Impact response of a finite crack in an orthotropic strip

Summary The elastodynamic response of a finite crack in an infinite orthotropic strip under normal impact is investigated in this study. The crack is situated symmetrically and oriented in a direction normal to the edges of the strip. Laplace and Hankel transforms are used to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution to the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor for some fiber-reinforced composite materials are obtained and the results are graphed to display the influence of the material orthotropy.     

Impact response of an anti-plane crack in a FGPM bonded to a homogeneous piezoelectric strip

A finite crack under transient anti-plane shear loads in a functionally graded piezoelectric material (FGPM) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential functions along the thickness of the strip, and that the two layered strips is under combined anti-plane shear mechanical and in-plane electrical impact loads. The analysis is conducted on the electrically unified crack boundary condition. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Fredholm integral equations of the second kind in the Laplace transform domain. Then, a numerical Laplace inversion is performed and the dynamic intensities are obtained as functions of time and geometric parameters, which are displayed graphically.     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

Anti-plane shear problem for an edge crack in a finite orthotropic plate

The problem of an edge crack in a finite orthotropic plate under anti-plane shear is considered. The boundary collocation method is used to calculate the mode III stress intensity factor (SIF). For the case in which the material is isotropic, the present results agree very well with those obtained by using the integral equation method. Furthermore, the method can be extended readily for general cases with arbitrary geometrical and boundary loading conditions and material properties.     

Impact response of a strip composite with a finite crack

The dynamic response of a central crack in a strip composite under normal impact is analyzed. The crack is oriented normally to the interfaces. Laplace and Fourier transform techniques are used to reduce the elastodynamic problem to a pair of dual integral equations. The integral equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometrical parameters is discussed.     

Plastic zones in an orthotropic plate of finite width containing a Griffith crack

The problem of an orthotropic infinite plate of finite width containing a centrally located stressed Griffith crack is considered. The crack is located perpendicular to the edges of the orthotropic plate. The crack tips are fully yielded and in the inelastic zones the material carries only constant normal stresses equal to the yield stress. Dugdale's model is employed to find the effects of the material anisotropy on the size of the plastic zones around the crack tips. Graphical results showing the effects of anisotropy on the length of the plastic zone are also presented.     

Closed form solutions for a finite length crack moving in an orthotropic layer of finite thickness

In this short note, an integral transform technique is used for the elastodynamic plane problem of a crack of fixed length propagating at a constant speed parallel to the surfaces of an orthotropic layer. Two problems are considered; (i) the anti-plane shear load is applied on the surfaces of the layer, and (ii) the surfaces of the layer are clamped and displaced equally in opposite directions to produce anti-plane shear motion of the crack. Closed form solutions are obtained for the stress intensity factors and numerical values for the stress intensity factors are graphically presented for several orthotropic materials.     

Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear

Summary Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.     

On saturation-strip model of a permeable crack in a piezoelectric ceramic

Summary. The saturation-strip model for piezoelectric crack is re-examined in a permeable environment to analyze fracture toughness of a piezoelectric ceramic. In this study, a permeable crack is modeled as a vanishing thin but finite rectangular slit with surface charge deposited along crack surfaces. This permeable saturation crack model reveals that there exists a possible leaky mode for electrical field, which allows applied electric field passing through the dielectric medium inside a crack. By taking into account the leaky mode effect, a first-order approximated solution is obtained with respect to slit height, h ₀, in the analysis of electrical and mechanical fields in the vicinity of a permeable crack tip. The permeable saturation crack model presented here also considers the effect of charge distribution on crack surfaces, which may be caused by any possible charge-discharge process in the dielectric medium inside the crack. A closed form solution is obtained for the permeable crack perpendicular to the poling direction under both mechanical as well electrical loads. Both local and global energy release rates are calculated. Remarkably, the global energy release rate for a permeable crack has an expression, where M is elastic modulus, a is the half crack length, is permittivity constant, and e is piezoelectric constant. This result is in a broad agreement with some experimental observations and may be served as the fracture criterion for piezoelectric materials. This contribution elucidates how an applied electric field affects crack growth in piezoelectric ceramic through its interaction with permeable environment surrounding a crack. The author would like to acknowledge the support from the Academic Senate Committee on Research at University of California (Berkeley) through the fund of BURNL-07427-11503-EGSLI.     

Effects of remanent field on an elliptical flaw and a crack in a poled piezoelectric ceramic

Commonly used piezoelectric ceramics such as PZT and PLZT are polarized ferroelectric polycrystals. After poling, remanent strains and a remanent polarization exist in a ceramic material. Remanent field can affect the electroelastic field and consequently plays a critical role in fracture of poled ceramics. Based on a linear constitutive law, the electroelastic field and the energy release rate of an elliptical cavity (or a crack) in a poled piezoelectric are re-examined in this study by including the effects of remanent field. It is noted that the remanent field generally has a minor effect on the stress field and a pronounced effect on the electric field at the apex of the major axis of an elliptical flaw. When the permittivity of the cavity is small, the effect of remanent polarization is similar to that of a very strong electric field applied along the poling direction. However, for the case of a conducting flaw, the remanent field does not influence the electroelastic field and energy release rate. Energy release rate of a flaw in a poled ferroelectric ceramic with and without the remanent polarization is generally different.     

Fracture of a finite piezoelectric layer with a penny-shaped crack

This paper studies a penny-shaped crack in a finite thickness piezoelectric material layer. The piezoelectric medium is subjected to a thermal flux on its top and bottom surfaces. Both thermally insulated crack and heated crack are considered. Numerical solution for the finite layer thickness is obtained through the solution of a pair of dual integral equations. The result reduces to the closed form solution when the thickness of the piezoelectric layer becomes infinite. Exact expressions for the stress and electric displacement at the crack border are given as a function of the stress intensity factor, which is determined by the applied thermal flux. This paper is useful for the reliability design of piezoelectric materials in thermal environments.     

A dislocation interacts with a finite crack in piezoelectric media

Analytical solutions for the generalized two-dimensional problem of a dislocation interacting with a finite crack in piezoelectric media are formulated via Stroh formalism. Two kinds of electrically boundary conditions along the crack surface, namely, electrically impermeable and electrically permeable are studied in the analysis. Based on the complex variable method and the perturbation technique, the closed form solutions are obtained. The field intensity factors at the crack tip and the image forces on the dislocation due to the crack are computed and discussed.     

Impact response of a cracked orthotropic medium-revisited

Elastodynamics response of an infinite orthotropic medium containing a central crack under impact loading has been investigated. Laplace and Fourier transforms have been employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform domain which has finally been solved by the method of iteration in the low frequency case. Analytic expressions for the stress intensity factors and crack opening displacement are also obtained for low frequency.     

Strain energy release rates for an interface crack in orthotropic media--a finite element investigation

Strain energy release rate (SERR) components for an interface crack in two-dimensional orthotropic media were obtained using finite element (FE) analysis. The elastic analysis of interface cracks results in oscillatory singularity. This is prevalent over a very small zone near the crack-tip, where the traction free crack faces undergo unacceptable deformations resulting in the interpenetration of crack faces. The individual and total strain energy release rates are calculated using modified crack closure integral (MCCI) method. Although the total SERR converges, it is observed that the individual SERR components are dependent on the values of the smallest element size (Δa) at the crack-tip. It is observed that both the crack opening and sliding displacements are oscillatory when the interpenetration is allowed in the contact zone. The contact zone length (r_c) calculated using Suo's analytical expression [Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. Royal Soc. London, Ser A427 (1990) 331] is in good agreement with the results from FE analysis and MCCI calculations. However, for the chosen material properties, the estimated contact zone length based on the analytical expression proposed by Ni and Nemat-Nasser [J. Mech. Phys. Solids 39 (1991) 113] exhibits a large deviation from the present FE results. It is seen that the mode-II behavior dominates the crack growth, even under mode-I loading.     

Cruciform crack in an orthotropic elastic plane

The problems of determining the stress and displacement fields in an infinite orthotropic plane containing a cruciform crack 387-1, y=0 and 387-2, x=0 when (I) the shape of the crack is prescribed and (II) the cracks are opened by given normal pressures, are reduced to mixed boundary value problems for the quarter plane. Using integral transform techniques, a closed form solution is obtained for problem I, whereas the solution of problem II has been reduced to solving a Fredholm integral equation of second kind with non-singular kernel. Numerical calculation of the stress intensity factor and crack energy in the case of a linear loading function for various crack lengths are presented for problem II, using the values of material constants for a Boron-Epoxy composite.     

Moving Griffith crack in an orthotropic material

In this paper, an integral transform technique is employed to solve the plane elastodynamic problem of a crack of fixed length propagating at a constant speed in a uniformly stressed medium. It is assumed that the crack is located in a plane of elastic symmetry of the material. The stresses and strains ahead of the crack tip are determined explicitly and the conditions governing the initial growth of the crack are investigated using two current theories of fracture (Maximum normal stress and Minimum strain-energy density). Based on these theories, it appears that, depending upon the particular orthotropy of the material, the crack may extend in a straight line for all velocities or may immediately branch out at low velocities (compared to the shear wave velocity of the material) or may start propagating along its initial position for small velocities and then, as the velocity increases, may curve and branch out.     

Scattering of antiplane shear waves by a finite crack in piezoelectric laminates

Summary Following the dynamic theory of linear piezoelectricity, we consider the scattering of horizontally polarized shear waves by a finite crack in a composite laminate containing a piezoelectric layer. The piezoelectric layer is bonded between two half-spaces of a different elastic solid. The crack is normal to the interfaces and is placed at an equal distance away from them. Both cases of a partially broken layer and a completely broken layer are studied. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a singular integral equation. The propagation of symmetric first mode is studied numerically, and the dynamic stress intensity factor and the dynamic energy release rate are obtained for some piezoelectric laminates.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.