Generalização das curvas de forças de materiais com relaxação exponencial e lei de potência

Abstract

We studied the behavior of the relaxation function of viscoelastic materials subjected to rigid indenters. Viscoelastic materials generally exhibit exponential or power-law decays that depend on the geometry of the mechanical contact between the indenter and the sample. We propose a mechanical circuit composed of a fractional element and a standard linear solid connected in parallel. The standard linear solid element exhibits exponential relaxations as those found in soft materials like foam or synthetic materials. Fractional calculus has been applied to various physical phenomena, sometimes with more success than traditional calculus, which presents integer order derivatives. Fractional elements, for example, can describe power-law viscoelastic materials, as occurs, for example, in living cells. Our analytical model leads to generalized equations capable of reproducing the two most common types of temporal decay found in viscoelastic materials. In addition, our model considers indenters with different geometries, the most common being conical, spherical, and cylindrical. We analyze the contact force curves while the sample is deformed (called the loading force curve) or the indentation is kept constant (called the dwell force curve). In the latter case, the material relaxes due to an energy minimization process that decreases the response force. Force measurements are used to investigate different soft materials. The relaxation decay is related to the composition of those materials, which can identify faults. In the case of cells, for instance, the change in the rheological behavior is associated with the emergence of pathologies such as cancer and other diseases. Our work allows a generalized analytical interpretation of the force curves, where the sample's mechanical properties and the indenter's geometry are parameters regulated in the same equation.