2,968 research outputs found
Anisotropic and dispersive wave propagation within strain-gradient framework
In this paper anisotropic and dispersive wave propagation within linear
strain-gradient elasticity is investigated. This analysis reveals significant
features of this extended theory of continuum elasticity. First, and contrarily
to classical elasticity, wave propagation in hexagonal (chiral or achiral)
lattices becomes anisotropic as the frequency increases. Second, since
strain-gradient elasticity is dispersive, group and energy velocities have to
be treated as different quantities. These points are first theoretically
derived, and then numerically experienced on hexagonal chiral and achiral
lattices. The use of a continuum model for the description of the high
frequency behavior of these microstructured materials can be of great interest
in engineering applications, allowing problems with complex geometries to be
more easily treated
Invariant-based approach to symmetry class detection
In this paper, the problem of the identification of the symmetry class of a
given tensor is asked. Contrary to classical approaches which are based on the
spectral properties of the linear operator describing the elasticity, our
setting is based on the invariants of the irreducible tensors appearing in the
harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that
aim we first introduce a geometrical description of the space of elasticity
tensors. This framework is used to derive invariant-based conditions that
characterize symmetry classes. For low order symmetry classes, such conditions
are given on a triplet of quadratic forms extracted from the harmonic
decomposition of the elasticity tensor , meanwhile for higher-order classes
conditions are provided in terms of elements of , the higher irreducible
space in the decomposition of . Proceeding in such a way some well known
conditions appearing in the Mehrabadi-Cowin theorem for the existence of a
symmetry plane are retrieved, and a set of algebraic relations on polynomial
invariants characterizing the orthotropic, trigonal, tetragonal, transverse
isotropic and cubic symmetry classes are provided. Using a genericity
assumption on the elasticity tensor under study, an algorithm to identify the
symmetry class of a large set of tensors is finally provided.Comment: 32 page
Generic separating sets for 3D elasticity tensors
We define what is a generic separating set of invariant functions (a.k.a. a
weak functional basis) for tensors. We produce then two generic separating sets
of polynomial invariants for 3D elasticity tensors, one made of 19 polynomials
and one made of 21 polynomials (but easier to compute) and a generic separating
set of 18 rational invariants. As a byproduct, a new integrity basis for the
fourth-order harmonic tensor is provided
Symmetry classes in piezoelectricity from second-order symmetries
The piezoelectricity law is a constitutive model that describes how
mechanical andelectric fields are coupled within a material. In its linear
formulation this law comprises threeconstitutive tensors of increasing order:
the second order permittivity tensor S, the third orderpiezoelectricity tensor
P and the fourth-order elasticity tensor C. In a first part of the paper,the
symmetry classes of the piezoelectricity tensor alone are investigated. Using a
new approachbased on the use of the so-called clips operations, we establish
the 16 symmetry classes of thistensor and provide their associated normal
forms. Second order orthogonal transformations(plane symmetries and -angle
rotations) are then used to characterize and classify directly 11out of the 16
symmetry classes of the piezoelectricity tensor. An additional step to
distinguishthe remaining classes is proposedComment: Mathematics and Mechanics of Complex Systems, mdp, In pres
Parabolic reciprocity gap for heat source identification
The deformation of solid materials is nearly always accompanied with temperature variations. These variations, governed by the heat diffusion equation stemming from the first and second laws of thermodynamics, are induced by intrinsic dissipation of energy and thermomechanical coupling. Infrared thermography techniques provide an experimental means for measuring thermal fields on specimen boundaries. But even if thermal fields are related to the material behavior they are not intrinsic to it as they also depend on external factors such as boundary conditions. Inverting boundary thermal fields is thus needed to obtain valid insight into the specimen thermomechanical behavior. Such an operation belongs to the class of source inverse problem. Inverse source problems are known to be ill-posed in the sense of Hadamard: their solution does not depend continuously on the data, and is not unique for a general source distribution when using only boundary measurements. Modeling hypotheses on the sought sources are thus needed to properly retrieve information
Invariant-based approach to symmetry class detection
32 pagesIn this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is based on the invariants of the irreducible tensors appearing in the harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that aim we first introduce a geometrical description of the space of elasticity tensors. This framework is used to derive invariant-based conditions that characterize symmetry classes. For low order symmetry classes, such conditions are given on a triplet of quadratic forms extracted from the harmonic decomposition of the elasticity tensor , meanwhile for higher-order classes conditions are provided in terms of elements of , the higher irreducible space in the decomposition of . Proceeding in such a way some well known conditions appearing in the Mehrabadi-Cowin theorem for the existence of a symmetry plane are retrieved, and a set of algebraic relations on polynomial invariants characterizing the orthotropic, trigonal, tetragonal, transverse isotropic and cubic symmetry classes are provided. Using a genericity assumption on the elasticity tensor under study, an algorithm to identify the symmetry class of a large set of tensors is finally provided
Identification of transient heat sources using the reciprocity gap
International audienceThe deformation of solid materials is nearly always accompanied with temperature variations, induced by intrinsic dissipation and thermomechanical coupling. Heat sources give precious information on the thermomechanical behavior of materials. They can be indirectly observed from thermal measurements on the specimen boundary, obtained e.g. via infrared thermography. To solve the inverse problem of identifying heat sources from such observations, a non-iterative algebraical method based on the Reciprocity Gap Method is proposed. This approach, used elsewhere mainly for time-independent identification, is applied here to transient measurements. Under appropriate modelling assumptions the number of heat sources, their spatial locations and energies are retrieved, as demonstrated on numerical experiments where the robustness of the method to measurement noise is also studied
Démonstration du théorème d'Hermann à partir de la méthode Forte–Vianello
International audienceUne nouvelle dérivation du théorème d'Hermann est proposée. La démonstration s'appuiera sur les outils introduit par Forte et Vianell
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