In K. G¨ urlebeck L. Hempel C. K¨onke (Eds.) IKM2003: Digital Proceedings of 16th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering, ISSN 1611-4086. Mechanical Models with Interval Parameters ⋆ E. D. Popova 1 , M. Datcheva 2,3 , R. Iankov 2 , and T. Schanz 3 1 Inst. of Mathematics & Informatics, Bulgarian Academy of Sciences 2 Institute of Mechanics, Bulgarian Academy of Sciences Acad. G. Bonchev str., block 8 1 , block 4 2 , BG-1113 Sofia, Bulgaria epopova@bio.bas.bg, {iankovr, datchevam}@yahoo.com 3 Bauhaus-Universit¨atWeimar,Germany, tom.schanz@bauing.uni-weimar.de Abstract. In this paper we consider modelling of composite material with inclusions where the elastic material properties of both matrix and inclusions are uncertain and vary within prescribed bounds. Such mechanical systems, involving interval uncertainties and modelled by finite element method, can be described by parameter dependent systems of linear interval equations and process variables depending on the system solution. A newly developed hybrid interval approach for solving parametric interval linear systems is applied to the considered model and the results are compared to other interval methods. The hybrid approach provides very sharp bounds for the process variables – element strains and stresses. The sources for overestimation when dealing with interval computations are demonstrated. Based on the element strains and stresses, we introduce a definition for the values of nodal strains and stresses by using a set-theoretic approach. 1 Introduction All engineering design problems involve imprecision, approximation, or uncertainty to varying degrees [2,4]. In particular, mathematical models in environmental geomechanics cover a broad class of problems involving uncertainties of different types. Since soil and rock materials are natural ones, there is uncertainty in the material properties [7]. When the information about an uncertain parameter in form of a preference or probability function is not available or not sufficient then the interval analysis can be used most conveniently [4]. Many mechanical systems, modelled by finite element method (FEM), can be described by parameter dependent systems of linear equations. If some of the parameters are uncertain but bounded, the problem can be transformed into a parametric interval linear system which should be solved appropriately to bound the mechanical system response. This technique is usually called Interval Finite Element Method. The efforts for developing suitable interval FE methods started at mid nineties and attract considerable attention [2]. Here, we consider a a 2D plane strain problem with two inclusions and interval parameters related to the elastic material properties of both matrix and inclusions. Since safety is an issue in environmental geomechanics, the goal is to describe the response of the system under a worst case scenario of uncertain parameters varying within prescribed bounds. 2 FEM model with interval uncertainties Let us consider an elastic material model based on the following assumptions: small strain theory is applied to describe the deformation in material; the latter is deformed elastic; material properties are isotropic; the temperature, creep and time dependent effects are not taken into account. Let V be a representative volume and S be the boundary of this volume. S = S ¯ u  S σ , where S ¯ u is a boundary part with prescribed ⋆ This work was supported by the NATO CLG 979541 and the Bulgarian National Science Fund under grant No. I-903/99.