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 [1]), 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. [5] and Kouznetsova et al [6].

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 [7]. 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 [8]. 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.

1

P. M. Suquet, "Local and global aspects in the mathematical theory of plasticity",Plasticity today: modelling methods and applications, Elsevier Applied Science Publishers, 279-310,1985.

2

C. Miehe, J. Schotte and J. Schroder, "Computational micro-macro transitions and overall tangent moduli in the analysis of polycrystals at large strains", Computational Material Science, 16, 372-382, 1999. doi:10.1016/S0927-0256(99)00080-4

3

C. Miehe, J. Schroder and J. Schotte, "Computational homogenization analysis in finite plasticity. Simulation of texture developtment in polycrystalline materials", Computer methods in applied mechanics and engineering, 171, 387-418, 1999. doi:10.1016/S0045-7825(98)00218-7

4

C. Miehe and A. Koch, "Computational micro-to-macro transitions of discretized microstructures undergoing small strains", Archive of Applied Mechanichs, 72, 300-317, 2002. doi:10.1007/s00419-002-0212-2

5

R.J.M. Smit, W.A.M. Brekelmans and H.E.H. Meijer, "Prediction of the mechanichal behaviour of nonlinear heterogeneous syatems by multi-level finite element modeling", Computer methods in applied mechanics and engineering, 155, 181-192, 1998. doi:10.1016/S0045-7825(97)00139-4

6

V. G. Kouznetsova, W. A. M. Brekelmans and F. P. T. Baaijens, "An Approach to micro-macro modelling of heterogeneous materials", Computational Mechanics, 27, 37-48, 2001. doi:10.1007/s004660000212

7

C. C. Swan, "Techniques for stress- and strain-controlled homogenization of inelastic periodic composites", Computer methods in applied mechanics and engineering, 117, 249-267, 1994. doi:10.1016/0045-7825(94)90117-1

8

R. Hill, "On constitutive macro-variables for heterogeneous solids at finite strain", Proceeding of the Royal Society of London, 326, 131-147, 1972. doi:10.1098/rspa.1972.0001

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