Stability of ideal infinite two-dimensional crystal lattice

Podolskaya, E. A.; Кривцов, А. М.; Panchenko, A.Yu.; Tkachev, P. V.

doi:10.1134/s1028335812020115

Cited by 6 publications

(4 citation statements)

References 5 publications

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“…Note that on the strong ellipticity domains' boundaries positivity is lost either by one of the instantaneous Young's moduli or by one of the instantaneous shear moduli [19]. Black dots in Fig.…”

Section: Strong Ellipticity Domainsmentioning

confidence: 97%

“…Taking into account that • ∇ = • ∇ r · ∇ and denoting v = δu, we obtain the following wave equation [19] (see the Appendix):v…”

Section: Strong Ellipticity Of Simple Latticesmentioning

confidence: 99%

“…It is possible to exclude the components of k from (11), so that the strong ellipticity conditions are expressed in terms of the components of tensor 4 Q only [19], i.e.,…”

Section: Strong Ellipticity Domainsmentioning

confidence: 99%

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Loss of ellipticity and structural transformations in planar simple crystal lattices

et al. 2015

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This work focuses on investigation of structural (phase) transformations in crystal lattices from continuum and discrete points of view. Namely, the continuum, which is equivalent to a simple lattice in the sense of the Cauchy-Born energy, is constructed using long-wave approximation, and its strong ellipticity domains in finite strain space are obtained. It is shown that various domains correspond to variants of triangular and square lattices, and the number of the domains depends on the interaction potential parameters. Non-convex energy profiles and stress-strain diagrams, which are typical for materials allowing twinning and phase transformations, are obtained on the straining paths which connect the domains and cross nonellipticity zones. The procedures of the lattice stability examinations and estimation of energy relaxation by means of molecular dynamical (MD) simulation are developed, and experimental construction of the envelope of the energy profiles, corresponding to the energy minimizer, is done on several straining paths. The MD experiment also allows to observe the energy minimizing microstructures, such as twins and two-phase structures.Mathematics Subject Classification 74B20 · 74N05 · 74N15 · 74A50

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Section: Strong Ellipticity Domainsmentioning

confidence: 97%

“…Taking into account that • ∇ = • ∇ r · ∇ and denoting v = δu, we obtain the following wave equation [19] (see the Appendix):v…”

Section: Strong Ellipticity Of Simple Latticesmentioning

confidence: 99%

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Loss of ellipticity and structural transformations in planar simple crystal lattices

et al. 2015

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“…The stability and fracture of 2D periodic structures was investigated in the presentation by Artem Yuryevich Panchenko (‘On stability of 2D lattices under finite strain within moment interaction approach using molecular dynamics’) where molecular dynamics (MD) has been used [28].…”

Section: Thematic Sessionsmentioning

confidence: 99%

Euromech 563 Cisterna di Latina 17–21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: A review of presentations and discussions

Placidi

Giorgio

Corte

et al. 2016

Mathematics and Mechanics of Solids

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In the present paper, a rational report on Euromech 563, Generalized continua and their applications to the design of composites and metamaterials (Cisterna di Latina 17–21 March 2014), is provided. The frank and constructive spirit which animated the workshop by Dell’Isola et al. ( ZAMM 2014; 94(5): 367–372) also characterized Euromech Colloquium 563. All presentations were video-recorded and are freely available online at the address http://www.memocsevents.eu/euromech563/?page_id=1013 . The topics treated were selected by the organizers in order to allow a comparison of the available experimental evidence with the predictive capability of current theoretical models. The numerical investigations selected and presented aimed to make more effective the aforementioned comparison. The interested reader will find more details about the colloquium at the dedicated webpage http://www.memocsevents.eu/euromech563 . The design and use of advanced materials and structural elements requires an extensive and rigorous process of mathematical modeling. The organizers of Euromech 563, being aware of this need, have chosen as participants renowned and reputed scientists (and some promising young researchers) who all agree on recognizing to mathematical sciences their role of unifying and coordinating tool in the effort for the advance of technology. The Colloquium has indeed surely contributed to the development of more advanced technological possibilities and to the theoretical conception of completely new ones.

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