Perturbation bounds for the largest C-eigenvalue of piezoelectric-type tensors

In this paper, we focus on the perturbation analysis of the largest C-eigenvalue of the piezoelectric-type tensor which has concrete physical meaning which determines the highest piezoelectric coupling constant. Three perturbation bounds are presented, theoretical analysis and numerical examples show that the third perturbation bound has high accuracy when the norm of the perturbation tensor is small

Symmetry classes for even-order tensors

The purpose of this article is to give a complete and general answer to the recurrent problem in continuum mechanics of the determination of the number and the type of symmetry classes of an even-order tensor space. This kind of investigation was initiated for the space of elasticity tensors. Since then, different authors solved this problem for other kinds of physics such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. All the aforementioned problems were treated by the same computational method. Although being effective, this method suffers the drawback not to provide general results. And, furthermore, its complexity increases with the tensorial order. In the present contribution, we provide general theorems that directly give the sought results for any even-order constitutive tensor. As an illustration of this method, and for the first time, the symmetry classes of all even-order tensors of Mindlin second strain-gradient elasticity are provided.Comment: Mathematics and Mechanics of Complex Systems (2013) (Accepted

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 $C$, meanwhile for higher-order classes conditions are provided in terms of elements of $H^{4}$, the higher irreducible space in the decomposition of $C$. 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

Lamination exact relations and their stability under homogenization

Relations between components of the effective tensors of composites that hold regardless of composite's microstructure are called exact relations. Relations between components of the effective tensors of all laminates are called lamination exact relations. The question of existence of sets of effective tensors of composites that are stable under lamination, but not homogenization was settled by Milton with an example in 3D elasticity. In this paper we discuss an analogous question for exact relations, where in a wide variety of physical contexts it is known (a posteriori) that all lamination exact relations are stable under homogenization. In this paper we consider 2D polycrystalline multi-field response materials and give an example of an exact relation that is stable under lamination, but not homogenization. We also shed some light on the surprising absence of such examples in most other physical contexts (including 3D polycrystalline multi-field response materials). The methods of our analysis are algebraic and lead to an explicit description (up to orthogonal conjugation equivalence) of all representations of formally real Jordan algebras as symmetric $n\times n$ matrices. For each representation we examine the validity of the 4-chain relation|a 4th degree polynomial identity, playing an important role in the theory of special Jordan algebras

Structural energy flow optimization through adaptive shunted piezoelectric metacomposites

International audienceIn this article, a numerical approach for modeling and optimizing two-dimensional smart metacomposites is presented. The proposed methodology is based on the Floquet-Bloch theorem in the context of elastodynamics including distributed shunted piezoelectric patches. The dedicated numerical technique is able to cope with the multimodal wave dispersion behavior over the whole first Brillouin zone for periodically distributed two-dimensional shunted piezomechanical systems. Some indicators allowing the optimization of the shunt impedance for specific performance objectives are presented and applied for illustration purposes on the design of an adaptive metacomposite with specific functionalities. In order to validate the strategy, the designed metacomposite is integrated in a support structure, and a full three dimensional model is derived to illustrate the efficiency of the approach

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 $C$, meanwhile for higher-order classes conditions are provided in terms of elements of $H^{4}$, the higher irreducible space in the decomposition of $C$. 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

Dynamic Response of Tunable Phononic Crystals and New Homogenization Approaches in Magnetoactive Composites

This research investigates dynamic response of tunable periodic structures and homogenization methods in magnetoelastic composites (MECs). The research on tunable periodic structures is focused on the design, modeling and understanding of wave propagation phenomena and the dynamic response of smart phononic crystals. High amplitude wrinkle formation is employed to study a one-dimensional phononic crystal slab consists of a thin film bonded to a thick compliant substrate. Buckling induced surface instability generates a wrinkly structure triggered by a compressive strain. It is demonstrated that surface periodic pattern and the corresponding large deformation can control elastic wave propagation in the low thickness composite slab. Simulation results show that the periodic wrinkly structure can be used as a smart phononic crystal which can switch band diagrams of the structure in a transformative manner. A magnetoactive phononic crystal is proposed which its dynamic properties are controlled by combined effects of large deformations and an applied magnetic field. Finite deformations and magnetic induction influence phononic characteristics of the periodic structure through geometrical pattern transformation and material properties. A magnetoelastic energy function is proposed to develop constitutive laws considering large deformations and magnetic induction in the periodic structure. Analytical and finite element methods are utilized to compute dispersion relation and band structure of the phononic crystal for different cases of deformation and magnetic loadings. It is demonstrated that magnetic induction not only controls the band diagram of the structure but also has a strong effect on preferential directions of wave propagation. Moreover, a thermally controlled phononic crystal is designed using ligaments of bi-materials in the structure.Comment: PhD mechanical engineering, University of Nevada, Reno (2015

First-principles calculations of electric field gradients in complex perovskites

Various experimental and theoretical work indicate that the local structure and chemical ordering play a crucial role in the different physical behaviors of lead-based complex ferroelectrics with the ABO 3 perovskite structure. First-principles linearized augmented plane wave (LAPW) with the local orbital extension method within local density approximation (LDA) are performed on structural models of Pb(Zr1/2Ti1/2 )O3 (PZT), Pb(Sc1/2Ta1/2)O3 (PST), Pb(Sc2/3W1/3)O3 (PSW), and Pb(Mg 1/3Nb2/3)O3 (PMN) to calculate electric field gradients (EFGs). In order to simulate these disordered alloys, various structural models were constructed with different imposed chemical orderings and symmetries. Calculations were carried out as a function of B-site chemical ordering, applied strain, and imposed symmetry. Large changes in the EFGs are seen in PZT as the electric polarization rotates between the tetragonal and rhombohedral directions. The onset of polarization rotation in monoclinic Cm symmetry strongly correlates with the shearing of the TiO6 octahedron, and there is a sharp change in slope in plots of Ti EFGs versus octahedral distortion index. The same changes in EFGs and the BO6 shearing corresponding to the change of off-centering direction are also seen in PST. In PSW and PMN, the calculated B cation EFGs showed more sensitivity to the surrounding nearest B neighboring environments. Calculated B atom EFGs in all alloys are considerably larger than those inferred from the NMR measurements. Based on comparisons with experiments, the calculated results are interpreted in terms of static and dynamic structural models of these materials