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Spectral Decomposition of Compliance and Stiffness Fourth‐Rank Tensors Suitable for Orthotropic Materials
Author(s) -
Theocaris P. S.,
Philippidis T. P.
Publication year - 1991
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.19910710308
Subject(s) - orthotropic material , positive definiteness , isotropy , eigenvalues and eigenvectors , tensor (intrinsic definition) , stiffness matrix , rank (graph theory) , stiffness , mathematics , mathematical analysis , matrix (chemical analysis) , pure mathematics , physics , materials science , positive definite matrix , composite material , combinatorics , quantum mechanics , finite element method , thermodynamics
The spectral decomposition of the compliance and stiffness tensors related with tranversely isotropic materials was developed and their characteristic values were calculated by using the components of these fourth‐rank tensors in a Cartesian frame defining the principal material directions. Imposing the eigenvalues of the 6 × 6 matrix associated with the contracted 4th rank symmetric tensor to be strictly positive, as implied by the positive definiteness of the elastic potential, bounds of the values of Poisson 's ratios were established. They were shown to restrain considerably the existing limits of variations of this fundamental property of the orthotropic materials. Energy orthogonal states of stress for the tranversely isotropic material were also established decomposing the elastic potential in distinct parts associated with the deformation eigen‐states of the material symmetry. Thus, the unsolved as yet important problem of extension of separation of the elastic energy to anisotropic materials was achieved by an effective manner which is useful in practical applications.