Solution of the Stochastic Boundary-Value Problem of Steady-State Creep for a Thick-Walled Tube Using the Small-Parameter Method

Abstract

The physically and statistically nonlinear problem of steady-state creep for a thick-walled tube loaded by internal pressure is solved in the third approximation using the small-parameter method. The variances of random creep strain rates and displacements are calculated. The results obtained are compared with the solution of the same problem in the first and second approximations. A reliability assessment method for the tube using the strain failure criteria is proposed. 1.The substantial effect of random perturbations of the mechanical characteristics of materials on the stress and strain fields and the need for developing the corresponding stochastic models for strength analysis were discussed in many papers (see, e.g., [1][2][3]). This problem is of utmost importance for creep strain, for which the spread of experimental values is as high as 50-70% and one has to consider these results as acceptable [3][4][5].Determining the strains and stresses of structural members subjected to nonlinear-creep conditions is a very difficult problem even in the deterministic formulation. The necessity of considering the microheterogeneities of the material leads to stochastic boundary-value problems, in which statistical nonlinearity should be taken into account in addition to the physical nonlinearity of the governing equations. Because of these difficulties, stochastic boundary-value creep problems admit analytical solutions only in the simplest cases [6-9].To solve stochastic boundary-value problems in elastic and creep regions, the small-parameter method is used [6-10]. However, owing to substantial difficulties that arise in calculating the second and higher order moments of a random function, this method provides solutions of the stochastic boundary-value problems of steady-state creep only in the first approximation [6,8].In the present paper, the analytical solution of the boundary-value problem of steady-state creep of a thickwalled tube loaded by internal pressure is constructed to the third approximation using the small-parameter method.We consider this problem in cylindrical coordinates for plane strain [ε z (r, t) = 0 orε z (r, t) = 0], assuming that the stochastic heterogeneities of the tube material are described by a function of one variable -the radius r. In this case, the components of the strain and stress tensors are random functions of the radius r.In accordance to the theory of viscous flow (steady-state creep), the creep strains ε r and ε ϕ are described by the following rheological relations in stochastic form [9]:Here σ r and σ ϕ are the radial and hoop stresses, respectively, U (r) is the random function governing the stochastic heterogeneity of the tube material, whose statistical characteristics are known: U = 0 and U 2 = 1, α is the

Nonlinear stochastic creep problem for an inhomogeneous plane with the damage to the material taken into account

“…Under creep conditions, the development of analytical methods for solving stochastic boundary-value problems is significantly complicated, primarily because of the physical and stochastic nonlinearity of the governing equations. One-dimensional stochastic problems of steady-state creep (for example, a tube subjected to internal pressure) can be solved with any degree of accuracy using the small parameter method [2]. As regards plane and spatial creep problems, they have been solved only as a first approximation using steady-state creep theory [3][4][5].…”