A curvilinear high order finite element framework for electromechanics: From linearised electro-elasticity to massively deformable dielectric elastomers

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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.

Introduction

In recent years, exploiting actuation and harvesting through the heterogeneous class of Electro-Active-Polymers (EAP) has received considerable research focus. In particular, the electronic subgroup of EAP such as Dielectric Elastomers (DE) and electrostrictive relaxor ferroelectric polymers or Piezoelectric Polymers (PP) have become the subject of intensive mathematical and numerical analyses. Typically, the physically insightful information that could be gained from the numerical studies, depends on the capability of the underlying mathematical model utilised to simulate EAP. This has led to the development of a diverse range of mathematical models ranging from simplified to high fidelity models [[1], [2]].

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 [[1], [3], [4]]. Particularly more popular in the experimental physics community, these models have been used successfully to characterise actuation and energy harvesting capabilities of a range of materials, inherently due to their proximity to the actual experimental set-up. The potential of a material, in exhibiting electrostriction is typically exploited using such simplified formulations [[5], [6], [7], [8], [9], [10]].

On the other end of the spectrum lies the class of mathematically more sophisticated formulations that exploit the large deformation characteristics of EAP [[2], [11], [12], [13], [14], [16], [17], [18]]. The point of departure for such formulations is an assumed energy functional for the coupled electromechanical system. Conceptually, essential and suitable mathematical requirements for the energy functional such as ellipticity [[19], [20]], multi-variable convexity [[13], [14]], coercivity [21] and material frame indifference [22] can only be studied in a large deformation context. From a phenomenological point of view, these requirements or rather restrictions have important physical implications, in particular in guaranteeing the positive definiteness of the generalised electromechanical acoustic tensor, existence of real wave speeds in the material in the vicinity of an equilibrium configuration and the electromechanical stability of the material [[14], [23]]. Apart from these requirements other forms of physical instabilities present in dielectric elastomers such as pull-in instability, snap-through and the formation, propagation and nucleation of wrinkles have also been reported for these models, numerically as well experimentally [[24], [25], [26], [27], [28], [29], [30], [31]].

In an important intermediate class for electromechanics, the large deformation characteristics of the system are neglected, whereas the nonlinearity still present in the material emanates from the electrostriction of the material through the Maxwell (for vacuum V) or Minkowski (for material V) stress tensors [[32], [33], [34], [35]]. Theoretical aspects of these formulations were first introduced in Landau and Lifschitz [33]. The practical relevance of Maxwell stress tensor has led to a widespread utilisation of these formulations for exploiting electrostriction and magnetostriction. Unfortunately, electrostrictive models based on the utilisation of Minkowski stress tensor, in the generic case of anisotropy do not satisfy material frame indifference (i.e. objectivity or invariance of the energy with respect to rotations) of the electromechanical (total) stress tensor, due to the inherent non-symmetric nature of the Minkowski stress. Several authors in the past have used ad-hoc solutions, such as symmetrisation of the total stress tensor, or consideration of the conservation of angular momentum in the formulation, as a remedy [[34], [36]]. Nevertheless, the extended electromechanical Hessian still remains non-symmetric, which dictates the development of specialised non-symmetric finite element frameworks. Recently, Bustamente [37] has shown that physically admissible energy functionals can be constructed by choosing suitable constitutive restrictions such that their linearisation yields objective Minkowski-type stresses.

The present manuscript presents a computational framework suitable for both geometrically linearised and large deformation large electric field electromechanics. A convex multi-variable strain energy description based on the works of Gil and Ortigosa [[13], [14], [15]] is chosen for modelling EAPs under actuation and energy harvesting scenarios. For the case of small strains, following Bustamente [37], the present manuscript extends the framework developed by Gil and Ortigosa [[13], [14]] to the case of geometrically linearised electrostriction, to redress the aforementioned inconsistencies for the class of intermediate formulations. Importantly, all the aforementioned mathematical requirements are imposed at a large deformation level to arrive at a physically admissible energy functional. In this context, convex multi-variable energies typically expressed in terms of fundamental kinematic measures {F,H,J} are re-expressed in terms of a set of symmetric kinematics {C,G,C} to guarantee the objectivity of the energy functional. Linearisation with respect to geometrical fields is then performed by perturbing the energy in the vicinity of the reference configuration. Analogous to [38], this is achieved through a staggered scheme where the equations of electrostatics are solved for in a nonlinear fashion whereas the linearised mechanical equations are updated incrementally.

Admittedly, all the three class of formulations have been primarily applied to simplified geometries where the aim has been to verify the computational framework rather than to simulate realistic electromechanical components. In the present work, emphasis is put on accurate geometrical representation of electromechanical components by utilising an a posteriori curvilinear mesh deformation technique to accurately represent the true CAD boundaries. This is achieved by placing the high order nodes in the computational mesh on the true CAD boundary (curves/surfaces) through the solid mechanics based mesh deformation technique [38], but the underlying finite element functional spaces are not modified, i.e. a standard isoparametric finite element discretisation is employed. By relying on accurate geometrical representation and standard finite element technology, the current framework takes finite element analysis of electromechanical systems beyond the verification stage through virtual prototyping of a series of electromechanical components with potential applications in pattern forming, soft-robotics and other medical applications.

The paper is organised as follows. In Section 2, nonlinear continuum electromechanics is described. In Section 3, a variational framework for displacement-potential electromechanics is described. Aspects of particularisation to small strains and the corresponding staggered scheme are also discussed in this section. In Section 4, a series of numerical examples pertaining to the capability of the current framework in modelling DEs are analysed, starting from the h and p convergence properties of the curvilinear finite element framework presented in Section 4.1. The effect of accurate boundary representation using high order curvilinear finite elements is analysed in Section 4.2 and compared to high order planar elements (elements with planar faces/edges). In Section 4.3, examples of electromechanical actuation for linearised electromechanics using the presented staggered scheme are discussed. In Section 4.4, the performance of the current high order finite element displacement-potential approach is compared to those of mixed Hu–Washizu formulations presented in [[13], [14], [15]]. In Section 4.5, impact of code transformation and data parallelism for an efficient implementation of convex multi-variable electro-elasticity in the context of high order finite elements is presented. Finally, a series of examples pertaining to the massive deformation of dielectric elastomers are presented in Section 4.6. The inherent instabilities in DEs such as pull-in instability and the formation of wrinkles are studied with detailed precision using h and p refinements, pinpointing the robustness and the high performant capability of the current framework.

Section snippets

Kinematics: motion and deformation

Let us consider the motion of an electro-active body which in its initial or material configuration is defined by a domain V of boundary V with outward unit normal N. After the motion, the body occupies a spatial configuration defined by a domain v of boundary v with outward unit normal n, as shown in Fig. 1. The motion of the electro-active polymer V is defined by a pseudo-time (t) dependent mapping field ϕ which links a material particle from material configuration X to spatial

Displacement-electric potential based variational formulation

A variational principle can be established by the total energy minimisation defined in terms of the internal energy of the system esym=esym(C,D0). In this case, the total potential energy Πe(x,φ,D0) can be written as Πe(x,φ,D0)=infx,D0supφ{Vesym(C,D0)dV+VD00φdVΠext(x,φ)},where (x,φ,D0) denotes the exact solution and Πext(x,φ) represents the external coupled electromechanical work additively decomposed into the purely mechanical Πextm(x) and electrical Πexte(φ) components Πextm(x)=

Numerical examples

In this section a series of numerical examples for electromechanics are presented. These include (a) (mesh refinement) h and (polynomial enrichment) p convergence studies for high order displacement-potential formulation for convex multi-variable internal energies presented in Section 3, (b) the impact of accurate geometrical representations on the solution of large deformation electromechanical problems, (c) comparison of monolithic (nonlinear) approach with the incrementally linearised

Concluding remarks

A high order finite element implementation of the convex multi-variable electro-elasticity for large deformations large electric fields simulations and its particularisation to the case of small strains through a staggered scheme is presented. Accurate geometrical representation through a high performance curvilinear finite element framework based on a posteriori mesh deformation technique is developed to accurately discretise the underlying displacement-potential variational formulation. The

Acknowledgements

The first author acknowledges the financial support received through the European Commission EACEA Agency, Framework Partnership Agreement 2013-0043 Erasmus Mundus Action 1b, as a part of the EM Joint Doctorate Simulation in Engineering and Entrepreneurship Development (SEED). The second author acknowledges the financial support received through the European Training Network AdMoRe (Project ID: 675919).

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