Dielectric elastomer composites: analytical and numerical non-convex homogenization methods and applications

With the practical objective of shedding light on promising experimental results that have recently identified dielectric elastomer composites as potential enablers of new technologies (essentially, as the next generation of soft sensors and actuators), this work puts forth analytical and numerical methods to determine the macroscopic elastic dielectric behavior of this class of soft electroactive materials directly in terms of their microscopic behavior. The macroscopic behavior of dielectric elastomer composites is first investigated within the classical asymptotic context of small deformations and moderate electric fields. Specifically, by a combination of analytical and numerical techniques, rigorous homogenization solutions are constructed for dielectric elastomer composites with general (possibly anisotropic) classes of two-phase particulate microstructures. Aimed at identifying what types of filler particles lead to enhanced elastic dielectric behaviors, these solutions are deployed to examine dielectric elastomers filled with stiff high-permittivity particles, high-permittivity particles that are liquid-like in mechanical behavior, and vacuous pores. In addition to generalizing the fundamental purely elastic and purely dielectric solutions of Eshelby and Maxwell to the coupled and nonlinear realm of electroelastostatics, the above-outlined rigorous asymptotic solutions turn out to be essential in the development of corresponding homogenization solutions for finite deformations and finite electric fields. Indeed, it is shown that they can be utilized as building blocks for the derivation of a general approximate homogenization solution for non-Gaussian dielectric elastomers filled with nonlinear elastic dielectric particles that may exhibit polarization saturation. By construction, this approximate solution is exact in the limit of small deformations and moderate electric fields. For finite deformations and finite electric fields, its accuracy is assessed by direct comparisons with full-field hybrid finite-element simulations, as well as with numerical solutions generated via a new WENO finite-difference scheme developed specifically for this class of problems. With the object of scrutinizing recent experimental results, the specializations of the proposed solution to various cases wherein the filler particles are of poly- and mono-disperse sizes and exhibit different types of elastic dielectric behaviors are discussed in detail. Stark disagreement between the theoretical results outlined above and a plurality of experimental results indicates that the basic point of view that dielectric elastomer composites can be idealized as two-phase particulate elastic dielectric composites is fundamentally incomplete, especially for cases involving stiff filler particles which (as opposed to what the theory predicts) have been reported to exhibit extreme enhancements in their electrostriction capabilities. It is posited that such extreme enhancements are the manifestation of interphasial phenomena. In particular, the presence of interphasial free charges that oscillate rapidly in space at the length scale of the microstructure of elastic dielectric composites is shown to have a significant and even dominant effect on their macroscopic response, possibly leading to extreme behaviors ranging from unusually large permittivities and electrostriction coefficients to metamaterial-type properties featuring negative permittivities. These results suggest a promising strategy to design deformable dielectric composites --- such as electrets and dielectric elastomer composites --- with exceptional electromechanical properties

Time-dependent dielectric response of polymer nanoparticulate composites containing rapidly oscillating source terms

This thesis presents the derivation of the homogenized equations for the macroscopic response of time-dependent dielectric composites that contain space charges varying spatially at the length scale of the microstructure and that are subjected to alternating electric fields. The focus is on dielectrics with periodic microstructures and two fairly general classes of space charges: passive (or fixed) and active (or locally mobile). With help of a standard change of variables, in spite of the presence of space charges, the derivation amounts to transcribing a previous two-scale-expansion result introduced in Lefevre and Lopez-Pamies (2017a) for perfect dielectrics to the realm of complex frequency-dependent dielectrics. With the objectives of illustrating their use and of showcasing their ability to describe and explain the macroscopic response of emerging materials featuring extreme dielectric behaviors, the derived homogenization results are deployed to examine dielectric spectroscopy experiments on various polymer nanoparticulate composites. It is found that so long as space charges are accounted for, the proposed theoretical results are able to describe and explain all the experimental results. By the same token, more generally, these representative comparisons with experiments point to the manipulation of space charges at small length scales as a promising strategy for the design of materials with exceptional macroscopic properties

Homogenization Theory: Periodic and Beyond (online meeting)

The objective of the workshop has been to review the latest developments in homogenization theory for a large category of equations and settings arising in the modeling of solid, fluids, wave propagation, heterogeneous media, etc. The topics approached have covered periodic and nonperiodic deterministic homogenization, stochastic homogenization, regularity theory, derivation of wall laws and detailed study of boundary layers,..

Effective balance equations for electrostrictive composites

This work concerns the study of the effective balance equations governing linear elastic electrostrictive composites, where mechanical strains can be observed due to the application of a given electric field in the so-called small strain and moderate electric field regime. The formulation is developed in the framework of the active elastic composites. The latter are defined as composite materials constitutively described by an additive decomposition of the stress tensor into a purely linear elastic contribution and another component, which is assumed to be given and quadratic in the applied electric field when further specialised to electrostrictive composites. We derive the new mathematical model by describing the effective mechanical behaviour of the whole material by means of the asymptotic (periodic) homogenisation technique. We assume that there exists a sharp separation between the micro-scale, where the distance among different sub-phases (i.e. inclusions and/or fibres and/or strata) is resolved, and the macro-scale, which is related to the average size of the whole system at hand. This way, we formally decompose spatial variations by assuming that every physical field and material property are depending on both the macro-scale and the micro-scale. The effective governing equations encode the role of the micro-structure, and the effective contributions to the global stress tensor are to be computed by solving appropriate linear-elastic-type cell problems on the periodic cell. We also provide analytic formulae for the electrostrictive tensor when the applied electric field is either microscopically uniform or given by a suitable multiplicative decomposition between purely microscopically and macroscopically varying components. The obtained results are consistently compared with previous works in the field, and can pave the way towards improvement of smart active materials currently utilised for engineering (possibly bio-inspired) purposes

Advances in Multiscale and Multifield Solid Material Interfaces

Interfaces play an essential role in determining the mechanical properties and the structural integrity of a wide variety of technological materials. As new manufacturing methods become available, interface engineering and architecture at multiscale length levels in multi-physics materials open up to applications with high innovation potential. This Special Issue is dedicated to recent advances in fundamental and applications of solid material interfaces

Magnetic Hybrid-Materials

Externally tunable properties allow for new applications of suspensions of micro- and nanoparticles in sensors and actuators in technical and medical applications. By means of easy to generate and control magnetic fields, fluids inside of matrices are studied. This monnograph delivers the latest insigths into multi-scale modelling, manufacturing and application of those magnetic hybrid materials

Some analytical techniques for partial differential equations on periodic structures and their applications to the study of metamaterials

The work presented in this thesis is a study of homogenisation problems in electromagnetics and elasticity with potential applications to the development of metamaterials. In Chapter 1, I study the leading order frequency approximations of the quasi-static Maxwell equations on the torus. A higher-order asymptotic regime is used to derive a higher-order homogenised equation for the solution of an elliptic second-order partial differential equation. The equivalent variational approach to this problem is studied which leads to an equivalent higher-order homogenised equation. Finally, the derivation of higher-order constitutive laws relating the fields to their inductions is presented. In Chapter 2, I study the governing equations of linearised elasticity where the periodic composite material of interest is made up of a "critically" scaled "stiff" rod framework with the voids in between filled in with a "soft" material which is in high-contrast with the stiff material. Using results from two-scale convergence theory, a well posed homogenised model is presented with features reminiscent of both high-contrast and thin structure homogenised models with the additional feature of a linking relation of Wentzell type. The spectrum of the limiting operator is investigated and the establishment of the convergence of spectra from the initial problem is derived. In the final chapter, I investigate brie y three additional homogenisation problems. In the first problem, I study a periodic dielectric composite and show that there exists a critical scaling between the material parameter of the soft inclusion and the period of the composite. In the second problem, I use of two-scale convergence theory to derive a homogenised model for Maxwell's equations on thin rod structures and in the final problem I study Maxwell's equations in R^3 under a chiral transformation of the coordinates and derive a homogenised model in this special geometry

Magnetic Hybrid-Materials

Externally tunable properties allow for new applications of suspensions of micro- and nanoparticles in sensors and actuators in technical and medical applications. By means of easy to generate and control magnetic fields, fluids inside of matrices are studied. This monnograph delivers the latest insigths into multi-scale modelling, manufacturing and application of those magnetic hybrid materials

Performances of passive electric networks and piezoelectric transducers for beam vibration control

This thesis is focused on beam vibration control using piezoelectric transducers and passive electric networks. The first part of this study deals with the modeling and the analysis of stepped piezoelectric beams. A refined one-dimensional model is derived and experimentally validated. The modal properties are determined with four numerical methods. A homogenized model of stepped periodic piezoelectric beams is derived by using two-scale convergence. The second part deals with the performance analysis of three passive circuits in damping structural vibrations: the piezoelectric shunting, the second order transmission line and the fourth order transmission line. The effects of uncertainties of the electric parameters on the system performances are analyzed. Theoretical predictions are validated through different experimental setup

Performances of passive electric networks and piezoelectric transducers for beam vibration control

This thesis is focused on beam vibration control using piezoelectric transducers and passive electric networks. The first part of this study deals with the modeling and the analysis of stepped piezoelectric beams. A refined one-dimensional model is derived and experimentally validated. The modal properties are determined with four numerical methods. A homogenized model of stepped periodic piezoelectric beams is derived by using two-scale convergence. The second part deals with the performance analysis of three passive circuits in damping structural vibrations: the piezoelectric shunting, the second order transmission line and the fourth order transmission line. The effects of uncertainties of the electric parameters on the system performances are analyzed. Theoretical predictions are validated through different experimental setup