A numerical investigation and experimental verification of size effects in loaded bovine cortical bone

Abstract

In this paper, we present 2-and 3-dimensional finite element-based numerical models of loaded bovine cortical bone that explicitly incorporate the dominant microstructural feature: the vascular channel or Haversian canal system. The finite element models along with the representation of the microstructure within them are relatively simple: 2-dimensional models, consisting of a structured mesh of linear elastic planar elements punctuated by a periodic distribution of circular voids, are used to represent beam samples of cortical bone in which the channels are orientated perpendicular to the sample major axis, while 3-dimensional models, using a corresponding mesh of equivalent solid elements, represent those samples in which the canals are aligned with the axis. However, these models are exploited in an entirely novel approach involving the representation of material samples of different sizes and surface morphology. The numerical results obtained for the virtual material samples when loaded in bending indicate that they exhibit size effects not forecast by either classical (Cauchy) or more generalized elasticity theories. However, these effects are qualitatively consistent with those that we observed in a series of carefully conducted experiments involving the flexural testing of bone samples of different sizes. Encouraged by this qualitative agreement, we have identified appropriate model parameters, primarily void volume fraction but also void separation and matrix modulus by matching the computed size effects to those we observed experimentally. Interestingly, the parameter choices that provide the most suitable match of these effects broadly concur with those we actually observed in cortical bone.KEYWORDS cortical bone, finite element analysis, generalized continuum, mechanical behaviour, micropolar (Cosserat) elasticity, size effect

Size effect anomalies in the behaviour of loaded 3D mechanical metamaterials

“…Ordinarily, these more involved theories forecast a positive size effect in which material stiffness increases as the overall size reduces to that associated with the internal structure, behaviour that has indeed been observed in a variety of heterogeneous materials [8,9,10,11,12,13,14,15,16]. Nevertheless, contradictory behaviour in which compliance increases with decreasing size, a contrasting negative effect, has been observed in other materials, notably hard biological tissues [17,18]. Very recently, both conventionally anticipated [19] and more involved [20] size effects have been reported in additively manufactured mechanical metamaterials.…”

On the question of the sign of size effects in the elastic behavior of foams

“…Though, this effect in regular mesostructures can be explained by the shift of material away from the neutral axis [27,45,47] according to Steiner's theorem. And consequently, the opposite size effect appears if the surface is realized in another way within the same metamaterial [1,10]. Recently, Reasa and Lakes [31] formulated the hypothesis that there might be different regimes regarding specimen size relative to the cell size.…”