This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green’s functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving bi-Helmholtz-type operators are not applicable. More specifically we establish the analytical fundamental solution for the linearized equations governing the equilibrium of pantographic 2D continua in the neighbourhood of the reference configuration. Moreover, by means of found novel Green’s functions, it is shown that it is possible to solve aforesaid equilibrium equations by using Fredholm integral equations. It is seen that an approximated analytical solution for the standard bias test for pantographic 2D continua can be found by using judiciously the found analytical fundamental solutions. The micro-macro-asymptotic identification allows for a clear and satisfactory physical interpretation of the obtained analytical results.

Green’s functions and integral representation of generalized continua: the case of orthogonal pantographic lattices

F. dell'Isola
2021

Abstract

This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green’s functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving bi-Helmholtz-type operators are not applicable. More specifically we establish the analytical fundamental solution for the linearized equations governing the equilibrium of pantographic 2D continua in the neighbourhood of the reference configuration. Moreover, by means of found novel Green’s functions, it is shown that it is possible to solve aforesaid equilibrium equations by using Fredholm integral equations. It is seen that an approximated analytical solution for the standard bias test for pantographic 2D continua can be found by using judiciously the found analytical fundamental solutions. The micro-macro-asymptotic identification allows for a clear and satisfactory physical interpretation of the obtained analytical results.
File in questo prodotto:
File Dimensione Formato  
Boutin-DellIsola2021_GreenSFunctions.pdf

solo utenti autorizzati

Descrizione: Articolo principale
Tipologia: Documento in Post-print
Licenza: Creative commons
Dimensione 2.36 MB
Formato Adobe PDF
2.36 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/175316
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 0
social impact