Homeostasis of soft tissues within the human body, such as ligaments and tendons, is largely controlled by the mechanical environment which surrounds the fibroblasts (cells) in these tissues. For this reason, it is important to understand the stress and strain fields which develop on specific microstructural levels within these tissues. While it is possible to determine these stress and strain fields directly using a detailed microstructural model and the finite element method, this approach can become computationally prohibitive for large regions of soft tissue. An alternative approach is provided by homogenization theory, which utilizes the periodicity of the microstructural unit and the microstructural deformation in an asymptotic expansion of the deformation field to simplify the problem. This allows the micromechanical environment for multiple locations within a larger region of soft tissue to be determined with a minimum of computational effort. To the present time, however, homogenization techniques have only been developed and applied for linearized elastic media, such as traditional composite materials or bone.

This thesis describes the development of homogenization theory to encompass finite elastic deformation, such that it may be applied to soft tissues. As will be shown, this requires several new developments, particularly with regard to the relationship between microstructural and continuum levels and the consideration of derivatives across these levels. Notably, the application of ideas utilized for linearized elastic media will not lead to an appropriate treatment for the case of finite elastic deformation. New (and necessary) notation is presented to handle the multiple levels of the finite deformation problem. Using new relations between the microstructural and continuum levels, the total deformation gradient is shown to be a multiplicative combination of the microstructural and continuum level deformation gradients.

The theory is shown to reduce appropriately to the accepted forms for linearized elastic media, and more importantly, to properly reduce for the case of finite deformation of a single continuum (the case of no microstructure). The governing equations of the overall homogenization problem, including a kinematic constraint placed on the relationship between the microstructural and continuum levels, are then formulated via a Galerkin weighted residual method. The treatment of additional topics of interest for finite deformation of multiple-scale media, including consideration of frame invariance of the developed deformation relations and the incorporation of material symmetries (e.g. transverse isotropy) are also presented.