The rigidity of a network of elastic beams crucially depends on the specificdetails of its structure. We show both numerically and theoretically that thereis a class of isotropic networks which are stiffer than any other isotropicnetwork with same density. The elastic moduli of these \textit{stiffest elasticnetworks} are explicitly given. They constitute upper-bounds which compete orimprove the well-known Hashin-Shtrikman bounds. We provide a convenient set ofcriteria (necessary and sufficient conditions) to identify these networks, andshow that their displacement field under uniform loading conditions is affinedown to the microscopic scale. Finally, examples of such networks with periodicarrangement are presented, in both two and three dimensions.
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