# 04/07/2019 – M. Merkel – A minimal-length approach unifies rigidity in under-constrained materials

Séminaire exceptionnel – 4 juil. 2019 – 11h salle 250

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A minimal-length approach unifies rigidity in under-constrained materials
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Matthias Merkel – CPT, Marseille
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What do a guitar string and a balloon have in common? They are both floppy unless rigidified by geometric incompatibility. The same kind of rigidity transition in under-constrained materials has more recently been discussed in the context of disordered biopolymer networks like collagen and models for biological tissues. Here we show that these materials exhibit generic elastic behavior close to the rigidity transition, which is independent of the microscopic structure and the disorder in 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 under-constrained spring networks as well as 2D and 3D vertex models for dense biological tissues. For instance, we analytically 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 also 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.