The objective of this paper is to provide a new irreducible nonpolynomial representation for elastic constitutive equations of cubic crystals with the material symmetry group O or T
 d
 or O
 h
 . The presented result is expressed in terms of a generating set composed of nine polynomial tensor generators. It is simpler and more compact than a recent result in terms of a generating set composed of ten tensor generators.

A new representation theorem for elastic constitutive equations of cubic crystals

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

Crystals in the monoclinic system and composites reinforced with families of fibres in two non-orthogonal and mechanically non-equivalent directions are referred to as being clinotropic materials. In this paper, the complete and irreducible representations are established for clinotropic scalar-, vector-, second-order symmetric tensor- and second-order skew-symmetric tensor-valued functions of any finite number of second-order symmetric tensors, second-order skew-symmetric tensors and vectors. These representations constitute a rational basis for modelling the complex mechanical behaviour of clinotropic materials. As illustrative applications, the formulation of thermoelastic constitutive laws of clinotropic materials is developed.

Tensor function representations as applied to formulating constitutive laws for clinotropic materials

This paper is concerned with invariants of a symmetric second–order tensor relative to crystal classes and quasicrystal classes and some non–continuous infinite orthogonal subgroups. Simple irreducible functional bases in unified forms, each of which consists of eight polynomial invariants only, are presented for all kinds of finite subgroups of the maximal transverse isotropy group, D ∞ h , except the subgroups of the orthotropy group D 2 h . Results are also provided for other kinds of orthogonal subgroups including icosahedral quasicrystal classes, etc. For the trigonal, tetragonal and hexagonal crystal classes without two–fold axis, the results given here are even more compact than the corresponding results recently derived by this author (Xiao 1996 a ). For all non–crystal classes except the transverse isotropy groups, the presented results are the first ones.

On anisotropic invariants of a symmetric tensor: crystal classes, quasi-crystal classes and others

Es werden die Vorgange wahrend der Abkuhlung eines thermisch gefugten Querpresverbandes von zwei Kreisringscheiben aus dem gleichen Material mit linearer Verfestigung im plastischen Bereich betrachtet. Bei uberschreitung eines kritischen ubermases tritt zu Fugebeginn Plastizierung mit gleichzeitig einsetzender Entlastung in einem der beiden Ringe auf. Die instationare Spannungsverteilung und der zeitliche Verlauf der elastisch-plastischen Grenzradien werden untersucht und durch numerische Ergebnisse illustriert.

Spannungen im thermisch gefügten elastisch-plastischen Querpreßverband mit elastischer Entlastung

An irreducible, non-polynomial representation is established for an anisotropic constitutive equation being an explicit relation between two symmetric, second-order tensors. The considered material anisotropy is characteristic of two non-orthogonal and mechanically non-equivalent privileged directions in plane. Particular cases of orthotropy and of cubic symmetry are analyzed in detail. An illustrative example referring to infinitesimal elasticity is presented in order to compare the developed representation with the classical Hooke law.



Es wird eine nichtreduzierbare Darstellung, die keine Polynomdarstellung ist, fur ein anisotropes Stoffgesetz, welches eine explizite Beziehung zweier symmetrischer Tensoren zweiter Stufe herstellt, formuliert. Die Anisotropie des Materials ist charakterisiert durch zwei nichtorthogonale, mechanisch ungleich bevorzugte Richtungen in der Ebene. Einzelne Falle von Orthotropie und kubisher Symmetrie sind im einzelnen untersucht. Um die entwickelte Darstellung mit dem klassischen Hookeschen Gesetz zu vergleichen, ist ein erlauterndes Beispiel zur infinitesimalen Elastizitat angefugt.

M. Basista

Papers

Tensor Functions Representations as Applied to Deriving Constitutive Relations for Skewed Anisotropy