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# Energy formulations of nonlinear elasticity including electric / magnetic couplings

Equilibrium theories for a continuum body may be formulated by either of the

following the classic paradigms: (1) We begin with the *stress postulation* *(Cauchy’s formulation)* and write down the kinematics, conservation laws, and

constitutive relations. In this way, one can obtain a system of field equations

which, presumably, can be solved upon specifying boundary conditions and

determine the equilibrium state of the body. (2) A second way is to start from

the *energy postulation (Green’s
formuation).* The constitutive relations are then lumped into the stored

energy function, and the equilibrium states shall minimize the total free

energy of the system.

There are applications that require our knowledge beyond the field equations

and solutions to the boundary value problem (BVP) associated with the field

equations. Two examples: (1) Stability of a solution. It may occur that our BVP

admits multiple solutions. Then the question is whether the solution is stable

or not. A simple example is the classic Euler buckling theory. Our system

admits two solutions: one pure compression and one bending solution. To show

the compression solution being an unstable solution has to involve the concept of

free energy of the system. (2) The quasistatic evolution problem. The "driving

force" on defects including crack tips, dislocation lines and interfaces

can only be identified from an energy consideration (Eshelby's energy-moment

tensor and J-integral). And an additional constitutive law (kinetic law) has to

be supplemented to close the system. For these applications, an

energy formulation will be much more convenient, if not indispensable.

This motivates us to reconsider the continuum theory including electric and

magnetic. Undoubtedly the field equations are given by the Maxwell equations

and mechanical balance law. However, the free energy of the system is somewhat

subtle because of the arbitrariness in choosing the state variables and

decomposing the total energy into the “mechanical part” and “electrostatic part”.

Also, the Maxwell stress is still in debate whether it is a useful or meaningful

concept for electro-elasticity. In form, the Maxwell stress looks very much

like the Eshelby's energy-moment tensor. One may wonder if we can interpret the

Maxwell stress as the "configurational stress". To proceed with such

a calculation we need to fix the total free energy of the system and possible

variations of the state variables.

In my opinion, the free energy of a system is meaningless unless the

boundary conditions are precisely specified. Physicists are often vague,

tacitly assuming an infinite body among others. In the first paper , for

simplicity I assume a rigid body and explore relations between different energy

formulations for a body in electrostatics. The goal is to have a unified

treatment of different materials including conductors, dielectrics,

ferroelectrics, etc. Also, general boundary conditions including Robin-type BC

are considered. Applications of the proposed energy formulation include the

effective free energy of a dielectric body interacting with a conductor and

rigorous bounds on effective properties of nonlinear dielectric composites.

In the second paper , I include deformation and magnetics. To avoid the muddy

issues associated with the Maxwell stress, we start from the energy postulation

and verify that the Maxwell stress can indeed be regarded as the “configurational

force” associated with the electric/magnetic parts of the total free energy. (This

calculation has been done in a 2007 Caltech thesis by L. Tian.) Also, the stored / internal energy functions

are found for typical materials including nonlinear dielectrics, photoelastics,

piezoelectrics, flexoelectrics among others. Novel applications of the proposed

energy formulation include explicit solution of deformation of a nonlinear

dielectric ellipsoids in an electric field, magnetoelectric effects of soft

elastomers and formal derivation of the bending theory of flexoelectric thin

films.

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## nonlinear elasticity, energy

nonlinear elasticity, energy formulation, electric coupling, magnetic coupling