D/BEM implementation of Robinson's viscoplastic model in creep analysis of metals using a complex variable numerical technique

Abstract

A new boundary element approach for the solution of time-dependent inelastic problems arising in creeping metallic structures subjected to the combined action of high temperature gradient and quasi-static mechanical loading conditions is investigated. The new approach allows the use of complex variable techniques in the boundary element procedure for the evaluation of stress components as derivatives of the displacement integral equations. This methodology makes faster and more accurate the conventional boundary element method. To validate the efficiency of the proposed method in the implementation of Robinson's viscoplastic model, the results obtained using the present methodology are compared with those obtained by using known analytical and finite element solution for the analysis of a thick-walled internally pressurized cylinder and an experimental cylindrical thrust chamber in plane strain under general thermomechanical loading histories.

Introduction

Recently, remarkable progress has been made in the technological applications of metals and alloys at elevated temperatures and thus, in order to improve the high temperature inelastic behavior of structures, especially those used in nuclear and aerospace industries, some new and more realistic constitutive models (called viscoplastic models) have been proposed. A very comprehensive review of numerous viscoplastic models was developed by Walker [1] and Lindholm et al. [2], Lemaitre and Chaboche [3], Freed and Walker [4] and Saleeb et al. [5].

The main mathematical feature of these newer viscoplastic models is that the non-elastic strain rates can be expressed as functions of the current values of stress, temperature and some certain well defined state variables. The mathematical formulation of these viscoplastic models became very complex since they incorporate constitutive equations, which were highly non-linear and mathematically stiff. Thus, it is concluded that many practical problems, which involve complex geometries as well as complex thermomechanical loading histories such as those arising in hot gas-path components of gas turbine engines and rocket engines, have to be solved in the environment of such numerical solution methodologies which could make these viscoplastic models adaptable for realistic structural and life analyses of this kind of structural components. Robinson's model [6], [7] is one of these viscoplastic models using state variables that aim to represent inelastic behavior of metallic structures more faithfully than is possible with the traditional inelastic models in a variety of applications at elevated temperature thermomechanical loading.

Arya [8], [9], Arya and Arnold [10] and Arya and Kaufman [11] proposed a finite element methodology (FEM) for the inelastic analysis of structures with material behavior governed by the Robinson's constitutive viscoplastic model. They used a rate formulation of the governing differential equation of the problem. This implementation was exercised first on several uniaxial problems involving isothermal and non-isothermal cyclic loading and later more complex structural problems were treated. This FEM was found to work efficiently for the selected numerical examples.

In Providakis and Kourtakis [12] demonstrated that the domain/boundary element method (D/BEM) is a very powerful method with several potential advantages over the FEM for solving non-linear time-dependent inelastic deformation problems in the presence of high temperature gradients. One of the main advantages is that the number of unknowns in the final algebraic system of equations is proportional to the number of boundary elements in D/BEM as opposed to the total number of nodes in FEM. However, one of the most difficult problem in BEM-related problems was the numerical simulation of domain-based effects (e.g. inertial and interior loading effects, inelastic and thermal strain effects). This is caused by the existence of domain integrals in the formulation, which can only seldom be directly transferred into boundary integrals forms. In high temperature and time-dependent creeping problems, domain integrals arise due to thermal and inelastic term effects. However, some inherent difficulties, such as the strong singularities arising in the derivation of the domain integrals, have stymied the further development of this method.

Several approaches have been developed to deal with the removal of the strong singularities arising in the domain integrals such as those referenced in Gao and Davies [13]. The present paper could be considered as an extension of the previous work presented in Providakis and Kourtakis [12] where a boundary element methodology was proposed for the implementation of Robinson's viscoplastic model to the analysis of time-dependent non-linear inelastic deformations of metallic structural components subjected to high temperature thermomechanical loading histories. The present formulation is based in the same as the above work boundary element implementation of Robinson's viscoplastic model but now adopts, actually for the first time, the use of a complex variable numerical technique to eliminate the strong singularity involving in the evaluation of internal stresses in this kind of complicated inelastic problems in the existence of high temperature gradients. This is achieved by an appropriate introduction of complex variables into the displacement integral equations. This numerical approach which was proposed by Lyness and Moler [14] was applied to the computation of the derivatives of any complicated function by using a step size in the imaginary part of the variable. The proposed new approach of D/BEM method can be considered numerically as an ‘exact derivative method’ since its numerical accuracy can be guaranteed by taking a sufficiently small step size. Following this concept, the stresses at any internal point could be accurately obtained by using only the numerical derivative of the displacement integral equations which, however, involve only weakly singular integrals. This technique makes faster and more accurate the conventional boundary element algorithm since the complicated and error sensitive methodologies proposed by other researchers for evaluating the internal stresses could be now eliminated.

In the present boundary element formulation the time rates of change of displacements and stresses are generated at any time step. The time histories of the quantities of interest are then obtained by using a time integration scheme with automatic time-step control. For the implementation of the proposed boundary element formulation isoparametric linear boundary elements were used. Due to the lack of existence of other available solutions, to compare with the present D/BEM approach results an analytical solution is developed and discussed herein which is, however, valid only for the case of the axisymmetric analysis of a thick-walled thermoviscoplastic cylinder. Then, a comparison of the two solutions is achieved by developing two different programs, the first one for the present viscoplastic D/BEM and the second one for the analytical axisymmetric approach. In order to avoid programming bias and make the comparison as meaningful as possible both of the programs are created by the same research group. Finally, numerical examples are presented and compared with analytical and FEM solutions in the last section of this paper for plane strain deformation problems of cylinders subjected to mechanical and high temperature loading histories.