Positive definiteness in coupled strain gradient elasticity
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O R I G I NA L A RT I C L E
Lidiia Nazarenko
· Rainer Glüge · Holm Altenbach
Positive definiteness in coupled strain gradient elasticity
Received: 22 September 2020 / Accepted: 28 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The linear theory of coupled gradient elasticity has been considered for hemitropic second gradient materials, specifically the positive definiteness of the strain and strain gradient energy density, which is assumed to be a quadratic form of the strain and of the second gradient of the displacement. The existence of the mixed, fifth-rank coupling term significantly complicates the problem. To obtain inequalities for the positive definiteness including the coupling term, a diagonalization in terms of block matrices is given, such that the potential energy density is obtained in an uncoupled quadratic form of a modified strain and the second gradient of displacement. Using orthonormal bases for the second-rank strain tensor and third-rank strain gradient tensor results in matrix representations for the modified fourth-rank and the sixth-rank tensors, such that Sylvester’s formula and eigenvalue criteria can be applied to yield conditions for positive definiteness. Both criteria result in the same constraints on the constitutive parameters. A comparison with results available in the literature was possible only for the special case that the coupling term vanishes. These coincide with our results. Keywords Strain gradient elasticity · Coupling fifth-rank tensor · Positive definiteness of the potential energy 1 Introduction The classical theory of elasticity is one of the most important tools of engineering suitable to describe many phenomena in bodies deformed under action of external forces. However, as any theory, it has a limited range of application. It is scale insensitive, and its solutions contain singularities when the boundary conditions contain singularities or the boundary geometry has sharp corners, as known from the Flamant–Boussinesq problem, the Kelvin problem, the crack tip problem and others [35]. Indeed, in order to take into account size effects (cf. [3,26,27]), to remove singularities in the stresses and displacements, when discontinues appear in the boundary conditions (e.g., [5,16,34,35]), to describe phenomena in the micro- and nanometer range like Communicated by Andreas Öchsner. L. Nazarenko (B) · R. Glüge · H. Altenbach Fakultät für Maschinenbau, Instititut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany E-mail: [email protected]
R. Glüge E-mail: [email protected]
H. Altenbach E-mail: [email protected]
L. Nazarenko et al.
dislocations [14], to catch some relevant phenomena in regions with a stress concentration [4], to generalize theories of plates [12] and to include boundary and surface energies [13,21], more encompassing models are required. A natural generalization of classical elasticity is the strain gradient elasticity, in which higher derivatives of
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