Rights statement: This is the author’s version of a work that was accepted for publication in Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Mathematical Analysis and Applications, 504, 1, 2021 DOI: 10.1016/j.jmaa.2021.125404
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - The first-order flexibility of a crystallographic framework
AU - Kastis, Eleftherios
AU - Power, Stephen
N1 - This is the author’s version of a work that was accepted for publication in Journal of Mathematical Analysis and Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Mathematical Analysis and Applications, 504, 1, 2021 DOI: 10.1016/j.jmaa.2021.125404
PY - 2021/12/1
Y1 - 2021/12/1
N2 - 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.
AB - 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.
KW - Crystallographic framework
KW - Rigidity
KW - Crystal
KW - RUM spectrum
KW - Aperiodic phase transition
U2 - 10.1016/j.jmaa.2021.125404
DO - 10.1016/j.jmaa.2021.125404
M3 - Journal article
VL - 504
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
M1 - 125404
ER -