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Inverse Hooke's law and complementary strain energy in coupled strain gradient elasticity
Author(s) -
Nazarenko Lidiia,
Glüge Rainer,
Altenbach Holm
Publication year - 2021
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.202100005
Subject(s) - strain energy density function , isotropy , mathematical analysis , decoupling (probability) , infinitesimal strain theory , elasticity (physics) , tensor (intrinsic definition) , finite strain theory , lateral strain , classical mechanics , mathematics , inverse , density gradient , positive definite matrix , physics , geometry , materials science , thermodynamics , quantum mechanics , eigenvalues and eigenvectors , finite element method , control engineering , engineering , composite material
The inverse Hooke's law and complementary strain energy density has been examined in the context of the theory of coupled gradient elasticity for second gradient materials. To this end, it was assumed that the potential energy density is a quadratic form of the strain and of the second gradient of displacement. Existence of the coupling term significantly complicates the problem. To avoid this complication the equation for the potential energy density was transformed in order to present it as an uncoupled quadratic form of a modified strain and the second gradient of displacement or of the strain and a modified second gradient of displacement. These transformations, which is in essence a block matrix diagonalization, lead to a decoupling of strains and strain gradient in the potential energy density and makes it possible to determine tensorial relations for the compliance tensors of fourth‐, fifth‐ , and sixth‐rank. Both modifications result in the same compliance tensors and are valid for an arbitrary material symmetry class. In the case of hemitropic materials, the compliance tensors have the same symmetry and the same form as the stiffness tensors and are characterized by eight independent constants, namely the two classical isotropic constants, five constants in the strain gradient part and one constant in the coupling term. Explicit expressions for these eight parameters are obtained from the tensorial relations for the compliance tensors and are compared with the direct solution of a linear system for the compliance's. All three solutions are identical, what we consider as a verification of the presented results.