Stress concentrations in the particulate composite with transversely isotropic phases. (Q1430795)

In this paper, the Podilchuk's method developed for a solid with a single inclusion subjected to polynomial external load and based on the separation of variables in the governing equations describing steady stress in transversely isotropic solids, using the properly introduced curvilinear coordinates, is expanded to the many-particle models of composite with transversely isotropic phases. The essence of the method proposed is the multipole expansion technique, reducing the complicated original boundary value problem for three-dimensional multiple-connected domain to a set of ordinary linear algebraic equations. First, the author obtains, for one-particle problem, an exact analytic solution written in compact matrix-vector form, in the case of arbitrary oriented anisotropy axes of the matrix and inclusion materials. The theory developed is valid, and the solution remains exact for arbitrary polynomial external load. The obtained series expansions contain the higher (\(t > 1\)) harmonics. This result is used to derive accurate, asymptotically exact solutions of many-particle problems. Then, the method developed is applied to obtain accurate solutions for a solid containing a finite array and infinite periodic, lattice type array of inclusions. By using the re-expansion formulae, the problem for finite array of \(N\) spherical inclusions is reduced to a coupled set of \(N\) ``a medium with one inclusion in the inhomogeneous external field'' problems. The third model considered is an unbounded medium containing a spatially periodic array of inclusions. The numerical results for the cases: (i) a medium with a single spherical cavity; (ii) a solid containing two particles/cavities; (iii) a simple cubic array of particles embedded in the transversely isotropic matrix, -- demonstrate the accuracy and numerical efficiency of the method. Finally, the author discusses how the mismatch of elastic properties, anisotropy degree, orientation of anisotropy axes and interaction between the inclusions can influence the stress concentration at the inclusion interface.