Hi Experts,
This is very strange: The book I am reading is doing some simple math with mistake? The load p(x) is applied at the surfaces of a crack (from 0 to a), the released enagy U = 0.5int[p(x)v(x,a)]dx (from 0 to a), fine. However, the book then differentiate the equation with respect to a and got dU/da = 0.5int[p(x)dv(x,a)/da]dx (from 0 to a). Is this forgot 'a' is a variable now in the integration! Why?
Mistakes do happen. Even in grammar and spelling...
Because there is a variable a and a variable x in V(x,a), then you have to differentiate with respect to a as well as to x simultaneously - and because there are two variables being differentiated "at the same time" , look to one of those rules (chain rule, I think) to properly differentiate.
I am not going to do the differentiation for you, (even in my HP) but the final form looks right for da in a dx and vice-versa.
No mistake in the book.
I order not to get mixed up with the english article ´a´ let´s change boundary a to b:
U = 0.5int[p(x)v(x,b)]dx (from 0 to b)
b is the depth of the crack, x is the displacement of the load along the depth. The crack will deepen under load. Therefore the depth b will vary, i.e. increase, and this is what changes the ´fixed boundary´ b into a ´moving boundary´ b which is to be treated like a variable. Since integration and differentiation (even with respect to different variables) is interchangeable we yield
d/db U = d/db 0.5int[p(x)v(x,b)]dx = 0.5int{d/db[p(x)v(x,b)]}dx = 0.5int{[p(x)d/db v(x,b)]}dx
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