Ageing societies suffer from an increasing incidence of bone fractures. Bone strength depends on the amount of mineral measured by clinical densitometry, but also on the micromechanical properties of the hierarchical organization of bone. Here, we investigate the mechanical response under monotonic and cyclic compression of both single osteonal lamellae and macroscopic samples containing numerous osteons. Micropillar compression tests in a scanning electron microscope, microindentation and macroscopic compression tests were performed on dry ovine bone to identify the elastic modulus, yield stress, plastic deformation, damage accumulation and failure mechanisms. We found that isolated lamellae exhibit a plastic behaviour, with higher yield stress and ductility but no damage. In agreement with a proposed rheological model, these experiments illustrate a transition from a ductile mechanical behaviour of bone at the microscale to a quasi-brittle response driven by the growth of cracks along interfaces or in the vicinity of pores at the macroscale.

The growing incidence of skeletal fractures poses a significant challenge to ageing societies. Since a major part of physiological loading in the lower limbs is carried by cortical bone, it would be desirable to better understand the structure-mechanical property relationships and scale effects in this tissue. This study aimed at assessing whether microindentation properties combined with chemical and morphological information are usable to predict macroscopic elastic and strength properties in a donor- and site-matched manner. Specimens for quasi-static macroscopic tests in tension, compression, and torsion and microindentation were prepared from a cohort of 19 male and 20 female donors (46 to 99 years). All tests were performed under fully hydrated conditions. The chemical composition of the extra-cellular matrix was investigated with Raman spectroscopy. The results of the micro-mechanical tests were combined with morphological and compositional properties using a power law relationship to predict the macro-mechanical results. Microindentation properties were not gender dependent, remarkably constant over age, and showed an overall small variation with standard deviations of approximately 10 %. Similar results were obtained for chemical tissue composition. Macro-mechanical stiffness and strength were significantly related to porosity for all load cases (p<0.05). In case of macroscopic yield strain and work-to-failure this was only true in torsion and compression, respectively. The correlations of macro-mechanical with micro-mechanical, morphological, and chemical properties showed no significance for cement line density, mineralisation, or variations in the microindentation results and were dominated by porosity with a moderate explanatory power of predominately less than 50 %. The results confirm that age, with minor exceptions gender, and small variations in average mineralisation have negligible effect on the tissue microindentation properties of human lamellar bone in the elderly. Furthermore, our findings suggest that microindentation experiments are suitable to predict macroscopic mechanical properties in the elderly only on average and not on a one to one basis. The presented data may help to form a better understanding of the mechanisms of ageing in bone tissue and of the length scale at which they are active. This may be used for future prediction of fracture risk in the elderly.

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Nonlinear computational analysis of materials showing elasto-plasticity or damage relies on knowledge of their yield behavior and strengths under complex stress states. In this work, a generalized anisotropic quadric yield criterion is proposed that is homogeneous of degree one and takes a convex quadric shape with a smooth transition from ellipsoidal to cylindrical or conical surfaces. If in the case of material identification, the shape of the yield function is not known a priori, a minimization using the quadric criterion will result in the optimal shape among the convex quadrics. The convexity limits of the criterion and the transition points between the different shapes are identified. Several special cases of the criterion for distinct material symmetries such as isotropy, cubic symmetry, fabric-based orthotropy and general orthotropy are presented and discussed. The generality of the formulation is demonstrated by showing its degeneration to several classical yield surfaces like the von Mises, Drucker-Prager, Tsai-Wu, Liu, generalized Hill and classical Hill criteria under appropriate conditions. Applicability of the formulation for micromechanical analyses was shown by transformation of a criterion for porous cohesive-frictional materials by Maghous et al. In order to demonstrate the advantages of the generalized formulation, bone is chosen as an example material, since it features yield envelopes with different shapes depending on the considered length scale. A fabric- and density-based quadric criterion for the description of homogenized material behavior of trabecular bone is identified from uniaxial, multiaxial and torsional experimental data. Also, a fabric- and density-based Tsai-Wu yield criterion for homogenized trabecular bone from in silico data is converted to an equivalent quadric criterion by introduction of a transformation of the interaction parameters. Finally, a quadric yield criterion for lamellar bone at the microscale is identified from a nanoindentation study reported in the literature, thus demonstrating the applicability of the generalized formulation to the description of the yield envelope of bone at multiple length scales.

A new anisotropic elastic-viscoplastic damage constitutive model for bone is proposed using an eccentric elliptical yield criterion and nonlinear isotropic hardening. A micromechanics-based multiscale homogenization scheme proposed by Reisinger et al. is used to obtain the effective elastic properties of lamellar bone. The dissipative process in bone is modeled as viscoplastic deformation coupled to damage. The model is based on an orthotropic ecuntric elliptical criterion in stress space. In order to simplify material identification, an eccentric elliptical isotropic yield surface was defined in strain space, which is transformed to a stress-based criterion by means of the damaged compliance tensor. Viscoplasticity is implemented by means of the continuous Perzyna formulation. Damage is modeled by a scalar function of the accumulated plastic strain [Formula: see text] , reducing all element s of the stiffness matrix. A polynomial flow rule is proposed in order to capture the rate-dependent post-yield behavior of lamellar bone. A numerical algorithm to perform the back projection on the rate-dependent yield surface has been developed and implemented in the commercial finite element solver Abaqus/Standard as a user subroutine UMAT. A consistent tangent operator has been derived and implemented in order to ensure quadratic convergence. Correct implementation of the algorithm, convergence, and accuracy of the tangent operator was tested by means of strain- and stress-based single element tests. A finite element simulation of nano- indentation in lamellar bone was finally performed in order to show the abilities of the newly developed constitutive model.

Trabecular bone is a porous mineralized tissue playing a major load bearing role in the human body. Prediction of age-related and disease-related fractures and the behavior of bone implant systems needs a thorough understanding of its structure-mechanical property relationships, which can be obtained using microcomputed tomography-based finite element modeling. In this study, a nonlinear model for trabecular bone as a cohesive-frictional material was implemented in a large-scale computational framework and validated by comparison of μFE simulations with experimental tests in uniaxial tension and compression. A good correspondence of stiffness and yield points between simulations and experiments was found for a wide range of bone volume fraction and degree of anisotropy in both tension and compression using a non-calibrated, average set of material parameters. These results demonstrate the ability of the model to capture the effects leading to failure of bone for three anatomical sites and several donors, which may be used to determine the apparent behavior of trabecular bone and its evolution with age, disease, and treatment in the future.

Ageing and associated skeletal diseases pose a significant challenge for health care systems worldwide. Age-related fractures have a serious impact on personal, social and economic wellbeing. A significant proportion of physiological loading is carried by the cortical shell. Its role in the fracture resistance and strength of whole bones in the ageing skeleton is of utmost importance. Even though a large body of knowledge has been accumulated on this topic on the macroscale, the underlying micromechanical material behaviour and the scale transition of bone's mechanical properties are yet to be uncovered. Therefore, this review aims at providing an overview of the state-of-the-art of the post-yield and failure properties of cortical bone at the extracellular matrix and the tissue level.