The first-order flexibility of a crystallographic framework

Kastis, Eleftherios and Power, Stephen (2021) The first-order flexibility of a crystallographic framework. Journal of Mathematical Analysis and Applications, 504 (1): 125404. ISSN 0022-247X

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Abstract

Four sets of necessary and sufficient conditions are obtained for the first-order rigidity of a periodic bond-node framework $\C$ in $\bR^d$ which is of crystallographic type. In particular, an extremal rank characterisation is obtained which incorporates a multi-variable matrix-valued transfer function $\Psi_\C(z)$ defined on the product space $\bC^d_* = (\bC\backslash \{0\})^d$. In general the first-order flex space is the closed linear span of polynomially weighted geometric velocity fields whose geometric multi-factors in $\bC^d_*$ lie in a finite set. It is also shown that, paradoxically, a first-order rigid crystal framework may possess a nontrivial continuous motion. Examples of this phenomenon are given which are associated with aperiodic displacive phase transitions between periodic states.