Maximizing Electro-Momentum Coupling in Generalized 2D Willis Metamaterials
H.D. Huynh, X. Zhuang, H.S. Park, S.S. Nanthakumar, Y. Jin and T. Rabczuk
Extreme Mechanics Letters 2023; 61:101981
Abstract
The coupling of momentum to strain in elastic metamaterials, known as the Willis coupling, has been widely studied in recent years for its potential in enabling
novel phenomena in wave propagation. More recent work has shown that in piezoelectric composites, the momentum can also be coupled to the electrical stimulus,
resulting in a new form of electro-momentum coupling, which offers a new approach to controlling elastic wave phenomena through a non-mechanical stimulus.
In this study, we present a topology optimization approach to maximize the electro-momentum coupling in piezoelectric composites, where dynamic homogenization
is utilized to obtain the effective mechanical, electrical, and electro-mechanical constitutive relations. We first validate the approach in one-dimension,
then demonstrate that the electro-momentum coupling can enable non-reciprocal wave propagation in two-dimensions, both through mechanical and electrical loadings.
This approach can enable the design of piezoelectric composites that support novel wave phenomena that can be excited through non-mechanical means.
This paper is available in PDF form
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Enhanced Converse Flexoelectricity in Piezoelectric Composites by Coupling Topology Optimization with Homogenization
X. Chen, J. Yvonnet, H.S. Park and S. Yao
Journal of Applied Physics 2021; 129:245104
Abstract
We demonstrate that large apparent converse flexoelectric properties can be obtained in piezoelectric composites using theoretical approaches. To do so,
we first present a numerical homogenization method accounting for all electromechanical terms related to strain and electric field gradient. We then evaluate
the coefficients of the model by numerical simulations on periodic piezoelectric composites. After combining the homogenization approach with topology
optimization to enhance the converse properties of the composite, we present numerical results that reveal that the apparent converse flexoelectric coefficients,
as well as those associated with the higher order coupling terms involving the electric field gradient, are of the same order as the direct flexoelectric
properties of the local constituents. These results suggest that both converse and higher order electromechanical coupling effects may contribute strongly to the
flexoelectric response and properties of piezoelectric composites. Finally, we show that it is theoretically possible to obtain optimized designs of composites
with apparent converse flexoelectric properties {1-2 orders of magnitude} larger than ones obtained with naive guess designs.
This paper is available in PDF form
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Flexoelectric Electricity Generation by Crumpling Graphene
B. Javvaji, R. Zhang, X. Zhuang and H.S. Park
Journal of Applied Physics 2021; 129:225107
Abstract
We utilize atomistic simulations that account for point charges and dipoles to demonstrate that flexoelectricity, which arises from strain gradients,
can be exploited to generate electricity from crumpled graphene sheets. Indentation of a circular graphene sheet generates localized developable (d)-cones,
for which we verify the core radius and azimuthal angle with established theoretical models. We determine the voltage that can be generated based on the
resulting electrostatic fields, and compare the voltage generation to previous theoretical predictions that are scaled down to the nanoscale. In doing so,
we find that the voltage generated from crumpling graphene exceeds, by about an order of magnitude, the expected voltage generation, indicating the benefit of
exploiting the large strain gradients that are possible at the nanoscale. Finally, we demonstrate that crumpling may be a superior mechanism of flexoelectric
energy generation as compared to bending of two-dimensional nanomaterials.
This paper is available in PDF form
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Topology Optimization of Flexoelectric Composites Using Computational Homogenization
X. Chen, J. Yvonnet, S. Yao and H.S. Park
Computer Methods in Applied Mechanics and Engineering 2021; 381:113819
Abstract
We present a topology optimization framework to design periodic composites comprised of piezoelectric constituents that exhibit large flexoelectric constants.
The novelty of the approach is that it leverages a representative volume element (RVE)-based computational homogenization approach that enables the analysis
of periodic composites where the characteristic dimensions of the microstructure are significantly smaller than those of the structure, and as such requires
only the optimization of a single RVE rather than that of the entire structure. We utilize this approach to analyze the enhancement in flexoelectric
constants that can be achieved in different types of PZT-based composites, including hard-hard (PZT-PZT), and hard-soft (PZT-polymer composite, and porous PZT)
structures. In all cases, significant enhancements are observed, with improvements between 2 and 15 times those of a naive guess, with some designs reaching
a factor of one order of magnitude larger than BTO. We identify different mechanisms governing the enhanced electromechanical couplings, which can arise either
from an enhancement of effective piezoelectricity in the RVE for PZT-PZT composites, or from a more subtle interplay involving the enhancement of effective
piezoelectric and dielectric properties coupled with a reduction in mechanical compliance for PZT-polymer and porous PZT RVEs.
This paper is available in PDF form
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A Staggered Explicit-Implicit Isogeometric Formulation for Large Deformation Flexoelectricity
T.Q. Thai, X. Zhuang, H.S. Park and T. Rabczuk
Engineering Analysis with Boundary Elements 2021; 122:1-12
Abstract
Flexoelectricity is an electromechanical coupling occurring in dielectric materials that has recently attracted significant attention. The
flexoelectric effect is described by a coupled, higher-order electromechanical set of equations that have typically been solved using a
computationally expensive monolithic formulation. In the present work, we propose a staggered, explicit-implicit formulation that both
significantly reduces the computational expense, while enabling the capturing of electromechanical instabilities through the usage of inertia.
The higher order equations are discretized using an isogeometric formulation, and we demonstrate via two numerical examples the combination of
increased computational efficiency with comparable accuracy that is gained from the proposed formulation as compared to the standard monolithic
approaches.
This paper is available in PDF form
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High Flexoelectric Constants in Janus Transition-Metal Dichalcogenides
B. He, B. Javvaji, X. Zhuang and H.S. Park
Physical Review Materials 2019; 3:125402
Abstract
Due to their combination of mechanical stiffness and flexibility, two-dimensional (2D) materials have received significant interest as potential
electromechanical materials. Flexoelectricity is an electromechanical coupling between strain gradient and polarization. Unlike piezoelectricity,
which exists only in non-centrosymmetric materials, flexoelectricity theoretically exists in all dielectric materials. However, most work on the
electromechanical energy conversion potential of 2D materials has focused on their piezoelectric, and not flexoelectric behavior and properties.
In the present work, we demonstrate that the intrinsic structural asymmetry present in monolayer Janus transition metal dichalcogenides (TMDCs)
enables significant flexoelectric properties. We report these flexoelectric properties using a recently developed charge-dipole model that couples
with classical molecular dynamics simulations. By employing a prescribed bending deformation, we directly calculate the flexoelectric constants
while eliminating the piezoelectric contribution to the polarization. We find that the flexoelectric response of a Janus TMDC is positively
correlated to its initial degree of asymmetry, which contributes to stronger σ-σ interactions as the initial degree of asymmetry rises.
In addition, the high transfer of charge across atoms in Janus TMDCs leads to larger electric fields due to π-σ coupling. These enhanced
σ-σ and π-σ interactions are found to cause the flexoelectric coefficients of the Janus TMDCs to be several times higher than
traditional TMDCs such as MoS2, whose flexoelectric constant is already ten times larger than graphene.
This paper is available in PDF form
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Intrinsic Bending Flexoelectric Constants in Two-Dimensional Materials
X. Zhuang, B. He, B. Javvaji and H.S. Park
Physical Review B 2019; 99:054105
Abstract
Flexoelectricity is a form of electromechanical coupling that has recently emerged because, unlike piezoelectricity, it is theoretically
possible in any dielectric material. Two-dimensional (2D) materials have also garnered significant interest because of their unusual
electromechanical properties and high flexibility, but the intrinsic flexoelectric properties of these materials remain unresolved.
In this work, using atomistic modeling accounting for charge-dipole interactions, we report the intrinsic flexoelectric constants for a
range of two-dimensional materials, including graphene allotropes, nitrides, graphene analogs of group-IV elements, and the transition
metal dichalcogenides (TMDCs). We accomplish this through a proposed mechanical bending scheme that eliminates the piezoelectric contribution
to the total polarization, which enables us to directly measure the flexoelectric constants. While flat 2D materials like graphene have low
flexoelectric constants due to weak π-σ interactions, buckling is found to increase the flexoelectric constants in monolayer group-IV
elements. Finally, due to significantly enhanced charge transfer coupled with structural asymmetry due to bending, the TMDCs are found to have
the largest flexoelectric constants, including MoS2 having a flexoelectric constant ten times larger than graphene.
This paper is available in PDF form
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A Multi-Material Level Set-Based Topology Optimization of Flexoelectric Composites
H. Ghasemi, H.S. Park and T. Rabczuk
Computer Methods in Applied Mechanics and Engineering 2018; 332:47-62
Abstract
We present a computational design methodology for topology optimization of multi-material-based flexoelectric composites. The methodology
extends our recently proposed design methodology for a single flexoelectric material. We adopt the multi-phase vector level set (LS) model
which easily copes with various numbers of phases, efficiently satisfies multiple constraints and intrinsically avoids overlap or vacuum
among different phases. We extend the point wise density mapping technique for multi-material design and use the B-spline elements to
discretize the partial differential equations (PDEs) of flexoelectricity. The dependence of the objective function on the design variables
is incorporated using the adjoint technique. The obtained design sensitivities are used in the Hamilton-Jacobi (H-J) equation to update the
LS function. We provide numerical examples for two, three and four phase flexoelectric composites to demonstrate the flexibility of the
model as well as the significant enhancement in electromechanical coupling coefficient that can be obtained using multi-material topology
optimization for flexoelectric composites.
This paper is available in PDF form
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Topology Optimization of Flexoelectric Structures
S.S. Nanthakumar, X. Zhuang, H.S. Park and T. Rabczuk
Journal of the Mechanics and Physics of Solids 2017; 105:217-234
Abstract
We present a mixed finite element formulation for flexoelectric nanostructures that is coupled with topology optimization to maximize their
intrinsic material performance with regards to their energy conversion potential. Using Barium Titanate (BTO) as the model flexoelectric material,
we demonstrate the significant enhancement in energy conversion that can be obtained using topology optimization. We also demonstrate that non-smooth
surfaces can play a key role in the energy conversion enhancements obtained through topology optimization. Finally, we examine the relative benefits of
flexoelectricity, and surface piezoelectricity on the energy conversion efficiency of nanobeams. We find that the energy conversion efficiency of
flexoelectric nanobeams is comparable to the energy conversion efficiency obtained from nanobeams whose electromechanical coupling occurs through
surface piezoelectricity, but are ten times thinner. Overall, our results not only demonstrate the utility and efficiency of flexoelectricity as a
nanoscale energy conversion mechanism, but also its relative superiority as compared to piezoelectric or surface piezoelectric effects.
This paper is available in PDF form
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A Level-Set Based IGA Formulation for Topology Optimization of Flexoelectric Materials
H. Ghasemi, H.S. Park and T. Rabczuk
Computer Methods in Applied Mechanics and Engineering 2017; 313:239-258
Abstract
This paper presents a design methodology based on a combination of isogeometric analysis (IGA), level set and point wise density mapping techniques
for topology optimization of a continuum considering piezoelectric and flexoelectric effects. The fourth order partial differential equations (PDEs)
of flexoelectricity, which require at least C1 continuous approximations, are discretized by using Non-Uniform Rational B-spline (NURBS). The point
wise density mapping technique with consistent derivatives is directly used in the weak form of the governing equations. The boundary of the design
domain is clearly and implicitly represented by a level set function. The accuracy of the IGA model is confirmed through numerical examples including
a cantilever beam under a point load and a truncated pyramid under compression with different electrical boundary conditions. Finally, we provide
numerical examples demonstrating the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization.
This paper is available in PDF form
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Topology Optimization of Piezoelectric Nanostructures
S.S. Nanthakumar, T. Lahmer, X. Zhuang, H.S. Park and T. Rabczuk
Journal of the Mechanics and Physics of Solids 2016; 94:316-335
Abstract
We present an extended finite element formulation for piezoelectric nanobeams and nanoplates that is coupled with topology optimization to study the
energy harvesting potential of piezoelectric nanostructures. The finite element model for the nanoplates is based on the Kirchoff plate model, with a
linear through the thickness distribution of electric potential. Based on the topology optimization, the largest enhancements in energy harvesting are
found for closed circuit boundary conditions, though significant gains are also found for open circuit boundary conditions. Most interestingly, our
results demonstrate the competition between surface elasticity, which reduces the energy conversion efficiency, and surface piezoelectricity, which
enhances the energy conversion efficiency, in governing the energy harvesting potential of piezoelectric nanostructures.
This paper is available in PDF form
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Surface Effects on the Piezoelectricity of ZnO Nanowires
S. Dai and H.S. Park
Journal of the Mechanics and Physics of Solids 2013; 61:385-397
Abstract
We utilize classical molecular dynamics to study surface effects on the piezoelectric properties of ZnO nanowires as calculated under uniaxial loading.
An important point to our work is that we have utilized two types of surface treatments, those of charge compensation and surface passivation, to
eliminate the polarization divergence that otherwise occurs due to the polar (0001) surfaces of ZnO. In doing so, we find that if appropriate
surface treatments are utilized, the elastic modulus and the piezoelectric properties for ZnO nanowires having a variety of axial and surface
orientations are all reduced as compared to the bulk value as a result of polarization-reduction in the polar [0001] direction. The reduction in
effective piezoelectric constant is found to be independent of the expansion or contraction of the polar (0001) surface in response to surface
stresses. Instead, the surface polarization and thus effective piezoelectric constant is substantially reduced due to a reduction in the bond
length of the Zn-O dimer closest to the polar (0001) surface. Furthermore, depending on the nanowire axial orientation, we find in the
absence of surface treatment that the piezoelectric properties of ZnO are either effectively lost due to unphysical transformations from the wurtzite
to non-piezoelectric d-BCT phases, or also become smaller with decreasing nanowire size. The overall implication of this study is that if
enhancement of the piezoelectric properties of ZnO is desired, then continued miniaturization of square or nearly square cross section
ZnO wires to the nanometer scale is not likely to achieve this result.
This paper is available in PDF form
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Surface Piezoelectricity, Size-effects in Nanostructures and the Emergence of Piezoelectricity in Non-piezoelectric Materials
S. Dai, M. Gharbi, P. Sharma and H.S. Park
Journal of Applied Physics 2011; 110:104305
Abstract
In this work, using a combination of a theoretical framework and atomistic calculations, we highlight the concept of surface
piezoelectricity that can be used to interpret the piezoelectricity of nanostructures. Focusing on three specific material systems
(ZnO, SrTiO3 and BaTiO3), we discuss the renormalization of apparent piezoelectric behavior at small scales. In a rather interesting
interplay of symmetry and surface effects, we show that nanostructures of certain non-piezoelectric materials may also exhibit
piezoelectric behavior. Finally, for the case of ZnO, using a comparison with first principles calculations, we also comment on the
fidelity of the widely-used core-shell interatomic potentials to capture non-bulk electro-mechanical response.
This paper is available in PDF form
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A New Multiscale Formulation for the Electromechanical Behavior of Nanomaterials
H.S. Park, M. Devel and Z. Wang
Computer Methods in Applied Mechanics and Engineering 2011; 200:2447-2457
Abstract
We present a new multiscale, finite deformation, electromechanical formulation to capture the response of surface-dominated nanomaterials
to externally applied electric fields. To do so, we develop and discretize a total energy that combines both mechanical and electrostatic
terms, where the mechanical potential energy is derived from any standard interatomic atomistic potential, and where the electrostatic
potential energy is derived using a Gaussian-dipole approach. By utilizing Cauchy-Born kinematics, we derive both the bulk and surface
electrostatic Piola-Kirchoff stresses that are required to evaluate the resulting electromechanical finite element equilibrium equations,
where the surface Piola-Kirchoff stress enables us to capture the non-bulk electric field-driven polarization of atoms near the surfaces
of nanomaterials. Because we minimize a total energy, the present formulation has distinct advantages as compared to previous approaches,
where in particular, only one governing equation is required to be solved. This is in contrast to previous approaches which require
either the staggered or monolithic solution of both the mechanical and electrostatic equations, along with coupling terms that link
the two domains. The present approach thus leads to a significant reduction in computational expense both in terms of fewer equations
to solve and also in eliminating the need to remesh either the mechanical or electrostatic domains due to being based on a total Lagrangian
formulation. Though the approach can apply to three-dimensional cases, we concentrate in this paper on the one-dimensional case. We first
derive the necessary formulas, then give numerical examples to validate the proposed approach in comparison to fully atomistic
electromechanical calculations.
This paper is available in PDF form
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Piezoelectric Constants for ZnO Calculated Using Classical Polarizable Core-Shell Potentials
S. Dai, M.L. Dunn and H.S. Park
Nanotechnology 2010; 21:445707
Abstract
We demonstrate the feasibility of using classical atomistic simulations, i.e. molecular dynamics and molecular statics, to study the
piezoelectric properties of ZnO using core-shell interatomic potentials. We accomplish this by reporting piezoelectric constants for
ZnO as calculated using two different classical interatomic core-shell potentials, that originally proposed by
Binks et al., and that proposed by Nyberg et al. We demonstrate that the
classical core-shell potentials are able to qualitatively reproduce the piezoelectric constants as compared to benchmark \emph{ab initio}
calculations. We further demonstrate that while the presence of the shell is required to capture the electron polarization effects that
control the clamped ion part of the piezoelectric constant, the major shortcoming of the classical potentials is a significant
underprediction of the clamped ion term as compared to previous ab initio results. However, the present results suggest
that overall, these classical core-shell potentials are sufficiently accurate to be utilized for large scale atomistic
simulations of the piezoelectric response of ZnO nanostructures.
This paper is available in PDF form
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