Abstract:

Bones are inhomogeneous solids with highly complex structures that require a multiscale modeling to understand its electromechanical behavior and its remodeling mechanisms. Experimental evidence indicates that bones present the piezoelectric effect and that this effect accelerates bone growth and remodeling, which is very useful in the consolidation of bone fracture. Mechanical and piezoelectrical properties of bone tissue and associated theoretical models have been studied in the last five decades. However, a comprehensive theoretical analysis of the electromechanical modeling to assess all effective properties of bone is not yet available. In this paper, the Asymptotic Homogenization Method (AHM) yields a two-scale procedure to analyze the electromechanical behavior of cortical bones. To obtain their effective properties, they are modeled as a two-phase composite material containing a periodic distribution of unidirectional circular cylindrical holes in a linearly piezoelectric and transversely isotropic homogeneous matrix whose constituent material belongs to the symmetry crystal class 622. The holes are centered in a periodic cell of square cross section and the periodicity is the same in two perpendicular directions. The composite is under a coupled state of both out-of-plane shear deformation and in-plane electric field. A set of local problems arises from the two-scale analysis using the AHM. Complex variable methods are used to obtain solutions to these problems. These solutions are expanded in a power series of Weierstrass elliptic functions containing coefficients that are determined from the solutions of systems of infinite linearly algebraic equations. Truncating the infinite systems, we obtain analytical formulae for elastic, piezoelectric, and dielectric effective properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. To compare these formulae with formulae available in the literature, the effective properties are plotted against volume fraction, showing a good agreement between the corresponding formulae.