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Plane shear waves in a fully saturated granular medium with velocity and stress controlled boundary conditions
Pergamon
Vol. 32, No. 3. pp. 489 503. 1997 ‘( 1997 Elsevm Saence Ltd Pnnted m Great Britarn. All rights reserved OOZO-7462197f 17.00 + 0.00
Int. .I Non-Lmear Mechnnm.
PII : SOO2CL7462(96)0008&7
PLANE SHEAR WAVES IN A FULLY SATURATED GRANULAR MEDIUM WITH VELOCITY AND STRESS CONTROLLED BOUNDARY CONDITIONS Michael S. Gordon*?, *Center
Michael Shearer*$ and David Schaeffer$/)
for Research
in Scientific Computation and Department of Mathematics, North State University, Raleigh, NC 27695, U.S.A.; #Department of Mathematics, Duke University, Durham, NC 27708, U.S.A. (Received 29 February
Carolina
1996; accepted 14 May 1996)
Abstract-A one-dimensional system describing small shearing disturbances in a semi-infinite, fully saturated granular medium is studied. The system is fully non-linear as a result of the incrementally non-linear constitutive law for the material. In particular, there are two different wave speeds corresponding to loading or unloading of the material. Global solutions are constructed with boundary data consisting of a single pulse in either velocity or stress. In the case of velocity controlled boundary conditions, the solution is a traveling pulse of increasing kinetic energy which eventually unloads the material, regardless of whether the initial pulse loads or unloads. The solution with stress controlled boundary conditions has these features only if the initial stress pulse unloads the material. If the initial stress pulse is loading then the solution is a slowly traveling pulse of decreasing kinetic energy which is also loading. c 1997 Elsevier Science Ltd. All rights reserved. Keywords: plane shear waves, saturated granular material, hypoplastic flow rule, scale-invariant problem. characteristic wave, shock wave, rarefaction wave, wave curve. state space, kinetic energy
1. INTRODUCTION
The following system of equations was derived by Osinov and Gudehus [l] as the key equations in a simplified model for plane shear waves in a saturated granular body:
ata =
uaxl: + blaxuj.
(1.1)
Here a and b are constants satisfying 0 c b < a. The dependent variables are r and o; L‘is velocity and CJis a component of stress. Osinov and Gudehus derive the system (1.1) from a full three-dimensional system of equations for the deformation of a saturated granular material with a hypoplastic flow rule (see [225]). They linearize this system about a static state (i.e., zero strain rate) with constant stress tensor To; further, they assume that the incremental variables depend only on t and x, and that u2 is the only non-zero component of velocity. The equations for v = rZ and c = T,2 - Ty2 (the perturbation of the shear stress) decouple from the other equations, leading to the system (1.1). We should point out that the constant b is positive because we assume that TF2 < 0; the sign of b changes if TF2 > 0 (see Cl]). A consequence of Tp2 < 0 is that increasing 0 decreases the magnitude of the total shear stress, and thus unloads the material. Similarly, decreasing ~7loads the material. A physical context for this model is shown in Fig. 1. The figure shows a saturated granular material resting on an inclined solid mass. Plane shear waves described by (1.1) propagate in the direction perpendicular to the interface between the solid and the granular
TResearch supported by National Science Foundation grants DMS 9201115 and DMS 9504583, which include funds from AFOSR. $Adjunct Professor of Mathematics at Duke University. Research supported by National Science Foundation grants DMS 9201115 and DMS 9504583, which include funds from AFOSR, and by Army Research Office grant DAAH04-94-G-0043. /IResearch supported by National Science Foundation grants DMS 9201034 and DMS 9504577, which include funds from AFOSR. Contributed by W. F. Ames. 489
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et a/
material, while the velocity vector is parallel to it. Boundary disturbances could be created by waves propagating through the solid. A negative shear stress TfZ is created by the weight of the granular material. We consider the system (1.1) in the quarter-plane x > 0, t > 0 with initial data u(x, 0) = cJ(x, 0) = 0,
and an appropriate boundary condition a velocity controlled boundary condition,
(1.2)
at x = 0; specifically, we consider separately
$0, 0 = i(t),
(1.3)
and a stress controlled boundary condition a(09 r) = V(L).
(1.4)
The functions c(t) and r(t) are the piecewise linear pulse functions shown in Fig. 2. In Section 3, we construct a continuous, piecewise linear solution of (l.l), (1.2), (1.3); in Section 4, we similarly solve (l.l), (1.2), (1.4). Note that although (1.1) is a fully non-linear system, nonetheless, in regions where the solution satisfies 8,~ > 0, the system (1.1) reduces to the linear wave equation with wave speed m; and in regions where the solution satisfies d,v < 0, system (1.1) reduces to the linear wave equation with wave speed m. Thus signals for which a,v has a constant sign propagate without distortion. However, the data in Fig. 2 give rise to pulses in which a,u changes sign, so there is not a uniform wave speed. We solve the velocity controlled problem (l.l), (1.2), (1.3) for the trapezoidal boundary data i(t) with exactly the parameters shown in Fig. 2(a). The same construction would suffice to solve (l.l), (1.2), (1.3) with similar boundary data obtained by varying the parameters in Fig. 2(a) (i.e., the height of the pulse, the width of the flat top, the slopes of the leading and trailing edges). We believe that focusing on specific data makes the somewhat technical presentation easier to follow. Although scaling the height of the pulse i(t) makes no qualitative difference to the solution, changing its sign makes a great difference. Osinov and Gudehus [l] solved (Ll),
Fig. 1. Physical
(a)
setting
for the model equations.
(b)
u
Fig. 2. (a) Velocity-controlled
boundary
conditions;
CJ
(b) Stress-controlled
boundary
conditions.
Plane shear waves in a fully saturated
granular
491
medium
(1.2), (1.3) in the case where i(t) has the opposite sign, as illustrated in Fig. 3(a). This case admits a simpler analysis because the leading edge of the pulse propagates at the fast speed m while the trailing edge propagates at the slow speed fl. Thus, after some time t, [specifically, to = 3 for the data of Fig. 3(a)], the solution has a profile as illustrated in Fig.3(b). As t evolves, the width of the flat top increases but otherwise the pulse is undistorted. By contrast, if i(t) has the sign illustrated in Fig. 2(a), then the trailing edge of the pulse overtakes the leading edge, and a complicated set of wave interactions ensues. Following these, at large time the solution assumes an inverted profile which greatly resembles the preceding case (cf. Fig. 12). As a result of inversion, the leading edge of the pulse consists of two line segments rather than one, but this is the only distortion. In light of the complicated interactions that we find in Section 3, it seems remarkable that so little distortion occurs in the asymptotic form of the solution. After this sign inversion, the leading edge propagates faster than the trailing edge, so no further wave interactions occur. Osinov and Gudehus [l] note that, as the small wavespeed goes to zero, the trailing edge of the pulse shown in Fig. 3(b) fails to propagate. Thus a boundary pulse like that shown in Fig. 3(a) results in the entire medium being set in motion. A natural question is whether this also occurs for the boundary data shown in Fig. 2(a). At the end of Section 3, we consider this question for the slightly simplified boundary data shown in Fig. 4. Consider the pulse v(x, t) for t > t*, with t* the time at which the last wave interaction occurs. In Theorem 3.1. we show that the maximum max, lu(x, t)j of u for t > t* tends to zero as the slow wave speed approaches zero, i.e., as h/a + 1. Thus, as the trailing edge of the pulse slows down to zero speed, the pulse’s amplitude decreases to zero. This behavior is in contrast to that for the boundary data shown in Fig. 2(a), for which the resulting pulse has amplitude independent of the parameters a and b. Since t* is bounded as bJa + 1, this result also implies that the kinetic energy of the pulse at a fixed time tends to zero as the slow wave speed approaches zero. For stress controlled boundary conditions (1.4), we also study only initial data with the “difficult” sign; i.e., the sign for which the trailing edge of the resulting pulse propagates faster than the leading edge. For simplicity, we consider only the case of a triangular pulse, as shown in Fig. 2(b); the results for a trapezoidal pulse would be similar but slightly more complicated. Very dij‘krent behavior emerges for the stress controlled problem. In contrast to velocity controlled boundary conditions, there are an infinite number of interactions between waves with 2,~ > 0 and waves with a,v < 0. As a result of these, the solution never achieves an asymptotic profile, but rather it decays to zero! This result is stated as Theorem 4. I. In Theorem 4.2, we prove the related stronger result that the kinetic energy decays to zero.
(a)
Fig. 3. (a) OsinovjGudehus
boundary
conditions;
(b) Solutions
for
t>3
u(O. 0
1
t 1
2
kFig. 4. Velocity-controlled
boundary
conditions
without
neutral
region.
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We solve (l.l), (1.2) with piecewise linear boundary data by repeated application of the solution of what we call the “scale-invariant” problem; i.e., an initial value problem on the whole line - co < x < co with piecewise linear initial data having one jump in the derivative [cf. (2.91. The scale-invariant problem is analysed separately in Section 2.
2. PRELIMINARIES
In this section, we show how to solve scale-invariant problems using piecewise linear solutions. In Subsection 2.1, we analyse such solutions as individual waves; in Subsection 2.2, we show how to combine the individual waves to solve more general scale-invariant problems.
2.1. Piecewise linear solutions As preparation for solving (l.l), (1.2), (1.3) we find conditions under which a continuous, piecewise linear function of x and t, linear on either side of a line in xt-space, is a solution of (1.1). First, notice that linear solutions of (1.1) have the form u=px+qt+K1 0 =
qx + (up + blpl)t + K1,
where p, q, ICI, and K2 are arbitrary constants. In a region where the solution is linear, we shall refer to the state of the solution in that region as the values of p = 8,~ and q = 8,~. The solution in a linear region may be recovered from the state and from the values of u and 0 at any one point in the region. Now consider a continuous, piecewise linear solution of (1.1) with a discontinuity in the first derivative (henceforth referred to as a wauefiont) propagating with speed s (which is necessarily constant). Let (po, qO) and (p, q) be the states ahead of and behind the wavefront. Continuity of u and 0 across the wavefront implies that (a, + s8,)u and (a, + sd,)a are the same on either side of the wavefront. Combining this with (l.l), we have the following conditions relating the states on either side of a wavefront and the speed of the wavefront. (a) 4 + so = q. + spa (b) ap + 4
+ sq = ape +
(2.1)
WPOI+ wo.
We assume in addition an entropy condition, analogous to that of Lax [6] for shock waves. The entropy condition requires that at any point on a wavefront, there are three characteristics pointing into (or parallel to) the wavefront and one pointing out [see Fig. 5(a)]. This is necessary in order to have a well-posed boundary value problem at a wavefront. More specifically, there are five unknowns for a boundary value problem at a wavefront: the speed of the wavefront and u and r~on either side of the wavefront. Having three characteristics entering the wavefront, along with the two relations (2.1) completely determine these unknowns. Notice that, in regions where 8,~ > 0, the characteristics propagate with speed + Ja+b and, in regions where a,u< 0,they propagate with speed + m.
(4
Fig. 5. (a) Entropy-satisfying
front; (b) Entropy-violating
front.
Plane shear waves in a fully saturated
granular
medium
493
We break (2.1) into four cases: (i) (9
PO,
PO f
PO,P
0,
2%
(iii) p. < 0 < p,
(2.2)
(iv) P < 0 < po.
If (2.2(i)) holds then (2.1) implies that (a - b - s’)(p - po) = 0 and 4 - q. = - s(p - po). Since p - p. and q - q. are not both zero, it follows that s = + m. We shall refer to such a wavefront as a slow characteristic wave. If (2.2(ii)) holds, a similar argument shows that s = f m9 and we refer to such a wavefront as a fast characteristic wave. Notice that, in both these cases, 8Xv does not change sign across the wavefront, so that the wave front is a solution of the linear wave equation, and thus propagates with characteristic speed. We now consider the cases (2.2(iii)) and (2.2(iv)), in which d,c changes sign across the wavefront. If (2.2(iii)) holds then (2.1) implies
(4 P = - a (b)
q =
:yl
s2
+
p.
2bpos+ 40.
a+b-s2
(2.3)
It also follows from (2.2(iii)) and (2.3(a)) that 1 - 2b/(a + b - s2) < 0 which implies that a-bbs’
(2.4)
We refer to such a wavefront as a shock wave.* If (2.2(iv)) holds, a similar argument shows that 1 + 2b/(a - b - s2) < 0 which again implies (2.4). However, in this case, three characteristics point out of the wavefront while one points in, violating the entropy condition [see Fig. 5(b)]. We therefore reject (2.2(iv)) as a possibility. 2.2. Scale-invariant problems In order to solve (1.1) (1.2), (1.3), we must be able to continue the solution of (1.1) past a time when two wavefronts meet. This is done by solving an initial value problem for (1.1) with different initial states on either side of the point where the two wavefronts meet. Translating this point to the origin of xr-space, this amounts to solving (1.1) with piecewise linear initial data of the form u(x, 0) = 0(x, 0) =
pLx if x < 0 i pRx if x > 0,
qrx if x d 0 i qRx if x > 0.
(2.5)
This is a scale-invariant problem in the s&se of [7]. As in that paper, we construct a continuous solution of (l.l), (2.5) which is linear on wedges bounded by wavefronts emanating from the origin of xr-space (see Fig. 7, for example). First, it is necessary to find the states which may be joined to (pR, qR) in the right quarter-plane of xr-space using forward moving wavefronts. We consider pR < 0 and pR 3 0 as separate cases. pR < 0. From our previous consideration
of (2.2(i)), we know that (pR, qR) may be joined to a state for which 8,~ d 0 using a slow characteristic wave. By (2.1(a)), the locus of such states is a line, in the left half-plane of state space, of slope - m passing through (pR, qR) [see Fig. 6(a)]. If this is the case, no more forward moving wavefronts are possible because they
*More precisely, if we differentiate equation (1.1) with respect to x to obtain a system of conservation laws for (?,D and ?,o. the wavefront becomes a shock wave solution in which d,o and 8,~ have a jump discontinuity.
494
M. S. Gordon
er al
a20 (a>
(PRY QR) \
(0,qR + PRm)
azv
\
i Fig. 6. States for right moving
waves.
would have to travel with speed slower than m. Alternatively, from our consideration of (2.2(iii)), we know that (pa, qR) may be joined to a state for which d,v > 0 using a shock wave. The locus of such states is a hyperbola in the right half-plane given parameterically by (2.3) with pa, qR in place ofp,, qOand m < s< m [see Fig. 6(a)]. As s + m, this curve approaches the a,o-axis at the right endpoint of the set (i.e., line) of possible states in the left half-plane. The curve is decreasing and concave up, and as s -+ &?%, the slope approaches - $%& the slope of the dotted line in Fig. 6(a). If such a shock were to be joined to another forward moving wavefront, its speed would have to be slower than s. From our consideration of (2.2(i)) and (2.2(iv)), such a wave could be neither a slow characteristic nor a shock, and so no further wavefronts are possible. pR > 0. From our consideration of (2.2(ii)), we know that (pR, qR) may be joined to a state for which d,v > 0 using a fast characteristic wave. By (2.1(a)), the locus of such states is a line, in the right half-plane of state space, of slope - m passing through (pR, qR) [see Fig. 6(b)]. If d,v > 0 for this state, no further wavefronts are possible, however if a,v = 0 then this state may be joined to a state for which a,v < 0 using a slow characteristic wave. The locus of such states is a line, in the left half-plane of state space, of slope - m passing through (0, qa + pa m) [ see Fig. 6(b)]. We shall refer to a region for which a,v = 0, bounded by a fast characteristic wave and a slow characteristic wave, as a rarefaction.*
A similar argument shows that the states which may be joined to (pL, qL) in the left quarter-plane of xt-space using backward moving wavefronts are given by the same curves (with pL, qL in place of pa, qa) reflected through the line d,a = qL, We shall refer to the set of states which may be joined to (pa, qR) using forward moving waves as a forward wave curve and those which may be joined to (pL, qL) using backward moving waves as a backward wave curve. In the solution of (1.1) (24 there must be a single state (p,, q,) which follows (pa, qa) in the right quarter-plane of xt-space and (pL, qL) in the left quarter-plane. This state is given by the intersection of the forward and backward wave curves passing through (pa, qa) and (pL, qL), respectively. A specific example is illustrated in Figs 7(a), 7(b), and 8. Figure 7(a) shows how (p,, q,) is found, given (pa, qa) and (pL, qL), Fig. 7(b) shows the structure of the solution in xt-space, and Fig. 8 shows a cross section of the solution for v and 0 at some positive fixed time to (the initial data are indicated by dashed lines). Notice that in this case the solution involves a forward moving rarefaction and a backward moving slow characteristic wave.
*If the absolute values respect to x to obtain a rarefaction wave solutions as the limit of rarefaction
in (1.1) were replaced by a smooth function, we could differentiate the equation with system of conservation laws for 8,~ and 2,~. The conservation laws would possess in which d,v and 3,~ vary continuously. A rarefaction solution of(l.1) may be obtained solutions of such smooth approximations of (1.1).
Plane shear waves in a fully saturated
Fig. 7. (a) States for the scale-invariant
_____------
problem;(b)
____-----
-.
granular
Wave structure
medium
495
for the scale-invariant
problem
-x --__
ci(x, td
‘. --._ --._ -. ---
Fig. 8. Solution
of the scale-invariant
0(x,0)
problem.
A4
Fig. 9. States for the velocity-controlled
3. VELOCITY-CONTROLLED
problem.
PROBLEM
We now construct the solution of (l.l), (1.2), (1.3). The construction is shown in state space in Fig. 9, and the solution is represented in Fig. 10 in the xt-plane. Corresponding graphs of u and 0 at various times are shown in Fig. 11.
M. S. Gordon
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et al.
t
t3
t: 3
t 2
1
Fig. 10. Wave structure
for the velocity-controlled
problem.
To begin with, we determine the form of the solution adjacent to the t-axis determined by the boundary conditions (i.e., until wave reflections affect the solution on the t-axis.) First note that the initial conditions imply that the state bordering the x-axis must be (0,O). This state is labeled 0 in state space in Fig. 9, as is the region which it occupies in xt-space in Fig. 10. The boundary conditions imply that &u = 3,~ = 1 on the t-axis between t = 0 and t = 1, so the state bordering the t-axis there must satisfy 8,~ = 1. There must be some forward moving wavefront emanating from the origin of the xt-plane which separates these two states. Notice that the forward wave curve passing through 0 intersects the line d,a = 1 at (- l/J=% I), so the wavefront is a slow characteristic (labeled cur in Fig. 10) and iY,v= - l/m for the state following it (labeled AI in Fig. 9). The boundary conditions between t = 1 and t = 2 imply that A, must then be joined to a state for which 8,~ = a,a = 0 using a wavefront emanating from the t-axis at t = 1. The forward wave curve passing through AI intersects d,o = 0 at 0, and so 0 follows AI using a slow characteristic t12.Now 0 must be joined to a state for which 8,~ = &a = - 1 using a wavefront emanating from the t-axis at t = 2. The forward wave curve passing through 0 intersects ~?,a = - 1 at
Plane shear waves in a fully saturated granular medium
Fig. 11. Solution of the velocity-controlled
491
problem.
A2 = (l/m, - l), and so AZ follows 0 using a fast characteristic PI. Finally, A2 must be joined to a state for which 8,~ = 8,~ = 0 using a wavefront emanating from the t-axis at t = 3. The forward wave curve passing through A2 intersects a,o = 0 at 0, and so 0 follows A2 using a fast characteristic p2. Notice that, since /I1 is traveling faster than c(~,these wavefronts will eventually meet. To determine what happens after this meeting, we must solve the initial value problem (1.1) (2.5) with A, as the right state and A2 as the left state. The forward wave curve through A, is a hyperbola (labeled h in Fig. 9) in the right side of state space, and intersects the backward wave curve through A2 at a state A3 (see Fig. 9). Thus the intersection of /3r and a2 produces a backward moving fast characteristic p3 and a forward moving shock s1 with the state A, between them. We should point out that it is possible for B3 to reach the t-axis before f = 3, thereby altering the wave emanating from that point. This leads to more complex wave interactions, however the qualitative behavior of the solution is the same. We avoid this complication here by assuming that a>--.
5b
4
(3.1)
It is not hard to show that p3 will meet )(j2before striking the t-axis if and only if (3.1) holds. We then solve (l.l), (2.5) with A3 as the right state and 0 as the left state resulting in a backward moving slow characteristic LX~and a forward moving rarefaction, with new states A4 and A5 as shown in Figs 9 and 10. The meeting of (Yewith the t-axis is handled in the same manner as the boundary value problems at t = 0, t = 1, and t = 2. This results in the new state A6 adjacent to the t axis. Note that A6 has 8,~ = d,v = 0, as required by the boundary condition. We claim that the next wave interaction will be the meeting of sr and cur. From Fig. 10, it is clear that the only other possibility is that sr and fi2 intersect first. (Note that (x4and a5 are parallel since they are both slow characteristic waves.) To rule out this possibility, we could compute the points of intersection of sr and a1 and then of s1 and p2 and compare the results. This is rather tedious, so we give the following simpler argument. Suppose the next wave interaction is the meeting of s1 and p2 at some time t = to. Inspection of Figs 9 and 10 reveals that C?,V< 0 for all states at t = to (i.e., Al, Ah, A,, A6, and possibly 0). Since v(x, to) is continuous in x and has finite support, this implies u(x, to) E 0, which is a contradiction since iJ,v(x, to) is not identically zero. Having verified that sr intersects aI, we now solve (1.1). (2.5) with 0 as the right state and As, as the left state. The solution has backward and forward moving fast characteristics and a middle state A2 (that already appeared).
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The remaining wavefronts and states may be found by continuing the above procedure of solving initial value problems of the form (l.l), (2.5). This is illustrated in Figs 9 and 10. Figure 11 shows profiles of the solution for u and e at the specific times tl , t2, t3 labeled on the t-axis in Fig. 10. Notice that there are only a finite number of wave interactions and that after t = t, there are five forward moving wavefronts: three fast characteristics, p4, p2, and p6, joined to two slow characteristics, tx5 and c(,. Hence there are no further wave interactions. Figure 12 shows the profiles of u and 0 at some time after t = t,. Notice that though u is initially non-negative, it is non-positive after t = t,. It is not hard to use Figs 9 and 10 to verify that this is true even if we alter the boundary pulse by changing the slopes of the front and back edges of the pulse and/or the length of the neutral region. We would now like to consider the question of what happens to the solution of the velocity-controlled problem when the small wavespeed goes to zero (while the large wavespeed remains fixed). Notice, however, that this would result in (3.1) being violated, complicating the analysis. We avoid this by instead considering the aforementioned question for the simplified velocity-controlled boundary condition shown in Fig. 4. Notice that the only difference between this boundary pulse and (1.3) is the absence of a neutral region between the loading and unloading fronts. The result is that there is no lag time between the interaction of the slow leading wave and the fast trailing wave, resulting in fewer overall wave interactions. The solution for large time, however, is essentially unchanged: the profile of the solution is inverted and consists of a fast leading edge with two line segments and a slow trailing edge, separated by a growing neutral region. This means that, as with the boundary data considered by Osinov and Gudehus, the trailing edge of the pulse tends to remain stationary as the small wavespeed goes to zero. There is, however, one significant difference; the amplitude of the pulse also tends to zero as the small wavespeed goes to zero, unlike the Osinov/Gudehus solution pulse, whose amplitude is independent of either wavespeed. This is made precise in Theorem 3.1 below. The solution of (l.l), (1.2) with the boundary data shown in Fig. 4 may be obtained in the manner described above. The states for the solution are shown in Fig. 13 and the structure of the solution in the xt-plane is illustrated in Fig. 14. Arguing as before, we can show that the shock s1 meets al before meeting Dr. Also, a routine calculation shows that this interaction occurs at some time after t = 2. Thus the solution must have the form shown in Fig. 14. As before, we let t, be the time at which the last wave interaction occurs (see Fig. 14). We define t = 2 + 4Jm = 2 + 8cd/(d2 - c’), where c = F and d = m. Figure 14 shows how t”is found geometrically. Notice that t”is an upper bound for t,. Remark 1. Notice that t”+ 2 as c -+ 0 or a/b -+ 1, i.e., as the slow wavespeed goes to zero. Also, t”-+ co as b -+ 0 or c - d -P 0, i.e., as the equations become linear. THEOREM
3.1. Let v, r~be the solution of (l.l), (1.2) with the boundary data shown in Fig. 4.
Then max {I@, x
t)l, I+G t)l> <
4JZ-x m
+ J&$
4cd = c+d
for t ’ t:
(3.2)
Proof: Since the maximum values of (vi and )o( for t > t”occur in the neutral region between a2 and p4, the theorem will be proven if we show that the increase in (VIand 101across regions A5 and A6 is bounded by the right side of (3.2). Notice from Fig. 13 that the states
Fig. 12. Large-time
solution
of the velocity-controlled
problem
499
Plane shear waves in a fully saturated granular medium
a.v
Fig. 13. States for the simplified velocity-controlled
problem.
Fig. 14. Wave structure for the simplified velocity-controlled
problem.
A, and A6 satisfy Id,vJ = la,al < 1 and la,o( = (a + b)(8,u(
< m.
It is not difficult to show from Fig. 14 that the width of the region AswAg is 4-/(-
+ Ja+b).
Combining
(3.3) in the t-direction
this with (3.3) gives the result.
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500
4. STRESS-CONTROLLEDPROBLEM
We now construct the solution of (l.l), (1.2), (1.4). To simplify the analysis, we consider boundary data without a neutral region between loading and unloading fronts, so there is no lag time before the interaction of the slow leading wave and the fast trailing wave. As for the velocity-controlled problem, the solution with a time lag generates more wave interactions at first, but it is qualitatively the same as time increases. Again, the state bordering the x-axis is 0. The boundary conditions imply that d,a = (a - b)d,v = - 1 on the t-axis between t = 0 and t = 1, so the state following 0 is using a slow characteristic CI~(see Figs 15 and 16). The Al = ( - l/(a - b), 1/m, boundary conditions also imply that d,o = (a + b)8,u = 1 on the t-axis between t = 1 and t = 2, so the state A2 following Al, using a shock sl, is the intersection of the forward wave curve through Al and the line i3,v = l/(a + b). Since 8,a = 8,~ = 0 on the t-axis for t > 2, the state A3 following AZ, using a fast characteristic pl, is the intersection of 8,v = 0 and the line of slope - l/Ja+b through AZ. The next wave interaction will be the meeting of s1 and ,81 (rather than s1 and crl). The proof is identical to the one given for the velocity controlled boundary conditions, replacing v with 0. We then solve (l.l), (2.5) with A1 as the right state and A2 as the left state resulting in a backward moving fast characteristic p2 and a forward moving shock s2. Notice that the state A4 between them is on the same wave curve as A, and that s2 is slower than sl. When p2 reaches the t-axis, a forward moving fast characteristic p3 is produced, which will meet s2 before s2 reaches cxl (by the same argument as before), producing a shock s3 which is slower than s2. It is not hard to see from Figs 15 and 16 that this pattern continues indefinitely, with fast characteristics interacting alternately with the t-axis and the forward moving shocks, decreasing the shock speed with each interaction. Moreover, the shock never reaches CI~.Notice also from Fig. 15 that the states A, are approaching the origin of state space as n ---f00. Profiles of u and (T containing a forward moving fast characteristic are sketched in Fig. 17. In the following theorem, we show that the amplitude of these profiles approaches zero as t -+ 00. THEOREM
4.1. Let u, (T be the solution
of(l.l),
(1.2), (1.4). Then
lim max {/0(x, t)l, 10(x, t)j} = 0. f’rn x Proof Let t, be the time when pzn meets the t-axis (see Fig. 16). We know from Figs 15 and 16 that the maximum values of IuJand ((~1at a fixed time occur at the shock. Also, since the
a.0
Fig. 15. States for the stress-controlled problem.
Plane shear waves in a fully saturated granular medium
Fig. 16. Wave structure for the stress-controlled problem.
Fig. 17. Large-time solution of the stress-controlled
problem.
502
M. S. Gordon et al.
Fig. 18. Definition of functionsfand
y.
shocks are approaching al, where both u and u are zero, these maximums must be decreasing as t increases. Also, if max, lfl.1-+ 0 as t + co then the distance between the shocks and tli must be approaching zero as well, which implies that max, 1u) -+ 0 also. Thus we need only show that lim max 10(x, t,)l = 0. (4.1) n+oo X Let K, = la,c~l for the state A2n+2. Clearly
max 10(x, t,)l d X
K,,
(supp (T(x,t,)) = ~,t,
,/a- b,
(4.2)
so we prove (4.1) by showing that q,t,-+ 0 as n + co. Define a functionf(z) as follows [see Fig. 18(a)]. Through the point (0, z) in the xt-plane, draw a line of slope l/m. Through the point of intersection of this line and the line of slope l/m emanating from the origin, draw a line of slope - l/m. Definef(z) to be the value oft at the point of intersection of the second line and the t-axis. It is not hard to show that (4.3) It is clear from Fig. 16 that t n+l <“ml)
(4.4)
for any n. Define a function g(lc) as follows [see Fig. 18(b)]. Through the point of intersection of d,a = - K and the line tangent to h at the origin in state space [a routine calculation using (2.3) shows this line has slope - a/m], draw a line of slope - m. Through the point of intersection of this line and the a,a-axis, draw a line of slope m. Define - g(rc) to be the value of d,a at the point of intersection of the second line and the line tangent to h at the origin. It is not hard to show that g(K) =
a-J_ (4.5)
a+JZP.
It is clear from Fig. 15 that %+ I < &o)
(4.6)
for any n. Combining (4.3), (4.4), (4.9, and (4.6), we have K n+l
t ntl
Kntn
from which (4.1) follows.
2
Plane shear waves in a fully saturated
In the following time.
theorem,
SO3
medium
we show that the kinetic energy of the solution
4.2. Let u, (T be the solution
THEOREM
granular
of(l.l),
decays for large
(1.2), (1.4). Then
lim Oc’V(X, t)’ dx = 0. t-ac s 0 Proqf
Let t, be defined
as in Theorem
(4.7)
4.1. Then
7 D(X, t,,)’ dx d (supp u(x, t,)) max u(x, t,)’ = t, G X s0 Notice
(4.8)
that max Iu(x, t,)l = max \a(~, t,))/JZC * x
because
max t’(x, t,)2. *
the maximum
d tint,
(4.9)
of u and 0 occur at the shock S, and because d,o(x, t,) = - &o(x, t,)/&zi
for the state Al. Combining
(4.9) with (4.2) and (4.8), we have that
s m
u(x, t,)’ dx 6 K,2t:G,
0
Combining
(4.3), (4.4), (4.5), and (4.6) we have a-Jz3<1
Jz+ 1t,3+1 lc2t3 n n
which implies
<
b
that X lim u(x, t,)’ dx = 0, n-r, s 0
from which (4.7) follows.
H
Acknowledyements-The velocity controlled problem was posed to us by V. A. Osinov. We thank him for his interest in our results, and for the many constructive comments he made concerning earlier drafts of this paper.
REFERENCES 1. V. A. Osinov and G. Gudehus, Plane shear waves and loss of stability in a saturated granular body. Mechunics of Cohesive Frictional Materials and Structures 1, 25-44 (1996). 2. E. Bauer, Calibration of a comprehensive hypoplastic model for granular materials. Soils and Foundations. Jup. Sot. Soil Mech. Found. Eny. 36(l), 13-26 (1996). 3. G. Gudehus, A comprehensive constitutive equation for granular materials. Soils und Foundations, Jup. Sot,. Soil Me& Found. Eny. 36(l), 1-12 (1996). 4. D. Kolymbas, and W. Wu, Introduction to hypoplasticity. In Modern Approaches ro Plasticity. D. Kolymbas (ed.), pp. 213-223. Elsevier, Amsterdam (1993). 5. D. Kolymbas, I. Herle and P. A. von Wolffersdorff, Hypoplastic constitutive equation with internal variables. Inl. J. Num. Anal. Meth. Geomech. 19, 415-436 (1995). 6. P. D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10. 537-566 (1957). I. D. Schaefler and M. Shearer, Scale-invariant initial value problems in one-dimensional dynamic elastoplasticity, with consequences for multidimensional nonassociative plasticity. Euro. Jnl of Appl. Math. 3, 225 254 (1992).
Vol. 32, No. 3. pp. 489 503. 1997 ‘( 1997 Elsevm Saence Ltd Pnnted m Great Britarn. All rights reserved OOZO-7462197f 17.00 + 0.00
Int. .I Non-Lmear Mechnnm.
PII : SOO2CL7462(96)0008&7
PLANE SHEAR WAVES IN A FULLY SATURATED GRANULAR MEDIUM WITH VELOCITY AND STRESS CONTROLLED BOUNDARY CONDITIONS Michael S. Gordon*?, *Center
Michael Shearer*$ and David Schaeffer$/)
for Research
in Scientific Computation and Department of Mathematics, North State University, Raleigh, NC 27695, U.S.A.; #Department of Mathematics, Duke University, Durham, NC 27708, U.S.A. (Received 29 February
Carolina
1996; accepted 14 May 1996)
Abstract-A one-dimensional system describing small shearing disturbances in a semi-infinite, fully saturated granular medium is studied. The system is fully non-linear as a result of the incrementally non-linear constitutive law for the material. In particular, there are two different wave speeds corresponding to loading or unloading of the material. Global solutions are constructed with boundary data consisting of a single pulse in either velocity or stress. In the case of velocity controlled boundary conditions, the solution is a traveling pulse of increasing kinetic energy which eventually unloads the material, regardless of whether the initial pulse loads or unloads. The solution with stress controlled boundary conditions has these features only if the initial stress pulse unloads the material. If the initial stress pulse is loading then the solution is a slowly traveling pulse of decreasing kinetic energy which is also loading. c 1997 Elsevier Science Ltd. All rights reserved. Keywords: plane shear waves, saturated granular material, hypoplastic flow rule, scale-invariant problem. characteristic wave, shock wave, rarefaction wave, wave curve. state space, kinetic energy
1. INTRODUCTION
The following system of equations was derived by Osinov and Gudehus [l] as the key equations in a simplified model for plane shear waves in a saturated granular body:
ata =
uaxl: + blaxuj.
(1.1)
Here a and b are constants satisfying 0 c b < a. The dependent variables are r and o; L‘is velocity and CJis a component of stress. Osinov and Gudehus derive the system (1.1) from a full three-dimensional system of equations for the deformation of a saturated granular material with a hypoplastic flow rule (see [225]). They linearize this system about a static state (i.e., zero strain rate) with constant stress tensor To; further, they assume that the incremental variables depend only on t and x, and that u2 is the only non-zero component of velocity. The equations for v = rZ and c = T,2 - Ty2 (the perturbation of the shear stress) decouple from the other equations, leading to the system (1.1). We should point out that the constant b is positive because we assume that TF2 < 0; the sign of b changes if TF2 > 0 (see Cl]). A consequence of Tp2 < 0 is that increasing 0 decreases the magnitude of the total shear stress, and thus unloads the material. Similarly, decreasing ~7loads the material. A physical context for this model is shown in Fig. 1. The figure shows a saturated granular material resting on an inclined solid mass. Plane shear waves described by (1.1) propagate in the direction perpendicular to the interface between the solid and the granular
TResearch supported by National Science Foundation grants DMS 9201115 and DMS 9504583, which include funds from AFOSR. $Adjunct Professor of Mathematics at Duke University. Research supported by National Science Foundation grants DMS 9201115 and DMS 9504583, which include funds from AFOSR, and by Army Research Office grant DAAH04-94-G-0043. /IResearch supported by National Science Foundation grants DMS 9201034 and DMS 9504577, which include funds from AFOSR. Contributed by W. F. Ames. 489
490
M. S. Gordon
et a/
material, while the velocity vector is parallel to it. Boundary disturbances could be created by waves propagating through the solid. A negative shear stress TfZ is created by the weight of the granular material. We consider the system (1.1) in the quarter-plane x > 0, t > 0 with initial data u(x, 0) = cJ(x, 0) = 0,
and an appropriate boundary condition a velocity controlled boundary condition,
(1.2)
at x = 0; specifically, we consider separately
$0, 0 = i(t),
(1.3)
and a stress controlled boundary condition a(09 r) = V(L).
(1.4)
The functions c(t) and r(t) are the piecewise linear pulse functions shown in Fig. 2. In Section 3, we construct a continuous, piecewise linear solution of (l.l), (1.2), (1.3); in Section 4, we similarly solve (l.l), (1.2), (1.4). Note that although (1.1) is a fully non-linear system, nonetheless, in regions where the solution satisfies 8,~ > 0, the system (1.1) reduces to the linear wave equation with wave speed m; and in regions where the solution satisfies d,v < 0, system (1.1) reduces to the linear wave equation with wave speed m. Thus signals for which a,v has a constant sign propagate without distortion. However, the data in Fig. 2 give rise to pulses in which a,u changes sign, so there is not a uniform wave speed. We solve the velocity controlled problem (l.l), (1.2), (1.3) for the trapezoidal boundary data i(t) with exactly the parameters shown in Fig. 2(a). The same construction would suffice to solve (l.l), (1.2), (1.3) with similar boundary data obtained by varying the parameters in Fig. 2(a) (i.e., the height of the pulse, the width of the flat top, the slopes of the leading and trailing edges). We believe that focusing on specific data makes the somewhat technical presentation easier to follow. Although scaling the height of the pulse i(t) makes no qualitative difference to the solution, changing its sign makes a great difference. Osinov and Gudehus [l] solved (Ll),
Fig. 1. Physical
(a)
setting
for the model equations.
(b)
u
Fig. 2. (a) Velocity-controlled
boundary
conditions;
CJ
(b) Stress-controlled
boundary
conditions.
Plane shear waves in a fully saturated
granular
491
medium
(1.2), (1.3) in the case where i(t) has the opposite sign, as illustrated in Fig. 3(a). This case admits a simpler analysis because the leading edge of the pulse propagates at the fast speed m while the trailing edge propagates at the slow speed fl. Thus, after some time t, [specifically, to = 3 for the data of Fig. 3(a)], the solution has a profile as illustrated in Fig.3(b). As t evolves, the width of the flat top increases but otherwise the pulse is undistorted. By contrast, if i(t) has the sign illustrated in Fig. 2(a), then the trailing edge of the pulse overtakes the leading edge, and a complicated set of wave interactions ensues. Following these, at large time the solution assumes an inverted profile which greatly resembles the preceding case (cf. Fig. 12). As a result of inversion, the leading edge of the pulse consists of two line segments rather than one, but this is the only distortion. In light of the complicated interactions that we find in Section 3, it seems remarkable that so little distortion occurs in the asymptotic form of the solution. After this sign inversion, the leading edge propagates faster than the trailing edge, so no further wave interactions occur. Osinov and Gudehus [l] note that, as the small wavespeed goes to zero, the trailing edge of the pulse shown in Fig. 3(b) fails to propagate. Thus a boundary pulse like that shown in Fig. 3(a) results in the entire medium being set in motion. A natural question is whether this also occurs for the boundary data shown in Fig. 2(a). At the end of Section 3, we consider this question for the slightly simplified boundary data shown in Fig. 4. Consider the pulse v(x, t) for t > t*, with t* the time at which the last wave interaction occurs. In Theorem 3.1. we show that the maximum max, lu(x, t)j of u for t > t* tends to zero as the slow wave speed approaches zero, i.e., as h/a + 1. Thus, as the trailing edge of the pulse slows down to zero speed, the pulse’s amplitude decreases to zero. This behavior is in contrast to that for the boundary data shown in Fig. 2(a), for which the resulting pulse has amplitude independent of the parameters a and b. Since t* is bounded as bJa + 1, this result also implies that the kinetic energy of the pulse at a fixed time tends to zero as the slow wave speed approaches zero. For stress controlled boundary conditions (1.4), we also study only initial data with the “difficult” sign; i.e., the sign for which the trailing edge of the resulting pulse propagates faster than the leading edge. For simplicity, we consider only the case of a triangular pulse, as shown in Fig. 2(b); the results for a trapezoidal pulse would be similar but slightly more complicated. Very dij‘krent behavior emerges for the stress controlled problem. In contrast to velocity controlled boundary conditions, there are an infinite number of interactions between waves with 2,~ > 0 and waves with a,v < 0. As a result of these, the solution never achieves an asymptotic profile, but rather it decays to zero! This result is stated as Theorem 4. I. In Theorem 4.2, we prove the related stronger result that the kinetic energy decays to zero.
(a)
Fig. 3. (a) OsinovjGudehus
boundary
conditions;
(b) Solutions
for
t>3
u(O. 0
1
t 1
2
kFig. 4. Velocity-controlled
boundary
conditions
without
neutral
region.
M. S. Gordon
492
et al
We solve (l.l), (1.2) with piecewise linear boundary data by repeated application of the solution of what we call the “scale-invariant” problem; i.e., an initial value problem on the whole line - co < x < co with piecewise linear initial data having one jump in the derivative [cf. (2.91. The scale-invariant problem is analysed separately in Section 2.
2. PRELIMINARIES
In this section, we show how to solve scale-invariant problems using piecewise linear solutions. In Subsection 2.1, we analyse such solutions as individual waves; in Subsection 2.2, we show how to combine the individual waves to solve more general scale-invariant problems.
2.1. Piecewise linear solutions As preparation for solving (l.l), (1.2), (1.3) we find conditions under which a continuous, piecewise linear function of x and t, linear on either side of a line in xt-space, is a solution of (1.1). First, notice that linear solutions of (1.1) have the form u=px+qt+K1 0 =
qx + (up + blpl)t + K1,
where p, q, ICI, and K2 are arbitrary constants. In a region where the solution is linear, we shall refer to the state of the solution in that region as the values of p = 8,~ and q = 8,~. The solution in a linear region may be recovered from the state and from the values of u and 0 at any one point in the region. Now consider a continuous, piecewise linear solution of (1.1) with a discontinuity in the first derivative (henceforth referred to as a wauefiont) propagating with speed s (which is necessarily constant). Let (po, qO) and (p, q) be the states ahead of and behind the wavefront. Continuity of u and 0 across the wavefront implies that (a, + s8,)u and (a, + sd,)a are the same on either side of the wavefront. Combining this with (l.l), we have the following conditions relating the states on either side of a wavefront and the speed of the wavefront. (a) 4 + so = q. + spa (b) ap + 4
+ sq = ape +
(2.1)
WPOI+ wo.
We assume in addition an entropy condition, analogous to that of Lax [6] for shock waves. The entropy condition requires that at any point on a wavefront, there are three characteristics pointing into (or parallel to) the wavefront and one pointing out [see Fig. 5(a)]. This is necessary in order to have a well-posed boundary value problem at a wavefront. More specifically, there are five unknowns for a boundary value problem at a wavefront: the speed of the wavefront and u and r~on either side of the wavefront. Having three characteristics entering the wavefront, along with the two relations (2.1) completely determine these unknowns. Notice that, in regions where 8,~ > 0, the characteristics propagate with speed + Ja+b and, in regions where a,u< 0,they propagate with speed + m.
(4
Fig. 5. (a) Entropy-satisfying
front; (b) Entropy-violating
front.
Plane shear waves in a fully saturated
granular
medium
493
We break (2.1) into four cases: (i) (9
PO,
PO f
PO,P
0,
2%
(iii) p. < 0 < p,
(2.2)
(iv) P < 0 < po.
If (2.2(i)) holds then (2.1) implies that (a - b - s’)(p - po) = 0 and 4 - q. = - s(p - po). Since p - p. and q - q. are not both zero, it follows that s = + m. We shall refer to such a wavefront as a slow characteristic wave. If (2.2(ii)) holds, a similar argument shows that s = f m9 and we refer to such a wavefront as a fast characteristic wave. Notice that, in both these cases, 8Xv does not change sign across the wavefront, so that the wave front is a solution of the linear wave equation, and thus propagates with characteristic speed. We now consider the cases (2.2(iii)) and (2.2(iv)), in which d,c changes sign across the wavefront. If (2.2(iii)) holds then (2.1) implies
(4 P = - a (b)
q =
:yl
s2
+
p.
2bpos+ 40.
a+b-s2
(2.3)
It also follows from (2.2(iii)) and (2.3(a)) that 1 - 2b/(a + b - s2) < 0 which implies that a-bbs’
(2.4)
We refer to such a wavefront as a shock wave.* If (2.2(iv)) holds, a similar argument shows that 1 + 2b/(a - b - s2) < 0 which again implies (2.4). However, in this case, three characteristics point out of the wavefront while one points in, violating the entropy condition [see Fig. 5(b)]. We therefore reject (2.2(iv)) as a possibility. 2.2. Scale-invariant problems In order to solve (1.1) (1.2), (1.3), we must be able to continue the solution of (1.1) past a time when two wavefronts meet. This is done by solving an initial value problem for (1.1) with different initial states on either side of the point where the two wavefronts meet. Translating this point to the origin of xr-space, this amounts to solving (1.1) with piecewise linear initial data of the form u(x, 0) = 0(x, 0) =
pLx if x < 0 i pRx if x > 0,
qrx if x d 0 i qRx if x > 0.
(2.5)
This is a scale-invariant problem in the s&se of [7]. As in that paper, we construct a continuous solution of (l.l), (2.5) which is linear on wedges bounded by wavefronts emanating from the origin of xr-space (see Fig. 7, for example). First, it is necessary to find the states which may be joined to (pR, qR) in the right quarter-plane of xr-space using forward moving wavefronts. We consider pR < 0 and pR 3 0 as separate cases. pR < 0. From our previous consideration
of (2.2(i)), we know that (pR, qR) may be joined to a state for which 8,~ d 0 using a slow characteristic wave. By (2.1(a)), the locus of such states is a line, in the left half-plane of state space, of slope - m passing through (pR, qR) [see Fig. 6(a)]. If this is the case, no more forward moving wavefronts are possible because they
*More precisely, if we differentiate equation (1.1) with respect to x to obtain a system of conservation laws for (?,D and ?,o. the wavefront becomes a shock wave solution in which d,o and 8,~ have a jump discontinuity.
494
M. S. Gordon
er al
a20 (a>
(PRY QR) \
(0,qR + PRm)
azv
\
i Fig. 6. States for right moving
waves.
would have to travel with speed slower than m. Alternatively, from our consideration of (2.2(iii)), we know that (pa, qR) may be joined to a state for which d,v > 0 using a shock wave. The locus of such states is a hyperbola in the right half-plane given parameterically by (2.3) with pa, qR in place ofp,, qOand m < s< m [see Fig. 6(a)]. As s + m, this curve approaches the a,o-axis at the right endpoint of the set (i.e., line) of possible states in the left half-plane. The curve is decreasing and concave up, and as s -+ &?%, the slope approaches - $%& the slope of the dotted line in Fig. 6(a). If such a shock were to be joined to another forward moving wavefront, its speed would have to be slower than s. From our consideration of (2.2(i)) and (2.2(iv)), such a wave could be neither a slow characteristic nor a shock, and so no further wavefronts are possible. pR > 0. From our consideration of (2.2(ii)), we know that (pR, qR) may be joined to a state for which d,v > 0 using a fast characteristic wave. By (2.1(a)), the locus of such states is a line, in the right half-plane of state space, of slope - m passing through (pR, qR) [see Fig. 6(b)]. If d,v > 0 for this state, no further wavefronts are possible, however if a,v = 0 then this state may be joined to a state for which a,v < 0 using a slow characteristic wave. The locus of such states is a line, in the left half-plane of state space, of slope - m passing through (0, qa + pa m) [ see Fig. 6(b)]. We shall refer to a region for which a,v = 0, bounded by a fast characteristic wave and a slow characteristic wave, as a rarefaction.*
A similar argument shows that the states which may be joined to (pL, qL) in the left quarter-plane of xt-space using backward moving wavefronts are given by the same curves (with pL, qL in place of pa, qa) reflected through the line d,a = qL, We shall refer to the set of states which may be joined to (pa, qR) using forward moving waves as a forward wave curve and those which may be joined to (pL, qL) using backward moving waves as a backward wave curve. In the solution of (1.1) (24 there must be a single state (p,, q,) which follows (pa, qa) in the right quarter-plane of xt-space and (pL, qL) in the left quarter-plane. This state is given by the intersection of the forward and backward wave curves passing through (pa, qa) and (pL, qL), respectively. A specific example is illustrated in Figs 7(a), 7(b), and 8. Figure 7(a) shows how (p,, q,) is found, given (pa, qa) and (pL, qL), Fig. 7(b) shows the structure of the solution in xt-space, and Fig. 8 shows a cross section of the solution for v and 0 at some positive fixed time to (the initial data are indicated by dashed lines). Notice that in this case the solution involves a forward moving rarefaction and a backward moving slow characteristic wave.
*If the absolute values respect to x to obtain a rarefaction wave solutions as the limit of rarefaction
in (1.1) were replaced by a smooth function, we could differentiate the equation with system of conservation laws for 8,~ and 2,~. The conservation laws would possess in which d,v and 3,~ vary continuously. A rarefaction solution of(l.1) may be obtained solutions of such smooth approximations of (1.1).
Plane shear waves in a fully saturated
Fig. 7. (a) States for the scale-invariant
_____------
problem;(b)
____-----
-.
granular
Wave structure
medium
495
for the scale-invariant
problem
-x --__
ci(x, td
‘. --._ --._ -. ---
Fig. 8. Solution
of the scale-invariant
0(x,0)
problem.
A4
Fig. 9. States for the velocity-controlled
3. VELOCITY-CONTROLLED
problem.
PROBLEM
We now construct the solution of (l.l), (1.2), (1.3). The construction is shown in state space in Fig. 9, and the solution is represented in Fig. 10 in the xt-plane. Corresponding graphs of u and 0 at various times are shown in Fig. 11.
M. S. Gordon
496
et al.
t
t3
t: 3
t 2
1
Fig. 10. Wave structure
for the velocity-controlled
problem.
To begin with, we determine the form of the solution adjacent to the t-axis determined by the boundary conditions (i.e., until wave reflections affect the solution on the t-axis.) First note that the initial conditions imply that the state bordering the x-axis must be (0,O). This state is labeled 0 in state space in Fig. 9, as is the region which it occupies in xt-space in Fig. 10. The boundary conditions imply that &u = 3,~ = 1 on the t-axis between t = 0 and t = 1, so the state bordering the t-axis there must satisfy 8,~ = 1. There must be some forward moving wavefront emanating from the origin of the xt-plane which separates these two states. Notice that the forward wave curve passing through 0 intersects the line d,a = 1 at (- l/J=% I), so the wavefront is a slow characteristic (labeled cur in Fig. 10) and iY,v= - l/m for the state following it (labeled AI in Fig. 9). The boundary conditions between t = 1 and t = 2 imply that A, must then be joined to a state for which 8,~ = a,a = 0 using a wavefront emanating from the t-axis at t = 1. The forward wave curve passing through AI intersects d,o = 0 at 0, and so 0 follows AI using a slow characteristic t12.Now 0 must be joined to a state for which 8,~ = &a = - 1 using a wavefront emanating from the t-axis at t = 2. The forward wave curve passing through 0 intersects ~?,a = - 1 at
Plane shear waves in a fully saturated granular medium
Fig. 11. Solution of the velocity-controlled
491
problem.
A2 = (l/m, - l), and so AZ follows 0 using a fast characteristic PI. Finally, A2 must be joined to a state for which 8,~ = 8,~ = 0 using a wavefront emanating from the t-axis at t = 3. The forward wave curve passing through A2 intersects a,o = 0 at 0, and so 0 follows A2 using a fast characteristic p2. Notice that, since /I1 is traveling faster than c(~,these wavefronts will eventually meet. To determine what happens after this meeting, we must solve the initial value problem (1.1) (2.5) with A, as the right state and A2 as the left state. The forward wave curve through A, is a hyperbola (labeled h in Fig. 9) in the right side of state space, and intersects the backward wave curve through A2 at a state A3 (see Fig. 9). Thus the intersection of /3r and a2 produces a backward moving fast characteristic p3 and a forward moving shock s1 with the state A, between them. We should point out that it is possible for B3 to reach the t-axis before f = 3, thereby altering the wave emanating from that point. This leads to more complex wave interactions, however the qualitative behavior of the solution is the same. We avoid this complication here by assuming that a>--.
5b
4
(3.1)
It is not hard to show that p3 will meet )(j2before striking the t-axis if and only if (3.1) holds. We then solve (l.l), (2.5) with A3 as the right state and 0 as the left state resulting in a backward moving slow characteristic LX~and a forward moving rarefaction, with new states A4 and A5 as shown in Figs 9 and 10. The meeting of (Yewith the t-axis is handled in the same manner as the boundary value problems at t = 0, t = 1, and t = 2. This results in the new state A6 adjacent to the t axis. Note that A6 has 8,~ = d,v = 0, as required by the boundary condition. We claim that the next wave interaction will be the meeting of sr and cur. From Fig. 10, it is clear that the only other possibility is that sr and fi2 intersect first. (Note that (x4and a5 are parallel since they are both slow characteristic waves.) To rule out this possibility, we could compute the points of intersection of sr and a1 and then of s1 and p2 and compare the results. This is rather tedious, so we give the following simpler argument. Suppose the next wave interaction is the meeting of s1 and p2 at some time t = to. Inspection of Figs 9 and 10 reveals that C?,V< 0 for all states at t = to (i.e., Al, Ah, A,, A6, and possibly 0). Since v(x, to) is continuous in x and has finite support, this implies u(x, to) E 0, which is a contradiction since iJ,v(x, to) is not identically zero. Having verified that sr intersects aI, we now solve (1.1). (2.5) with 0 as the right state and As, as the left state. The solution has backward and forward moving fast characteristics and a middle state A2 (that already appeared).
498
M. S. Gordon
et al.
The remaining wavefronts and states may be found by continuing the above procedure of solving initial value problems of the form (l.l), (2.5). This is illustrated in Figs 9 and 10. Figure 11 shows profiles of the solution for u and e at the specific times tl , t2, t3 labeled on the t-axis in Fig. 10. Notice that there are only a finite number of wave interactions and that after t = t, there are five forward moving wavefronts: three fast characteristics, p4, p2, and p6, joined to two slow characteristics, tx5 and c(,. Hence there are no further wave interactions. Figure 12 shows the profiles of u and 0 at some time after t = t,. Notice that though u is initially non-negative, it is non-positive after t = t,. It is not hard to use Figs 9 and 10 to verify that this is true even if we alter the boundary pulse by changing the slopes of the front and back edges of the pulse and/or the length of the neutral region. We would now like to consider the question of what happens to the solution of the velocity-controlled problem when the small wavespeed goes to zero (while the large wavespeed remains fixed). Notice, however, that this would result in (3.1) being violated, complicating the analysis. We avoid this by instead considering the aforementioned question for the simplified velocity-controlled boundary condition shown in Fig. 4. Notice that the only difference between this boundary pulse and (1.3) is the absence of a neutral region between the loading and unloading fronts. The result is that there is no lag time between the interaction of the slow leading wave and the fast trailing wave, resulting in fewer overall wave interactions. The solution for large time, however, is essentially unchanged: the profile of the solution is inverted and consists of a fast leading edge with two line segments and a slow trailing edge, separated by a growing neutral region. This means that, as with the boundary data considered by Osinov and Gudehus, the trailing edge of the pulse tends to remain stationary as the small wavespeed goes to zero. There is, however, one significant difference; the amplitude of the pulse also tends to zero as the small wavespeed goes to zero, unlike the Osinov/Gudehus solution pulse, whose amplitude is independent of either wavespeed. This is made precise in Theorem 3.1 below. The solution of (l.l), (1.2) with the boundary data shown in Fig. 4 may be obtained in the manner described above. The states for the solution are shown in Fig. 13 and the structure of the solution in the xt-plane is illustrated in Fig. 14. Arguing as before, we can show that the shock s1 meets al before meeting Dr. Also, a routine calculation shows that this interaction occurs at some time after t = 2. Thus the solution must have the form shown in Fig. 14. As before, we let t, be the time at which the last wave interaction occurs (see Fig. 14). We define t = 2 + 4Jm = 2 + 8cd/(d2 - c’), where c = F and d = m. Figure 14 shows how t”is found geometrically. Notice that t”is an upper bound for t,. Remark 1. Notice that t”+ 2 as c -+ 0 or a/b -+ 1, i.e., as the slow wavespeed goes to zero. Also, t”-+ co as b -+ 0 or c - d -P 0, i.e., as the equations become linear. THEOREM
3.1. Let v, r~be the solution of (l.l), (1.2) with the boundary data shown in Fig. 4.
Then max {I@, x
t)l, I+G t)l> <
4JZ-x m
+ J&$
4cd = c+d
for t ’ t:
(3.2)
Proof: Since the maximum values of (vi and )o( for t > t”occur in the neutral region between a2 and p4, the theorem will be proven if we show that the increase in (VIand 101across regions A5 and A6 is bounded by the right side of (3.2). Notice from Fig. 13 that the states
Fig. 12. Large-time
solution
of the velocity-controlled
problem
499
Plane shear waves in a fully saturated granular medium
a.v
Fig. 13. States for the simplified velocity-controlled
problem.
Fig. 14. Wave structure for the simplified velocity-controlled
problem.
A, and A6 satisfy Id,vJ = la,al < 1 and la,o( = (a + b)(8,u(
< m.
It is not difficult to show from Fig. 14 that the width of the region AswAg is 4-/(-
+ Ja+b).
Combining
(3.3) in the t-direction
this with (3.3) gives the result.
n
M. S. Gordon et al.
500
4. STRESS-CONTROLLEDPROBLEM
We now construct the solution of (l.l), (1.2), (1.4). To simplify the analysis, we consider boundary data without a neutral region between loading and unloading fronts, so there is no lag time before the interaction of the slow leading wave and the fast trailing wave. As for the velocity-controlled problem, the solution with a time lag generates more wave interactions at first, but it is qualitatively the same as time increases. Again, the state bordering the x-axis is 0. The boundary conditions imply that d,a = (a - b)d,v = - 1 on the t-axis between t = 0 and t = 1, so the state following 0 is using a slow characteristic CI~(see Figs 15 and 16). The Al = ( - l/(a - b), 1/m, boundary conditions also imply that d,o = (a + b)8,u = 1 on the t-axis between t = 1 and t = 2, so the state A2 following Al, using a shock sl, is the intersection of the forward wave curve through Al and the line i3,v = l/(a + b). Since 8,a = 8,~ = 0 on the t-axis for t > 2, the state A3 following AZ, using a fast characteristic pl, is the intersection of 8,v = 0 and the line of slope - l/Ja+b through AZ. The next wave interaction will be the meeting of s1 and ,81 (rather than s1 and crl). The proof is identical to the one given for the velocity controlled boundary conditions, replacing v with 0. We then solve (l.l), (2.5) with A1 as the right state and A2 as the left state resulting in a backward moving fast characteristic p2 and a forward moving shock s2. Notice that the state A4 between them is on the same wave curve as A, and that s2 is slower than sl. When p2 reaches the t-axis, a forward moving fast characteristic p3 is produced, which will meet s2 before s2 reaches cxl (by the same argument as before), producing a shock s3 which is slower than s2. It is not hard to see from Figs 15 and 16 that this pattern continues indefinitely, with fast characteristics interacting alternately with the t-axis and the forward moving shocks, decreasing the shock speed with each interaction. Moreover, the shock never reaches CI~.Notice also from Fig. 15 that the states A, are approaching the origin of state space as n ---f00. Profiles of u and (T containing a forward moving fast characteristic are sketched in Fig. 17. In the following theorem, we show that the amplitude of these profiles approaches zero as t -+ 00. THEOREM
4.1. Let u, (T be the solution
of(l.l),
(1.2), (1.4). Then
lim max {/0(x, t)l, 10(x, t)j} = 0. f’rn x Proof Let t, be the time when pzn meets the t-axis (see Fig. 16). We know from Figs 15 and 16 that the maximum values of IuJand ((~1at a fixed time occur at the shock. Also, since the
a.0
Fig. 15. States for the stress-controlled problem.
Plane shear waves in a fully saturated granular medium
Fig. 16. Wave structure for the stress-controlled problem.
Fig. 17. Large-time solution of the stress-controlled
problem.
502
M. S. Gordon et al.
Fig. 18. Definition of functionsfand
y.
shocks are approaching al, where both u and u are zero, these maximums must be decreasing as t increases. Also, if max, lfl.1-+ 0 as t + co then the distance between the shocks and tli must be approaching zero as well, which implies that max, 1u) -+ 0 also. Thus we need only show that lim max 10(x, t,)l = 0. (4.1) n+oo X Let K, = la,c~l for the state A2n+2. Clearly
max 10(x, t,)l d X
K,,
(supp (T(x,t,)) = ~,t,
,/a- b,
(4.2)
so we prove (4.1) by showing that q,t,-+ 0 as n + co. Define a functionf(z) as follows [see Fig. 18(a)]. Through the point (0, z) in the xt-plane, draw a line of slope l/m. Through the point of intersection of this line and the line of slope l/m emanating from the origin, draw a line of slope - l/m. Definef(z) to be the value oft at the point of intersection of the second line and the t-axis. It is not hard to show that (4.3) It is clear from Fig. 16 that t n+l <“ml)
(4.4)
for any n. Define a function g(lc) as follows [see Fig. 18(b)]. Through the point of intersection of d,a = - K and the line tangent to h at the origin in state space [a routine calculation using (2.3) shows this line has slope - a/m], draw a line of slope - m. Through the point of intersection of this line and the a,a-axis, draw a line of slope m. Define - g(rc) to be the value of d,a at the point of intersection of the second line and the line tangent to h at the origin. It is not hard to show that g(K) =
a-J_ (4.5)
a+JZP.
It is clear from Fig. 15 that %+ I < &o)
(4.6)
for any n. Combining (4.3), (4.4), (4.9, and (4.6), we have K n+l
t ntl
Kntn
from which (4.1) follows.
2
Plane shear waves in a fully saturated
In the following time.
theorem,
SO3
medium
we show that the kinetic energy of the solution
4.2. Let u, (T be the solution
THEOREM
granular
of(l.l),
decays for large
(1.2), (1.4). Then
lim Oc’V(X, t)’ dx = 0. t-ac s 0 Proqf
Let t, be defined
as in Theorem
(4.7)
4.1. Then
7 D(X, t,,)’ dx d (supp u(x, t,)) max u(x, t,)’ = t, G X s0 Notice
(4.8)
that max Iu(x, t,)l = max \a(~, t,))/JZC * x
because
max t’(x, t,)2. *
the maximum
d tint,
(4.9)
of u and 0 occur at the shock S, and because d,o(x, t,) = - &o(x, t,)/&zi
for the state Al. Combining
(4.9) with (4.2) and (4.8), we have that
s m
u(x, t,)’ dx 6 K,2t:G,
0
Combining
(4.3), (4.4), (4.5), and (4.6) we have a-Jz3<1
Jz+ 1t,3+1 lc2t3 n n
which implies
<
b
that X lim u(x, t,)’ dx = 0, n-r, s 0
from which (4.7) follows.
H
Acknowledyements-The velocity controlled problem was posed to us by V. A. Osinov. We thank him for his interest in our results, and for the many constructive comments he made concerning earlier drafts of this paper.
REFERENCES 1. V. A. Osinov and G. Gudehus, Plane shear waves and loss of stability in a saturated granular body. Mechunics of Cohesive Frictional Materials and Structures 1, 25-44 (1996). 2. E. Bauer, Calibration of a comprehensive hypoplastic model for granular materials. Soils and Foundations. Jup. Sot. Soil Mech. Found. Eny. 36(l), 13-26 (1996). 3. G. Gudehus, A comprehensive constitutive equation for granular materials. Soils und Foundations, Jup. Sot,. Soil Me& Found. Eny. 36(l), 1-12 (1996). 4. D. Kolymbas, and W. Wu, Introduction to hypoplasticity. In Modern Approaches ro Plasticity. D. Kolymbas (ed.), pp. 213-223. Elsevier, Amsterdam (1993). 5. D. Kolymbas, I. Herle and P. A. von Wolffersdorff, Hypoplastic constitutive equation with internal variables. Inl. J. Num. Anal. Meth. Geomech. 19, 415-436 (1995). 6. P. D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10. 537-566 (1957). I. D. Schaefler and M. Shearer, Scale-invariant initial value problems in one-dimensional dynamic elastoplasticity, with consequences for multidimensional nonassociative plasticity. Euro. Jnl of Appl. Math. 3, 225 254 (1992).