Computational & Technology Resources
an online resource for computational,
engineering & technology publications
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Multiscale Finite Element Simulation for Heterogeneous Materials with Reference to the Effective Tangent Modulus Computation
A.J. Carneiro Molina, E.A. de Souza Neto and D. Peric
School of Engineering, University of Wales Swansea, United Kingdom
A.J. Carneiro Molina, E.A. de Souza Neto, D. Peric, "Multiscale Finite Element Simulation for Heterogeneous Materials with Reference to the Effective Tangent Modulus Computation", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 181, 2006. doi:10.4203/ccp.83.181
Keywords: multiscale analysis, small strains, homogenization, non-linear behaviour, RVE, finite elements, tangent modulus.
This paper presents a multiscale homogenization numerical procedure required for computation of the overall tangent moduli of microstructures with non-linear material behaviour undergoing small strains. Such procedures are important for computer modelling of heterogeneous materials when the length-scale of heterogeneities is small compared to dimensions of the body. In this instance applying a single mesh becomes too costly.
A homogenized macro-continuum is considered with locally attached microstructures. The microstructures define representative volume elements (RVE) of heterogeneous materials such as elasto-plastic composites. The homogenization procedure is based on the standard finite element (FE) discretisation of the microstructure. The tangent modulus is obtained as a function of stiffness, material properties of the components and geometrical distribution of heterogeneities.
Since the basic principles for the micro-macro modelling of heterogeneous materials were introduced (see Suquet ), this technique has proved to be the most effective way to deal with arbitrary physically non-linear and time dependent material behaviour at micro-level. A number of recent works deal with various approaches and techniques for the micro-macro simulation of heterogeneous materials. Among these we highlight the contributions by Miehe and coworkers in [2,3,4], Smit et al.  and Kouznetsova et al .
In this work an alternative and efficient numerical procedure to compute the overall tangent moduli for evolving microstructures is described, based on the multiscale finite element homogenization. The main objective is to obtain a consistent numerical procedure incorporating the appropriate tangent operators in order to perform a Newton-Raphson based solution of the discrete boundary valued problem in the FE context.
Two dimensional macro and micro structures are considered. The paper focuses on deformation-driven microstructures, which have proven to provide the most convenient format . In order to compute the tangent moduli, two types of boundary conditions are imposed over the unit cell: (a) linear displacements on the boundary and (b) periodic displacements and antiperiodic tractions on the boundary. These boundary conditions satisfy the fundamental Hill-Mandel averaging condition, which equates microscopic and macroscopic virtual work . We note that these boundary conditions, in general, generate two different values of the effective tangent modulus.
Numerical tests have been performed for a material with voids. The quadratic rate of convergence obtained by a Newton-type solution method procedure for the macroscopic boundary value problem confirms the success of the tangent modulus computation and efficient solution of the discrete problem.
purchase the full-text of this paper (price £20)