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Generalized proper states for anisotropic elastic materials

Ostrowska-Maciejewska J., Rychlewski J.

Archives of Mechanics

2001

Vol. 53, nr 4-5

501-518

The main aim of this paper is to determine all the unit stresses ... for which the stored elastic energy ... has the local extrema in some classes of stresses. Our consideration is restricted to two classes: K1 - uniaxial tensions and then the directions for which the Young modulus assumes its extremal value are determined, and K2 - pure shears in physical space. The problem is then reduced to the determination of the planes of minimal and maximal shear modulus. The idea of a generalized proper state for Hooke's tensor is introduced. It is shown that a mathematical treatment of the considered problem comes down to the problem of the generalized proper elastic states for the compliance tensor C. The problem has been effectively solved for cubic symmetry.

A qualitative approach to Hooke's tensors. Pt. 2

Rychlewski J.

Archives of Mechanics

2001

Vol. 53, nr 1

45-63

A straightforward and complete description of all possible invariant linear decompositions of the space of Hooke's tensors has been given in Part I, [1]. In this Part II we demonstrate various elaborations and consequences of these decompositions. This gives a qualitative description of the anisotropy of Hooke's tensors. In particular, we demonstrate examples A through G, not only important but also astonishing. When reference is made to the formulae in [1], we shall add "Part I" to the number. The notions and notations are the same (see Appendices 1, 2 in [1]).

A qualitative approach to Hooke's tensors. Part 1

Rychlewski J.

Archives of Mechanics

2000

Vol. 52, nr 4-5

737-759

The main qualitative properties of Hooke's tensors can be found in their invariant decompositions, both linear and nonlinear. The invariant nonlinear spectral decompositions are presented in the review [7] and the papers quoted therein. This paper deals with linear invariant decompositions initiated in [12] - [20]. A straightforward and complete description of all such possible decompositions is presented here in Part I. The main results are given in formulae (7.1), (7.3). The next part (to appear), Part II, will contain derivations, conclusions and unexpected applications.