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.     

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。      

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.     

Impact response of a crack in a semi-infinite body with a surface layer under longitudinal shear

Summary The impact response of a crack in a semi-infinite body with a surface layer which is subjected to antiplane shear deformation is considered in this study. The semi-infinite body contains a crack near an interface. Using Laplace and Fourier transforms, the case of a crack perpendicular to the interface is reduced to a set of triple integral equations in the Laplace transform plane. The solution to the triple integral equations is then expressed in terms of a singular integral equation of the first kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factors at the crack tips are obtained for several values of time, material constants, and geometrical parameters.With 8 Figures     

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.     

Dynamic response for non-homogeneous piezoelectric medium with multiple cracks

A general procedure to analyze the dynamic response of non-homogeneous piezoelectric medium containing some non-collinear cracks is developed. It is assumed that all the material properties only depend on the coordinates y (along the thickness direction). The assumption is made that the non-homogeneous medium is composed of numerous laminae with their surfaces perpendicular to the thick direction. The solution method is based upon the Fourier and Laplace transforms to reduce the boundary value problem to a system of generalized singularity equations in the Laplace transform domain. The singular integral equations for the problem are derived and numerically solved by weight residual value method. The time-dependent full field solutions are obtained in the time domain. As numerical illustration, the stress and electric displacement intensity factors for a three-layer plate specimen with two cracks are presented. It is found that the stress and electric fields are coupled in the crack plane ahead of the crack tip for non-homogenous piezoelectric materials.     

Dynamic stress intensity factors studied by boundary integro-differential equations

The boundary integro-differential equation method is illustrated by two numerical examples concerning the study of the dynamic stress intensity factor around a penny-shaped crack in an infinite elastic body. Harmonic and impact load on the crack surface has been considered. Applying the Laplace transform with respect to time to the governing equations of motion the problem is solved in the transformed domain by the boundary integro-differential equations. The Laplace transformed general transient problem can be used to solve the steady-state problem as a special case where no numerical inversion is involved.     

Modeling method for the crack problem of a functionally graded interfacial zone with arbitrary material properties

A new multi-layered model is developed for the fracture analysis of a functionally graded interfacial zone with arbitrary material properties. It is assumed that the interfacial zone is divided into sub-layers with the material properties of each sub-layer varying in a power-law function. The model is used to study the crack problem in the functionally graded interfacial zone between two homogeneous half-planes under a dynamic anti-plane load. Using Fourier–Laplace transforms and the transfer matrix method, the mixed boundary value problem is reduced to a Cauchy singular integral equation, which is solved numerically in the Laplace transform domain. Laplace numerical inversion transform is employed to obtain the stress intensity factors. The results show that the new model is general and effective for the crack problem of the functionally graded interfacial zone with arbitrary properties.     

Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

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。     

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.     

Sudden twisting of an external circular crack in an infinite medium with a cylindrical inclusion

In this paper the torsional impact response of an external circular crack in an infinite medium bonded to a cylindrical inclusion has been investigated. The infinite medium and cylindrical inclusion are assumed to be of different homogeneous isotropie elastic materials. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the results are expressed in terms of a Fredhol integral equation of the second kind. By solving Fredholm integral equation of the second kind the numerical results for the dynamic stress-intensity factor are obtained which measure the load transmission on the crack.     

Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.     

Dynamic Stress Intensity Factor of Interfacial Finite Cracks in Orthotropic Materials

The elastodynamic response of an infinite non-homogeneous orthotropic material with an interfacial finite crack under distributed normal and shear impact loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some materials are obtained. Interfacial cracks between two different materials and between two pieces of the same material but different fiber orientation are considered. Bimaterial formulation of a crack problem is shown to converge to the mono-material formulation, derived independently, in the limiting case when both materials are the same.     

Singular integral equation-boundary element method to solve the Saint-Venant bending problem of cracked cylinders

Using the single crack solutions and the regular solution of harmonic function, the bending problem of a cracked cylinder is reduced to solving two sets of mixed-type integral equations which can be solved by combining the numerical method of the singular integral equation with the boundary element method. Several numerical examples are calculated and the stress intensity factors are obtained.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

Fracture analysis of a functionally graded coating-substrate structure with a crack perpendicular to the interface - Part II: Transient problem

In Part I of this paper, the static problem of a functionally graded coating-substrate system with an internal or edge crack perpendicular to the interface has been studied. In this part, the transient response of the structure is considered under an in-plane impact. Laplace and Fourier transforms are applied to reduce the mixed boundary value problem to a singular integral equation which is solved in the Laplace domain numerically. The dynamic stress intensity factors (DSIFs) are obtained by numerical Laplace inversion technique. The influences of material constants and geometry parameters on the dynamic stress intensity factors are studied. It is found that the DSIFs for an internal crack rise rapidly to a peak and then tend to the steady value without obvious oscillations, but the DSIFs for an edge crack have more obvious oscillations after rising to a peak with the increasing of the nonhomogeneity constant.     

Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads

The problem of a semi-infinite impermeable mode-III crack in a piezoelectric material is considered under the action of impact loads. For the case when a pair of concentrated anti-plane impact loads and electric displacements are exerted symmetrically on the upper and lower surfaces of the crack, the asymptotic electroelastic field ahead of the crack tip is determined in explicit form. The dynamic intensity factors of electroelastic field and dynamic mechanical strain energy release rate are obtained. The obtained results can be taken as fundamental solutions, from which general results may directly be evaluated by integration. The method adopted is to reduce the mixed initial-boundary value problem, by using the Laplace and Fourier transforms, into two simultaneous dual integral equations. One may be converted into an Abel's integral equation and the other into a singular integral equation with Cauchy kernel, and the solutions of both equations can be determined in closed-form, respectively. For some particular cases, the present results reduce to the previous results.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.     

Dynamic fracture behavior of an internal interfacial crack between two dissimilar magneto-electro-elastic plates

In this paper, the anti-plane problem for an interfacial crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical impact loadings is investigated. Four kinds of crack surface conditions are adopted: magneto-electrically impermeable (Case 1), magnetically impermeable and electrically permeable (Case 2), magnetically permeable and electrically impermeable (Case 3), and magneto-electrically permeable (Case 4). The position of the interfacial crack is arbitrary. The Laplace transform and finite Fourier transform techniques are employed to reduce the mixed boundary-value problem to triple trigonometric series equations in the Laplace transform domain. Then the dislocation density functions and proper replacements of the variables are introduced to reduce the series equations to a standard Cauchy singular integral equation of the first kind. The resulting integral equation together with the corresponding single-valued condition is approximated as a system of linear algebra equations, which can easily be solved. Field intensity factors and energy release rates are determined and discussed. The effects of loading combination parameters on dynamic energy release rate are plotted for Cases 1-3. On the other hand, since the magneto-electrically permeable condition is perhaps more physically reasonable for type III crack, the effect of the crack configuration on the dynamic fracture behavior of the crack tips is studied in detail for Case 4. The results could be useful for the design of multilayered magneto-electro-elastic structures and devices.