# yade-users team mailing list archive

## Re: Elastic energy

```> I wanted to say that I was actually doing a numerical integration
> with trapezoidal rule instead of using analytical method.
>
Yes, this was clear to me.

>
> >  Using trapezoidal rule, hence working out
> > the net work done by the spring in an incremental way, or either taking
> the
> > value of the integral should give you the same result.
>
> This might be not true for slipping - we are entering plastic
> behaviour regime. See below.
>
Jan, I think there is some misunderstanding here. Or maybe you are not
updated with the way we calculate plastic dissipation. We do it
files I sent to Bruno a while ago which explain more analytically the
concept.
The elastic part can be always computed in a once analytically since anyway
the friction force cannot overcome the maximum value. Therefore what we
compute in elasticEnergy is only elastic part, both for shear and normal
direction.

>
>
> > Perhaps the mistake is in the computation of the plastic dissipation. But
> I
>
>
>
> Let me rewrite this line from elasticEnergy():
>
>  energy += 0.5*(phys->normalForce.squaredNorm()/phys->kn +
> phys->shearForce.squaredNorm()/phys->ks);
>
>
>  E += (Fn²⁄kn + Fs²⁄ks)⁄2
>
> The normal part Wn =∑ (Fnᵢ+Fnᵢ₋₁)*(xᵢ-xᵢ₋₁)⁄2 should be rather
> equivalent to En+=(Fn²⁄kn)⁄2
>
>
> But slipping might be actually different in those two approaches:
>
>  Ws =∑ (Fsᵢ+Fsᵢ₋₁)*(sᵢ-sᵢ₋₁)⁄2
>
> and this: Es+=(Fs²⁄ks)⁄2
>
> That is because the slip distance is implicitly calculated
> in formula (Fs²⁄ks)⁄2, because after all:
>
>  Fs²⁄ks = (ks*s)²/ks = ks*s²
>
> and I'm afraid that maybe s≠∑(sᵢ-sᵢ₋₁)  , because sᵢ is supposed
> to follow the slipping path (or slip "trace") on the surface of a
> sphere, while s is calculated from current value Fs.
>
Sure, see above.

>
> To say in other words, I think that when spheres start to slip on
> each other and Fs stays constant, the increment of s which stays
> inside constant Fs differs from the path increment on the sphere's
> surface slipping path sᵢ-sᵢ₋₁.
>
> I'm not sure if I worded myself clearly.... I hope that you can
> understand what I mean?

It is clear and, in fact, it is also what we are doing. That is why I am
struggling because it should give the expected result, total energy
constant. I will try again..

>
>
> IIRC Vaclav was calculating somewhere the total accumulated
> _geometrical_ path of one sphere on another sphere.
>
> best regards
> --
> Janek Kozicki                               http://janek.kozicki.pl/  |
>
> _______________________________________________
>
```

Attachment: Energy_1.pdf