A complete constitutive theory that describes the nonlinear elastic behavior of foams and concentrated emulsions with two‐dimensional, polydisperse hexagonal structure is derived. The individual bubbles are irregular hexagons that can vary widely in size and shape. The straight thin films and threefold symmetric Plateau borders exhibited by perfectly ordered monodisperse systems are retained by polydisperse hexagonal structures. The constitutive theory is expressed as explicit relations between the Cauchy stress and the Cauchy–Green strain tensors, or in terms of a strain energy function. The important material characteristics are surface tension, number density of bubbles, and if the dispersed phase is gas, an equation of state. The specific spatial arrangement and size distribution of the bubbles does not affect the elastic response. The volume fraction of the phases only influences the stress through the dispersed phase pressure. Weaire, Kermode, and Fu have conjectured that the elastic response associated with monodisperse structure provides an upper bound for all two‐dimensional fluid–fluid networks; we extend this conjecture to polydisperse hexagonal structure, not just monodisperse structure.