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Message #03373
Re: Elastic energy

To:
yadeusers@xxxxxxxxxxxxxxxxxxx

From:
Janek Kozicki <janek_listy@xxxxx>

Date:
Sun, 4 Jul 2010 18:38:38 +0200

Face:
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Inreplyto:
<AANLkTilsTzlq9iVTJ1vYrlGsCNrYzWflzewJHt6fOF_8@mail.gmail.com>
chiara modenese said: (by the date of Sun, 4 Jul 2010 09:15:03 +0000)
> Thanks Janek for explanation! And thanks Bruno for remind me I should set to
> True the bool traceEnergy ;) I did it. But I do not understand the results.
> You can also reproduce them setting traceEnergy=True in the script I
> attached in this thread. Please just see the attached plot. The code of
> plasticDissipation() I am sure it is correct. There is sliding as expected
> in fact. But what is wrong now?
hmm. I remember calculating internal work, and in fact I remember
that we stumbled upon this problem.
(I hope that your UTF8 works good, because I'm using UTF below)
What is the work of a typical spring?
W=∫F*dx
F=k*x
W=∫k*x*dx = k*x²/2
, see that squared distance factor?
But we need to integrate that in a little different manner, because
we are summing from one iteration to another. Let us make a plot of
force versus displacement and calculate the surface area below the
force. To make it simple we are integrating using the Trapezoidal Rule:
Wᵢ = (Fᵢ+Fᵢ₋₁)*Δxᵢ⁄2
, where ᵢ is the iteration, Fᵢ₋₁ is previous force, Fᵢ is
current force, Δxᵢ=xᵢxᵢ₋₁ is the displacement that happened from
iteration ᵢ₋₁ to ᵢ. Now if we perform summation over all
iterations, like that:
W += Wᵢ
we would get:
i="now"
W = ∑ (Fᵢ+Fᵢ₋₁)*(xᵢxᵢ₋₁)⁄2
i=1
In the end we were using that formula for all DOFs, to get a total
work, as a sum of normal, shearing and rotational displacements over
respective forces (and moment).
W=Wn+Ws+Wr
Wn =∑ (Fnᵢ+Fnᵢ₋₁)*(xᵢxᵢ₋₁)⁄2
Ws =∑ (Fsᵢ+Fsᵢ₋₁)*(sᵢsᵢ₋₁)⁄2
Wr =∑ (Mᵢ+Mᵢ₋₁)*quaternion_to_Euler_angle(qᵢ*(qᵢ₋₁)⁻¹).angle_in_radians⁄2
I don't have the implementation at hand right now, I'll find it
tomorrow. I might have made some mistake above, but you should get
the general idea.
If I find a mistake I'll let you know :)
best regards

Janek Kozicki http://janek.kozicki.pl/ 
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