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Nonlinear compliance of elastic layers to indentation

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Biomechanics and Modeling in Mechanobiology Aims and scope Submit manuscript


Thin samples adherent to a rigid substrate are considerably less compliant to indentation when compared to specimens that are not geometrically confined. Analytical corrections to this so-called substrate effect exist for various types of indenters but are not applicable when large deformations are possible, as is the case in biological materials. To overcome this limitation, we construct a nonlinear scaling model characterized by one single exponent, which we explore employing a parametric finite element analysis. The model is based on asymptotes of two length scales in relation to the sample thickness, i.e., indentation depth and radius of the contact area. For small indentation depth, we require agreement with analytical, linear models, whereas for large indentation depth and extensive contact area, we recognize similarity to uniaxial deformation, indicating a divergent force required to indent nonlinear materials. In contrast, we find linear materials not to be influenced by the substrate effect beyond first order, implying that nonlinear effects originating from either the material or geometric confinement are clearly separated only in thin samples. Furthermore, in this regime the scaling model can be derived by following a heuristic argument extending a linear model to large indentation depths. Lastly, in a large indentation setting where the contact is small in comparison with sample thickness, we observe nonlinear effects independent of material type that we attribute to a higher-order influence of geometrical confinement. In this regime, we define a scalar as the ratio of strains along principal axes as obtained by comparison with the case of a point force on a half-space. We find this scalar to be in quantitative agreement with the scaling exponent, indicating an approach to distinguish between nonlinear effects in the scaling model. While we conjecture our findings to be applicable to other flat-ended indenters, we focus on the case of a flat-ended cylinder in normal contact with a thin layer. The analytical solution for small indentation associated with this problem has been given by Hayes et al. (J Biomech 5:541–551, 1972), for which we provide a convenient numerical implementation.

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  1. In addition, it is usually feasible to probe cells in thin lamellipodial regions, as measurement near the center would require treatment of the cell as a composite structure due to mechanical contributions from the nucleus Cao et al. (2013) and Gavara and Chadwick (2012).

  2. We note, that for \(\nu \rightarrow 0.5\), \(\lambda _1\lambda _2\lambda _3\approx 1\) in conjunction with the second invariant \(I_2\) implies a dependence on \(\lambda _i^{-4}\).

  3. Other hyperelastic material models can be analytically evaluated in a similar fashion as addressed in Supplement E.


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The authors would like to thank Holger Stark for bringing the issue of substrate effects to their attention.

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Correspondence to Adrian Fessel or Hans-Günther Döbereiner.

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The authors declare that they have no conflict of interest.

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Fessel, A., Döbereiner, HG. Nonlinear compliance of elastic layers to indentation. Biomech Model Mechanobiol 17, 419–438 (2018).

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