Optical acquisition and polar decomposition of the full-field deformation gradient tensor within a fracture callus

Tracking tissue deformation is often hampered by material inhomogeneity, so local measurements tend to be insufficient thus lending to the necessity of full-field optical measurements. This study presents a novel approach to factoring heterogeneous deformation of soft and hard tissues in a fracture callus by introducing an anisotropic metric derived from the deformation gradient tensor (F). The deformation gradient tensor contains all the information available in a Green-Lagrange strain tensor, plus the rigid-body rotational components. A recent study [Bottlang et al., Journal of Biomechanics 41(3), 2008] produced full-field strains within ovine fracture calluses acquired through the application of electronic speckle pattern interferometery (ESPI). The technique is based on infinitesimal strain approximation (Engineering Strain) whose scheme is not independent of rigid-body rotation. In this work, for rotation extraction, the stretch and rotation tensors were separately determined from F by the polar decomposition theorem. Interfragmentary motions in a fracture gap were characterized by the two distinct mechanical factors (stretch and rotation) at each material point through full-field mapping. In the composite nature of bone and soft tissue, collagen arrangements are hypothesized such that fibers locally aligned with principal directions will stretch and fibers not aligned with the principal direction will rotate and stretch. This approach has revealed the deformation gradient tensor as an appropriate quantification of strain within callus bony and fibrous tissue via optical measurements.