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3 Continuum Mechanics and Nonlinear Elasticity
where C is a constant of integration. Using the initial condition x(X, 0) = X gives
C = ln X, and since ln x − ln X = ln(x/X), the above equation gives
x = X(1 + αt
2 ),
which agrees with Eq. (3.1).
3.2.2 Motion in 3D
Consider the motion of a three-dimensional continuum relative to a set of Cartesian
coordinate axes. At t = 0, an arbitrary particle is located at the material coordinates
(X 1 , X 2 , X 3 ). At some later time t > 0, it occupies the spatial coordinates
(x 1 , x 2 , x 3 ). Relative to the origin, the respective position vectors are given by
R = X i e i
r = x i e i ,
(3.10)
where the e i are Cartesian base vectors.
As the particle travels through space and time, its motion can be described by the
relation
x i = x i (X j , t),
(3.11)
which represents three equations, one for each spatial coordinate (i = 1, 2, 3). Here,
x i on the right-hand side represents a function of the three material coordinates and
time, i.e., x 1 = x 1 (X 1 , X 2 , X 3 , t), etc. These equations can be inverted to give
X i = X i (x j , t).
(3.12)
Equations (3.11) and (3.12) are the 3D generalizations of the specific 1D relations
given by Eqs. (3.1) and (3.2). In direct notation, they can be written as
r = r(R, t)
R = R(r, t),
(3.13)
which are valid for any coordinate system. Note that we must have r(R, 0) = R.
The displacement vector of a particle from its initial position is
u = r − R,
(3.14)