M- and L-integrals enclosing two circular holes in an infinite plate

An analytic solution is presented for stresses induced in an infinite plate with two unequal circular holes by remote uniform loadings and arbitrary internal pressures in the holes. The solution has been obtained by using the general expression for a biharmonic function in bipolar coordinates provided by Jeffery (1921). The Airy stress function is decomposed in the sum of a fundamental stress function for an infinite plate remotely loaded, which gives non vanishing tractions on the circular boundaries, and an auxiliary stress function required to satisfy the boundary conditions on the pressures at the edges of the holes, which produces vanishing stresses at infinity. By using the Jeffery solution, the problem of a circular disk containing a sliding eccentric circular inclusion has been recently solved by Radi and Strozzi (2009).Once the stress and displacement fields are obtained in closed form, the path independent Jk- (k = 1, 2), M- and L-integrals introduced by Knowles and Sternberg (1972) and Budiansky and Rice (1973) are analytically calculated on a closed contour encircling both holes by considering traction free hole surfaces. These integrals play an important role in the description of multiple defects damaged brittle materials. Physically, the Jk-, M- and L-integrals can be interpreted as the energy release rate for uniform movements, expansion, and rotation of the defects, respectively. Results are here presented for varying loading orientation angle ζ and holes geometry. The J1- and J2-integrals calculated for a closed contour enclosing both holes are found to vanish, whatever be the remote loading orientation and holes geometry. These results confirm the conservation laws proposed by Chen and Hasabe (1998) and then proved by Chen (2001) for multiple discontinuities, such as cracks, voids and inclusions, subject to remote uniform loading conditions, when the integration contour encloses all the discontinuities. Differently from the J1- and J2-integrals, the M- and L-integrals do not vanish when the integration contour encloses both holes. In particular, the M-integral attains a maximum for a certain loading orientation angle ζ0, and, correspondingly, the L-integral becomes vanishing small. For ζ < ζ0 the L-integral turns out to be negative, whereas for ζ > ζ0 it assumes positive values. Chen (2001) and Hu and Chen (2009, 2011) observed that an implicit relation exists between the M-integral and the reduction in the effective elastic modulus. These authors showed that the loading direction along which the M-integral becomes maximum coincides with the direction corresponding to the minimum of the effective elastic modulus, due to the presence of the holes. Conversely, the loading direction along which the M-integral becomes minimum is just the direction of the maximum effective elastic modulus. This occurrence allowed these authors to conjecture the possiblity of formulating the effective elastic properties and describing the damage level induced by interacting holes in terms of the M-integral, although the proper mathematical formulation has not been adequately investigated yet.The purpose of the present contribution is to provide some basic understanding for the role played by conservation laws in multiple defects analysis.