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Suppose I have some material like graphene that I shaped in some structure (honeycomb, triangular lattice), then it would have a different rigidity according to the structure. My question is, how can we relate rigidity to the structure of the material? Is there any approximate formula that can give us the relation between rigidity and some property of the structure? In this paper, they present rigidity for different structures, but I want to know whether we can predict the rigidity of a non-trivial structure.

P.S: With structure, I mean, the way you form a rigid solid out of that material.

Thanks in advance.

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In general there isn't a simple relationship between the structure of an object and its rigidity because the mechanical properties of whatever material you want to make the object out of also come into play, and describing the intrinsic mechanical properties of a material can get quite complicated. Consider graphene, the material you mentioned. Graphene is extremely strong when subjected to tension stresses (i.e. when trying to be stretched apart), but at the same time it is extremely weak under compression stresses (i.e., it buckles and collapses when the edges of a graphene sheet are pushed inward). There are also shearing stresses, and I don't offhand know how strong graphene is to those.

The bottom line is if you make an object entirely out of graphene, in general a detailed analysis will be required to predict how rigid the object will be when subjected to external stresses. Under external stresses, some parts of the object may be under tensile stresses, other parts may be under compressive stresses, and there will be shearing stresses all around. Also, for an anisotropic material like graphene, the overall rigidity will also depend on how the graphene sheets are oriented with respect to the principal stress axes throughout the object. You can probably start to see now why multiple materials with different mechanical properties are used together in many engineering applications.

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