Dynamic behavior of unequal parallel permeable interface multi-cracks in a piezoelectric layer bonded to two piezoelectric materials half planes

This study is concerned with the treatment of the dynamic behavior of interacting cracks in a piezoelectric layer bonded to two dissimilar piezoelectric half planes subjected to harmonic anti-plane shear waves. The permeable electric boundary condition is considered. By use of the Fourier transform technique, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in two series of Jacobi polynomials. The electromechanical behavior of two pairs of unequal parallel cracks was determined. Numerical examples are provided to show the effects of the geometry of the cracks, the frequency of the incident waves and materials properties upon the dynamic stress intensity factors (DSIFs) and the electric displacement intensity factors.     

Coupled field state around three parallel non-symmetric cracks in a piezoelectric/piezomagnetic material plane

In this paper, the behavior of three parallel non-symmetric permeable cracks in a piezoelectric/piezomagnetic material plane subjected to anti-plane shear stress loading was studied by the Schmidt method. The problem was formulated through Fourier transform into three pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric displacement, the magnetic flux and the stress fields near the crack tips can be obtained. The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the lengths and spacing of cracks. It was also revealed that the crack shielding effect is present in piezoelectric/piezomagnetic materials.     

Basic solution of two parallel non-symmetric permeable cracks in piezoelectric materials

The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading was studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations ip which the unknown variables are the jumps of displacements across crack surfaces. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.     

Interactions of multiple parallel symmetric permeable mode-III cracks in a piezoelectric material plane

In this paper, the interactions of multiple parallel symmetric and permeable finite length cracks in a piezoelectric material plane subjected to anti-plane shear stress loading were studied by the Schmidt method. The problem was formulated through Fourier transform into dual integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relation between the electric field and the stress field near the crack tips was obtained. The results show that the stress and the electric displacement intensity factors at the crack tips depend on the lengths and spacing of the cracks. It is also revealed that the crack shielding effect presents in piezoelectric materials.     

Dynamic behavior of two unequal parallel permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes

IntroductionDuetotheintrinsicelectro_mechanicalcouplingbehavior,piezoelectricmaterialsareveryusefulinelectronicdevices.However,mostpiezoelectricmaterialsarebrittlesuchasceramicsandcrystals.Therefore ,piezoelectricmaterialshaveatendencytodevelopcriticalcracksduringthemanufacturingandthepolingprocesses.So ,itisimportanttostudytheelectro_elasticinteractionandfracturebehaviorsofpiezoelectricmaterials.Theincreasingattentiontothestudyofcrackproblemsinpiezoelectricmaterialshasledtoalotofsignificantw…     

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials subjected to harmonic antiplane shear waves is investigated using the Schmidt method. The present problem can be solved using the Fourier transform and the technique of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric, magnetic flux, and dynamic stress fields near crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between the two parallel cracks, and the circular frequency of the incident waves upon the stress, electric displacement, and magnetic flux intensity factors at crack tips.     

MULTIPLE PARALLEL SYMMETRIC PERMEABLE MODEL-Ⅲ CRACKS IN A PIEZOELECTRIC/PIEZOMAGNETIC COMPOSITE MATERIAL PLANE

In this paper, the interactions of multiple parallel symmetric and permeable finite length cracks in a piezoelectric/piezomagnetic material plane subjected to anti-plane shear stress loading are studied by the Schmidt method.The problem is formulated through Fourier transform into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces.To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials.Finally, the relation between the electric field, the magnetic flux field and the stress field near the crack tips is obtained.The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the length and spacing of the cracks.It is also revealed that the crack shielding effect presents in piezoelectric/piezomagnetic materials.     

Scattering of harmonic anti-plane shear waves by an interface crack in magneto-electro-elastic composites

IntroductionCompositematerialconsistingofapiezoelectricphaseandapiezomagneticphasehasdrawnsignificantinterestinrecentyears,duetotherapiddevelopmentinadaptivematerialsystems .Itshowsaremarkablylargemagnetoelectriccoefficient,thecouplingcoefficientbetweenst…     

Basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials

The basic solution of two parallel mode-I permeable cracks in functionally graded piezoelectric materials was studied in this paper using the generalized Almansi’s theorem. To make the analysis tractable, it was assumed that the shear modulus varies exponentially along the horizontal axis parallel to the crack. The problem was formulated through a Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the gradient of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

The nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves

In this paper, the dynamic interaction of two parallel cracks in functionally graded materials (FGMs) is investigated by means of the non-local theory. To make the analysis tractable, the shear modulus and the material density are assumed to vary exponentially with the coordinate vertical to the crack. To reduce mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the dynamic problem to obtain stress fields near the crack tips. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips. The present result provides theoretical references helpful for evaluating relevant strength and preventing material failure of FGMs with initial cracks. The magnitude of the finite stress field depends on relevant parameters, such as the crack length, the distance between two parallel cracks, the parameter describing the FGMs, the frequency of the incident waves and the lattice parameter of materials. The project supported by the National Natural Science Foundation of China (90405016, 10572044) and the Specialized Research Fund for the Doctoral Program of Higher Education (20040213034). The English text was polished by Yunming Chen.     

Two parallel limited-permeable mode-I cracks or four parallel limited-permeable mode-I cracks in the piezoelectric materials

In this paper, the basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials were investigated by means of the Schmidt method for the limited-permeable electric boundary conditions. The electric permittivity of air in the crack was considered. Through the Fourier transform, the problems can be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces, not the dislocation density functions. To solve the dual integral equations, the jumps of the displacements across the crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the effects of the distance between two parallel cracks, the distance between two collinear cracks and the electric boundary conditions on the stress and the electric intensity factors in the piezoelectric materials are analyzed. These results can be used for the strength evaluation of the piezoelectric materials with multi-cracks. The crack shielding effect is also present in the piezoelectric materials.     

The dynamic behavior of two collinear interface cracks in magneto-electro-elastic materials

In this paper, the dynamic behavior of two collinear symmetric interface cracks between two dissimilar magneto-electro-elastic material half planes under the harmonic anti-plane shear waves loading is investigated by Schmidt method. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. To solve the triple integral equations, the jump of the displacements across the crack surface is expanded in a series of Jacobi polynomials. Numerical solutions of the stress intensity factor, the electric displacement intensity factor and the magnetic flux intensity factor are given. The relations among the electric filed, the magnetic flux field and the stress field are obtained.     

Basic solution of a mode-I limited-permeable crack in functionally graded piezoelectric materials

In this paper, the basic solution of a mode-I crack in functionally graded piezoelectric materials was investigated by using the generalized Almansi’s theorem. In the analysis, the electric permittivity of air inside the crack were considered. To make the analysis tractable, it was assumed that the shear modulus, piezoelectric constants and dielectric constants vary exponentially with coordinate parallel to the crack. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the effects of the electric boundary conditions on the electric displacement fields near the crack tips can not be ignored. Simultaneously, the solution of the present paper will revert to a closed form one when the functionally graded parameter equals to zero.     

The behavior of permeable multi-cracks in a piezoelectric material

In this paper, the behavior of four parallel symmetric cracks in a piezoelectric material under anti-plane shear loading is studied by the Schmidt method for the permeable crack surface boundary conditions. By use of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations that the unknown variables are the jumps of the displacement across the crack surfaces. These equations are solved by means of the Schmidt method. The results show that the stress and the electric displacement intensity factors of cracks depend on the geometry of the crack. Contrary to the impermeable crack surface condition solution, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.     

The interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material

In this paper, the interaction of two parallel Mode-I limited-permeable cracks in a functionally graded piezoelectric material was investigated by using the generalized Almansi's theorem. In the analysis, the electric permittivity of the air inside the crack was considered. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surface. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the electric permittivity of the air inside the crack and the gradient parameter of functionally graded piezoelectric material properties. It was also revealed that the crack shielding effect is also present in functionally graded piezoelectric materials.     

Solutions to a limited-permeable crack or two limited-permeable collinear cracks in piezoelectric/piezomagnetic materials

The solutions of a limited-permeable crack (case I) or two collinear limited-permeable cracks (case II) in piezoelectric/piezomagnetic materials subjected to a uniform tension loading were investigated in this paper using the generalized Almansi’s theorem. At the same time, the electric permittivity and the magnetic permeability of air in crack were firstly considered. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables were jumps of displacements across crack surfaces, not the dislocation density functions or the complex variable functions. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials to obtain the relations among electric displacement intensity factors, magnetic flux intensity factors and stress intensity factors at crack tips.     

THE BEHAVIOR OF TWO COLLINEAR CRACKS IN MAGNETO-ELECTRO-ELASTIC COMPOSITES UNDER ANTI-PLANE SHEAR STRESS LOADING

In this paper, the behavior of two collinear cracks in magneto-electro-elastic composite material under anti-plane shear stress loading is studied by the Schmidt method for permeable electric boundary conditions. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of displacements across the crack surfaces. In solving the triple integral equations, the unknown variable is expanded in a series of Jacobi polynomials. Numerical solutions are obtained. It is shown that the stress field is independent of the electric field and the magnetic flux.     

Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane

The Schmidt method is adopted to investigate the fracture problem of multiple parallel symmetric and permeable finite length mode-III cracks in a functionally graded piezoelectric/piezomagnetic material plane. This problem is formulated into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. In order to obtain the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The results show that the stress, the electric displacement, and the magnetic flux intensity factors of cracks depend on the crack length, the functionally graded parameter, and the distance among the multiple parallel cracks. The crack shielding effect is also obviously presented in a functionally graded piezoelectric/piezomagnetic material plane with mul- tiple parallel symmetric mode-III cracks.     

The closed form solution of a Mode-I crack in the piezoelectric/piezomagnetic materials

In this paper, the behavior of a Mode-I crack in the piezoelectric/piezomagnetic materials subjected to a uniform tension loading is investigated by the generalized Almansi’s theorem. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps are directly expanded in a series of Jacobi polynomials. Then the closed form solution of this problem can be obtained.     

Crack bifurcation predicted for dynamic anti-plane collinear cracks in piezoelectric materials using a non-local theory

A non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in the piezoelectric materials plane under anti-plane shear waves for the permeable crack surface boundary conditions. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses and the electric displacements finite at the crack tip. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and electric displacement near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations in which the unknown variable is the jump of the displacement across the crack surface. The solutions are obtained by means of the Schmidt method. Crack bifurcation is predicted using the strain energy density criterion. Minimum values of the strain energy density functions are assumed to coincide with the possible locations of fracture initiation. Bifurcation angles of ±5° and ±175° are found. The result of possible crack bifurcation was not expected before hand.