Tensile behavior of cortical bone: Dependence of organic matrix material properties on bone mineral content

(a) A representative unit cell for stress transfer, (b) representative unit cell after the organic matrix at the ends of mineral platelets has yielded due to shear stress. The microporosity (osteocytes, lacunae, Volkmann canals etc.) is not shown. For detail of nomenclature, see Kotha et al. (2001).

(a) Variation in the elastic modulus of bone as a function of aspect ratio (length to thickness ratio) of bone mineral platelets (bone mineral content, BMC=0.40). The dashed horizontal line shows elastic modulus of control bone (19.3GPa). (b) Variation in the elastic modulus of bone as a function of mineral volume fraction {aspect ratio=20.0}; Bone Mineral Content (BMC=0.40)}. The organic matrix elastic modulus is kept constant in each line and does not vary with mineral content. These figures indicate that an increase in the elastic modulus of the organic matrix, the aspect ratio of the bone mineral, and bone mineral content leads to increases in the elastic modulus of bone tissue.

Variation in the elastic modulus of the organic matrix with respect to bone mineral content of moc. Four different relationships are considered in Eq. (4) (given in Supplementary data). In all cases, the elastic modulus of the organic matrix without any voids is accepted as 7.5GPa (i.e. when V_m-moc=0.588; Hypermineralization). To obtain the elastic modulus of control bone (19.3GPa) when the mineral volume fraction within the moc is 0.471, elastic modulus of the organic matrix must increase with smaller aspect ratios of mineral platelets (V_m-moc=0.471 is equivalent to BMC=0.40). In the case of very large aspect ratios (32.0), elastic modulus of the organic matrix in moc becomes independent of the volume fraction of the mineral phase in the moc which is shown with the horizontal line passing through the E_3moc=1.31GPa point.

(a) The microscopic yield strain in control bone tissue normalized by organic matrix shear yield stress (τ_oy). (b) The microscopic yield stress of the control bone tissue normalized by organic matrix shear yield stress (τ_oy). Variations in both parameters are presented with respect to different mineral volume fractions and aspect ratios. Elastic modulus of the organic matrix (E_3moc) is constant (4.7GPa) in both figures.

(a) The macroscopic yield strain in control bone tissue that is normalized by shear yield stress of the organic matrix (τ_oy). (b) The macroscopic yield stress of control bone tissue that is normalized by shear yield stress of the organic matrix (τ_oy). Variations in both parameters are presented with respect to different mineral volume fractions and aspect ratios. Elastic modulus of the organic matrix E_3moc is constant (4.7GPa) in both figures.

(a) Elastic moduli of bone (E_bone). (b) Organic matrix shear yield stress (τ_oy). (c) Ultimate stress of bone (σ_{bone ultimate}). (d) Ultimate strain of bone (ε_{ultimatestrain}). Variations in these parameters are shown with respect to different mineral aspect ratios and corresponding organic matrix elastic modulus E_3moc which is a function of mineral volume fraction. Elastic modulus of the organic matrix of control bone varies between 5.5 and 1.31GPa resulting in aspect ratios of bone mineral between 17.7 and 32. For 3-day fluoride treated samples, elastic modulus of the organic matrix varies between 4.2 and 1.31GPa for aspect ratios between 17.7 and 32. For 12-day fluoride treated samples, elastic modulus of the organic matrix varies between 3.57 and 1.31GPa for aspect ratios between 17.7 and 32. Shear yield stress of the organic matrix is assumed so that theoretical and experimental 0.002 yield stress values are matched as summarized in Table 2. The best match for experimental parameters is provided when the elastic modulus of the organic matrix in control bone is 3.0GPa and the aspect ratio is 27.7. It is noted that failure occurs before microscopic yield, when the organic matrix elastic modulus is 5.5GPa. Lines connecting different points show the constant aspect ratio solutions. Bars over experimental points indicate standard deviations.

Theoretically predicted organic matrix normal stresses in the y direction as a function of mineral volume fraction for different aspect ratios. Stresses in the y direction are obtained by matching the theoretically predicted ultimate strains to experimental values for control and fluoride treated bone samples and assuming the principal failure stress criteria (Eq. (13) in Supplementary data). This figure indicates that the variation in the normal stresses in the y direction is minimum in all three groups, when we use the organic matrix elastic modulus and the aspect ratio 3.0GPa and 27.7, respectively, in control bone. The a, b and h parameters are computed from the control bone. The elastic modulus of the organic matrix phase for 3- and 12-day fluoride treated samples were computed from Eq. (4) (given in Supplementary data) with corresponding bone tissue mineral volume fraction by using a, b and h values from the control bone.

Experimental and theoretical stress–strain curves of plexiform cortical bovine bone tissue with different mineral contents. All the theoretically predicted mechanical parameters (elastic modulus, yield stresses and strains, ultimate stresses and strains) are within 15% of the experimental values.

Tensile properties of the organic matrix for control, 3- and 12-day fluoride treated samples.

Stress–strain curve for shear for the organic matrix in control, 3- and 12-day fluoride treated samples.