Cell-wise Stress-Strain Relation in 2D Polygonal Tessellation Systems

Randomized polygonal tessellations are frequently observed as nature’s preferred geometric pattern and engineering platform. Mechanical aspects such as constitutive relations and elasticity have been the topics of extensive study in the past decades in various contexts such as foams and more recently, in confluent biological tissues. In this work we present a theory for the cell-level stress-strain relationship. Both analytical derivation and numerical simulation indicate a consistent stress-strain relation across different energy models. An analytical expression for cell-wise shear modulus is derived, validated numerically, and compared with prior work in the foam literature. This result finds broad applicability in tessellated structures at mechanical equilibria.