We investigate the effective conductivity tensor for unidirectionally arbitrary distributed cylinders when the contact between the components is not perfect. The corresponding boundary value problem is reduced to a system of functional-differential equations, which is solved using symbolic computations. The major advantage of the method presented here is to provide approximate analytical expressions for the local thermal flux and for the effective conductivity in a systematic manner in terms of the concentration of inclusions, of the contrast parameter, of the thermal resistance and of the location of the parallel fibres.

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S ₂ (for conductivity) and T ₂ (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S ₂ and T ₂ for different lattices are established. In particular, the value S ₂ = π for the square and hexagonal arrays is discussed and T ₂ = π /2 for the hexagonal is deduced.

We study the mechanical properties of unidirectional fibers with circular section embedded into a matrix by the method of complex potentials. First, a finite number $n$ of inclusions is considered ...

According to Muskhelishvili’s approach, two-dimensional elastic problems for media with non-overlapping inclusions are reduced to boundary value problems for analytic functions in multiply connected domains. Using a method of functional equations developed by Mityushev, we reduce such a problem for a circular multiply connected domain to functional-differential equations. It is proved that the operator corresponding to the functional-differential equations is compact in the Hardy–Sobolev space. Moreover, these equations can be solved by the method of successive approximation under some natural conditions.