An energy–momentum time integration scheme based on a convex multi-variable framework for non-linear electro-elastodynamics

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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.

Introduction

ElectroActive Polymers (EAPs) [[1], [2], [3], [4]] represent an important family of smart materials where dielectric elastomers and piezoelectric polymers are some of their most iconic integrants. Dielectric elastomers are very well-known for their outstanding actuation capabilities and low stiffness properties, which makes them ideal for their use as soft robots [5]. For instance, electrically induced area expansions of over 380% on dielectric elastomer thin films placed on the verge of snap-through configurations have been reported by Li et al. [6]. Other applications for dielectric elastomers include Braille displays, deformable lenses, haptic devices and energy generators, to name but a few [7]. Piezoelectric polymers have similar dielectric properties to dielectric elastomers but, on the other hand, have much larger stiffness. As a result, piezoelectric polymers cannot in principle exhibit large electrically induced deformations. Instead, they can be used as moderately deformable actuators. Other important type of applications for these materials include tactile sensors, energy harvesters, acoustic transducers and inertial sensors [[5], [7]].

The variational formulation of the governing equations of these materials is well established. In the most standard formulation, displacements and the scalar electric potential [[8], [9], [10], [11], [12], [13]] are modelled as the unknown fields. In this formulation, the constitutive information is encapsulated in the Helmholtz energy functional via its invariant-based representation depending upon kinematic strain measures and the electric field [[14], [15]]. However, for more complex constitutive models than that of an ideal dielectric elastomer, the saddle point nature of the Helmholtz functional (convex with respect to the deformation gradient tensor and concave with respect to the electric field in the small strain/small electric field regime), makes in general impossible to define a priori constitutive models which satisfy the ellipticity condition [[16], [17], [18], [19]]. This is a necessary condition that ensures the well-posedness of the problem. Motivated by the possible loss of ellipticity of the Helmholtz functional, Gil and Ortigosa [[20], [21], [22]] advocated for the use of the internal energy functional for the definition of constitutive models in nonlinear electro-mechanics. In essence, the authors postulated a definition of the internal energy convex with respect to an extended set of arguments, namely the deformation gradient tensor F, its co-factor H, its Jacobian J, the Lagrangian electric displacement field D0 and d, defined as d=FD0 and proved that this definition satisfies the ellipticity condition unconditionally. Crucially, it is then possible to safely introduce the Helmholtz energy via a Legendre transformation.

The objective of this paper is to derive a new and robust EM time integrator scheme, tailor-made for nonlinear electro-elastodynamics [[23], [24]]. The proposed time integrator is extremely suitable for long-term numerical simulations of EAPs in both actuation and energy harvesting scenarios. An example that showcases the applicability of the method in actuation scenarios is the consideration of EAP-based actuators subjected to a time dependent electric field in order to achieve user-defined operational configurations.

Consistent implicit EM time integration schemes (EM schemes in the sequel) inherit the conservation laws of total energy, linear momentum and angular momentum. The consistency of these types of algorithms refer to their ability to preserve or dissipate the total energy of a system in agreement with the laws of thermodynamics [25], hence not restricting their applicability to (reversible) hyperelastic isotropic material models. For instance, EM consistent integrators can also be applied in the context of mixed variational formulations [26], anisotropic material behaviour [27], non-linear visco-elastodynamics [[25], [28]] and non-linear elasto-thermodynamics [[29], [30]]. A great overview of the development of EM schemes is provided by the textbook [31].

In the purely mechanical case, the thermodynamical consistency of these methods is ensured by virtue of replacing the (exact) derivative of the strain energy with respect to the right Cauchy–Green deformation tensor (i.e. the consistent second Piola–Kirchhoff stress tensor) with its algorithmic counterpart. The latter, denoted as discrete derivative [[24], [26], [30], [32]] must be defined in compliance with the so-called directionality property. Specially interesting is the recent work by Betsch et al. [33], where a new consistent EM time integrator scheme has been developed in the context of polyconvex elasticity. Essentially, these authors proposed the consideration of three discrete derivatives of the strain energy (as opposed to the single discrete derivative used in the standard case). Each discrete derivative represents the algorithmic counterpart of the work conjugates of the right Cauchy–Green deformation tensor, its co-factor and its determinant. In a recent work [30], this formulation has been extended to the field of thermo-elasticity. In comparison to previously proposed discrete derivative expressions (see e.g. [[24], [34]]), the new stress formula in [[30], [33]] assumes a remarkably simple form. A key factor for that simplification is the use of a tensor cross product pioneered by the Boer [35] and employed for the first time by Bonet et al. [[36], [37]] in the case of nonlinear electromechanics [[20], [21], [22], [38]].

The outline of this paper is as follows: in Section 2, some basic principles of kinematics are presented. The governing equations in nonlinear electro-elastodynamics are also presented in this section. The concept of multi-variable convexity and its importance from the material stability point of view is presented in Section 3. Section 4 starts with the three-field mixed formulation presented in Ref. [20] in the context of static electromechanics. Its extension to electro-elastodynamics is then carried out by defining the appropriate action integral. After derivation of the stationary conditions of the action integral, Section 5 introduces a new one-step implicit EM time integrator scheme for electro-elastodynamics. Section 6 briefly describes the finite element implementation of the new time integrator scheme and Section 7 presents five numerical examples in order to validate the conservation properties and robustness of the new scheme. Finally, Section 8 provides some concluding remarks. The Appendix outlines the definition of the discrete derivative expressions featuring in the proposed time integrator in Section 5.

Section snippets

Nonlinear continuum electromechanics

A brief introduction into nonlinear continuum electromechanics and the relevant governing equations will be presented in this section.

Constitutive equations in nonlinear electro-elasticity

The governing equations presented in Section 2 are coupled by means of a suitable constitutive law. The objective of the following section is to introduce some notions on constitutive laws in nonlinear electro-elasticity.

Electro-elastodynamics

The objective of this section is to present the variational formulation that will be used in order to develop an EM time integration scheme in Section 5.

Energy–momentum integration scheme for electro-elastodynamics

Following the work of Simo [23], Gonzalez [24], Romero [32] and Betsch et al. [[26], [33]] in the context of nonlinear elasticity and Franke et al. [30] in the context of thermoelasticity, the objective of this section is to propose an EM preserving time discretisation scheme for the set of weak forms in (35).

Finite element implementation

As standard in finite elements, the domain B0 described in Section 2.1 and representing the EAP is sub-divided into a finite set of non-overlapping elements eE such that B0B0h=eEB0e.

The unknown fields {v,ϕ,φ,D0} in the semi-discrete weak forms Wv, Wϕ, Wφ and WD0 in (47) are discretised employing the following functional spaces Vϕh×Vϕh×Vφh×VD0h defined as Vϕh={ϕVϕ;ϕhB0e=a=1nnodeϕNaϕϕa};Vφh={φVφ;φhB0e=a=1nnodeφNaφφa};VD0h={D0VD0;D0hB0e=a=1nnodeD0NaD0D0a},where for any field

Numerical examples

The objective of this section is to study the performance of the newly proposed (internal energy-based) EM time integration scheme presented in Eq. (47) in a variety of examples.

Conclusions

A new consistent energy–momentum one-step time integrator scheme is presented in the context of nonlinear electro-elastodynamics. Both a two-field Helmholtz’s-based scheme and a three-field internal energy-based scheme have been developed. Following the work of Gil and Ortigosa [[20], [21], [22]], the internal energy functional is preferred in this paper over the Helmholtz energy functional for the definition of electromechanical constitutive models, due to considerations of material stability.

Acknowledgements

The first and fourth authors acknowledge the support provided by the Sêr Cymru National Research Network under the Ser Cymru II Fellowship “Virtual engineering of the new generation of biomimetic artificial muscles”, funded by the European Regional Development Fund . The third and fifth authors acknowledge the support provided by the Deutsche Forschungsgemeinschaft (DFG) under Grant BE 2285/9-2. The fourth author acknowledges the financial support received through the European Training Network

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      With the variational formulation in (64) the balance laws for the considered thermo-electro-elastic continuum are examined. For the above weak form (64) we aim at the development of an EM consistent time integration scheme following the publications [26,27,30,32,34,35,43,45,46]. The definition of the discrete derivatives, which are presented in detail in Appendix A, must satisfy two crucial properties for the design of EM time integrators, namely:

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