18 Compression of a Poroelastic Sample Mandel s Problem

Transcription

1 Compression of a Poroelastic Sample Mandel s Problem Compression of a Poroelastic Sample Mandel s Problem 18.1 Problem Statement During compression of a poroelastic specimen under constant boundary conditions, the pore pressure will display a non-monotonic variation with consolidation time. At initial consolidation times, an increase in the pore pressure will be induced near the center of the sample when it is subjected to a constant vertical load and drained laterally. Subsequently, the pore pressure falls. This effect was pointed out by Mandel (1950). It was also predicted by Cryer (1963), and thus is also known as the Mandel-Cryer effect, and was demonstrated experimentally by Verruijt (1965). In Mandel s problem, a sample of saturated poroelastic material is loaded under plane-strain conditions by a constant compressive force applied on rigid impervious platens (see Figure 18.1). The width of the sample is 2a, its height is 2b, and the force intensity is 2F. The application of the load is instantaneous, the platens are impervious, and the sample is free to drain laterally. The short-term response of the material corresponds to a uniform vertical stress across the sample. As lateral drainage takes place, the non-uniform dissipation of induced pore pressure causes an apparent softening of the material near the edges of the sample. The resulting stress concentration in the stiffer (still undrained) core is then responsible for an additional increase in pore pressure in the middle of the sample. As drainage proceeds and the pore pressure gradient decreases, the vertical load is again transmitted in a uniform manner. The non-monotonic variation of pore pressure with time observed in Mandel s problem serves to illustrate the main difference in prediction between the Biot and Terzaghi theories.. > O N =. Figure 18.1 Mandel s problem

2 18-2 Verification Problems 18.2 Analytical Solution The solution to Mandel s problem, generalized for the case of compressible constituents, is given by Cheng and Detournay (1988). The expression for the pore pressure is p = 2FB(1 + ν u) 3a i=1 sin α [ i cos α ix α i sin α i cos α i a cos α i ] exp( α 2 i ct/a2 ) (18.1) where B is Skempton s pore pressure coefficient (the ratio of induced pore pressure to variation of confining pressure under undrained conditions), ν and ν u are drained and undrained Poisson s ratio, c is the true diffusivity (or generalized consolidation coefficient), t is time and α i, i = 1,, are the roots of the equation: tan α i = 1 ν ν u ν α i (18.2) If the solid grains that form the material are assumed to be incompressible, as is the case in FLAC, the following relations between material constants apply: B = K w K w + nk K u = K + K w n c = k/( n K w + 1 K + 4G/3 ) (18.3) where K w is fluid bulk modulus, n is porosity, K and K u are drained and undrained bulk moduli, G is shear modulus and k is mobility coefficient. The drained and undrained Poisson s ratios are related to the moduli G, K and K u, as follows: ν = 3K 2G 6K + 2G ν u = 3K u 2G 6K u + 2G (18.4)

3 Compression of a Poroelastic Sample Mandel s Problem 18-3 The formulae for horizontal displacement at x = a and vertical displacement at y = b are: u x (a, t) = Fν 2G + F(1 ν u) G i=1 F(1 ν)b u y (b, t) = + F(1 ν u)b 2Ga Ga sin α i cos α i α i sin α i cos α i exp( α 2 i ct/a2 ) i=1 sin α i cos α i α i sin α i cos α i exp( α 2 i ct/a2 ) (18.5) According to the exact solution, the initial (instantaneous) and final vertical displacements of the upper platen are: u y (b, 0) = Fb 1 ν u 2Ga u y (b, ) = Fb 1 ν 2Ga (18.6) The lateral boundaries of the sample stay plane during drainage; they first move outward and then inward. The initial (instantaneous) and final lateral displacements of the right boundary are: u x (a, 0) = F ν u 2G u x (a, ) = F ν 2G (18.7) The degrees of consolidation in the horizontal and vertical directions, D h and D v, are defined as: D h = u x(a, t) u x (a, 0) u x (a, ) u x (a, 0) D v = u y(b, t) u y (b, 0) u y (b, ) u y (b, 0) (18.8)

4 18-4 Verification Problems These degrees are found to be identical, and the analytic expression for D = D h = D v is D = 1 4(1 ν u) 1 2ν i=1 sin α i cos α i α i sin α i cos α i exp( α 2 i ct/a2 ) (18.9) Because the discharge has only a horizontal component in Mandel s problem, the pore pressure, stress and strain solutions are indepent of the y-coordinate FLAC Model By symmetry, only a quarter of the sample is considered in the numerical model.* The grid has 20 zones in the x-direction and 2 in the y-direction. Mandel s problem is solved for the case drained Poisson s ratio (ν) 0.2 Skempton coefficient (B) 0.9 The FLAC simulation is carried out to produce results in terms of normalized pore pressure, ˆp, distance, ˆx, and time, ˆt, defined as: ˆp = ap F ˆx = x a ˆt = ct a 2 (18.10) The normalized results are not affected by the absolute magnitude of material properties used, as long as their combination yields the values specified above for ν u and B. The FLAC grid dimensions, applied force and property values used in the simulation may be viewed as scaled for the purpose of producing the normalized results. They are selected as follows: * CAUTION: When running this example in the GIIC, be sure to turn off the listing to the Console pane using the File/Preference Settings menu item. Otherwise, the size of the save files will become excessive. Check Do not save console pane in the Resources tab of this menu item.

5 Compression of a Poroelastic Sample Mandel s Problem 18-5 model width (a) 1 model height (b) 0.1 applied force (F ) 1 drained bulk modulus (K) 1 shear modulus (G) 0.75 fluid bulk modulus (K w ) 9 porosity (n) 0.5 With the above values, the true diffusivity in the FLAC model is unity, and the model time is ˆt. The model mechanical boundary conditions correspond to roller boundaries along the x- and y-axes of symmetry. The sample is initially stress-free, and the pore pressure is equal to zero. Undrained conditions are established first by applying a constant unit mechanical pressure at the top boundary of the model. In the second part of the simulation, the pore pressure is fixed at zero on the right side of the model to allow drainage to occur. The rigid plate condition is enforced by applying a vertical velocity at the top of the model. The velocity magnitude is derived from the exact displacement solution (Eq. (18.5)). As a verification to the numerical solution, the reaction force on the top platen is monitored to check whether it remains constant and equal to one Results The numerical simulation is carried out for a total value of normalized time equal to 4, with intermediate results at ˆt = 0.01, 0.1, 0.5, 1 and 2. (Results at ˆt = 0 correspond to the undrained response.) The pore pressure profiles at those times are checked against exact solutions in Figure At early times, the pore pressure at x = 0 (center of sample) is seen to rise above the undrained value, before decreasing as drainage evolves. This is also shown in Figure 18.3, which compares FLAC to the analytical solution for the actual pore pressure versus consolidation time at the center of the sample. As may be seen in Figure 18.4, the reaction force stays equal to unity throughout the simulation. (The approach taken to apply the force boundary condition is thus equivalent to a servo-controlled velocity.) Numerical values for the horizontal and vertical degrees of consolidation are plotted versus the analytical solution values (Eq. (18.9))inFigure 18.5.

6 18-6 Verification Problems 18.5 References Cheng, A. H. D., and E. Detournay. A Direct Boundary Element Method for Plane Strain Poroelasticity, Int. J. Num. Methods and Anal. in Geomechanics, 12, (1988). Cryer, C. W. A Comparison of the Three-Dimensional Consolidation Theories of Biot and Terzaghi, Quart. J. Mech. and Appl. Math., XVI, 4, (1963). Mandel, J. Consolidation des sols (étude mathématique), Géotechnique, 3, (1953). Verruijt, A. Discussion, Proc. 6th Int. Conf. Soil Mechanics and Foundation Engineering, 3, Montréal, (1965).

7 Compression of a Poroelastic Sample Mandel s Problem 18-7 JOB TITLE : Mel s Problem FLAC (Version 6.00) LEGEND 5-May-08 22:46 step Flow Time E+00 Table Plot anal (t = 0.0) FLAC (t = 0.0) anal (t = 0.01) FLAC (t = 0.01) anal (t = 0.1) FLAC (t = 0.1) anal (t = 0.5) FLAC (t = 0.5) anal (t = 1.0) FLAC (t = 1.0) anal (t = 2.0) FLAC (t = 2.0) anal (t = 4.0) FLAC (t = 4.0) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA -01 (10 ) (10 ) Figure 18.2 Pore pressure profile comparison: ˆp vs ˆx JOB TITLE : Mel s Problem FLAC (Version 6.00) LEGEND 5-May-08 22:49 step Flow Time E+00 HISTORY PLOT Y-axis : 1 ana_pp (FISH) 2 num_pp (FISH) X-axis : 3 Groundwater flow time -01 (10 ) Itasca Consulting Group, Inc. Minneapolis, Minnesota USA -01 (10 ) Figure 18.3 Pore pressure versus consolidation time at the center of the sample

8 18-8 Verification Problems JOB TITLE : Mel s Problem FLAC (Version 6.00) LEGEND 7-May-08 22:50 step Flow Time E+00 Table Plot y-reaction force Itasca Consulting Group, Inc. Minneapolis, Minnesota USA -01 (10 ) Figure 18.4 History of y-reaction force on top platen JOB TITLE : Mel s Problem FLAC (Version 6.00) LEGEND 5-May-08 22:51 step Flow Time E+00 Table Plot Table 7 Table 8 Table Itasca Consulting Group, Inc. Minneapolis, Minnesota USA -01 (10 ) Figure 18.5 Degree of consolidation versus log time

9 Compression of a Poroelastic Sample Mandel s Problem Data File MANDEL.DAT ;Project Record Tree export ;... State: man0.sav... config gw ; --- definitions --- def ini man c F = 1. ; applied force c a = 1. ; sample half width c b = 0.1 ; sample half height c wb = 4.5 ; water bulk modulus c wk = 11./18. ; mobility coefficient c po = 0.5 ; porosity c bu = 1. ; drained bulk modulus c sh = 0.75 ; shear modulus c ub = cbu + cwb/c po ; undrained bulk modulus c sk = cwb/(c wb + cpo*c bu) ; Skempton coefficient val = c bu/c sh c nu = (3.*val-2.)/(6.*val+2.) ; drained Poisson ratio val = c ub/c sh c un = (3.*val-2.)/(6.*val+2.) ; undrained Poisson ratio stor = c po/c wb+1./(c bu+4.*c sh/3.); storativity diff = c wk/stor ; diffusivity coe = diff/(c a*c a) v co = -(1.-c un)*c F*c b/(c sh*c a) ; coeff. for applied vel v co = v co*coe c nm = 100 ; number of mech steps per flow step ini man ; --- grid --- grid 20,2 gen c b c a c b c a 0 def f gp figp = igp fjgp = jgp f gp ; --- properties --- m e water bulk=c wb prop perm=c wk por=c po prop dens 1 bu=c bu sh=c sh ; --- fish function--- def check ppu ; undrained pore pressure profile ini man

10 18-10 Verification Problems pcoe = c F*c sk*(1.+c un)/(3.*c a) tabn = taba + 10 loop ip (1,igp) xval = x(ip,2) xtable(taba,ip) = xval ytable(taba,ip) = pcoe xtable(tabn,ip) = xval ytable(tabn,ip) = gpp(ip,2) loop ; --- boundary conditions --- fix x i=1 fix y j=1 apply pr 1 j=fjgp ; --- undrained response --- set flow off solve sratio 1.e-3 set taba=10 check ppu save man0.sav ;... State: man1.sav... ; --- fish functions --- def ini root tabroot = 100 nroot = 50 tab1 = tabroot + 1 tab2 = tabroot + 2 tab3 = tabroot + 3 tab4 = tabroot + 4 tab5 = tabroot + 5 c coe = (1. - c nu)/(c un - c nu) pcoe = 2.*c F*c sk*(1.+c un)/(3.*c a) tbi = 0 ini root ; calculate and store roots ca froot.fis def func func = tan(c x0)/(c coe*c x0) - 1. def store root val root = -pi c eps2 = 1.e-2 tol = 1.e-6 c int = pi*0.5

11 Compression of a Poroelastic Sample Mandel s Problem loop ii (1,nroot) val = float(ii-1)*pi c eps1 = c eps2*(val + c int - val root - pi) if ii > 50 then c eps1 = c eps1*10. c x1 = val - c eps1 c x2 = val + c int - c eps1 val root = froot xtable(tabroot,ii) = ii ytable(tabroot,ii) = val root sr = sin(val root) cr = cos(val root) den = val root-sr*cr r2 = val root * val root xtable(tab1,ii) = ii ytable(tab1,ii) = sr/den xtable(tab2,ii) = ii ytable(tab2,ii) = cr/den xtable(tab3,ii) = ii ytable(tab3,ii) = sr*cr/den xtable(tab4,ii) = ii ytable(tab4,ii) = r2 xtable(tab5,ii) = ii ytable(tab5,ii) = r2*sr*cr/den loop store root def rf ; reaction force sum = 0. loop ii(1,igp) sum = sum + yforce(ii,jgp) loop rf = sum def qout ; outflow outflow = 0.0 loop j (1,jgp) outflow = outflow - gflow(igp,j) loop qout = outflow def ana yv ; velocity per mechanical step velo = 0. veln = 0. jjsav = 0

12 18-12 Verification Problems loop jj (1,nroot) al2 = ytable(tab4,jj) veln = velo + ytable(tab5,jj)*exp(-al2*coe*gwtime) if veln = velo then jjsav = jj jj = nroot velo = veln loop ana yv = veln*v co*gwtdel def check pp ; pore pressure profile ini man pcoe = 2.*c F*c sk*(1.+c un)/(3.*c a) tabn = taba + 10 loop ip (1,igp) sumo = 0. sumn = 0. jjsav = 0 xval = x(ip,2) loop jjp (1,nroot) al2 = ytable(tab4,jjp) al = ytable(tabroot,jjp) term = cos(al*xval/c a)-cos(al) sumn = sumo + ytable(tab1,jjp)*term*exp(-al2*coe*gwtime) if sumn = sumo then jjsav = jjp jjp = nroot sumo = sumn loop xtable(taba,ip) = xval ytable(taba,ip) = sumn*pcoe xtable(tabn,ip) = xval ytable(tabn,ip) = gpp(ip,2) loop def ana pp ; pore pressure history sumo = 0. sumn = 0. jjsav = 0 xval = x(1,2) loop jjp (1,nroot) al2 = ytable(tab4,jjp) al = ytable(tabroot,jjp) term = cos(al*xval/c a)-cos(al)

13 Compression of a Poroelastic Sample Mandel s Problem sumn = sumo + ytable(tab1,jjp)*term*exp(-al2*coe*gwtime) if sumn = sumo then jjsav = jjp jjp = nroot sumo = sumn loop ana pp = sumn*pcoe num pp = gpp(1,2) def deg cons ; degrees of consolidation history ini man ccoe = 2.*(1.-c un)/(c un-c nu) uxa0 = c F*c un/(2.*c sh) duxa = c F*(c nu - c un)/(2.*c sh) uyb0 = -c F*c b*(1. - c un)/(2.*c sh*c a) duyb = -c F*c b*(c un - cnu)/(2.*c sh*c a) taba1 = taba+1 taba2 = taba+2 loop ip (1,tbi) sumo = 0. sumn = 0. jjsav = 0 tt = xtable(3,ip) loop jjp (1,nroot) al2 = ytable(tab4,jjp) sumn = sumo + ytable(tab3,jjp)*exp(-al2*coe*tt) if sumn = sumo then jjsav = jjp jjp = nroot sumo = sumn loop xval = log(tt) xtable(taba,ip) = xval ytable(taba,ip) = 1.-ccoe*sumn xtable(taba1,ip) = xval ytable(taba1,ip) = (ytable(4,ip)-uxa0)/duxa xtable(taba2,ip) = xval ytable(taba2,ip) = (ytable(3,ip)-uyb0)/duyb loop def big step loop kk (1,nfstep) command set flow on mech off

14 18-14 Verification Problems step 1 set flow off mech on command val yv = ana yv command ini yv val yv j=fjgp step 1 ini yv 0 j=fjgp set echo on mess on solve sratio 5.e-2 set echo off mess off command tbi = tbi + 1 xtable(1,tbi) = gwtime ytable(1,tbi) = rf ; reaction force xtable(2,tbi) = gwtime ytable(2,tbi) = val yv ; velocity increment per step xtable(3,tbi) = gwtime ytable(3,tbi) = ydisp(1,jgp) ; top y-displacement xtable(4,tbi) = gwtime ytable(4,tbi) = xdisp(igp,1) ; right x-displacement loop ; --- drained conditions --- apply remove mech j=fjgp fix pp i=figp ini pp 0 i=figp fix y j=fjgp ini yv 0 j=fjgp set flow on hist ana pp hist num pp hist gwtime ; --- drained test --- set gwdt=8e-5 set nfstep = 125 big step set taba=11 check pp save man1.sav ;... State: man2.sav... set nfstep = 1125 big step set taba=12 check pp

15 Compression of a Poroelastic Sample Mandel s Problem save man2.sav ;... State: man3.sav... set nfstep = 5000 big step set taba=13 check pp save man3.sav ;... State: man4.sav... set nfstep = 6250 big step set taba=14 check pp save man4.sav ;... State: man5.sav... set nfstep = big step set taba=15 check pp save man5.sav ;... State: man6.sav... set nfstep = big step set taba=16 check pp save man6.sav ;... State: man7.sav... set taba=7 deg cons set echo on mess on save man7.sav ;*** plot commands **** ;plot name: Pore pressure versus consolidation time plot hold history 1 line 2 line vs 3 ;plot name: Pore pressure profile label table 10 anal (t = 0.01) label table 20 FLAC (t = 0.01) label table 11 anal (t = 0.01)

16 18-16 Verification Problems label table 21 FLAC (t = 0.01) label table 12 anal (t = 0.1) label table 22 FLAC (t = 0.1) label table 13 anal (t = 0.5) label table 23 FLAC (t = 0.5) label table 14 anal (t = 1.0) label table 24 FLAC (t = 1.0) label table 15 anal (t = 2.0) label table 25 FLAC (t = 2.0) label table 16 anal (t = 4.0) label table 26 FLAC (t = 4.0) plot hold table 26 cross 16 line 25 cross 15 line 24 cross 14 line 23 cross 13 line 22 cross 12 line 21 cross 11 line 20 cross 10 line ;plot name: Degree of consolidation versus log time label table 7 D (anal) label table 8 Dh (FLAC) label table 9 Dv (FLAC) plot hold table 9 line 8 line 7 line ;plot name: Y-reaction force history label table 1 y-reaction force plot hold table 1 line &

17 Compression of a Poroelastic Sample Mandel s Problem Data File FROOT.FIS ; --- froot.fis --- ; Using Brent s method, find the root of a function FUNC=f(x) ; bracketed in the interval [c x1,c x2] ; (i.e., such that f(c x1)*f(c x2) < 0) ; input: c x1, c x2 interval bounds ; func fish function with input c x ; tol ; output: froot root value ; ; Numerical Recipes: function zbrent ; def froot itmax = 100 c eps = 3.e-8 c x = c x1 fa = func c x = c x2 fb = func c a = c x1 c b = c x2 if fa*fb > 0. then toto=out( root must be bracketed for froot ) command pause command exit fc = fb loop iter (1,itmax) if fb*fc > 0. then c c = c a fc = fa c d = c b - c a c e = c d if abs(fc) < abs(fb) then c a = c b c b = c c c c = c a fa = fb fb = fc fc = fa tol1 = 2.*c eps*abs(c b)+0.5*tol

18 18-18 Verification Problems xm = 0.5*(c c-c b) if abs(xm) <= tol1 then froot = c b exit if fb = 0. then froot = c b exit if abs(c e) >= tol1 then if abs(fa) > abs(fb) then c s = fb/fa if c a = c c then c p = 2.*xm*c s c q = 1. - c s else c q = fa/fc c r = fb/fc c p = c s*(2.*xm*c q*(c q-c r)-(c b-c a)*(c r-1.)) c q = (c q-1.)*(c r-1.)*(c s-1.) if c p > 0. then c q = -c q c p=abs(c p) if 2.*c p < min(3.*xm*c q-abs(tol1*c q),abs(c e*c q)) then c e = c d c d = c p/c q else c d = xm c e = c d else c d = xm c e = c d else c d = xm c e = c d c a = c b fa = fb if abs(c d) > tol1 then c b = c b + c d else c b = c b + sign(tol1,xm)

19 Compression of a Poroelastic Sample Mandel s Problem c x = c b fb = func loop toto = out( froot exceeding maximum iteration ) froot = c b command pause command

20 18-20 Verification Problems