CONTINUUM MECHANICS for ENGINEERS

where the notation in the last form indicates that the variable X is held
constant in taking the partial derivative of χ. Also, as is standard practice,
the super-positioned dot has been introduced to denote differentiation with
respect to time. In an obvious generalization, we may define the velocity field
of the total body as the derivative
dx
∂χ
( X, t)
∂ xX ( , t)
v= x˙=
= =
dt ∂t
∂t
Similarly, the acceleration field is given by
2
d x ∂χ
( X,
t)
a = v˙ = x˙˙
= = 2 2
dt ∂t
2
(4.2-12)
(4.2-13)
and the acceleration of any particular particle determined by substituting its
material coordinates into Eq 4.2-13.
Of course, the individual particles of a body cannot execute arbitrary
motions independent of one another. In particular, no two particles can
occupy the same location in space at a given time (the axiom of impenetrability),
and furthermore, in the smooth motions we consider here, any two
particles arbitrarily close in the reference configuration remain arbitrarily
close in all other configurations. For these reasons, the function χ in Eq 4.2-6
must be single-valued and continuous, and must possess continuous derivatives
with respect to space and time to whatever order is required, usually
to the second or third. Moreover, we require the inverse function χ –1 in the
equation
X = χ –1 (x, t) (4.2-14)
to be endowed with the same properties as χ. Conceptually, Eq 4.2-14 allows
us to “reverse” the motion and trace backwards to discover where the particle,
now at x, was located in the reference configuration. The mathematical
condition that guarantees the existence of such an inverse function is the
non-vanishing of the Jacobian determinant J. That is, for the equation
∂χi
J = ≠0
∂X
to be valid. This determinant may also be written as
A
i J =
X
∂ x
∂
A
(4.2-15)
(4.2-16)