Coordinate free representation of the hierarchically symmetric tensor of rank 4 in determination of symmetry

R. Baerheim

Research output: ThesisDoctoral thesis 1 (Research UU / Graduation UU)


General theory of elasticity treats the anisotropic behaviour of media i.e. that property-dependence on spatial direction is taken care of. Examples of elastic media are rocks, building- and biological materials. The tensor concept is the most fundamental concept in the description of elastic anisotropy. Although a tensor describes a physical property and as such is independent of coordinate systems, the tensor can be represented by components referred to a coordinate system. A vector - which is a first rank tensor - is the most familiar quantity where the components are dependent on the coordinate system. The set of components is a representation of the vector. A scalar quantity like the length of a vector is independent of the coordinate system to which the vector is referred to. A main subject in my thesis, is a representation of the anisotropic elastic tensor by means of coordinate-free - or invariant - quantities to describe symmetry properties of the medium. A question in elastic anisotropy is how material symmetry can be determined from the components of the elastic tensor. This is of main concern in my thesis. In an arbitrary coordinate system an elastic tensor has 21 non-zero components. The exception is the isotropic tensor with some vanishing components in all coordinate systems and only two independent components. For ideal media with specified symmetry, there exist coordinate systems where some of the components are zero. How can a coordinate system which reduces the number of constants be determined, and what can be said about symmetry? The previous questions motivated investigating general theories which could represent a tool in solving such and related problems. For experimentally observed tensors, there are no vanishing components due to deviation from ideal symmetry and inaccuracy in measurements. Thus there are 21 components for real media in any coordinate system. A theory for representing elastic tensors geometrically was given in Backus (1970). The theory is based on a specified decomposition of the elastic tensor into harmonic tensors, and Maxwell multipoles are the geometrical representation of the harmonic tensor in 3-dimensional space. Harmonic decomposition is also done with different approaches by e.g. Mochizuki (1988) and Cowin (1989) (see Ch. 2). Kelvin (1856) and Sutcliffe (1992) present decompositions of the elastic tensor by means of eigentensors and eigenstiffnesses (see Ch. 1). The concept was proposed by Kelvin (1856), but does not appear to have been accepted at that time. It was independently discovered by Pipkin (1976), Rychlewsky (1984), Mehrabadi and Cowin (1990), and Sutcliffe(1992). A thorough discussion is given in Helbig (1994). My work on coordinate-free representation is mainly based on Backus's theory including Maxwell multipoles. This thesis contains applications and further developments of his theory. In the literature, the notation 'elastic tensor' is normally used for tensors of rank four and dimension three, describing real i.e., stable, media. Backus's theory, however, is valid regardless of the stability conditions. A tensor of rank four in three dimensions satisfying" elastic" symmetry is here defined as a hierarchically symmetric tensor. Stability conditions need not be satisfied. According to the introduction of the concept 'hierarchically symmetric tensor', this concept was progressively taken into use in my work (Chapters 3, 4, 5 and 6). However in my first publication (Ch. 2) hierarchically symmetric tensors were not defined. I have kept it in my thesis as it was published, except for minor changes. Theoretical analysis of elastic anisotropy has a much wider range of application than in earth sciences, and is presented in scientific diciplines like mathematics, physics, material sciences and biomechanics as well. Most of my work is related to ideal media, with symmetry properties as in crystal physics. In Ch. 6 modeling of real media are performed by perturbation of tensors of ideal symmetry. Perturbation of a triclinic tensor models the inaccuracy of physical measurements. Perturbation of a tensor of ideal higher symmetry, models deviation from ideal symmetry. Further work has to be done in applying the fundamentals to obtain tools for determine symmetry elements for experimentally determined tensors.

Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Utrecht University
  • Helbig, K., Primary supervisor, External person
Award date2 Feb 1998
Place of PublicationUtrecht
Print ISBNs90-5744-017-2
Publication statusPublished - 2 Feb 1998


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