Symmetry classes in piezoelectricity from second-order symmetries

Olive, Marc; Auffray, Nicolas

doi:10.2140/memocs.2021.9.77

Cited by 8 publications

(5 citation statements)

References 47 publications

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“…As a consequence of lemma 2.4, the isotropy classes J (Piez) can be deduced from isotropy classes Ela, Piez and S: J (Piez) = (J (Ela) ⊚ J (Piez)) ⊚ J (S). Recall from [23] the isotropy classes of Piez (the notations and definitions of O(3)-subgroups have been moved to Appendix A) (3). In that case, there exists γ ∈ Γ \ Γ + such that −Γ − = γΓ + and Γ = Γ + ∪ −γΓ + .…”

Section: Application To Piezoelectricitymentioning

confidence: 99%

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Clips operation between type II and type III O(3)-subgroups with application to piezoelectricity

Perla¹,

Olive²

2022

Preprint

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Clips is a binary operation between conjugacy classes of closed subgroups of a compact group, which has been introduced to obtain the isotropy classes of a direct sum of representations E1 ⊕ E2, if the isotropy classes are known for each factor E1, E2. This allows, in particular, to compute the isotropy classes of any reducible representation, once one knows the decomposition into irreducible factors and the symmetry classes for the irreducible factors. In the specific case of the three dimensional orthogonal group O(3), clips between some types of subgroups of O(3) have already been calculated. However, until now, clips between type II and type III subgroups were missing. Those are encountered in 3D coupled constitutive laws. In this paper, we complete the clips tables by computing the missing clips. As an application, we obtain 25 isotropy classes for the standard O(3) representation on the full 3D Piezoelectric law, which involves the three Elasticity, Piezoelectricity and Permittivity constitutive tensors. Contents 10 4.4. Clips with O − 12 4.5. Clips with O(2) − 13 5. Application to Piezoelectricity 14 Appendix A. Closed O(3)-subgroup 15 References 18

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Section: Application To Piezoelectricitymentioning

confidence: 99%

“…A systematic way to calculate the isotropy classes was proposed by Chossat and Guyard in [2] using a binary operation between conjugacy classes. This operation was named the clips operation in [21,22,20,23], where it was generalized and used to determine isotropy classes for reducible representations.…”

mentioning

confidence: 99%

Clips operation between type II and type III O(3)-subgroups with application to piezoelectricity

Perla¹,

Olive²

2022

Preprint

Self Cite

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“…Substitution of the elastic and electric expressions of Equation (30) into Equation (22) and utilization of orthogonality of sin nθ and cos nθ terms, one harmonic at a time, yield the following relations between the fiber and matrix coefficients.…”

Section: Interfacial Continuitiesmentioning

confidence: 99%

“…In this section, we validate the proposed LEHT against several other analytical results (including Eshelby solutions, analytical asymptotic solutions, and classical M−T model), FE‐, and finite volume (FV)‐based numerical simulations, as well as experimental measurements in the literature, to demonstrate the method's computational efficiency and accuracy. Several microstructural parameters, such as fiber volume fraction, fiber arrangement, [ 24,30 ] etc., are tested on their influence of the effective and localized responses of piezoelectric heterogeneous materials. Piezoelectric composite systems consisting of several distinct constituents, including PZT‐7 A/BaTiO 3 , PZT‐7 A/PVDF, and porous BaTiO 3 , are investigated over a wide range of fiber volume fractions from 0 to 0.65.…”

Section: Numerical Validations and Investigationsmentioning

confidence: 99%

Extended Locally Exact Homogenization Theory for Effective Coefficients and Localized Responses of Piezoelectric Composites

Wang

Liu³

et al. 2021

Adv Eng Mater

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Herein, the elasticity‐based locally exact homogenization theory (LEHT) is extended to study the effective and localized responses of piezoelectric fiber composites. Based on the Trefftz concept, coupled internal solutions for electric fields and elastic displacements are developed for repeating unit cells (RUCs) that are characterized from the composite domain. The unknown coefficients of the internal fields can be determined by imposing interfacial continuity conditions between fiber and matrix and implementing the newly proposed multiphysics periodic boundary conditions. Finally, the generalized constitutive relations are established to predict the effective coefficients. The accuracy of the proposed technique is tested by validating the present simulations against the analytical Eshelby solution and asymptotic method, classical Mori−Tanaka (M−T) model, finite element (FE)‐ and finite volume (FV)‐based methods, as well as the experimental measurement in the literature with excellent agreement. What is more, several microstructural effects, such as phase volume fraction, fiber arrangement, etc., are varied to test their effects on the piezoelectric composites at both homogenized and localized levels. Based on the presented computational efficiency, the multiphysics LEHT is encapsulated into a blackbox with input/output data constructions for easy implementation by professionals or nonprofessionals.

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“…The effect of viscoelastic damping on the natural frequencies of piezoelectric nanobeams have been investigated recently in [4]. Symmetry classes of the second-order permittivity tensor, the third-order piezoelectricity tensor, and the fourth-order elasticity tensor have been determined using clips operations in [5]. More specific piezoelectric models have been proposed, such as for layered piezoelectric beams in [6], and the exploitation of distributed arrays of piezoelectric transducers for the control of vibrations, energy harvesting of health monitoring in the general context of smart materials and electromechanical systems is a concept expanded on in the review paper [7].…”

Section: Introductionmentioning

confidence: 99%

Construction of piezoelectric and flexoelectric models of composites by asymptotic homogenization and application to laminates

Nasimsobhan

Ganghoffer

Shamshirsaz

2021

Mathematics and Mechanics of Solids

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The effective piezoelectric and flexoelectric properties of heterogeneous solid bodies with constituents obeying a piezoelectric behavior are evaluated in full generality, based on the asymptotic expansion method. The successive situations of materials obeying a piezoelectric and flexoelectric behavior at the macroscale is envisaged in the present work. Closed-form expressions for the effective flexoelectric properties are obtained for stratified materials. A general theory for laminated piezoelectric plates is formulated on the basis of the formulated asymptotic models, and the response of the homogeneous substitution plate is evaluated for a loading consisting of a pure bending moment, triggering electric fields and strain and electric fields gradients within the plate thickness. The local mechanical and electric fields at the microscopic level within the initial heterogeneous stratified domain are evaluated by solving unit cell boundary value problems for the localization operators. An effective flexoelectric plate model for a stratified composite is constructed, showing the generation of the gradient of an electric field under application of a pure bending moment.

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Symmetry classes in piezoelectricity from second-order symmetries

Cited by 8 publications

References 47 publications

Clips operation between type II and type III O(3)-subgroups with application to piezoelectricity

Clips operation between type II and type III O(3)-subgroups with application to piezoelectricity

Extended Locally Exact Homogenization Theory for Effective Coefficients and Localized Responses of Piezoelectric Composites

Construction of piezoelectric and flexoelectric models of composites by asymptotic homogenization and application to laminates

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