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Abstract

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.
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... Forte and Vianello's paper [2] gave a more precise definition. There are seven non-isotropic symmetry types for the elasticity tensor C [2][3][4][5][6][7] and seven distinct systems for crystals. ...
... where is the Kronecker symbol, which is isotropic for any orthogonal tensor . This decomposition contains two independent scalars, two second-order and one fourth-order deviator, which was widely used in the symmetry problem of the elasticity tensor [2,[4][5][6][7]. Meanwhile, the irreducible decomposition was used for the symmetry classification of the other physical tensors [9][10][11][12][13], and it was also applied in tensor analysis. ...
... Forte and Vianello [2] proved that the number of symmetry types of the elasticity tensor is eight for the first time. Their study greatly promoted a comprehensive understanding and resulted in a series of studies on the symmetry of elasticity tensor [3][4][5][6][7]. The relevant methods were also applied to the other fourth-order physical tensors like the photo-elasticity tensor [9] and the flexoelectric tensor [10]. ...
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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.
... where c i,j,k,l is the fourth-rank elasticity tensor of the lattice. The non-zero components of c i,j,k,l are derived considering the point group of the lattice [38], and are given in Appendix A for cubic and hexagonal lattices. ...
... Here we give results for two common classes of materials used in polar optics, namely cubic and hexagonal crystal structures. A full discussion of the symmetries of the elasticity tensor can be found in Ref. [38] and the unique components for a variety of symmetries can be found in Ref. ...
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Polar dielectrics are a promising platform for mid-infrared nanophotonics, allowing for nanoscale electromagnetic energy confinement in oscillations of the crystal lattice. We recently demonstrated that in nanoscopic polar systems a local description of the optical response fails, leading to erroneous predictions of modal frequencies and electromagnetic field enhancements. In this Paper we extend our previous work providing a scattering matrix theory of the nonlocal optical response of planar, anisotropic, layered polar dielectric heterostructures. The formalism we employ allows for the calculation of both reflection and transmission coefficients, and of the guided mode spectrum. We apply our theory to complex AlN/GaN superlattices, demonstrating the strong nonlocal tuneability of the optical response arising from hybridisation between photon and phonon modes. The numerical code underlying these calculations is provided in an online repository to serve as a tool for the design of phonon-based mid-infrared optoelectronic devices.
... where c i,j,k,l is the fourth-rank elasticity tensor of the lattice. The non-zero components of c i,j,k,l are derived considering the point group of the lattice [38], and are given in Appendix A for cubic and hexagonal lattices. ...
... Here we give results for two common classes of materials used in polar optics, namely cubic and hexagonal crystal structures. A full discussion of the symmetries of the elasticity tensor can be found in Ref. [38] and the unique components for a variety of symmetries can be found in Ref. [55]. ...
Preprint
Polar dielectrics are a promising platform for mid-infrared nanophotonics, allowing for nanoscale electromagnetic energy confinement in oscillations of the crystal lattice. We recently demonstrated that in nanoscopic polar systems a local description of the optical response fails, leading to erroneous predictions of modal frequencies and electromagnetic field enhancements. In this Paper we extend our previous work providing a scattering matrix theory of the nonlocal optical response of planar, anisotropic, layered polar dielectric heterostructures. The formalism we employ allows for the calculation of both reflection and transmission coefficients, and of the guided mode spectrum. We apply our theory to complex AlN/GaN superlattices, demonstrating the strong nonlocal tuneability of the optical response arising from hybridisation between photon and phonon modes. The numerical code underlying these calculations is provided in an online repository to serve as a tool for the design of phonon-based mid-infrared optoelectronic devices.
... 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. ...
... There are two ways to express the symmetries of the elasticity tensor: one is by the symmetry groups (Bona et al. [2004], Forte and Vianello [1996]), another is by the admitted sets of symmetry planes (Chadwick et al. [2001], Ting [2003]) Cowin and Mehrabadi [1995] showed that the operations associated with the symmetry groups of the elasticity tensor, including the center of symmetry, the n-fold rotation axis and the n-fold inversion axis is equivalent to a symmetry transformation of reflection. Therefore the classification of symmetry group can be developed from combinations of symmetry planes, which imply the equivalence of the two routes to analyse the symmetry of elasticity tensor. ...
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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.
... The hierarchy of the crystal symmetry systems is an important issue for a lot of subjects in elasticity, in particular, for the problem of averaging the elasticity tensor of a low-symmetry crystal by a higher symmetry prototype -generalized Fedorov problem [22], see also [27] for recent study. Different non-equivalent hierarchy diagrams often appear in elasticity and acoustic literature, see for instance [11], [22], [4]. To our knowledge, there is not yet a generally accepted agreement on this subject. ...
... This pass is forbidden in our approach because of the different structures of the R-matrices. Our diagram is different from the schemes given in [4], [], and [27] where the trigonal system is considered as a sub-family of the monoclinic one. ...
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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.
... This can be captured to a certain degree in linear elasticity by imposing symmetry on the elasticity tensor; however, the elasticity tensor is only capable of describing a rather limited number of symmetry classes. In fact, it has been shown that the elasticity tensor has exactly eight symmetry classes in three dimensions [3,4,6,11] and four symmetry classes in two dimensions [1,8]. In addition to being important for describing material response, imposing symmetry on the elasticity tensor is also necessary for dimension-reducing approximations such as plane strain, plane stress, and axisymmetry [10]. ...
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We present an elementary and self-contained proof that there are exactly four symmetry classes of the elasticity tensor in two dimensions: oblique, rectangular, square, and isotropic. In two dimensions, orthogonal transformations are either reflections or rotations. The proof is based on identification of constraints imposed by reflections and rotations on the elasticity tensor, and it simply employs elementary tools from trigonometry, making the proof accessible to a broad audience. For completeness, we identify the sets of transformations (rotations and reflections) for each symmetry class and report the corresponding equations of motions in classical linear elasticity.
... There are eight types of anisotropic symmetries in continuous elastic materials (e.g. Cowin & Mehrabadi, 1985;Mehrabadi & Cowin, 1990;Bona et al. 2004Bona et al. , 2007Bona et al. , 2008, including the isotropic and triclinic media. In this study, we consider polar anisotropic (TTI) model which is commonly used in seismic methods for approximating wave propagation in realistic structural sedimentary (e.g. ...
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In Part I of this study, we obtained the ray (group) velocity gradients and Hessians with respect to the ray locations, directions and the anisotropic model parameters, at nodal points along ray trajectories, considering general anisotropic (triclinic) media and both, quasi-compressional and quasi-shear waves. Ray velocity derivatives for anisotropic media with higher symmetries were considered particular cases of general anisotropy. In this part, Part II, we follow the computational workflow presented in Part I, formulating the ray velocity derivatives directly for polar anisotropic (transverse isotropy with tilted axis of symmetry, TTI) media for the coupled qP and qSV waves and for SH waves. The acoustic approximation for qP waves is considered a special case. The medium properties, normally specified at regular three-dimensional fine grid points, are the five material parameters: the axial compressional and shear velocities and the three Thomsen parameters, and two geometric parameters: the polar angles defining the local direction of the medium symmetry axis. All the parameters are assumed spatially (smoothly) varying, where their gradients and Hessians can be reliably computed. Two case examples are considered; the first represents compacted shale/sand rocks (with positive anellipticity) and the second, unconsolidated sand rocks with strong negative anellipticity (manifesting a qSV triplication). The ray velocity derivatives obtained in this part are first tested by comparing them with the corresponding numerical (finite difference) derivatives. Additionally, we show that exactly the same results (ray velocity derivatives) can be obtained if we transform the given polar anisotropic model parameters (five material and two geometric) into the twenty-one stiffness tensor components of a general anisotropic (triclinic) medium, and apply the theory derived in Part I.
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Polar dielectrics are a promising platform for midinfrared nanophotonics, allowing for nanoscale electromagnetic energy confinement in oscillations of the crystal lattice. We recently demonstrated that in nanoscopic polar systems a local description of the optical response fails, leading to erroneous predictions of modal frequencies and electromagnetic field enhancements. In this paper we extend our previous work providing a scattering matrix theory of the nonlocal optical response of planar, anisotropic, layered polar dielectric heterostructures. The formalism we employ allows for the calculation of both reflection and transmission coefficients, and of the guided mode spectrum. We apply our theory to complex AlN/GaN superlattices, demonstrating the strong nonlocal tunability of the optical response arising from hybridization between photon and phonon modes. The numerical code underlying these calculations is provided in an online repository to serve as a tool for the design of phonon-based midinfrared optoelectronic devices.
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In the literature, there is an ambiguity in defining the relationship between trigonal and cubic symmetry classes of an elasticity tensor. We discuss the issue by examining the eigensystems and symmetry groups of trigonal and cubic tensors. Additionally, we present numerical examples indicating that the sole verification of the eigenvalues can lead to confusion in the identification of the elastic symmetry.
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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.
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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.
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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.
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Backus (Rev. Geophys. Space Res. 8 (1970) 633) presents a theory on decomposition of the elasticity tensor and its application in several problems in anisotropy. The theory is supposed to be relatively difficult. In this article, an illustration of the theory by examples is presented. Special attention is paid to the problem of deciding which kind of symmetry a material has when the elastic constants are measured relative to an arbitrary coordinate system. A second-order symmetric tensor associated to the elasticity tensor can be used to verify if the coordinate axes are the symmetry axes of the medium, and determine a symmetry coordinate system. Also a comparison of Backus's theory with Cowins's decomposition (Q. Jl Mech. appl. Math. 42 (1989) 249) is presented. Uniqueness of the decompositions is specially diseussed. Backus's decomposition is expressed here by means of the Voigt tensor, the dilatational modulus tensor and the traces of those two. Some misprints in Backus's expressions are indicated.
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The Cowin–Mehrabadi theorem is generalized to allow less restrictive and more flexible conditions for locating a symmetry plane in an anisotropic elastic material. The generalized theorems are then employed to prove that the number of linear elastic symmetries is eight. The proof starts by imposing a symmetry plane to a triclinic material and, after new elastic symmetries are found, another symmetry plane is imposed. This process exhausts all possibility of elastic symmetries, and shows that there are only eight elastic symmetries. At each stage when a new symmetry plane is added, explicit results are obtained for the locations of the new symmetry plane that lead to a new elastic symmetry. It takes as few as three, and at most five, symmetry planes to reduce a triclinic material (which has no symmetry plane) to an isotropic material for which any plane is a symmetry plane.
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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.
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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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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.
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The problem considered here is that of identifying the type of elastic material symmetry of a material, given the values of the components of the fourth-rank elasticity tensor of the material relative to a known, but arbitrary, coordinate system. Four simple eigenvalue problems are posed for the determination of the normals to the planes of reflective material symmetry of the elastic material. The solution of the eigenvalue problems will determine the number and orientation of the normals to the planes of reflective material symmetry. This information is then used to determine the elastic material symmetry possessed by the material.
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It is well known that elastic tensors are classified according to higher symmetry as monoclinic, orthorhombic etc. A classification into these classes by means of bouquets of space directions, called Maxwell's multipole, is given, and explicit expressions for the magnitudes of the directions are developed. The analysis is based on harmonic decomposition of the hierarchically symmetric tensor presented in Backus ( Rev. Geophys. Space Phys . 8 (1970) 633-671), and further developed in Baerheim ( Q. Jl Mech. appl. Math . 46 (1993) 391-418). Hierarchically symmetric tensors are defined as fourth rank tensors in three dimensions, satisfying the symmetry conditions E ijkl = E jikl = E ijlk = E klij . Software is developed to calculate the bouquets of space directions, and MATHEMATICA is used for displaying the results. As an example, multipoles are calculated for a specific tensor of monocline symmetry.