneous piezoelectric strip

A finite crack under transient anti-plane shear loads in a functionally graded piezoelectric material (FGPM) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential functions along the thickness of the strip, and that the two layered strips is under combined anti-plane shear mechanical and in-plane electrical impact loads. The analysis is conducted on the electrically unified crack boundary condition. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Fredholm integral equations of the second kind in the Laplace transform domain. Then, a numerical Laplace inversion is performed and the dynamic intensities are obtained as functions of time and geometric parameters, which are displayed graphically.     

Impact response of a strip composite with a finite crack

The dynamic response of a central crack in a strip composite under normal impact is analyzed. The crack is oriented normally to the interfaces. Laplace and Fourier transform techniques are used to reduce the elastodynamic problem to a pair of dual integral equations. The integral equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometrical parameters is discussed.     

加层电磁弹性材料界面裂纹瞬态响应

研究加层电磁弹性材料界面裂纹在反平面剪切冲击载荷和面内电磁冲击载荷作用下的动态响应问题。假设裂纹面是电磁不导通的。采用Laplace变换、Fourier变换和位错密度函数将混合边值问题转化为求解Laplace域内Cauchy奇异积分方程。讨论了磁冲击载荷、电冲击载荷、材料参数及加层厚度对能量释放率的影响。该问题的解有助于分析含裂纹电磁弹性材料的动态断裂特性。     

Response of gradient-viscoelastic bar to static and dynamic axial load

Summary. The response of a bar to static or dynamic axial load is studied analytically on the basis of a simple linear theory of gradient viscoelasticity. The governing equations of axial equilibrium and motion are first obtained by combining the basic equations. They are also obtained by a variational statement, which provides in addition all possible boundary conditions. A correspondence principle between the gradient elastic and gradient viscoelastic formulation and solution is established. Thus, the Laplace transformed with respect to time viscoelastic solution is obtained from the corresponding elastic one by simply replacing the elastic modulus by its Laplace transform times the Laplace transform parameter. The time domain response is finally obtained by a numerical inversion of the transformed solution. Two boundary value problems, one quasi-static and one dynamic, are studied and the gradient viscoelasticity effect on the solutions is assessed.     

Inverse fracture problems in piezoelectric solids by local integral equation method

The meshless local Petrov–Galerkin (MLPG) method is used to solve the inverse fracture problems in two-dimensional (2D) piezoelectric body. Electrical boundary conditions on the crack surfaces are not specified due to unknown dielectric permittivity of the medium inside the crack. Both stationary and transient dynamic boundary conditions are considered here. The analyzed domain is covered by small circular subdomains surrounding nodes spread randomly over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIE) on the boundaries of the chosen subdomains. The Laplace-transform technique is applied to eliminate the time variation in the governing equation. The local integral equations are nonsingular and take a very simple form. The spatial variation of the Laplace transforms of displacements and electrical potential are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. The singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion to retrieve the time-dependent solutions.     

Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack

In this paper the plane thermo-mechanical behavior of a crack in a viscoelastic functionally graded materials (FGMs) coating with arbitrary material properties bonded to a homogeneous substrate is studied. In order to avoid the complex forms that describe the viscoelastic properties of FGMs, a multi-layered model for the FGMs coating is developed. The compliance and thermal conductivity in the multi-layered model linearly vary in each layer. In this mixed boundary value problem, the system is reduced to singular integral equations and solved numerically with the Lobatto-Chebyshev collocation technique. Using the correspondence principle and Laplace transform, the problem of an interface crack between a homogeneous substrate and a viscoelastic FGMs is solved. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the multi-layered model. The numerical results confirm that the fracture toughness of materials can be greatly improved by the graded variation of material parameters. It is also confirmed that the specific variation of material parameters greatly influences the fracture behavior of viscoelastic FGMs coating.