We consider the old problem of finding a basis of polynomial invariants of the fourth rank tensor C of elastic moduli of an anisotropic material. Decomposing C into its irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are traceless symmetric second rank tensors, and D is completely symmetric and traceless fourth rank tensor (D e T~). We obtain by reinterpreting the results of classical invariant theory a polynomial basis of invariants for D which consists of 9 invariants of degrees 2 to 10 in components of D. Finally we use this result together with a well-known description of joint invariants of a number of second-rank symmetric tensors to obtain joint invariants of the triplet (a, b, D) for a generic D.
O. Introduction. Motivation for the workThe main purpose of the work is to construct a classification of linear anisotropic elastic materials. In colloquial terms we ask the question: How do we give distinct names to distinct anisotropic elastic materials? Clearly a designation based on the 21 components of the tensor C of elastic moduli in a fixed reference frame is not good for this purpose because it provides, generically, different names for different orientations of a given material. As the material is rotated the tensor C moves on its orbit in the space T~, of elasticity tensors. Thus what is needed is a parametrization of distinct orbits. The set O c of the distinct orbits of elasticity moduli is a manifold of (21-3) = 18 dimensions that has a fairly complicated boundary. The problem of naming the distinct orbits would be solved if Or, the manifold of distinct orbits, can be mapped in a one-to-one and continuous manner into the linear space Rn; the coordinates of an image point would then serve as the name of the associated orbit. It is of interest to know what minimal dimension n is needed for this purpose.The following examples may be of help in thinking about this idea:If the manifold of interest is a circle which is one dimensional then the minimal dimension n = 2. If the manifold of interest is the group SO(3) of rigid
The paper is concerned with the representation of the relationship that exists, for a given material and temperature and for small deformations, between histories of applied stress and the observed strain and the accompanying changes in internal structure of the material. Emphasis is given to creep damage in metals as a vehicle for illustration of the main ideas introduced in the paper. In particular, the role played by irreducible even rank tensors in the representation of internal structure is discussed and clarified. The restrictions placed by thermodynamics on constitutive equations are considered and the use of potentials in these equations is examined and criticized.
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