Cauchy Tetrahedron Argument Applied to Higher Contact Interactions

Dell’Isola, Francesco; Madeo, Angela; Seppecher, Pierre

doi:10.1007/s00205-015-0922-6

Cited by 74 publications

(59 citation statements)

References 46 publications

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“…We emphasize that it has been shown in [28,33] how an expression of type S ins (D, V) can be transformed in an expression of type S fro (D, V), while in [120,130] the converse is shown. Actually, the necessity for mechanicians to divide the mechanical interactions into these subclasses comes from their desire to find constitutive laws for the tensors τ and F which only involve local quantities.…”

Section: (B) Frontier and Inside-the-body Interactionsmentioning

confidence: 98%

“…While inequality (2.5) could seem a very weak assumption, it has been emphasized in [130] that it rules out some possible stress states, as for instance those occurring in continua including material surfaces or continua including interfaces with Laplace surface tension.…”

Section: (C) Alternative: D'alembert Versus Cauchymentioning

confidence: 99%

“…only on its normal n). Although many authors (among which Richard Toupin [26,121]) were aware of this fact, no effort has been attempted to generalize the tetrahedron construction in order to encompass theories of higher gradient continua until the works [115,120,130] (see also Maugin G. MathReview MR1437786 (98d:73003) 73A05 (73B18 73S10) on the paper [120]). The reason is probably due to the mathematical difficulties, as explicitly remarked in [115,120,141,162], which are encountered when dealing with the double dependence of power functional S fro (D, V) on velocity fields and on subbodies of the considered continuum.…”

Section: Postulation Of the Mechanics Of Continuous Bodies à La Cauchymentioning

confidence: 99%

“…The idea of the proof (which can be found in [166,167] or [130]) is again to apply the Quasi-Balance inequality to a shrinking family of domains made by the intersection of D and suitable polyhedra.…”

Section: Postulation Of the Mechanics Of Continuous Bodies à La Cauchymentioning

confidence: 99%

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

2015

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Section: (B) Frontier and Inside-the-body Interactionsmentioning

confidence: 98%

Section: (C) Alternative: D'alembert Versus Cauchymentioning

confidence: 99%

Section: Postulation Of the Mechanics Of Continuous Bodies à La Cauchymentioning

confidence: 99%

Section: Postulation Of the Mechanics Of Continuous Bodies à La Cauchymentioning

confidence: 99%

See 2 more Smart Citations

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

2015

Self Cite

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“…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]). …”

Section: Micromodeling Of Pantographic Latticesmentioning

confidence: 99%

Linear pantographic sheets: Asymptotic micro-macro models identification

Boutin

Dell’Isola

Giorgio

et al. 2017

Math. Mech. Compl. Sys.

165

120

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In this paper we consider linear pantographic sheets, which in their natural configuration are constituted by two orthogonal arrays of straight fibers interconnected by internal pivots. We introduce a continuous model by means of a micro-macro identification procedure based on the asymptotic homogenization method of discrete media. The rescaling of the mechanical properties and of the deformation measures is calibrated in order to comply with the specific kinematics imposed by the quasi-inextensibility of the fibers together with the large pantographic deformability. The obtained high-order continuum model shows interesting and exotic features related to its extreme anisotropy and also to the subcoercivity of its deformation energy. Some initial numerical simulations are presented, showing that the model can account for experimental uncommon phenomena occurring in pantographic sheets. The paper focuses on the precise analysis and the understanding of the effective behavior based on a well-calibration of the extension and bending phenomena arising at the local scale. In an upcoming work, the analysis will be extended to oblique arrays, some analytical solutions to proposed equations and some further applications.

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Trefftz collocation method for two‐dimensional strain gradient elasticity

Solyaev

Lurie

2020

Numerical Meth Engineering

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Indirect Trefftz method is proposed for solving two‐dimensional boundary value problems of the strain gradient elasticity theory (SGET). A system of trial functions satisfying the fourth‐order equilibrium equations of SGET are developed based on the generalized Papkovich‐Neuber potentials. The classical part of the displacement solution is represented through the T‐complete system of functions satisfying the Laplace equation. The gradient part of the solution is represented through the system of heuristic functions satisfying the Helmholtz equation. The least squares collocation method is used to enforce the boundary conditions. Numerical examples are presented for the square domain under non‐uniform tensile and bending loads. It is shown, that the advantage of the presented method is that it allows to directly control the accuracy of the fulfillment of all nonstandard boundary conditions, that are prescribed in SGET on the surfaces and edges of the body.

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Cauchy Tetrahedron Argument Applied to Higher Contact Interactions

Cited by 74 publications

References 46 publications

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

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

Linear pantographic sheets: Asymptotic micro-macro models identification

Trefftz collocation method for two‐dimensional strain gradient elasticity

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