# WIPP Model: Cylindrical Cavity

Note

To view this project in 3DEC, use the menu command Help ► Examples…. The project’s main data files are shown at the end of this example.

The WIPP-reference creep model in 3DEC 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 3DEC 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 3DEC 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

CylindricalCavityWIPP.dat

;------------------------------------------------------------
;       cylindrical cavity -- power law model
;-------------------------------------------------------------------
model new
fish automatic-create off
model title "Power-Law Creep Model --- Cylindrical Cavity"

model configure creep
model large-strain off

block tolerance 0.001
;
program call 'cylinder.fis'
[r_inner = 1.0]
[r_outer = 20.0]
[nr = 20]
[nd = 1]
[nc = 10]
[ratio = 1.1]
[length = 0.1]
[cylinder]

block join on

;block zone generate center 0 0 0 edgelength-center 0.1 edgelength-dist 2.5 distance 20
block zone generate hexahedra

block zone cmodel wipp
; Properties and stresses in Pascal units (not MPa)
block zone property density 2700 bulk 1e9 shear 3e8
block zone property constant-gas 1.987 activation-energy 12e3 ...
exponent 3 constant-d 5.5299e-17
block zone property constant-a 0 constant-b 0 creep-rate-critical 5.39e-8 ...
temperature 300
;
; apply stress first!
block face apply stress -100e6 -100e6 -100e6 0 0 0 ...
range cylinder end-1 0 0 0 end-2 0 0 -1 radius 19 21

block gridpoint apply velocity-x 0 range position-x 0
block gridpoint apply velocity-y 0 range position-y 0
block gridpoint apply velocity-z 0 range position-z -1 0

block zone initialize stress xx -100e6 yy -100e6 zz -100e6
;
model creep active off
model solve ratio 1e-7

model save 'cyl_1'

model history mechanical unbalanced-maximum
block history velocity-x position 1 0 0
block history velocity-x position 20 0 0
block history displacement-x position 1 0 0
block history displacement-x position 20 0 0
block history stress-xx position 1 0 0
block history stress-yy position 1 0 0
block history stress-zz position 1 0 0
model history creep time-total
model history timestep
;
block gridpoint initialize displacement (0,0,0)
block gridpoint initialize velocity (0,0,0)
model creep active on
model creep timestep starting 1.0e-6
model creep timestep minimum 1.0e-6
model creep timestep maximum 0.1

model solve time-total 200
model save 'cyl_2'

; compare analytic solution......
program call 'solution.fis' suppress
table  '1' label '3DEC x-velocity'
table '11' label 'Analytical x-velocity'
table  '2' label '3DEC radial stress'
table '12' label 'Analytical radial stress'
table  '3' label '3DEC hoop stress'
table '13' label 'Analytical hoop stress'
table  '4' label '3DEC out-of-plane stress'
table '14' label 'Analytical out-of-plane stress'
model save 'cylindrical-cavity-wipp'
program return