Asymptotic analysis of a solution of the problem of the theory of elasticity for a shallow cone

Construction of a homogeneous solution to the elasticity theory problem for an inhomogeneous truncated transversally isotropic cone

“…The results reported here are a generalization of certain results from [9,10,14,15,[17][18][19]. In the case b 12 =b 23 =λ, b 44 =G, b 11 =b 22 =2G+λ, all the solutions derived fully coincide with the solutions for the radial-inhomogeneous isotropic cone [14,19].…”

“…In the case b 12 =b 23 =λ, b 44 =G, b 11 =b 22 =2G+λ, all the solutions derived fully coincide with the solutions for the radial-inhomogeneous isotropic cone [14,19]. When G and λ are constant, the solutions derived coincide with the solutions for the isotropic cone [9,10,18]. In particular, when b The current work is theoretical in nature.…”

“…Asymptotic methods have made a significant contribution to solving the three-dimensional problems of the theory of elasticity for the cone. In [8][9][10], a method of homogeneous solutions was used to study the axisymmetric problem of the theory of elasticity for a truncated isotropic cone, when the lateral part of the cone is free of stresses. However, the system of homogeneous solutions constructed in [8] is not complete and therefore does not make it possible to meet arbitrary boundary conditions at the ends of the cone.…”

“…However, the system of homogeneous solutions constructed in [8] is not complete and therefore does not make it possible to meet arbitrary boundary conditions at the ends of the cone. In [9,10], the roots of the characteristic equation are classified with a relatively small parameter characterizing the thinness of the cone; solutions were constructed depending on the roots of the characteristic equation. It is shown that the stress-deformed state in a cone of small thickness consists of three types: a penetrating stressed state, a simple edge effect, and a boundary layer.…”