# WIPP Model: Cylindrical Cavity

Note

To view this project in FLAC3D, use the menu command Help ► Examples…. Choose “Creep/CylindricalCavityWIPP” and select “CylindricalCavityWIPP.prj” to load. The project’s main data files are shown at the end of this example.

The WIPP-reference creep model in FLAC3D is used to solve the problem of radial creep of an infinitely long, thick-walled cylinder subjected to a pressure on its outer surface. The analytical steady-state solution, assuming that creep is defined by a single-component power law, is provided in Power Model: Cylindrical Cavity. The WIPP model can be converted to a power law formulation by using only the secondary creep-strain component. This is achieved by setting the WIPP properties a_wipp and b_wipp to zero. The WIPP model is then reduced to the form

$\dot \epsilon_{cr} = A \bar \sigma^n$

where $$A = D \exp(-Q/RT)$$.

For this problem, $$Q$$ = 12,000 cal/mol, $$R$$ = 1.987 cal/mol K, $$T$$ = 300° K, $$D$$ = 5.5299 × 10-17 (or $$A$$ = 1 × 10-25 Pa-3 yr-1), and $$n$$ = 3. The elastic properties of the material are $$E$$ = 820 MPa and $$\nu$$ = 0.3636.

The model uses the same FLAC3D grid and the same boundary conditions as as those described in Power Model: Cylindrical Cavity.

The results of this example are summarized in Figure 1 and Figure 2. Figure 1 compares the analytical solution for the radial velocity in the steady-state condition with the FLAC3D results. Figure 2 shows the comparison of radial and hoop stresses. The agreement between results is similar to that for the power law test in Power Model: Cylindrical Cavity.

Data File

;------------------------------------------------------------
;       cylindrical cavity -- WIPP model
;-------------------------------------------------------------------
model new
model large-strain off
fish automatic-create off
model title "WIPP Creep Model --- Cylindrical Cavity"
model configure creep
;
zone create brick  point 0 1 0 0               point 1 20 0 0           ...
point 3 0.98769 0.15643 0   point 6 19.754 3.129 0   ...
point 2 1 0 -1              point 4 20 0 -1          ...
point 5 0.98769 0.15643 -1  point 7 19.754 3.129 -1  ...
size 20 1 1  ratio 1.2 1 1
zone face skin
zone cmodel wipp
; Properties and stresses in Pascal units (not MPa)
zone property bulk=1e9 shear=3e8
zone property constant-gas 1.987 activation-energy 12e3 exponent 3 ...
constant-d 5.5299e-17
zone property constant-a 0 constant-b 0 creep-rate-critical 5.39e-8 ...
temperature 300
;
zone gridpoint fix velocity-y range position-y 0
zone gridpoint fix velocity-z range position-y 0
zone face apply velocity-normal 0 range group 'North'
zone face apply velocity-dip    0 range group 'North'
zone face apply stress-normal -100e6 range group 'East'
zone initialize stress xx -100e6 yy -100e6 zz -100e6
;
model step 2000
model save 'cyl_wl'
;
model history mechanical unbalanced-maximum
zone history velocity-x gridpointid 1
zone history velocity-x gridpointid 81
zone history displacement-x gridpointid 1
zone history displacement-x gridpointid 81
zone history stress-xx gridpointid 1
zone history stress-yy gridpointid 1
zone history stress-zz gridpointid 1
model history creep time-total
model history timestep
;
zone gridpoint initialize velocity (0,0,0)
model creep active on
model creep timestep starting 1.0e-3
model creep timestep minimum 1.0e-4
model creep timestep maximum 1.0
model solve time-total 2000
model save 'cyl_w2'
; compare analytic solution......
program call 'solution' suppress
table  '1' label 'Flac3d x-velocity'
table '11' label 'Analytical x-velocity'
table  '2' label 'Flac3d radial stress'
table '12' label 'Analytical radial stress'
table  '3' label 'Flac3d hoop stress'
table '13' label 'Analytical hoop stress'
model save 'cylindrical-cavity-wipp'
program return