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Nanoscience and Technology: An International Journal
ESCI SJR: 0.228 SNIP: 0.484 CiteScore™: 0.37

ISSN Druckformat: 2572-4258
ISSN Online: 2572-4266

Nanoscience and Technology: An International Journal

Formerly Known as Nanomechanics Science and Technology: An International Journal

DOI: 10.1615/NanomechanicsSciTechnolIntJ.v6.i1.60
pages 65-85

STRUCTURE OF GENERALIZED THEORIES OF ELASTICITY OF MEDIA WITH DEFECTIVE FIELDS AND OF GRADIENT THEORIES

P. A. Belov
Scientific Innovative Center "Institute of Research, Development and Transfer of Technologies", Moscow, Russia; Institute of Applied Mechanics, Russian Academy of Sciences, Moscow, Russia
Sergey A. Lurie
Institute of Applied Mechanics, Russian Academy of Sciences, 7 Leningradsky Ave., Moscow, 125040, Russia; Dorodnicyn Computing Centre FIC IU of the Russian Academy of Sciences, 40 Vavilov Str., Moscow, 119333, Russia
C. Qi
Beijing University of Civil Engineering and Architecture, Beijing, PR China

ABSTRAKT

Generalized theories of elasticity, including theories of media with defective fields and gradient theories, are considered. In contrast to classical elasticity, which does not have scaling parameters characterizing the inner structure of material, such parameters are natural in nonlocal theories of elasticity and theories of media with defective fields. Therefore, they are applied in the solution of numerous applied problems for inhomogeneous structures when scaling effects are to be considered. Generalized models of continuous media are especially attractive in simulating properties of various micro/nanostructures, elastic properties of composite materials and structured materials with submicron and nanosized internal structures wherein effective properties are significantly defined by scaling effects (effects of close interaction (cohesion) and adhesion). Nonclassical physical properties of generalized media are determined in terms of a sixth-rank tensor of gradient modulus of elasticity, which should obey some symmetry conditions. In the present work we discuss general fundamental properties and structure of nonclassical sixth-rank tensor of elasticity moduli in the theories of generalized media and propose a classification of models that provides grounds for building correct models of continua with account for scaling effects. An orthogonal basis of fifteen tensors of "moment" moduli is built and investigated in the sixth-rank tensor space. The structure of reversible models of deformation of solid media, including, both ideal (nondefective) media and media with conserved dislocation fields, is indicated. It is demonstrated that eleven basic tensors determine reversible properties of gradient media, whereas the remaining four tensors determine their dissipative properties. Eleven basic tensors, defining the properties of reversible deformations processes, were used to build five tensors of gradient moduli, substantial for nonlocal gradient theories. As a result, a structure of tensors for correct versions of gradient theories of elasticity is presented, and the type of tensors of elasticity for a very simple fully symmetric gradient two-parameter theory that can be proposed as an applied theory for simulating micro/nanosized effects is established. The notion of the media space (models) has been introduced in accordance with the number of nonclassical moduli. It is demonstrated that in a general case the density of potential energy of curvatures is represented as a sum of densities of potential energies of subspaces of the Mindlin–Toupin gradient theories, theories of media with the Mindlin defective fields, and a bilinear form defining their interaction.