Keywords: maximum entropy method, tensorial Fourier expansion.

Phenomenological models seem to be generally unable to adequately represent the evolution of the crystallite orientation distribution. Therefore, if the evolving crystallographic texture has to be taken into account in the context of finite element simulations in most cases the Taylor model is applied [7,6,8]. In this case, the texture evolution is described by a system of ordinary or algebro differential equations. The dimension of such a system is in between 1000 - 10000. Hence, at each integration point of the finite element mesh large systems of differential equations have to be integrated. This fact considerably limits the number of degrees of freedom that can be handled by standard finite element codes. Therefore, there is a need for homogenization strategies which allow to condense the number of degrees of freedom and nevertheless accurately describe the crystallite orientation distribution function.

We present an approach that is based on a tensorial Fourier expansion of the CODF [1,5]. The tensorial Fourier coefficients or texture coefficients can be considered as micro-mechanically based tensorial internal variables [3,4]. They are defined in terms of the CODF, which can be determined by texture measurements. Based on the conservation law of the CODF the differential equation of the tensorial texture coefficients has been derived and formulated in a coordinate-free setting. The evolution equation of each coefficient depends on the complete CODF and the lattice spin, which is a constitutive quantity. Hence, for the solution of the differential equation for a finite number of coefficients, the CODF has to be estimated. The problem of estimating the CODF based on a finite number of coefficients is generally ill-posed. In [2] it has been shown how the CODF can be estimated by means of the maximum entropy method. In the present paper this maximum entropy approach is used for the solution of the evolution equation governing the tensorial texture coefficients. The differential equation is numerically solved for the simple shear deformation for three cases with an increasing number of coefficients. It is shown that the evolution of the CODF and of the Taylor factor is reproduced for moderate deformations if at least the 6th-order coefficients are taken into account. In contrast to formulations of the Taylor model, which are based on discrete crystal orientations, the approach suggested here allows to model the evolution of the CODF by a much smaller number of internal variables.

1

B.L. Adams, J.P. Boehler, M. Guidi, E.T. Onat. "Group theory and representation of microstructure and mechanical behavior of polycrystals", J. Mech. Phys. Solids, 40(4):723-737, 1992. doi:10.1016/0022-5096(92)90001-I

2

T. Böhlke. "Application of the maximum entropy method in texture analysis", To appear in: Computational Materials Science, 2004. doi:10.1016/j.commatsci.2004.09.041

3

T. Böhlke, A. Bertram. "Crystallographic texture induced anisotropy in copper: An approach based on a tensorial Fourier expansion of the codf", J. Phys. IV, 105:167-174, 2003. doi:10.1051/jp4:20030184

4

T. Böhlke, A. Bertram. "Modeling of deformation induced anisotropy in free-end torsion", Int. J. Plast., 19:1867-1884, 2003. doi:10.1016/S0749-6419(03)00043-3

5

M. Guidi, B.L. Adams, E.T. Onat. "Tensorial representation of the orientation distribution function in cubic polycrystals", Textures Microstruct, 19:147-167, 1992. doi:10.1155/TSM.19.147

6

S.R. Kalidindi, C.A. Bronkhorst, L. Anand. "Crystallographic texture evolution in bulk deformation processing of fcc metals", J. Mech. Phys. Solids, 40(3):537-569, 1992. doi:10.1016/0022-5096(92)80003-9

7

K.K. Mathur, P.R. Dawson. "On modeling the development of crystallographic texture in bulk forming processes", Int. J. Plast., 5:67-94, 1989. doi:10.1016/0749-6419(89)90020-X

8

C. Miehe, J. Schröder, J. Schotte. "Computational homogenization in finite plasticity. Simulation of texture development in polycrystalline materials", Comp. Meth. Appl. Mech. Engng., 171:387-418, 1999. doi:10.1016/S0045-7825(98)00218-7

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