A Note From Philippe on Viscosities in SOPALE


I would like to provide here a few words of explanation on SOPALE's
viscosities, in response to some comments heard recently.

in Sopale, viscosities are computed in the routine 'viscosities1'

The basic method used for computing viscosities in sopale is a 'parallel
assemblage' of mechanisms,

which means that we add stresses (or forces) due to the individual
component mechanisms.

So the primary quantity is STRESS. when the total stress has benn
assembled we can compute a DERIVED quantity: the viscosity. This
viscosity is the effective or equivalent viscosity of our parallel

'rheological circuit'. We have to keep in mind though that the
parametrization of the individual rheological component may be different
. This is what occurs in Sopale where a cocktail of 4 processes is used
(frictional-cohesive plasticity, nonlinear dislocation creep, grain size
sensitive creep, and Peierls creep). I explain here HOW these are
parametrized and why all the supposed problems of consistency are
imaginary if you do understand inputs and formula.

 

I have coded 3 models of creep, which in my mind were different: (i will
omit peierls creep and Mohr-Coulomb here)

-nonlinear creep model : uses a nonlinear relation between to
'mechanical/kinematical'

quantities : stress (say s) and strain-rate say e . The basic form of the
equation is:

e=a*(s^n)

the formula i read in one of sopale's versions (zopale) for the
computation of

nonlinear creep STRESSES are :

if(nlcreep_updown_T.eq.-1)then

sigma2=nlcreep_sc*((edot/nlcreep_eps)**(1.d0/nlcreep_n))*

1dexp((nlcreep_T+tshift)/(nlcreep_n*(tshift+tlg(l))))

endif

if(nlcreep_updown_T.eq.1)then

sigma2=nlcreep_sc*((edot/nlcreep_eps)**(1.d0/nlcreep_n))*

1dexp((tshift+tlg(l))/(nlcreep_n*(tshift+nlcreep_T)))

endif

if(nlcreep_updown_T.eq.0)then

sigma2=nlcreep_sc*((edot/nlcreep_eps)**(1.d0/nlcreep_n))

endif

from which you will all agree that nlcreep_sc must be homogeneous to

sigma2 ( a STRESS !) if nlcreep_n is dimensionless (the nonlinear creep
exponent) and nlcreep_eps is homegeneous to a strain rate (as edot is)

THERE SHOULD THEREFORE BE NO FACTOR OF 2 HERE. We are just computing a
stress, and there is no VISCOSITY used in this formula !

-the grain size sensitive creep model is different. It is NOT(not purely
: the grain size is a non-kinematical quantity ) a relation between
mechanical and kinematical quantities . The nlcreep model was implemented
to mimic the physics of dislocation creep

while the gscreep model was intended to mimic the physics of diffusion
creep ,where diffusive mass transport is couples to the deformation. I
have considered this category of processes as viscous with

a dependence of viscosity on a 'texture or micro-structural parameter:
the grain size (which itself evolves see Jean's work e.g.).

For these reasons, SOPALE PROVIDES A VISCOSITY and NOT a STRESS MODEL,
indeed formula for gscreep are : (zopale) :

if(gscreep_updown_T.eq.-1)then

sigma1=gscreep_mu*((gscreep_a/gscreep_a0)**(gscreep_m))*

1dexp((gscreep_T+tshift)/(tlg(l)+tshift))*2.d0*edot

endif

if(gscreep_updown_T.eq.1)then

sigma1=gscreep_mu*((gscreep_a/gscreep_a0)**(gscreep_m))*

1dexp((tlg(l)+tshift)/(tshift+gscreep_T))*2.d0*edot

endif

if(gscreep_updown_T.eq.0)then

sigma1=gscreep_mu*((gscreep_a/gscreep_a0)**(gscreep_m))*

1edot*2.d0

endif

Reading these formula you will all agree that we were computing stresses
( sigma1) but that what we have parametrized is the VISCOSITY: i.e. e.g.
gscreep_mu*((gscreep_a/gscreep_a0)**(gscreep_m))

mu is the greek letter used ubiquitously to note viscosities. gscreep_mu
refers quite logically to the viscosity (scale) associated to Grain Size
sensitive creep (gscreep). Just compare this to the name that was used
for nonlinear creep : nlcreep_sc which stands for the STRESS (scale)
associated with nonlinear creep.

You will recognize that if gscreep_a/gscreep_a0 is adimensional (does
this not look so?) (this by the way is a dimensionless grain size) and if
gscreep_m stands for an exponent indeed ALL PREVIOUS FORMULA have the
right expression : stress= viscosity*2*edot and there neither an extra or
a missing factor of 2 !

if now you want to relate this to the inputs it is easy:

read(8,*)gscreep_mu(i),gscreep_m(i),gscreep_T(i)

1,gscreep_strain1(i),gscreep_strain2(i),gscreep_a0(i),

1gscreep_a1(i),gscreep_a2(i),gscreep_updown_T(i)

read(8,*)nlcreep_eps(i),nlcreep_sc1(i),nlcreep_sc2(i),

1nlcreep_n(i),

1nlcreep_T(i),nlcreep_strain1(i),nlcreep_strain2(i),

1nlcreep_updown_T(i)

so i read this for you :

-gscreep models viscosities and you enter a viscosity scale gscreep_mu(i)

-nlcreep models stresses and you enter stress scales :

nlcreep_sc1(i),nlcreep_sc2(i)

 

when the total stress has been computed the VISCOSITY (effective) is

computed as : (zopale) :

viscos(l,n)=sleast/(2.d0*edot)

where sleast is the min(
sigma1(gscreep)+sigma2(nlcreep)+sigma3(peierls),sigma_mohr_coulomb)

again this is correct.

cheers

pf