ORTHOGONAL POLYNOMIAL SOLUTIONS FOR PRESSURIZED ELLIPTICAL CRACKS

Consider an infinite elastic solid containing a flat elliptical crack, which is opened symmetrically by a prescribed pressure p. We expand p and the crack-face displacement w as Fourier series in φ, and expand each Fourier component as a series of orthogonal polynomials in ρ, where (in Cartesian coordinates) the crack occupies the surface {(x, y, z): x=aρ cos φ, y=bρ sin φ, z=0, 0<=p<1, 0<=φ<2π). We obtain explicit relations between the coefficients in the series for w and p, and derive a formula for the stress-intensity factor. As an example, we consider the quadratic pressure p(x, y)=A+Bx+Cy+Dx²+Exy+Fy² in detail, and compare our solution with those of other authors.