In this paper, the static problem of several cracks in a functionally graded piezoelectric– piezomagnetic (FGPP) rectangular plane subjected to concentrated anti-plane mechanical and in-plane electric and magnetic fields is described. The material properties are assumed to vary continuously according to exponential functions along the transverse of the FGPP rectangular plane. The dislocation method and integral transforms technique are applied to obtain a set of Cauchy singular integral equations. The field intensity factors for cracks, are obtained by using the corresponding solution to these equations. The numerical examples of mode-III problem are presented to illustrate the interesting mechanical and electromagnetic coupling phenomena induced by multi-crack interactions. Finally, the effects of material non-homogeneity constant, the cracks length and the cracks configuration upon the field intensity factors are investigated. The obtained conclusions seem useful for design of the magneto-electro-elastic structures and devices of high performance.

The solution to the problem of an orthotropic long cylinder subjected to torsional loading is first obtained by means of separation valuables. The cylinder is twisted by two lateral shear tractions, and the ends of the cylinder surface of the cylinder are stress-free. First, the domain under consideration is weakened by an axisymmetric rotational Somigliana ring dislocation. The dislocation solution is employed to derive a set of Cauchy singular integral equations for the analysis of Multiple axisymmetric planner cracks. The numerical solution to these integral equations is used to determine the stress intensity factors for the tips of the concentric planar cracks. A preliminary comparison between results of this study and those available in the literature is performed to confirm the validity of the proposed technique. Several examples of Multiple concentric planner cracks are solved and graphically displayed. Furthermore, the configuration of the cracks and the interaction between cracks is studied.

In this paper, the analytical solution of an electric and Volterra edge dislocation in a functionally graded piezoelectric (FGP) medium is obtained by means of complex Fourier transform. The system is subjected to inplane mechanical and electrical loading. The material properties of the medium vary exponentially with coordinating parallel to the crack. In this study, the rate of the gradual change of the shear moduli and mass density is assumed to be same. At first, the Volterra edge dislocation solutions are employed to derive singular integral equations in the form of Cauchy singularity for an FGP plane containing Multiple horizontal moving cracks. Then, these equations are solved numerically to obtain dislocation density functions on moving crack surfaces. Finally, the effects of the crack moving velocity, material properties, electromechanical coupling factor and cracks arrangement on the normalized mode I and mode II stress intensity factors and electric displacement intensity factor are studied.

In this article, stress analysis of an isotropic infinite cylinder weakened by Multiple concentric cylindrical cracks subjected to shear and radial loadings on the lateral surface of the cylinder is accomplished. First, relations for stress and displacement components are given in terms of the Galerkin biharmonic vector potential. Next, using these relations, the dislocation solutions for both Somigliana and Volterra dislocation are derived. Using the distributed dislocation technique, integral equations for the infinite cylinder with arbitrary number of the cylindrical cracks are constructed. The numerical solution of resulting integral equations which are of the Cauchy type singular equations leads to evaluation of Somiglianaand Volterra dislocation densities on the crack surfaces. Using related dislocation densities, the stress intensity factors of crack tips for some examples are attained and variations of them with lengths of cracks are discussed. For some special cases, validation of the results of this study with others available in the literature is done.

Based on the distributed dislocation technique, an analytical solution for the orthotropic layer with functionally graded material (FGM) orthotropic coating containing Multiple axisymmetric interfacial cracks subjected to torsional loading is investigated. It is assumed that the material properties of the FGM orthotropic coating vary power-law form along the thickness of the layer. At first, by using the Hankel transform, the solution for Somigliana-type rotational ring dislocation in the layer and its coating is obtained. Then, the dislocation solution is used to derive a set of singular integral equations for a system of coaxial axisymmetric interface cracks, including penny-shaped and annular cracks. cracks with Cauchy type kernel. The integral equations are of Cauchy singular type, which are solved by Erdogan’, s collocation method for dislocation density on the surface of interfacial crack and the results are used to determine stress intensity factors (SIFs) for axisymmetric interface cracks. Finally, several examples are provided to study the effects of the non-homogeneity constant, orthotropy parameter and thickness of FGM coating on the SIFs for interfacial cracks. The effects of the non-homogeneity constant, orthotropy parameter and thickness of FGM coating as well as the interaction of Multiple interfacial cracks on the SIFs are investigated. The results reveal that the value of the SIFs decreases with increasing the non-homogeneity parameter, orthotropy and thickness of FGM coating. The SIFs for inner tips of annular interface crack are larger than the outer tips.

In this paper, numerical solutions of Multiple cracks problems in an infinite plate are studied. Hypersingular integral equations (hieq) for the cracks are formulated using the complex potential method. For all kernels such as regular or hypersingular kernels, we are using the appropriate quadrature formulas to solve and evaluate the unknown functions numerically. Furthermore, by using this equation the stress intensity factor (SIF) was calculated for crack tips. For two serial cracks (horizontal) and two dissimilar cracks (horizontal and inclined), our numerical results agree with the previous works.

All materials are made up of sub-bodies, which constitute their substructure or microstructure. The size of a sub-body may vary from atomic dimensions to a macroscopic scale, such as grain size. Depending upon the nature and accuracy of the physical phenomena to be modeled, the average distance of the sub-bodies plays a central role. This distance may vary from the order of the lattice parameter (10-8 cm in perfect crystals), to a few millimeters, as in granular solids. The boundary and initial conditions bring into play another characteristic, length. Such models, entitled nonlocal theories, have been proposed for the past four decades. The solutions of various critical problems have veried our hopes and expectations in that, by means of nonlocal models, it will be possible to make accurate predictions of physical phenomena at submicroscopic scale. In the present study, the anti-plane stress field of Multiple cracks is obtained using the solution of screw dislocation in an infinite elastic plane, based on nonlocal elasticity. The distribution dislocation technique is used to model the crack problem with screw dislocation distribution. Unlike the classical elasticity solution, a lattice parameter enters into the problem, which makes the stresses finite in the screw dislocation solution in the infinite elastic plane in nonlocal theory, which has no singularity at the dislocation tip, and which is consistent with theoretical results. Similarly, the crack problem using the distribution dislocation theory is solved with no singularity at the crack tip. The kernel in the related equation is of the Cauchy type, and to determine the distribution of dislocations, which generates traction along the crack line, the Gauss-Chebyshev quadrature has been used. Several numerical examples to illustrate the accuracy and capability of the solution have been calculated, where the effect of crack length, lattice parameter and constant is calculated as a variable parameter, which includes all of them. Stress at the crack tip and its graphs are depicted and the results obtained are compared with classical results in this field.

This paper presents a general formulation for an isotropic wedge reinforced by an orthotropic coating involving Multiple arbitrarily oriented defects under out of plane deformation. The exact closed form solution of the problem weakened by a screw dislocation in the isotropic wedge is obtained by making use of finite Fourier cosine transform. Also, the closed-form solutions of the out of plane stress and displacement fields are obtained. After that, by making use of a distributed dislocation approach, a set of singular integral equations of the domain involving smooth cavities and cracks subjected to out of plane external loading are achieved. The cracks and cavities are considered to be only in the isotropic wedge. The presented integral equations have Cauchy singularity and must be evaluated numerically. Multiple numerical examples will be presented to show the applicability and efficiency of the presented solution. The geometric and point load singularities of the stress components are obtained and compared with the available data in the literature.

The analytical method is developed to examine the fracture behavior of a functionally graded piezoelectric rectangular plane (FGPRP) with finite geometry under impact loads. The material properties of the FGPRP vary continuously in the transverse direction. Two different types of boundary conditions are examined and discussed in the analyses. The finite Fourier cosine and Laplace transforms are employed to obtain stress and electric displacement fields in the finite plane containing electro-elastic screw dislocation. Based on the distributed dislocation technique, a set of integral equations for the finite plane is weakened by Multiple parallel cracks under electromechanical impact loads. By solving numerically, the resulting singular integral equation, the dynamic stress intensity factor (DSIF) is obtained for the electrically impermeable case. The new results are provided to show the applicability of the proposed solution. The effects of the geometric parameters including plate length, width, crack position, crack length, loading parameter, and FG exponent on the dynamic stress intensity factors are shown graphically and discusseThe analytical method is developed to examine the fracture behavior of a functionally graded piezoelectric rectangular plane (FGPRP) with finite geometry under impact loads. The material properties of the FGPRP vary continuously in the transverse direction. Two different types of boundary conditions are examined and discussed in the analyses. The finite Fourier cosine and Laplace transforms are employed to obtain stress and electric displacement fields in the finite plane containing electro-elastic screw dislocation. Based on the distributed dislocation technique, a set of integral equations for the finite plane is weakened by Multiple parallel cracks under electromechanical impact loads. By solving numerically, the resulting singular integral equation, the dynamic stress intensity factor (DSIF) is obtained for the electrically impermeable case. The new results are provided to show the applicability of the proposed solution. The effects of the geometric parameters including plate length, width, crack position, crack length, loading parameter, and FG exponent on the dynamic stress intensity factors are shown graphically and discussed.