13 Plastic Flow in a Punch Problem

Transcription

1 Plastic Flow in a Punch Problem Plastic Flow in a Punch Problem 13.1 Problem Statement Difficulties are sometimes reported in the modeling of plastic flow where large velocity gradients exist. The use of interfaces embedded within a continuum finite-element mesh (e.g., van Langen and Vermeer 1991) has been suggested at locations where singular behavior is expected. The velocity field at the corners of a punch driven into a cohesive, frictionless soil is shown to be discontinuous. Figure 13.1 shows the problem geometry and boundary conditions. This example is used to demonstrate that FLAC can produce accurate results without introducing interfaces, provided that the expected singular point is not located at a gridpoint. One reason for avoiding interfaces is that, for some cases, the internal rupture surfaces may be unknown in advance. free surface imposed velocity 1 unit 1 unit rough, rigid punch 2 units 1 unit Figure 13.1 Boundary conditions and dimensions for the numerical simulation of a punch problem

2 13-2 Verification Problems 13.2 FLAC Model Figure 13.2 shows the grid used for the simulation. There are eight zones and nine nodes under the punch. The properties of the material are: bulk modulus (K) MPa shear modulus (G) 1.0 MPa density (p) 1000 kg/m 3 cohesion (c) 10kPa friction angle (φ) 0 A velocity loading condition, shown in Figure 13.3, is applied. The gradual application of the boundary velocity reduces the tendency for initial oscillation in the loading curve, but does not affect the collapse load. X = 1.0 Controlled Nodes Figure 13.2 FLAC grid for 8-zone punch

3 Plastic Flow in a Punch Problem 13-3 JOB TITLE : Punch Problem FLAC (Version 6.00) LEGEND 7-May-08 20:12 HISTORY PLOT Y-axis : 4 vel_his (FISH) X-axis : Number of steps -06 (10 ) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA 03 (10 ) Figure 13.3 Applied punch velocity 13.3 Results and Discussion The FISH function load calculates the numerical and analytical values of pressure beneath the punch. Note that for the numerical simulation, the total pressure is taken as the sum of vertical forces on the velocity-controlled nodes, divided by the width of the punch (unity, in this case). The width of the punch extends to one-half the zone at which the velocity jump occurs. The load is normalized by dividing by the cohesion, c, and the displacement is normalized by multiplying by the factor G/c. The resulting normalized load/displacement curve is given in Figure 13.4, and the steady-state velocity field is given in Figure The numerical value calculated for the steadystate load is c, which is only 0.12% in error of the exact load of (2 + π)c. The pattern of shear strain rate is illustrated in Figure The observed collapse mechanism is defined quite well, even for a material that does not soften. Table 13.1 records the steady-state loads for various mesh densities, expressed in terms of the number of FLAC zones under the punch. The accuracy is within 3.5% when only one element represents the punch. (Recall that FLAC uses constant-strain elements.) The overestimate of load for the finer discretizations can be reduced if the simulation is run with zero damping or at a slower velocity.

4 13-4 Verification Problems JOB TITLE : Punch Problem FLAC (Version 6.00) LEGEND 7-May-08 20:12 HISTORY PLOT Y-axis : 1 load (FISH) 2 anal (FISH) X-axis : 3 disp (FISH) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 13.4 Normalized load/displacement for 8-zone punch JOB TITLE : Punch Problem FLAC (Version 6.00) LEGEND 7-May-08 20: E+00 <x< 3.000E E+00 <y< 5.000E Boundary plot 0 1E 0 Velocity vectors scaled to max = 1.000E-05 max vector = 4.027E E Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 13.5 Steady-state velocity field for 8-zone punch

5 Plastic Flow in a Punch Problem 13-5 JOB TITLE : Punch Problem FLAC (Version 6.00) LEGEND 7-May-08 20: E+00 <x< 3.000E E+00 <y< 5.000E Boundary plot 0 1E 0 Max. shear strain-rate 0.00E E E E E E E E-05 Contour interval= 2.00E-06 Extrap. by averaging Itasca Consulting Group, Inc. Minneapolis, Minnesota USA Figure 13.6 Contours of maximum shear strain rate for 8-zone punch Table 13.1 Steady-state punch pressures (normalized) for various discretizations Punch Zones Pressure Error % % % % % 13.4 Reference van Langen, H., and P. A. Vermeer. Interface Elements for Singular Plasticity Points, Int. J. Num. Anal. Methods Geomech., 15, (1991).

6 13-6 Verification Problems 13.5 Data File PUNCH.DAT ;Project Record Tree export ;... State: m9.sav... ; measure load on moving plate def load sum = 0.0 loop i (1,9) sum = sum + yforce(i,17) end loop load = sum / (cohesion(1,1)*(x(9,17)+x(10,17))/2.) disp = -ydisp(1,17)*shear mod(1,1)/cohesion(1,1) anal = (2.0+pi) end ; gradual increase in starting velocity def ramp while stepping if step <= 3000 then ud app = 4e-7 + step *3.6e-6/3000 vel his = udapp loop i (1,9) yvel(i,17) = - ud app end loop end if end config grid 17,24 model mohr gen (0,-3) (0,0) (2,0) (2,-3) prop dens=1000 bulk= e6 shear=1e6 coh=1e4 tens=1e10 mod null i=1,9 j=17,24 ; boundary condition fix x i=1 fix x i=18 fix x,y j=1 fix x i=10 j=17,25 fix x,y i=1,9 j=17 ; histories his load his anal his disp his vel his his sratio his nstep 100

7 Plastic Flow in a Punch Problem 13-7 save m9.sav ;*** plot commands **** ;plot name: grid plot hold grid ;plot name: Applied punch velocity plot hold history 4 line ;plot name: Normalized load/displacement plot hold history 1 line 2 line begin 3000 skip 30 vs 3 ;plot name: Steady-state velocity field plot hold bound velocity max 1.0E-5 ;plot name: Contours of max. shear strain rate plot hold bound ssr fill

8 13-8 Verification Problems