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It is shown that all symmetries possible for the elastic tensors can be reduced to the twelve symmetries already used in the description of the crystal classes. Each symmetry can be characterized by a group of rotations generated by no more than two rotations. The use of a canonical basis related to such rotations considerably simplifies the component forms of the elasticity tensor. This result applies to non-symmetric tensors; for symmetric tensors, the number of independent symmetries reduces from twelve to ten.
After the present work was submitted, the following paper came to our attention: 14. S.C. Cowin and M.M. Mehrabadi, On the identification of material symmetry for anisotropic elastic materials. Q. Jl. Mech. appl. Math.40 (1987) 451–476. This paper contains an independent analysis of the partial ordering ≺ among the crystallographic elastic symmetries. However, it does not deal with the problem of the completeness of these symmetries.

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... One of the interesting results in [11] is that elastic symmetry under two generic rotations in three dimensions implies isotropy. (The term " generic " is made precise in Definition 1 below .) ...

... This requires verifying the theorem's irreducibility hypotheses which involves lengthy and detailed calculations. We rely on the analysis in [11] for some of the technical details. ...

... where w and γ can be computed as follows (see the Appendix in [11] for an algebraic proof). Let ...

We derive minimal conditions on a symmetry group of a linearly elastic material that implies its isotropy. A natural setting
for the formulation and analysis is provided by the group representation theory where the necessary and sufficient conditions
for isotropy are expressed in terms of the irreducibility of certain group representations. We illustrate the abstract results
by (re)deriving several old and new theorems within a unified theory.
KeywordsLinear elasticity–Isotropy–Symmetry groups–Group representation–Schur’s Lemma–Random media

... More specifically, one may answer the following question: how to characterize a given set C sym of elasticity with sym material symmetries, so that the properties thus identified can be used in the construction of the probabilistic model? In fact, several approaches for elastic material symmetry classification and characterization have been proposed by numerous authors, among which [13] [7] [22] [47] and [10] [3] [33] to name a few. ...

... The constraints to be taken into account in the MaxEnt-based derivation are given by Eqs. (14), (15), (22) and (23) (together with normalization condition). Note that the deterministic boundedness constraint implies an uniform ellipticity condition on the random bilinear form arising in the weak formulation of the elasticity stochastic boundary value problem. ...

... Following sections 3.1 and 3.2, let [Λ] ∈ M S n (R), {τ k ∈ R} m k=1 , (1 − L ) ∈ R and (1 − L u ) ∈ R be the Lagrange multipliers associated with constraints (14), (15), (22) and (23). From Eq. (13), it can be deduced that p.d.f. ...

In this work, we address the stochastic modeling of apparent elasticity tensors, for which both material symmetry and stochastic boundedness constraints have to be taken into account, in addition to the classical constraint of invertibility. We first introduce a stochastic measure of anisotropy, which is defined using metrics in the set of elasticity tensors and used for quantitatively characterizing the fulfillment of material symmetry constraints. After having defined a numerical approximation for the stochastic boundedness constraint, we then propose a methodology allowing one to unify maximum entropy based models that have been previously derived by considering some of these constraints and which consists in constructing a probabilistic model for an auxiliary random variable. The latter can be interpreted as a stochastic compliance tensor, for which the available information to be used in the maximum entropy formulation can be readily deduced from the one considered for the elasticity tensor. A numerical illustration of the approach to an elastic microstructure is finally provided.

... ; C 11 (Coleman and Noll, 1964). Then, in linear elasticity, the 11 types of elastic energy reduce to 9 (Huo and Del Piero, 1991) and the relevant anisotropic symmetry classes to 8, including isotropy (Forte and Vianello, 1996;Chadwick et al., 2001). On the basis of such classification it is spontaneous to evaluate the behavior of the engineering elastic constants for the various elastic symmetries. ...

... The material symmetries considered here are two of those symmetries such that, when the solid is subjected to a unit dipole acting along a material principal direction, no shear strains arise. The material symmetries fulfilling this requirement are those corresponding to the groups (Coleman and Noll, 1964;Gurtin, 1972;Huo and Del Piero, 1991): C 6 C 7 (cubic symmetry, characterized by three elastic constants), C 10 C 11 (hexagonal symmetry, five elastic constants: transverse isotropy), C 5 (tetragonal, six elastic constants) and C 3 (orthorhombic, nine elastic constants: orthotropy). ...

... The cubic case represents the symmetry with the lesser number, 3, of independent elastic constants among the crystallographic classes, excluding, of course, the case of isotropy. It corresponds to the symmetry group C 6 coinciding, in the case of a symmetric S (hyperelasticity), with the symmetry group C 7 (Huo and Del Piero, 1991). For the symmetry under investigation, the matrix representation of the elasticity tensor in (9), written in the principal reference system, and taking into account also the symmetries on S has this simpler form: ...

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.

... Eq. (22) has been obtained from a random perturbation on the elasticity matrix (in GPa) of a carbon-epoxy unidirectional composite and can be interpreted as the mean value of a mesoscale representation for a heterogeneous material that is almost transversely isotropic from a macroscopic point of view. Consequently, let us consider the random variable µ T I M , corresponding to the stochastic measure of anisotropy (see Eq. (10) It is readily seen that both the mean distance to C T I and the level of statistical fluctuations increase together with dispersion parameter δ [C] , no matter the distance that is used. ...

... A tremendous amount of work has been devoted to the classification [14] [8] [22] [51] and characterization (see [11] [3] [31] and the references therein, for instance) of material symmetries. Among the developed methodologies, the eigensystem-based characterization [42] [4] turns out to be especially suitable for the present work, since it allows for an interpretation of the above properties within the framework of random matrix theory. ...

... Let us consider the mean model defined by Eq. (22) and let = 20. Consequently, following Section 3.3, the level of statistical fluctuations of random elasticity matrix [C] is almost fixed. ...

The aim of this paper is to provide a general overview on random matrix ensembles for modeling stochastic elasticity tensors that exhibit uncertainties on material symmetries. Such an issue is of primal importance in many practical situations involving either a computational or experimental analysis on random heterogeneous materials (such as bones, reinforced composites, etc.). For this purpose, we first define a stochastic measure of anisotropy, the definition of which relies on the use of distances in the set of fourth-order elasticity tensors. We subsequently describe two random matrix ensembles that have been proposed within the framework of information theory and making use of a MaxEnt approach. In particular, we discuss the relevance of each of those with respect to constraints on the proposed anisotropy measure. It is shown that the capability of prescribing the mean distance to a given symmetry class depends, in view of the eigensystem-based characterization, on the behavior of the random eigenvalues. Finally, we propose a procedure allowing for the identification of the stochastic representation, should a set of experimental data be available. The approach, which is based on the use of the maximum likelihood principle, is exemplified in the case of experimental realizations that are almost transversely isotropic.

... The obtained tensor C Sym is then the closest to C symmetric tensor. As symmetry groups for four-rank tensors are ordered (Cowin, 1987;Yong-Zongh, 1991) we can, for each level of symmetry, determine the best symmetry for C. Choice of the level remains arbitrary. For our specimen, the discrepancy between C and C Sym for every symmetry is given in table 3; the superscript corresponds to the symmetry level according to Cowin. ...

... On obtient ainsi la meilleure approximation C Sym de C parmi les tenseurs invariants par Sym ainsi que la base associée B Sym . Les groupes de symétrie forment un ensemble ordonné(Cowin, 1987 ;Yong-Zongh, 1991) et peuvent être rangés par niveaux dans l'arbre associé à l'ordre. On peut donc pour chaque niveau déterminer le meilleur groupe de symétrie Sym pour C. Mais le choix du niveau reste arbitraire en fonction des imprécisions sur les mesures et/ou des applications ultérieurement envisagées. ...

... The characterization and classification of the material symmetries 1 for an anisotropic fourth order tensor have been investigated by several researchers, notably by Love [12], Voigt [30], Gurtin [10] and Thurston [28], Nye [21], Hou and Del Pierro [11], Forte and Vianello [8], Chadwick et al. [4] among many others. In these studies, the authors express the different forms of the elasticity tensor into a matrix of dimension 6 × 6 (by means of Voigt type notations) in the cartesian system. ...

... We derive an irreducible basis, 1 There are various and non equivalent definitions of the symmetry classes. Forte and Vianello [8], Chadwick et al. [4] introduced only eight symmetry classes while Hou and Del Pierro [11] introduced ten symmetry classes. 2 The definition of minor symmetries for sixth order tensors is given in section 2 and is defined by Monchiet and Bonnet [15] for tensors of order higher than 6. constituted of 31 elements, for any transversely isotropic sixth order tensors. The proposed representation is coordinate free since all these tensors are constructed as the outer products of elementary tensors attached to the direction of transverse isotropy. ...

In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.

... Le Quang et al. (2012) proved that the elasticity tensor B possesses 17 symmetry classes (including symmetry), whereas Auffray et al. (2013) provided a matrix representation for everyone of the 16 effectively anisotropic symmetry classes. These results together with the analogous ones relative to C (8 symmetry classes after Huo and Del Pietro, 1991;Forte and Vianello (1996)) constitute a basic reference to study anisotropy. ...

... orthotropic materials (9 independent constants), transverse isotropic materials (5 independent constants), isotropic materials (2 independent Lamé constants). See Huo and Del Pietro (1991); Forte and Vianello (1996); Olive and Auffray (2013) for an exhaustive treatment of this issue. ...

Anisotropy of centro-symmetric (first) strain gradient elastic materials is addressed and the role there played by the dual gradient directions (i.e. directions of strain gradient and of double stress lever arm) is investigated. Anisotropy manifests itself not only through the classical fourth-rank elasticity tensor C (21 independent constants) in the form of moduli anisotropy, but also through a sixth-rank elasticity tensor B (171 independent constants) in a unified non-separable form as compound internal length/moduli anisotropy. Depending on the microstructure properties, compound anisotropy may also manifest itself in a twofold separable form through a decoupled tensor B=LC, consisting of a moduli anisotropy attached to C, and an internal length anisotropy attached to a symmetric positive definite second-rank internal length tensor L (6 independent constants). Tensor L confers a tensorial character to the concept of internal length and is the basis of a mutual one-to-one relationship between the dual gradient directions. Indeed, at every point of a material with separable compound anisotropy, a characteristic ellipsoid exists whereby the generic radius and the associated normal to the ellipsoidal surface constitute dual gradient directions. Therefore, at every point, there are at least three mutually orthogonal directions in each of which the dual gradient directions are collinear, but infinite in number when the ellipsoid is rotational or even spherical. With restrictions on dual gradient directions, simplified models with a reduced number of independent constants are derived, namely: i) Gradient symmetric materials which obey a reciprocity principle and generally exhibit a non-separable compound anisotropy (21 + 126 independent constants in total); ii) Materials with separable compound anisotropy, a subclass of gradient symmetric materials (21 + 6 independent constants in total). A typical boundary-value problem for materials with separable compound anisotropy is presented and a way to transform the three governing fourth-order PDEs into six second-order ones is suggested. An application to an Euler–Bernoulli beam is analytically worked out.

... I cristalli sono classificabili in 32 classi, in base a proprietà fisiche omogenee; Smith e Rivlin [1] hanno mostrato che queste classi dispongono complessivamente di 11 (più l'isotropia) tipi differenti di energia elastica, indicati successivamente con C 1 , C 2 , . . . , C 11 da Coleman e Noll [2]; in elasticità lineare, infine, gli 11 tipi di energia di deformazione si riducono a 9 (Yong-Zhong e Del Piero [3]). Sulla base di questa classificazioneè spontaneo quindi pensare di valutare il comportamento delle costanti elastiche ingegneristiche per le diverse classi di simmetria. ...

... Corrisponde al gruppo di simmetria C 6 coincidente, nel caso di S simmetrico, con il gruppo di simmetria C 7 (v. Yong-Zhong e Del Piero [3]). Ora e nel seguito risulta conveniente esprimere le componenti del tensore elastico inverso S nella notazione contratta a due indici di Voigt (v. ...

SOMMARIO Per un solido omogeneo, elastico lineare e anisotropo, si determina l'espressionepì u gen-erale del modulo di Young E(n), valida per qualunque classe d'anisotropia. Si formula quindi il problema di estremo vincolato per la determinazione delle direzioni n secondo le quali E(n) assume valori stazionari. Ci si riferisce, in particolare, alle classi di simme-tria cubica e trasversalmente isotropa, determinando soluzioni esplicite per le direzioni n corrispondenti a punti di estremo per E(n). Per ciascun caso vengono discusse le propri-e a delle direzioni estremali e dei corrispondenti valori del modulo, in dipendenza delle caratteristiche meccaniche del materiale, espresse mediante opportune combinazioni dei coefficienti del tensore elastico, che ne esprimono la deviazione dall'isotropia. ABSTRACT For a homogeneous anisotropic linearly elastic solid, the general expression of Young's modulus E(n), embracing all classes that characterize the anisotropy, is given. A con-strained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are dis-cussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy.

... The characterization and classification of the material symmetries 1 for an anisotropic fourth order tensor have been investigated by several researchers, notably by Love [12], Voigt [30], Gurtin [10] and Thurston [28], Nye [21], Hou and Del Pierro [11], Forte and Vianello [8], Chadwick et al. [4] among many others. In these studies, the authors express the different forms of the elasticity tensor into a matrix of dimension 6 × 6 (by means of Voigt type notations) in the cartesian system. ...

... We derive an irreducible basis, 1 There are various and non equivalent definitions of the symmetry classes. Forte and Vianello [8], Chadwick et al. [4] introduced only eight symmetry classes while Hou and Del Pierro [11] introduced ten symmetry classes. 2 The definition of minor symmetries for sixth order tensors is given in section 2 and is defined by Monchiet and Bonnet [15] for tensors of order higher than 6. constituted of 31 elements, for any transversely isotropic sixth order tensors. The proposed representation is coordinate free since all these tensors are constructed as the outer products of elementary tensors attached to the direction of transverse isotropy. ...

Algebra of transversely isotropic sixth order tensors and solution to higher order inhomogeneity problems V. Monchiet · G. Bonnet Abstract In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.

... Herein, we use the definition according to which conjugate symmetry groups belong to the same class, as will be discussed in Section 3. In accordance with this definition, there are eight symmetry classes, as shown by Forte and Vianello [10], Chadwick et al. [5], Ting [22], Bóna et al. [4]. Another definition of the symmetry class has been proposed by Hou and Del Piero [26] and discussed by Forte and Vianello [11]; this definition results in ten symmetry classes. The ability of identifying the symmetry class of a given medium without the knowledge of the orientation of its symmetry axes is of practical importance. ...

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these
conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a
priory knowledge of the symmetry axes.

... This can be seen by noting that, in the most general case, F contains 54 independent material parameters, whereas C comprises 21 ones. The problem of determining the number and types of all rotational symmetries for elastic tensors was, for the first time, explicitly and rigorously formulated and treated in a seminal paper of Huo & Del Piero (1991) about 20 years ago. The fundamental problem posed by Huo & Del Piero in the context of elasticity has then been approached in different ways and captured the attention of scientists in the fields of mechanics, physics, applied mathematics and engineering (e.g. ...

Flexoelectricity is due to the electric polarization generated by a non-zero strain gradient in a dielectric material withou or with centrosymmetric microstructure. It is characterized by a fourth-order tensor, referred to as flexoelectric tensor which relates the electric polarization vector to the gradient of the second-order strain tensor. This paper solves the fundamenta problem of determining the number and types of all possible rotational symmetries for flexoelectric tensors and specifie the number of independent material parameters contained in a flexoelectric tensor belonging to a given symmetry class. Thes results are useful and even indispensable for experimentally identifying or theoretically/numerically estimating the flexoelectri coefficients of a dielectric material.

... Notice that to prove this fact one does not need (nor would it be correct) to look at the crystal classes of, say, the material components of a composite. A proof of this theorem, mentioned in Prof. Helbig's book on seismic anisotropy [7], was provided a few years ago by Huo and del Piero [8]. A final remark: for non-simple (even if elastic) materials, the symmetry group choice is a much larger set and, in mundane terms, a completely different animal (the composition law, for instance, is not just a simple multiplication of matrices). ...

In this paper certain basic concepts of continuum mechanics are presented and discussed in a manner that may be suggestive for applications in geophysics. In particular, ideas such as Cosserat media, continuous distribution of inhomogeneities, and the role of material symmetries are emphasized. In the spirit of a live meeting, the style of the presentation has been kept informal.
The Geomechanics Project has been established this year as a joint venture between four companies and the University of Calgary. The first action taken by the consortium has been to establish an industrial research professor position at the University of Calgary, a position now occupied by Dr. Michael Slawinski, who was instrumental in getting the consortium started in the first place. Curiously, perhaps, the partnership was implemented with the Department of Mechanical Engineering, where I act as the University representative. So, it is this project that has brought a geophysicist (Michael) and a professor of mechanical engineering (myself) together, hopefully for the benefit of the sponsoring companies and beyond. One of our first collaborative efforts pertains to raytracing in anisotropic media following some ideas based on the calculus of variations and the application of Noether's theorem.

... The classification of material symmetries has been investigated by many researchers and was historically based on crystallographic considerations. Quite recently, other approaches, in which the elasticity tensors were classified either by considering the set of admitted (minor) symmetry planes (see Chadwick et al., 2001;Cowin and Mehrabadi, 1987) or with respect to symmetry groups (see Forte and Vianello, 1996;Huo and Del Piero, 1991), were proposed. Both approaches basically result in the definition of eight symmetry classes, as shown in Chadwick et al. (2001). ...

In this paper, we consider the probabilistic modeling of media exhibiting uncertainties on material symmetries. More specifically, we address both the construction of a stochastic model and the definition of a methodology allowing the numerical simulation (and consequently, the inverse experimental identification) of random elasticity tensors whose mean distance (in a sense to be defined) to a given class of material symmetry is specified. Following the eigensystem characterization of the material symmetries, the proposed approach relies on the probabilistic model derived in Mignolet and Soize (2008), allowing the variance of selected eigenvalues of the elasticity tensor to be partially prescribed. In this context, a new methodology (regarding in particular the parametrization of the model) is defined and illustrated in the case of transversely isotropic materials. The efficiency of the approach is demonstrated by computing the mean distance of the random elasticity tensor to a given material symmetry class, the distance and projection onto the space of transversely isotropic tensors being defined by considering the Riemmanian metric and the Euclidean projection, respectively. It is shown that the methodology allows the above distance to be (partially) reduced as the overall level of statistical fluctuations increases, no matter the initial distance of the mean model used in the simulations. A comparison between this approach and the initial nonparametric approach introduced in Soize (2008) is finally provided.

... As for the symmetry types of arbitrary order even-type deviators, they directly referred to the results of Ihrig and Golubitsky [16]. The modern definition of symmetry classification of tensors was introduced by Huo and Del Piero [17] in 1991 and further modified by Forte and Vianello [2] in 1996. The latter one is now accepted and applied extensively. ...

The number of distinct components of a high-order material/physical tensor might be remarkably reduced if it has certain symmetry types due to the crystal structure of materials. An nth-order tensor could be decomposed into a direct sum of deviators where the order is not higher than n, then the symmetry classification of even-type deviators is the basis of the symmetry problem for arbitrary even-order physical tensors. Clearly, an nth-order deviator can be expressed as the traceless symmetric part of tensor product of n unit vectors multiplied by a positive scalar from Maxwell’s multipole representation. The set of these unit vectors shows the multipole structure of the deviator. Based on two steps of exclusion, the symmetry classifications of all even-type deviators are obtained by analyzing the geometric symmetry of the unit vector sets, and the general results are provided. Moreover, corresponding to each symmetry type of the even-type deviators up to sixth-order, the specific multipole structure of the unit vector set is given. This could help to identify the symmetry types of an unknown physical tensor and possible back-calculation of the involved physical coefficients.

... The study of C had experienced a long history (Love (1944)) before a complete understanding of C was achieved quite recently. In fact, only about 20 years ago and for the first time, Huo and Del Piero (1991) explicitly posed, rigorously formulated and treated the fundamental problem of determining all the rotational symmetry classes that the fourth-order elastic tensor C can possess. This problem has then received the attention of researchers from mechanics, materials science, physics, applied mathematics and engineering (see, e.g., Zheng and Boehler (1994); Vianello (1996, 1997); He and Zheng (1996); Xiao (1997); Chadwick et al. (2001); Bóna et al. (2004Bóna et al. ( , 2007; Moakher and Norris (2006)). ...

The theory of first strain gradient elasticity (SGE) is widely used to model
size and non-local effects observed in materials and structures. For a material
whose microstructure is centrosymmetric, SGE is characterized by a sixth-order
elastic tensor in addition to the classical fourth-order elastic tensor. Even
though the matrix form of the sixth-order elastic tensor is well-known in the
isotropic case, its complete matrix representations seem to remain unavailable
in the anisotropic cases. In the present paper, the explicit matrix
representations of the sixth-order elastic tensor are derived and given for all
the 3D anisotropic cases in a compact and well-structured way. These matrix
representations are necessary to the development and application of SGE for
anisotropic materials

... (For references to the literature and a discussion of this development see Forte and Vianello (1996, Section 6) and Cowin and Mehrabadi (1989)). A modern approach which makes the treatment of symmetry in classical anisotropic elasticity self-contained and independent of crystallography was introduced by Huo and Del Piero (1991) and by Forte and Vianello (1996) who investigated, albeit using different deÿnitions, the sets of symmetry groups which are possible for elasticity tensors without further restrictions. Here we follow a third and apparently diierent route, ÿrst explored by Cowin and Mehrabadi (1987), in which elasticity tensors are classiÿed according to the set of symmetry planes which they admit. ...

It is shown here that there are exactly eight different sets of symmetry planes that are admissible for an elasticity tensor. Each set can be seen as the generator of an associated group characterizing one of the traditional symmetry classes.

... Conversely, if the material possesses some planes or axes of elastic symmetry, the number of independent elastic coefficients is accordingly reduced. Constraints imposed by material symmetry on the elasticity tensor, classification of symmetry classes, and number of the different types of anisotropy, are topics widely discussed in the literature (see, among others, Love, 1994; Hearmon, 1961; Gurtin, 1972; Ting, 1996; Forte and Vianello, 1996; Huo and Del Piero, 1991; Cowin and Mehrabadi, 1995; Chadwick et al., 2001). For any material symmetry, it is customary to define, at each point P of the body, a ÔprincipalÕ, or ÔmaterialÕ, orthogonal reference system x 1 x 2 x 3 in which the elasticity tensor shows the fewest number of independent non-vanishing components. ...

Homogeneous, anisotropic and linearly elastic solids, subjected to a given state of strain (or stress), are considered. The problem dealt with consists in finding the mutual orientations of the principal directions of strain to the material symmetry axes in order to make the strain energy density stationary. Such relative orientations are described through three EulerÕs angles. When the stationarity problem is formulated for the generally anisotropic solid, it is shown that the necessary condition for stationarity demands for coaxiality of the stress and the strain tensors. From this feature, a procedure which leads to closed form solutions is proposed. To this end, tetragonal and cubic symmetry classes, to-gether with transverse isotropy, are carefully dealt with, and for each case all the sets of EulerÕs angles corresponding to critical points of the energy density are found and discussed. For these symmetries, three material parameters are then defined, which play a crucial role in ordering the energy values corresponding to each solution.

... This number might be reduced if it exhibits material symmetries, which could arise from crystal structure, microstructure, etc. In the context of linear elasticity, the number of material symmetries has been proven to be eight (Huo and del Piero, 1991;Zheng and Boehler, 1994;He and Zheng, 1996;Forte and Vianello, 1996;Chadwick et al., 2001;Ting, 2003;Bóna et al., 2004). Namely a material is either isotropic or anisotropic, and that an anisotropic material is either triclinic (generally anisotropic), monoclinic, trigonal, orthogonal, tetragonal, cubic or trans-versely isotropic. ...

We develop a method through the mirror plane (MP) to identify the symmetry type of linear elastic stiffness tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the irreducible decomposition of high-order tensor into a set of deviators and the multipole representation of a deviator into a scalar and a unit-vector set. Since a unit-vector depends on two Euler angles, we can illustrate the MP normals of the elastic tensor as zeros of a characteristic function on a unit disk and identify its symmetry immediately, which is clearer and simpler than the methods proposed before. Furthermore, by finding the common MPs of three unit-vector sets using Fortran recipes, we can also analytically recognize the symmetry type first and then recover the natural coordinate system associated with the linear elastic tensor. The structures of linear elastic stiffness tensors of real materials with all possible anisotropies are investigated in detail.

... This problem has a long history, recalled by Forte and Vianello in [18]. These authors have definitively clarified the mathematical problem about the symmetry classes of an elasticity tensor and removed the link with crystallographic point groups which was extremely confusing and lead to the false assumption that there were ten, rather than eight, symmetry classes [26,13,25]. These eight classes were confirmed in 2001, using an alternative approach [12], where symmetry planes rather than rotations play the central role. ...

We produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor, using an original generalized cross product on totally symmetric tensors. This allows us to formulate coordinate-free conditions using polynomial covariant tensors for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Besides, we produce a new minimal set of 297 generators for the invariant algebra of the Elasticity tensor, using these tensorial covariants.

... In linear elasticity, the number of material symmetry classes has been proven to be eight (Huo and Del Piero [1991], Baerheim [1993], Baerheim [1998], Zheng and Boehler [1994], He and Zheng [1996], Forte and Vianello [1996], Chadwick et al. [2001], Ting [2003], Bona et al. [2004]). The two opposite cases of symmetry class are triclinic, that mean without any symmetry, and isotropic, that fulfil all possible symmetries. ...

Recent advances in additive manufacturing (polymer or
metal) have revived the interest in lattice materials. We
have chosen to study the simplest regular twodimensional
lattices made up of triangles. The sides of
the triangles are modeled by bars assuming articulated
connections or beams for rigid connections.
A lattice structure can be defined as the combination of
a network and a pattern where the pattern represents
the thickness of the bars at the vertices of the triangle.
All possible combinations of triangular arrays and 2D
patterns are studied.
In 2D, elasticity tensor has 4 groups of symmetry that
can be distinguished using the Viannello ’s invariants.
Using these invariants, we have calculated the
geometric and mechanical relations that the bars and
the beams must satisfy for each group of symmetry.
The thesis confirms the known result that a bar
structure can only represent the Cauchy elasticity
(materials for which C1122 = C1212) while a structure
of beams is most general.
It is finally shown that ,by choosing appropriate stiffness
of bars or beams, it is possible to obtain an elastic
symmetry class greater than the symmetry of the lattice
alone.

... Generalizing the discrete symmetry classification of crystals ( [1][2][3]), the notion of continuous symmetries, in particular, Lie groups of point symmetries introduced by Sophus Lie (e.g., [4]), with multiple further generalizations (e.g., [5,6] and references therein). Noether [7] provided a connection between conservation laws (energy, momentum, etc.) and local symmetries for models arising from a variational principle (see, e.g, [5,6]). ...

A methodology based on Lie analysis is proposed to investigate the mechanical behavior of materials exhibiting experimental master curves. It is based on the idea that the mechanical response of materials is associated with hidden symmetries reflected in the form of the energy functional and the dissipation potential leading to constitutive laws written in the framework of the thermodynamics of irreversible processes. In constitutive modeling, symmetry analysis lets one formulate the response of a material in terms of so-called master curves, and construct rheological models based on a limited number of measurements. The application of symmetry methods leads to model reduction in a double sense: in treating large amounts number of measurements data to reduce them in a form exploitable for the construction of constitutive models, and by exploiting equivalence transformations extending point symmetries to efficiently reduce the number of significant parameters, and thus the computational cost of solving boundary value problems (BVPs). The symmetry framework and related conservation law analysis provide invariance properties of the constitutive models, allowing to predict the influence of a variation of the model parameters on the material response or on the solution of BVPs posed over spatial domains. The first part of the paper is devoted to the presentation of the general methodology proposed in this contribution. Examples of construction of rheological models based on experimental data are given for setting up a reduced model of the uniaxial creep and rupture behaviour of a Chrome-Molybdenum alloy (9Cr1Mo) at different temperatures and stress levels. Constitutive equations for creep and rupture master responses are identified for this alloy, and validated based on experimental data. Equivalence transformations are exemplified in the context of parameter reduction in fully nonlinear anisotropic fiber-reinforced elastic solids.

... These authors have definitively clarified the mathematical problem of classifying the symmetry classes of the representation of SO(3) on Ela, the 21-dimensional space of elasticity tensors. They removed the link with crystallographic point groups which was extremely confusing and lead to the false assumption that there were ten, rather than eight, symmetry classes [36,25,47]. These eight classes were confirmed in 2001, using an alternative approach [20], where symmetry planes rather than rotations play the central role. ...

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.

... From theorem 10, the symmetry group of the elasticity tensor is the intersection of the symmetry groups of its projectors. These ones have the structure of an elasticity tensor (with 6 − M ip null eigenvalues) then their symmetry group belongs to one of the eight possible between isotropic, cubic, transverse isotropic, tetragonal, orthotropic, trigonal, monoclinic and tricilinic) (Huo and Del Piero, 1991;Forte and Vianello, 1996). From theorem 7.3, the symmetry group of a projector is the set of orthogonal transformations internal in the corresponding subspace. ...

A general anisotropic damage model is developed that accounts for the thermodynamics of irreversible processes in the framework of generalized standard materials and Kelvin tensor decomposition. Damage is described by fourth-rank tensors, one per eigenspace of the initial stiffness tensor. Their number thus ranges from two for an initially isotropic material to six for an initially triclinic material. The yield criteria are expressed in terms of a limiting energy for each eigenspace. The second-rank eigentensors (at most six) of the fourth-rank damage tensors define the direction of influence of the damage, while the associated eigenvalues characterize its intensity. These eigentensors evolve during loading, inducing an evolution of the symmetry group of the elastic tensor subject to the constraints of the Curie principle.

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.

The linear, planar, anisotropic elastic equilibrium equations are transformed to canonical form, through linear transformations of both coordinates and unknown displacement functions, together with a linear combination of equations. Correspondingly, the six original material moduli are replaced by two canonical elastic moduli. Similar results have been reached by Olver in 1988. However, the method demonstrated in this paper is more concise and direct. As an example, the general solution to the canonical equations is obtained in the case of a pair of double roots.

Harmonic and Cartan decompositions are used to prove that there are eight symmetry classes of elasticity tensors. Recent results in apparent contradiction with this conclusion are discussed in a short history of the problem.

A simple, exact, locally convergent series solution of the Navier-Stokes and continuity equations for steady, incompressible, axisymmetric fluid flow is derived which exhibits the qualitative features of vortex breakdown of the bubble or B-type. This solution, which converges in a neighborhood of the axis of symmetry of the flow, is shown to produce vortex breakdown flows which are in good qualitative agreement with both experimentally and numerically observed B-type vortex breakdown phenomena.

We reconsider the definitions of both material symmetries and physical symmetries which are described in terms of point groups, i.e. subgroups of the full orthogonal group, because these two concepts are often confused and the classical descriptions of physical symmetry for inelastic behaviour of materials are impracticable. All two- and three-dimensional point groups are classified into two types: compact and non-compact. The reality of every compact point group in the description of a material or a physical symmetry is justified in four aspects, that is: (i) point groups characterized by a finite set of tensors, (ii) Hilbert's theorem for integrity bases, (iii) correlation between integrity bases and function bases (generalization of Wineman and Pipkin's theorem), and (iv) physical reality. The unreality of any non-compact point group in the description of a material or a physical symmetry is proposed as a new principle of continuum physics. As applications, the complete sets of all classes of two-and three-dimensional point groups which describe physical symmetries for linear physical properties (such as thermal expansion, piezoelectricity, elasticity, etc.) and for more general mechanical constitutive laws are given.

A symmetry class of an elasticity tensor, c, is determined by the variance of this tensor with respect to a subgroup of the special orthogonal group, SO(3). Using the double covering of SO(3) by the special unitary group, SU(2), we determine the subgroups of SU(2) that correspond to each of the eight symmetry classes. A family of maps between C2 and R3 that preserve the action of the two groups is constructed. Using one of these maps and three associated polynomials, we derive new methods for characterizing the symmetry classes of elasticity tensors.

The material symmetry of the constitutive law of a continuum material is described by the Kronecker powers of the orthogonal tensors which belong to the so-called material symmetry group, a subgroup of the full orthogonal tensor group, of the material. The properties, especially the canonical representations, of Kronecker powers of orthogonal tensors may be applied to deal with material symmetry problems. In this paper, we obtain the basic recurrence formulae in order to determine the canonical representations for finite order Kronecker powers of any given orthogonal tensor; and by using the recurrence formulae we derive the canonical representations for first, second, third and fourth order Kronecker powers of any two- or three-dimensional orthogonal tensor. Finally, we apply these results to construct the micropolar elasticity matrices for micropolar elastic tensors under the 13 anisotropic mechanics symmetry groups as well as the isotropic symmetry group ; and we also explain how to find an appropriate orthogonal tensor subgroup which may be regarded as the idealized material symmetry group for a given tensor.

An ultrasonic measurement of the stiffness tensor of a spruce specimen is made for different moisture content. We show that a known scalar indicator describres correctly the quality of the wood and that the distance from the stiffness tensor to the tetragonal symmetry group describes the quality of the measurements. -- Une mesure ultrasonore du tenseur d'\'elasticit\'e d'un \'epic\'ea de lutherie dont on fait varier la teneur en eau est pr\'esent\'ee. Il est montr\'e qu'un indicateur scalaire connu d\'ecrit bien l'\'evolution de la qualit\'e du bois et que la distance du tenseur d'\'elasticit\'e au groupe de sym\'etrie t\'etragonal permet de quantifier la qualit\'e des mesures.

We prove that there are eight subgroups of the orthogonal group O(3) that determine all symmetry classes of an elasticity tensor. Then, we provide the necessary and sufficient conditions that allow us to determine the symmetry class to which a given elasticity tensor belongs. We also give a method to determine the natural coordinate system for each symmetry class.

Le présent mémoire de thèse propose de nouvelles méthodes de simulation pour la prédiction du comportement dynamique des bâtiments sous séisme. En s'appuyant sur la similitude architecturale des étages d'un immeuble, une méthode d'homogénéisation des milieux à périodicité unidirectionnelle est élaborée afin de réduire son modèle structural tridimensionnel à un modèle simplifié (poutre équivalente). La structure simplifiée est ensuite étudiée en interaction avec le sol. La dispersion significative observée sur les propriétés du sol est introduite sous la forme d'une matrice d'impédance aléatoire construite à l'aide du maximum d'entropie. L'ensemble est soumis à un signal sismique naturel et la réponse dynamique est calculée dans le domaine linéaire (cas déterministe et stochastique) et dans le domaine non-linéaire (décollement de fondation). Dans le cas non-linéaire, une sous structuration de type Craig-Bampton permet de déterminer la réponse de la structure par analyse modale

By extending and developing the characteristic notion of the classical linear elasticity initiated by Lord Kelvin, a new type of representation for classical and micropolar anisotropic elasticity tensors is introduced. The new representation provides general expressions for characteristic forms of the two kinds of elasticity tensors under the material symmetry restriction and has many properties of physical and mathematical significance. For all types of material symmetries of interest, such new representations are constructed explicitly in terms of certain invariant constants and unit vectors in directions of material symmetry axes and hence they furnish invariants which can completely characterize the classical and micropolar linear elasticities. The results given are shown to be useful. In the case of classical elasticity, the spectral properties disclosed by our results are consistent with those given by similar established results.

A complete and unified study of symmetries and anisotropies of classical and micropolar elasticity tensors is presented by
virtue of a novel method based on a well-chosen complex vector basis and algebra of complex tensors. It is proved that every
elasticity tensor has nothing but 1-fold, 2-fold, 3-fold, 4-fold and ∞-fold symmetry axes. From this fact it follows that
the crystallographic symmetries plus the isotropic symmetry are complete in describing the symmetries of any kind of classical
elasticity tensors and micropolar elasticity tensors. Further, it is proved that for each given integer m>>2 every classical Green elasticity tensor with an m-fold symmetry axis must have at least m elastic symmetry planes intersecting each other at this symmetry axis. From this fact and the aforementioned fact it follows
that for all possible material symmetry groups, there exist only eight distinct symmetry classes for classical Green elasticity
tensors, which correspond to the isotropy group and the seven crystal classes S
2, C
2h
, D
2h
, D
3d
, D
4h
, D
6h
and O
h
, while it is shown that there exist twelve distinct symmetry classes for any other kind of elasticity tensors, including
the classical Cauchy elasticity tensor and the micropolar elasticity tensors, which correspond to the eight subgroup classes
just mentioned and the four crystal classes S
6, C
4h
, C6h
and T
h
. From these results, it turns out that all possible elasticity symmetry groups are nothing but the full orthogonal group,
the transverse isotropy groups C
∞ h
and D
∞ h
, and the nine centrosymmetric crystallographic point groups except C
6h
and D
6h
.

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.

We give a new proof of Kolodner's result that longitudinal waves can propagate in at least three directions in a hyperelastic
anisotropic medium. We give examples of an orthotropic hyperelastic tensor with exactly three such directions, of a monoclinic
elastic (but not hyperelastic) tensor with only one, and of a monoclinic elastic (elliptic, but not uniformly elliptic) tensor
with no direction for longitudinal waves.

We find and classify all conservation laws in transversely isotropic linear elastic materials. The complete family of laws is generated by the family of laws which exist for all linear elastic materials and two laws which are generalizations of those found in isotropic materials. One of these latter conservation laws is new. In the case that a certain degeneracy condition exists, there are two more previously unknown laws. There are no further conservation laws unless the material is equivalent under a general linear change of variables to an isotropic material.

First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.

Representations in complete and irreducible forms for tensor functions allow general consistent invariant forms of the nonlinear constitutive equations and specify the number and type of the scalar variables involved. They have proved to be even more pertinent in attempts to model mechanical behavior of anisotropic materials, since here invariant conditions predominate and the number and type of independent scalar variables cannot be found by simple arguments. In the last few years, the theory of representations for tensor functions has been well established, including three fundamental principles, a number of essential theorems and a large amount of complete and irreducible representations for both isotropic and anisotropic tensor functions in three- and two-dimensional physical spaces. The objective of the present monograph is to summarize and recapitulate the up-to-date developments and results in the theory of representations for tensor functions for the convenience of further applications in contemporary applied mechanics. Some general topics on unified invariant formulation of constitutive laws are investigated.

A novel and rational approach based on Lie analysis is proposed to investigate the mechanical behaviour of materials exhibiting experimental master curves. Our approach relies on the idea that the mechanical response of materials is associated with hidden symmetries; the general objective of this contribution is to reveal those symmetries from measurements and to construct constitutive laws from them. This approach provides a priori two ways of formulating constitutive laws from data as well as the possibility of predicting new master curves and material charts. The first part of the paper is devoted to the presentation of the general methodology. Afterwards, the strategy is applied to the uniaxial creep and rupture behaviour of a Chrome-Molybdene alloy (9Cr1Mo) at different temperatures and stress levels. Constitutive equations for creep and rupture master responses are identified for this alloy and validated on experimental data.

The in-plane behaviour of tetrachiral lattices should be characterized by bi-dimensional orthotropic material owing to the existence of two orthogonal axes of rotational symmetry. Moreover, the constitutive model must also represent the chirality inherent in the lattices. To this end, a bi-dimensional orthotropic chiral micropolar model is developed based on the theory of irreducible orthogonal tensor decomposition. The obtained constitutive tensors display a hierarchy structure depending on the symmetry of the underlying microstructure. Eight additional material constants, in addition to five for the hemitropic case, are introduced to characterize the anisotropy under Z 2 invariance. The developed continuum model is then applied to a tetrachiral lattice, and the material constants of the continuum model are analytically derived by a homogenization process. By comparing with numerical simulations for the discrete lattice, it is found that the proposed continuum model can correctly characterize the static and wave properties of the tetrachiral lattice.

In a paper, which has been recently published in Applied Mathematics and Computations, Khan and Ahmad (2012) [1] deal with the detection of the extrema of Young’s modulus, E, in hexagonal materials.
A few issues presented in that paper, which deserve being outlined and thoroughly discussed, are tackled.
Moreover, in the case of hexagonal materials, a suitable classification is suggested, an exhaustive panoramic view of the possible shape of the surface E(n)E(n) generated by Young’s modulus for all possible orientations nn is illustrated, and some meaningful numerical examples are proposed.

Two different definitions of symmetries for photoelasticity tensors are compared. Earlier for such symmetries the existence of exactly 12 classes was proved based on an equivalence relation induced on the set of subgroups of SO(3). Here, an another viewpoint is chosen, and photoelasticity tensors themselves are divided into symmetry classes, according to a different definition. By use of group-theoretical techniques, such as harmonic and Cartan decomposition, it is shown that this approach again leads to 12 classes.

This study addresses the stochastic modeling of media whose elasticity ten-sor exhibits uncertainties on the material symmetry class to which it belongs. More specifically, we focus on the construction of a probabilistic model which allows realizations of random elasticity tensors to be simulated, under the constraint that the mean distance (in a sense to be defined) to a given class of material symmetry is specified. Following the eigensystem characterization of the material symmetries, the proposed approach relies on the probabilis-tic model derived in [6] which allows the variance of selected eigenvalues of the elasticity tensor to be partially prescribed. A new methodology and parameterization of the model are then defined. The proposed approach is exemplified considering the mean to transverse isotropy. The efficiency of the methodology is demonstrated by computing the mean distance of the random elasticity tensor to this material symmetry class, the distance and projection onto the space of transversely isotropic tensors being defined by considering the Riemannian metric and the Euclidean projection, respec-tively. It is shown that the methodology allows the above distance to be (partially) reduced as the overall level of statistical fluctuations increases, no matter the initial distance of the mean model used in the simulations. A comparison between this approach and the nonparametric probabilistic approach (with anisotropic fluctuations) proposed in [12] is finally provided.

A classical topic is the determination of the form of a tensor expressing a physical property and invariant under the point group of a crystal. For example, one finds listings showing 16 distinct forms of the piezomagnetic tensor that can arise in a crystal. Alongside each of these distinct forms is listed the point groups which give rise to that specific form [see R. Birss, Symmetry and magnetism. North Holland (1966)]. This is a classification of the form of piezomagnetic tensors in crystals. We claim, however, that this classification is quite ambiguous in the sense that it hides the very difference between material and behaviour symmetries and a classification of piezomagnetic tensors in behaviour equivalence classes, found to be 15.

The measure of the complete stiffness tensor of an anisotropic elastic material bring experimental difficulties when the possible symmetries are formerly unknowns. As described mechanical measurements show great complexity, we prefer the use of an ultrasonic experimentation. These are done by using the direct contact method on a 26-faces specimen. The stiffness tensor is obtained by the minimization of a functionnal based on the discrepancy between two expression of the acoustic (Christoffel's) tensor. In order to reveal the (exact or not) symmetries of the obtained stiffness tensor expressed in the specimen's axis, pole figures show the correlation between the stiffness tensor and it's symmetric by every plane of the space. These maps are revealing every possible material symmetry. An intrinsic operator determine the stiffness tensor having the chosen symmetry the nearest from the experimentally obtained stiffness tensor. The discrepancy between the two tensors and the order of the symmetry levels allow to choose the best symmetry for the studied material. Finally, damage-induced anisotropy is investigated with Kelvin's decomposition.

While it is demonstrated here that the symmetry plane classification criteria and the symmetry group classification criteria are equivalent in the case of symmetric second order tensors, it is also demonstrated that the two classification criteria are not equivalent in the case of asymmetric second order tensors.

The anisotropy matrices (tensors) of quasielastic (Cauchy-elastic) materials were obtained for all classes of crystallographic symmetries in explicit form. The fourth-rank anisotropy tensors of such materials do not have the main symmetry, in which case the anisotropy matrix is not symmetric. As a result of introducing various bases in the space of symmetric stress and strain tensors, the linear relationship between stresses and strains is represented in invariant form similar to the form in which generalized Hooke’s law is written for the case of anisotropic hyperelastic materials and contains six positive Kelvin eigen moduli. It is shown that the introduction of modified rotation-induced deformation in the strain space can cause a transition to the symmetric anisotropy matrix observed in the case of hyperelasticity. For the case of transverse isotropy, there are examples of determination of the Kelvin eigen moduli and eigen bases and the rotation matrix in the strain space. It is shown that there is a possibility of existence of quasielastic media with a skew-symmetric anisotropy matrix with no symmetric part. Some techniques for the experimental testing of the quasielasticity model are proposed.

We construct an eighteen-dimensional orbifold that is in a one-to-one correspondence with the space of SO (3)-orbits of elasticity tensors. This allows us to obtain a local parametrization of SO (3)-orbits of elasticity tensors by six SO (6)-invariant and twelve SO (3)-invariant parameters. This process unravels the structure of the space of the orbits of the elasticity tensors.

If we consider a body of perfectly elastic material to undergo deformation in which a point initially at X
i
′
in the rectangular Cartesian coordinate system x
i
′
moves to x
i
′
in the same coordinate system, then the strain-energy function W is a single-valued function of the quantities g
ij
′
defined by.

As a newly minted Ph.D., Paul Halmos came to the Institute for Advanced Study in 1938--even though he did not have a fellowship--to study among the many giants of mathematics who had recently joined the faculty. He eventually became John von Neumann's research assistant, and it was one of von Neumann's inspiring lectures that spurred Halmos to write Finite Dimensional Vector Spaces. The book brought him instant fame as an expositor of mathematics. Finite Dimensional Vector Spaces combines algebra and geometry to discuss the three-dimensional area where vectors can be plotted. The book broke ground as the first formal introduction to linear algebra, a branch of modern mathematics that studies vectors and vector spaces. The book continues to exert its influence sixty years after publication, as linear algebra is now widely used, not only in mathematics but also in the natural and social sciences, for studying such subjects as weather problems, traffic flow, electronic circuits, and population genetics. In 1983 Halmos received the coveted Steele Prize for exposition from the American Mathematical Society for "his many graduate texts in mathematics dealing with finite dimensional vector spaces, measure theory, ergodic theory, and Hilbert space."

There can be few books on mathematical mechanics as famous as this, a
work that forms a comprehensive account of all the classical results of
analytical dynamics.

A representation formula for the elasticity tensor of a linearly elastic, transversely isotropic material is obtained, depending on eight constants. If, besides rotations about the axis of symmetry, reflections with respect to planes through that axis are also regarded as admissible symmetry transformations for the material, it is shown that the number of constants reduces to six. It is also shown that, no matter whether reflections belong to the collection of admitted symmetry transformations or not, only five constants are needed for hyperelastic materials.

Die Beschrnkungen, welche die Symmetrie des Materials den Nachgiebigkeitsmatrizen anisotroper viskoelastischer Stoffe auferlegt, werden ohne Bezug auf thermodynamisch bedingte Symmetrien abgeleitet. Experimente zur berprfung der thermodynamischen Theorien lassen sich auf die Unterschiede grnden, die zwischen den mit oder ohne Annahmen ber Matrix-Symmetrie erhaltenen Formen bestehen. Transversale Isotropie und kubische Symmetrie werden im Detail behandelt.

This paper is concerned with elastic materials; these are substances for which the present stress S on a material point depends on only the present local configuration1M of that point: $$ S = S(M).$$ (1).

"First edition, 1892-1893... Fourth edition, 1927. First American Printing 1944" Incluye bibliografía

After the present work was submitted, the following paper came to our attention

- Addendum

Addendum. After the present work was submitted, the following paper came to our attention: