Vol. 6, No. 4, 2018

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Heterogeneous directions of orthotropy in three-dimensional structures: finite element description based on diffusion equations

Rachele Allena and Christophe Cluzel

Vol. 6 (2018), No. 4, 339–351

Heterogeneous materials such as bone or woven composites show mesostructures whose constitutive elements are all oriented locally in the same direction and channel the stress flow throughout the mechanical structure. The interfaces between such constitutive elements and the matrix are regions of potential degradations. Then, when building a numerical model, one has to take into account the local systems of orthotropic coordinates in order to properly describe the damage behavior of such materials. This can be a difficult task if the orthotropic directions constantly change across the complex three-dimensional geometry as is the case for bone structures or woven composites. In the present paper, we propose a finite element technique to estimate the continuum field of orthotropic directions based on the main hypothesis that they are mainly triggered by the external surface of the structure itself and the boundary conditions. We employ two diffusion equations, with specific boundary conditions, to build the radial and the initial longitudinal unit vectors. Then, to ensure the orthonormality of the basis, we compute the longitudinal, the circumferential, and the radial vectors via a series of vector products. To validate the numerical results, a comparison with the average directions of the experimentally observed Haversian canals is used. Our method is applied here to a human femur.

orthotropic materials, cortical bone, diffusion equations
Mathematical Subject Classification 2010
Primary: 00AXX
Received: 9 April 2018
Revised: 14 June 2018
Accepted: 25 June 2018
Published: 1 October 2018

Communicated by Francesco dell'Isola
Rachele Allena
Institute de Biomécanique Humaine George Charpak
Arts et Métiers ParisTech
Christophe Cluzel
Laboratoire de Mécanique et Technologie
Ecole Normale Supérieure Paris Saclay
Département Science et Genie des Materiaux
Institut Universitaire de Technologie d’Evry Val d’Essonne