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158 Current Configuration as the Reference Configuration

3.31 Current Configuration as the Reference Configuration

Let x' be the position at time r of a material point which is at the spatial position x at
time t, then the kinematic equations of motion (the pathline equations) take the form of

Equations (3.31.1) describe the motion using the current configuration as the reference
configuration. The subscript t in x,' indicates that the current time t is the reference time and
as such in addition to x and r, it is also a function of t.

Example 3.31.1
Given the velocity field

Find the pathline equations using the current configuration as the reference configuration.
Solution. LetJt'!(x,r), *' 2(x,r), x'-$(\, r) be the position at time r then

The second and the third equation state that both x'i andx'$ are constants. Since they must
be KI and x$ at time t, therefore,

we
Now from the first equation, since x'2 - *2» have