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[Vincent Richefeu <richefeu@xxxxxxxxxxxxxxx>]

Le 14 sept. 2009 à 18:30, Bruno Chareyre a écrit :

> > I wrote Shop::totalForceInVolume, which computes stiffness and force
> > just for that purpose. I am not sure it is physically correct, but
> > should be: if you sum forces on all interactions (abs values of the
> > components, that is), you get some summary force in the whole
> > simulation. Dividing that force e.g. z-component by xy area of the
> > periodic cell, you have average stress in the z sense, right?
> After 1min of thinking :
> mmmh.... probably not. Suppose you have the same stress in a sample  
> that is twice higher (y-length) with the same x/z length, the sum of  
> forces will be multiplied by 2, you divide by the same area. YOUR  
> stress is twice bigger when it should be the same.
> You need to divide by a volume like in Love-Weber : Sum(fi*lj)/V  // 
> component (ij) of the stress tensor
> Not a big deal to compute that (its somewhere in a recorder already,  
> it needs adaptation for periodic volume though).

Maybe it should be more clear with that:

sigma components (xx, xy,..., zz) = 0
FOREACH_INTERACTIONS
    - Compute lx, ly and lz (components of the branch vector that  
connects the mass centers)
    - Compute fx, fy and fz (components of interaction force, don't  
mind about the constitutive law and the sign if we just want to  
compute the mean stress tensor)

    sxx += fx * lx;
    sxy += fx * ly;
    sxz += fx * lz;
    // ...
    syx += fy * lx;
    // ...
    szz += fz * lz;
END_LOOP

- Compute volume V of the cell by using cellMin and cellMax vectors.
- Multiply the mean stress components by 1/V.

(obviously, it should be shortly written with wm3...)

Note also that a dynamic contribution (usually neglected in quasi  
static problems) can be added to this expression.

> > I would be glad if you could check that Shop::totalForceInVolume.
> >
> > Some control code is in PeriIsoCompressor, you can take that as
> > inspiration, I think it is quite readable:
> >
> > Vector3r cellSize=rb->cellMax-rb->cellMin;
> > Vector3r cellArea=Vector3r(cellSize[1]*cellSize[2],cellSize[0] 
> > *cellSize[2],cellSize[0]*cellSize[1]);
> > Vector3r sumForces=Shop::totalForceInVolume(avgStiffness,rb);
> > Vector3r sigma=Vector3r(sumForces[0]/cellArea[0],sumForces[1]/ 
> > cellArea[1],sumForces[2]/cellArea[2]);
> > // etc.
> If I change the value of, say, rb->cellMax, will it instantly  
> translate all particles in proportion? (you remember this old  
> discussion?)

With tri-periodic conditions, the cell can be considered (and can be  
treated) as a bodies defined by a mass, a matrix 3x3 for the position,  
and two other matrixes 3x3 for velocity and acceleration.
Any transformation of the cell should instantly be translate to the  
particles ! This is actually what I do in a 2D code (tapio-K).

I hope it helps.

VR