8th Annual Symposium
Physics of Cancer
Leipzig, Germany
October 4-6, 2017
Contributed Talk
Non-linear compliance of elastic layers to indentation
Adrian Fessel, Christina Oettmeier, Hans-Günther Döbereiner
Institut für Biophysik, Universität Bremen, Bremen, Germany
Contact: 
In cells and tissues, diseased states are often accompanied by changes in elastic behavior, motivating diagnostic application. Methods targeting the elasticity of soft matter are as diverse as the requirements they need to meet. While other methods focus on e.g., throughput, we address here indentation testing, a method that stands out in the sense that it enables spatial mapping of material properties, and may impose well-defined deformations prompting even a non-linear elastic response if desired. Indentation testing is applicable on various length scales, starting at the sub-cellular level when implemented via an atomic force microscope. However, researches performing indentation tests are faced with the challenge of having to adapt their analyses to sample geometry, as biological samples quite often cannot be shaped to fit the needs of the experiment without altering their properties.

In detail, thin samples, such as cells 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 non-linear 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 non-linear materials. In contrast, we find linear materials not to be influenced by the substrate effect beyond first order, implying that separation between non-linear effects originating either from the material or geometric confinement is only possible in thin samples. In a large indentation setting where the contact is small in comparison to sample thickness, we observe non-linear effects independent of material type that we attribute to a higher order influence of geometrical confinement.

We apply a viscoelastic extension of the scaling model to data obtained from indentation experiments performed on microplasmodia of the unicellular slime mold Physarum polycephalum that serve as a model system. We identify the multi modal distribution of parameters such as Young’s moduls, Poisson’s ratio and relaxation times associated with viscous processes that cover five orders of magnitude. Results suggest a characterization of microplasmodia as porous, compressible structures that act like elastic solids with high Young’s modulus on short time scales, whereas on long time-scales and upon repeated indentation viscous behavior dominates and the effective modulus is significantly decreased.
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