### Abstract:

The Green’s tensor for the first (displacement given) boundary value problem of elasticity for a circular domain is computed under a closed form expression. The method of solution uses the “incompressible influence element” for which the Green’s tensor is given by representation using Green’s function for Poisson’s equation. Using such a representation, is shown that the main problem is to find the dilatation along the boundary induced by the displacements Green’s function. The volume dilatation is than obtained by solving an integral equation along the circular boundary. Explicit expressions are obtained for the Green’s displacements tensor and for the traction along the circular boundary, allowing expressing the solution for any kind of “displacement” boundary condition and body forces. On the basis of the constructed Green’s tensor is given the integral formula which presents a generalization of the well known Poisson’s integral formula from the theory of harmonic potentials onto the theory of elasticity. An example of application of Green’s tensor in micromechanics of defects in solids as radial Volterra’s slip dislocation in an elastic circle is presented. These results were obtained in explicit form and for the first time. Applied here the “incompressible influence element method” (IIEM) can be used to derive the Green’s tensor for a wide classis of different boundary value problems for canonical domains of many systems of coordinates. So, IIEM will increase considerable the possibilities to solve new complicate boundary value problems in bounded and “unbounded” solids, acted by different inner actions: body forces, temperature dislocations, Volterra’s dislocations, eigenstrains, inclusions etc and any boundary displacements.