On the symmetries of 2D elastic and hyperelastic tensors

A thorough investigation is made of the independent point-group symmetries and canonical matrix forms that the 2D elastic and hyperelastic tensors can have. Particular attention is paid to the concepts relevant to the proper definition of the independence of a symmetry from another one. It is shown that the numbers of all independent symmetries for the 2D elastic and hyperelastic tensors are six and four, respectively. In passing, a symmetry result useful for the homogenization theory of 2D linear elastic heterogeneous media is derived.     

On singularities associated with the curvilinear anisotropic elastic symmetries

The objective of this paper is to describe a different approach to modeling the material symmetry associated with singularities that can occur in curvilinear anisotropic elastic symmetries. In this analysis, the intrinsic non-linearity of a cylindrically anisotropic problem is demonstrated. We prove that a simple homogenization process applied to a representative volume element containing the cylindrical anisotropic singularity removes the singularity. This geometric and interpretive approach is an aid to better modeling of material symmetry associated with these singularities.     

On Symmetries and Anisotropies of Classical and Micropolar Linear Elasticities: A New Method Based upon a Complex Vector Basis and Some Systematic Results

A complete and unified study of symmetries and anisotropies of classical and micropolar elasticity tensors is presented by virtue of a novel method based on a well-chosen complex vector basis and algebra of complex tensors. It is proved that every elasticity tensor has nothing but 1-fold, 2-fold, 3-fold, 4-fold and ∞-fold symmetry axes. From this fact it follows that the crystallographic symmetries plus the isotropic symmetry are complete in describing the symmetries of any kind of classical elasticity tensors and micropolar elasticity tensors. Further, it is proved that for each given integer m>>2 every classical Green elasticity tensor with an m-fold symmetry axis must have at least m elastic symmetry planes intersecting each other at this symmetry axis. From this fact and the aforementioned fact it follows that for all possible material symmetry groups, there exist only eight distinct symmetry classes for classical Green elasticity tensors, which correspond to the isotropy group and the seven crystal classes S ₂, C _2h , D _2h , D _3d , D _4h , D _6h and O _h , while it is shown that there exist twelve distinct symmetry classes for any other kind of elasticity tensors, including the classical Cauchy elasticity tensor and the micropolar elasticity tensors, which correspond to the eight subgroup classes just mentioned and the four crystal classes S ₆, C _4h , C_6h and T _h . From these results, it turns out that all possible elasticity symmetry groups are nothing but the full orthogonal group, the transverse isotropy groups C _{∞ h} and D _{∞ h} , and the nine centrosymmetric crystallographic point groups except C _6h and D _6h . This revised version was published online in August 2006 with corrections to the Cover Date.     

On the construction of linearly independent tensors

We consider various methods for constructing linearly independent isotropic, gyrotropic, orthotropic, and transversally isotropic tensors. We state assertions and theorem that permit one to construct these tensors. We find linearly independent above-mentioned tensors up to and including rank six. The components of the tensor may have no symmetry or have symmetries of various types.     

Characterization of Hamiltonian symmetries and their first integrals

We provide explicit criteria when the Hamiltonian symmetries for a finite dimensional canonical Hamiltonian system correspond to their first integrals. There are two approaches used for the construction of the first integrals once the symmetry is known. In the standard classical approach the first integrals are obtained up to a distinguished function of time t. In the second, which is recent, the integrals are given by a formula which involves the determination of the divergence terms. In both methods utilized, the first integrals are not determined uniquely. Firstly we show what conditions need to be imposed on the Hamiltonian symmetry in order that it constructively and uniquely yields a first integral. Secondly we provide the extra condition on the first integral for the first approach and the integrability conditions on the divergence term for the second. As a consequence, we show that both methods are in fact equivalent. Furthermore, it is shown that when the Hamiltonian symmetries provide first integrals they form a Lie algebra. Moreover, we prove that the Hamilton first integral is invariant under the Hamilton action symmetry. Several examples taken from the literature are given to illustrate our results and conditions.     

A stochastic model for elasticity tensors with uncertain material symmetries

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

Stationarity of the strain energy density for some classes of anisotropic solids

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

A Simple Explicit Formula for the Voigt-Reuss-Hill Average of Elastic Polycrystals with Arbitrary Crystal and Texture Symmetries

The Voigt-Reuss-Hill (VRH) average provides a simple way to estimate the elastic constants of a textured polycrystal in terms of its crystallographic texture and the elastic constants of the constituting crystallites. Empirically the VRH estimates were found in most cases to have an accuracy comparable to those obtained by more sophisticated techniques such as self-consistent schemes. In this paper we determine, in the space of fourth-order tensors with major and minor symmetries, a special set of irreducible basis tensors, with which we obtain a simple explicit formula for the VRH average for elastic polycrystals with arbitrary crystal and texture symmetries. Our formula is correct to first order in the texture coefficients.     

On the Statistical Dependence for the Components of Random Elasticity Tensors Exhibiting Material Symmetry Properties

This work is concerned with the characterization of the statistical dependence between the components of random elasticity tensors that exhibit some given material symmetries. Such an issue has historically been addressed with no particular reliance on probabilistic reasoning, ending up in almost all cases with independent (or even some deterministic) variables. Therefore, we propose a contribution to the field by having recourse to the Information Theory. Specifically, we first introduce a probabilistic methodology that allows for such a dependence to be rigorously characterized and which relies on the Maximum Entropy (MaxEnt) principle. We then discuss the induced dependence for the highest levels of elastic symmetries, ranging from isotropy to orthotropy. It is shown for instance that for the isotropic class, the bulk and shear moduli turn out to be independent Gamma-distributed random variables, whereas the associated stochastic Young modulus and Poisson ratio are statistically dependent random variables.     

Irreducible tensors and their applications in problems of dynamics of solids

One difficulty encountered in solving mechanical problems with complicated interaction is to express either the moments of forces or the force function via the phase variables of the problem. Here various transformations of coordinate systems are used, because interactions are determined by a relation between tensor variables one of which refers to the body and the other refers to the field. In this connection, the usual definition of a tensor in Cartesian coordinates is inconvenient because of the fact that the components of a tensor of rank l ≥ 2 can be arranged as several linear combinations that behave differently under rotations of the coordinate system. Naturally, one needs to define tensors in such a way that their components and linear combinations of these be transformed in a unified manner under rotations of the coordinate system. This requirement is satisfied by irreducible tensors. The mathematical apparatus of irreducible tensors was created to satisfy the requirements of quantum mechanics and turned out to be rather universal. As far as the author knows, this apparatus was first used in mechanics by G. G. Denisov and the author of the present paper [1]. Using this apparatus, one can see the clear physical meaning of complicated interactions, express these interactions in invariant form, easily perform transformations from one coordinate system to another coordinate system turned relative to the first, consider rather complicated types of interactions writing them in compact form explicitly depending on the phase variables of the problem, easily use the symmetry of both the rigid body and the force field structure, and perform the averaging procedure for the entire object rather than componentwise. The present paper further develops the paper [1]. We present a brief introduction to the theory of irreducible tensors. We show that the force function of various interactions between a rigid body and a force field can be represented as the scalar product of irreducible tensors. We study general properties of evolution motions of a rigid body in axisymmetric and nonsymmetric force fields under the action of moments caused by various harmonics of the force function.     

Invariant analysis and exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation by Lie group analysis

This paper is concerned with the time fractional Sharma–Tasso–Olver (FSTO) equation, Lie point symmetries of the FSTO equation with the Riemann–Liouville derivatives are considered. By using the Lie group analysis method, the invariance properties of the FSTO equation are investigated. In the sense of point symmetry, the vector fields of the FSTO equation are presented. And then, the symmetry reductions are provided. By making use of the obtained Lie point symmetries, it is shown that this equation can transform into a nonlinear ordinary differential equation of fractional order with the new independent variable ξ=xt ^?α/3. The derivative is an Erdélyi–Kober derivative depending on a parameter α. At last, by means of the sub-equation method, some exact and explicit solutions to the FSTO equation are given.     

On the Averaging of Symmetric Positive-Definite Tensors

In this paper we present properly invariant averaging procedures for symmetric positive-definite tensors which are based on different measures of nearness of symmetric positive-definite tensors. These procedures intrinsically account for the positive-definite property of the tensors to be averaged. They are independent of the coordinate system, preserve material symmetries, and more importantly, they are invariant under inversion. The results of these averaging methods are compared with the results of other methods including that proposed by Cowin and Yang (J. of Elasticity 46 (1997) pp. 151–180.) for the case of the elasticity tensor of generalized Hooke's law.     

On the non-conservativeness of a class of anisotropic damage models with unilateral effectsSur le caractère non-conservatif de quelques modèles d'endommagement anisotrope avec conditions unilatérales

The aim of this Note is to show that a class of anisotropic elastic-damage models including unilateral effects can be considered, for constant damage values, as non-linear and non-conservative elastic. The conservative character of corresponding constitutive models is related to the symmetry of the Hessian tensor. For the models under consideration, it is shown that the condition of conservativeness (existence of the elastic potential energy function) is obtained only when there is coaxiality of the strain and damage tensors. To cite this article: N. Challamel et al., C. R. Mecanique 334 (2006).     

Generalized lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator

Generalized Lie symmetries and the integrability of generalized Holmes-Rand non-linear oscillator (GHRNO) are considered. The constraint which the variable-coefficient functions must satisfy for the GHRNO to have infinite-dimensional symmetry algebras is derived. The integral of motion for the GHRNO under this condition can be read off from the symmetry vector fields. The structure of the symmetry algebras is also presented.     

Hamiltonian structures of dynamics of a gyrostat in a gravitational field

The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, SO(2) and SO(3) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.     

Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin-Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. A more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. This approach is very similar in spirit but differs in its details from earlier work of Willis, showing how Hashin-Shtrikman bounds and certain classes of self-consistent estimates may be related. These self-consistent estimates always lie within the bounds for physical choices of the crystal elastic constants and for all the choices of crystal symmetry considered. For cubic symmetry, the present method reproduces the self-consistent estimates obtained earlier by various authors, but the formulas for both bounds and estimates are generated in a more symmetric form. Numerical values of the estimates obtained this way are also very comparable to those found by the Gubernatis and Krumhansl coherent potential approximation (or CPA), but do not require computations of scattering coefficients.     

Nonlocal symmetries of evolution equations

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution equations in one spatial variable invariant under Lie algebras of the dimension up to three. As a result, we construct the broad families of new nonlinear evolution equations possessing nonlocal symmetries which in principle cannot be obtained within the classical Lie approach.     

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demonstrates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.