The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

“…[2,65] for classical references and [66,67] for more recent results) are good candidates for this purpose, as shown in [68][69][70]. A review of results on the theoretical foundation of a variational approach for higher gradient theories is [71]. Besides, the use of the methods of metamaterials [72][73][74] like those for pantographic structures [75][76][77] could be considered even to take into account the effects of damage [78][79][80][81][82].…”

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“…Following Lagrange's ideas, we start with the mechanical definition of forces and consider them as quantities dual to the virtual displacements. As a consequence and as stated for instance in [dell'Isola et al 2015b;2011;2012], we therefore accept the idea of considering forces as distributions in the sense of L. Schwartz [1951]. This definition already uses the concept of duality and motivates the formulation of the equilibrium conditions in variational form, which is the first axiom, the principle of virtual work.…”

“…We refer to [dell'Isola et al 2015b] for more technical details about this result. Remark that (4) gives a much larger spectrum of force representations than required for the classical continuum theory.…”

“…The second method, which traces back to Lagrange [1788] and Piola [1832;1848;2014;dell'Isola et al 2015a], is of a variational nature and defines forces in a generalized sense as the quantities dual to the virtual displacement field and the gradients thereof. The variational approach has gained much attention during the last half century, especially in the field of generalized continua like gradient materials and polar and micromorphic media; see for instance [dell'Isola et al 2015b;2011;Toupin 1964;Germain 1973b]. Most of these generalized theories cannot be stated straightforwardly by using the nonvariational approach.…”

Linear pantographic sheets: Asymptotic micro-macro models identification

“…We used these microscopic floppy modes to prove that, in general, the class of first-gradient continua (those introduced by Cauchy and usually considered in continuum mechanics; see, e.g., [dell'Isola et al 2016a;2015c]) is not enough to describe, at a macrolevel, all conceivable physical systems. In particular, when there is high contrast (see [Camar-Eddine and Seppecher 2001;Pideri and Seppecher 1997;Hans and Boutin 2008;Boutin and Soubestre 2011;Cecchi and Rizzi 2001]) in physical properties at the microlevel, it may become necessary at the macrolevel to introduce higher-order continua (see 2016e]).…”