deformation (continuum mechanics)

[sara_87]

1. The problem statement, all variables and given/known data

A body which in the reference configuration is a unit cube with its edges parallel to the coordinate axes undergoes the following deformation:

x1=a1(X1+sX2), x2=a2X2, and x3=a3X3
(where a1,a2,a3,s are constants).

determine the lengths of its edges after the deformation

2. Relevant equations



3. The attempt at a solution

I think i need to know the x1, x2, and x3 before the deformation, as in when it was a unit cube. so if x1=1 then the new length (length after deformation) will be:
sqrt[(1-a1(X1+sX2))^2]

but this is not working for me since the answer is:
lengths: a1, sqrt[(s^2)(a1^2)+a2^2], a3

[tiny-tim]

Hi sara_87! :smile:

(try using the X2 tag just above the Reply box :wink:)
I think i need to know the x1, x2, and x3 before the deformation, as in when it was a unit cube. so if x1=1 then the new length (length after deformation) will be:
sqrt[(1-a1(X1+sX2))^2]

but this is not working for me since the answer is:
lengths: a1, sqrt[(s^2)(a1^2)+a2^2], a3

You're getting very confused :confused: …

X1 and X2 will not appear in the final result …

and the ends of your edges are {X1 = 1, X2 = X3 = 0} etc :wink:

[sara_87]

how did you know that X1=1 and X2=X3=0 ?
and are these for the reference configuration or after the deformation?
I am confused :(

[tiny-tim]

Because it's "a unit cube with its edges parallel to the coordinate axes" …

so the ends of three edges (before the deformation) are {1 0 0} {0 1 0} and {0 0 1} :smile:

[sara_87]

thanks.
so say for one of the edges, it used to be (1,0,0) then after deformation, it's
(a1, s*a1, 0)
so the length is: sqrt[(a1-1)^2+(sa1)^2]

i think im making a big mistake but i dont know what it is.

[tiny-tim]

so say for one of the edges, it used to be (1,0,0) then after deformation, it's
(a1, s*a1, 0)

No, you're not applying the formula …

for example, the new X2 should be a2(old X2), = a20 :smile:

[sara_87]

oh ok, thank you.
also, how do i find the angles between these edges (after deformation)?

[tiny-tim]

dot product divided by moduli = … ? :smile:

[sara_87]

=cos(theta), thank you very much :)