Analytical relations between effective material properties and microporosity: Application to bone mechanics

This work deals with the problem of reconstruction of microstructural parameters of viscoelastic composite material from measurements of its effective properties. The Stieltjes representation of the effective shear complex modulus of two-component composite material is exploited to recover information about structural parameters. This representation is derived from problem of torsion of a heterogeneous cylinder using asymptotic expansions method. The microstructural information is contained in the spectral measure in this analytical representation. The spectral function can be recovered from the measurements over a range of frequencies. The problem of reconstruction of the spectral measure is very ill-posed. Regularized algorithm is derived to ensure stability of the results. We apply the proposed method to recovery of porosity of cancellous bone from measurements of its effective shear modulus. Bone is modeled as a medium with a microstructure composed of two viscoelastic isotropic or/and transversely isotropic materials, with the components representing trabeculae (elastic component) and bone marrow (viscoelastic component). Porosity is understood as volume fraction of bone marrow. To verify the approach we apply it to analytically and numerically simulated response of a cylinder filled with a composite material with hexagonal microstructure. The proposed method does not use any specific assumptions about the microgeometry of the composite and is applicable to any two-phase composite medium.