A computational framework for incompressible electromechanics based on convex multi-variable strain energies for geometrically exact shell theory
Introduction
Electro Active Polymers (EAPs) belong to a special class of smart materials with very attractive actuator and energy harvesting capabilities [1]. Piezoelectric polymers and dielectric elastomers are some of the most representative examples of this kind of materials. The latter have shown electrically induced area expansions of up to 1980% [2], demonstrating their outstanding capabilities as soft robots, wing morphing actuators for remotely controlled micro air vehicles, adaptive optics, balloon catheters and Braille displays among others [2], [3], [4], [5], [6].
In this paper, a computational framework for the simulation of incompressible EAPs using a geometrically exact shell theory in scenarios characterised by large strains and/or large electric fields is presented. A particularisation/degeneration of the variational and constitutive frameworks developed by the authors in previous publications [7], [8], [9], [10] to the case of shells is carried out. This is motivated by the large number of applications [6] 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 [11], [12], [13], [14], [15], [16], [17], [18], 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 interpolation 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 Refs. [13], [20], where the in-extensibility of the director field (which guarantees that this field follows an orthogonal transformation [12]) is enforced 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 typically 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 Ref. [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 complex 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, Schröder et al. [12] have explored the consideration of complex polyconvex [22], [23], [24], [25], [26] 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 models, satisfying the ellipticity condition and hence, material stability [27], [28], [29] 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 [31], [32], [33], [34] in the context of solid mechanics. This tensor cross product 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 concept of multi-variable convexity is extended to the context of nonlinear shell theory. The tangent operators of the (convex multi-variable) internal energy and the Helmholtz’s energy are also presented in this section. Section 4 presents a classical three-field mixed variational principle [10], [35] used for incompressible three-dimensional electromechanics and its degeneration to the particular case of a shell. Section 5 discusses aspects regarding the Finite Element implementation of the proposed formulation. Section 6 presents some numerical examples to demonstrate the applicability of the formulation and its comparison with well-established continuum formulations. Finally, Section 7 provides some concluding remarks and a summary of the key contributions of this paper.
Section snippets
Continuum kinematics
Consider the three dimensional deformation of a possible EAP from its initial configuration occupying a volume , of boundary , into a final configuration occupying a volume , of boundary . The motion of the EAP is described by a pseudo-time dependent mapping which relates a material particle from the material configuration to the spatial configuration according to (refer to Fig. 1). Virtual and incremental variations of will be denoted by and , respectively. It will
Shells kinematics
Let us assume that the continuum (EAP) described in Section 2.1 can be kinematically described as a shell. This is the case for the majority of applications of EAPs, where they feature as thin shell-like components. Let the reference configuration of the shell be characterised by a mid surface , parametrised in terms of convective coordinate as . For clarity, the reader is referred to Fig. 2. Let the mid-surface be perpendicular to the thickness of the shell,
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. This stems from the following standard three-field total energy minimisation variational principle [10], [35] where denotes the exact solution for the geometry , the electric potential and the pressure field , respectively, and , the work done by external contributions, defined as
Finite element discretisation
The objective of this section is to describe relevant aspects regarding the discretisation of the different fields in the set , featuring in the variational principle in Eq. (37).
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 [7], [8], [9], [10], 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 in Eq. (11), enables to show
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 [7], [8], [9], [10], degenerated in this paper to the case of a nonlinear shell theory. Moreover, the kinematics of the shell allows for the possibility of
Acknowledgement
Both authors acknowledge the financial support provided by the Sêr Cymru National Research Network for Advanced Engineering and Materials.
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