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     

Torsional impact response of a penny-shaped interface crack in a layered composite

The torsional impact response of a penny-shaped interface crack in a layered composite is considered in this study. The geometry of the composite consists of two bonded dissimilar elastic layers which are sandwiched between two half-spaces made of a different material. 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 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 geometry parameters is discussed.     

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.     

Torsional impact response of a penny-shaped crack lying on a bimaterial interface

The torsional impact response of a penny-shaped crack lying on a bimaterial interface is considered in this study. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution to the dual integral equations is expressed in terms of a Fredholm integral equation of the second kind with a finite integral kernel. 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 and material constants is discussed.     

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.     

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.     

Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

Impact response of a finite crack in an orthotropic piezoelectric ceramic

Summary The transient dynamic stress intensity factor and dynamic energy release rate were determined for a cracked piezoelectric ceramic under normal impact in this study. A plane step pulse strikes the crack and stress wave diffraction takes place. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used to compute the values of the dynamic stress intensity factor and the dynamic energy release rate for some piezoelectric ceramics, and the results are graphed to display the electroelastic interactions.     

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.     

Effect of Couple-Stresses on the Transient Dynamic Stress Intensity Factors for a Crack in an Infinite Elastic Medium Under an Impact Stress Wave

This study applies linearized couple-stress theory to evaluate the dynamic stresses around a crack in an infinite elastic medium that is subjected to an incoming shock stress wave impinging normal to the crack. The boundary conditions with respect to the crack are reduced to dual integral equations using a Fourier transform in the Laplace domain. To solve these equations, the differences in the displacement and rotation at the crack are expanded by a series of functions that are zero-valued outside the crack in the Laplace domain.     

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.     

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 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.     

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.     

Impact behavior of an inclined edge crack in a layered medium with a graded nonhomogeneous interfacial zone: antiplane deformation

Summary The impact response of an inclined edge crack in a layered medium with a functionally graded interfacial zone is investigated under the state of antiplane deformation. The interfacial zone is modeled by a nonhomogeneous interlayer having the power-law variations of shear modulus and mass density between the coating and the substrate of dissimilar homogeneous properties. Based on the Laplace and Fourier integral transform technique and the coordinate transformations of basic field variables, the transient crack problem is reduced to the solution of a singular integral equation with a generalized Cauchy kernel in the Laplace transform domain. The crack-tip response in the physical domain is recovered through the inverse Laplace transform to evaluate the dynamic mode III stress intensity factors as functions of time. The peak values of the dynamic stress intensity factors are further obtained versus the crack orientation angle, addressing the effects of crack obliquity on the overshoot characteristics of the transient crack-tip behavior for various combinations of material and geometric parameters of the layered medium.     

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。      

Dynamic stress intensity factors around a rectangular crack in an infinite plate under impact load

A three-dimensional solution is presented for the transient response of an infinite plate which contains a rectangular crack. The Laplace and Fourier transforms are used to reduce the problem to a pair of dual integral equations. These equations are solved with the series expansion method. The stress intensity factors are defined in the Laplace transform domain, and they are inverted numerically in the physical space.     

Displacement Discontinuity Method for Fracture Mechanics Analysis of Reissner Plates: Static and Dynamic

This paper is concerned with the displacement discontinuity method applied to the shear deformable plates (Reissner’s and Mindlin’s theories) with cracks subjected to static and dynamic loads. Fundamental solutions of dislocation are derived using the Fourier transform method and the Laplace transformation technique. Boundary integral equations are presented in terms of rotations/displacement on the crack surfaces. The Chebyshev polynomials of the second kind are used to evaluate the integral equations with hypersingular kernels on the crack boundaries and determine the stress intensity factors at the crack tips. Comparisons are made with other numerical solutions to demonstrate the proposed method is accurate both for static and dynamic problems.     

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.     

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.