A Minimal-length Approach Unifies Rigidity in Under-constrained Materials

Orateur : Matthias MERKEL (Alan Turing Center for Living Systems & Centre de Physique Théorique, Marseille, France)

What do a guitar string and a balloon have in common ? They are both floppy unless rigidified by geometric incompatibility. This is generally true for under-constrained systems ; they are floppy, but can be rigidified by being forced into a regime of geometric incompatibility. This kind of rigidity transition has more recently been discussed in the context of disordered biopolymer networks like collagen and models for biological tissues. We show here that the elastic behavior close to this rigidity transition is generic, i.e., its functional form is independent of the microscopic structure of the system. Phrasing the condition of geometric incompatibility in terms of a minimal length function, we obtain analytic expressions for the elastic stresses and moduli. We numerically verify our findings by simulations of sub-isostatic spring networks as well as 2D and 3D vertex models for dense biological tissues. For instance, we obtain exact expressions for the magnitudes of bulk modulus and shear modulus discontinuities at the rigidity transition, several scaling relations of the shear modulus, and the magnitude of the anomalous Poynting effect. Moreover, we show that the ratio of the excess shear modulus to the shear stress is inversely proportional to the critical shear strain with a prefactor of three, which we expect to be a general hallmark of rigidity in under-constrained materials induced by geometric incompatibility. This could be used in experiments to distinguish whether strain-stiffening as observed for instance in biopolymer networks arises from nonlinear characteristics of the microscopic material components or from effects of geometric incompatibility.

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