Deformation (continuum mechanics)

[sara_87]

Homework Statement

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

Homework Equations

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!

(try using the X₂ tag just above the Reply box )

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 …

X₁ and X₂ will not appear in the final result …

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

 

[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}

 

[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 I am making a big mistake but i don't 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

 

[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 = … ?

 

[sara_87]

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