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.     

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.     

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     

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.     

Elastodynamic response of a finite strip with two coplanar cracks under impact load

The elastodynamic response of two coplanar Griffith cracks in a finite elastic strip under in-plane compression and anti-plane shear impact is considered in this paper. Laplace and Fourier transforms are used to reduce two mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform plane, which are solved numerically. The elastodynamic stress-intensity factors are obtained as functions of time and geometry parameters.     

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.     

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.     

Diffraction of transient horizontal shear waves by a finite crack at the interface of two bonded dissimilar elastic solids

Scattering of transient horizontal shear waves by a finite crack located at the interface of two bonded dissimilar elastic solids is investigated in this study. Laplace and Fourier transform technique is used to reduce the problem to a pair of dual integral equations. The solution of the dual integral equation is expressed in terms of the Fredholm integral equation of the second kind having the kernel of a finite integration. Dynamic stress intensity factor is obtained as a function of the material and geometric properties and time.     

Crack tip stress intensities in viscoelastic anisotropic bimaterials and the use of the M-integral

The time evolution of the stress intensity factors (S.I.F's) at both tips of a finite crack lying near the interface of a viscoelastic anisotropic bimaterial, is studied. The simultaneous integral equations for the dislocation density of the crack developed in [2], are now used in the Laplace transformed domain. Their numerical solution and the solution via Neumann series are used for the determination of the Laplace trasformed S.I.F's. In the case of rectilinear anisotropies the latter are extracted from the M(p) integral which has been evaluated along a circle at infinity and along the interface. Numerical results for the real time dependence of the S.I.F's for two different anisotropies and geometries are also discussed.     

Heat Conduction Analysis of 3-D Axisymmetric and Anisotropic FGM Bodies by Meshless Local Petrov–Galerkin Method

The meshless local Petrov–Galerkin method is used to analyze transient heat conduction in 3-D axisymmetric solids with continuously inhomogeneous and anisotropic material properties. A 3-D axisymmetric body is created by rotation of a cross section around an axis of symmetry. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem. The cross section is covered by small circular subdomains surrounding nodes randomly spread over the analyzed domain. A unit step function is chosen as test function, in order to derive local integral equations on the boundaries of the chosen subdomains, called local boundary integral equations. These integral formulations are either based on the Laplace transform technique or the time difference approach. The local integral equations are nonsingular and take a very simple form, despite of inhomogeneous and anisotropic material behavior across the analyzed structure. Spatial variation of the temperature and heat flux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions.     

Transient dynamic stress intensity factors around two rectangular cracks in a nonhomogeneous interfacial layer between two dissimilar elastic half-spaces under impact load

Summary Transient dynamic stresses around two rectangular cracks in a nonhomogeneous interfacial layer sandwiched between two dissimilar elastic half-spaces are examined. The material properties vary continuously in the layer within a range from those of the upper half-space to those of the lower half-space. An incoming shock stress wave impinges perpendicular on the crack surfaces. In order to solve the problem, the interfacial layer is divided into several homogeneous layers that have different material properties. Application of Laplace and Fourier transforms reduces the problem to the solution of a pair of dual integral equations. To solve the equations, the differences in the crack surface displacements are expanded into a series of functions that vanish outside the crack. The unknown coefficients in the series are solved using the Schmidt method. The stress intensity factors are defined in the Laplace transform domain and these are inverted numerically in physical space. Numerical calculations are carried out for composite materials made of a ceramic half-space and a steel half-space.     

The plane elastostatic solution for a symmetrically loaded crack in a strip composite

The plane elastostatic problem for a crack in a strip composite loaded with normal or shearing traction is reduced to a single integral equation. The dependence of the solution on the material parameters is exhibited explicitly in the integral equation through two composite parameters. The integral equation is solved numerically and the dependence of the stress intensity factors on the material parameters is displayed graphically for all physically relevant composites for each of several chosen values of the crack length to strip width ratio.     

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.     

Dynamic stress intensity factors around four rectangular cracks in an infinite elastic medium under impact load

The 3-D dynamic problem is presented for an infinite elastic medium weakened by four plane rectangular cracks of equal size. The surfaces of the cracks are loaded by a uniform pressure with Heaviside-function time dependence. Fourier-Laplace transform technique is utilized to reduce the problem to a solution of two simultaneous integral equations which can be solved by using the series expansion method. The Laplace transformed stress intensity factors are defined and are inverted numerically in the physical space.     

Thermal calculation of rotary regenerators with a dispersed packing

The integral Laplace transformation and the reduction of differential equations to Volterra integral equations are used to obtain a solution to the equation of the packing and gas temperature distribution over the thickness of the section of a rotary radially sectioned regenerator with a dispersed packing in relation to time for the initial period of operation of the regenerator.     

A time-domain symplectic method for finite viscoelastic cylinders

The symplectic method is introduced for boundary-condition problems of finite viscoelastic cylinders. On the basis of the state space formalism and the use of the Laplace integral transform, the general solution of the governing equations, zero- and nonzero-eigenvalue eigenvectors, are obtained. Since the eigenvectors are expressed in concise analytical forms, the adjoint symplectic relation of the Laplace domain is generalized to the time domain. Therefore, the particular solution and the eigenvector expansion method can be discussed directly in the eigenvector space of the time domain, without employing the iterative application of the inverse Laplace transformation. Using this method, various boundary conditions, the particular solution of nonhomogeneous equations, especially the interfacial continuity conditions of composite materials, can be conveniently described by combinations of the eigenvectors.     

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.     

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。     

Torsional impact of a layer or a cylinder bonded to an elastic half-space

In this paper two torsional impact problems are considered. The first problem deals with the solution of a layer bonded to an elastic half-space when the layer is driven by the torsional impact over a bonded rigid circular disc. In the second problem sudden torsion by a rigid disc attached over the plane face of a circular cylinder is considered and the rest of the plane surface of the cylinder is stress free. The cylinder is bonded to the half-space, making use of Laplace and Hankel transforms the solution of each problem is reduced into Fredholm integral equations of the second kind. A numerical Laplace inversion technique is then used to recover the time depencence of the solution. The numerical values for the applied torque at the surface of rigid disc are calculated for each problem and then are displayed graphically.     

Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation

In this paper the diffusion equation is solved in two-dimensional geometry by the dual reciprocity boundary element method (DRBEM). It is structured by fully implicit discretization over time and by weighting with the fundamental solution of the Laplace equation. The resulting domain integral of the diffusive term is transformed into two boundary integrals by using Green's second identity, and the domain integral of the transience term is converted into a finite series of boundary integrals by using dual reciprocity interpolation based on scaled augmented thin plate spline global approximation functions. Straight line geometry and constant field shape functions for boundary discretization are employed. The described procedure results in systems of equations with fully populated unsymmetric matrices. In the case of solving large problems, the solution of these systems by direct methods may be very time consuming. The present study investigates the possibility of using iterative methods for solving these systems of equations. It was demonstrated that Krylov-type methods like CGS and GMRES with simple Jacobi preconditioning appeared to be efficient and robust with respect to the problem size and time step magnitude. This paper can be considered as a logical starting point for research of iterative solutions to DRBEM systems of equations. © 1998 John Wiley & Sons, Ltd.