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# The elastic tensor of given symmetry nearest to an anisotropic elastic tensor

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... Given a generally anisotropic tensor c, we wish to examine its relations to the seven nontrivial symmetry classes. Such relations can be understood in terms of the concept of distance to a given symmetry class, which was proposed by Gazis et al. (10), and allow us to infer information about properties of the material represented by c, such as its layering or fractures. Following several papers, notably (11)(12)(13)(14), we consider the Frobenius norm to find the closest tensors that belong to particular symmetry classes. ...
... where the integration is over g ∈ G sym , with respect to the invariant measure, μ, normalized so that μ(G sym ) = 1, as described by Gazis et al. (10). Integral (16) reduces to a finite sum for the classes whose symmetry groups are finite, which are all classes except isotropy and transverse isotropy. ...
... We note that the global minimum of distance is also attained with other rotations, as expected from the normalizer of the group G ortho ; see expression (10). They merely yield certain permutations of the rows and columns of matrix (25). ...
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We consider the problem of representing a generally anisotropic elasticity tensor, which might be obtained from physical measurements, by a tensor belonging to a chosen material symmetry class, so-called ‘effective tensor'. Following previous works on the subject, we define this effective tensor as the solution of a global optimization problem for the Frobenius distance function. For all nontrivial symmetry classes, except isotropy, this problem is nonlinear, since it involves all orientations of the symmetry groups. We solve the problem using a metaheuristic method called particle-swarm optimization and employ quaternions to parametrize rotations in 3-space to improve computational efficiency. One advantage of this approach over previously used plot-guided local methods and exhaustive grid searches is that it allows us to solve a large number of instances of the problem in a reasonable time. As an application, we can use Monte-Carlo method to analyze the uncertainty of the orientation and elasticity parameters of the effective tensor resulting from the uncertainty of the given tensor, which may be caused, for example, by measurement errors.
... The purpose of this paper is to examine a generally anisotropic elasticity tensor, expressed in terms of twenty-one elasticity parameters and obtained from vertical seismic profiling (VSP) measurements, to infer properties of materials represented by this tensor. Beginning with a generally anisotropic tensor, we are able to infer these properties by relating this tensor to its closest counterparts in the sense of Frobenius norm, as defined by Gazis et al. (1963), in all material symmetries of Hookean solids, as shown by Danek et al. (2013). Herein, we focus our attention on examining symmetries used in seismology: monoclinic, orthotropic, transversely isotropic, and isotropic tensors. ...
... To infer information about materials examined through VSP measurements, we consider relationships between the obtained tensor and its symmetric counterparts. Such a tensor was obtained by Dewangan and Grechka (2003) from multi-component and multi-azimuth walkaway VSP data, and such relationships are considered in terms of distance between tensors, as proposed by Gazis et al. (1963). The concept of such a distance is discussed by several researchers, including Norris (2006), Bόna (2009), and Kochetov and Slawinski (2009a. ...
... For a fixed coordinate system, we can relate a general elasticity tensor, c, to its counterpart, c sym , which belongs to a particular symmetry class. Tensor c sym is the orthogonal projection of c, in the sense of the Frobenius inner product (which is the sum of products of the corresponding components, a ijkl b ijkl ), onto the linear space containing all tensors of that symmetry class, as described by Gazis et al. (1963). ...
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A b s t r a c t A generally anisotropic elasticity tensor can be related to its closest counterparts in various symmetry classes. We refer to these counterparts as effective tensors in these classes. In finding effective tensors, we do not assume a priori orientations of their symmetry planes and axes. Knowledge of orientations of Hookean solids allows us to infer properties of materials represented by these solids. Obtaining orientations and parameter values of effective tensors is a highly nonlinear process involving finding absolute minima for orthogonal projections under all three-dimensional rotations. Given the standard deviations of the components of a generally anisotropic tensor, we examine the influence of measurement errors on the properties of effective tensors. We use a global optimization method to generate thousands of realizations of a generally anisotropic tensor, subject to errors. Using this optimization, we perform a Monte Carlo analysis of distances between that tensor and its counterparts in different symmetry classes, as well as of their orientations and elasticity parameters.
... Six among these classes form a partial ordering between general anisotropy and isotropy; the former is described by twenty-one parameters, the latter by two. The concept of distance of a given elasticity tensor to a material-symmetry class was introduced by Gazis et al. [13] using orthogonal projections of elasticity tensors on the space of a particular class, and by Fedorov [11], who minimized the mean-square difference of the slowness surface. ...
... In general, since Hookean solids are mathematical analogies for physical materials, we might choose to represent a material by a solid exhibiting a particular symmetry. Several researchers-among them, Gazis et al. [13], Moakher and Norris [18], Kochetov and Slawinski [16,17], Danek et al. [9]-examined relations between a generally anisotropic Hookean solid and its symmetric counterparts, which invoke the concept of distance within the space of Hookean solids. Voigt [23] and Norris [19] examined, in particular, relations between a generally anisotropic solid and its most symmetric counterpart: the closest isotropic solid. ...
... First, consider m = 0, which means that μ j (α) ≡ d j and μ k (α) ≡ d k . Whether or not this is possible depends on the multiplicity of the eigenvalues of D, which, as shown in Eq. (13), are also the eigenvalues of C. Hence, for a fixed i, 1 ≤ i ≤ n, let ...
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In the physical realm, an elasticity tensor that is computed based on measured numerical quantities with resulting numerical errors does not belong to any symmetry class for two reasons: (1) the presence of errors, and more intrinsically, (2) the fact that the symmetry classes in question are properties of Hookean solids, which are mathematical objects, not measured physical materials. To consider a good symmetric model for the mechanical properties of such a material, it is useful to compute the distance between the measured tensor and the symmetry class in question. One must then of course decide on what norm to use to measure this distance. The simplest case is that of the isotropic symmetry class. Typically, in this case, it has been common to use the Frobenius norm, as there is then an analytic expression for the closest element and it is unique. However, for other symmetry classes this is no longer the case: there are no analytic formulas and the closest element is not known to be unique. Also, the Frobenius norm treats an n×n matrix as an n 2-vector and makes no use of the matrices, or tensors, as linear operators; hence, it loses potentially important geometric information. In this paper, we investigate the use of an operator norm of the tensor, which turns out to be the operator Euclidean norm of the 6×6 matrix representation of the tensor, in the expectation that it is more closely connected to the underlying geometry. We characterize the isotropic tensors that are closest to a given anisotropic tensor, and show that in certain circumstances they may not be unique. Although this may be a computational disadvantage in comparison to the use of the Frobenius norm—which has analytic expressions—we suggest that, since we work with only 6×6 matrices, there is no need to be extremely efficient and, hence, geometrical fidelity must trump computational considerations.
... In practice, appealing to Curie principle ("the symmetries of the causes are to be found in the effects"), their constitutive tensors shall inherit the material symmetry (orthotropy, cubic or monoclinic symmetry for example), so that the natural question is to determine the constitutive tensor with a given material symmetry the nearest to a given measured (triclinic) constitutive tensor. This question has been extensively studied, from both the theoretical and numerical points of view, since the pioneering work of Gazis, Tadjbakhsh and Toupin [30], and subsequent works in the 90s [8,7,26,28]. Most works focus on the elasticity tensor [30,24,26,27,33,28,46,39,23], a few ones on the piezoelectricity tensor [63]. ...
... This question has been extensively studied, from both the theoretical and numerical points of view, since the pioneering work of Gazis, Tadjbakhsh and Toupin [30], and subsequent works in the 90s [8,7,26,28]. Most works focus on the elasticity tensor [30,24,26,27,33,28,46,39,23], a few ones on the piezoelectricity tensor [63]. So far, we are not aware of some similar studies for the Hill plasticity tensor or the combination of several constitutive tensors. ...
... To solve the problem of the distance of a pair (E 0 , F 0 ) to cubic symmetry, we therefore have to find the critical points of the polynomial function (30) F (H, k, λ λ λ) := H 0 − H 2 + W K 0 − kH 2 + λ λ λ : g, with H ∈ H 4 an harmonic fourth-order tensor, k a scalar, and where the Lagrange multiplier λ λ λ ∈ H 2 is an deviatoric second-order tensor. Observe that the first-order Euler-Lagrange equations for this optimization problem can furthermore be recast in a similar form as (22). ...
Preprint
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).
... We show that the Backus (1962) equivalent-medium average, which is an average over a spatial variable, and the Gazis et al. (1963) effective-medium average, which is an average over a symmetry group, do not commute, in general. They commute in special cases, which we exemplify. ...
... These properties define w(x 3 ) as a probability-density function with mean 0 and standard deviation , explaining the use of the term "width" for . Gazis et al. (1963) average allows us to obtain the closest symmetric counterpart-in the Frobenius sense-of a chosen material symmetry to a generally anisotropic Hookean solid. The average is a Hookean solid, to which we refer as effective, whose elasticity parameters correspond to the symmetry chosen a priori. ...
... Integral (2) reduces to a finite sum for the classes whose symmetry groups are finite, which are all classes except isotropy and transverse isotropy. The Gazis et al. (1963) approach is reviewed and extended by Danek et al. (2013Danek et al. ( , 2015 in the context of random errors. Therein, elasticity tensors are not constrained to the same-or even different but known-orientation of the coordinate system. ...
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We show that the Backus (1962) equivalent-medium average, which is an average over a spatial variable, and the Gazis et al. (1963) effective-medium average, which is an average over a symmetry group, do not commute, in general. They commute in special cases, which we exemplify.
... where T is traction and n is the unit normal to the interface. No such equality is imposed on the other three components of this symmetric tensor; σ 11 , σ 12 and σ 22 can vary wildly along the x 3 -axis due to changes of elastic properties from layer to layer. Furthermore, regarding the strain tensor, we invoke the kinematic boundary conditions that require no slippage or separation between layers; in other words, the corresponding components of the displacement vector, u 1 , u 2 and u 3 , must be equal to one another across the interface (e.g., Slawinski [25], pp. ...
... tensor can be written as The other equivalent-medium elasticity parameters are zero. Thus, we have nine linearly independent parameters in the form of matrix (11). Hence, the equivalent medium exhibits the same symmetry as the individual layers. ...
... Another numerical study could examine whether the equivalent medium for a stack of strongly anisotropic layers, whose anisotropic properties are randomly different from each other, is weakly anisotropic. If so, we might seek-using the method proposed by Gazis et al. [11] and elaborated by Danek et al. [9]-an elasticity tensor of a higher symmetry that is nearest to that medium. For such a study, Kelvin's notation-used in this paper-is preferable, even though one could accommodate rotations in Voigt's notation by using the Bond [4] transformation (e.g., Slawinski [25], Sect. ...
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In this paper, following the Backus (1962) approach, we examine expressions for elasticity parameters of a homogeneous generally anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic layers. These expressions reduce to the results of Backus (1962) for the case of isotropic and transversely isotropic layers. In over half-a-century since the publications of Backus (1962) there have been numerous publications applying and extending that formulation. However, neither George Backus nor the authors of the present paper are aware of further examinations of mathematical underpinnings of the original formulation; hence, this paper. We prove that---within the long-wave approximation---if the thin layers obey stability conditions then so does the equivalent medium. We examine---within the Backus-average context---the approximation of the average of a product as the product of averages, and express it as a proposition in terms of an upper bound. In the presented examination we use the expression of Hooke's law as a tensor equation; in other words, we use Kelvin's---as opposed to Voigt's---notation. In general, the tensorial notation allows us to examine conveniently effects due to rotations of coordinate systems.
... show that, in general, the Backus (1962) average and the Gazis et al. (1963) average do not commute. Herein, we examine the extent of this noncommutativity. ...
... 1 Introduction Dalton and Slawinski (2016) show that-in general-the Backus (1962) average, which is an average over a spatial variable, and the Gazis et al. (1963) average, which is an average over a symmetry group, do not commute. These averages result in the so-called equivalent and effective media, respectively. ...
... Herein, B and G stand for the Backus (1962) average and the Gazis et al. (1963) Given monoclinic tensors, c ijk , in the upper left-hand corner of Diagram 1 and following the clockwise path, we have , according to Dalton and Slawinski (2016) and Bos et al. (2016), ...
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Dalton and Slawinski (2016) show that, in general, the Backus (1962) average and the Gazis et al. (1963) average do not commute. Herein, we examine the extent of this noncommutativity. We illustrate numerically that the extent of noncommutativity is a function of the strength of anisotropy. The averages nearly commute in the case of weak anisotropy.
... These properties define w(x 3 ) as a probability-density function, whose mean is zero and whose standard deviation is , thus explaining the use of the term "width" for . The Gazis et al. [8] average, which is an average over an anisotropic symmetry group, allows us to obtain the closest symmetric counterpart-in the Frobenius sense-of a chosen material symmetry to a generally anisotropic Hookean solid. The average is a Hookean solid, to which we refer as effective, and whose elasticity parameters correspond to a symmetry chosen a priori. ...
... The Gazis et al. [8] average is a projection given by ...
... Integral (2) reduces to a finite sum for the classes whose symmetry groups are finite, which are all classes in Fig. 1, except isotropy and transverse isotropy. The Gazis et al. [8] approach is reviewed and extended by Danek et al. [6,7] in the context of random errors. Therein, elasticity tensors are not constrained to the same-or even different but known-orientation of the coordinate system. ...
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We show that, in general, the translational average over a spatial variable---discussed by Backus \cite{backus}, and referred to as the equivalent-medium average---and the rotational average over a symmetry group at a point---discussed by Gazis et al. \cite{gazis}, and referred to as the effective-medium average---do not commute. However, they do commute in special cases of a given symmetry class, which correspond to particular relations among the elasticity parameters. We also examine the extent of this noncommutativity, and show that it is a function of the strength of anisotropy. Surprisingly, a perturbation of the elasticity parameters about a point of weak anisotropy results in the commutator of the two types of averaging being of the order of the {\it square} of this perturbation. Thus, these averages nearly commute in the case of weak anisotropy, which is of interest in such disciplines as quantitative seismology.
... One can distinguish a group of techniques, which are based on the separation of an additive part of the elasticity tensor (e.g., by way projecting), possessing one or another type of symmetry [34][35][36][37]. In this case, the identification problem is reduced to determination of the symmetric part, which is the closest in some metric to a given tensor, as e.g., in [38][39][40][41][42][43][44][45]. ...
... As shown in [35], projecting a tensor onto a subspace, which is invariant to some orthogonal subgroup, is equivalent to averaging [68] this tensor over the same subgroup. Thus, adhering to the symmetry classification of elasticity tensors, one can write: ...
... Such condition is desirable for approximating tensors as well. In [35], it is shown that positive definiteness of a 4-rank tensor holds during its averaging over the orthogonal subgroup. Due to equivalency of this operation to the "  "orthogonal projection, one can conclude that in the symmetry classes the ⋅ 4 3  -optimal approximations of a positive definite tensor are also positive definite. ...
Article
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The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of "elasticity class" the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.
... 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 result is in a correspondence with the diagrams given in [11] and in [22]. Notice that in [22], but not in [11], there is an additional inclusion of the cubic system into the non-reduced trigonal system. ...
... This result is in a correspondence with the diagrams given in [11] and in [22]. Notice that in [22], but not in [11], there is an additional inclusion of the cubic system into the non-reduced trigonal system. This pass is forbidden in our approach because of the different structures of the R-matrices. ...
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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 $6\times 6$ 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.
... 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 result is in a correspondence with the diagrams given in [11] and in [22]. Notice that in [22], but not in [11], there is an additional inclusion of the cubic system into the non-reduced trigonal system. ...
... This result is in a correspondence with the diagrams given in [11] and in [22]. Notice that in [22], but not in [11], there is an additional inclusion of the cubic system into the non-reduced trigonal system. This pass is forbidden in our approach because of the different structures of the R-matrices. ...
Preprint
Full-text available
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.
... where the integration is over the symmetry group, G sym , whose elements are g, with respect to the invariant measure, µ , normalized so that µ(G sym ) = 1 , as described by Gazis et al. [7]; q denotes the dependence of the value of integral (7) on the relative orientations of c andĉ . This integral reduces to a finite sum for the classes whose symmetry groups are finite, which are all classes except isotropy and transverse isotropy. ...
... This integral reduces to a finite sum for the classes whose symmetry groups are finite, which are all classes except isotropy and transverse isotropy. As shown by Gazis et al. [7], projection (7) ensures that a positive-definite tensor is projected to another positive-definite tensor, as required by Hookean solids. As shown by Moakher and Norris [9] and Bucataru and Slawinski [3]-in a fixed orientation of the rotation-symmetry axis that coincides with the x 3 -axis of the coordinate system-the Frobenius-norm effective transversely isotropic tensor derived from expression (7) has the form of tensor (3) with components given bŷ ...
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A generally anisotropic elasticity tensor, which might be obtained from physical measurements, can be approximated by a tensor belonging to a particular material-symmetry class; we refer to such a tensor as the effective tensor. The effective tensor is the closest to the generally anisotropic tensor among the tensors of that symmetry class. The concept of closeness is formalized in the notion of norm. Herein, we compare the effective tensors belonging to the transversely isotropic class and obtained using two different norms: the Frobenius norm and the L2 operator norm. We compare distributions of the effective elasticity parameters and symmetry-axis orientations for both the error-free case and the case of the generally anisotropic tensor subject to errors. © 2009 - 2014. Universita degli Studi di Padova - Padova University Press - All Rights Reserved.
... We show that, in this situation, the arithmetic average of the rotated tensors results in the so-called Gazis et al. [2] average, which is the isotropic tensor closest to C, according to the Frobenius norm, as discussed in Sect. 2. Thus, the Gazis et al. average is the closest-in the Frobenius sense-isotropic counterpart of C. In contrast, as we show in Theorem 5, the Backus average of the same randomly rotated tensors is, in general, only transversely isotropic, and thus provides a transversely isotropic counterpart for C. Now for some details. Each layer is expressed by Hooke's law, ...
... In such a context, Gazis et al. [2] prove the following theorem. ...
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As shown by Backus (J Geophys Res 67(11):4427–4440, 1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of randomly oriented anisotropic elasticity tensors, which—one might reasonably expect—would result in an isotropic medium. However, we show—by means of a fundamental symmetry of the Backus average—that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, an analogy between the Backus and Gazis et al. (Acta Crystallogr 16(9):917–922, 1963) averages.
... Böhlke (2001) ; Fedorov (1968) use the Frobenius norm to measure the distance between two stiffness tensors as we will do later on. It has been shown by Gazis et al. (1963) that the average group action gives a projection of a stiffness tetrad onto the subset of all stiffness tetrads with a given symmetry. In Glüge et al. (2012) , in the course of determining the anisotropy induced by a representative volume element (RVE) of cubic shape, a similar projection method is used to quantify the anisotropy of a stiffness tetrad and also the distance to the cubic symmetry class. ...
... In conjunction, the orthogonality and the uniqueness due to the convexity of the subset guarantee that the associated distance between K and 8 P K is minimal. This has also been found by Gazis et al. (1963) , Sec. 2. ...
Article
In the scope of linear anisotropic elasticity, the fourth-order elasticity tensor or tetrad has to be identified. This can be done either by measurements or by numerical simulations. An important task is then to identify a given tetrad, probably with some experimental or numerical scattering, with one of the symmetry classes. For this purpose one needs a distance function between a given tetrad and the class of all tetrads with a particular symmetry, which is zero if the tetrad obeys this symmetry, or non-zero otherwise. In this paper we present a fast method to solve these problems. We firstly introduce the 8th order projectors that map any stiffness tetrad into the part that is invariant under the action of a specific fixed symmetry group. For this purpose we consider the seven out of the eight symmetry classes that are distinguishable in linear elasticity. Secondly, since the symmetry axes of the specific stiffness tensor under consideration is generally not aligned with the used tensorial basis of the projector, we need to rotate the sample stiffness. The optimal orientation is obtained when the distance between the rotated stiffness and the rotated and projected stiffness is minimal. Thus, we need to apply only linear mappings and minimize over three Euler angles. The latter is quite simple, as the domain of the Euler angles is periodic, and the number of local minima is limited. This procedure has the advantage that it is applicable in an algorithmic manner, and does not require an a priori identification of symmetry planes, symmetry axes or component symmetries, which are only apparent under special choices for the tensorial basis.
... The determination of the symmetry class of a stiffness as well as other closely related topics have been discussed for a long time. In Gazis, Tadjbakhsh, and Toupin (1963) the closest cubic stiffness is calculated by projecting a stiffness onto a cubic base which has some degrees of freedom. These degrees of freedom are determined by minimizing the residual. ...
... It is introduced as an average of the orbit of a stiffness in a symmetry group. Earlier Gazis et al. (1963) introduced a similar calculation procedure using an integral average. Equation (25) is applicable for the triclinic, monoclinic, orthotropic, tetragonal, cubic, trigonal, and hexagonal symmetry classes. ...
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A completely algebraic algorithm is given to determin the distance of an elastic stiffness tensor to any of the symmetry classes.
... Underlying this question is the accuracy of the generally anisotropic tensor, and, hence, its effect on the reliability of information provided by the chosen model. Several researchers-among them, Voigt [25], Gazis et al. [15], Moakher and Norris [20], Norris [21], Bucataru and Slawinski [6], Kochetov and Slawinski [18,19], Diner et al. [11,12], Bos and Slawinski [5]-examined relations between a given generally anisotropic Hookean solid and its symmetric counterpart by invoking the concept of a norm in the space of elasticity tensors. In this approach, the term elasticity tensor is understood in a broader sense: the requirement of positive definiteness is omitted so the possible values of c form a vector space. ...
... Another nice mathematical property enjoyed by the Frobenius norm is the fact that the positive definiteness of a generally anisotropic elasticity tensor is inherited by its effective counterpart (Gazis et al. [15]). This is not necessarily so for the weighted Frobenius norm, as illustrated by Sergey Sadov [24, pers. ...
Article
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We introduce the effective elasticity tensor of a chosen material-symmetry class to represent a measured generally anisotropic elasticity tensor by minimizing the weighted Frobenius distance from the given tensor to its symmetric counterpart, where the weights are determined by the experimental errors. The resulting effective tensor is the highest-likelihood estimate within the specified symmetry class. Given two material-symmetry classes, with one included in the other, the weighted Frobenius distance from the given tensor to the two effective tensors can be used to decide between the two models—one with higher and one with lower symmetry—by means of the likelihood ratio test.
... The determination of the symmetry class of a stiffness as well as other closely related topics have been discussed for a long time. In [10] the closest cubic stiffness is calculated by projecting a stiffness onto a cubic base which has some degrees of freedom. These degrees of freedom are determined by minimizing the residual. ...
... It is introduced as an average of the orbit of a stiffness in a symmetry group. Earlier GAZIS et al. [10] introduced a similar calculation procedure using an integral average. Equation (4.1) is applicable for the triclinic, monoclinic, orthotropic, tetragonal, cubic, trigonal, and hexagonal symmetry classes. ...
Article
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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.
... Unfortunately, they are a priori useless in the common case of a triclinic/biclinic measured elasticity tensor. When experimental discrepancy has to be dealt with, the literature approaches are based on the concept of distance of a tensor to a considered symmetry class [17,14,16,15,21,19,11], starting from a given (usually measured) elasticity tensor C raw with no material symmetry, and sometimes from the additional quantification of the measurement errors [7,10,18]. ...
... a problem which has already been extensively studied [17,14,16,26,15]. Here, we choose to work with the Frobenius norm (see [21] for other norms) C = C, C , derived from the O(2)-invariant scalar product ...
Preprint
Constitutive tensors are of common use in mechanics of materials. To determine the relevant symmetry class of an experimental tensor is still a tedious problem. For instance, it requires numerical methods in three-dimensional elasticity. We address here the more affordable case of plane (bi-dimensional) elasticity, which has not been fully solved yet. We recall first Vianello's orthogonal projection method, valid for both the isotropic and the square symmetric (tetragonal) symmetry classes. We then solve in a closed-form the problem of the distance to plane elasticity orthotropy, thanks to the Euler-Lagrange method.
... The effective stiffness tensor is computed column-wise by prescribing six orthogonal macroscopic strains [112]. The isotropic approximation of the stiffness is computed by minimizing the Frobenius norm to the given stiffness tensor [113]. Upon increasing the QTT ranks, the Young's modulus increases as well. ...
Article
Phase-field models permit accounting for the underlying physics of the microstructure evolution process when simulating the emergent microstructure of a variety of multiscale materials. As an inherent characteristic, the individual phases (or states) of the material are not known in advance and only emerge upon evolution. To account for this matter, structured grids are typically used when solving phase-field models. In particular for large three-dimensional grids and long time scales, the used memory and computational time can be considerable, instigating research on alternative approaches. The work at hand studies the Tensor Train (TT) format as a possible remedy for phase-field simulations. More precisely, the TT format is a specific tensor format which permits – small so-called TT rank provided – to store huge arrays of data and comes with specific algorithms replacing the linear algebra operations familiar for fully stored arrays. We investigate the Cahn-Hilliard equation and use a semi-implicit discretization in time which is based on solving a single linear system per time step. We compare this strategy with a classic implicit Euler discretization using a Newton scheme as a nonlinear solver. Upon a finite-difference discretization in space, the resulting equations are recast in the Quantics Tensor Train (QTT) format and solved by dedicated linear solvers. We study the performance of the algorithms and investigate the effective properties of the resulting bicontinuous composites via computational homogenization. We show that the Cahn-Hilliard equation can be solved using the TT format and that runtimes scale well with the grid size.
... Un tenseur d'élasticité est isotrope si et seulement s'il est égal à sa partie isotrope. L'étude de la norme euclidienne du résidu isotrope ||C − C iso || permet, donc, de quantier la distance à la symétrie isotrope (Gazis et al., 1963;Moakher et Norris, 2006;Oliver-Leblond et al., 2021). La distance relative à l'isotropie mécanique d'un tenseur d'élasticité peut être dénie comme suit : ...
Thesis
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.
... where the integration is over the symmetry group G sym , whose elements are g, with respect to the invariant measure , normalized so that ͑G sym ͒ ‫ס‬ 1, as described by Gazis et al. ͑1963͒. Integral 3 reduces to a finite sum for the classes whose symmetry groups are finite; in the context of Hookean solids, this is true for all classes except isotropy and transverse isotropy. ...
Article
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We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular sym-metry that corresponds to measurable traveltime and polar-ization quantities. These quantities — the wavefront-slow-ness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization prob-lem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of partic-ular symmetry without assuming its orientation is challeng-ing. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we dis-tinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.
... We refer to [1,2] for a historical overview. In particular, we adopt similar mathematical techniques and in this respect we have found the following papers particularly illuminating [1,2,3,4,5,6]. ...
Article
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The orientational order of nematic liquid crystals is traditionally studied by means of the second-rank ordering tensor $\mathbb{S}$. When this is calculated through experiments or simulations, the symmetry group of the phase is not known \emph{a-priori}, but need to be deduced from the numerical realisation of $\mathbb{S}$, which is affected by numerical errors. There is no generally accepted procedure to perform this analysis. Here, we provide a new algorithm suited to identifying the symmetry group of the phase. As a by product, we prove that there are only five phase-symmetry classes of the second-rank ordering tensor and give a canonical representation of $\mathbb{S}$ for each class. The nearest tensor of the assigned symmetry is determined by group-projection. In order to test our procedure, we generate uniaxial and biaxial phases in a system of interacting particles, endowed with $D_{\infty h}$ or $D_{2h}$, which mimic the outcome of Monte-Carlo simulations. The actual symmetry of the phases is correctly identified, along with the optimal choice of laboratory frame.
... Accordingly, only C 22 and C 33 components of the anisotropic constitutive matrices are used to determine the equivalent isotropic constitutive parameters, E and ν. Other more general approaches exist in the literature to determine the closest isotropic matrix by minimizing an appropriately defined distance from the anisotropic matrix [34,75,81]. However, both the approach used here as well as the general case will lead to the same inference-that the distance of separation (between the anisotropic matrix and a uniquely defined isotropic matrix) is significantly higher for the damaged microstructure as opposed to the healthy case-thus indicating a departure from an isotropic description. ...
Article
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A probabilistic multiscale based computational scheme is developed to predict locations of microcracks and to estimate the associated macroscopic constitutive material properties for structural systems. The proposed scheme only requires a single realization of the macroscale response field (e.g., strain field, strain energy density field) that is often typically available in practice. Here, the macroscale is associated with structural systems of size of the order of 10--100 $m$, while microcracks are used to refer to micron-scale cracks of size 10--100 $\mu m$ (depending on materials) that have the potential to cause catastrophic failures when they coalesce and form larger cracks. The analysis of such microcracks, before a macrocrack visibly appears (say, at the size of a few $cms$), is beyond the scope of the classical fracture mechanics. The present work addresses this issue in a certain sense by incorporating the effects of microcracks into macroscopic constitutive material properties within a probabil...
... The explicit calculation of the symmetry class of a given elasticity tensor has been an active subject of research in mechanics of deformable solids. Besides, the problem becomes even more complicated if one considers that, in real life, a measured elasticity tensor (assuming that one can access to all of its components) is subject to experimental errors and has therefore no symmetry but is nevertheless close to a given theoretical tensor with a given symmetry [41,58]. Concerning this, it is worth to cite the excellent work of François and coauthors [39,40] who performed a deep experimental and numerical study of the problem using acoustic measurements on polyhedral testing samples of raw materials. ...
Article
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.
... In order to find the closest isotropic elasticity tensor to the orthotropic tensor given in Table 4, several different measures can be applied (e.g. Gazis et al., 1963;Bos and Slawinski, 2015). We perform a least squares inversion which leads to the following analytical solutions for the equivalent 1-D isotropic l and l c values Fig. 7. Obtained optic mode periods (1=xr ) in bridgmanite in lower mantle conditions as a function of characteristic length Lc. ...
Article
We provide a method to compute the Cosserat couple modulus for a bridgmanite (MgSiO3 silicate perovskite) solid from frequency gaps observed in Raman experiments. To this aim, we apply micropolar theory which is a generalization of the classical linear elastic theory, where each particle has an intrinsic rotational degree of freedom, called micro-rotation and/or spin, and which depends on the so-called Cosserat couple modulus that characterizes the micropolar medium. We investigate both wave propagation and dispersion. The wave propagation simulations in both potassium nitrate and bridgmanite crystal leads to a faster elastic wave propagation as well as to an independent rotational field of motion, called optic mode, which is smaller in amplitude compared to the conventional rotational field. The dispersion analysis predicts that the optic mode only appears above a cutoff frequency, , which has been observed in Raman experiments done at high pressures and temperatures on bridgmanite crystal. The comparison of the cutoff frequency observed in experiments and the micropolar theory enables us to compute for the first time the temperature and pressure dependency of the Cosserat couple modulus of bridgmanite. This study thus shows that the micropolar theory can explain particle motions observed in laboratory experiments that were before neglected and that can now be used to constrain the micropolar elastic constants of Earth’s mantle like material. This pioneer work aims at encouraging the use of micropolar theory in future works on deep Earth’s mantle material by providing Cosserat couple modulus that were not available before.
... It is worthy to remark that such projection preserves positive definiteness [5]. A canonical basis for a tensor from a symmetry class is not unique. ...
Article
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An approach to identify the elastic symmetry of anisotropic materials is considered and applied together with the multi-level constitutive models of polycrystals. By using the two-level model of elasto-visco-plasticity, the changes in the symmetry of the elasticity tensor for the representative volume element of a polycrystalline copper during inelastic deformation are studied. The two-link strain trajectories of the "simple shear – simple shear" type are analyzed and the problem of the compatibility of the properties under investigation with any symmetry classes is formulated and solved at various deformation stages.
... To examine the effects of anisotropy, we study dispersion curves for the closest isotropic counterpart, as formulated by Voigt [2]; this formulation is an isotropic case of the Gazis et al. [10] average. The two elasticity parameters of the isotropic counterpart of a Backus medium are (Slawinski [9, equations (4.77) and ( (16) henceforth, this result is referred to as the Voigt medium. ...
Article
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We examine the Backus average of a stack of isotropic layers overlying an isotropic halfspace to examine its applicability for the quasi-Rayleigh and Love wave dispersion curves, both of which apply to the same model. We compare these curves to values obtained for the stack of discrete layers using the propagator matrix. The Backus average is applicable only for thin layers or low frequencies. This is true for both weakly inhomogeneous layers resulting in a weakly anisotropic medium and strongly inhomogeneous alternating layers resulting in a strongly anisotropic medium. We also compare the strongly anisotropic and weakly anisotropic media, given by the Backus averages, to results obtained by the isotropic Voigt averages of these media. As expected, we find only a small difference between these results for weak anisotropy and a large difference for strong anisotropy. We perform the Backus average for a stack of alternating transversely isotropic layers that is strongly inhomogeneous to evaluate the dispersion curves for the resulting medium. We compare these curves to values obtained using a propagator matrix for that stack of discrete layers. Again, there is a good match only for thin layers or low frequencies. Finally, we perform the Backus average for a stack of nonalternating transversely isotropic layers that is strongly inhomogeneous, and evaluate the quasi-Rayleigh wave dispersion curves for the resulting transversely isotropic medium. We compare these curves to values obtained using the propagator matrix for the stack of discrete layers. In this case, the Backus average performs less well, but---for the fundamental mode---remains adequate for low frequencies or thin layers.
... In linear elasticity, which involves a fourth-order tensor E, the distance to an isotropy stratum has been formulated as the minimization problem [27,25,18,54,12] min ...
Preprint
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.
... According to Backus (1962), a medium obtained by Backus averaging is positive definite if the layers, prior to averaging, are also positive definite. Also, according to Gazis et al. (1963), a Frobenius-norm counterpart of a positive-definite tensor is positive definite. Thus, it suffices to ensure condition (6) for each layer. ...
Article
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In this paper, we discuss five parameters that indicate the inhomogeneity of a stack of parallel isotropic layers. We show that, in certain situations, they provide further insight into the intrinsic inhomogeneity of a Backus medium, as compared to the Thomsen parameters. Additionally, we show that the Backus average of isotropic layers is isotropic if and only if $\gamma=0$. This is in contrast to parameters $\delta$ and $\epsilon$, whose zero values do not imply isotropy.
... However, we show-by means of a fundamental symmetry of the Backus average-that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, a relationship between the Backus and Gazis et al. (1963) averages. ...
Article
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As shown by Backus (1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of a randomly oriented anisotropic elasticity tensor, which-one might expect-would result in an isotropic medium. However, we show-by means of a fundamental symmetry of the Backus average-that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, a relationship between the Backus and Gazis et al. (1963) averages.
... For any distance functions proposed to determine the symmetry type of the elastic tensor of a given material (cf. Diner et al., 2010;Gazis et al., 1963;Moakher and Norris, 2006), regardless of its validity, figure illustration and identification are unavoidable. ...
Article
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We develop a method through the mirror plane (MP) to identify the symmetry type of linear elastic stiffness tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the irreducible decomposition of high-order tensor into a set of deviators and the multipole representation of a deviator into a scalar and a unit-vector set. Since a unit-vector depends on two Euler angles, we can illustrate the MP normals of the elastic tensor as zeros of a characteristic function on a unit disk and identify its symmetry immediately, which is clearer and simpler than the methods proposed before. Furthermore, by finding the common MPs of three unit-vector sets using Fortran recipes, we can also analytically recognize the symmetry type first and then recover the natural coordinate system associated with the linear elastic tensor. The structures of linear elastic stiffness tensors of real materials with all possible anisotropies are investigated in detail.
... The proposed method of determining the reference transversely isotropic stiffness tensor for the given stiffness tensor of a generally anisotropic medium has been coded as a new option of program tiaxis.for of software package FORMS (Bucha and Bulant, 2016 ). Gazis et al. (1963) proposed a general but considerably involved method of the nearest media approximation consisting in the normal projection of the given stiffness tensor onto the subset of invariant tensors. It can be proved that the reference transversely isotropic medium (22) obtained here as the average of the rotated stiffness tensor is identical to the transversely isotropic medium obtained by the normal projection of the given stiffness tensor onto the subset of tensors invariant with respect to rotation (5) about a given reference symmetry axis. ...
Article
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For a given stiffness tensor (tensor of elastic moduli) of a generally anisotropic medium, we can estimate the extent to which the medium is transversely isotropic, and determine the direction of its reference symmetry axis. In this paper, we rotate the given stiffness tensor about this reference symmetry axis, and determine the reference transversely isotropic (uniaxial) stiffness tensor as the average of the rotated stiffness tensor over all angles of rotation. The obtained reference transversely isotropic (uniaxial) stiffness tensor represents an analytically differentiable approximation of the given generally anisotropic stiffness tensor. The proposed analytic method is compared with a previous numerical method in two numerical examples.
... Nevertheless, determining explicitly the symmetry class of a given Elasticity tensor is not an easy task and has been the subject of many researches, using different means. Moreover, the problem becomes even more complicated if one consider that, in real life, a measured elasticity tensor (assuming that one can access to all of its components) is subject to experimental errors and has therefore no symmetry but is nevertheless close to a given theoretical tensor with a given symmetry [21,32]. ...
Preprint
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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.
... Aside from pure theoretical interest by itself, this question is important for interpretation of observational data, see [17]. Recently this problem was studied intensively, see [8,18,28] and the references therein. Let in some coordinate system a material be described by a full set of 21 elastic parameters C ij kl . ...
Article
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We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.
Article
For elastic constant tensor, the norm concept, norm ratio and anisotropy degree are described. The norm of a tensor is used as a criterion for comparing the overall effect of the properties of anisotropic materials and norm ratios are used as a criterion to represent the anisotropy degree of the properties of these materials. Norm and norm ratios as well as the measure of “nearness” to the nearest isotropic tensor are computed for several examples from various anisotropic materials possessing elastic symmetries such as cubic, transversely isotropic, tetragonal, trigonal and orthorhombic. These computations are used to compare and assess the anisotropy in various anisotropic materials by means of strength or magnitude and also determine the “nearness” of the nearest isotropic tensor for the materials with lower symmetry types.
Article
It is often desirable to approximate a full anisotropic tensor, given by 21 independent parameters, by one with a higher symmetry. If one considers measurement errors of an elasticity tensor, the standard approaches of finding the best approximation by a higher symmetric tensor do not produce the most likely tensor. To find such a tensor, I replace the distance metric used in previous studies with one based on probability distribution functions of the errors of the measured quantities. In the case of normally distributed errors, the most likely tensor with higher symmetries coincides with the closest higher symmetric tensor, using a deviation-scaled Euclidean metric. © 2009 Society of Exploration Geophysicists. All rights reserved.
Article
We investigated the stress effects on nP yellow excitons in Cu2O thin films recrystallized epitaxially in a sample gap between paired MgO substrates. In such samples, it is expected that a two-dimensional compressive stress acts on Cu2O because of the slightly larger lattice constant of Cu2O than of MgO. To clarify such stress effects, we measured the X-ray diffraction and nP absorption transitions of the yellow excitonic system and analyzed the strain and stress effects. Although the detected lattice strain and the energy shift of the yellow excitonic band gap are smaller than the values expected from the lattice mismatch at the heterointerface, this can be explained self-consistently by considering strain and stress relaxations in Cu2O thin films with departing from the MgO heterointerface. Consequently, we can find that shallow trapping potentials for the yellow excitons are formed in the similar to 1.3-mu m-thick region interfaced with the MgO substrates.
Article
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We formulate a method of representing a generally anisotropic elasticity tensor by an elasticity tensor exhibiting a material symmetry: an effective tensor. The method for choosing the effective tensor is based on examining the features of the plot of the monoclinic-distance function of a given tensor, choosing an appropriate symmetry class, and then finding the closest tensor in that class. The concept of the effective tensor is not tantamount to the closest tensor since one always obtains a closer approximation using a monoclinic tensor than a tensor of any other nontrivial symmetry. Hence, we use qualitative features of the plot of the monoclinic-distance function to choose an effective symmetry class within which the closest tensor can be computed.
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In Pressurized Water Reactors (PWR, fuel is generally composed of uranium dioxide pellets piled up within a zirconium tubular cladding. Modeling of fuel behavior in nominal and accidental conditions requires multi-scale models in order to take into account both the thermo-mechanical behavior of the pellets and the effects of fission gases. Recent development of micromecanical tools of simulation has improved coupling possibilities. Our study proposes to set a full micromechanical model for uranium dioxide dealing with both mechanics and fission products transport at the scale of a polycristalline aggregate. Both the effective behavior of the RVE and stress localization effects are studied. Hydrostatic pressure, which directly controls the diffusion of fission gases, is given a particular focus. The numerical robustness of results is also debated in terms of mesh refinement. A series of simulations leads to a satisfying compromise between accuracy and calculation time. A study compares experimental measurement of intergranular crack opening and simulation results performed using cohesive models. The micromecanical behavior of uranium dioxide during irradiation is analysed by submitting the polycristalline RVE to transient irradiation. The local stress distribution leads to a debate on the consequences of intergranular strain incompatibility on fission gases diffusion.<br /
Chapter
Linear elasticity is one of the more successful theories of mathematical physics. Its pragmatic success in describing the small deformations of many materials is uncontested. The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson. The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lamé, Saint-Venant, Somigliana, Stokes, and others. These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations. In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness.
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A decomposition method[5] based upon orthonormal representations is reviewed and improved toexpress any anisotropic engineering tensor showing the effect of the material properties on the structures. A new decomposed form for the stress tensor (example for symmetric second rank tensor) different from the one available in the literature where the engineering understanding is improved, is presented. Numerical examples from different engineering materials serve to illustrate and verify the decomposition procedure. The norm concept of elastic constant tensor and norm ratios are used to study the anisotropy of these materials. It is shown that this method allows to investigate the elastic and mechanical properties of an anisotropic material possessing any material symmetry and determine anisotropy degree of that material. For a material given from an unknown symmetry, it is possible to determine its material symmetry type by this method.
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This work proposes an extension of the well-known random sequential adsorption (RSA) method in the context of non-overlapping random mono- and polydisperse ellipsoidal inclusions. The algorithm is general and can deal with inclusions of different size, shape and orientation with or without periodic geometrical constraints. Specifically, polydisperse inclusions, which can be in terms of different size, shape, orientation or even material properties, allow for larger volume fractions without the need of additional changes in the main algorithm. Unit-cell computations are performed by using either the fast Fourier transformed-based numerical scheme (FFT) or the finite element method (FEM) to estimate the effective elastic properties of voided particulate microstructures. We observe that an isotropic overall response is very difficult to obtain for random distributions of spheroidal inclusions with high aspect ratio. In particular, a substantial increase (or decrease) of the aspect ratio of the voids leads to a markedly anisotropic response of the porous material, which is intrinsic of the RSA construction. The numerical estimates are probed by analytical Hashin-Shtrikman-Willis (HSW) estimates and bounds.
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The aim of this short paper is to provide, for elasticity tensors, generalized Euclidean distances that preserve the property of invariance by inversion. First, the elasticity law is expressed under a non-dimensional form by means of a gauge, which leads to an expression of elasticity (stiffness or compliance) tensors without units. Based on the difference between functions of the dimensionless tensors, generalized Euclidean distances are then introduced. A subclass of functions is proposed, which permits the retrieval of the classical log-Euclidean distance and the derivation of new distances, namely the arctan-Euclidean and power-Euclidean distances. Finally, these distances are applied to the determination of the closest isotropic tensor to a given elasticity tensor.
Article
This paper presents in vivo mechanical characterization of the muscularis, submucosa and mucosa of the porcine stomach wall under large deformation loading. This is important for the development of gastrointestinal pathology-specific surgical intervention techniques. The study is based on testing the cardiac and fundic glandular stomach regions using a custom-developed compression elastography setup. Particular attention has been paid to elucidate the heterogeneity and anisotropy of tissue response. A Fung hyperelastic material model has been used to model the mechanical response of each tissue layer. A univariate analysis comparing the initial shear moduli of the three layers indicates that the muscularis (5.69±4.06 kPa) is the stiffest followed by the submucosa (3.04±3.32 kPa) and the mucosa (0.56±0.28 kPa). The muscularis is found to be strongly distinguishable from the mucosa tissue in the cardiac and fundic region based on a multivariate discriminant analysis. The cardiac muscularis is observed to be stiffer than the fundic muscularis tissue (shear moduli of 7.96±3.82 kPa vs. 3.42±2.96 kPa), more anisotropic (anisotropic parameter of 2.21±0.77 vs. 1.41±0.38), and strongly distinguishable from its fundic counterpart. Finally, a univariate comparison of the in vivo and ex vivo initial shear moduli for each layer shows that the muscularis and submucosa tissues are softer while in vivo, but the mucosa tissue is stiffer while in vivo. The mechanical properties highlight the inhomogeneity and anisotropy of multilayer stomach tissue.
Article
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.
Article
The problem of determining the transversely isotropic tensor closest in Euclidean norm to a given anisotropic elastic modulus tensor is considered. An orthonormal basis in the space of transversely isotropic tensors for any given axis of symmetry was obtained by decomposition of a transversely isotropic tensor in the general coordinate system into one isotropic part, two deviator parts, and one nonor part. The closest transversely isotropic tensor was obtained by projecting the general anisotropy tensor onto this basis. Equations for five coefficients of the transversely isotropic tensor were derived and solved. Three equations describing stationary conditions were obtained for the direction cosines of the axis of rotation (symmetry). Solving these equations yields the absolute minimum distance from the transversely isotropic tensor to a given anisotropic elastic modulus tensor. The transversely isotropic elastic modulus tensor closest to the cubic symmetry tensor was found.
Article
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.
Article
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It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor, it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the Frobenius 21-component norm and the operator norm on a general Hookean solid. We find that both Frobenius norms result in similar isotropic counterparts, and the operator norm results in a counterpart with a slightly larger discrepancy. The reason for this discrepancy is rooted in the very definition of that norm, which is distinct from the Frobenius norms and which consists of the largest eigenvalue of the elasticity tensor. If we constrain the elasticity tensor to values expected for modelling physical materials, the three norms result in similar isotropic counterparts of a generally anisotropic tensor. To examine this important case and without loss of generality, we illustrate the isotropic counterparts by commencing from a transversely isotropic tensor obtained from a generally anisotropic one. Also, together with the three norms, we consider the L 2 slowness-curve fit. Upon this study, we infer that-for modelling physical materials-the isotropic counterparts are quite similar to each other, at least, sufficiently so that-for values obtained from empirical studies, such as seismic measurements-the differences among norms are within the range of expected measurement errors.
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