Comput Mech (2010) 45:349–361 DOI 10.1007/s00466-009-0455-7 ORIGINAL PAPER A generalized stochastic perturbation technique for plasticity problems Marcin Marek Kami ´ nski Received: 20 June 2009 / Accepted: 24 November 2009 / Published online: 18 December 2009 © Springer-Verlag 2009 Abstract The main aim of this paper is to present an algorithm and the solution to the nonlinear plasticity problems with random parameters. This methodology is based on the finite element method covering physical and geometrical nonlinearities and, on the other hand, on the generalized nth order stochastic perturbation method. The perturbation approach resulting from the Taylor series expan- sion with uncertain parameters is provided in two different ways: (i) via the straightforward differentiation of the initial incremental equation and (ii) using the modified response surface method. This methodology is illustrated with the analysis of the elasto-plastic plane truss with random Young’s modulus leading to the determination of the probabilistic moments by the hybrid stochastic symbolic-finite element method computations. Keywords Elastoplasticity · Finite element method · Nonlinear problems · Stochastic perturbation technique 1 Introduction The complexity and not satisfactory accuracy of the existing non-statistical methods in solution of the nonlinear mechan- ical problems with random parameters as well as large time consumption in statistical and stochastic simulations still lead to a development of the new approaches in this area. Such nonlinear problems appear frequently in all the branches of modern engineering, where mechanical properties like M. M. Kami ´ nski (B ) Department of Mechanics of Materials, Faculty of Civil Engineering, Architecture and Environmental Engineering, Technical University of Lód´ z, Al. Politechniki 6, 90-924 Lód´ z, Poland e-mail: Marcin.Kaminski@p.lodz.pl URL: http://kmm.p.lodz.pl/pracownicy/Marcin_Kaminski/index.html Young’s modulus, Poisson’s ratio and plastic stress (for the simple elastoplasticity model) are determined experimen- tally using the statistical estimators [5]. On the other hand, there are several situations, where geometrical nonlinear- ity is important considering especially the uncertainties in the structural general dimensions, joining of the particular structural elements and the defects [4] or inclusions [1]. A lot of attention is paid to the plasticity models in com- posites in terms of probabilistic mechanics—in macro-scale [3], smaller scales [20] or even in the area of nano-com- posites [7]. The generalized stochastic perturbation theory [8] is an interesting alternative to the other existing theoret- ical and computational approaches in stochastic mechanics [14, 15, 22, 23]. The main reason of an introduction of the gen- eralized method is full Taylor expansion including as many probabilistic moments as it is necessary for an accurate solu- tion. One of the most important benefits is the opportunity to model any probability distributions like lognormal, Wei- bull or Gumbel being very important in the reliability mod- eling. The probabilistic characteristics of the increments of displacements, strains and stresses are computed now more precisely than in the second order second moment (SOSM) technique [11] and the method remains accurate for the input coefficients of variation significantly larger than the applica- bility limit for the SOSM—α = 0.10. At the same time, a computational procedure leading to those moments deter- mination still remains faster than the Monte-Carlo simula- tion method [9]. Furthermore, the response surface method (RSM) is employed here besides the classical straightfor- ward differentiation approach to find a solution to the initial problem with both material and geometrical random nonlin- earities. Both probabilistic methods displayed here are based on the Taylor series representation of the increments of the state functions using the moments of the input random vari- ables as well as up to the given order partial derivatives of 123