Papers dealing with the generalized Hooke’s law for linearly elastic anisotropic media are reviewed. The papers considered are based on Kelvin’s approach disclosing the structure of the generalized Hooke’s law, which is determined by six eigenmoduli of elasticity and six orthogonal eigenstates.

Anisotropy of elastic properties of materials

The spectral decomposition of elasticity tensor for all symmetry groups of a linearly elastic
material is reviewed. In the paper it has been derived in non-standard way by imposing the
symmetry conditions upon the orthogonal projectors instead of the stiffness tensor itself. The
numbers of independent Kelvin moduli and stiffness distributors are provided. The corresponding
representation of the elasticity tensor is specified.

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Review on spectral decomposition of Hooke’s tensor for all symmetry groups of linear elastic material

Using the conception of an extended object (Umezawa et al.) based on the continuum theory of elasticity, monovacancy formation energies are calculated. Anisotropy is included for cubic symmetry by means of a first order approximation to the Green's function tensor and some tensor algebraic calculations. The last considerations contribute to a better understanding of the meaning of “Voigt's average isotropic elastic constants”. Further lattice structure effects are taken into account by a common parameter for every type of structure. A comparison with experimental data or results from other theories shows either satisfactory agreement or the inconsistency of the first order approximation to the anisotropy for some materials with common properties.



Unter Benutzung des Konzepts eines ausgedehnten Objekts (Umezawa u. a.), das auf der Kontinuumstheorie der Elastizitat beruht, werden Monovakanzformierungsenergien berechnet. Die Anisotropie wird fur kubische Symmetrie mit Hilfe einer Naherung erster Ordnung fur den Green-funktionstensor und einigen Tensoralgebrarechnungen berucksichtigt. Die letzteren Betrachtungen tragen zu einem besseren Verstandnis der Bedeutung der „gemittelten isotropen elastischen Konstanten nach Voigt” bei. Weitere Gitterstruktureffekte werden durch einen gemeinsamen Parameter fur jeden Strukturtyp berucksichtigt. Ein Vergleich mit experimentellen Daten oder Ergebnissen anderer Theorien zeigt entweder eine befriedigende Ubereinstimmung oder die Inkonsistenz der Naherung erster Ordnung fur die Anisotropie bei einigen Materialien mit gemeinsamen Eigenschaften.

Monovacancy Formation Energies in Cubic Crystals

On propose d'utiliser le rapport du rayon de l'orbite d'un tenseur a sa norme comme caracteristique quantitative de son anisotropie

On evaluation of anisotropy of properties described by symmetric second-order tensors

Micromechanical model of polycrystalline materials with lamellar substructure is presented. The lamellar microstructure of grains is accounted for using the well-established framework developed for layered composites. Within the approach different scale transition rules between the level of lamellar grain and the polycrystalline sample can be employed. The model capabilities are tested using the example of α2 + γ-TiAl intermetallic. Elastic properties and the initial yield surface for the lamellar grain (PST crystal) and for the untextured polycrystal are estimated. Elastic and plastic anisotropy degree is analyzed.

J. Rychlewski

Papers

Zur Abschätzung der Anisotropie