Institute of Fundamental Technological Research

In this paper the problem of *nding all elastic anisotropic materials, that transmit longitudinal elastic waves in exactly the same manner as isotropic materials, is proposed and solved. The solution is given in Theorem 1. In such anisotropic materials elastic energy stored under any one-dimensional stretch does not depend on the stretching direction. In Theorem 3 a di1erent class of anisotropic materials is described, for which the elastic energy stored under pure shearing does not depend on the shear direction. Such materials transmit at least one transverse elastic wave in every direction. The tools used to solve the problems are invariant decompositions of Hooke’s tensors.

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]).

The main aim of this paper is to determine all the unit stresses w (w•w = 1) for which the stored elastic energy F(w) 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.

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 [8] 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.

The pure shear tensors play a significant role in continuum mechanics. Recent authors' results concerning some remarkable properties of pure shear tensors and their sets are presented. A list of the properties defining pure shears is quoted. The properties of the whole set of all pure shears are described. An extraordinary role of the pure shears in the linear theory of elasticity is disclosed. A new approach, in terms of Kelvin moduli and pure shears as proper elastic states, to the description of the symmetries of elastic materials is proposed. Finally, it is shown that a two-parametric set of orthonormal bases of deviatoric space consisting solely of pure shears can be constructed.

An anisotropy of a property described by a tensor becomes apparent in the variability
of the tensor under rotations. A representation of this variability is contained
in the notion of the orbit of a tensor, i.e. the set of all tensors obtained by rotating
the given tensor. We propose to use the ratio of the radius of the orbit of a tensor to
its norm as a quantitative characteristic of its anisotropy. The proposed invariant is expressed
in terms of the norm of the traceless part of the tensor and modular angle.

A novel method of describing elastic anisotropy based on the concept of an elastic eigen state is proposed. The structure of the rigidity tensor is determined. In particular, it is shown that the set of 21 constants describing continuously the manifold of elastic solids is composed of three different subsystems: 6 true rigidity moduli, 12 dimensionless rigidity distributors, and 3 invariant parameters defining the orientation of the body in question relative to the laboratory system of coordinates. It is shown that Hooke's Law can be written uniquely for an arbitrary anisotropic body in the form of several laws describing the direct proportionality of the corresponding parts of the stress and deformation tensors. The construction of these parts is illustrated using an example of a transversally isotropic solid.

The paper concerns materials and processes for which equations (2.1) and (2.2) are valid. Rigid perfectly plastic material is an example. Spatial motions with nonnegative stress power are considered. One of the properties of these motions is stated in section 6 in the form of a criterion. It links the sign of the velocity vector modulus increment along a stream line with the position of this line relative to Cauchy's cones produced by the stress state. In a pure geometrical form the criterion is given for complex lamellar motions. Suitable results obtained in [2–4] represent some particular cases of the criterion derived.

A number of practical problems require a consideration of the case when material constants change in a jump-like manner. In the present paper the plain plastic strain with a jump-like distribution of the yield limit is considered. The behaviour of solutions in the neighbourhood of the jump surfaces is investigated. An example of the limit stage of a body composed of two materials is considered in detail.

In many practical problems the need arises to consider bodies whose non-homogeneity is described by non-continuous functions. The subject of the paper is the determination of the carrying capacity of simply supported circular plates, loaded by uniform pressure and composed of concentric annuli with different mechanical properties. The above jump of mechanical properties may be caused by jump of plate thickness or by change of material properties. Depending on the values of non-homogeneity and the division parameters there exist six different solutions of the considered problem. In each particular case the moment field, velocity field and limit load are given. The ranges of validity of all solutions are established. Also, the problem of optimum design in the above class of plates is considered. In the second part an orthotropic plate is discussed. The special case concerning the uniform distribution of circumferential reinforcement in the case of a reinforced concrete plate is considered. Here again, depending on different parameters, there may be seven solutions. The above analysis has allowed some qualitative conclusions to be drawn concerning the design of isotropic and orthotropic plates with jump non-homogeneity.

The problems of plastic jump non-homogeneity i.e. the problems concerned with a jump-like change of the yield limit, were considered in previous papers by the author. In the present work, a rectangular bar composed of two materials with different yield limits and subjected to torsion is considered. It turns out that even for such a simple problem nine different solutions exist depending on the values of three parameters characterizing non-homogeneity, form, and partition of the cross-section.

Plastic torsion of bars composed of a finite number of prismatic parts with various yield limits is considered. This problem is equivalent to the mathematical problem of constructing a function whose gradient varies in a jump-like manner. Continuity conditions for the stress vector at the contact lines have been analysed, as well asNádai's analogies and local solutions. The results are applied to find the limit twisting moment of circular bars with jump non-homogeneity. It can be observed that homogeneous bars of multiply connected cross-section are a particular case of a class of bars studied in the present paper.

Dynamical System, Fluid Dynamics, Finite Number, Transport Phenomenon, Continuity Condition

The theory of plasticity represents one of the branches of rapidly developing continuum mechanics. The progress consists of both a careful analysis of the basic equations and physical relations and an extension of the range of problems and effective solutions covered by this theory. In real media the non-homogeneity of mechanical properties may be caused by numerous phenomena and its nature may be very diverse. First, it is evident that a universal property of bodies occurring in practice is their microscopic non-homogeneity. It is well- known that there have been many successful1 attempts to include microscopic non-homogeneity in the structure of Continuum Theory. The objective of the chapter, however, is the macroscopic nonhomogeneity that is taking place in regions whose linear dimensions are comparable with the characteristic dimensions of the body under consideration. The nonhomogeneity of mechanical properties may be caused, for instance, by phenomena such as the influence of flow of elementary particles, the action of temperature gradients, a nonhomogeneous hardening of the material, different types of surface working, nonhomogeneity of the composition, and so on.