Anti-plane elastodynamic analysis of planes with multiple defects

The solution of a screw dislocation under time-harmonic condition is obtained in an infinite isotropic plane by means of the Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a plane weakened by cracks and cavities. Cavities are considered as closed curved cracks without singularity. Several examples are solved and the stress intensity factor of cracks and hoop stress on cavities are obtained.     

Elastodynamic analysis of a cracked orthotropic half-plane

The solution of elastodynamic volterra-type dislocation in an orthotropic half-plane is obtained by means of the Fourier transforms. The distributed dislocation technique is used to construct integral equations for an orthotropic half-plane weakened by cracks where the domain is under time-harmonic anti plane traction. These equations are of Cauchy singular type at the location of dislocation which is solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determine stress intensity factors for multiple smooth cracks. Several examples are solved and the stress intensity factors for multiple cracks with different configuration are obtained.     

Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

Multiple moving cracks in a functionally graded strip

This paper considers several finite moving cracks in a functionally graded material subjected to anti-plane deformation. The distributed dislocation technique is used to carry out stress analysis in a functionally graded strip containing moving cracks under anti-plane loading. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. By utilizing the Fourier sine transformation technique the stress fields are obtained for a functionally graded strip containing a screw dislocation. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by several moving cracks. Numerical examples are provided to show the effects of material properties, the crack length and the speed of the crack propagating upon the stress intensity factor.     

Anti-plane elastodynamic analysis of orthotropic planes weakened by several cracks

Stress analysis is carried out in an orthotropic plane containing a Volterra-type dislocation, the distributed dislocation technique is employed to obtain integral equations for an orthotropic plane weakened by cracks under time-harmonic anti-plane traction. The integral equations are of Cauchy singular type at the location of dislocation which are solved numerically. Several examples are solved and the stress intensity factors for multiple cracks with different configuration are obtained.     

Anti-plane elastodynamic analysis of cracked graded orthotropic layers with viscous damping

Stress analysis is carried out in a graded orthotropic layer containing a screw dislocation undergoing time-harmonic deformation. Energy dissipation in the layer is modeled by viscous damping. The stress fields are Cauchy singular at the location of dislocation. The dislocation solution is utilized to derive integral equations for multiple interacting cracks with any location and orientation in the layer. These equations are solved numerically thereby obtaining the dislocation density function on the crack surfaces and stress intensity factors of cracks. The dependencies of stress intensity factors of cracks on the excitation frequency of applied traction and material properties of the layer are investigated. The analysis allows the determination of natural frequencies of a cracked layer. Furthermore, the interactions of two cracks having various configurations are studied.     

功能梯度材料有限宽板的反平面断裂问题研究

研究了功能梯度材料有限宽板中与板边平行的III型裂纹问题．假设材料的剪切模量沿板宽度方向呈指数规律变化,利用Fourier变换将问题描述为奇异积分方程,并进一步将未知的位错密度函数表示为Chebyshev多项式的级数式,从而将奇异积分方程化为线性代数方程组进行配点数值求解．基于数值结果,讨论了材料非均匀性参数、板和裂纹的几何参数等对应力强度因子（SIF）的影响．研究表明,SIF随裂纹长度的增大而增大,随裂纹所在区域材料刚度的增大而减小;板越窄,SIF对非均匀性参数的变化越敏感,且变化规律也越复杂．随着非均匀性参数的增大,SIF既可能增大也可能减小还可能基本保持不变,这主要取决于板的相对宽度和裂纹的相对位置．当裂纹位于板的中央或当板较宽时,SIF对非均匀性参数的变化都不太敏感．     

A general method for constructing Timoshenko-type theories

The equations of the plane theory of for the elasticity bending of a long strip are reduced by the method of simple iterations to the solution of a system of two equations for the displacement of the axis of the strip and the shear stress. If the transverse load varies slowly along the strip, the resolvent equations reduce to a single equation that is identical to the classical equation for the bend of a beam. When a local load is applied, the resolvent equation acquires an additional singular term that is the solution of the equation for the shear stresses under the assumption that the displacement (deflection) is a function of small variability. The convergence of the solution in an asymptotic sense is demonstrated. The application of the method of simple iterations to the dynamic equations for the bending of a strip also leads to a system of two resolvent equations in the displacement of the axis of the strip and the shear stress. These equations reduce to a single equation that is identical with the well-known Timoshenko equation. Hence, the procedure for using the method of simple iterations that has been developed can be classified as a general method for obtaining Timoshenko-type theories. An equation is derived for the bending of a strip on an elastic base with an isolated functional singular part with two bed coefficients, corresponding to the transverse and longitudinal springiness of the base.     

Anti-plane stress analysis of dissimilar sectors with multiple defects

In this article, the anti-plane deformation of a typical dissimilar sector consists of two sub-sectors attached to each other on one circular edge is studied. The solution of a Volterra type screw dislocation problem in the sector is obtained through finite Fourier cosine transform. Exact closed-form solutions for the displacement and stress fields are also presented. Next, using a distributed dislocation method, integral equations of the sectors weakened by cracks and cavities under an anti-plane traction are obtained. The defects are assumed to be located only in one of the sub-sector regions. The obtained equations for the latter problem are of the Cauchy singular type and have been solved numerically. Several examples are presented to demonstrate the efficiency and applicability of the proposed solution procedure. The geometric and force singularities of the stress field are studied and compared to those reported in the literature.     

The stress field problem for a pre-stressed plate-strip with finite length under the action of arbitrary time-harmonic forces

In the framework of the three-dimensional linearized theory of elastodynamics the finite element modeling of the stress field problem for the pre-stressed plate-strip with finite length resting on a rigid foundation under the action of inclined linearly located time-harmonic forces is developed. The numerical results involving the normal stress acting on the interface plane of the plate-strip and the rigid foundation are presented. Moreover, the dependencies between this stress, the frequency of the arbitrary inclined linearly located external force and the initial stretching of the plate-strip are analyzed.     

Some properties of the Cauchy-type integral for the time-harmonic Maxwell equations

We study the analog of the Cauchy-type integral for the theory of time-harmonic electromagnetic fields in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given pair of vector fields from such a surface up to a solution of the time-harmonic Maxwell equations in a domain. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between time-harmonic solutions of the Maxwell equations and some versions of quaternionic analysis.     

Stress analysis in a sheet with multiple cracks

The stress field due to the presence of a Volterra dislocation in an isotropic elastic sheet is obtained. The stress components exhibit the familiar Cauchy type singularity at dislocation location. The solution is utilized to construct integral equations for elastic sheets weakened by multiple embedded or edge cracks. The cracks are perpendicular to the sheet boundary and applied traction is such that crack closing may not occur. The integral equations are solved numerically and stress intensity factors (SIFs) are determined on a crack edges.     

Forced vibration of the pre-stressed bi-layered plate-strip with finite length resting on a rigid foundation

Within the scope of the piecewise homogeneous body model utilizing Three-Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies the time-harmonic dynamical stress field in the pre-stressed bi-layered plate-strip with finite length resting on the rigid foundation is investigated. The materials of the layers are assumed to be isotropic. The FEM modeling is developed for the solution to the corresponding boundary-value-contact problem. The numerical results regarding the influence of the finiteness of the layers’ length on the stress distribution on the interface planes are presented and discussed. In particular, it is shown that with increasing the plate length the results obtained for the considered case approach to the corresponding ones attained for the bi-layered plate with infinite length.     

Boundary integral equations for an interface linear crack under harmonic loading

The present study is devoted to application of boundary integral equations to the problem of a linear crack located on the bimaterial interface under time-harmonic loading. Using the Somigliana dynamic identity the system of boundary integral equations for displacements and tractions at the interface is derived. For the numerical solution the collocation method with piecewise constant approximation on each linear continuous boundary elements is used. The distributions of the displacements are computed for different values of the frequency of the incident tension-compression wave. Results are compared with static ones.     

Thermal fracture of functionally gradient ceramics

An edge crack in a strip of functionally gradient ceramics (FGC) is studied under thermal loading conditions. Two FGC materials are considered, i.e., one with a spatial variation of shear modulus and the other with a spatial variation of thermal conductivity. Thermal stress intensity factors (TSIF) are numerically calculated based on singular integral equations derived for the dislocation density along the crack faces. It is shown that: (a) for the FGC with a graded shear modulus, the TSIF are reduced for crack lengths longer thanl _c b and remain approximately the same as those of a homogeneous material for shorter crack lengths, wherel _c is about 0.065 andb is the width of the strip; and (b) for the FGC with a thermal conductivity gradient, the TSIF are generally lower compared with those for the bonded two-layer material.     

Are plastic yielding and work hardening sensitive to the boundary conditions?

The plane strain shear of a single crystal strip with one active slip system placed in a mixed device with one clamped and one free boundary is considered. Since dislocations pile up against only the clamped boundary, the plastic yielding and work hardening differ essentially from those of a hard device, showing clearly their sensitivity to the boundary conditions. An analytical solution to this problem within continuum dislocation theory is found explicitly which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effects. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

Steady state stresses induced in a transversely isotropic elastic solid by a moving dislocation

An unbounded, transversely isotropic, elastic solid, is subjected to a dislocation moving at constant speed. By means of an appropriate coordinate transformation, the transient version of this problem is used to obtain the steady state solution. The solution for the plane stress field is explicit and valid for dislocation speeds which are sub-, tran-, or super-sonic with respect to the material wave speeds. The previously discovered transonic speed at which the Mach head wave was annihilated for the transient problem, is found to be present in the steady state problem also.     

Contact Interaction of Faces of the Plane Crack under Harmonic Loading

The present work is devoted to the solution of the three-dimensional fracture mechanics problem for a linear elastic, homogeneous and isotropic solid with a stationary plane crack under normal time-harmonic loading. The problem has been solved by the method of boundary integral equations with the allowance for the contact interaction of the opposite faces of the crack. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dynamic contact problem which reduces to a singular integral equation with two fixed singularities

The problem of the harmonic sheat oscillations of an elastic strip, coupled to an elastic half-space is considered. Using the method of integral transformations, the problem is reduced to a singular integral equation in the contact stresses in the region where the strip and the half-space are coupled when there are two fixed singularities at points bounding the integration intervals. One of the main results of this paper is the method of solving this equation numerically, taking into account the true singularity of the solution and based on the use of special quadrature formulae for singular integrals. The approximate solution obtained provides the possibility of numerically investigating the effect of the oscillation frequency and the ratio of the elastic constants of the strip and the half-space on the stress distribution in the contact area.