The Mooney-Rivlin law, often used for elastic polymers, states that the stress
(
) in a material is given as
where λ, known as the stretch, is given as
The function
() is the displacement of the material, and
and
are positive constants. The objective of this exercise is, for a material which occupies the interval 0 ≤
≤ , to determine
from measured (known) values of
. This will be done by first finding λ from , and then determining
from λ. Also, note that the stretch is positive.
(a) Write down an algorithm that uses either Newton’s method or the secant method, for finding the value of λ for a given value of
.
(b) Show that
(c) Suppose the interval 0 ≤
≤
contains
equally spaced points
i
, where
1
= 0 and
n
=
. Also, suppose that the value of λ is known at each of these points, and designate these values as λ
i
, for
= 1, 2, ,
. Write down an algorithm that uses the trapezoidal rule, which can be used to determine the
i
values from the λ
i
values.
(d) Suppose that the values for the
i
’s are given as
Using your algorithms from parts (a) and (c), compute the
i
’s in the case of when
= 20 and
= 1, and then plot the values (
i
,
i
). In this calculation assume that
(0) = 0, and take
= 20 and
= 10, which are typical values for an elastomer.
(e) Note that the exact solution at
= 1 is
= 1/3. For your algorithm in part (d), what value for
do you need to take so the error in your computed answer at
= 1 is no more than 10−6?