Homogenization of periodic 1-3 piezocomposite using wave propagation: Toward an experimental method.

1-3 piezocomposites are first choice materials for integration in ultrasonic transducers due to their high electromechanical performance, particularly, in their thickness mode. The determination of a complete set of effective electroelastic parameters through a homogenization scheme is of primary importance for their consideration as homogeneous. This allows for the simplification of the transducer design using numerical methods. The method proposed is based on acoustic wave propagation through an infinite piezocomposite, which is considered to be homogeneous material. Christoffel tensor components for the 2 mm symmetry were expressed to deduce slowness curves in several planes. Simultaneously, slowness curves of a numerical phantom were obtained using a finite element method (FEM). Dispersive curves were initially calculated in the corresponding heterogeneous structure. The subsequent identification of the effective parameters was based on a fitting process between the two sets of slowness curves. Then, homogenized coefficients were compared with reference results from a numerical method based on a fast Fourier transform for heterogeneous periodic piezoelectric materials in the quasi-static regime. A relative error of less than 2% for a very large majority of effective coefficients was obtained. As the aim of this paper is to implement an experimental procedure based on the proposed homogenization scheme to determine the effective parameters of the material in operating conditions, it is shown that simplifications to the procedure can be performed and a careful selection of only seven slowness directions is sufficient to obtain the complete database for a piezocomposite containing square-shaped fibers. Finally, further considerations to adapt the present work to a 1-3 piezocomposite with a fixed thickness are also presented.

Maximum effect of the heterogeneity of tissue mineralization on the effective cortical bone elastic properties.

The mineralization level is heterogeneous in cortical bone extracellular matrix as a consequence of remodeling. Models of the effective elastic properties at the millimeter scale have been developed based on idealizations of the vascular pore network and matrix properties. Some popular models do not take into account the heterogeneity of the matrix. However, the errors on the predicted elasticity when the difference in elastic properties between osteonal and interstitial tissues is not modeled have not been quantified. This work provides an estimation of the maximum error. We compare the effective elasticity of a representative volume element (RVE) assuming (1) different elastic properties in osteonal and interstitial tissues vs. (2) average matrix properties. In order to account for the variability of bone microstructure, we use a collection of high resolution images of the pore network to build RVEs. In each RVE we assumed a constant osteonal wall thickness and we artificially varied this thickness between 35 and 140 [Formula: see text]m to create RVEs with different amounts of osteonal tissue. The homogenization problem was solved with a fast Fourier transform (FFT)-based numerical scheme. We found that the error depends on pore volume fraction and varies on average from 1 to [Formula: see text] depending on the assumed diameter of the osteons. The results suggest that matrix heterogeneity may be disregarded in cortical bone models in most practical cases.

Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity.

With ageing and various diseases, the vascular pore volume fraction (porosity) in cortical bone increases, and the morphology of the pore network is altered. Cortical bone elasticity is known to decrease with increasing porosity, but the effect of the microstructure is largely unknown, while it has been thoroughly studied for trabecular bone. Also, popular micromechanical models have disregarded several micro-architectural features, idealizing pores as cylinders aligned with the axis of the diaphysis. The aim of this paper is to quantify the relative effects on cortical bone anisotropic elasticity of porosity and other descriptors of the pore network micro-architecture associated with pore number, size and shape. The five stiffness constants of bone assumed to be a transversely isotropic material were measured with resonant ultrasound spectroscopy in 55 specimens from the femoral diaphysis of 29 donors. The pore network, imaged with synchrotron radiation X-ray micro-computed tomography, was used to derive the pore descriptors and to build a homogenization model using the fast Fourier transform (FFT) method. The model was calibrated using experimental elasticity. A detailed analysis of the computed effective elasticity revealed in particular that porosity explains most of the variations of the five stiffness constants and that the effects of other micro-architectural features are small compared to usual experimental errors. We also have evidence that modelling the pore network as an ensemble of cylinders yields biased elasticity values compared to predictions based on the real micro-architecture. The FFT homogenization method is shown to be particularly efficient to model cortical bone.