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Journal ArticleDOI

A computational framework for incompressible electromechanics based on convex multi-variable strain energies for geometrically exact shell theory

15 Apr 2017-Computer Methods in Applied Mechanics and Engineering (North-Holland)-Vol. 317, pp 792-816
TL;DR: In this paper, a new computational framework for the analysis of incompressible Electro Active Polymer (EAP) shells subjected to large strains and large electric fields is presented, based on a rotationless description of the kinematics of the shell, enhanced with extra degrees of freedom corresponding to the thickness stretch and the hydrostatic pressure.
About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2017-04-15 and is currently open access. It has received 15 citations till now. The article focuses on the topics: Shell (structure) & Hydrostatic pressure.

Summary (3 min read)

1. Introduction

  • Electro Active Polymers (EAPs) belong to a special class of smart materials with very attractive actuator and energy harvesting capabilities [5].
  • This rotationless approach, which complies with the principle of material frame indifference [21], avoids a well-known drawback associated with rotation-based formulations.
  • More specifically, convex multi-variable electromechanical constitutive models, satisfying the ellipticity condition and hence, material stability [27–29] for the entire range of deformations and electric fields, are used for the first time in the context of shell theory.
  • Section 3 presents the kinematical description of the proposed shell formulation.
  • Additionally, the 3 concept of multi-variable convexity is extended to the context of nonlinear shell theory.

3.1. Shells kinematics

  • This is the case for the majority of applications of EAPs, where they feature as thin shell-like components.
  • Notice that the spatial vector v does not have to be necessarily perpendicular to the plane Γ. Moreover, γ(ηα, s) in above equation (11) represents the thickness stretch [17], which accounts for possible deformations across the thickness of the shell.
  • Notice that this extra field γ might depend not only on the convective coordinates ηα, but also 7 upon s.

3.4. Tangent operators in incompressible electro-elasticity. Continuum degenerate shell formulation

  • As shown in Section 3.2, the kinematics of the shell leads to further geometrical non-linearities with respect to the continuum formulation.
  • As a result, these extra non-linearities will also be reflected in the tangent operators of the internal and Helmholtz’s energy functionals, e (20) and Φ (27), respectively4.

3.4.1. Tangent operator of the internal energy e

  • The tangent operator of both the isochoric and volumetric components of the internal energy, ê and U , respectively, can be defined for the continuum 4Refer to [4] for a comparison with the tangent operator of both the internal and Helmholtz’s energy functionals emerging in the continuum formulation.
  • Notice that these tensors introduce an additional geometrical nonlinearity, represented by the second terms on the right hand side of both tangent operators in equation (28), with respect to the tangent operators emerging in the continuum formulation, presented in Reference [4].

4. Variational formulation of nearly and incompressible dielectric elastomer shells

  • The objective of this Section is to present the variational framework for the proposed shell formulation.
  • An iterative6 Newton-Raphson process is usually preferred to converge 5The expression of the external virtual work DW ext[δu0, δv, δγ] is well known and, hence, omitted.
  • 6The letter k will indicate iteration number.
  • These recursive relationships (carried out at every Gauss point of the domain) between the Helmholtz’s energy Φ̂ and the internal energy.

5.2. Interpolation across the thickness of the shell

  • The interpolation of the uniparametric functions J a(s) is carried out via element-wise (e) continuous (or discontinuous) Lagrange polynomial interpolants of degree pJ .
  • When considering continuous (or discontinuous) interpolants, this will be denoted as Continuum-Based-Continuous (CBC) (or Continuum-Based-Discontinuous (CBD)) approach, both described as J a(s) = ns∑ e=1 pJ+1∑ b=1 J abe N bJ e(s), (47) where J abe represents a degree of freedom, N bJ e(s) its associated shape function and ns the number of elements in the discretisation of s.
  • The CBC approach has been used for the fields {ϕ, γ, p} and the CBD approach has been specifically used for the field γ when discontinuous strains are expected across the thickness.
  • In addition, CBC and CBD approaches have been compared against a truncated Taylor series expansion, as that in [43], denoted as Taylor-Expansion (TE) approach.

6. Numerical examples

  • The objective of this section is to demonstrate the applicability of the proposed shell formulation via a series of numerical examples, in which convex multi-variable electromechanical constitutive models, defined in the context of continuum formulations [1–4], will be considered.
  • In all the examples, a reconstruction of the continuum associated with the shell has been carried out at a post-processing level.
  • This reconstruction, based on the mapping x in equation (11), enables to show results not only in the mid surface of the shell but also across its thickness.

6.1. Bending actuators

  • This example considers the actuation device with geometry depicted in Figure 3. 6.1.1. Results for bending actuator configuration 1 Objective 2: The second objective is to test the performance of the formulation in scenarios characterised by the presence of discontinuities of the electric field distribution across the thickness of the shell.
  • Interestingly, Figures 7g−l show the purely mechanical and electrical contributions of the Cauchy stress tensor.
  • Regarding objective 2 and objective 3, the same conclusion as those obtained in the previous example are obtained and hence, omitted for brevity.

6.2. Helicoidal actuator

  • Regarding the boundary conditions, the degrees of freedom associated with the displacements of the mid surface of the shell and the director field d at X3 = 0m are completely constrained.
  • An electric charge per unit undeformed area of +ω0 and −ω0 is applied in both electrodes .
  • The value of the material parameters chosen for this particular example are shown in Table 2.
  • The first objective of this example is to demonstrate the applicability of the proposed formulation to scenarios where the reference configuration of the shell is curved, as that described by the cylindrically parametrised geometry (in the reference configuration) in equation (55), also known as Objective 1.
  • Figure 13 shows the contour plot of various stress and electric-like fields for a fixed value of the applied electric charge ω0.

6.3. Hyperboloid piezoelectric polymer

  • The hyperboloid with geometry described in Figure 15, presented in the context of pure elasticity in Reference [12], has been considered.
  • The material is transversely anisotropic, with the preferred axis of anisotropy N tangent to the surface of the hyperboloid as depicted in Figure 15.
  • The objective of this following example is to demonstrate the applicability of the proposed shell formulation to piezoelectric materials, where deformations can create a distribution of electric field in the material.
  • 8The area expansion has been computed as 1/γ, with γ the thickness stretch.
  • 34 35 Figures 16 displays contour plot of the (mechanically induced) electric field E3 for different values of the applied surface force q. Finally, Figure 17 shows the contour plot distribution of H22, σ33, p, ϕ, E1 and D03 for a given value of the applied surface force q.

7. Concluding remarks

  • This paper has provided a computational approach to formulate incompressible EAPs shells undergoing large strains and large electric field scenarios.
  • The proposed formulation, based upon a rotationless kinematical description of the shell, stems from the variational and constitutive framework proposed by the authors in previous publications [1–4], degenerated in this paper to the case of a nonlinear shell theory.
  • Two approaches have been considered for the interpolation of the electric potential across the thickness of the shell.
  • Specifically, the continuumbased-continuous (CBC) approach described in Section 5.2 and the Taylor expansion approach (TE) in [43].
  • A comparison of the results rendered by both approaches has been presented.

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Citations
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Journal ArticleDOI
TL;DR: In this paper, a geometrically nonlinear theory for circular cylindrical shells made of incompressible hyperelastic materials is developed, which is higher-order in both shear and thickness deformations.

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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.

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Cites background from "A computational framework for incom..."

  • ...A convex multi-variable strain energy description based on the works of Gil and Ortigosa [1, 2, 3] is chosen for modelling EAPs under actuation and energy harvesting scenarios....

    [...]

  • ...For the case of small strains, the staggered scheme is shown to capture the electrostrictive behaviour of EAPs fairly well with a threshold point in applied voltage beyond which the fully coupled nonlinear solver becomes computational more favourable....

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  • ...On the other end of the spectrum lies the class of mathematically more sophisticated formulations that exploit the large deformation characteristics of EAP [5, 14, 15, 1, 2, 16, 17, 18]....

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  • ...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....

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  • ...In recent years, exploiting actuation and harvesting through the heterogenous class of ElectroActive-Polymers (EAP) has received considerable research focus....

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TL;DR: In this article, the behavior of heterogeneous magnetorheological composites subjected to large deformations and external magnetic fields is studied and different types of boundary conditions based on the primary variables of the magneto-elastic enthalpy and internal energy functionals are applied to solve the problem at the micro-scale.

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TL;DR: Domain-aware expression templates combined with SIMD instructions are shown to provide a significant speed-up over the classical low-level style programming techniques.

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Cites background from "A computational framework for incom..."

  • ...Recently, Gil and Ortigosa [47, 48, 58, 59] have introduced the concept of multi-variable convexity, which satisfies the well-posedness of the governing equations described in subsection 2.2, and postulated as e(F ,D0) = W (F ,H , J,D0,d); d = FD0, (4) where W represents a convex multi-variable functional in terms of the extended set of arguments V = {F ,H , J,D0,d}....

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  • ...Recently, Gil and Ortigosa [47, 48, 58, 59] have introduced the concept of multi-variable convexity, which satisfies the well-posedness of the governing equations described in subsection 2....

    [...]

Journal ArticleDOI
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.

20 citations

References
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Journal ArticleDOI
TL;DR: In this paper, an extension of the shell theory and numerical analysis presented in Part I, II and III to include finite thickness stretch and initial variable thickness is presented, which plays a significant role in problems involving finite membrane strains, contact, concentrated surface loads and delamination (in composite shells).
Abstract: This paper in concerned with the extension of the shell theory and numerical analysis presented in Part I, II and III to include finite thickness stretch and initial variable thickness. These effects play a significant role in problems involving finite membrane strains, contact, concentrated surface loads and delamination (in composite shells). We show that a direct numerical implementation of the standard single extensible director shell model circumvents the need for rotational updates, but exhibits numerical ill-conditioning in the thin shell limit. A modified formulation obtained via a multiplicative split of the director field into an extensible and inextensible part is presented, which involves only a trivial modification of the weak form of the equilibrium equations considered in Part III, and leads to a perfectly well-conditioned formulation in the thin-shell limit. In sharp contrast with previous attempts in the context of the degenerated solid approach, the thickness stretch is an independent field, not a dependent variable updated iteratively via the plane stress condition. With regard to numerical implementation, an exact update procedure which automatically ensures that the thickness stretch remains positive is presented. For the present theory, standard displacement models would exhibit ‘locking’ in the incompressible limit as a result of the essentially three-dimensional character of the constitutive equations. A mixed formulation is described which circumvents this difficulty. Numerical examples are presented that illustrate the effects of the thickness stretch, the performance of the proposed mixed interpolation, and the well-conditioned response exhibited by the present approach in the thin-shell (inextensible director) limit.

452 citations

Journal ArticleDOI
TL;DR: In this article, a discrete canonical, singularity-free mapping between the five and the six degree of freedom formulation is constructed by exploiting the geometric connection between the orthogonal group (SO(3)) and the unit sphere (S2).
Abstract: Computational aspects of a linear stress resultant (classical) shell theory, obtained by systematic linearization of the geometrically exact nonlinear theory, considered in Part I of this work, are examined in detail. In particular, finite element interpolations for the reference director field and the linearized rotation field are constructed such that the underlying geometric structure of the continuum theory is preserved exactly by the discrete approximation. A discrete canonical, singularity-free mapping between the five and the six degree of freedom formulation is constructed by exploiting the geometric connection between the orthogonal group (SO(3)) and the unit sphere (S2). The proposed numerical treatment of the membrane and bending fields, based on a mixed Hellinger-Reissner formulation,provides excellent results for the 4-node bilinear isoparametric element. As an example, convergent results are obtained for rather coarse meshes in fairly demanding, singularity-dominated, problems such as the classical rhombic plate test. The proposed theory and finite element implementation are evaluated through an extensive set of benchmark problems. The results obtained with the present approach exactly match previous solutions obtained with state-of-the-art implementations based on the so-called degenerated solid approach.

450 citations

Journal ArticleDOI
TL;DR: In this article, a formulation of polyconvex anisotropic hyperelasticity at finite strains is proposed, where the authors represent the governing constitutive equations within the framework of the invariant theory, in order to guarantee the existence of minimizers.

415 citations

Proceedings ArticleDOI
10 Jul 2002
TL;DR: Testing with experimental biological techniques and apparatus has confirmed that these dielectric elastomer artificial muscles can indeed reproduce several of the important characteristics of natural muscle.
Abstract: To achieve desirable biomimetic motion, actuators must be able to reproduce the important features of natural muscle such as power, stress, strain, speed of response, efficiency, and controllability. It is a mistake, however, to consider muscle as only an energy output device. Muscle is multifunctional. In locomotion, muscle often acts as an energy absorber, variable-stiffness suspension element, or position sensor, for example. Electroactive polymer technologies based on the electric-field-induced deformation of polymer dielectrics with compliant electrodes are particularly promising because they have demonstrated high strains and energy densities. Testing with experimental biological techniques and apparatus has confirmed that these dielectric elastomer artificial muscles can indeed reproduce several of the important characteristics of natural muscle. Several different artificial muscle actuator configurations have been tested, including flat actuators and tubular rolls. Rolls have been shown to act as structural elements and to incorporate position sensing. Biomimetic robot applications have been explored that exploit the muscle-like capabilities of the dielectric elastomer actuators, including serpentine manipulators, insect-like flapping-wing mechanisms, and insect-like walking robots.

333 citations

Journal ArticleDOI
TL;DR: In this article, the authors place a dielectric elastomer near the verge of snap-through instability, trigger the instability with voltage, and bend the snapthrough path to avert electric breakdown.
Abstract: Dielectric elastomers are capable of large voltage-induced deformation, but achieving such large deformation in practice has been a major challenge due to electromechanical instability and electric breakdown. The complex nonlinear behavior suggests an important opportunity: electromechanical instability can be harnessed to achieve giant voltage-induced deformation. We introduce the following principle of operation: place a dielectric elastomer near the verge of snap-through instability, trigger the instability with voltage, and bend the snap-through path to avert electric breakdown. We demonstrate this principle of operation with a commonly used experimental setup—a dielectric membrane mounted on a chamber of air. The behavior of the membrane can be changed dramatically by varying parameters such as the initial pressure in the chamber, the volume of the chamber, and the prestretch of the membrane. We use a computational model to analyze inhomogeneous deformation and map out bifurcation diagrams to guide the experiment. With suitable values of the parameters, we obtain giant voltage-induced expansion of area by 1692%, far beyond the largest value reported in the literature.

302 citations

Frequently Asked Questions (2)
Q1. What have the authors contributed in "A computational framework for incompressible electromechanics based on convex multi-variable strain energies for geometrically exact shell theory" ?

In this paper, a new computational framework for the analysis of incompressible Electro Active Polymer ( EAP ) shells subjected to large strains and large electric fields is presented. Two novelties are incorporated in this work. First, the variational and constitutive frameworks developed by the authors in recent publications [ 1–4 ] in the context of three-dimensional electromechanics are particularised/degenerated to the case of geometrically exact shell theory. The proposed formulation follows a rotationless description of the kinematics of the shell, enhanced with extra degrees of freedom corresponding to the thickness stretch and the hydrostatic pressure, critical for the consideration of incompressibility. More specifically, convex multi-variable ( three-dimensional ) constitutive models, complying with the ellipticity condition and hence, satisfying material stability for the entire range of deformations and electric fields, Corresponding author: r. ortigosa @ swansea. Different approaches are investigated for the interpolation of these extra fields and that of the electric potential across the thickness of the shell. 

Moreover, the kinematics of the shell allows for the possibility of compression and stretch across the thickness of the shell [ 17 ], crucial for the consideration of incompressible behaviour. Two approaches have been considered for the interpolation of the electric potential across the thickness of the shell.