The first-order flexibility of a crystallographic framework

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

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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.

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Journal Article
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Journal of Mathematical Analysis and Applications
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Deposited On:
01 Jun 2021 09:17
In Press
Last Modified:
02 Jun 2021 14:30