A Quantitative Interpretation of the Response of Articular Cartilage to Atomic Force Microscopy-Based Dynamic Nanoindentation Tests

Schematic side-view of the AFM-DN setup representing the microcantilever, the spherical indenter, and the soft sample. The controlled displacement of the fixed end of the cantilever (h0), the cantilever deflection (hc), and the indentation depth (hs) are reported.

Overview of the numerical approach based on two models working in sequence and described in the central column: the abaqus model, where the equilibrium indentation depth is reached through a nonlinear simulation, and the matlab model, where the harmonic displacement is applied using a linearized constitutive relationship. The lateral columns report the AC constitutive equations used to feed the models: the transversely isotropic model and the continuum mixture model, respectively. It is worth noting that, for both the constitutive relationships, the two steps share the same mesh; in the case of the continuum mixture model, also the stiffness tangent matrix obtained at the end of the abaqus step is passed to the integration points in the matlab step.

Experimental data in terms of (a) the storage modulus versus frequency and (b) the tangent of the phase shift versus frequency for the three indentation depths analyzed. Mean values and standard deviation bars are plotted.

Best fitting results for all the analyzed experimental data, in terms of (a) E'*(f) and (b) tan(φf) when the transversely isotropic model is used. Data for the three penetration depths are reported: black lines indicate the numerical results and gray lines represent the experimental data (mean curves only).

Best fitting results for all the analyzed experimental data, in terms of (a) E'*(f) and (b) tan(φf) when the fibril distribution model is used. Gray lines define experimental data (mean curves only) whereas black lines indicate the numerical results obtained by using the parameters of model A_FIBERS.