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

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.

Abstract: 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 (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.

Summary

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.

Figures

Figure 5: Summary of FEM discretisation spaces for examples 6.1.1 and 6.1.2 used in (a), the MŴF formulation [4] and (b), in the proposed shell formulation. Notice that the superscripts C and D have been used to indicate the continuous or discontinuous character of a field, respectively.

Figure 15: Hyperboloid piezoelectric polymer. Geometry and boundary conditions. t = 12m, D = 9.95m, H = 0.05m and q, a surface load per unit undeformed area.

Figure 6: Bending actuator configuration 1. Contour plot of σ11 (a) for the shell formulation and (b) the MŴF, respectively. Incompressible model in (52). Total number of dofs for shell and continuum formulation ({x, ϕ, p}) of 14498 dofs (refer to Figure 4) and {3 × (8019), 8019, 800} (32876 dofs) (not including the discontinuous fields, refer to Figure 5a), respectively.

Figure 1: Deformation mapping of a continuum (EAP) and associated kinematics variables: F , H , J .

Figure 16: Hyperboloid piezoelectric polymer. Contour plot of the electric potential ϕ for the proposed shell formulation. Results obtained for a value of the surface load per unit undeformed area of q = λ×(16×106)Pa, with (a) λ = 0.0065; (b) λ = 0.0221; (c) λ = 0.0716; (d) λ = 0.25; (e) λ = 0.664; (f) λ = 1. Constitutive model in (58). Finite Element discretisation of 64 × 27 elements. CBC approach for ϕ and γ with ns = 1 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3× 7095, 3× 1820, 3× 7095, 1× 1728, 1× 1728, 1820}.

Figure 11: Helicoidal actuator. Geometry description and electrical boundary conditions.

Figure 13: Helicoidal actuator. Contour plot of the p for the proposed shell formulation. Incompressible model in (56). Results obtained for a value of ω0 = λ × 10−5Q/m2 with (a) λ = 200; (b) λ = 300; (c) λ = 500; (d) λ = 600; (e) λ = 800; (f) λ = 900; (g) λ = 1100; (h) λ = 1200; (i) λ = 1300. Finite Element discretisation of 29× 6 elements. CBC approach for ϕ and γ with ns = 2 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3 × 767, 3 × 210, 5 × 767, 2 × 174, 2 × 174, 210}. Reference configuration represented by shadowed region.

Figure 3: Bending actuator. Electrical boundary conditions for: (a) configuration 1 and (b), configuration 2. a = 10m, b = 1m and H = 0.05m.

Figure 17: Hyperboloid piezoelectric polymer. Contour plot of the H22, σ33, p, ϕ, E3 and D03 for the proposed shell formulation. Constitutive model in (58). Results obtained for a value of q = λ× (16× 106)Pa. Finite Element discretisation of 64× 27 elements. CBC approach for ϕ and γ with ns = 1 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3×7095, 3×1820, 3×7095, 1× 1728, 1× 1728, 1820}.

Figure 10: Bending actuator configuration 2. Contour plot of σ11 (a) for the shell formulation and (b) the MŴF, respectively. Incompressible model in (52). Total number of dofs for shell and continuum formulation ({x, ϕ, p}) of 14498 (refer to Figure 9) and {3 × (8019), 8019, 800}=32876 (not including the discontinuous fields, refer to Table 5a), respectively.

Figure 4: Bending actuator configuration 1. Contour plot of ΣF 11 . Results for an applied surface electric charge of ω0 = (λ/300)×(3× 10−3) Q/m2 with λ = 25 (shadowed configuration) and λ = 42. Incompressible model in (52). Finite Element discretisation of (40×5) elements. CBC approach for {ϕ, p} and CBD for γ with ns = 4 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3× 891, 3× 246, 11× 891, 8× 80, 5× 80, 246} (14498 dofs).

Figure 2: Kinematics of the shell: (a) Reference and current configurations of the mid surface of the shell describing an EAP. (b) Covariant basis {G01,G02,V } in the mid surface of the shell (s = 0) and covariant basis {G1,G2,V } at a point characterised by the same convective coordinates {ηα, α = 1, 2} and s 6= 0, all in the reference configuration.

Figure 14: Helicoidal actuator. Evolution of area expansion 1/γ for a point of the shell in the reference configuration at X1 = −0.75m, X2 = 0m and X3 = 2m.

Figure 9: Bending actuator configuration 2. Contour plot of ΣF 11 . Results for an applied surface electric charge of ω0 = (λ/300) × (3× 10−3) Q/m2 with (a) λ = 7; (b) λ = 14; (c) λ = 21; (d) λ = 28; (e) λ = 35; (f) λ = 42; (g) λ = 49; (h) λ = 56; (i) λ = 63; (j) λ = 70; (k) λ = 77; (l) λ = 86. Finite Element discretisation of (40 × 5) elements. CBC approach for ϕ and CBD for γ with ns = 2 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3× 891, 3× 246, 11× 891, 8× 80, 5× 80, 246} (14498 dofs).

Frequently asked questions

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. 

Q2. What are the future works mentioned in the paper "A computational framework for incompressible electromechanics based on convex multi-variable strain energies for geometrically exact shell theory" ?

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.