Some Basic Principles of Continuum Mechanics

In order to properly consider multifield coupling effects, we will need to draw on some of the tools of classical continuum mechanics.

The term deformation refers to a change in the shape of the continuum between a reference configuration and current configuration. In the reference configuration, a representative particle of the continuum occupies a point

p

in space and has the position vector

$$ {\bf X}=X_1{\bf e}_1+X_2{\bf e}_2+X_3{\bf e}_3\,$$

where

e

1

,

e

2

,

e

3

is a Cartesian reference triad, and

X

1

,

X

2

,

X

3

(with center

O

) can be thought of as labels for a point. Sometimes, the coordinates or labels (

X

1

,

X

2

,

X

3

,

t

) are called the referential coordinates. In the current configuration, the particle originally located at point

P

is located at point

P

′, and can also be expressed in terms of another position vector

x

, with the coordinates (

x

1

,

x

2

,

x

3

,

t

). These are called the current coordinates. It is obvious with this arrangement that the displacement is

u

 = 

x

–

${\emph \bf X}$

for a point originally at

${\emph \bf X}$

and with final coordinates

x

.