Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome the mathematical difficulties, 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 the electric displacement near at the crack tip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the thickness of the strip, the circular frequency of incident wave and the lattice parameter.     

Investigation of the dynamic behavior of two parallel symmetric cracks in piezoelectric materials use of non-local theory

In this paper, the dynamic behavior of two parallel symmetric cracks in piezoelectric materials under harmonic anti-plane shear waves is investigated by use of the non-local theory for permeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the problem to obtain the stress occurs near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations that the unknown variables are the jumps of the displacement along the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the frequency of the incident wave, the distance between two cracks and the lattice parameter of the materials, respectively. Contrary to the impermeable crack surface condition solution, it is found that the dynamic electric displacement for the permeable crack surface conditions is much smaller than the results for the impermeable crack surface conditions. The results show that the dynamic field will impede or enhance crack propagation in the piezoelectric materials at different stages of the dynamic load.     

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.     

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in the piezoelectric plate by using the non-local theory

In this paper, the dynamic interaction between two collinear cracks in a piezoelectric material plate under anti-plane shear waves is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved using the Schmidt method. This method is more reasonable and more appropriate. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The project supported by the Natural Science Foundation of Heilongjiang Province and the National Natural Science Foundation of China(10172030, 50232030)     

On anti-plane shear behavior of a Griffith permeable crack in piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials under anti-plane shear loading for permeable crack surface conditions. By means of the Fourier transform the problem can be solved with the help of a pair of dual integral equations with the unknown variable being the jump of the displacement across the crack surfaces. These equations are solved by the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length and the lattice parameter of the materials, respectively. The project supported by the National Natural Science Foundation of China (50232030 and 10172030)     

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

IntroductionCompositematerialconsistingofapiezoelectricphaseandapiezomagneticphasehasdrawnsignificantinterestinrecentyears,duetotherapiddevelopmentinadaptivematerialsystems .Itshowsaremarkablylargemagnetoelectriccoefficient,thecouplingcoefficientbetweenst…     

Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips

The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.     

The scattering of harmonic elastic anti-plane shear waves by a finite crack in infinitely long strip using the non-local theory

In this paper, the scattering of harmonic anti-plane shear waves by a finite crack in infinitely long strip is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress occurring at the crack tips. Contraty to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the width of the strip and the lattice parameter. Supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province and the National Foundation for Excellent Young Investigators.     

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in anisotropic material plane by using the non-local theory

In this paper, the dynamic behavior of two collinear cracks in the anisotropic elasticity material plane subjected to the harmonic anti-plane shear waves is investigated by use of the nonlocal theory. To overcome the mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The nonlocal elasticity solutions yield a finite hoop stress at the crack tips, thus allowing us to using the maximum stress as a fracture criterion. The magnitude of the finite stress field not only depends on the crack length but also on the frequency of the incident waves and the lattice parameter of the materials.     

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.     

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…     

DYNAMIC BEHAVIOR OF TWO COLLINEAR PERMEABLE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF SPACES

IntroductionIn the fracture mechanics studies for piezoelectric materials,differently electricboundary conditions at the crack surfaces have been proposed by many researchers.Forexample,for the sake of analytical simplification,the assumption that the cra…     

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.     

SCATTERING OF HARMONIC ANTI-PLANE SHEAR STRESS WAVES BY A CRACK IN FUNCTIONALLY GRADED PIEZOELECTRIC/PIEZOMAGNETIC MATERIALS

In this paper, the dynamic behavior of a permeable crack in functionally graded piezoelectric/piezomagnetic materials is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using the Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown is the jump of displacements across the crack surfaces. These equations are solved to obtain the relations between the electric filed, the magnetic flux field and the dynamic stress field near the crack tips using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter and the circular frequency of the incident waves upon the stress, the electric displacement and the magnetic flux intensity factors of the crack.     

Dynamic behavior of two parallel symmetric permeable cracks in a piezoelectric strip

The dynamic behavior of two parallel symmetric cracks in a piezoelectric strip under harmonic anti-plane shear waves is studied using the Schmidt method for permeable crack surface conditions. The cracks are parallel to the edge of the strip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the schmidt method. The results show that the stress and the electric displacement intensity factors depend on the geometry of the cracks, the frequency of incident waves, the distance between cracks and the thickness of the strip. It is also found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. Project supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province, the National Science Foundation with the Excellent Young Investigator Award (No. 19725209) and the Scientific Research Foundation of Harbin Institute of Technology (HIT.2000.30).     

DYNAMIC BEHAVIOR OF TWO PARALLEL SYMMETRY CRACKS IN MAGNETO-ELECTRO-ELASTIC COMPOSITES UNDER HARMONIC ANTI-PLANE WAVES

The dynamic behavior of two parallel symmetry cracks in magneto-electro-elastic composites under harmonic anti-plane shear waves is studied by Schmidt method. By using the Fourier transform, the problem can be solved with a pair of dual integral equations in which the unknown variable is the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surface were expanded in a series of Jacobi polynomials. The relations among the electric filed, the magnetic flux and the stress field were obtained. From the results, it can be obtained that the singular stresses in piezoelectric/piezomagnetic materials carry the same forms as those in a general elastic material for the dynamic anti-plane shear fracture problem. The shielding effect of two parallel cracks was also discussed.     

Non-local theory solution for a Mode I crack in piezoelectric materials

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials subjected to a uniform tension loading. The permittivity of the air in the crack is considered. By means 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 the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the materials constants, the electric boundary conditions and the lattice parameter on the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions on the electric displacement fields are large. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.     

Multi-interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading

The behavior of four parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading is investigated. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved by the Schmidt method. This process is quite different from that papers adopted previously. The normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions. Numerical examples are provided to show the effect of the geometry of the interacting cracks, the thickness and the materials constants of the piezoelectric layer upon the stress and electric displacement intensity factors of the cracks. 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.     

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.     

SCATTERING OF ANTI-PLANE SHEAR WAVES IN A FUNCTIONALLY GRADED MATERIAL STRIP WITH AN OFF-CENTER VERTICAL CRACK

The scattering problem of anti-plane shear waves in a functionally graded material strip with an off-center crack is investigated by use of Schmidt method. The crack is vertically to the edge of the strip. By using the Fourier transform, the problem can be solved with the help of a pair of dual integral equations that the unknown variable is the jump of the displacement across the crack surfaces. To solve the dual integral equations, the jump of the displacement across the crack surfaces was expanded in a series of Jacobi polynomials. Numerical examples were provided to show the effects of the parameter describing the functionally graded materials, the position of the crack and the frequency of the incident waves upon the stress intensity factors of the crack.