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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

AbstractIn 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 (Gil and Ortigosa, 2016; Ortigosa and Gil, 2016; Ortigosa et al., 2016) in the context of three-dimensional electromechanics are particularised/degenerated to the case of geometrically exact shell theory. This formulation is computationally very convenient as EAPs are typically used as thin shell-like components in a vast range of applications. 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. Different approaches are investigated for the interpolation of these extra fields and that of the electric potential across the thickness of the shell. Crucially, this allows for the simulation of multilayer and composite materials, which can display a discontinuous strain distribution across their thickness. As a second novelty, a continuum degenerate approach allows for the consideration of complex three-dimensional electromechanical constitutive models, as opposed to those defined in terms of the main strain measures of the shell. 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, are used for the first time in the context of shell theory.

Topics: Shell (structure) (54%), Hydrostatic pressure (53%)

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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Accepted Manuscript
A computational framework for incompressible electromechanics
based on convex multi-variable strain energies for geometrically exact
shell theory
Rogelio Ortigosa, Antonio J. Gil
PII: S0045-7825(16)31444-X
DOI: http://dx.doi.org/10.1016/j.cma.2016.12.034
Reference: CMA 11280
To appear in: Comput. Methods Appl. Mech. Engrg.
Received date: 27 October 2016
Revised date: 22 December 2016
Accepted date: 30 December 2016
Please cite this article as: R. Ortigosa, A.J. Gil, A computational framework for
incompressible electromechanics based on convex multi-variable strain energies for
geometrically exact shell theory, Comput. Methods Appl. Mech. Engrg. (2017),
http://dx.doi.org/10.1016/j.cma.2016.12.034
This is a PDF file of an unedited manuscript that has been accepted for publication. As a
service to our customers we are providing this early version of the manuscript. The manuscript
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A computational framework for incompressible
electromechanics based on convex multi-variable strain
energies for geometrically exact shell theory
Rogelio Ortigosa
1
, Antonio J. Gil
2
Zienkiewicz Centre for Computational Engineering, College of Engineering
Swansea University, Bay Campus, SA1 8EN, United Kingdom
Abstract
In this paper, a new computational framework for the analysis of incompress-
ible 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 re-
cent publications [14] in the context of three-dimensional electromechanics
are particularised/degenerated to the case of geometrically exact shell theory.
This formulation is computationally very convenient as EAPs are typically
used as thin shell-like components in a vast range of applications. The pro-
posed 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 incom-
pressibility. Different approaches are investigated for the interpolation of
these extra fields and that of the electric potential across the thickness of the
shell. Crucially, this allows for the simulation of multilayer and composite
materials, which can display a discontinuous strain distribution across their
thickness. As a second novelty, a continuum degenerate approach allows for
the consideration of complex three-dimensional electromechanical constitu-
tive models, as opposed to those defined in terms of the main strain measures
of the shell. More specifically, convex multi-variable (three-dimensional) con-
stitutive models, complying with the ellipticity condition and hence, satisfy-
ing material stability for the entire range of deformations and electric fields,
1
Corresponding author: r.ortigosa@swansea.ac.uk
2
Corresponding author: a.j.gil@swansea.ac.uk
Preprint submitted to Computational Mechanics December 22, 2016

are used for the first time in the context of shell theory.
Keywords: material stability, shell, geometrically exact shell theory,
Electro Active Polymers
1. Introduction
Electro Active Polymers (EAPs) belong to a special class of smart ma-
terials with very attractive actuator and energy harvesting capabilities [5].
Piezoelectric polymers and dielectric elastomers are some of the most repre-
sentative examples of this kind of materials. The latter have shown electri-
cally induced area expansions of up to 1980% [6], demonstrating their out-
standing capabilities as soft robots, wing morphing actuators for remotely
controlled micro air vehicles, adaptive optics, balloon catheters and Braille
displays among others [610].
In this paper, a computational framework for the simulation of incom-
pressible EAPs using a geometrically exact shell theory in scenarios charac-
terised by large strains and/or large electric fields is presented. A particulari-
sation/degeneration of the variational and constitutive frameworks developed
by the authors in previous publications [14] to the case of shells is carried
out. This is motivated by the large number of applications [10] where EAPs
feature as very thin shell-like components, for which this formulation is very
convenient from the computational standpoint.
In the context of geometrically exact shell theory [1118], some authors
[12, 19] follow an approach where interpolation of the director field (initially
perpendicular to the mid surface of the shell) is preferred over interpola-
tion of rotations. Nonetheless, it is ultimately rotations and not the director
field which are part of the unknowns of the problem. In contrast, we follow
in this paper a completely rotationless approach, similar to that presented
in Reference [13, 20], where the in-extensibility of the director field (which
guarantees that this field follows an orthogonal transformation [12]) is en-
forced as a constraint. Notice however that both approaches are equivalent
at least in the continuum case. This rotationless approach, which complies
with the principle of material frame indifference [21], avoids a well-known
drawback associated with rotation-based formulations. When considering
rotation-based formulations, rotations around the shell normal (known as
drilling rotations) do not introduce any stiffness contribution in this specific
direction and hence, lead to an ill-conditioning of the system. This is typ-
2

ically overcome via the addition of an appropriate (drilling) constraint or
penalty term [15, 16].
The proposed shell formulation shares some common features with the
classical Reissner-Mindlin theory. Specifically, sections initially straight in
the reference configuration remain straight after the motion of the shell and
hence, out of plane warping deformations are not considered. Furthermore,
in order to account for the incompressibility of the shell, two additional
unknown fields are included as in Reference [17], namely the pressure and
thickness stretch fields, the latter enhancing the kinematical description of
the classical Reissner-Mindlin theory. Regarding the interpolation across
the thickness of the shell, different strategies are considered in this paper.
Critically, these enable to capture discontinuities of the strain field across
the thickness of the shell, enabling to simulate the response of composite and
multilayered shells.
In the context of large strain elasticity, several authors have used com-
plex three-dimensional constitutive models for the Finite Element analysis
of shells, as opposed to simpler constitutive models (derived from the Saint-
Venant Kirchhoff model) in terms of the main strain measures of the shell.
In particular, Schr¨oder et al. [12] have explored the consideration of complex
polyconvex [2226] anisotropic constitutive models. This paper explores the
consideration of complex electromechanical three-dimensional constitutive
models to the particular case of a continuum degenerate shell formulation.
More specifically, convex multi-variable electromechanical constitutive mod-
els, satisfying the ellipticity condition and hence, material stability [2729]
for the entire range of deformations and electric fields, are used for the first
time in the context of shell theory.
The present formulation utilises the algebra based on the tensor cross
product operation pioneered in [30] and reintroduced and exploited for the
first time in [3134] in the context of solid mechanics. This tensor cross prod-
uct operation is particularly helpful when dealing with convex multi-variable
constitutive laws, where invariants of the co-factor and the determinant of
the deformation feature heavily in the representation of the internal energy
functional.
The paper is organised as follows. Section 2 briefly revises the main
strain measures and their directional derivatives in the three-dimensional
continuum formulation. Furthermore, the Faraday law, the electric field and
the electric potential are introduced in this Section. Section 3 presents the
kinematical description of the proposed shell formulation. Additionally, the
3

Figures (14)
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Abstract: In the present work, the behavior of heterogeneous magnetorheological composites subjected to large deformations and external magnetic fields is studied. Computational homogenization is used to derive the macroscopic material response from the averaged response of the underlying microstructure. The microstructure consists of two materials and is far smaller than the characteristic length of the macroscopic problem. 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. The overall responses of the RVEs with different sizes and particle distributions are studied under different loads and magnetic fields. The results indicate that the application of each set of boundary conditions presents different macroscopic responses. However, increasing the size of the RVE, solutions from different boundary conditions get closer to each other and converge to the response obtained from periodic boundary conditions.

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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

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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....

    [...]

  • ...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]....

    [...]

  • ...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....

    [...]

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

    [...]


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.
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.

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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.
Abstract: The paper presents aspects of implementation of a new high performance tensor contraction framework for the numerical analysis of coupled and multi-physics problems on streaming architectures. In addition to explicit SIMD instructions and smart expression templates, the framework introduces domain specific constructs for the tensor cross product and its associated algebra recently rediscovered by Bonet et al. (2015, 2016) in the context of solid mechanics. The two key ingredients of the presented expression template engine are as follows. First, the capability to mathematically transform complex chains of operations to simpler equivalent expressions, while potentially avoiding routes with higher levels of computational complexity and, second, to perform a compile time depth-first or breadth-first search to find the optimal contraction indices of a large tensor network in order to minimise the number of floating point operations. For optimisations of tensor contraction such as loop transformation, loop fusion and data locality optimisations, the framework relies heavily on compile time technologies rather than source-to-source translation or JIT techniques. Every aspect of the framework is examined through relevant performance benchmarks, including the impact of data parallelism on the performance of isomorphic and nonisomorphic tensor products, the FLOP and memory I/O optimality in the evaluation of tensor networks, the compilation cost and memory footprint of the framework and the performance of tensor cross product kernels. The framework is then applied to finite element analysis of coupled electro-mechanical problems to assess the speed-ups achieved in kernel-based numerical integration of complex electroelastic energy functionals. In this context, 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....

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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.