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The Maxwell-Calladine index theorem plays a central role in our currentunderstanding of the mechanical rigidity of discrete materials. By consideringthe geometric constraints each material component imposes on a set ofunderlying degrees of freedom, the theorem relates the emergence of rigidity toconstraint counting arguments. However, the Maxwell-Calladine paradigm issignificantly limited -- its exclusive reliance on the geometric relationshipsbetween constraints and degrees of freedom completely neglects the actualenergetic costs of deforming individual components. To address this limitation,we derive a generalization of the Maxwell-Calladine index theorem based onsusceptibilities that naturally incorporate local energetic properties such asstiffness and prestress. Using this extended framework, we investigate howlocal energetics modify the classical constraint counting picture to capturethe relationship between deformations and external forces. We then apply thisformalism to design mechanical metamaterials where differences in symmetrybetween local energy costs and structural geometry are exploited to controltheir response to external forces.

Integrating local energetics into Maxwell-Calladine constraint counting to design mechanical metamaterials