TY - JOUR
T1 - A differential approach to microstructure-dependent bounds for multiphase heterogeneous media
AU - Liu, Liping
N1 - Funding Information:
Acknowledgments The author gratefully acknowledges the support of NSF under Grant No. CMMI-1238835 and AFOSR (YIP-12). He also thanks the anonymous reviewer for pointing out the relevant references on the differential scheme with respect to volume fraction.
PY - 2014/4
Y1 - 2014/4
N2 - We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin-Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.
AB - We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin-Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.
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U2 - 10.1007/s00707-013-1060-6
DO - 10.1007/s00707-013-1060-6
M3 - Article
AN - SCOPUS:84898049479
SN - 0001-5970
VL - 225
SP - 1245
EP - 1266
JO - Acta Mechanica
JF - Acta Mechanica
IS - 4-5
ER -