| Abstract [eng] |
Evaluation of homogeneous elasticity parameters for heterogeneous material with periodic microstructure was discussed in this paper. The multiscale modeling was used – model was analyzed in two scales with different assumptions. A heterogeneous three-dimensional body was modeled in microscale and an equivalent homogenous two-dimensional body was modeled in macroscale. Elasticity parameters of heterogeneous body that should be used in modeling a two dimensional homogeneous body were evaluated in three ways – asymptotic homogenization, by modeling pure loads in LS-DYNA, mechanical approach. In case of asymptotic homogenization and modeling pure loads numerical finite element method was used. Analyzed structure was divided to finite number of simple shape elements. Stresses and strains were evaluated in the nodes of each element. The two-dimensional model with evaluated parameters was developed. Stresses of two-dimensional and three-dimensional models were compared. With the assumption that material is homogeneous and the two-dimensional body is analyzed, the amount of calculation is significantly reduced in comparison with three-dimensional model. The mathematical formulation of the asymptotic homogenization for linear elasticity problems, description of finite element method and multiscale modeling was presented in this paper. The reduction of three-dimensional stress-strain equations to two-dimensional stress-strain equations and evaluation of elasticity parameters in three ways was also introduced in this paper. Two numerical examples were analyzed for the model of a unidirectional fiber layer, linear static analysis was performed. Stresses of two dimensional models with elasticity parameters obtained from asymptotic homogenization method and modeling pure loads differed less than 10% compared with stresses of three-dimensional model if fiber volume fraction is less than 0.5. That is a result of the fact that the elasticity constants of composing materials differ widely. Universal mathematical and programming language MATLAB and finite element modeling software LS-DYNA was applied to deal with the calculations. |
