Investigation of multiphase composites via asymptotic homogenization and its application to the bone hierarchical structure

Raimondo Penta (Universidad Politécnica de Madrid)

Thursday 16th February, 2017 14:00-15:00 Maths 515


A multiscale approach is developed for three dimensional multiphase elastic composites via asymptotic
homogenization. Each phase is assumed to behave as a linear, possibly anisotropic and
inhomogeneous, elastic solid. Discontinuities of the elastic constants across the interface between the host
medium (matrix) and any subphase interface are allowed. The classical stress balance equations are stated in
each phase, where volume forces and inertia are neglected. Coupling among phases is enforced via continuity of
stresses and displacements across every interface. Asymptotic expansion of the displacements is carried out to
exploit the sharp length scale separation between the spatially periodic structure (fine scale) and the whole
material (coarse scale). The coarse scale mechanics is described by a standard anisotropic elastic model, where
the role of the fine scale geometry is encoded in the effective elasticity tensor, which is to be computed solving
elastic-­type problems on the appropriate periodic cell. The model is general with respect to the number of
subphases and periodic cell shapes. The cell problems are equipped with stress discontinuities, which are
proportional to the jumps of the elastic constants across interfaces. The effective elasticity tensor and the auxiliary
strains which arise from the cell problems computation are characterized by specific properties and
representations which lead to a consistent effective elasticity tensor definition, in terms of symmetries and
energetic bounds [1]. A novel three dimensional numerical study is performed assuming an isotropic and
homogeneous linear elastic rheology for each phase. The periodic cell geometrical setup is chosen to properly
compare the model response to that provided by Eshelby based techniques and point out analogies and
differences between the two approaches. The model is benchmarked by comparing our method to well established
semi-­analytical schemes [2]. An example of application to the hierarchical structure of the bone (see, e.g.,
[3]), where the host medium is identified with the collagen matrix and the subphases to the mineral inclusions is
provided. We account for the formation of a continuous mineral foam [5], which is represented extending the mineral
inclusions up to the periodic cell boundary. Such a physiological condition (which characterizes aged bone tissue)
cannot be captured by simple average field techniques, which are widely exploited in the bone literature (see


[1] Penta, Raimondo, and Alf Gerisch. "The asymptotic homogenization elasticity tensor properties for composites with material discontinuities." Continuum Mechanics and Thermodynamics 29.1 (2017): 187-206.

[2] Penta, Raimondo, and Alf Gerisch. "Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study."Computing and Visualization in Science 17.4 (2015): 185-201.

[3] S. Weiner and H. D. Wagner. The material bone: Structure-­mechanical function relations. Annual Reviews of
Materials Science, 28:271–298, 1998. 

[4] Penta, R., et al. "Can a continuous mineral foam explain the stiffening of aged bone tissue? A micromechanical approach to mineral fusion in musculoskeletal tissues."Bioinspiration & biomimetics 11.3 (2016): 035004.

[5] Sara Tiburtius, Susanne Schrof, Ferenc Molnar, Peter Varga, Franc¸oise
Peyrin, Quentin ´ Grimal, Kay Raum, and Alf Gerisch. On the elastic properties of mineralized turkey leg tendon tissue: multiscale model and experiment. Biomechanics and modeling in mechanobiology, 13:1003–1023, 2014.

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