Linearized Elastic Models for Degrading Materials

We investigate behavior of the stress tensor in the vicinity of elliptical hole cut-out from elastic material stretched on opposite sides by constant force. We perform the analysis on the linearized elastic model first with constant material coefficients and then with material coefficients depending on the concentration of diffusing fluid. In our model elastic coefficients are shrinking with increasing concentration of diffusing fluid. Models of this type are used to describe degrading elastic solid. Introduction If homogeneous linearized elastic solid plate is stretched by constant force, the stress is spread evenly inside. However, if there is cut-out in the material, the stress is no more uniform. There will be areas where the stress is concentrated (in the vicinity of cut-out), therefore its size being significantly greater than in the rest of the material. In this paper we examine how the concentration of the stress changes for different types of elliptical cut-out. First for the linearized elastic solid with constant elastic coefficients and then for degrading elastic materials whose material moduli μ and λ decreases with time in the vicinity of cut-out. In the first chapter we introduce the model of the linearized elasticity. We implement this model in the Comsol Multiphysics and prove accuracy of our implementation by solving benchmark problem introduced in Cai et al. [2005] and comparing our results with that study. In the second chapter we investigate behavior of the stress tensor for the variable half-axis ratio of an elliptical cut-out. Finally in the third chapter we present model of degrading material and compare the results for this model with the results for linearized elasticity. For the further description of the model of degrading material, see Muliana et al. [2009]. Formulation of the problem and benchmark tests DFG group presented a benchmark problem, see Cai et al. [2005], to find distribution of stress tensor for the elastic square solid with circular hole in the middle stretched by constant force on opposite sides, see Figure 1. For the purpose of the further analysis in this study we also consider geometries with an elliptical hole in the middle. Due to the symmetry it is possible to solve the problem on cut-out quarter of the original geometry. Let a and b be half-axes of the elliptic cut-out. Then Ω ⊂ R is defined by Ω = {x ∈ R : 0 < x1 < 10, 0 < x2 < 10, (10− x1) a2 + x22 b2 > 1} (1) (see Figure 2 for circular cut-out and Figure 3 for the general case of elliptic one). For circular cut-out inside the plate set a = 1 and b = 1. Figure 1. Model geometry. Figure 2. Computational domain. Figure 3. Computational domain. 218 WDS'09 Proceedings of Contributed Papers, Part III, 218–223, 2009. ISBN 978-80-7378-103-3 © MATFYZPRESS