Effective elastic properties of a composite containing multiple types of anisotropic ellipsoidal inclusions, with application to the attachment of tendon to bone

Estimates of the effective stiffness of a composite containing multiple types of inclusions are needed for the design and study of a range of material systems in engineering and physiology. While excellent estimates and tight bounds exist for composite systems containing specific classes and distributions of identical inclusions, these are not easily generalized to systems with multiple types of inclusions. The best estimate available for a composite containing multiple classes of inclusions arises from the Kanaun-Jeulin approach. However, this method is analogous to a generalized Benveniste approach, and therefore suffers from the same limitations: while excellent for low volume fractions of inclusions, the Kanaun-Jeullin and Benveniste estimates liebelow three-point bounds at higher volume fractions. Here, we present an estimate for composites containing multiple classes of aligned ellipsoidal inclusions that lies within known three-point bounds at relatively higher volume fractions of inclusions and that is applicable to many engineering and biological composites. The approach involves replacing the averaged strains used in the Kanaun-Jeulin method with an effective strain measure. We demonstrate application of the constitutive model to the graded tissue system at the attachment of tendon to bone.