Electroelastic Energy Research Articles

Instability of single polymer chain in an electroelastic problem

Under electric loading produced by compliant electrodes, a dielectric elastomer is prone to material instabilities which, in a microstructural level, may connect to single polymer chain instability. To reveal if such connection exists, we aim to use an electroelastic energy model for single polymer chain to study the chain instability. We approximate curvature shape of the chain under the electric loading by using trigonometry series in a spatial coordinate. The Rayleigh–Ritz method is then applied to solve the energy equation formed by the trigonometry series. We demonstrate in the numerical examples that the instability of the chain may occur at the value of electric field with corresponding configuration of the chain close to its full length.

Improving the energy density and flexibility of PMN-0.3PT based piezoelectric generator by composite designing

Ceramics based piezoelectric generators are known for their high energy density and poor flexibility. In this work, 0.7Pb(Mg1/3Nb2/3O3)-0.3PbTiO3 (PMN-0.3PT) and polydimethylsiloxane (PDMS) based 2–2 composite with optimum PMN-0.3PT content (vr) was designed that demonstrated enhanced output energy density and superior mechanical flexibility under dynamic mechanical excitation. vr-PMN-0.3PT/PDMS 2–2 composite with different PMN-PT reinforcement content (vr) and two different reinforcement configurations were fabricated and characterized for effective electro-elastic properties and energy harvesting response. Parallelly, using the finite element method and analytical models, effective electromechanical properties were calculated. Composites with parallel connectivity of the reinforcement phase demonstrated enhanced piezoelectric charge coefficient (d33c) even with low PMN-0.3PT content whereas the relative permittivity (κ33c) and elastic modulus (E33c) exhibited a linearly increasing trend with reinforcement volume fraction. At a compressive load of 50 N and 5 Hz frequency, a piezoelectric generator (PG) based on a vr = 0.2, vr- PMN-0.3PT/PDMS 2–2 composite with parallel connectivity produced a maximum short-circuit current density of 69 nA/cm2 and an open-circuit electric field of 189 V/cm, translating to a maximum output power density of ∼13 μW/cm3 higher than that of pristine PMN-0.3 PT based piezoelectric generator. Estimated mechanical flexibility was found to be ∼53 % higher than that of pristine PMN-0.3PT.

An electroelastic Kirchhoff rod theory incorporating free space electric energy

This work presents a geometrically exact Kirchhoff-like electroelastic rod theory wherein the contribution of free space energy is also factored in. In addition to the usual mechanical variables such as the rod’s centerline and cross-section orientation, three electric potential parameters are also introduced to account for the variation in electric potential within the rod’s cross-section as well as along the rod length. The free space energy is included through an electric flux-like variable acting on the lateral surface of the rod. The governing equations of this rod are derived through balance of forces and moments (arising from both mechanical and electric sources) for mechanical variables, weighted integration of the three-dimensional electric displacement equation in the rod’s cross-section for electric potential parameters and through the asymptotic expansion of a boundary integral equation for the electric flux variable. We also obtain the expressions of internal contact moment and conjugates of electric potential parameters in terms of the rod’s aforementioned mechanical and electric variables. A specific choice of the three-dimensional electroelastic stored energy density is further taken catering to piezoelectric materials and dielectric elastomers and the rod’s constitutive equations are rederived which exhibit interesting interplay between mechanical and electric variables. The presented work can help improve the design of soft electroelastic rod-like actuators and sensors.

Thermal fluctuation spectrum of flexoelectric viscoelastic semiflexible filaments and polymers: A line liquid crystal model

AbstractIn this paper, we generalize the internal viscosity model developed by Professor Williams to semiflexible polymers, biofilaments, and worm‐like micelles, where molecular dissipation is generated by bending. Current models for viscoelasticity with internal viscosity in semiflexible polymers and filaments are based on generalizations of the worm‐like model, but they neglect potential electromechanical couplings such as flexoelectricity. In this paper, inspired by the early work of Professor Williams, we develop a model for worm‐like viscoelastic flexoelectric filaments based on the “line liquid crystal model”. The electroelastic free energy and entropy production are formulated and used to derive the shape equation for these filaments undergoing thermal fluctuations. The resulting time relaxation spectrum is a useful tool to characterize experimentally viscoelastic material parameters. We show that flexoelectricity or polarization‐induced bending softens the filaments. The predicted time relaxation spectrum shows that at longer wavelength modes, the filament behaves like a rigid rod in a viscous solvent, but at shorter wavelengths, it reaches a plateau defined by the bending time scale. The key effect of flexoelectricity is to shift the entire spectrum to higher values, slowing down the response. The model itself is validated using the worm‐like chain and the viscoelasticity of liquid crystals, and the predictions are shown to be in qualitative consistency with the data. Since filament flexoelectricity is associated with 1D sensor‐actuator functionalities, the presented model has many potential novel applications in reduced geometries.

Bending control and stability of functionally graded dielectric elastomers

A rectangular plate of dielectric elastomer exhibiting gradients of material properties through its thickness will deform inhomogeneously when a potential difference is applied to compliant electrodes on its major surfaces, because each plane parallel to the major surfaces will expand or contract to a different extent. Here we study the voltage-induced bending response of a functionally graded dielectric plate on the basis of the nonlinear theory of electroelasticity, when both the elastic shear modulus and the electric permittivity change with the thickness coordinate. The theory is illustrated for a neo-Hookean electroelastic energy function with the shear modulus and permittivity varying linearly across the thickness. In general the bending angle increases with the potential difference, and this enables the material inhomogeneity to be tuned to control the bending shape. We derive the Hessian criterion that ensures stability of the bent configurations in respect of a general form of electroelastic constitutive law specialized for the considered geometry. This requires that the Hessian remains positive. For the considered model we show that the bent configuration is stable until the voltage reaches the value for which the cross section of the bent configuration forms a complete circle.

Composite dielectric elastomers modeling based on statistical mechanics

We propose in this work a hybrid microstructural-phenomenological modeling of composite dielectric elastomer based on statistical mechanics. The composite dielectric elastomer is made from high permittivity filler particles doped to elastomeric based material. We assume that the particles are uniformly distributed in the based material and form particle-chain structures. With the help of statistical mechanics, the electroelastic strain energy is formulated from the particle-chain reorientation induced by applied electric field. Moreover, the phenomenological dielectric constant model is used to eliminate the inverse Langevin function in the strain energy formulation. Our proposed constitutive model is validated with respect to two electroelastic experiments which use two composite dielectric elastomers with different material properties. From the validation, our proposed model works well to capture large stretch electroelastic deformations not only for different material properties but also for different particle fraction in the composite dielectric elastomer.

Bifurcation of finitely deformed thick-walled electroelastic spherical shells subject to a radial electric field

This paper is concerned with the bifurcation analysis of a pressurized electroelastic spherical shell with compliant electrodes on its inner and outer boundaries. The theory of small incremental electroelastic deformations superimposed on a radially finitely deformed electroelastic thick-walled spherical shell is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect spherical shape of the shell. Specifically, axisymmetric bifurcations are analyzed, and results are obtained for three different electroelastic energy functions, namely electroelastic counterparts of the neo-Hookean, Gent and Ogden elastic energy functions. For the neo-Hookean energy function it was reported previously that for the purely mechanical case axisymmetric bifurcations are possible under external pressure only, no bifurcation solutions being possible for internally pressurized spherical shells. In the case of an electroelastic neo-Hookean model bifurcation under internal pressure becomes possible when the potential difference between the electrodes exceeds a certain value, which depends on the ratio of inner to outer undeformed radii. Results obtained for the three classes of model are significantly different and are illustrated for a range of fixed values of the potential difference. Although of less practical significance, results are also shown for fixed charges, and these are both different between the models and different from the case of fixed potential difference.

Fine tuning the electro-mechanical response of dielectric elastomers

We propose a protocol to model accurately the electromechanical behavior of dielectric elastomer membranes using experimental data of stress-stretch and voltage-stretch tests. We show how the relationship between electric displacement and the electric field can be established in a rational manner from these data. Our approach demonstrates that the ideal dielectric model, prescribing linearity in the purely electric constitutive equation, is quite accurate at low-to-moderate values of the electric field and that, in this range, the dielectric permittivity constant of the material can be deduced from stress-stretch and voltage-stretch data. Beyond the linearity range, more refined couplings are required, possibly including a non-additive decomposition of the electro-elastic energy. We also highlight that the presence of vertical asymptotes in voltage-stretch data, often observed in the experiments just prior to failure, should not be associated with strain stiffening effects but instead with the rapid development of electrical breakdown.

The out-of-plane behaviour of dielectric membranes: Description of wrinkling and pull-in instabilities

Voltage controlled dielectric membranes exhibit two fundamental types of instability, strongly affecting their performances: the occurrence of wrinkling, which is due to membranal compressive stresses, and the onset of pull-in, a catastrophic thinning localisation that preludes electrical breakdown. In this manuscript we provide a unifying energetic description of both instabilities for large, out-of-plane and inhomogeneous deformations. By using the ideas of relaxation and regularisation of the energy, originally proposed by Pipkin (1986) and Hilgers and Pipkin (1992) for purely elastic membranes, we show that the onset and development of wrinkling can be effectively described by the relaxed electroelastic energy. For axially symmetric membranes and neo-Hookean materials, we show that pull-in corresponds to failure of the strong ellipticity condition of the regularised electroelastic energy, thus extending to out-of-plane deformations the validity of a previous estimate for planar systems (Zurlo et al., 2017). In agreement with ubiquitous experimental evidence, we also show that wrinkled states are always stable below the pull-in voltage. Our theoretical findings are assessed by the comparison with experiments on out-of-plane, voltage-actuated annular membranes, showing good agreement both in terms of description of wrinkled states, and for the prediction of the pull-in instability.

Bifurcation of finitely deformed thick-walled electroelastic cylindrical tubes subject to a radial electric field

This paper is concerned with the bifurcation analysis of a pressurized electroelastic circular cylindrical tube with closed ends and compliant electrodes on its curved boundaries. The theory of small incremental electroelastic deformations superimposed on a finitely deformed electroelastic tube is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect cylindrical shape of the tube. First, prismatic bifurcations are examined and solutions are obtained which show that for a neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This result contrasts with that for the purely elastic case for which prismatic bifurcation modes were found only for an externally pressurized tube. Second, axisymmetric bifurcations are analyzed, and results for both neo-Hookean and Mooney–Rivlin electroelastic energy functions are obtained. The solutions show that in the presence of a moderate electric field the electroelastic tube becomes more susceptible to bifurcation, i.e., for fixed values of the axial stretch axisymmetric bifurcations become possible at lower values of the circumferential stretches than in the corresponding problems in the absence of an electric field. As the magnitude of the electric field increases, however, the possibility of bifurcation under internal pressure becomes restricted to a limited range of values of the axial stretch and is phased out completely for sufficiently large electric fields. Then, axisymmetric bifurcation is only possible under external pressure.

A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics

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.

Finite deformations of an electroelastic circular cylindrical tube

In this paper the theory of nonlinear electroelasticity is used to examine deformations of a pressurized thick-walled circular cylindrical tube of soft dielectric material with closed ends and compliant electrodes on its curved boundaries. Expressions for the dependence of the pressure and reduced axial load on the deformation and a potential difference between, or uniform surface charge distributions on, the electrodes are obtained in respect of a general isotropic electroelastic energy function. To illustrate the behaviour of the tube, specific forms of energy functions accounting for different mechanical properties coupled with a deformation independent quadratic dependence on the electric field are used for numerical purposes, for a given potential difference and separately for a given charge distribution. Numerical dependences of the non-dimensional pressure and reduced axial load on the deformation are obtained for the considered energy functions. Results are then given for the thin-walled approximation as a limiting case of a thick-walled cylindrical tube without restriction on the energy function. The theory described herein provides a general basis for the detailed analysis of the electroelastic response of tubular dielectric elastomer actuators, which is illustrated for a fixed axial load in the absence of internal pressure and fixed internal pressure in the absence of an applied axial load.

Comportamiento ferroeléctrico de una nanoesfera de titanato de plomo debido a campos de despolarización y a esfuerzos mecánicos

A theorical model has been developed based on the theory of Ginzburg-Landau-Devonshire to study and predict the effects the decreasing of size particle in a nanosphere of PbTiO3 subjected to the action of depolarization fields and mechanical stress. It was considered that the nanosphere is surrounded by a layer of space charges on its surface, and containing 180° domains generated by minimizing free energy of depolarization. Energy density of depolarization, wall domain and electro-elastic energy have been incorporated into the free energy of the theory Ginzburg-Landau-Devonshire. Free energy minimization was performed to determine the spontaneous polarization and transition temperature system. These results show that the transition temperature for nanosphere is substantially smaller than the corresponding bulk material. Also, it has been obtained that the stability of the ferroelectric phase of nanosphere is favored for configurations with a large number of 180° domains, with the decreasing of thickness space charge layer, and the application of tensile stress and decreases with compressive stress.

Anomalous toughening in nanoscale ferroelectrics with polarization vortices

Unusual mechanical behavior of cracks in nanoscale ferroelectrics, where spontaneous polarization characteristically forms closed-flux vortices, is investigated using state-of-the-art real-space phase-field modeling based on the Ginzburg–Landau theory. An anomalously large toughening effect is revealed in nanoscale ferroelectrics due to drastic stress release near the crack tip, which is almost one order of magnitude larger in stress intensity than that observed in macroscale materials. Such anomalous toughening is attributed to an unusual switching behavior of the polarization vortices in nanoscale ferroelectrics, which is no longer localized near the crack tip as in macroscale ferroelectrics, but expands to the entire structure through the splitting and multiplication of vortices. We further demonstrate that this local-to-global switching is intrinsically induced by strong cross-coupling between the ferroelectric polarization and mechanical strain that concentrates to the electro-elastic energy, not only in the vicinity of crack tip, but also to each polarization vortex due to its inhomogeneous distribution. Our finding provides a novel insight into nanoscale polarization vortices and leads to an entirely new strategy for the tailoring and improvement of fracture toughness in ferroelectric materials by engineering the microstructure of polarization vortices.

Dielectric elastomers: Asymptotically-correct three-dimensional displacement field

Asymptotically-accurate dimensional reduction from three to two dimensions and recovery of 3-D displacement field of non-prestretched dielectric hyperelastic membranes are carried out using the Variational Asymptotic Method (VAM) with moderate strains and very small ratio of the membrane thickness to its shortest wavelength of the deformation along the plate reference surface chosen as the small parameters for asymptotic expansion. Present work incorporates large deformations (displacements and rotations), material nonlinearity (hyperelasticity), and electrical effects. It begins with 3-D nonlinear electro-elastic energy and mathematically splits the analysis into a one-dimensional (1-D) through-the-thickness analysis and a 2-D nonlinear plate analysis. Major contribution of this paper is a comprehensive nonlinear through-the-thickness analysis which provides a 2-D energy asymptotically equivalent of the 3-D energy, a 2-D constitutive relation between the 2-D generalized strain and stress tensors for the plate analysis and a set of recovery relations to express the 3-D displacement field. Analytical expressions are derived for warping functions and stiffness coefficients. This is the first attempt to integrate an analytical work on asymptotically-accurate nonlinear electro-elastic constitutive relation for compressible dielectric hyperelastic model with a generalized finite element analysis of plates to provide 3-D displacement fields using VAM. A unified software package ‘VAMNLM’ (Variational Asymptotic Method applied to Non-Linear Material models) was developed to carry out 1-D non-linear analysis (analytical), 2-D non-linear finite element analysis and 3-D recovery analysis. The applicability of the current theory is demonstrated through an actuation test case, for which distribution of 3-D displacements are provided.

Nonlinear response of an electroelastic spherical shell

In this paper the theory of nonlinear electroelasticity is used to examine radial deformations of a thick-walled spherical shell of soft dielectric material with compliant electrodes on its inner and outer surfaces combined with an internal pressure. A general expression for the dependence of the pressure on the deformation and a potential difference between the electrodes, or uniform surface charge distributions on the electrodes, is obtained in respect of a general incompressible isotropic electroelastic energy function. To illustrate the behavior of the shell specific forms of energy functions are used for numerical purposes, for either fixed potential difference or fixed charge distribution. Explicit general results are given for the limiting case of a thin-walled (or membrane) shell. Depending on the elastic part of the energy function (as distinct from that part which involves the electric field) different behaviors are obtained, including non-monotonicity of the pressure–radius relationship reminiscent of that known in the purely elastic situation, but moderated by the presence of an electric field.

Effect of the compressive stress on both polarization rotation and phase transitions in PMN-30%PT single crystal

In this paper, we have investigated the dependence of both the electromechanical effect and the electrostriction on the compressive stress in PMN-30%PT single crystal on the basis of single domain polarization rotation model. In the model, the electroelastic energy induced by the compressive stress is taken into account. The results have demonstrated that the compressive stress can lead to a significant change in the initial polarization state in the crystal. The reason lies in the stress induced anisotropy which is the coupling between the compressive stress and the electrostrictive coefficients. Thus, the initial polarization state in single crystal is determined by the combination of both electrocrystalline anisotropy and the stress induced anisotropy. The compressive stress along the [100] axis can make the polarization in the crystal be perpendicular to the stress direction, and make it difficult to be polarized to the saturation. This model is useful for better understanding both the polarization rotation and electromechanical effect in ferroelectric crystals with the compressive stress present.

A finite-strain constitutive theory for electro-active polymer composites via homogenization

This paper presents a homogenization framework for electro-elastic composite materials at finite strains. The framework is used to develop constitutive models for electro-active composites consisting of initially aligned, rigid dielectric particles distributed periodically in a dielectric elastomeric matrix. For this purpose, a novel strategy is proposed to partially decouple the mechanical and electrostatic effects in the composite. Thus, the effective electro-elastic energy of the composite is written in terms of a purely mechanical component together with a purely electrostatic component, this last one dependent on the macroscopic deformation via appropriate kinematic variables, such as the particle displacements and rotations, and the change in size and shape of the appropriate unit cell. The results show that the macroscopic stress includes contributions due to the changes in the effective dielectric permittivity of the composite with the deformation. For the special case of a periodic distribution of electrically isotropic, spherical particles, the extra stresses are due to changes with the deformation in the unit cell shape and size, and are of order volume fraction squared, while the corresponding extra stresses for the case of aligned, ellipsoidal particles can be of order volume fraction, when changes are induced by the deformation in the orientation of the particles.

Continuum electromechanical modeling of protein-membrane interactions

A continuum electromechanical model is proposed to describe the membrane curvature induced by electrostatic interactions in a solvated protein-membrane system. The model couples the macroscopic strain energy of membrane and the electrostatic solvation energy of the system, and equilibrium membrane deformation is obtained by minimizing the electroelastic energy functional with respect to the dielectric interface. The model is illustrated with the systems with increasing geometry complexity and captures the sensitivity of membrane curvature to the permanent and mobile charge distributions.

Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification

We propose and experimentally validate a first-principles based model for the nonlinear piezoelectric response of an electroelastic energy harvester. The analysis herein highlights the importance of modeling inherent piezoelectric nonlinearities that are not limited to higher order elastic effects but also include nonlinear coupling to a power harvesting circuit. Furthermore, a nonlinear damping mechanism is shown to accurately restrict the amplitude and bandwidth of the frequency response. The linear piezoelectric modeling framework widely accepted for theoretical investigations is demonstrated to be a weak presumption for near-resonant excitation amplitudes as low as 0.5 g in a prefabricated bimorph whose oscillation amplitudes remain geometrically linear for the full range of experimental tests performed (never exceeding 0.25% of the cantilever overhang length). Nonlinear coefficients are identified via a nonlinear least-squares optimization algorithm that utilizes an approximate analytic solution obtained by the method of harmonic balance. For lead zirconate titanate (PZT-5H), we obtained a fourth order elastic tensor component of c1111p=−3.6673×1017 N/m2 and a fourth order electroelastic tensor value of e3111=1.7212×108 m/V.