Quadratic Programming Method for Numerical Modeling of Elastoplastic Contact Problems: Theory, Software, Applications

This paper reviews some advances and applications in Parametric Quadratic Programming (PQP) method for numerical modeling of elastoplastic contact problems. The Parametric Variational Principle (PVP) and the corresponding finite element model for numerical simulation of 3D elastoplastic frictional contact problems with isotropic/anisotropic (orthotropic) friction law are presented. The finite element software JIFEX is then developed with the application oriented concept for nonlinear analysis of complex structures in general purposed engineering. Some typical engineering applications such as compressor impeller and the railway wheel/rail contact analysis are shown to illustrate the potential of the software developed. Introduction Elastoplastic contact problems are of particular importance in various engineering applications such as structure-structure interaction, machine design and metal forming process. They are hard to solve because of their geometric and material nonlinearalities. There are much more literatures on elastic contact problems than on elastoplastic contact problems, for the former, see the papers [1-3] and the references therein. The recent research work on elastoplastic contact problem is available in [4-6]. Some advances and applications in PQP method for numerical modeling of elastoplastic contact problems will be reviewed in this paper. Zhong et al. developed PVP for the analyses of contact problems and elastoplastic structures [7,8]. In Zhong's method, both contact and elastoplastic problems can be formulated as the same form of parametric programming problems by means of finite element method. Then they are reduced to linear complementarity problems. Zhang et al. further enhanced and applied the method in several areas, such as the combination of parametric programming method and iteration algorithm for the analysis of elastic or elastic-plastic contact problems [9, 10], the parametric variation principle used for coupled analysis of liquid and solid interaction occurred in saturated and partially saturated porous media [11,12], the implementation of gradient dependent model adopted for the strain localization analysis of the materials [12,13], the local and multi-scale analysis of periodic composite materials with mirco-construction of elastic or inelastic cohesive granular bodies [14,15], the mixed energy method for solution of quadratic programming problems and elastic-plastic analysis of truss structures [16], and the non-interior smoothing algorithm for frictional contact problems [17]. For elastoplastic contact analysis, PVP and PQP method are still used in this paper. Emphases are on the basic theories used for analysis of elastoplastic contact problems with isotropic/orthotropic friction law. Particular attention will be on the CAE software system JIFEX developed in Dalian University of Technology and its applications in engineering. Key Engineering Materials Online: 2004-10-15 ISSN: 1662-9795, Vols. 274-276, pp 65-72 doi:10.4028/www.scientific.net/KEM.274-276.65 © 2004 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications Ltd, www.scientific.net. (Semanticscholar.org-13/03/20,17:14:02) Constitutive Equations and Orthotropic Friction Law Constitutive equations of plasticity can be expressed as 0 , 0 ), / ( , 0 ) , , , ( ), ( 2 = ≥ ∂ ∂ = ≤ ∇ − = f g d f d d D d T kl p kl p kl ij p kl kl ijkl ij λ λ σ λ ε κ κ ε σ ε ε σ (1) where the notations of all the variables and parameters are used as in the usual way. A gradient dependent model is used to show the potential of the algorithm developed. Without loss of generality, we assume λ κ h d = , κ 2 / ∂∇ ∂ = f h c , and is the hardening/softening modulus. Applying Taylor expansion to Eq. (1b), we obtain the consistent equation h 0 d 2 0 ≤ ∇ + − + λ λ c M f ε W (2) where is the initial value of the yield function at instant, and 0 f ( ) ( ) ( ) ( ) ( ) T T T T T / / / / , / h f g f g M f p κ ∂ ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ = ∂ ∂ = σ ε σ W D σ W (3) Consider two candidate contact bodies, let the common elastoplastic contact surface be denoted by Sc. We denote the principal friction orthotropic axes on the tangential plane at a point on Sc locally by τ1 and τ2, and let pτ1, pτ2 be the components of tangential contact traction pτ , ∆uτ1, ∆uτ2 be the components of tangential contact relative displacement ∆uτ along these axes. The corresponding coefficients of friction are denoted by μ1 and μ2. Let 0 ≤ n p ( ) denote normal contact force. The form of the static friction law is introduced as: If ( ) [ ] < 2 / 1 n p p − 2 2 2 1 / / μ τ τ ≥ p + 2 1 μ 0 , then ∆uτ=0; If [ , then there exists ( ) ( ) n p p p − = + 2 / 1 2 2 2 2 1 1 / / μ μ τ τ ] γ such that , . So the following equality holds 2 1 / μ 1 γ τ p u − ∆ 1 τ = 2 2 2 / μ τ 2 γ τ p u − = ∆ ) / )( / ( / 2 2 2 2 2 1 2 1 τ τ τ τ μ μ p p u u = ∆ ∆ (4) Let θu and θf denote the inclination angles of the tangential relative displacement ∆uτ and tangential contact force pτ with respect to the τ1-axis, respectively, then Eq. (4) becomes ( ) f u θ μ μ θ tan / tan 2 2 1 = (5) which indicates that the direction of tangential contact force can be different from the direction of slip for the orthotropic friction law. The following closed set is then introduced ( ) ( ) [ ]       ≤ + + = = 0 / / ~ : ) , , ( 2 1 2 2 2 2 1 1 2 1 n n e p p p f p p p C μ μ τ τ τ τ (6) where Ce describes an elliptic cone, see Fig. 1a. For numerical simulation, a piecewise linearization approximation of the elliptic cone Ce is used in the finite element model. Fig. 1b shows the cross-sectional shapes of elliptic cone and polyhedral cone for some negative n p . The linearized approximation of elliptic cone Ce can be written as 66 Advances in Engineering Plasticity and Its Applications