Electroelastic analysis of a penny-shaped crack in a piezoelectric ceramic under mode I loading

Following the theory of linear piezoelectricity, we consider the electroelastic problem for a piezoelectric ceramic with a penny-shaped crack under mode I loading. The problem is formulated by means of Hankel transform and the solution is solved exactly. The stress intensity factor, energy release rate and energy density factor for the exact and impermeable crack models are expressed in closed form and compared for a P-7 piezoelectric ceramic. Based on current findings, we suggest that the energy release rate and energy density factor criteria for the exact crack model are superior to fracture criteria for the impermeable crack model.     

Axisymmetric problem of a nonhomogeneous elastic layer

Summary The paper deals with a theoretical treatment of elastic behavior for a medium with nonhomogeneous materials property, which is defined by the relation , i.e., shear modulus of elasticity G varies with the dimensionless axial coordinate by the power product form, arbitrarily. Fundamental differential equation for such nonhomogeneous medium has been already proposed in [5]. It is given by a second-order partial differential equation. However, it was found that the fundamental equation is not sufficient in general to solve several kinds of boundary-value problems. On the other hand, it is shown in the present paper making use of the fundamental equations system for a nonhomogeneous medium, which has been proposed in our previous paper [7], it is possible to solve axisymmetric problems for a thick plate (layer) subjected to an arbitrarily distributed load or a concentrated load on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacements stress and components are shown in graphical form. Accepted for publication 25 March 1997     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

A mixed electric boundary value problem for a two-dimensional piezoelectric crack

In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.     

Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation

Summary A hypersingular integral equation or a differential-integral equation is used to solve the penny-shaped crack problem. It is found that, if a displacement jump (crack opening displacement COD) takes the form of (a ²−x ²−y ²)^1/2 x ^m y ⁿ , where a denotes the radius of the circular region, the relevant traction applied on the crack face can be evaluated in a closed form, and the stress intensity factor can be derived immediately. Finally, some particular solutions of the penny-shaped crack problem are presented in this paper. Received 1 July 1997; accepted for publication 13 October 1997     

Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer

The assumptions of impermeable and permeable cracks give rise to significant errors in determining electro-elastic behavior of a cracked piezoelectric material. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an opening crack interior. Considering the presence of the dielectric of an opening crack interior and the permeability of the crack surfaces for electric field, this paper analyzes electro-elastic behavior induced by a penny-shaped dielectric crack in a piezoelectric ceramic layer. In the cases of prescribed displacement or prescribed stress at the layer surfaces, the Hankel transform technique is employed to reduce the problem to Fredholm integral equations with a parameter dependent nonlinearly on the unknown functions. For an infinite piezoelectric space, a closed-form solution can be derived explicitly, while for a piezoelectric layer, an iterative technique is suggested to solve the resulting nonlinear equations. Field intensity factors are obtained in terms of the solution of the equations. Numerical results of the crack opening displacement intensity factors are presented for a cracked PZT-5H layer and the effect of applied electric field on crack growth are examined for both cases. The results indicate that the fracture toughness of a piezoelectric ceramic is affected by the direction of applied electric fields, dependent on the elastic boundary conditions.     

反平面断裂问题的无单元伽辽金比例边界法

将比例边界法与无单元伽辽金法相结合,建立了反平面断裂分析的无单元伽辽金比例边界法。这是一种边界型无网格法,在环向方向上采用无单元伽辽金法进行离散,因此计算时仅需要边界上的节点信息,不需要边界元所要求的基本解。为了便于施加本质边界条件,通过建立节点值和虚拟节点值之间的关系给出了修正的移动最小二乘形函数。在径向方向上,该方法利用解析的方法求解,因此是一种半解析的数值方法。最后,给出了数值算例,并验证了所提方法后处理简单和计算精度高的特点,适合于求解反平面断裂问题。     

Phragmen–Lindelof alternative for the displacement boundary value problem in a theory of non-linear micropolar elasticity

In this note we investigate the spatial behavior of the solutions for the displacement boundary value problem in a theory of non-linear micropolar elasticity. Under suitable conditions on the non-linear terms we obtain estimates for the solutions. The main tool is the energy method.     

Eigenvalues of the antiplane-shear crack problem for a power-law material

This paper discusses the problem of finding the eigenvalue spectrum in determining the stress and strain fields at the tip of an antiplane-shear crack in a power-law material. It is shown that the perturbation method provides an analytical dependence of the eigenvalue on the material nonlinearity parameter and the eigenvalue of the linear problem. Thus, it is possible to find the entire spectrum of eigenvalues and not only the eigenvalue of the Hutchinson-Rice-Rosengren problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 173–180, January–February, 2008.     

SAINT-VENANT’S TORSION PROBLEM FOR A COMPOSITE CIRCULAR CYLINDER WITH ANINTERNAL EDGE CRACK

In this paper the writer uses Muskhelishvili single-layer potential function solutionand single crack solution for the torsion problem of a circular cylinder to discuss thetorsion problem of a composite cylinder with an internal crack,and the problem isreduced to a set of mixed-type integral equation with generalized Cauchy-kernel.Then,by using the integration formula of Gauss-Jacobi.the numerical method isestablished and several numerical examples are calculated.The torsional rigidity andthe stress intensity factors are obtained.The results of these examples fit the resultsobtained by the previous papers better.     

Contact problem for a plane crack under a normally incident antiplane shear wave

The contact-interaction problem for a stationary plane crack with friction between its edges under the action of a normal (to the crack plane) harmonic shear wave is addressed. Antiplane deformation conditions are considered. The distribution of contact forces and displacement discontinuity of crack edges are studied Published in Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 138–142, May 2007.     

Green's function of the displacement boundary value problem for a heat source in an infinite plane with an arbitrary shaped rigid inclusion

Summary Green's functions of the displacement boundary value problem are derived within two-dimensional thermoelasticity for a heat source in an infinite plane with an arbitrary shaped rigid inclusion. The following two cases are considered: either rigid-body displacements and rigid-body rotations of the inclusion are allowed or no rigid-body displacements and no rigid-body rotations of the inclusion are possible. To solve these problems, fundamental solutions are developed for a point heat source, for rigid-body rotations of the inclusion, and for concentrated loads acting on the inclusion. Complex stress functions, temperature function, a rational mapping function and the thermal dislocation method are used for the analysis. In analytical examples, distributions of stresses are developed for an infinite plane with a rectangular rigid inclusion. Received 5 August 1998; accepted for publication 1 December 1998     

SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＦＯＲＡＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＦＩＲＳＴＯＲＤＥＲＳＹＳＴＥＭＣｈｅｎＳｏｎｇｌｉｎ（陈松林）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ８,１９８４;ＲｅｖｉｓｅｄＡｐｒｉｌ１５,１９...     

Stress intensity factor in a contact problem for a plane crack under an antiplane shear wave

The frictional contact interaction of the finite edges of a plane crack under the action of a normally incident harmonic shear wave that produces antiplane deformation is studied. The influence of the forces of contact interaction on the stress intensity factor is analyzed Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 115–119, September 2007.     

On use of the Galerkin method to solve the fracture mechanics problem for a linear crack under normal loading

The problem of contact interaction of the opposite faces of a linear crack under a normally incident harmonic tension-compression wave is numerically solved by the Galerkin method with piecewise-linear continuous elements. The dependence of the stress intensity factor (opening mode) on the wave number is investigated Published in Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 137–142, November 2005.     

The fundamental solutions for the plane problem in piezoelectric media with an elliptic hole or a crack

1.IntroductionItiswell-knownthatthefundame,ltalsolutionsorGreen'sfunctionsplayanimportantroleilllinearelasticity.Forexample,theycanbeusedtoconstructmanyanalyticalsolutionsofpracticalproblems.Itismoreimportantthattheyareusedasthefundamentalsolutionsintheboundaryelementmethod(BEM)tosolvesomecomplicatedproblem.Withthewidely-increasingapplicationofpiezoelectricmaterialsinengineeringproblems,thestudyregardingtheGreen'sfLlnctionsinpiezoelectricsolidshasreceivedmuchinterest.The3DGreen'sfunctionsi…     

Singular perturbation of boundary value problem for a vector fourth order nonlinear differential equation

We study the vector boundary value problem with boundary perturbations: ε~2y~((4))=f(x,y,y″,ε, μ) ( μ<χ<1-μ) y(χ,ε,μ)l_(χ-μ)= A_1(ε,μ), y(χ,ε,μ)l_(χ-1-μ)=B_1(ε,μ) y″(χ,ε,μ)l_(χ-μ)=A_2(ε,μ),y″(χ,ε,μ)l_(χ-1-μ)=B_2(ε,μ)where yf, A_j and B_j (j=1,2) are n-dimensional vector functions and ε,μ are two small positive parameters. This vector boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated. Under appropriate assumptions, using the method of differential inequalities we find a solution of the vector boundary value problem and obtain the uniformly valid asymptotic expansions.     

The second initial-boundary value problem for a quasilinear parabolic equations with nonlinear boundary conditions

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆａｓｔｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｏｆｄｉｖｅｒｇｅｎｃｅｆｏｒｍａｓｕｔ =(ａ(ｕ)ｕｘ) ｘ ｂ(ｕ) ｘ ｃ(ｕ)　　 (ａ( 0 ) = ∞ ) ( 1 )ｈａｓｉｍｐｏｒｔａｎｔｐｈｙｓｉｃａｌｂａｃｋｇｒｏｕｎｄ ,ｓｕｃｈａｓ [1 ] .Ｉｎｒｅｃｅｎｔｙｅａｒｓ ,ｓｏｍｅｒｅｓｕｌｔｓａｂｏｕｔ ( 1 )ｈａｖｅｂｅｅｎｏｂｔａｉｎｅｄ .Ｆｏｒｅｘａｍｐｌｅ ,[2 ] ,[3]ｒｅｓｐｅｃｔｉｖｅｌｙｄｉｓｃｕｓｓｅｄｔｈｅＣａｕｃｈｙｐｒｏｂｌｅｍｓｆｏｒｅｑｕａｔｉｏｎ( 1 )ａｎｄｕｔ=(…     

Passages to the limit in the dynamic problem for a crack at the interface between elastic media

The paper analyzes numerically the passages to the limit in the dynamic problem for a penny-shaped crack at the interface between dissimilar linear elastic, homogeneous, isotropic materials as either the frequency of harmonic load or the difference between the properties of the materials decreases. It is shown that as the frequency decreases, the solution of the dynamic problem tends to that of the static problem, and as the physical and mechanical properties of the materials become less different, the original problem goes into the dynamic problem for a crack in a homogeneous body __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 26–34, July 2008.     

Refinement of the plastic-zone boundary in the vicinity of a crack tip for the quasiviscous and viscous types of fracture

A refined solution of the elastoplastic problem of an insulated mode I crack in a thin plate of reasonably large dimensions is obtained. Estimates of the plastic zone in the vicinity of the crack tip are given for quasiviscous and viscous types of fracture.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 126–132, January–February, 2005