On surface effects in model heterogeneous materials and the consequences for a real material: cortical bone

Experimental testing and finite element analysis of model heterogeneous materials consisting of regular periodic arrays of circular voids within metallic or polymeric matrices reveals that the materials exhibit mechanical behaviour consistent with the predictions of micropolar or Cosserat elasticity theory: for samples of similar geometry sample stiffness increases as size reduces. However, this behaviour is only observed in cases where the sample surface remains smooth and continuous, the circular voids constituting the heterogeneity do not intersect the surface. In this paper the results of finite element analyses of model heterogeneous materials in which the surfaces are corrugated because the voids intersect them are presented. The results indicate that rather than exhibiting micropolar behaviour in which sample stiffness increases with reducing size the opposite is observed; the samples become more compliant as size decreases. Interestingly, the rate of decrease in the latter case where the voids intersect the surface is exactly equal to the rate of increase in the former case where the surfaces are smooth. Mechanical testing of a real material, bovine cortical bone, reveals a stiffness variation consistent with that of a heterogeneous material with corrugated surfaces, the smaller bone samples are more compliant than their larger counterparts. The observed rate of change of stiffness is then used to determine the value of an additional elastic constant present within micropolar elasticity theory, the characteristic length. The value obtained for this parameter is shown to be consistent with the length scales associated with the largest scale heterogeneity present within the bone, the Herversian canal system.