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)     

Investigation of a griffith crack subject to uniform tension using the non-local theory by a new method

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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 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)     

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.     

Investigation of the scattering of antiplane shear waves by two griffith cracks using the non-local theory

In this paper, the scattering of harmonic antiplane shear waves by two finite cracks is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of triple integral equations is solved using a new method, namely Schmidt's method. This method is more exact and more reasonable than Eringen's for solving this kind of problem. The result of the stress near the crack tip was obtained. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip, which can explain the problem of macroscopic and microscopic mechanics.     

Nonlinear response of anti-plane shear crack

Investigated in this communication is the nonlinear response of an anti-plane shear crack. A nonlinear hardening/softening relation is assumed between the resultant shear stress and shear strain for determining the state of affairs near the crack tip. Amplification of the local stress fields is obtained in terms of the material parameters by assuming that nonlinearity is confined to the immediate vicinity of the crack tip region.     

Analysis of a Griffith crack at the interface of two piezoelectric materials under anti-plane loading

The main objective of this work is the contribution to the study of the piezoelectric structures which contain preexisting defect (crack). For that, we consider a Griffith crack located at the interface of two piezoelectric materials in a semi-infinite plane structure. The structure is subjected to an anti-plane shearing combined with an in-plane electric displacement. Using integral Fourier transforms, the equations of piezoelectricity are converted analytically to a system of singular integral equations. The singular integral equations are further reduced to a system of algebraic equations and solved numerically by using Chebyshev polynomials. The stress intensity factor and the electric displacement intensity factor are calculated and used for the determination of the energy release rate which will be taken as fracture criterion. At the end, numerical results are presented for various parameters of the problem; they are also presented for an infinite plane structure.     

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.     

Elastic-viscoplastic fields near the tip of a propagating crack under anti-plane shear

The elastic-viscoplastic model proposed by Bingham was used to analyse the stress and strain surrounding the tip of a propagating crack under antiplane shear. The proper displacement pattern was given ; the asymptotic equations were derived and solved numerically. The analysis and calculation show that for smaller viscosity the crack-tip possesses logarthmic singularity, and for larger viscosity it possesses power-law singularity.In critical case, the two kinds of singularity are consistent with each other. The result revealed the important role of viscosity for crack-tip field.     

Electrical nonlinear anti-plane shear crack in a functionally graded piezoelectric strip

The electrical nonlinear behavior of an anti-plane shear crack in a functionally graded piezoelectric strip is studied by using the strip saturation model within the framework of linear electroelasticity. The analysis is conducted on the electrically unified crack boundary condition with the introduction of the electric crack condition parameter that can describe all the electric crack boundary condition in accordance with the aspect ratio of an ellipsoidal crack and the permittivity inside the crack, in particular, including traditional permeable and impermeable crack boundary conditions. The resulting mixed boundary value problem is analysed and near tip field is obtained by using the integral transform techniques. Numerical results for the normalized five kinds of energy release rates under the small scale electric saturation condition are presented and compared to show the influences of the electric crack condition parameter with the variation of the ellipsoidal crack parameters, electric loads, functionally graded piezoelectric material gradation, crack length, electromechanical coupling coefficient, and crack location. It reveals that there are considerable differences between the results obtained from the traditional electric crack models and those obtained from the current unified crack model.     

Elastic-plastic mechanics of steady crack growth under anti-plane shear

For a crack with steady growth under anti-plane shear, analysis shows a primary plastic zone included in an angle of ±19.7° ahead of the crack tip, and two very thin secondary (reverse) plastic zones along the crack flanks, each included in an angle of 0.37°. Numerical solutions give the shape of the plastic zones which determine the active and residual plastic strains, and give the crack tip displacement, which is approximately 0.07 of that for monotonic loading without growth. The length of the primary plastic zone is almost the same as that without growth, but the thickness is about 3/5 as great. Coupled with ductile fracture criteria, the present results predict initially stable crack growth, whereas analyses based on the simplification of yielding on just one plane predict unstable fracture immediately following initiation.     

Unloading characteristics of anti-plane shear crack in softening materials

The local characteristics of the anti-plane shear stress and strain field are determined for a material where the stress increases linearly with strain up to a limit and then softens nonlinearly. Two unloading models are considered such that the unloading path always returns to the origin while the other assumes the unloading modulus to be that of the initial shear modulus. As the applied shear increases, an unloading zone is found to prevail between a zone in which the material softens and another zone in which the material is linear-elastic even though the crack does not propagate. The divisions of these zones are displayed graphically.     

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M_c1 < M < M_c2, the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Discontinuous deformation gradients away from the tip of a crack in anti-plane shear

Summary This paper examines the finite elastostatic problem for a crack under anti-plane shear conditions. A class of incompressible, homogeneous, isotropic, elastic materials is considered for which the governing displacement equation of equilibrium is elliptic at both sufficiently small and sufficiently large shearing strains but is hyperbolic at an intermediate level of strains. The small-scale nonlinear crack problem is considered and an exact solution is found through use of the hodograph method.The principal feature of the solution is the presence of two pairs of curves issuing from points on the crack-faces, different from the crack-tips, across which the displacement gradients and stresses suffer jump discontinuities.The results communicated in this paper were obtained in the course of an investigation supported by the NSF Materials Research Laboratory at Brown University.     

Analysis of anisotropic sector with a radial crack under anti-plane shear loading

In this paper, the anti-plane shear deformation of an anisotropic sector with a radial crack is investigated. The traction–traction boundary conditions are imposed on the radial edges and the traction-free condition is considered on the circular segment of the sector. A novel mathematical technique is employed for the solution of the problem. This technique consists of the use of some recently proposed finite complex transforms (Shahani, 1999), which have complex analogies to the standard finite Mellin transforms of the first and second kinds. However, it is essential to state the traction-free condition of the crack faces in the form of a singular integral equation which is done in this paper by describing an exact analytical method. The resultant dual integral equations are solved numerically to determine the stress intensity factors at the crack tips. In the special cases, the obtained results coincide with those cited in the literature.