# On the nonlinear equations of thermo-electroelasticity

TL;DR: In this article, the authors derived the differential equations and boundary conditions describing the behavior of an electrically polarizable, finitely deformable, heat conducting continuum in interaction with the electric field.

Abstract: The differential equations and boundary conditions describing the behavior of an electrically polarizable, finitely deformable, heat conducting continuum in interaction with the electric field are derived by means of a systematic application of the laws of continuum physics to a macroscopic model consisting of an electronic charge continuum coupled to a lattice continuum. The resulting rotationally invariant description of thermoelectroelasticity consists of five differential equations in five dependent variables and, when thermal considerations are omitted, reduces to four differential equations in four dependent variables. These same four differential equations of electroelasticity in the same four dependent variables along with the associated boundary conditions can readily be obtained, by means of a Legendre transformation, from existing consistent variational treatments of electroelasticity, which yield a system of seven equations in seven dependent variables.

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TL;DR: In this paper, the authors present a theory of dielectric elastomers, developed within continuum mechanics and thermodynamics, and motivated by molecular pictures and empirical observations, which couples large deformation and electric potential, and describes nonlinear and nonequilibrium behavior, such as electromechanical instability and viscoelasticity.

Abstract: In response to a stimulus, a soft material deforms, and the deformation provides a function. We call such a material a soft active material (SAM). This review focuses on one class of soft active materials: dielectric elastomers. When a membrane of a dielectric elastomer is subject to a voltage through its thickness, the membrane reduces thickness and expands area, possibly straining over 100%. The dielectric elastomers are being developed as transducers for broad applications, including soft robots, adaptive optics, Braille displays, and electric generators. This paper reviews the theory of dielectric elastomers, developed within continuum mechanics and thermodynamics, and motivated by molecular pictures and empirical observations. The theory couples large deformation and electric potential, and describes nonlinear and nonequilibrium behavior, such as electromechanical instability and viscoelasticity. The theory enables the finite element method to simulate transducers of realistic configurations, predicts the efficiency of electromechanical energy conversion, and suggests alternative routes to achieve giant voltage-induced deformation. It is hoped that the theory will aid in the creation of materials and devices.

838 citations

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TL;DR: In this article, a new formulation of the field theory of dielectric solids is proposed, which does not start with Newton's laws of mechanics and Maxwell-Faraday theory of electrostatics, but produces them as consequences.

Abstract: Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell–Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both elastic and inelastic dielectrics. As applications of the theory, we discuss material laws for elastic dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.

485 citations

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388 citations

### Cites background from "On the nonlinear equations of therm..."

...The classical papers [118, 31, 117 ] do not a priori make use of the first assumption, as they study the implications of a nonlinear geometric setting with finite strains and rotations for the theory of elastic dielectric bodies....

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TL;DR: In this article, a simple negative velocity feed back control algorithm coupling the direct and converse piezoelectric effects is used to actively control the dynamic response of an integrated structure through closed loop control.

Abstract: Theoretical formulations, the Navier solutions and finite element models based on the classical and shear deformation plate theories are presented for the analysis of laminated composite plates with integrated sensors and actuators and subjected to both mechanical and electrical loadings. A simple negative velocity feed back control algorithm coupling the direct and converse piezoelectric effects is used to actively control the dynamic response of an integrated structure through closed loop control. The paper contains a detailed theoretical formulation, derivation of the Navier solutions of simply supported rectangular laminates, and displacement finite element models.

353 citations

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TL;DR: In this article, the principle of virtual power is applied to the theory of coupled fields in deformable continua, all states of magnetism and dielectricity being representable if one makes the appropriate adjustments.

Abstract: Using as main tools the principle of virtual power — in the form recently favored by French mechanicians - and continuum thermodynamics, this work of a synthetic nature develops in a “rational” manner and from a unified viewpoint the fully dynamical (albeit not relativistic) theory of electromagnetic continua. The resulting equations are those to be used by theoretical mechanicians, applied physicists and electronic engineers alike to study coupled electro-magneto-mechanical effects in electronic components. A fairly long account of the formal structure underlying the principle of virtual power is first given. Following then a simple purely mechanical application, a full illustration is given of the application of this principle to the theory of coupled fields in deformable continua, all states of magnetism and dielectricity being representable if one makes the appropriate adjustments. Other illustrations of the method concern the case of complicated schemes of elastic dielectrics, liquid crystals and ferrofluids. To end with a comparison with other energy approaches used nowadays in continuum physics is given.

273 citations

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01 Jan 1960

TL;DR: Theory of thermal stresses, Theory of Thermal Stresses, this paper, Theory of thermal stress, and thermal stresses theory, thermal stresses and thermal stress theory in literature, 2015.

Abstract: Theory of thermal stresses , Theory of thermal stresses , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

2,319 citations