## No full-text available

To read the full-text of this research,

you can request a copy directly from the author.

A geometrical picture of fourth-order, three-dimensional elastic tensors in terms of Maxwell multipoles is developed and used to obtain the elastic tensors appropriate to various crystal symmetry groups. Simply examining the picture shows whether an elastic tensor, described by its 21 independent components relative to an ill-chosen coordinate system, has an axis of symmetry of any order. The picture also facilitates obtaining the elastic tensor from the observed dependence of the three body-wave phase velocities on the direction of the propagation vector κ. In particular, for q = 2, 4, and 6, is a linear combination of the surface spherical harmonics of even orders up to and including q. Since determine uniquely, all the body-wave phase-velocity dependence on can be summarized by the 6 coefficients of spherical harmonics in P(2), the 15 coefficients in P(4), and the 28 coefficients in P(6). For nearly isotropic media, the anisotropy in the P velocity determines 15 of the 21 elastic coefficients, whereas determines the other 6 elastic coefficients. Our description of elastic tensors is generalized to all fourth-order tensors in three dimensions and certain fourth-order tensors in higher dimensions. The problem in higher dimensions produces simple examples of unitary representations of the rotation group ON+ with N ≥ 4 which contain no harmonic irreducible components.

To read the full-text of this research,

you can request a copy directly from the author.

... In this approach [61], the problem of higher order tensors in 3D is recast in the realm of binary forms, which are complex homogeneous polynomials in two variables. Using a powerful tool called the Cartan map [18,8,28,65], an integrity basis for the binary form of degree 2 can then be translated into an integrity basis for the harmonic tensor of degree . The gain is that invariant theory of binary forms (also know as Classical Invariant Theory) is an area of mathematics which has been extensively studied by a wide number of prestigious mathematicians such as Gordan or Hilbert and in which an impressive number of results has already been produced. ...

... The gain is that invariant theory of binary forms (also know as Classical Invariant Theory) is an area of mathematics which has been extensively studied by a wide number of prestigious mathematicians such as Gordan or Hilbert and in which an impressive number of results has already been produced. Combining these results with the use of the harmonic decomposition [8,83], integrity bases for the third order totally symmetric tensor and for the fourth-order elasticity tensor have been obtained recently [63,61,64]. In this approach, Gordan's algorithm for binary forms [36,37,38,39] is used first to generate a (non necessarily minimal) integrity basis and then, a reduction process using modern computational means is achieved to obtain minimality [62]. ...

... Note that (T) = (T ) and that when restricted to S (R 2 ), this correspondence S ↦ → p is a bijection, the inverse operation being given by the polarization of p (see [97,8,9,65]). Making use of this correspondence, the three tensorial operations defined above are recast into polynomial operations as follows. Let p := (S ) for = 1, 2, be the homogeneous polynomials associated with S ∈ S (R 2 ). ...

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and -- possibly coupled -- laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any $n$-uplet of bidimensional constitutive tensors.

... The decomposition (44) is constructed from two scalars S and A, two second-order traceless tensors P ij and Q ij , and a totally traceless and a completely symmetric 4th order tensor R ijkl . Tensors of the same types emerge in the harmonic decomposition that is widely used in elasticity literature, see for instance [2], [3], [6], [14], [15] and [16]. The harmonic decomposition is generated from the harmonic polynomials, i.e., the polynomial solutions of Laplace's equation. ...

... It is straightforwardly to show that five individual terms in this expression are reducible and do not represent elasticities by themselves. An alternative expression of Backus [2], [3], ...

... Thus they are irrelevant for the classification problem. The tensor parts (2) S ijkl and (2) A ijkl are completely described by two second-order tensors P ij and Q ij . For every symmetry class, the dimensions of the P -spaces and the Q-spaces are the same. ...

In linear elasticity, a fourth order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties of a material. Due to Voigt, this tensor is conventionally represented by a 6 × 6 symmetric matrix. This classical matrix representation does not conform with the irreducible decomposition of the elasticity tensor. In this paper, we construct two alternative matrix representations. The 3 × 7 matrix representation is in a correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by three 3 × 3 matrices is suitable for description the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

... Taking into account this observation, it has been tried by some authors to use the harmonic factorization, according to Sylvester's theorem [29] and Maxwell's multipoles [30]. However, this involves roots' computations of polynomials of degree 4 and 8 [3,5,9], in order to build a set of 8 unit vectors (Maxwell's multipoles), without any clue of how to organize such data. Besides, Maxwell's multipoles are not, strictly speaking, first-order covariants of E and are moreover very sensitive to conditioning. ...

... In this list, the notation r(n n n, θ ) denotes a rotation of angle θ around axis n n n , with the convention that r(e e e 3 There exists a partial order on symmetry classes, induced by inclusion between subgroups, and defined as follows: ...

... of an Elasticity tensor (see [3,27]), where d and v are the deviatoric parts of d and v, defined as ...

We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor. We produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants. An algorithm to detect the symmetry class of an Elasticity tensor is finally formulated.

... The underlying calculations go back to Maxwell [5] and were summarized by Backus [6] who also provided further information and some mathematical background. Zou et al. [7] gave explicit formulae to calculate the multipoles of a tensor. ...

... We call the remainder aT = T − sT (13) the asymmetric part of T. Obviously, the asymmetric part of any tensor has no totally symmetric part which creates a vector space of all asymmetric tensors which is actually orthogonal to the totally symmetric tensor space, see [6]. The symmetric and traceless part of a tensor T will be described by ⌊T⌋. ...

... There is a well known relation between totally symmetric tensors and spherical harmonics. This section will describe this connection, following the representation in Backus' [6] paper. ...

The analysis and visualization of tensor fields is a very challenging task. Besides the cases of zeroth- and first-order tensors, most techniques focus on symmetric second-order tensors. Only a few works concern totally symmetric tensors of higher-order. Work on other tensors of higher-order than two is exceptionally rare. We believe that one major reason for this gap is the lack of knowledge about suitable tensor decompositions for the general higher-order tensors. We focus here on three dimensions as most applications are concerned with three-dimensional space. A lot of work on symmetric second-order tensors uses the spectral decomposition. The work on totally symmetric higher-order tensors deals frequently with a decomposition based on spherical harmonics. These decompositions do not directly apply to general tensors of higher-order in three dimensions. However, another option available is the deviatoric decomposition for such tensors, splitting them into deviators. Together with the multipole representation of deviators, it allows to describe any tensor in three dimensions uniquely by a set of directions and non-negative scalars. The specific appeal of this methodology is its general applicability, opening up a potentially general route to tensor interpretation. The underlying concepts, however, are not broadly understood in the engineering community. In this article, we therefore gather information about this decomposition from a range of literature sources. The goal is to collect and prepare the material for further analysis and give other researchers the chance to work in this direction. This article wants to stimulate the use of this decomposition and the search for interpretation of this unique algebraic property. A first step in this direction is given by a detailed analysis of the multipole representation of symmetric second-order three-dimensional tensors.

... (1) two different explicit harmonic decompositions [8,18] of the piezoelectricity tensor; ...

... As a very classical result, any second-order symmetric tensor can be decomposed into a deviatoric part (which is symmetric and traceless) and a spherical part. The generalization of this decomposition to tensor spaces of any order is known as the harmonic decomposition [8,18], where we need to introduce the spaces of higher-order deviators, that are the spaces H of -th order harmonic tensors (definition 2.1). To define these spaces, let us first introduce the space S of totally symmetric tensors on R 3 . ...

... (2) The Schur-Weyl harmonic decomposition, consists in first decomposing the piezoelectricity tensor according to its index symmetries before proceeding to the harmonic decomposition of the each parts. This decomposition follow the lines of the method used by Backus in the case of the elasticity tensor [8]. This decomposition will find interesting application in section 4 for symmetry classes identification. ...

The piezoelectricity law is a constitutive model that describes how mechanical and electric fields are coupled within a material. In its linear formulation this law comprises three constitutive tensors of increasing order: the second order permittivity tensor S, the third order piezoelectricity tensor P and the fourth-order elasticity tensor C. In a first part of the paper, the symmetry classes of the piezoelectricity tensor alone are investigated. Using a new approach based on the use of the so-called clips operations, we establish the 16 symmetry classes of this tensor and provide their associated normal forms. Second order orthogonal transformations (plane symmetries and-angle rotations) are then used to characterize and classify directly 11 out of the 16 symmetry classes of the piezoelectricity tensor. An additional step to distinguish the remaining classes is proposed.

... (1) two different explicit harmonic decompositions [8,18] of the piezoelectricity tensor; ...

... As a very classical result, any second-order symmetric tensor can be decomposed into a deviatoric part (which is symmetric and traceless) and a spherical part. The generalization of this decomposition to tensor spaces of any order is known as the harmonic decomposition [8,18], where we need to introduce the spaces of higher-order deviators, that are the spaces H of thorder harmonic tensors (definition 2.1). To define these spaces, let us first introduce the space S of th-order totally symmetric tensors on R 3 . ...

... (2) The Schur-Weyl harmonic decomposition, consists in first decomposing the piezoelectricity tensor according to its index symmetries before proceeding to the harmonic decomposition of each part. This decomposition follow the lines of the method used by Backus in the case of the elasticity tensor [8]. This decomposition will find interesting application in section 4 for symmetry classes identification. ...

... Other approaches have used the harmonic decomposition (H, d ′ , v ′ , λ, µ) of the elasticity tensor E. For instance, following [25], some authors [26,49,8,20] have extracted partial information about the symmetry class of E from its second-order harmonic components d ′ and v ′ . In the same spirit, but to avoid loosing important information present in the harmonic fourth-order component H, Baerheim [8] has used the harmonic factorization introduced by Sylvester [81] (see also [6] and [67] for a more modern treatment). This factorization allows to decompose an harmonic tensor of order n as an n-tuple of vectors, the Maxwell multipoles [6]. ...

... In the same spirit, but to avoid loosing important information present in the harmonic fourth-order component H, Baerheim [8] has used the harmonic factorization introduced by Sylvester [81] (see also [6] and [67] for a more modern treatment). This factorization allows to decompose an harmonic tensor of order n as an n-tuple of vectors, the Maxwell multipoles [6]. Baerheim has used these multipoles to detect the different symmetry classes of E. This approach requires, however to solve polynomial equations of degree eight. ...

... The harmonic decomposition of the elasticity tensor was first obtained by Backus [6] (see also [26,7]) and is given by ...

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.

... Before providing a proof of theorem 4.3, we explicit these conditions in certain cases. S · ν ν ν − 5ν ν ν ⊙ S (2) · ν ν ν 2 + 20 3 ν ν ν 2 ⊙ S ...

... S · ν ν ν − 9ν ν ν ⊙ S (2) · ν ν ν 2 + 216 7 ν ν ν 2 ⊙ S (3) · ν ν ν 3 −48ν ν ν 3 ⊙ S (4) · ν ν ν 4 + 144 5 ν ν ν 4 ⊙ S (5) · ν ν ν 5 × ν ν ν = 0. ...

... is equivariant [2] and we have the following result. Theorem 6.1. ...

In this article, we formulate necessary and sufficient polynomial equations for the existence of a symmetry plane or an order-two axial symmetry for a totally symmetric tensor of order n $\ge$ 1. These conditions are effective and of degree n (the tensor's order) in the components of the normal to the plane (or the direction of the axial symmetry). These results are then extended to obtain necessary and sufficient polynomial conditions for the existence of such symmetries for an Elasticity tensor, a Piezo-electricity tensor or a Piezo-magnetism pseudo-tensor.

... The decomposition (44) is constructed from two scalars S and A, two second-order traceless tensors P ij and Q ij , and a totally traceless and a completely symmetric 4th order tensor R ijkl . Tensors of the same types emerge in the harmonic decomposition that is widely used in elasticity literature, see for instance [2], [3], [6], [14], [15] and [16]. The harmonic decomposition is generated from the harmonic polynomials, i.e., the polynomial solutions of Laplace's equation. ...

... It is straightforwardly to show that five individual terms in this expression are reducible and do not represent elasticities by themselves. An alternative expression of Backus [2], [3], ...

... Thus they are irrelevant for the classification problem. The tensor parts (2) S ijkl and (2) A ijkl are completely described by two second-order tensors P ij and Q ij . For every symmetry class, the dimensions of the P -spaces and the Q-spaces are the same. ...

In linear elasticity, a fourth order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties of a material. Due to Voigt, this tensor is conventionally represented by a $6\times 6$ symmetric matrix. This classical matrix representation does not conform with the irreducible decomposition of the elasticity tensor. In this paper, we construct two alternative matrix representations. The $3\times 7$ matrix representation is in a correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by three $3\times 3$ matrices is suitable for description the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

... Likewise, the Peter-Weyl theorem, which leads to the conclusion that the normalized Wigner D-functions form a Hilbert basis in the space of square-integrable, complex-valued functions on SO(3), is proved in Chap. 15. Nevertheless, in Chap. ...

... We label each point in C by the location x ∈ E 3 that it occupies under κ 0 . A transplacement h : κ 0 (C) → E 3 of C (i.e., h : E 3 → E 3 , x → h(x)) is said to be rigid (or isometric) if it is distance-preserving, 15 i.e., ...

... 45, 344-345]) is well known: 14 Here we adopt the terminology in continuum mechanics as regards the terms body, placement, and transplacement; see Truesdell [322]. 15 What we call rigid transplacements here are given various other names in the literature, e.g., rigid motions ( [66, p. 25], [232, p. 23]), rigid transformations [268, p. 16], Euclidean motions [304, p. 309], etc. As time is not involved, referring to a distance-preserving bijective mapping as "motion" is a misnomer. ...

This exposition consists of three parts. Part I is an introduction to classical texture analysis. The harmonic method and the approach initiated by Roe, where the orientation distribution function (ODF) is always defined on the rotation group SO(3), is emphasized and given a systematic treatment. Basic concepts (e.g., the orientation density function) are made precise through their mathematical definition. The active view of rotations is implemented throughout. A conscientious effort is made to use machinery already available in mathematics and physics. The Wigner D-functions, whose properties are familiar in physics, are used instead of Bunge’s and Roe’s versions of generalized spherical harmonics. By including three mathematical appendices, it is hoped that engineering students would find Part I readable. The objectives of Parts II and III are threefold, namely: (i) To delve deeper into the mathematical foundations of the harmonic method. The Weyl method is used to prove that the Wigner D-functions are the matrix elements of a complete set of pairwise-inequivalent, continuous, irreducible unitary representation of SO(3). General formulas of the Wigner D-functions, valid for any parametrization of SO(3), are derived. An elementary proof (attributed to Wigner) of the Peter-Weyl theorem is presented. (ii) To provide mathematical prerequisites in group representations for research on representation theorems that delineate the effects of crystallographic texture on material properties defined by tensors or pseudotensors. (iii) To introduce tensorial Fourier expansion of the ODF and the tensorial texture coefficients. The classical ODF expansion in Wigner D-functions is recast as a special tensorial Fourier series. The relation between the tensorial and classical texture coefficients in this context is derived.

... Note however that this approach, using symmetry planes, cannot be generalized to find the symmetry classes of higher order tensorial representations due to the fact that not all closed subgroups of O(3) can be generated by plane reflections. Finally, in 2014, a definitive and systematic way to determine the symmetry classes of any finite dimensional representation of the groups SO(2), SO (3), O (2) or O(3) was formulated (see [34,37,38,36]). This method uses clip's tables and the decomposition of the representation into irreducible components. ...

... In the same spirit, but to avoid loosing the information contained in the harmonic fourth-order component H, Baerheim [4] has used the harmonic factorization introduced by Sylvester [48] (see also [2] and [40] for a more modern treatment). This factorization allows to decompose an harmonic tensor of order n as an n-tuple of vectors, the so-called Maxwell multipoles [2]. ...

... In the same spirit, but to avoid loosing the information contained in the harmonic fourth-order component H, Baerheim [4] has used the harmonic factorization introduced by Sylvester [48] (see also [2] and [40] for a more modern treatment). This factorization allows to decompose an harmonic tensor of order n as an n-tuple of vectors, the so-called Maxwell multipoles [2]. Baerheim has formulated criteria on the multipoles to characterize the different symmetry classes of E. The difficulties with this approach is that the multipoles are not uniquely defined [40] and that the only way to obtain them is to solve a polynomial equation of degree 2n (hence, one degree eight and two degree four polynomials for the Elasticity tensor). ...

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.

... Moreover, the formulation of Kelvin projectors for cubic materials will be based on the harmonic decomposition of the elasticity tensor E E E having cubic material symmetry (Backus, 1970, Onat, 1984, Olive et al, 2017b). This will allow us to compute Kelvin projectors without using the spectral decomposition. ...

... In order to model the cubic anisotropy encountered for single crystal superlloys, such as CMSX-4, we now use the elasticity framework. We first present adequate mathematical tools: harmonic decomposition (Schouten, 1951, Backus, 1970, Spencer, 1970, Kelvin stresses and projectors (Kelvin, 1856, 1878, Rychlewski, 1984, Cowin et al, 1991, Biegler and Mehrabadi, 1995, François, 1995, Bertram and Olschewski, 1996, Arramon et al., 2000, Mahnken, 2002, Desmorat and Marull, 2011. We describe their use in continuum mechanics and define stress and plastic strain tensors dedicated to cubic material symmetry. ...

... with I I I the fourth order unit tensor. Eq. (10) is the harmonic decomposition of E E E cubic (Backus, 1970, Cowin et al, 1991, Baerheim, 1993, Forte and Vianello, 1996. Generalized Lamé constants λ = λ(E E E), µ = µ(E E E) are two invariants of the elasticity tensor. ...

A phenomenological damage model is proposed to account for the strain rate dependency of the damaging processes at high temperature. Mechanical softening during tertiary creep and monotonic tension are modeled by an isotropic scalar internal variable D, whose evolution is described using a rate damage law dD/dt = · · · governed by visco-plasticity and accounting for the enhancement by stress triaxiality. A novel rate sensitive damage threshold is introduced in order to reproduce the rate dependency of the onset of damaging processes. Damage evolution is coupled with the visco-plasticity model developed by the same authors for single crystal superalloys and accounting for microstructural evolution (like γ-rafting and coarsening) in Desmorat et al (2017). The curves presented in this article are identified at 1050 • C for the Ni base single crystal superalloy CMSX-4 but the proposed rate sensitive threshold modeling can be applied to other alloys showing a rate sensitive damage onset, as for example the single crystal superalloys MC2 but also the polycristalline aggregate AD 730 TM .

... In this paper, the goal is to derive a tensorial damage variable from discrete simulations. The tools for the intrinsic analysis of tensors, introduced by Backus (1970) in elasticity and by Leckie and Onat (1981) in damage mechanics, mixing harmonic analysis and the notion of covariants (generalizing that of invariants, Olive et al. (2018b)), are used here in order to analyze the effective elasticity tensors obtained through discrete simulations, without reference to a particular basis, and to achieve a general tensorial representation of damage. Discrete simulations will be used to obtain the evolution of the effective elasticity tensor during the rupture of a numerical specimen under different mechanical loads. ...

... The harmonic decomposition of tensors is a powerful mathematical tool (Schouten, 1989;Spencer, 1970), that has first been applied to three-dimensional elasticity tensors C ∈ la(ℝ 3 ) by Backus (1970). Formally in 2D, it is the equivariant decomposition ...

In this contribution, the use of discrete simulations to formulate an anisotropic damage model is investigated. It is proposed to use a beam-particle model to perform numerical characterization tests. Indeed, this discrete model explicitly describes cracking by allowing displacement discontinuities and thus capture crack induced anisotropy. Through 2D discrete simulations, the evolution of the effective elasticity tensor for various loading tests, up to failure, is obtained. The analysis of these tensors through bi-dimensional harmonic decomposition is then performed to estimate the tensorial damage evolution. As a by-product of present work we obtain an upper bound of the distance to the orthotropic symmetry class of bi-dimensional elasticity.

... The vector space Ela decomposes into a direct sum of SO(3)-irreducible subspaces (so-called harmonic decomposition [7,68]) ...

... Appendix A. Explicit harmonic decomposition of an elasticity tensor An elasticity tensor E ∈ Ela admits the following explicit harmonic decomposition [7]: (26) E = E + q ⊗ (4) a + q ⊗ (2,2) b + H. ...

We give a detailed description of a polynomial optimization method allowing to solve a problem in continuum mechanics: the determination of the elasticity or the piezoelectricity tensor of a specific isotropy stratum the closest to a given experimental tensor, and the calculation of the distance to the given tensor from the considered isotropy stratum. We take advantage of the fact that the isotropy strata are semialgebraic sets to show that the method, developed by Lasserre and coworkers which consists in solving polynomial optimization problems with semialgebraic constraints, successfully applies.

... The harmonic decomposition [Schouten 1954;Backus 1970;Spencer 1970;Cowin 1989;Baerheim 1993;Forte and Vianello 1996;Auffray et al. 2014] is a well-known tool for the study of the symmetry classes of tensors. The harmonic decomposition of an elasticity tensor defines three symmetric harmonic tensor spaces: a real scalar space H 0 , a space of second-order harmonic tensors h ∈ H 2 (i.e. ...

... The two harmonic second order tensors c and b derive from the dilatation tensor di = tr 12 T and from the Voigt tensor vo = tr 13 T [Baerheim 1993;Auffray et al. 2014]. [Backus 1970] used a different harmonic decomposition involving rari-constant (totally symmetric) and anti-rari-constant (asymmetric) terms. Rather in this work, we wish to express the elastic energy explicitly with respect to spherical and deviatoric parts of the stress (or strain) tensors involved. ...

This thesis deals with the lightweight design of aeronautical structures. On the one hand, topology optimization, that determines the optimal distribution of the material, is a response to this concern. On the other hand, anisotropic materials offer new degrees of freedom for structural optimization. The aim of this study is to propose a methodology to find concurrently the material spatial distribution and the material anisotropy repartition, for 3D structures using a transversely isotropic material. For this purpose an optimization strategy is developed for 2D structures and extended for 3D structures. In order to handle in the optimization the complexity brought by the anisotropy of the material behavior, the elasticity tensor is parameterized by invariants. The shape of the structure is parameterized by the SIMP method using a density variable that determines the presence or absence of material. The problem is solved using the alternate direction algorithm which is well- suited to take into account concurrently topology and material anisotropy. The algorithm alternates between local minimizations and global minimizations. Thanks to the use of invariants, the local minimizations are solved analytically. The global minimizations correspond to finite element calculations. The method is applied to global structural stiffness maximization problems for classical test cases. A complex industrial test case is also considered with the optimization of a fuselage door surrounding lintel modeled by a detailed finite element model (DFEM) provided by STELIA Aerospace.

... · ν ν ν 3 + 16ν ν ν 4 ⊙ S (4) · ν ν ν 4 = 0; n = 10: S − 10 ν ν ν ⊙ (S · ν ν ν) + 40ν ν ν 2 ⊙ S (2) · ν ν ν 2 − 80ν ν ν 3 ⊙ S (3) · ν ν ν 3 + 80ν ν ν 4 ⊙ S (4) · ν ν ν 4 − 32ν ν ν 5 ⊙ S (5) · ν ν ν 5 = 0. Example 4.6. Equation (4.6) defining a plane/axial symmetry of a totally symmetric pseudotensor or an axial symmetry of a totally symmetric tensor writes, for orders n = 2r + 1 < 10, n = 1: ...

... is equivariant [2], where S 4 (R 3 ), H 2 (R 3 ) and H 0 (R 3 ) are respectively equipped with the representations ρ 4 , ρ 2 and ρ 0 . We have the following result. ...

In this article, we formulate necessary and sufficient polynomial equations for the existence of a symmetry plane or an order-two axial symmetry for a totally symmetric tensor of order [Formula: see text]. These conditions are effective and of degree [Formula: see text] (the tensor’s order) in the components of the normal to the plane (or the direction of the axial symmetry). These results are then extended to obtain necessary and sufficient polynomial conditions for the existence of such symmetries for an elasticity tensor, a piezo-electricity tensor or a piezo-magnetism pseudo-tensor.

... An approach to determine the symmetry class of a stiffness based on Maxwell multipoles is shown in Backus (1970). Here symmetry axes are found by finding Maxwell multipoles of the parts of the harmonic decomposition of the stiffness. ...

... An algorithm to determine this tensor has been provided by Baerheim (1993) based on Backus (1970) and Cowin (1989) which applies only for the non-cubic case as we will discuss in the sequel. It makes use of the fact that every tensor can be decomposed into a totally symmetric part and a residual. ...

A completely algebraic algorithm is given to determin the distance of an elastic stiffness tensor to any of the symmetry classes.

... Taking account this observation, it has been tried by some authors to use the harmonic factorization, according to Sylvester's theorem [26] and Maxwell's multipoles [27]. However, this involves roots' computations of polynomials of degree 4 and 8 [4,5,8], in order to build a set of 8 unit vectors (Maxwell's multipoles), without any clue of how to organize such data. Besides, Maxwell's multipoles are not, strictly speaking, first-order covariant of E and are very sensitive to conditioning. ...

... of an elasticity tensor (see [4,24,2]), where d ′ and v ′ are the deviatoric parts of d and v, defined as (·) ′ := (·) − 1 3 tr(·)1. ...

We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor, once we know its symmetry class. In other words, we produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants.

... C ijkl = C jikl = C klij , it admits the following irreducible form (see e.g. [60,61]): ...

... In 3D, the irreducible form of the elasticity tensor is given by [60]: ...

A data-driven approach is proposed to construct anisotropic damage models with a minimal number of internal variables from numerical simulations on Representative Volume Elements (RVEs) of quasi-brittle materials. The approach resorts in particular to a harmonic analysis of damage. The orientation distribution functions of two elastic moduli are numerically determined while accounting for the effects of the nucleation and propagation of microcracks by the phase-field method. Given these two functions, the effective elastic tensor of a material without or with microcracks is uniquely determined. The expansions into two Fourier series of the relative variations of these two functions related to an undamaged reference state and to a damage state make appear damage internal variables naturally. The number and natures of these variables can be optimized by truncating the Fourier series according to the degree of approximation desired. Thus, 2D and 3D anisotropic damage models can be constructed without resorting to usual assumptions made in damage mechanics. This construction holds for complex microstructures including image-based ones and for arbitrary loading history. Two- and three-dimensional applications are provided to evaluate the accuracy of the damage models constructed and to show the potential of the approach proposed.

... In this paper, the goal is to derive a second order tensorial damage variable from discrete simulations. The tools for the intrinsic analysis of tensors, introduced by Backus (1970) in elasticity and by Leckie and Onat (1981) in damage mechanics, mixing harmonic analysis and the notion of covariants (generalizing that of invariants, Olive et al. (2018b)), are used here in order to analyze the effective elasticity tensors obtained through discrete simulations, without reference to a particular basis, and to achieve a general tensorial representation of damage. Discrete simulations will be used to obtain the evolution of the effective elasticity tensor during the rupture of a numerical specimen under different mechanical loads. ...

... The harmonic decomposition of tensors is a powerful mathematical tool (Schouten, 1989;Spencer, 1970), that has first been applied to three-dimensional elasticity tensors C ∈ la(ℝ 3 ) by Backus (1970). Formally in 2D, it is the equivariant decomposition C = ( , , d ′ , H) ∈ la(ℝ 2 ), into two scalars (invariants) , ∈ ℍ 0 (ℝ 2 ) ≃ ℝ, being the shear modulus, the bi-dimensional bulk modulus, one harmonic (deviatoric) second order covariant d ′ = d ′ (C) ∈ ℍ 2 (ℝ 2 ) and one harmonic fourth order covariant H = H(C) ∈ ℍ 4 (ℝ 2 ), such as ...

In this contribution, the use of discrete simulations to formulate an anisotropic damage model is investigated. It is proposed to use a beam-particle model to perform numerical characterization tests. Indeed, this discrete model explicitly describes cracking by allowing displacement discontinuities and thus capture crack induced anisotropy of quasi-brittle materials such as concrete. Through 2D discrete simulations, the evolution of the effective elasticity tensor for various loading tests, up to failure, is obtained. The analysis of these tensors through bi-dimensional harmonic decomposition is then performed to estimate the tensorial damage evolution. It is shown in a quantitative manner that a second order –instead of a fourth order– damage tensor is sufficient in practice, even when the micro-cracks are strongly interacting. As a by-product of present work we obtain an upper bound of the distance to the orthotropic symmetry class of bi-dimensional elasticity.

... Based on these definitions, Backus [Bac70] defined a deviator as a traceless, totally symmetric tensor. Next to the total symmetry, there are other types of symmetry. ...

... The method is tested by representing the deformation with the help of line segments. Backus [Bac70] gave a derivation of the deviatoric decomposition and represented the deviators of the stiffness tensor by five multipole set presentations (one set of normalized vectors for each deviator). Based on the deviatoric decomposition, Zou et al. [ZTL13] calculated a characteristic function. ...

Tensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large‐scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application‐oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio‐, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context. Tensors are used to describe complex physical processes in many applications.

... Cubic symmetry is of most importance for Ni-based single crystal superalloys, such as 51,53], the material of aircrafts gas turbine blades (subject to (visco-)plasticity [44,13]). Thanks to the harmonic decomposition [10,55,18,11], the geometry of cubic fourth order tensors is now well understood. This will make it possible to formulate the calculation of the distance to cubic symmetry as a polynomial optimization problem, not only for a single elasticity tensor, but also for a pair (E, P) of two fourth-order constitutive tensors. ...

... Remark 3.1. The decomposition (8) of E into λ, µ, and H (with H cubic), is the so-called harmonic decomposition of a cubic elasticity tensor (see [10,18]). ...

Generically, a fully measured elasticity tensor has no material symmetry. For single crystals with a cubic lattice, or for the aeronautics turbine blades superalloys such as Nickelbased CMSX-4, cubic symmetry is nevertheless expected. It is in practice necessary to compute the nearest cubic elasticity tensor to a given raw one. Mathematically formulated, the problem consists in finding the distance between a given tensor and the cubic symmetry stratum. It is known that closed symmetry strata (for any tensorial representation of the rotation group) are semialgebraic sets, defined by polynomial equations and inequalities. It has been recently shown that the closed cubic elasticity stratum is moreover algebraic, which means that it can be defined by polynomial equations only (without requirement to polynomial inequalities). We propose to make use of this mathematical property to formulate the distance to cubic symmetry problem as a polynomial (in fact quadratic) optimization problem, and to derive its quasi-analytical solution using the technique of Gr{\"o}bner bases. The proposed methodology also applies to cubic Hill elasto-plasticity (where two fourth-order constitutive tensors are involved).

... The first one is that the reference interval velocity model influences the angle coverage for a given acquisition length, limiting the range of illumination angles available to characterize the elastic coefficients. The second reason is that the longitudinal traveltime perturbation is influenced by 15 parameters only, where the six constants or combinations (c 11 , c 22 , c 33 , c 12 þ 2c 66 , c 23 þ 2c 44 , and c 13 þ 2c 55 Þ are the most sensitive parameters, with the remaining ones ðc 16 , c 15 , c 26 , c 24 , c 35 , c 34 , c 14 þ 2c 56 , c 25 þ 2c 46 , and c 36 þ 2c 45 Þ having a much smaller influence (Backus, 1970;Every and Sachse, 1992;Kazei and Alkhalifah, 2018a). These 15 parameters correspond to the totally symmetric part of the elastic tensor s ijkl ¼ ðc ijkl þ c iklj þ c iljk Þ∕3, which can be expressed with the Voigt notation (Love, 1944) Because the following tomography is performed along inlines, the 2D approach is developed. ...

In the petroleum industry, time-lapse (4D) studies are commonly used for reservoir monitoring, but are also useful to perform risk assessment for potential overburden deformations (e.g., well shearing, cap rock integrity). Although complex anisotropic velocity changes are predicted in the overburden by geomechanical studies, conventional time-lapse inversion workflows only deal with vertical velocity changes. To retrieve the geomechanically induced anisotropy, we propose to use a reflection traveltime tomography method coupled with a time-shift estimation algorithm of prestack data of the baseline and monitor simultaneously. For the 2D approach, we parameterize the anisotropy using five coefficients, enough to cover any type of anisotropy. Before applying the workflow to a real dataset, we first study a synthetic dataset based on the real dataset and include velocity variations between baseline and monitor found in the literature (vertical P-wave velocity decrease in the cap rock and isotropic P-wave velocity change in the reservoir). On the synthetics we measure the angular ray coverage necessary to retrieve the target anisotropy and observe that the retrieved anisotropies depend on the offset range. Based on a synthetic experiment, we believe that the acquisition of the real case study is suitable for performing tomographic inversion. The anisotropic velocity changes obtained on three inlines separated by 200 m are consistent and show a strong positive anomaly in the cap rock along the 45° direction (δ parameter in Thomsen notation) while the vertical velocity change is surprisingly almost negligible. We propose a rock-physics explanation compatible with these observations and geological considerations: a reactivation of water-filled subvertical cracks.

... A linear elasticity tensor E ∈ Ela is defined as a fourth-order tensor having the major and the minor index symmetries, E ijkl = E jikl = E klij . Let E s ∈ S 4 (R 3 ) be its totally symmetric part and A be its asymmetric part (in the sense of Backus [1]). Their components write as follows ...

We formulate necessary and sufficient conditions for a unit vector n to generate a plane or axial symmetry of a constitutive tensor. For the elasticity tensor, these conditions consist of two polynomial equations of degree lower than four in the components of n. Compared to Cowin-Mehrabadi conditions, this is an improvement, since these equations involve only the normal vector n to the plane symmetry (and no vector perpendicular to n). Similar reduced algebraic conditions are obtained for linear piezo-electricity and for totally symmetric tensors up to order 6.

... For higher order tensors, the determination of an integrity basis is much more complicated and one way to compute such a basis requires first to decompose the tensor space V into irreducible representations called also an harmonic decomposition of V (see [4,16,5,3,2,29] for more details). In this decomposition, the irreducible factors are isomorphic to the spaces H (R 3 ), of th-order harmonic tensors. ...

We define what is a generic separating set of invariant functions (a.k.a. a weak functional basis) for tensors. We produce then two generic separating sets of polynomial invariants for 3D elasticity tensors, one made of 19 polynomials and one made of 21 polynomials (but easier to compute) and a generic separating set of 18 rational invariants. As a byproduct, a new integrity basis for the fourth-order harmonic tensor is provided.

... For higher-order tensors, the determination of an integrity basis is much more complicated and one way to compute such a basis requires first to decompose the tensor space V into irreducible representations called also an harmonic decomposition of V (see [4,16,5,3,2,29] for more details). In this decomposition, the irreducible factors are isomorphic to the spaces H n (R 3 ), of nth-order harmonic tensors. ...

We define a generic separating set of invariant functions (a.k.a. a weak functional basis ) for tensors. We then produce two generic separating sets of polynomial invariants for three-dimensional elasticity tensors, one consisting of 19 polynomials and one consisting of 21 polynomials (but easier to compute), and a generic separating set of 18 rational invariants. As a by-product, a new integrity basis for the fourth-order harmonic tensor is provided.

... In 3D the space of elasticity tensors Ela admits the following harmonic decomposition which was first obtained by Backus [1970] (see also Baerheim [1993], Forte and Vianello [2006]): ...

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.

... Following Backus (Backus, 1970) and Pike and Sabatier (Pike and Sabatier, 2001), the elasticity tensor of an isotropic elastic medium only depends on the two parameters λ and µ, usually referred to as Lamé parameters: ...

The spatial accuracy of source localization by dolphins has been observed to be equally accurate independent of source azimuth and elevation. This ability is counter-intuitive if one considers that humans and other species have presumably evolved pinnae to help determine the elevation of sound sources, while cetaceans have actually lost them. In this work, 3D numerical simulations are carried out to determine the influence of bone-conducted waves in the skull of a short-beaked common dolphin on sound pressure in the vicinity of the ears. The skull is not found to induce any salient spectral notches, as pinnae do in humans, that the animal could use to differentiate source elevations in the median plane. Experiments are conducted in a water tank by deploying sound sources on the horizontal and median plane around a skull of a dolphin and measuring bone-conducted waves in the mandible. Their full waveforms, and especially the coda, can be used to determine source elevation via a correlation-based source localization algorithm. While further experimental work is needed to substantiate this speculation, the results suggest that the auditory system of dolphins might be able to localize sound sources by analyzing the coda of biosonar echoes. 2D numerical simulations show that this algorithm benefits from the interaction of bone-conducted sound in a dolphin's mandible with the surrounding fats.

... where c is the phase velocity, T is the period, Y is the azimuth, c 0 (T) is the isotropic term, and c 1-4 are the azimuthal coefficients [Backus, 1970]. The directions Q of fast propagation for Rayleigh waves and the amplitude A of the anisotropy can be obtained using . ...

We develop a three-dimensional model of shear wave velocity and anisotropy for the Mexico subduction zone using Rayleigh wave phase velocity dispersion measurements. This region is characterized by both steep and flat subduction and a volcanic arc that appears to be oblique to the trench. We give a new interpretation of the volcanic arc obliqueness and the location of the Tzitzio gap in volcanism based on the subduction morphology. We employ the two-station method to measure Rayleigh phase velocity dispersion curves between periods of 16 s to 171 s. The results are then inverted to obtain azimuthally anisotropic phase velocity maps and to model 3-D variations in upper mantle velocity and anisotropy. Our maps reveal lateral variations in phase velocity at all periods, consistent with the presence of flat and steep subduction. We also find that the data are consistent with two layers of anisotropy beneath Mexico: a crustal layer, with the fast directions parallel to the North American absolute plate motion, and a deeper layer that includes the mantle lithosphere and the asthenosphere, with the fast direction interpreted in terms of toroidal mantle flow around the slab edges. Our combined azimuthal anisotropy and velocity model enables us to analyze the transition from flat to steep subduction and to determine whether the transition involves a tear resulting in a gap between segments or is a continuous deformation caused by a lithospheric flexure. Our anisotropy results favor a tear, which is also consistent with the geometry of the volcanic belt.

... An approach to determine the symmetry class of a stiffness based on MAXWELL multipoles is shown in [1]. Here symmetry axes are found by finding MAXWELL multipoles of the parts of the harmonic decomposition of the stiffness. ...

For a given elastic stiffness tetrad an algorithm is provided to determine the distance of this particular tetrad to all tetrads of a prescribed symmetry class. If the particular tetrad already belongs to this class then the distance is zero and the presentation of this tetrad with respect to the symmetry axes can be obtained. If the distance turns out to be positive, the algorithm provides a measure to see how close it is to this symmetry class. Moreover, the closest element of this class to it is also determined. This applies in cases where the tetrad is not ideal due to scattering of its measurement. The algorithm is entirely algebraic and applies to all symmetry classes, although the isotropic and the cubic class need a different treatment from all other classes.

... To measure the anisotropy, we used a minimization algorithm and fitted the first three parameters of Smith and Dahlen's (42) anisotropy model for surface wave velocity v(T, θ) = a 0 (T) + a 1 (T) cos(2θ) + a 2 (T) sin(2θ) + a 3 (T) cos(4θ) + a 4 (T) sin(4θ) (1) where v is the velocity, T is the period, is the azimuth, a 0 is the isotropic velocity, and a 1−4 are the azimuthal coefficients (43) to every beamformer output. The reason why we only kept up to the 2 terms is that the azimuthal dependence of the fundamental-mode Rayleigh waves is practically insensitive to the 4 terms (44). ...

Buoyancy anomalies within Earth’s mantle create large convective currents that are thought to control the evolution of the lithosphere. While tectonic plate motions provide evidence for this relation, the mechanism by which mantle processes influence near-surface tectonics remains elusive. Here, we present an azimuthal anisotropy model for the Pacific Northwest crust that strongly correlates with high-velocity structures in the underlying mantle but shows no association with the regional mantle flow field. We suggest that the crustal anisotropy is decoupled from horizontal basal tractions and, instead, created by upper mantle vertical loading, which generates pressure gradients that drive channelized flow in the mid-lower crust. We then demonstrate the interplay between mantle heterogeneities and lithosphere dynamics by predicting the viscous crustal flow that is driven by local buoyancy sources within the upper mantle. Our findings reveal how mantle vertical load distribution can actively control crustal deformation on a scale of several hundred kilometers.

... The first one is that the reference interval velocity model influences the angle coverage for a given acquisition length, limiting the range of illumination angles available to characterize the elastic coefficients. The second one is that the longitudinal traveltime perturbation is influenced by 15 parameters only, where the 6 constants or combinations: c 11 , c 22 , c 33 , c 12 + 2c 66 , c 23 + 2c 44 , and c 13 + 2c 55 are the most sensitive parameters and the remaining ones: c 16 , c 15 , c 26 , c 24 , c 35 , c 34 , c 14 + 2c 56 , c 25 + 2c 46 , and c 36 + 2c 45 have a much smaller influence (Backus, 1970;Every and Sachse, 1992;Kazei and Alkhalifah, 2018). These 15 parameters correspond to the totally symmetric part of the elastic tensor s ijkl = 1 3 (c ijkl + c iklj + c iljk ) which can be expressed with the Voigt notation (Love, 1944) as Each quarter is subdivided into three parts such as each one represents 30°. ...

Petroleum reservoir production modifies the stress state of the subsurface. Petroleum is drawn out, the reservoir pore pressure decreases, and the reservoir rock supports an additional loading. The reservoir is not the only affected area of the field during the production: the layers above (the overburden) and below (the underburden) move. To understand the effect of the reservoir compaction on the overburden and on the subsidence, geomechanical models are built. The rock elastic behavior and the pore pressure change are used to compute the displacement field and the stress changes. In any case, the models of petroleum fields are riddled with uncertainties because the data are sparse and fuzzy and often the results of a long and complex processing. The data uncertainties make the interpretation difficult, or the usage of the prediction made by reservoir models. To reduce these uncertainties, seismic data such as time-lapse seismic (4D) is used in the geomechanical modelling workflow. 4D seismic are imprinted by the fluid flow in the reservoir, by the strain in the reservoir and in the overburden, and by the rock damaging. In most of the produced fields, the amount of time shifts (the reflectors time-arrival difference) in the overburden is significant. While only a small part of the overburden time shifts is explained by the strain, the velocity change plays a major role. But the relationship between the velocity change and the stress state change due to the reservoir compaction is not known. Microcracks opening is known to have a strong effect on the velocity change. Crack-based models can explain the stress-dependent velocity change in rock samples. Based on synthetics models, different authors predict that the velocity change is anisotropic because the stress change in the overburden is anisotropic. However, the velocity change anisotropy is not considered in the classic time-lapse seismic inversion as only the vertical direction is considered. The object of this thesis is to extract more information from the data to create a more precise description of the subsurface model. A Mechanical History Matching of time-lapse inverted time strains using Ensemble methods is performed. For this, we first propose to invert for pressure variation in the reservoir to match the time-lapse seismic data. Using the empirical R-factor law, time-lapse time strains are predicted by the mechanical vertical strains. The reservoir pore pressure is updated to match the geomechanical model predictions with the time-lapse inversion results, suggesting that a new compartmentalization of the reservoir is needed to match the 4D information in the overburden. This first approach interrogates on the usage of reservoir simulation data in the geomechanical model. Second, we propose to recover the velocity change anisotropy from prestack time-lapse seismic data. A tomographic reconstruction of the velocity change is performed by realigning the baseline to the monitor. The method is applied to a real case study. The retrieved anisotropic velocity change in the overburden corresponds to a large decrease in epsilon and delta in terms of Thomsen parameters while the vertical velocity change is tiny. Classically, geomechanical models show an overburden stretching corresponding to an opening of horizontal cracks; this is not what we observe in the velocity change anisotropy. To explain the data, we propose a rock-physics model corresponding to vertical cracks with a small aspect ratio (between 10-2 and 10-3) in a water-saturated rock. This second approach interrogates on the assumptions made when the geomechanical model simulation results are compared to the time-lapse seismic inversion results. As shown in the two approaches, a great potential for improving the consistency and reduce uncertainty of the models exists when mixing various data. This work is preliminary and much more work is required to integrate all sources of data into the shared earth model.

... Papers [4] and [15] use the eigensystems of elastic maps to find symmetries. Papers [2,3,13] are algebraic and are more sophisticated than the present paper. The papers [5,10,12,14] have applications to acoustics and seismology. ...

An elastic map $\mathbf {T}$ T describes the strain-stress relation at a particular point $\mathbf {p}$ p in some material. A symmetry of $\mathbf {T}$ T is a rotation of the material, about $\mathbf {p}$ p , that does not change $\mathbf {T}$ T . We describe two ways of inferring the group $\mathcal {S} _{ \mathbf {T} }$ S T of symmetries of any elastic map $\mathbf {T}$ T ; one way is qualitative and visual, the other is quantitative. In the first method, we associate to each $\mathbf {T}$ T its “monoclinic distance function” "Equation missing" on the unit sphere. The function "Equation missing" is invariant under all of the symmetries of $\mathbf {T}$ T , so the group $\mathcal {S} _{ \mathbf {T} }$ S T is seen, approximately, in a contour plot of "Equation missing" . The second method is harder to summarize, but it complements the first by providing an algorithm to compute the symmetry group $\mathcal {S} _{ \mathbf {T} }$ S T . In addition to $\mathcal {S} _{ \mathbf {T} }$ S T , the algorithm gives a quantitative description of the overall approximate symmetry of $\mathbf {T}$ T . Mathematica codes are provided for implementing both the visual and the quantitative approaches.

... Où l'exposant s désigne la symétrisation du tenseur assurant les symétries majeures du tenseur d'ordre 4 (les symétries mineures étant fournies par la symétrisation des tenseurs de contrainte et de déformation). Le tenseur d'élasticité C d'ordre 4 peut être ré-écrit en utilisant la décomposition harmonique (Backus, 1970). En 3D, tout tenseur d'élasticité peut être décomposé en trois parties : ...

Les systèmes d'ancrage chevillé sont utilisés pour assurer la fixation de nombreuses structures externes et d’Éléments Importants pour la Protection dans les installations nucléaires françaises. Ces systèmes servent à transmettre l'effort de l’élément fixé vers la structure porteuse. L'évaluation du comportement des ancrages existants représente un enjeu majeur pour la sûreté, notamment en cas de séisme.Généralement, l’étude du comportement des ancrages dans le béton se fait par des campagnes expérimentales. Cependant, celles-ci sont coûteuses et limitées par le nombre d’essais réalisés. De plus, elles ne sont pas réalisables pour requalifier les ancrages déjà installés et en arrêt de production. C’est pourquoi la simulation numérique est récemment devenue de plus en plus utile dans le domaine des fixations. Dans ce contexte, une modélisation numérique à deux échelles est proposée. La première modélisation est à l’échelle de l’ancrage où l’utilisation d’un modèle de type particulaire-lattice, nommé DEAP, est proposée pour mieux comprendre les mécanismes de rupture. Ce type de modèle permet une description fine et détaillée du comportement de l’interface entre l’ancrage et le béton ainsi que de la fissuration de ce dernier. Ensuite, une modélisation à l’échelle de la structure est réalisée. Pour ce faire, un modèle simplifié en variables généralisées est formulé et identifié à partir des résultats expérimentaux et des résultats obtenus par DEAP. Ce type de modèle macroscopique permet de simplifier la représentation du comportement non-linéaire de l’ancrage et de réduire conséquemment le temps de calcul, ce qui permet de réaliser un nombre important de calculs pour les analyses de vulnérabilité des ouvrages de génie civil sous chargement sismique.Au cours de ce travail de thèse, plusieurs contributions ont été réalisées notamment sous forme de développements numériques. Premièrement, une méthode de génération d’un maillage d’éléments discrets pour des géométries complexes et bien adaptée au cas de l’ancrage est proposée et développée. Deuxièmement, une nouvelle stratégie simplifiée pour la détection du contact entre l’acier et le béton en 2D ainsi qu’en 3D est mise en œuvre pour améliorer le temps de calcul. Tous ces développements ajoutés au modèle DEAP ont permis de réaliser des modélisations bidimensionnelles et tridimensionnelles d’un essai d’arrachement à l’échelle de l’ancrage. Les résultats ont permis de valider la capacité d’un modèle particulaire-lattice à reproduire le faciès de fissuration d’un test d’arrachement d’ancrage et à déterminer la force maximale de l’ancrage avant la rupture. Ensuite, sur la base des résultats expérimentaux et des simulations discrètes, une loi de comportement en variables généralisées a été formulée et identifiée. Les principaux mécanismes non-linéaires sont pris en compte dans cette loi afin de représenter le comportement réel d’un ancrage présent dans les ouvrages de génie civil. Ce modèle macroscopique simplifié est suffisamment flexible et simple pour être adapté à différents types d’ancrages. Les travaux et les contributions réalisés durant ces trois années de thèse constituent une étape importante pour des études plus approfondies sur différents types d’ancrages sous différents types d’exigences.

... The first step, when studying the geometry of elasticity tensors, consists in splitting Ela into stable, irreducible vector spaces (under the action of SO (3)). This is the so-called harmonic decomposition [4]. Introducing the second-order dilatation tensor (2.4) and the harmonic (i.e., totally symmetric and traceless) fourth-order tensor ...

Functional bases, synonymous with separating sets, are usually formulated for an entire vector space, such as the space Ela\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{E}\mathrm{la}$\end{document} of elasticity tensors. We propose here to define functional bases limited to symmetry strata, i.e., sets of tensors of the same symmetry class. We provide such low-cardinality minimal bases for tetragonal, trigonal, cubic or transversely isotropic symmetry strata of the elasticity tensor.

... Therefore, linear invariant decompositions can be applied to N to study its structure. Literature on invariant decompositions of fourth-order tensors is extensive, see, e.g., [30,[37][38][39][40][41] or those focusing on Hooke tensors and addressing an engineering audience [42][43][44]. Following [42], a Hooke tensor H with 21 independent components can be split into five parts (K, G, H 1 , H 2 , dev (H)) leading to ...

Fiber orientation tensors are established descriptors of fiber orientation states in (thermo-)mechanical material models for fiber-reinforced composites. In this paper, the variety of fourth-order orientation tensors is analyzed and specified by parameterizations and admissible parameter ranges. The combination of parameterizations and admissible parameter ranges allows for studies on the mechanical response of different fiber architectures. Linear invariant decomposition with focus on index symmetry leads to a novel compact hierarchical parameterization, which highlights the central role of the isotropic state. Deviation from the isotropic state is given by a triclinic harmonic tensor with simplified structure in the orientation coordinate system, which is spanned by the second-order orientation tensor. Material symmetries reduce the number of independent parameters. The requirement of positive-semi-definiteness defines admissible ranges of independent parameters. Admissible parameter ranges for transversely isotropic and planar cases are given in a compact closed form and the orthotropic variety is visualized and discussed in detail. Sets of discrete unit vectors, leading to selected orientation states, are given.

... ( ) = 0( ) + 1( )cos(2 ) + 2( )sin(2 ) + 3( )cos(4 ) + 4( )sin(4 ) (6) where T represents the period, θ the backazimuth, C 0 the isotropic velocity, and C 1−4 the azimuthal coefficients (Backus, 1970). This parametrization allows us to translate the wavefield's azimuthal dependence into a fast direction term and an amplitude term, which we can then visualize on a horizontal plane to determine whether any spatial patterns exists, and what association they have with the regional geologic structures. ...

The metropolitan Los Angeles region represents a zone of high-seismic risk due to its proximity to several fault systems, including the San Andreas fault. Adding to this problem is the fact that Los Angeles and its surrounding cities are built on top of soft sediments that tend to trap and amplify seismic waves generated by earthquakes. In this study, we use three dense petroleum industry surveys deployed in a 16x16-km area at Long Beach, California, to produce a high-resolution model of the top kilometer of the crust and investigate the influence of its structural variations on the amplification of seismic waves. Our velocity estimates reveal substantial lateral contrasts and correlate remarkably well with the geological background of the area, illuminating features such as the Newport-Inglewood fault, the Silverado aquifer, and the San Gabriel river. We then use computational modeling to show that the presence of these small-scale structures have a clear impact on the intensity of the expected shaking, and can cause ground-motion motion acceleration to change by several factors over a sub-kilometer horizontal scale. These results shed light onto the scale of variations that can be expected in this type of tectonic settings and highlight the importance of resolution in modern-day seismic hazard estimates.

... Thus , , ⋯ , are corresponded to 2p poles of a unit sphere, which are called multipole structures of deviators. This simple geometric picture was originally suggested by Maxwell [22] and further executed by Backus [23] and Baerheim [24,25]. It is worth noting that Zou and Zheng [19] provided a direct and constructive establishment of Maxwell's multipole representation. ...

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 first step, when studying the geometry of elasticity tensors, consists in splitting Ela into stable, irreducible vector spaces (under the action of SO(3)). This is the so-called harmonic decomposition [4]. Introducing the second-order dilatation tensor ...

Functional bases, synonymous with separating sets, are usually formulated for an entire vector space, such as the space Ela of elasticity tensors. We propose here to define functional bases limited to symmetry strata, i.e. sets of tensors of the same symmetry class. We provide such lowcardinal minimal bases for tetragonal, trigonal, cubic or transversely isotropic symmetry strata of the elasticity tensor.

... Many authors have treated the problem of identifying the symmetries of given elastic maps. See Backus (1970); Helbig (1994); Baerheim (1998); Bóna et al. (2007); Abramian et al. (2019). ...

The elastic map, or generalized Hooke’s Law, associates stress with strain in an elastic material. A symmetry of the elastic map is a reorientation of the material that does not change the map. We treat the topic of elastic symmetry conceptually and pictorially. The elastic map is assumed to be linear, and we study it using standard notions from linear algebra—not tensor algebra. We depict strain and stress using the “beachballs” familiar to seismologists. The elastic map, whose inputs and outputs are strains and stresses, is in turn depicted using beachballs. We are able to infer the symmetries for most elastic maps, sometimes just by inspection of their beachball depictions. Many of our results will be familiar, but our versions are simpler and more transparent than their counterparts in the literature.

Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

We formulate necessary and sufficient conditions for a unit vector \(\pmb{\nu }\) to generate a plane or axial symmetry of a constitutive tensor. For the elasticity tensor, these conditions consist of two polynomial equations of degree lower than four in the components of \(\pmb{\nu }\). Compared to Cowin–Mehrabadi conditions, this is an improvement, since these equations involve only the normal vector \(\pmb{\nu }\) to the plane symmetry (and no vector perpendicular to \(\pmb{\nu }\)). Similar reduced algebraic conditions are obtained for linear piezo-electricity and for totally symmetric tensors up to order 6.

We consider the natural \(\mathrm {SO}(3,\Bbbk )\) linear representation, \(\Bbbk =\mathbb {C}\) or \(\mathbb {R}\), on the \(\Bbbk\) vector space \({\mathsf {V}}=n\mathbb {S}^{2}(\mathbb {R}^3)\) of n second-order symmetric tensors, the associated invariant field \(\Bbbk ({\mathsf {V}})^{\mathrm {SO}(3,\Bbbk )}\) being known to be a purely transcendental extension in the complex case. We give an explicit tensorial form of a minimal generating set of the field of invariants, in both the complex and the real cases, showing that the invariant field is also a purely transcendental extension in the real case. The present results rely on some octahedral polynomial invariants obtained from Clebsch–Gordan projectors defined by a fourth-order octahedral covariant. Thanks to Cartan’s map, we also obtain a minimal set of generators for the \(\mathrm {SL}(2,\mathbb {C})\)-rational invariant field of n binary quartics.

Odd rank properties are, in Newnham’s words, null properties, meaning that they may vanish for certain point groups (like all centrosymmetric ones, see Sect. 3.6). As a result, not all materials will display third rank properties. Also as a consequence of being of odd rank, the RS will consist of overlapping positive and negative lobes, as shown in Fig. 7.1.

Plain Language Summary
While it has long been hypothesized that the main forces driving tectonic deformation result from the flow in the convecting mantle, there is new evidence suggesting that mantle vertical loading also plays an important role in crustal tectonics. Here, we illuminate the deformation field of the lower crust in different regions of the United States to search for evidence of mantle forcing of crustal flow. Our findings reveal that mantle gravitational loads can uplift or downwarp the crust‐mantle boundary and create lateral pressure gradients that drive the lower crustal away from upwellings and toward downwellings, causing thinning and thickening of the crust. This mechanism differs from conventional models in which crustal thickness variations are associated with topographic variations and suggests that mantle vertical loading might be key to maintaining isostatic equilibrium and a main driver of crustal deformation. Our results bring us closer to understanding how crustal motions respond to mantle‐derived forces and how these influence tectonic evolution processes.

In this paper, we propose a three‐dimensional meshing algorithm for discrete models combining the lattice approach with the polyhedral particle approach. Our aim is to be able to leverage readily available, well‐supported meshers to handle various geometries. We use them to generate a tetrahedral mesh, which our mesher then converts to a polyhedral mesh. The input mesh serves as geometrical support to generate the nodes of the discrete mesh. The desired shape is obtained with an assembly of convex polyhedral particles ‐ without having to clip them. We show that this approach enables meshing convex or concave geometries with sharp edges, curved features, and more. Several three‐dimensional geometries are presented to support this claim and illustrate the capabilities of the mesher. We provide a detailed analysis of its isotropy. The geometric isotropy is studied by analyzing the orientation of the generated beams. The mechanical isotropy is verified by assessing the properties of the elasticity tensor. Finally, we show that the new mesh retains its ability to be a good support for the generation of realistic cracking patterns.

The eastern sector of the Trans‐Mexican Volcanic Belt (TMVB) is an enigmatic narrow zone that lies just above where the Cocos plate displays a sharp transition in dipping angle in central Mexico. Current plate models indicate that the transition from flat to steeper subduction is continuous through this region, but the abrupt end of the TMVB suggests that the difference in subduction styles is more likely to be accommodated by a slab tear. Based on a high‐resolution shear wave velocity and radial anisotropy model of the region, we argue that a slab tear within South Cocos can explain the abrupt end of the TMVB. We also quantify the azimuthal anisotropy beneath each seismic station and present a well‐defined flow pattern that shows how mantle material is being displaced from beneath the slab to the mantle wedge through the tear in the subducted Cocos plate. We suggest that the toroidal mantle flow formed around the slab edges is responsible for the existence of the volcanic gap in central Mexico. Moreover, we propose that the temperature increase caused by the influx of hot, less‐dense mantle material flowing through the tear to the Veracruz area may have significant implications for the thermomechanical state of the subducted slab, and explain why the intermediate‐depth seismicity ends suddenly at the southern boundary of the Veracruz basin. The composite mantle flow formed by the movement of mantle material through the slab tears in western and southern Mexico may be allowing the Cocos plate to rollback in segments.

In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

Plane elastic waves traveling in an arbitrary direction in a crystal are, in general, neither transverse nor longitudinal and in fact show polarization angles up to 11° in nickel. A method for deducing the elastic constants of a cubic crystal is well known. The method is here extended to other crystal classes where the method may be used in order to deduce all the elastic constants of a crystal.

The qualitative effects of anisotropy on elastic waves propagating in a solid medium were well known to Lord Kelvin. Having been recognized, however, these effects were neglected as being of secondary importance in the dynamics of elastic mediums. This relegation of anisotropy to a secondary role in the dynamics of elastic mediums was undoubtedly justified, particularly in view of the relatively primitive state of experimental elasticity and seismology during Kelvin's time. It was not until after World War 2 that the effects of anisotropy again received serious attention. This was primarily because of the development of ultrasonic techniques for the measurement of dynamic elastic constants of pure crystals. In such experimental problems anisotropy no longer plays a secondary role. The study of how a disturbance, generated by a transducer on the surface of a crystal, spreads through the crystal led to the discovery, by Musgrave, that the wave surface, which forms the boundary of the spreading disturbance, could have cuspidal singularities. This had not been previously predicted, although it could have been predicted by Kelvin had he been more familiar with algebraic geometry. In another area of research (seismology), the post World War 2 years also saw a rise of interest in anisotropy, particularly in the effect of possible continental anisotropy on the propagation of Rayleigh waves. The increased experimental activity in crystal dynamics and the improvement of experimental seismology to the point where secondary effects became important resulted in a number of theoretical investigations into the propagation of plane, time harmonic waves in anisotropic mediums. By 1959 the state of the theoretical and experimental understanding of anisotropic elastic wave propagation had advanced to the point where rigorous wave theoretical calculations were in order. All the simple, solvable problems of isotropic dynamic elasticity, i.e. the initial value problem for an unbounded homogeneous medium, the mixed initial and boundary value problem for a surface line source on a half-space, the normal mode problem for an elastic wave guide, etc., can be formulated and solved in detail in the anisotropic case. Furthermore, the solutions can be obtained by extensions of the usual transform methods used in isotropic problems. The results can be physically interpreted by means of classical differential geometry. This review summarizes those anisotropic problems treated since 1959 and the techniques developed to solve them.

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

Standard and special seismic refraction profiles resulting in 1147 critically refracted mantle travel times were conducted across the Hawaiian arch to investigate the velocity characteristics of the upper mantle and the depth of the M discontinuity. The data are analyzed by a modified three-dimensional delay-time or time-term method in which the refractor velocity is assumed to possess anisotropy and lateral variations in velocity. The results show that the upper mantle immediately beneath the M discontinuity is anisotropic with respect to compressional waves. The maximum velocity occurs in an east-west direction and is 0.6 km/sec greater than the minimum, which occurs in a north-south direction. Lateral variations in the mean mantle velocity, which averages 8.16 km/sec for the entire area, are also indicated, although the evidence is not as definitive as for the anisotropy. The crustal thicknesses reflect the topographic arch, being relatively thin along the crest of the Hawaiian arch and thick in the Hawaiian deep. In the area on the arch directly north of the island of Hawaii, the crust is about 6.0 km thick, but it thins to about 5.3 km in both the southeast and northwest directions along the crest. The crust thickens beneath the southwestern flank of the arch and reaches thicknesses of 7.5–8.0 km in the Hawaiian deep.

The apparent body wave velocities vp of the oceanic mantle immediately below the Mohorovicic discontinuity (M), obtained from oceanic refraction shooting, can depend on azimuthbecause of real mantle anisotropy or horizontal variations in an isotropic crust and mantle. A small anisotropy of the uppermost mantle leads to 4 2 The Adetermine 5 of the 21 elastic parameters of the uppermost mantle. Curvature in the M discontinuity has the same effect on vp(), but the azimuth dependence found by Hess in Raitt's and Shor's velocities, if real, is too large to be due to curvature. A sloping M discon- tinuity adds to v() a term a cos -)- b sin 4. In a refraction survey of anisotropy, only two lines need be reversed to find a and b. If thedependence of vis due to a small mantle anisotropy and a small slope of the M discontinuity, v() is determined for allby shoot- ing five different lines through a point and reversing two of them or simply by shooting seven different lines. The theory is compared with Raitt's and Shor's data as reported by Hess. 1. Introduction. Hess (1964) has recently called attention to the possibility that anisot- ropy in the oceanic upper mantle is detectable in the seismic refraction work of Raiti (1963) near the Mendocino fracture zone and of Shot (1964) near Maui. The observation is that the speed of the P headwave just below the Moho- rovicic discontinuity (M) appears to depend on the azimuth of the line joining shot point and receiver. It is my purpose in the present note to in- dicate how an arbitrary small anisotropy in the upper mantle will affect the azimuth depend- ence of the measured speeds of P and S head- waves just below the M discontinuity. I hope this information may be useful in the design of experiments to study possible horizontal anisot- ropy in more detail. 2. Formulaitoh of ihe problem. In princi- ple, refraction shooting can give the azimuth dependence of the P and S wave velocities just below the M discontinuity. In practice, usually only the P wave velocity is measurable (Raitt, private communication, 1964), but it is of some interes.t to see what could be learned from the S wave velocities if they were available.

OLIVINE in rock-forming aggregates has a strong tendency during flow or deformation to develop a preferred orientation of its crystallographic axes. Turner1 describes a number of olivine fabrics developed under a variety of conditions. The pertinent examples, his figures numbers 7, 10 and 13, show fabrics for banded dunites in which the b crystallographic axis is strongly concentrated perpendicular to the banding, and figure 16, in which b is perpendicular to a plane of pronounced fissility. The (010) plane of olivine is the plane of the best cleavage and most probable glide plane for this mineral.

An approximation method for determining the elastic constants of cubic single crystals

- J R Neighbors
- C S Smith
- Neighbors