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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