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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 85
Edited by: B.H.V. Topping
Paper 23

A Framework for Stochastic Micromechanics

S.C. Baxter1 and T.O. Williams2

1Department of Mechanical Engineering
University of South Carolina, Columbia SC, United States of America
2Los Alamos National Laboratory, Theoretical Division T-3
Los Alamos NM, United States of America

Full Bibliographic Reference for this paper
S.C. Baxter, T.O. Williams, "A Framework for Stochastic Micromechanics", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 23, 2007. doi:10.4203/ccp.85.23
Keywords: stochastic micromechanics, random heterogeneous materials.

As applications for composite materials, in particular those with a random microstructural geometry, increase it has becomes increasingly important to recognize that these materials are, in fact, stochastic at the microstructural level. These stochastic effects arise predominantly from variability of the microstructure and can have a significant impact on the local and global history-dependent response of these systems up through failure.

A methodology using a combination of both a deterministic micromechanics model and a stochastic model based on a stochastic transformation field analysis (STFA) [1,2] is presented. The STFA provides general equations governing the behaviour of micromechanical concentration tensors; these tensors characterize the relationship between local and global mechanical or transformational (eigen) strains (or analogously stresses, eigenstresses). The general governing equations of the STFA are specifically designed to incorporate the behaviour of the spatially varying material microstructure. The resulting probability density functions (PDFs) can then be used to predict local and bulk behaviour of the composite. When calculating the history-dependent behaviour of the bulk material the question arises as to how sensitive the predictions of the STFA theory must be to the detail of its embedded statistics. Obviously, highly resolved representations for the PDFs are more computationally demanding; perhaps capturing more relevant detail, but potentially including more noise. Thus, it is important to determine what level of simplified statistical representation, and its influence for various predictions, should be used to accurately predict the inelastic response of the bulk material.

An example problem of a T300/2510, a graphite epoxy continuous fibre matrix composite is used to consider this question. The Generalized Method of Cells [3] micromechanics model was used to analyze realizations of a material. Numerically generated samples were used to create an ensemble description of the statistics for the material. The results show that the distributions of the associated PDFs for the mechanical concentration tensors in the fibres and the matrix were not Gaussian and exhibit significant tails. The predicted elastic properties resulting from the use of these tensors however, are shown to be relatively insensitive to the degree of discretization. Similarly, the level of approximation of the PDFs appears to have a negligible impact on the predicted in-plane inelastic response. However, the out-of plane response exhibited a very different behaviour. In particular, the hysteretic effects predicted with various degrees of discretization were significant, especially over the first cycle. In this case the level of representation had a definite impact on the predicted response of the composite.

Williams, T.O., "A General, Stochastic Transformation Field Theory", J. Eng. Mech., Vol. 132, No. 11, pp. 1224-1240, 2006. doi:10.1061/(ASCE)0733-9399(2006)132:11(1224)
Williams, T.O. and Baxter, S.C., "A Framework for Stochastic Mechanics, Prob. Eng. Mech., Vol. 21, No. 3, pp. 247-255, 2006. doi:10.1016/j.probengmech.2005.10.002
Paley, M. and Aboudi, J., "Micromechanical Analysis of Composites by the Generalized Cell Model", Mech. Matl., Vol. 14, pp. 127-139, 1992. doi:10.1016/0167-6636(92)90010-B

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