Critically elastic materials—those that are rigid with a single state of self-stress—can be generated from parent systems with two states of self-stress by the removal of one of many constraints. We show that the elastic moduli of the resulting homogeneous and isotropic daughter systems are interrelated by a universal functional form parametrized by properties of the parent. In simulations of both spring networks and packings of soft spheres, judicious choice of parent systems and bond removal allows for the selection of a wide variety of moduli and Poisson's ratios in the critically elastic systems, providing a framework for versatile deterministic selection of mechanical properties.

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• Elastic modulus
• Elasticity
• Network structure
• First-principles calculations