Generalized Hooke's law for isotropic second gradient materials

By Francesco dell'Isola, G. Sciarra, S. Vidoli. Proceedings of the Royal Society A, 2009

306 citations recorded by the source corpus.

In the spirit of Germain the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortuné & Vallée. Linear isotropic constitutive relations for stress and hyperstress in terms of strain and strain-gradient are then obtained proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski. Using a suitable decomposition of the strain-gradient, it is found a necessary and sufficient condition, to be verified by the elastic moduli, assuring positive definiteness of the stored elastic energy. The problem of warping in linear torsion of a prismatic second gradient cylinder is formulated, thus obtaining a possible measurement procedure for one of the second gradient elastic moduli.

Subfields: Materials Chemistry, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/213beb1d80c53046f654ad0fd4315ce39da28e1e

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Citations in the corpus

A unifying perspective: the relaxed linear micromorphic continuum (2013, unknown): “109) Let us consider the energy from the isotropic second gradient poromechanics model [38,42] E(∇u, ∇(tr(sym ∇u))) = μ ‖ dev sym ∇u‖2 + 2μ + 3λ 6 [tr(sym ∇u)]2 + μL(2)c ‖∇(tr(sym ∇u)‖2.”
Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps (2013, methodology): “d hence P !ru: Thismeansthattheenergy(14)reducesto W 2G(ru;rru) = hksymruk 2 + h 2 (trru)2 + g 2 krruk2 (44) which is indeed a second gradient energy for linear-elastic, isotropic media (see e.g. [15, 64]). Indeed, governing equations for second (and higher) gradient continua can be also obtained ”
Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials (2014, background): “In the remainder of this paper we will consider a particular constitutive form for which is valid in the isotropic, second-gradient, linear–elastic case [63]:”
Material symmetry group and constitutive equations of micropolar anisotropic elastic solids (2016, background): “Examples of proposed constitutive equations of generalized media can be found in [18, 19] for secondgradient materials and in [20–22] for micromorphic media.”
Granular micromechanics based micromorphic model predicts frequency band gaps (2015, result): “The nonzero components of the sixth rank tensor A are same as that obtained in [43].”
At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola (2013, background): “Many important applications can be conceived for higher gradient materials, as for instance those involving the phenomena described for instance in [1], [27], [30], [51], [71], [72], [103], [112], [113], [114], [115], [116], [117], [118], [119], [141], [142], [143], [144].”
Mechanical metamaterials: a state of the art (2019, background): “The ideas presented by Toupin [106], Mindlin [107] and Germain [108] are the basis of the following [114, 115].”
Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof (2015, methodology): “, [42–52]), no general method or technique is available yet for characterizing the discrete microstructures which must be modeled in the homogenized limit by means of generalized continua.”
Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids (2013, unknown): “[52]).”