Journal
CONTINUUM MECHANICS AND THERMODYNAMICS
Volume 33, Issue 3, Pages 713-725Publisher
SPRINGER
DOI: 10.1007/s00161-020-00949-2
Keywords
Strain gradient elasticity; Coupling fifth-rank tensor; Positive definiteness of the potential energy
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Funding
- German Research Foundation (DFG) [AL 341/51-1]
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The study examines the linear theory of coupled gradient elasticity for hemitropic second gradient materials. By diagonalization and matrix representation, inequalities for positive definiteness are obtained, with results matching those in the literature under certain special conditions.
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.
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