On wrinkled penny-shaped cracks

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Abstract

A nominally flat penny-shaped crack is subjected to a static loading. A perturbation method is developed for calculating the stress-intensity factors, based on an asymptotic analysis of the governing hypersingular boundary integral equation for the crack-opening displacement. Comparisons with known exact solutions for an inclined flat elliptical crack and for a crack in the shape of a spherical cap are made.

Introduction

A basic problem in classical fracture mechanics is the determination of the stress-intensity factors for a crack in an elastic solid. For three-dimensional problems, there are very few explicit solutions: one example is a penny-shaped crack under arbitrary loads. Recently, there has been some interest in calculating corrections to these stress-intensity factors when the crack is perturbed (Rice, 1989; Leblond and Mouchrif, 1996; Movchan et al., 1998). Taking the penny-shaped crack as the reference crack, two classes of perturbation emerge. First, the crack can be perturbed in its own plane. Such perturbations have been studied by Gao and Rice (1987) and by Martin (1996) for pressurized cracks (a scalar problem) and by Gao (1988) and Martin (1995) for shear-loaded cracks (a vector problem); comparisons with known exact solutions for flat elliptical cracks were made.

Second, the penny-shaped crack can be perturbed out of its own plane, giving a ‘wrinkled’ crack Ω. This is the problem studied here. Specifically, let us define Ω byΩ:z=εf(x,y),(x,y)∈D,where (x,y,z) are Cartesian coordinates, f is a given function, D is a flat circular disc in the xy-plane and ε is a small parameter. We have previously considered analogous scalar problems, such as calculating the added mass for potential flow past a wrinkled rigid disc (Martin 1998a, Martin 1998b). We have also studied the plane-strain problem of a slightly curved crack (Martin, 2000). We use a similar method here, although the details are more complicated. Thus, we first reduce the boundary-value problem to a boundary integral equation over the crack Ω; we choose to use a hypersingular integral equation for the crack-opening displacement (COD). Next, we project this equation onto the disc D, leading to a two-dimensional hypersingular integral equation over D. At this stage, the analysis is exact. Then, we introduce Eq. (1.1), leading to a sequence of hypersingular integral equations for each term in the regular expansion of the COD in powers of ε. These equations are just the corresponding hypersingular integral equation for a penny-shaped crack, with various forcing functions. Such equations can be solved exactly. We do this, in detail, for two particular crack geometries, namely, an inclined flat elliptical crack and a shallow spherical-cap crack. We obtain agreement with the known exact solutions, correct to order ε. These are stringent tests of the theory, because the solution for the spherical-cap crack, in particular, is very complicated. Indeed, one may regard the asymptotic approximations as validating the exact solution!

The use of hypersingular integral equations leads to a simpler formulation than would follow from the use of regularized integral equations; see, for example, the equations derived by Le Van and Royer (1986). Shliapoberskii (1978) and Xu et al. (1994) have developed perturbation theories based on regularized integral equations for dislocation densities (tangential gradient of the COD). Shliapoberskii (1978) assumed that the crack edge ∂Ω was in the plane z=0 (which is a severe restriction for three-dimensional problems) and did not give any explicit applications. Xu et al. (1994) were able to find the first-order correction for a semi-infinite crack.

To conclude this introduction, let us compare our approach with a more direct treatment, in which one tries to perturb the boundary-value problem itself. This would inevitably be a singular perturbation (unless the crack edge is fixed), because the perturbed fields are singular at the actual crack edge, not at the edge of the unperturbed crack. This leads to various technical complications, such as the introduction of a boundary layer near the crack edge. Our approach is less direct, perhaps, but it leads to regular perturbations: this happens because we work on the crack faces only, not within the solid, and, physically, we do not expect the COD to be much different for the perturbed and unperturbed cracks.

Section snippets

An integral equation

In this section, we derive an exact hypersingular integral equation for the COD across a crack Ω in a three-dimensional homogeneous isotropic elastic solid. We assume that Ω is modelled by a smooth open surface with a simply connected edge ∂Ω. The basic ingredient is Kelvin's fundamental solutionGij(P,Q)=[16πμ(1−ν)R0]−1(3−4ν)δij+RiRj.Here, i,j=1,2,3, μ is the shear modulus, ν is Poisson's ratio, the points P and Q have Cartesian coordinates (x1,x2,x3) and (x1′,x2′,x3′), respectively, δij is the

Projection

The integral equation (2.4) is exact and it holds on the surface Ω. It is more convenient to write (2.4) on a fixed reference surface D. Thus, we suppose for the moment that the surface Ω is given byΩ:x3≡az=aF(x,y),(x,y)∈D,where D is the unit disc in the xy-plane, centred at the origin, a is a length-scale, and the (dimensionless) function F gives the shape of Ω. Explicitly, the integration point q and the field point p are specified byq:(x1′,x2′,x3′)=(ax,ay,aF(x,y)),(x,y)∈Dandp:(x1,x2,x3)=(ax0

The flat penny-shaped crack

For a penny-shaped crack, we have F(x,y)=c, a constant. Hence, Γ=1 and all of F1, F10, F2, F20, Λ, Φ, Φ0 and Ψ are zero. It follows from the expressions given in Appendix A thatS̃ij=(1−ν)−1Sij0,whereS110=14(2−ν+3νcos2Θ),S220=14(2−ν−3νcos2Θ),S120=S210=32νsin2Θ,S330=12and S130=S230=S310=S320=0. Thus, we obtain the following integral equations:18πD{(2−ν+3νcos2Θ)ũ1+3νũ2sin2Θ}dAR3=(1−ν)t̃1(x0,y0),18πD{3νũ1sin2Θ+(2−ν−3νcos2Θ)ũ2}dAR3=(1−ν)t̃2(x0,y0),14πDũ3dAR3=(1−ν)t̃3(x0,y0)for (x0,y0)∈D. These

Wrinkled penny-shaped cracks

Suppose thatF(x,y)=εf(x,y),where ε is a small dimensionless parameter and f is independent of ε. SettingΛ=ελwithλ={f(x,y)−f(x0,y0)}/R,we find from the expressions given in Appendix A thatS̃ij=(1−ν)−1{Sij0+εSij1+O(ε2)}as ε→0, whereS131=32λcosΘ−14(2−ν+3νcos2Θ)f134νf2sin2Θ−14{1+2ν+3(1−2ν)cos2Θ}f1034(1−2ν)f20sin2Θ,S231=32λsinΘ−34νf1sin2Θ−14(2−ν−3νcos2Θ)f234(1−2ν)f10sin2Θ−14{1+2ν−3(1−2ν)cos2Θ}f20,S311=32λcosΘ−14{1+2ν+3(1−2ν)cos2Θ}f134(1−2ν)f2sin2Θ−14(2−ν+3νcos2Θ)f1034νf20sin2Θ,S321=32λsinΘ−34

Example 1: inclined elliptical crack

Suppose that Ω is a flat elliptical crack, lying in the plane z=xtanγ. Let X and Y be Cartesian coordinates on this plane, so thatX=xcosγ+zsinγ,Y=yandZ=zcosγ−xsinγ,where Z is a coordinate perpendicular to the plane. Then, the ellipse Ω with ∂Ω given byX2cos2γ+Y2=1can be specified byz=F(x,y)=xtanγ,(x,y)∈D.

For small inclinations of the ellipse to the plane z=0, set ε=tanγ and f(x,y)=x. Thus, f1=f10=1, f2=f20=0 and λ=cosΘ, whence , , , giveS131=S311=14ν(−1+3cos2Θ)andS231=S321=34νsin2Θ.

Next, we

Example 2: spherical-cap crack

Consider a crack in the shape of a spherical cap, given byx3=c−c2−x12−x22,x12+x22≤a2,where c is the radius of the sphere. The cap subtends a solid angle of 2π(1−cosα) at the centre of the sphere, where sinα=a/c. In dimensionless variables, the cap is given byz=F(x,y)=(c/a)−(c/a)2−x2−y2,(x,y)∈D.

We shall consider a shallow spherical cap, given approximately by z=εf(x,y) withf(x,y)=12(x2+y2)=12r2andε=a/c=sinα.We have f1=x, f2=y and λ=12{(x+x0)cosΘ+(y+y0)sinΘ}, whenceS131=(−1+3cos2Θ)P(x;x0)+3sinP

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