Boundary element formulation for plane problems in size-dependent piezoelectricity

Hajesfandiari, Arezoo; Hadjesfandiari, Ali R.; Dargush, Gary F.

doi:10.1002/nme.5227

Cited by 22 publications

(4 citation statements)

References 45 publications

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“…The boundary integral formulations and boundary element developments based on C-CST clearly demonstrate the consistency and practicality of this theory (Hadjesfandiari and Dargush, 2012;Hadjesfandiari et al, 2013;Hajesfandiari et al, 2016Hajesfandiari et al, , 2017Mikulich et al, 2018).…”

Section: Discussionmentioning

confidence: 62%

An Assessment of Couple Stress Theories

Hadjesfandiari¹,

Dargush²

2018

Preprint

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In this paper, we examine the mathematical and physical consistencies of the three primary couple stress theories: original Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), modified couple stress theory (M-CST) and consistent couple stress theory (C-CST).  As has been known for many years, MTK-CST suffers from some fundamental inconsistencies, such as the indeterminacy of the couple-stress tensor.  Therefore, despite the fact that MTK-CST has a fundamental position in the evolution of size-dependent continuum mechanics, it is not a reliable theory within continuum mechanics, for example, in developing new size-dependent multi-physics formulations.  We also observe that M-CST not only inherits all inconsistences from the original MTK-CST, but also suffers from new additional inconsistencies, such as the introduction of a new non-physical governing equation.  These inconsistencies refute the claim of those who state that the couple-stress tensor may be chosen symmetric.  Therefore, the apparent success of MTK-CST and M-CST in describing a size-effect for some problems, such as two-dimensional plate and beam bending, is not enough to justify these theories as suitable for general cases.  In fact, the symmetric couple-stresses in M-CST create torsional or anticlastic deformation, not bending.  On the other hand, C-CST, with a skew-symmetric couple-stress tensor, is the consistent continuum mechanics suitable for solving different size-dependent solid, fluid and multi-physics problems. 

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Section: Discussionmentioning

confidence: 62%

An Assessment of Couple Stress Theories

Hadjesfandiari¹,

Dargush²

2018

Preprint

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“…In the present work, boundary element method is used to solve a biomaterial flexoelectric problem with a simple geometry. The details on the integral representation of the formulation using the reciprocal theorem is provided in [3]. These integral representations are then discretized and implemented as a boundary element method formulation.…”

Section: Boundary Element Implementationsmentioning

confidence: 99%

Boundary Element Solution to Size-Dependent Piezoelectric Bimaterials

Hajesfandiari¹

2021

14th WCCM-ECCOMAS Congress

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A boundary element formulation based on the consistent couple stress theory is used to analyze two-dimensional size-dependent piezoelectric response in isotropic dielectric materials. In this approach, there exist a size-dependent piezoelectricity or flexoelectricity effect for centrosymmetric materials. The size-dependent effect is specified by one characteristic length scale parameter l, and the electromechanical effect is specified by one flexoelectric coefficient f. This phenomenon is a coupled problem involving mechanical and electrical effects. A boundary-only formulation is used for which the primary variables are displacements, rotations, force-tractions, couple-tractions, electric potential, and normal electric displacement. This BEM formulation is applied to a bimaterial computational problem to confirm the validity of the numerical implementation and to explore the physics of the flexoelectric coupling.

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“…To keep the presentation succinct, the details of finite element implementations are not discussed here. Finite element implementations of couple stress models for purely mechanical continuum elements are discussed in Chakravarty et al [2017]; Garg and Han [2015] for penalty formulation and in Deng et al [2018]; Kwon and Lee [2017] for Lagrange multiplier formulation (see also Hajesfandiari et al [2016Hajesfandiari et al [ , 2017 for boundary element treatment of the problem). The previous work of the authors also describe computational implementation of a series of mixed and high order finite element disretisations based on an enhanced set of variables in electromechanics for continuum and beam elements Ortigosa and Gil [2016a]; Poya et al [2018]; ; Poya et al [2015].…”

Section: Description Isotropic Anisotropicmentioning

confidence: 99%

On a family of numerical models for couple stress based flexoelectricity for continua and beams

Poya

Gil

Ortigosa

et al. 2019

Journal of the Mechanics and Physics of Solids

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A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew-symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material's centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the C 0 continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation.

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Boundary element formulation for plane problems in size-dependent piezoelectricity

Cited by 22 publications

References 45 publications

An Assessment of Couple Stress Theories

An Assessment of Couple Stress Theories

Boundary Element Solution to Size-Dependent Piezoelectric Bimaterials

On a family of numerical models for couple stress based flexoelectricity for continua and beams

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