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The Young's modulus, bending strength, apparent density, and ash content of 155 human compact bone bending specimens were determined. Both Young's modulus (E) and bending strength (S) were strongly correlated to tissue dry apparent density (rho a). Based upon the correlation coefficient (R) and the percent deviation of the data from the regression curve (% dev.), these correlations were best described by power law relationships: E infinity rho a 1.54 (R2 = 0.79, % dev. = 2.4) and S infinity rho a 2.18 (R2 = 0.80, % dev. = 6.4). Bending strength was related to Young's modulus raised to the 1.26 power, implying a nonlinear relationship for these variables. We found a weak correlation between ash content and the mechanical behavior of the compact bone specimens, particularly Young's modulus, but could not statistically justify formulation of a more complex multivariate power model incorporating both density and ash content. Regional variations in strength and stiffness along the femoral shaft and within the cortex were also noted and were attributed primarily to differences in apparent density. The relationships formulated for the mechanical behavior of human compact bone are discussed in terms of the results of previous investigations of the mechanical behavior of nonhuman compact bone and human cancellous bone.

We derive a Fractional Cahn-Hilliard Equation (FCHE) by considering a gradient flow in the negative order Sobolev space H −α , α ∈ [0, 1] where the choice α = 1 corresponds to the classical Cahn-Hilliard equation whilst the choice α = 0 recovers the Allen-Cahn equation. It is shown that the equation preserves mass for all positive values of fractional order α and that it indeed reduces the free energy. The well-posedness of the problem is established in the sense that the H 1 -norm of the solution remains uniformly bounded. We then turn to the delicate question of the L∞ boundedness of the solution and establish an L∞ bound for the FCHE in the case where the non-linearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier-Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semi-discrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order α. It is observed that the nature of the solution of the FCHE with a general α > 0 is qualitatively (and quantitatively) closer to the behaviour of the classical Cahn-Hilliard equation than to the Allen-Cahn equation, regardless of how close to zero be the value of α. An examination of the coarsening rates of the FCHE reveals that the asymptotic rate is rather insensitive to the value of α and, as a consequence, is close to the well-established rate observed for the classical Cahn-Hilliard equation.1991 Mathematics Subject Classification. 65N12, 65N30, 65N50.

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