Solution of the Axisymmetric Problem of Creep and Damage for a Piecewise Homogeneous Body with Meridional Section of Any Shape

Abstract

UDC 539.3 We consider the axisymmetric problem of creep and creep-induced damage for piecewise homogeneous bodies with meridional sections of any shape. We develop a method for the solution of the initial boundary-value problem based on the combined application of the R -function method and the Runge– Kutta–Merson method. The structures of the solution for the main types of boundary conditions are constructed. We present an example of calculation of creep and long-term strength for a three-layer cylinder used as a computational scheme of a solid-oxide fuel element. State-of-the-Art of the Problem. Statement of the Initial Boundary-Value Problem of Creep The problems of determination of the stress-strain state and strength of piecewise homogeneous cylindrical bodies are thoroughly described in the Ukrainian and foreign literature. At the same time, the nonlinear deformation of piecewise homogeneous bodies, in particular, the problems of creep and damage have not been adequately studied. This is connected with the complexity of solution of nonlinear initial boundary-value problems for piecewise homogeneous systems and with difficulties connected with the construction of determining relations, which must take into account various effects of deformation of contemporary materials. Moreover, the practical analyses of creep, damage, and long-term strength often require taking into account the complex geometric shape of the body or interfaces of its components. Consider a body of revolution of finite sizes referred to a cylindrical coordinate system Orzϕ , which consists of M components V1,V2,…,VM (V = V1 V2 …VM ) rigidly connected with each other. The body is under the action of external surface loads applied to a part Sp of its surface and a temperature field T = T(r,z,t) . The distribution of loads on Sp and given kinematically possible displacements on the surface Su are such that the desired solution is independent of ϕ . The external forces and temperature very slowly vary with time that and, hence, the inertial terms in equations of motion can be neglected. The strains in the body remain small in the process of creep. We denote by ∂Vab the interface of the neighboring parts of the body Va and Vb . The axis Oz coincides with the axis of revolution. The section of the body in the plane rOz has the shape of the domain Ω with boundary ∂Ω . The domain Ω is the union of constituent domains Ωk , k = 1,…, M , with boundaries ∂Ωk . The rates of displacements and external loads are given on the parts of the boundary ∂Ωu and ∂Ωp , respectively. We denote by ∂Ωab the interface of the neighboring domains Ωa and Ωb . By ∂Ωab and ∂Ωba , we denote the sides of the surface ∂Ωab that belong to Ωa and Ωb , respectively. Assume that the materials