citable using Digital Object Identifier – DOI) Early View publication on wileyonlinelibrary.com (issue and page numbers not yet assigned; ZAMM · Z. Angew. Math. Mech., 1 – 21 (2013) / DOI 10.1002/zamm.201300112 Discretization and numerical realization of contact problems for elastic-perfectly plastic bodies. PART I – discretization, limit analysis S. Sysala 1, ∗ , J. Haslinger 1,2 , I. Hlav´ aˇ cek 3 , and M. Cermak 1,4 1 Institute of Geonics AS CR, IT4Innovations Department, Ostrava, Czech Republic 2 Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic 3 Institute of Mathematics AS CR, Prague, Czech Republic 4 V ˇ SB–Technical University of Ostrava, Centre of Excellence IT4Innovations, Ostrava, Czech Republic Received 7 May 2013, revised 26 September and 21 October 2013, accepted 5 November 2013 Published online 25 November 2013 Key words Frictionless contact, elasto-perfect plasticity, limit analysis. The paper deals with a static case of discretized contact problems for bodies made of materials obeying Hencky’s law of perfect plasticity. The main interest is focused on the analysis of the formulation in terms of displacements. This covers the study of: i) a structure of the solution set in the case when the problem has more than one solution ii) the dependence of the solution set on the loading parameter ζ . The latter is used to give a rigorous justification of the limit load approach based on work of external forces as a function of ζ . A model example illustrates the efficiency of the method. c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Contact problems represent a branch of mechanics of solids whose goal is to study the behavior of a system of loaded deformable bodies being in mutual contact. This paper deals with 3D frictionless contact problems for two bodies made of materials obeying the Hencky constitutive law of perfect plasticity. The Hencky model has been investigated e.g. in [9, 18, 20, 21, 27] from the theoretical and in [3, 10, 11, 19] from the computational point of view. If, in addition unilateral, contact conditions are involved in the model then we refer to [14,15,17]. From the physical point of view this is one of the simplest models of plasticity. Its drawback is the fact that it does not take into account the loading history. To get a more realistic model one has to use quasi-static formulations [1, 7, 13]. Despite its simplicity, the analysis of the Hencky model is worthwhile since a time discretization of quasi-static problems leads to a sequence of problems of the type considered in this paper. On the contrary, the rigorous analysis of the mathematical model of Hencky’s plasticity in terms of displacements (primal formulation) is far from to be simple. Due to a linear growth of the respective inner energy functional, the variational formulation on the Sobolev space W 1,1 (Ω) is not well posed since the existence of minimizers is not guaranteed there. To make the problem well posed and to preserve the exact lower bound of the total potential energy functional, it is necessary to extend the space W 1,1 (Ω). This was done in [23, 24], where the space of functions with bounded deformation (BD) was introduced, and in [27] containing the detailed analysis of the BD spaces. On the other hand, the variational formulation in terms of stresses (dual formulation) is the classical one. This formulation however is not usually used in computations due to a high number of linear and nonlinear constraints characterizing the discretized set of plastically and statically admissible stresses. The number is proportional to the size of a used finite element discretization [14, 15, 17]. The existence of a solution to both the primal and the dual formulation depends among other on the existence of stresses which are at the same time statically and plastically admissible. To verify this property it is necessary to determine the so-called limit load corresponding to the applied load L. The load is considered in the form ζL, where ζ> 0 is a loading parameter. One seeks maximal ζ , denoted as ζ lim , for which the above mentioned set is non-empty [3, 5, 27]. The paper is focused on the numerical realization of the problem using a conventional finite element method. It consists of two parts. The main goal of present PART I is to analyze the discrete problem in details. The numerical realization itself is the subject of PART II [4]. We use the continuous, piecewise linear approximation of displacements and the piecewise constant approximation of stresses. This choice of the spaces is simple and commonly used in engineering community. The convergence analysis for ∗ Corresponding author E-mail: stanislav.sysala@ugn.cas.cz c  2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim