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.

工程和生理学中一系列材料系统的设计和研究需要对包含多种类型夹杂物的复合材料的有效刚度进行估计。虽然对于包含相同夹杂物的特定类别和分布的复合系统存在良好的估计和严格的界限，但这些并不能轻易推广到具有多种类型夹杂物的系统。对于包含多类夹杂物的复合材料，可用的最佳估计来自 Kanaun-Jeulin 方法。然而，该方法类似于广义的 Benveniste 方法，因此受到相同的限制：虽然对于低体积分数的夹杂物非常有效，但 Kanaun-Jeullin 和 Benveniste 的估计值在较高体积分数时低于三点界限。在这里，我们提出了对包含多类对齐椭圆体夹杂物的复合材料的估计，这些夹杂物位于已知三点边界内，夹杂物的体积分数相对较高，并且适用于许多工程和生物复合材料。该方法涉及用有效的应变测量代替 Kanaun-Jeulin 方法中使用的平均应变。我们演示了本构模型在肌腱与骨附着处的分级组织系统中的应用。