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Cylindrical Hole in an Infinite Mohr-Coulomb Material 1 - 1

1 Cylindrical Hole in an Infinite Mohr-Coulomb Material

1.1 Problem Statement

Stresses and displacements are determined numerically for the case of a cylindrical hole in aninfinite elasto-plastic material subjected to in-situ stresses. The material is assumed to be linearlyelastic, perfectly plastic, with a failure surface defined by the Mohr-Coulomb criterion, and bothassociated (dilatancy = friction angle) and nonassociated (dilatancy = 0) flow rules are used. Theresults of the simulation are compared with an analytic solution.

This problem tests the Mohr-Coulomb plasticity model with plane-strain conditions imposed inFLAC3D.

The Mohr-Coulomb material is assigned several properties:

shear modulus (G) 2.8 GPa

bulk modulus (K) 3.9 GPa

cohesion (c) 3.45 MPa

friction angle () 30

dilation angle () 0 and 30

The isotropic in-situ stress has a magnitude of30 MPa, and the pressure inside the hole may beneglected. (As a convention, compressive stresses are negative.) The radius, a, of the hole is smallcompared to the length of the cylinder, so that plane-strain conditions are applicable.

1.2 Closed-Form Solution

The analytic solution for this problem may be found in Salenon (1969). The yield zone radius,R0, may be expressed, in general terms, as

R0

a=

2

Kp+1

1+ qP0

kp

PiP0

+ qP0

kp

kp(1.1)

in which a is the hole radius,P0is the absolute value of the in-situ isotropic stress,Piis the pressureinside the hole (0 Pa, in our case), and

Kp =1+sin

1sin

kp =1

Kp1

q = 2c Kp

(1.2)

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Cylindrical Hole in an Infinite Mohr-Coulomb Material 1 - 5

,rp, U Ol g

ee,p8G,/G,rp, p,:SS:rr E

U: O E

LTZET

U Ol gl E

Figure 1.3 FLAC3D grid quarter symmetry

1.4 Results and Discussion

Figures 1.4through1.7show comparisons between FLAC3D results and the analytic solution along

a radial line. Normalized stresses,r/P0 and /P0, are plotted versus normalized radius,r/a , inFigures 1.4and1.5, while normalized displacements,ur/a, are represented versusr/ainFigures 1.6 and 1.7. Note that numerical stress values for the two flow rules cannot be differentiatedon the graphs.

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1 - 6 Verification Problems

,rp, D l g

ee,p8G,/G,rp, p,:SS:rp F

p i

, i. FD.NiS FD.Ni

D l gl F

Figure 1.4 Stress solution comparison associated(analytical values = lines; numerical values = crosses)

,rp, L l g

e5S59G,4G,rp, p,/SS/,e 3

p i, i.U3L.DiSU3L.Di

L l gl 3

Figure 1.5 Stress solution comparison nonassociated(analytical values = lines; numerical values = crosses)

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Cylindrical Hole in an Infinite Mohr-Coulomb Material 1 - 7

,rp, 3 l g

ee,p8G,/G,rp, p,:SS:rp F

e i

5UF3.Di

3 l gl F

Figure 1.6 Radial displacement solution comparison associated(analytical values = lines; numerical values = crosses)

,rp, 3 l g

e5S59G,4G,rp, p,/SS/,e F

e i5UF3.Di

3 l gl F

Figure 1.7 Radial displacement solution comparison nonassociated(analytical values = lines; numerical values = crosses)

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1 - 8 Verification Problems

The average relative error on the stresses and displacements is less than 2.0% (except 4.4% fordisplacements for the nonassociated model) throughout the grid. (Note that the error could bereduced by more appropriate handling of the far field conditions using, for example, the Lamsolution for a thick ring.)

Displacement contours and displacement vectors for the associated case are presented in Figure 1.8,as an illustration:

,rp, U P 3l Ug

ee,p8G,/G,rp, p,:SS:rpN

: Crlrgplro: Crgrrrlpgrrrlrgrrro

:

,g8,8/-ir,,g9err-ir,,gerrr-ir,,g,err-ir,

,grrrr-ir,pg9err-ir,pgerrr-ir,pg,err-ir,pgrrrr-ir,9gerrr-ir.egrrrr-ir.,gerrr-ir.pg.prS-ir.

: : rgr,8,898: S

U P 3l Ugl N

Figure 1.8 FLAC3Ddisplacement contours and displacement vectors asso-ciated

The files sal-associated.f3dat and sal-non-associated.f3dat are used to generate the numericalsolutions to the associated and nonassociated cases. (The dilatation angle is 30 in the associatedcase.) The files sal-associated-nmd.f3dat and sal-non-associated-nmd.f3dat generate the sametwo cases, but using FLAC3Ds nodal mixed discretization (NMD) feature. For the NMD models,an all-tet grid is used (sal nmd.Flac3dD, which is imported intoFLAC3D not listed here). Thetet grid has gridpoints identical to the hex grid.

The file sale.f3dat compares the numerical solutions to the analytical solution using two FISH

functions:

(1) nastr calculates (i) numerical and corresponding analytical values ofr/P0and/P0at the centroid of zones closest to the x -axis, and (ii) average relative error ofr/P0and/P0 throughout the grid;

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Cylindrical Hole in an Infinite Mohr-Coulomb Material 1 - 9

(2) nadiscalculates (i) numerical and analytical values ofur/a at gridpoints located on thex-axis, and (ii) the average relative error of the displacement ur/a throughout the grid.

The resulting errors are within 2% of the analytical solution for stresses and displacements for both

associated and nonassociated flow rules. Figures 1.91.13show the NMD results, and corresponddirectly withFigures 1.4 1.8 (results with an all-hex grid). As can be seen, the results are verysimilar.

,rp, L l g

e777/G,9G,rp, p,8SS8pe 3

p i, i.U3L.DiSU3L.Di

L l gl 3

Figure 1.9 NMD stress solution comparison associated (analytical values= lines; numerical values = crosses)

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1 - 1 0 Verification Problems

,rp, D l g

ee9r/G,9G,rp, p,:SS:.9F

p i

, i. FD.NiS FD.Ni

D l gl F

Figure 1.10 NMD stress solution comparison nonassociated (analytical val-ues = lines; numerical values = crosses)

,rp, L Ul g

e777/G,9G,rp, p,8SS8pe 6

e i5N6L.3i

LUl gl 6

Figure 1.11 NMD radial displacement solution comparison associated (an-alytical values = lines; numerical values = crosses)

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Cylindrical Hole in an Infinite Mohr-Coulomb Material 1 - 1 1

,rp, L l g

ee9r/G,9G,rp, p,:SS:./6

e i

5U6L.Di

L l gl 6

Figure 1.12 NMD radial displacement solution comparison nonassociated(analytical values = lines; numerical values = crosses)

,rp, U N -l Ug

e777prG,5G,rp, /8.r8pS (

8Crlrgplro8Crgrrrlpgrrrlrgrrro

8

.grrr5Pir,

.grrrrPir,,g7errPir,,gerrrPir,,g,errPir,,grrrrPir,pg7errPir,pgerrrPir,pg,errPir,pgrrrrPir,7gerrrPir.egrrrrPir.,gerrrPir.pg.p/pPir.

88rgr.rrr5p

8S

U N-l Ugl (

Figure 1.13 NMD FLAC3D displacement contours and displacement vectors associated

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1 - 1 2 Verification Problems

1.5 Reference

Salenon, J. Contraction Quasi-Statique Dune Cavite a Symetrie Spherique Ou Cylindrique DansUn Milieu Elastoplastique, Annales Des Ponts Et Chaussees,4, 231-236 (1969).

1.6 Listing of Data Files

The project file for this problem is located in mechanical\salencon.

Example 1.1 SAL-ASSOCIATED.F3DAT

;---------------------------------------------------------------------

; numerical solution for a long tunnel in pre-stressed

; Mohr Coulomb material (salencon problem)

; associated and non-associated plastic flow

;---------------------------------------------------------------------

new

gen zone radcyl size 1 1 30 30 rat 1 1 1 1.1 p1 10 0 0 p2 0 0.2 0 &

p 3 0 0 1 0 d i m 1 1 1 1

model mech mohr

prop bul 3.9e9 shea 2.8e9 cohesion 3.45e6

prop friction 30. dilation 30. tension 1.e10 ; associated flow

;prop friction 30. dilation 0. tension 1.e10 ; non-associated flow

ini sxx -30e6 syy -30e6 szz -30e6

fix z range z -.001 .001

fix x range x -.001 .001

fix y

apply sxx -30e6 range x 9.9 10.1

apply szz -30e6 range z 9.9 10.1

hist add gp xdisp 1 0 0

hist a dd g p xvel 1 0 0

hist add gp yf 1 0 0

hist add gp zf 1 0 0

hist n 20

solve

save associated

return

Example 1.2 SAL-ASSOCIATED-NMD.F3DAT

;---------------------------------------------------------------------

; Nodal Mixed Discretization (NMD) solution

; for a long tunnel in pre-stressed

; Mohr Coulomb material (salencon problem)

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Cylindrical Hole in an Infinite Mohr-Coulomb Material 1 - 1 3

; Associated plastic flow

;---------------------------------------------------------------------

new

impgrid sal_nmdset nmd on

model mech mohr overlay 1

prop bulk 3.9e9 shea 2.8e9 cohesion 3.45e6

prop friction 30. dilation 30. tension 1.e10 ; associated flow

;prop friction 30. dilation 0. tension 1.e10 ; non-associated flow

ini sxx -30e6 syy -30e6 szz -30e6

fix z range z -.001 .001

fix x range x -.001 .001

fix y

apply sxx -30e6 range x 9.9 10.1

apply szz -30e6 range z 9.9 10.1

hist add gp xdisp 1 0 0hist a dd g p xvel 1 0 0

hist add gp yf 1 0 0

hist add gp zf 1 0 0

hist n 20

solve

save associated_nmd

return

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