An original force-displacement relationship for spherical inclusions in multilayered viscoelastic finite media

This paper presents original solutions of the force-displacement relationships for a rigid spherical bead embedded in a composite medium made of n-isotropic linearly viscoelastic finite layers. Analytical solutions were provided for both compressible and incompressible elastic and viscoelastic solids, assuming no-slip conditions between the rigid spherical inclusion and its adjacent medium as well as between each layer of the composite medium. Thanks to these general formulas, we investigated the effect of finite size media on the force-bead displacement response and derived the exact relationship linking apparent and intrinsic elastic moduli of the medium. Such theoretical solutions can be interestingly applied to identify layer's heterogeneities and to characterize accurately the mechanical properties of living material like cells when using translational microrheology assays. This point is especially illustrated by modeling animal cell cytoskeleton as a bilayer composite medium probed by magnetic tweezers. Interestingly, our results highlighted the influence of finite cell size effects, while allowing to distinguish viscoelastic properties of deep cell cytoskeleton from those of cellular cortex. Moreover, we established that translational microrheology experiments are well suited to characterize locally the viscoelasticity properties of the layer in contact with the probe as soon this layer thickness is larger than ten bead diameters.