Micro and macroscopic stress-strain relations in disordered tessellated networks

Tessellated soft networks, such as liquid foams and epithelial tissues, are broadly present in applications in the materials, physical, and biological contexts. Mechanical aspects such as constitutive relations and elasticity have been the topics of extensive studies, yet predictive analyses are challenging due to geometric complexity and embedded randomness. However, one can often explore patterns, universal scaling laws, and consistent correlations to help illuminate the intrinsic physical behavior embedded in the network. Here we discover one simple “hidden” correlation between the microscopic, cellwise stress and strain in planar tessellated networks. We demonstrate that for a rigid and incompressible network in mechanical equilibrium, the microscopic stress and strain follows a simple relation, σ=pE, where σ is the deviatoric stress, E is a mean-field strain tensor, and p is the hydrostatic pressure. This relationship arises as the natural consequence of energy minimization or equivalently, mechanical equilibration. The result suggests not only that the microscopic stress and strain are aligned in the principal directions, but also microscopic deformations are predominantly affine. The relationship holds true regardless of the different (foam or tissue) energy model considered, and directly leads to a simple prediction for the shear modulus, μ=〈p〉/2, where 〈p〉 is the mean pressure of the tessellation, for general randomized lattices.