Calculation boundary terms of a functional

  • #1

Main Question or Discussion Point

Dear all,

I am stuck with the problem which is given below;
attachment.php?attachmentid=62140&stc=1&d=1380043019.png


In this problem the equilibrium equations of the given functional must be derived in u, v, and w directions from which the boundary terms must be found. I think that i have derived the equilibrium equations( 5 equations), but i dont know how to proceed. Does anyone maybe know how to do it??


Thanks in advance,
 

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  • #2
pasmith
Homework Helper
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Dear all,

I am stuck with the problem which is given below;
attachment.php?attachmentid=62140&stc=1&d=1380043019.png


In this problem the equilibrium equations of the given functional must be derived in u, v, and w directions from which the boundary terms must be found. I think that i have derived the equilibrium equations( 5 equations), but i dont know how to proceed. Does anyone maybe know how to do it??
I think you're missing a name for the integrand. I will use [itex]L[/itex].

To determine [itex]\delta I[/itex], integrate [itex]L(u + h, \dots, v + k, \dots, w + l, \dots) - L(u, \dots, v, \dots, w, \dots)[/itex] term by term. You can swap the order of integration so that terms involving derivatives with respect to x are integrated with respect to x first, and terms involving derivatives with respect to y are integrated with respect to y first.

For example, for the [itex]u_{,x}[/itex] term you get:
[tex]
\int_a^b \int_c^d h_{,x} \frac{\partial L}{\partial u_{,x}}\,\mathrm{d}x \,\mathrm{d}y
= \int_a^b \left[ h \frac{\partial L}{\partial u_{,x}}\right]_c^d\,\mathrm{d}y - \int_a^b \int_c^d h \frac{\partial}{\partial x} \left(\frac{\partial L}{\partial u_{,x}}\right)\,\mathrm{d}x\,\mathrm{d}y[/tex]

(The antiderivative with respect to x of [itex]h_{,x}[/itex] is [itex]h + A(y)[/itex] for an arbitrary function [itex]A[/itex]; but it will be the same function at both [itex]x = c[/itex] and [itex]x = d[/itex], so it cancels out.)

Ideally the conditions of your problem are such that the perturbation [itex]h[/itex] vanishes everywhere on the boundary, so the boundary term vanishes.

With the second derivatives of w you have to integrate by parts twice:
[tex]
\int_a^b \int_c^d l_{,xx} \frac{\partial L}{\partial w_{,xx}}\,\mathrm{d}x \,\mathrm{d}y
= \int_a^b \left[ l_{,x} \frac{\partial L}{\partial w_{,xx}} \right]_c^d\,\mathrm{d}y
- \int_a^b \int_c^d l_{,x} \frac{\partial}{\partial x}\left(\frac{\partial L}{\partial w_{,xx}}\right)\,\mathrm{d}x \,\mathrm{d}y
[/tex]
Again, one would hope that the conditions of your problem require that [itex]l_{,x}[/itex] vanishes on the boundary, so the boundary term vanishes. The remaining term is dealt with as for the other first derivatives.

Incidentally, I believe there should be three equilibrium equations (one each for u, v and w) so if you have five then you have made an error somewhere.
 
  • #3
I will try to solve the question now with your explanation. Thank you very much!!
 
  • #4
Regarding your explanation I ended up with the following equation to be solved.

attachment.php?attachmentid=62162&stc=1&d=1380059658.jpg


Is this correct or did I get your explanation totally wrong??(btw i forgot the dxdy term at the end of the equation)
 

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