Complex Aleksandrovich Representation of Solutions in Displacements in the Three-Dimensional Theory of Elasticity

The analytical possibilities of the representation of the solution in displacements in the three-dimensional theory of elasticity in the form of a two-dimensional complex structure, proposed in the 70s of the XX century in the works of A.I. Aleksandrovich, are discussed. Complex-valued displacements are sought in the form of a holomorphic expansion as series in powers of complex variables with antiholomorphic coefficients and in powers of conjugate complex variables with holomorphic coefficients. All holomorphic and antiholomorphic functions are expressed in terms of four arbitrary holomorphic functions. As test special cases leading to classical solutions known in the theory of elasticity, a plane strain state, antiplanar strain, a three-dimensional strain state in a thin plate of variable thickness, axisymmetric displacement fields, which are realized, in particular, under a linear combination of internal (external) pressure, (r theta)-torsion and axial (rz)-shear in a cylindrical layer and (theta z)-torsion of a solid cylinder are considered. In terms of complex-valued displacements, a system of equations of the axisymmetric theory of elasticity is written, the fundamental solution of which is a general representation of the displacement field in the axisymmetric case, similarly to the Kolosov-Muskhelishvili formulas in the plane problem.