Stimuli-Responsive Shell Theory
J-H Lee, H.S. Park and D.P. Holmes
Accepted for publication in Mathematics and Mechanics of Solids 2023
Abstract
The mechanics of soft matter generally involve finite deformations and instabilities of structures in response to a wide range of mechanical and non-mechanical
stimuli. Modeling plates and shells is generally a challenge due to their geometrically nonlinear response to loads, however non-mechanical loads further
complicate matters as it is often not clear how they modify the shell's energy functional. In this work, we demonstrate how to form a mechanical interpretation
of these non-mechanical stimuli, in which the standard shell strain energy can be augmented with potentials corresponding to how a non-mechanical stimulus acts
to change the shell's area and curvature via the natural stretch and curvature. As a result, the effect of non-mechanical stimuli to deform shells is transformed
into effective external loadings, and this framework allows for the application of analytical and computational tools that are standard within the mechanics
community. Furthermore, we generalize the effect of mass change during biological growth to account for its effect on the stress constitution. The theory is
formulated based on a standard, stress-free reference configuration which is known a priori, meaning that it can be physically observed, and only requires
the solution of a single-field equation, the standard mechanical momentum or equilibrium equation, despite capturing the effects of non-mechanical stimuli.
We validate the performance of this model by several benchmark problems, and finally we apply it to complex examples including the snapping of the Venus flytrap,
leaf growth, and the buckling of electrically active polymer plates. Overall, we expect that mechanicians and non-mechanicians alike can use the approach presented
here to easily modify existing computational tools with the effective external loadings calculated in this novel theory, in order to study how various types of
non-mechanical stimuli impact the mechanics and physics of thin shell structures.
A NURBS-Based Inverse Analysis of Swelling Induced Morphing of Thin Stimuli-Responsive Polymer Gels
N. Vu-Bac, T. Rabczuk, H.S. Park, X. Fu and X. Zhuang
Computer Methods in Applied Mechanics and Engineering 2022; 397:115049
Abstract
Gels are a mixture of cross-linked polymers and solvents, and have been widely studied in recent years for a diverse range of biomedical applications.
Because gels can undergo large, reversible shape changes due to swelling, their complex physical response must be modeled by coupling large reversible
deformation and mass transport. An ongoing challenge in this field is the ability to capture swelling or residual swelling-induced of such stimuli-responsive
gels from initially flat two-dimensional (2D) to three-dimensional (3D) curved shapes. Specifically, because such shape changes typically involve large
deformations, shape changes, and the exploitation of elastic instabilities, it remains an open question as to what external stimulus should be prescribed
to generate a specific target shape. Therefore, we propose a novel formulation that tackles, using both nonlinear kinematics and material models,
the coupling between elasticity and solvent transport using Kirchhoff-Love shell theory discretized using isogeometric analysis (IGA). Second, we propose an
inverse methodology that chemomechanically couples large deformation and mass transport to identify the external stimuli prescribed to generate a specific
target shape. Our numerical examples demonstrate the capability of identifying the required external stimuli, with the implication that the reconstructed
target shapes are accurate, including cases where the shape changes due to swelling involve elastic instabilities or softening. Overall, our study can be
used to effectively predict and control the large morphological changes of an important class of stimuli-responsive materials.
This paper is available in PDF form
.
Efficient Snap-Through of Spherical Caps by Applying a Localized Curvature Stimulus
L. Stein-Montalvo, J-H Lee, Y. Yang, M. Landesberg, H.S. Park and D.P. Holmes
The European Physical Journal E 2022; 45:3
Abstract
In bistable actuators and other engineered devices, a homogeneous stimulus (e.g., mechanical, chemical, thermal, or magnetic) is often applied to an entire
shell to initiate a snap-through instability. In this work, we demonstrate that restricting the active area to the shell boundary allows for a large reduction
in its size, thereby decreasing the energy input required to actuate the shell. To do so, we combine theory with 1D finite element simulations of spherical
caps with a non-homogeneous distribution of stimulus- responsive material. We rely on the effective curvature stimulus, i.e., the natural curvature induced
by the non-mechanical stimulus, which ensures that our results are entirely stimulus-agnostic. To validate our numerics and demonstrate this generality, we
also perform two sets of experiments, wherein we use residual swelling of bilayer silicone elastomers—a process that mimics differential growth—as well as a
magneto-elastomer to induce curvatures that cause snap-through. Our results elucidate the underlying mechanics, offering an intuitive route to optimal design
for efficient snap-through.
This paper is available in PDF form
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Elastic Instabilities Govern the Morphogenesis of the Optic Cup
J-H Lee, H.S. Park and D.P. Holmes
Physical Review Letters 2021; 127:138102
Abstract
Because the normal operation of the eye depends on sensitive morphogenetic processes for its eventual shape, developmental flaws can lead to wide-ranging ocular defects. However, the
physical processes and mechanisms governing ocular morphogenesis are not well understood. Here, using analytical theory and nonlinear shell finite-element simulations, we show for
optic vesicles experiencing matrix-constrained growth that elastic instabilities govern the optic cup morphogenesis. By capturing the stress amplification owing to mass increase
during growth, we show that the morphogenesis is driven by two elastic instabilities analogous to the snap-through in spherical shells, where the second instability is sensitive to
the optic cup geometry. In particular, if the optic vesicle is too slender, it will buckle and break axisymmetry, thus preventing normal development. Our results shed light on
the morphogenetic mechanisms governing the formation of a functional biological system, and the role of elastic instabilities in the shape selection of soft biological structures.
This paper is available in PDF form
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The Nonlinear Buckling Behavior of a Complete Spherical Shell under Uniform External Pressure and Homogeneous Natural Curvature
D.P. Holmes, J-H Lee, H.S. Park and M. Pezzulla
Physical Review E 2020; 102:023003
Abstract
In this work, we consider the stability of a spherical shell under combined loading from a uniform external pressure and a homogenous natural
curvature. Non--mechanical stimuli, such as one that tends to modify the rest curvature of an elastic body, are prevalent in a wide range of
natural and engineered systems, and may occur due to thermal expansion, changes in pH, differential swelling, and differential growth. Here,
we investigate how the presence of both an evolving natural curvature and an external pressure modifies the stability of a complete spherical
shell. We show that due to a mechanical analogy between pressure and curvature, positive natural curvatures can severely destabilize a thin shell,
while negative natural curvatures can strengthen the shell against buckling, providing the possibility to design shells that buckle at or above
the theoretical limit for pressure alone, i.e. a strengthening factor. These results extend directly from the classical analysis of the
stability of shells under pressure, and highlight the important role that non--mechanical stimuli can have on modifying the membrane state of
stress in a thin shell.
This paper is available in PDF form
.
A NURBS-Based Inverse Analysis of Thermal Expansion Induced Morphing of Thin Shells
N. Vu-Bac, T.X. Duong, T. Lahmer, P. Areias, R.A. Sauer, H.S. Park and T. Rabczuk
Computer Methods in Applied Mechanics and Engineering 2019; 350:480-510
Abstract
Soft, active materials have been widely studied due their ability to undergo large, complex shape changes in response to both mechanical
and non-mechanical external stimuli. However, the vast majority of such studies has focused on investigating the forward problem, i.e.
determining the shape changes that result from the applied stimuli. In contrast, very little work has been done to solve the inverse problem,
i.e. that of identifying the external loads and stimuli that are needed to generate desired shapes and morphological changes. In this work,
we present a new inverse methodology to study residual thermal expansion induced morphological changes in geometric composites made of soft,
thin shells. In particular, the method presented in this work aims to determine the prescribed external stimuli needed to reconstruct a specific
target shape, with a specific focus and interest in morphological changes from two-dimensional (2D) to three-dimensional (3D) shapes by
considering the external stimuli within a thermohyperelastic framework. To do so, we utilize a geometrically exact, rotation-free Kirchhoff-Love
shell formulation discretized by NURBS-based shape functions. We show that the proposed method is capable of identifying the stimuli, including
cases where thermal expansion induced shape changes involving elastic softening occur in morphing from the initially flat 2D to non-planar 3D
shapes. Validation indicates that the reconstructed shapes are in good agreement with the target shape.
This paper is available in PDF form
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A Methodology for Modeling Surface Effects on Stiff and Soft Solids
J. He and H.S. Park
Computational Mechanics 2018; 61:687-697
Abstract
We present a computational method that can be applied to capture surface stress and surface tension-driven effects in both stiff,
crystalline nanostructures, like size-dependent mechanical properties, and soft solids, like elastocapillary effects. We show that
the method is equivalent to the classical Young-Laplace model. The method is based on converting surface tension and surface elasticity
on a zero-thickness surface to an initial stress and corresponding elastic properties on a finite thickness shell, where the
consideration of geometric nonlinearity enables capturing the out-of-plane component of the surface tension that results for curved
surfaces through evaluation of the surface stress in the deformed configuration. In doing so, we are able to use commercially available
finite element technology, and thus do not require consideration and implementation of the classical Young-Laplace equation. Several
examples are presented to demonstrate the capability of the methodology for modeling surface stress in both soft solids and
crystalline nanostructures.
This paper is available in PDF form
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A Staggered Explicit-Implicit Finite Element Formulation for Electroactive Polymers
S. Seifi, K.C. Park and H.S. Park
Computer Methods in Applied Mechanics and Engineering 2018; 337:150-164
Abstract
Electroactive polymers such as dielectric elastomers (DEs) have attracted significant attention in recent years. Computational
techniques to solve the coupled electromechanical system of equations for this class of materials have universally centered around
fully coupled monolithic formulations, which while generating good accuracy requires significant computational expense. However, this
has significantly hindered the ability to solve large scale, fully three-dimensional problems involving complex deformations and
electromechanical instabilities of DEs. In this work, we provide theoretical basis for the effectiveness and accuracy of staggered
explicit-implicit finite element formulations for this class of electromechanically coupled materials, and elicit the simplicity of
the resulting staggered formulation. We demonstrate the stability and accuracy of the staggered approach by solving complex
electromechanically coupled problems involving electroactive polymers, where we focus on problems involving electromechanical
instabilities such as creasing, wrinkling, and bursting drops. In all examples, essentially identical results to the fully monolithic
solution are obtained, showing the accuracy of the staggered approach at a significantly reduced computational cost.
This paper is available in PDF form
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A NURBS-Based Inverse Analysis for Reconstruction of Nonlinear Deformations of Thin Shell Structures
N. Vu-Bac, T.X. Duong, T. Lahmer, X. Zhuang, R.A. Sauer, H.S. Park and T. Rabczuk
Computer Methods in Applied Mechanics and Engineering 2018; 331:427-455
Abstract
This article presents original work combining a NURBS-based inverse analysis with both kinematic and constitutive nonlinearities to
recover the applied loads and deformations of thin shell structures. The inverse formulation is tackled by gradient based optimization
algorithms based on computed and measured displacements at a number of discrete locations. The proposed method allows accurately
recovering the target shape of shell structures such that instabilities due to snapping and buckling are captured. The results obtained
show good performance and applicability of the proposed algorithms to computer-aided manufacturing of shell structures.
This paper is available in PDF form
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Kirigami Actuators
M.A. Dias, M.P. McCarron, D. Rayneau-Kirkhope, P.Z. Hanakata, D.K. Campbell, H.S. Park and D.P. Holmes
Soft Matter 2017; 13:9087-9092
Abstract
Thin elastic sheets bend easily and, if they are patterned with cuts, can deform in sophisticated ways. Here we show that carefully
tuning the location and arrangement of cuts within thin sheets enables the design of mechanical actuators that scale down to
atomically--thin 2D materials. We first show that by understanding the mechanics of a single, non--propagating crack in a sheet
we can generate four fundamental forms of linear actuation: roll, pitch, yaw, and lift. Our analytical model shows that these
deformations are only weakly dependent on thickness, which we confirm with experiments at centimeter scale objects and molecular
dynamics simulations of graphene and MoS2 nanoscale sheets. We show how the interactions between non--propagating cracks
can enable either lift or rotation, and we use a combination of experiments, theory, continuum computational analysis, and molecular
dynamics simulations to provide mechanistic insights into the geometric and topological design of kirigami actuators.
This paper is available in PDF form
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Nanomechanical Probing of Thin-Film Dielectric Elastomer Transducers
B. Osmani, S. Seifi, H.S. Park, V. Leung, T. Töpper and B. Müller
Applied Physics Letters 2017; 111:093104
Abstract
Dielectric elastomer transducers (DETs) have attracted interest as generators, actuators, sensors, and even as self-sensing actuators
for applications in medicine, soft robotics, and microfluidics. Their performance crucially depends on the elastic properties of the
electrode-elastomer sandwich structure. The compressive displacement of a single-layer DET can be easily measured using atomic force
microscopy (AFM) in contact mode. While polymers used as dielectric elastomers are known to exhibit significant mechanical stiffening
for large strains, their mechanical properties when subjected to voltages are not well understood. To examine this effect, we measured
the depths of 400 nanoindentations as a function of the applied electric field using a spherical AFM probe with a radius of (522 ± 4) nm.
Employing a field as low as 20 V/µm, the indentation depths increased by 42% at a 100 nN load with respect to the field-free condition,
implying an electromechanically driven elastic softening of the DET. This at-a-glance surprising experimental result agrees with related
nonlinear, dynamic finite element model simulations. Furthermore, the pull-off forces rose from (23.0 ± 0.4) to (49.0 ± 0.7) nN implying
a nanoindentation imprint after unloading. This embossing effect is explained by the remaining charges at indentation site.
The root-mean-square roughness of the Au electrode raised by 11% increasing the field from zero to 12 V/µm demonstrating that the
electrode's morphology change is an undervalued factor in the fabrication of DET structures.
This paper is available in PDF form
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Electro-elastocapillary Rayleigh-Plateau Instability in Dielectric Elastomer Films
S. Seifi and H.S. Park
Soft Matter 2017; 13:4305-4310
Abstract
We demonstrate, using both finite element simulations and a linear stability analysis, the emergence of an electro-elastocapillary Rayleigh-Plateau instability
in dielectric elastomer (DE) films under 2D, plane strain conditions. When subject to an electric field, the DEs exhibit a buckling instability for small
elastocapillary numbers. For larger elastocapillary numbers, the DEs instead exhibit the Rayleigh-plateau instability. The stability analysis demonstrates
the critical effect of the electric field in causing the Rayleigh-plateau instability, which cannot be induced solely by surface tension in DE films.
Overall, this work demonstrates the effects of geometry, boundary conditions, and multi-physical coupling on a new example of Rayleigh-Plateau
instability in soft solids.
This paper is available in PDF form
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Computational Modeling of Electro-Elasto-Capillary Phenomena in Dielectric Elastomers
S. Seifi and H.S. Park
International Journal of Solids and Structures 2016; 87:236-244
Abstract
We present a new finite deformation, dynamic finite element model that incorporates surface tension to capture elastocapillary effects on the
electromechanical deformation of dielectric elastomers. We demonstrate the significant effect that surface tension can have on the deformation
of dielectric elastomers through three numerical examples: (1) surface tension effects on the deformation of single finite elements with
homogeneous boundary conditions; (2) surface tension effects on instabilities in constrained dielectric elastomer films, and (3) surface tension
effects on bursting drops in solid dielectrics. Generally, we find that surface tension creates a barrier to instability nucleation. Specifically,
we find in agreement with recent experimental studies of constrained dielectric elastomer films a transition in the surface instability mechanism
depending on the elastocapillary length. The present results indicate that the proposed methodology may be beneficial in studying the electromechanical
deformation and instabilities for dielectric elastomers in the presence of surface tension.
This paper is available in PDF form
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Coarse-Grained Model of the J-integral of Carbon Nanotube Reinforced Polymer Composites
B. Arash, H.S. Park and T. Rabczuk
Carbon 2016; 96:1084-1092.
Abstract
The J-integral is recognized as a fundamental parameter in fracture mechanics that characterizes the inherent resistance of materials to crack growth.
However, the conventional methods to calculate the J-integral, which require knowledge of the exact position of a crack tip and the continuum fields
around it, are unable to precisely measure the J-integral of polymer composites at the nanoscale. This work aims to propose an effective calculation
method based on coarse-grained (CG) simulations for predicting the J-integral of carbon nanotube (CNT)/polymer composites. In the proposed approach,
the J-integral is determined from the load displacement curve of a single specimen. The distinguishing feature of the method is the calculation of
J-integral without need of information about the crack tip, which makes it applicable to complex polymer systems. The effects of the CNT weight fraction
and covalent cross-links between the polymer matrix and nanotubes, and polymer chains on the fracture behavior of the composites are studied in detail.
The dependence of the J-integral on the crack length and the size of representative volume element (RVE) is also explored.
This paper is available in PDF form
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Tensile Fracture Behavior of Short Carbon Nanotube Reinforced Polymer Composites: A Coarse-Grained Model
B. Arash, H.S. Park and T. Rabczuk
Composite Structures 2015; 134:981-988
Abstract
Short-fiber-reinforced polymer composites are increasingly used in engineering applications and industrial products owing to their unique combination of
superior mechanical properties, and relatively easy and low-cost manufacturing process. The mechanical behavior of short carbon nanotube (CNT) polymer
composites, however, remains poorly understood due to size and time limitations of experiments and atomistic simulations. To address this issue, the tensile
fracture behavior of short CNT reinforced poly (methyl methacrylate) (PMMA) matrix composites is investigated using a coarse-grained (CG) model. The
reliability of the CG model is demonstrated by reproducing experimental results on the strain-stress behavior of the polymer material. The effect of the
nanotube weight fraction on the mechanical properties, i.e. the Young's modulus, yield strength, tensile strength and critical strain, of the CNT/polymer
composites is studied in detail. The dependence of the mechanical properties of the composites on the orientation and length-to-diameter aspect ratio of
nanotube reinforcements is also examined.
This paper is available in PDF form
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The Temperature Dependent Viscoelastic Behavior of Dielectric Elastomers
J. Guo, R. Xiao, H.S. Park and T.D. Nguyen
Journal of Applied Mechanics 2015; 82:091009
Abstract
In this paper, we investigated the temperature dependent viscoelastic behavior of dielectric elastomers and the effects of viscoelasticity on the
electro-actuation behavior. We performed dynamic thermo-mechanical analysis to measure the master curve of the stress relaxation function and the temperature
dependence of the relaxation time of VHB 4905, a commonly used dielectric elastomer. The master curve was applied to calculate the viscoelastic spectrum for a
discrete multi-process finite deformation viscoelastic model. In addition, we performed uniaxial creep and stress relaxation experiments and electrical actuation
experiments under different prestretch conditions. The measured spectrum was applied to predict the experimental results. Generally, the model produced good
quantitative agreement with both the viscoelastic and electro-actuation experiments, which shows the necessity of using a multi-process relaxation model to
accurately capture the viscoelastic response for VHB. However, the model under-predicted the electro-actuated creep strain for high voltages near the pull-in
instability. We attributed the discrepancies to the complex boundary conditions that were not taken into account in the simulation. We also investigated the
failure of VHB membrane caused by viscoelastic creep when prestretched and subjected to constant voltage loading. The experimental time to failure for the
specimens decreased exponentially with voltage, which agreed well with the predictions of the model.
This paper is available in PDF form
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Mechanical Properties of Carbon Nanotube Reinforced Polymer Nanocomposites: A Coarse-Grained Model
B. Arash, H.S. Park and T. Rabczuk
Composites: Part B 2015; 80:92-100
Abstract
In this work, a coarse-grained (CG) model of carbon nanotube (CNT) reinforced polymer matrix composites is developed. A distinguishing feature of the CG model
is the ability to capture interactions between polymer chains and nanotubes. The CG potentials for nanotubes and polymer chains are calibrated using the strain
energy conservation between CG models and full atomistic systems. The applicability and efficiency of the CG model in predicting the elastic properties of
CNT/polymer composites are evaluated through verification processes with molecular simulations. The simulation results reveal that the CG model is able to
estimate the mechanical properties of the nanocomposites with high accuracy and low computational cost. The effect of the volume fraction of CNT reinforcements
on the Young's modulus of the nanocomposites is investigated. The application of the method in the modeling of large unit cells with randomly distributed CNT
reinforcements is examined. The established CG model will enable the simulation of reinforced polymer matrix composites across a wide range of length scales
from nano to mesoscale.
This paper is available in PDF form
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Electrostatically-Driven Creep in Viscoelastic Dielectric Elastomers
J. Wang, T.D. Nguyen and H.S. Park
Journal of Applied Mechanics 2014; 81:051006
Abstract
We utilize a nonlinear, dynamic finite element model coupled with a finite deformation viscoelastic constitutive law to study the inhomogeneous deformation and
instabilities resulting from the application of a constant voltage to dielectric elastomers. The constant voltage loading is used to study electrostatically-driven
creep and the resulting electromechanical instabilities for two different cases that have all been experimentally observed, i.e. electromechanical snap-through
instability and bursting drops in a dielectric elastomer. We find that in general, increasing the viscoelastic relaxation time leads to an increase in time needed
to nucleate the electromechanical instability. However, we find for these two cases that the time needed to nucleate the instability scales with the relaxation time.
This paper is available in PDF form
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Electromechanical Instability on Dielectric Polymer Surface: Modeling and Experiment
H.S. Park, Q. Wang, X. Zhao and P.A. Klein
Computer Methods in Applied Mechanics and Engineering 2013; 260:40-49
Abstract
We present a dynamic finite element formulation for dielectric elastomers that significantly alleviates the problem of volumetric locking that occurs due to the
incompressible nature of the elastomers. We accomplish this by modifying the Q1P0 formulation of Simo et al. [1], and adapting it to the electromechanical
coupling that occurs in dielectric elastomers. We demonstrate that volumetric locking has a significant impact on the critical electric fields that are
necessary to induce electromechanical instabilities such as creasing and cratering in dielectric elastomers, and that the locking effects are most severe in
problems related to recent experiments that involve significant constraints upon the deformation of the elastomers. We then compare the results using the new
Q1P0 formulation to that obtained using standard 8-node linear and 27-node quadratic hexahedral elements to demonstrate the capability of the proposed approach.
Finally, direct comparison to the recent experimental work on the creasing instability on dielectric polymer surface by Wang et al. [2] is presented.
The present formulation demonstrates good agreement to experiment for not only the critical electric field for the onset of the creasing instability, but also
the experimentally observed average spacing between the creases.
This paper is available in PDF form
.
Viscoelastic Effects on Electromechanical Instabilities in Dielectric Elastomers
H.S. Park and T.D. Nguyen
Soft Matter 2013; 9:1031-1042
Abstract
We present a computational study of the effect of viscoelasticity on the electromechanical behavior of dielectric elastomers. A dynamic, finite deformation
finite element formulation for dielectric elastomers is developed that incorporates the effects of viscoelasticity using the nonlinear viscoelasticity theory
previously proposed by Reese and Govindjee (IJSS, 1998). The finite element model features a three-field Q1P0 formulation to alleviate volumetric locking effects caused
by material incompressibility. We apply the formulation to first perform a fundamental examination of the effects of the viscoelastic deviatoric and volumetric
response on dielectric elastomers undergoing homogeneous deformation. Specifically, we evaluate the effects of the shear and bulk relaxation times on the
electromechanical instability, and demonstrate that while the bulk relaxation time has a negligible impact, the shear relaxation time substantially increases
the critical electric field needed to induce electromechanical instability. We also demonstrate a significant increase in the critical voltage needed to
induce electromechanical instability in the presence of a distribution of relaxation times, compared to a single relaxation time, where the former is more
representative of viscoelastic behavior of polymers. We then study the effects of viscoelasticity on crack-like electromechanical instabilities that have
recently been observed in constrained dielectric films with a small hole containing a conductive liquid. Viscoelasticity is shown again to not only
significantly increase the critical electric field to induce the electromechanical instability, but also to substantially reduce the crack propagation
speeds in the elastomer.
This paper is available in PDF form
.
A Dynamic Finite Element Method for Inhomogeneous Deformation and Electromechanical Instability of Dielectric Elastomer Transducers
H.S. Park, Z. Suo, J.X. Zhou and P.A. Klein
International Journal of Solids and Structures 2012; 49:2187-2194.
Abstract
We present a three-dimensional nonlinear finite element formulation for dielectric elastomers. The mechanical and electrical governing equations are solved
monolithically using an implicit time integrator, where the governing finite element equations are given for both static and dynamic cases. By accounting for
inertial terms in conjunction with the Arruda-Boyce rubber hyperelastic constitutive model, we demonstrate the ability to capture the various modes of
inhomogeneous deformation, including pull-in instability and wrinkling, that may result in dielectric elastomers that are subject to various forms of
electrostatic loading. The formulation presented here forms the basis for needed computational tools that can elucidate the electromechanical behavior
and properties of dielectric elastomers that are used for engineering applications.
This paper is available in PDF form
.