A curvilinear high order finite element framework for electromechanics: From linearised electro-elasticity to massively deformable dielectric elastomers
01 Feb 2018-Computer Methods in Applied Mechanics and Engineering (North-Holland)-Vol. 329, pp 75-117
TL;DR: In this paper, a high-order finite element implementation of the convex multi-variable electro-elasticity for large deformations large electric fields analyses and its particularisation to the case of small strains through a staggered scheme is presented.
Abstract: This paper presents a high order finite element implementation of the convex multi-variable electro-elasticity for large deformations large electric fields analyses and its particularisation to the case of small strains through a staggered scheme With an emphasis on accurate geometrical representation, a high performance curvilinear finite element framework based on an a posteriori mesh deformation technique is developed to accurately discretise the underlying displacement-potential variational formulation The performance of the method under near incompressibility and bending actuation scenarios is analysed with extremely thin and highly stretched components and compared to the performance of mixed variational principles recently reported by Gil and Ortigosa (2016) and Ortigosa and Gil (2016) Although convex multi-variable constitutive models are elliptic hence, materially stable for the entire range of deformations and electric fields, other forms of physical instabilities are not precluded in these models In particular, physical instabilities present in dielectric elastomers such as pull-in instability, snap-through and the formation, propagation and nucleation of wrinkles and folds are numerically studied with a detailed precision in this paper, verifying experimental findings For the case of small strains, the essence of the approach taken lies in guaranteeing the objectivity of the resulting work conjugates, by starting from the underlying convex multi-variable internal energy, whence avoiding the need for further symmetrisation of the resulting Maxwell and Minkowski-type stresses at small strain regime In this context, the nonlinearity with respect to electrostatic counterparts such as electric displacements is still retained, hence resulting in a formulation similar but more competitive with the existing linearised electro-elasticity approaches Virtual prototyping of many application-oriented dielectric elastomers are carried out with an eye on pattern forming in soft robotics and other potential medical applications
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254 citations
TL;DR: In this paper, a review of the recent advances in the theory of dielectric elastomer and demonstrates some examples of using theory to design DE transducers is presented, from the simplest homogeneous deformation of a flat membrane to the highly complex bifurcations of a tube.
Abstract: In the past decade, the development of theory has deeply revealed the electromechanical coupling deformation mechanism of dielectric elastomer (DE). Many theoretical predictions on highly nonlinear deformation of dielectric elastomer have been verified by experiments. With the guidance of theory, the voltage-induced areal strain of dielectric elastomer has been increased from 100% in the pioneering work to the current record of 2200% and the energy density of a dielectric elastomer generator has reached 780 mJ/g. Much more developments have been realized on the applications of DE transducers in the fields of bioinspired artificial muscles, soft robotics, tunable lenses, and haptic interfaces. However, there is a gap between theory and application. Great potentials of developing DE transducers with the aid of a systematic theory have yet to be explored. This paper reviews the recent advances in the theory of dielectric elastomer and demonstrates some examples of using theory to design DE transducers. 9 boundary value problems of DE structures are analyzed from the simplest homogeneous deformation of a flat membrane to the highly complex bifurcations of a tube. Comparisons between theory and experiment are discussed. The viscous effect and the vibration of dielectric elastomer are reviewed. The developments of DE transducers and new dielectric materials are also reviewed. We summarize the performance of existing DE transducers with different configurations and make some discussions. It is hoped that the mechanics of DE structures can help to develop high-performance DE transducers in the future.
88 citations
TL;DR: A new one-step second order accurate energy–momentum (EM) preserving time integrator for reversible electro-elastodynamics is shown to be extremely useful for the long-term simulation of electroactive polymers (EAPs) undergoing massive strains and/or electric fields.
Abstract: This paper introduces a new one-step second order accurate energy–momentum (EM) preserving time integrator for reversible electro-elastodynamics. The new scheme is shown to be extremely useful for the long-term simulation of electroactive polymers (EAPs) undergoing massive strains and/or electric fields. The paper presents the following main novelties. (1) The formulation of a new energy–momentumtime integrator scheme in the context of nonlinear electro-elastodynamics. (2) The consideration of well-posed ab initio convex multi-variable constitutive models. (3) Based on the use of alternative mixed variational principles, the paper introduces two different EM time integration strategies (one based on the Helmholtz’s and the other based on the internal energy). (4) The new time integrator relies on the definition of four discrete derivatives of the internal/Helmholtz energies representing the algorithmic counterparts of the work conjugates of the right Cauchy–Green deformation tensor, its co-factor, its determinant and the Lagrangian electric displacement field. (6) Proof of thermodynamic consistency and of second order accuracy with respect to time of the resulting algorithm is included. Finally, a series of numerical examples are included in order to demonstrate the robustness and conservation properties of the proposed scheme, specifically in the case of long-term simulations.
20 citations
TL;DR: In this article, the axial curvature vector is used as a strain gradient measure and a skew-symmetric couple stress theory is proposed to model the electric enthalpy in terms of curvature and electric field.
Abstract: 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 C0 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.
20 citations
TL;DR: In this article, a parameter identification procedure has been held for characterizing the widely used dielectric elastomer VHB, which has been performed using various experimental setups and parameters.
Abstract: In this work, a parameter identification procedure has been held for characterizing the widely used dielectric elastomer VHB. The calibration procedure has been performed using various experimental...
18 citations
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"A curvilinear high order finite ele..." refers background in this paper
...On one end of the spectrum lies the class of simplified formulations, where one-dimensional idealisation in the form of rod and beam structures with mass-spring-damper-capacitor support using small strains and linear electrostatic assumption is utilised [4, 6, 7]....
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2,329 citations
"A curvilinear high order finite ele..." refers background in this paper
...Conceptually, essential and suitable mathematical requirements for the energy functional such as ellipticity [19, 20], multi-variable convexity [1, 2], coercivity [21] and material frame indifference [22] can only be studied in a large deformation context....
[...]
Book•
01 Nov 1982TL;DR: In this article, the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis are discussed. But the authors do not discuss the application of functional analysis to the problem of elasticity.
Abstract: [Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute.
2,115 citations
"A curvilinear high order finite ele..." refers background in this paper
...For instance, the maximum hydrostatic pressure at the tip of the circular hole within the plate located at [5, 20, 2] is phyd = −4....
[...]
...Conceptually, essential and suitable mathematical requirements for the energy functional such as ellipticity [19, 20], multi-variable convexity [1, 2], coercivity [21] and material frame indifference [22] can only be studied in a large deformation context....
[...]