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Dynamic response of normal moving load in an initially stressed transversely isotropic half-space
0020-7683&f, 53 00+ 00 C 1986PergamonPressLid
fnr J SohJsS~rucr,rrrsVol 22. No 3, PP 283-291. 1986 Prmed I” Great Bntam
DYNAMIC RESPONSE OF NORMAL MOVING LOAD IN AN INITIALLY STRESSED TRANSVERSELY ISOTROPIC HALF-SPACE S. DEY, S. IL CHAKRABORTV and M. Departmenl
of Physics and Mathematws,
CHAKRABORTY
lndlan School of Mines, Dhanbad-826004,
India
Abstract-In thts paper the stresses developed m a transversely ~sotroptc half-space under compressive prestressed condittons have been ohtamed due to a movrng normal load on the rough surface. It has been shown that the exrsttng prestresses have much effect on the development of mcremental stresses due to normal load. The numerical calculations for developed stresses have been done for the case of ice for different values of prestressed parameters and have been presented m tabuiar form The condition under whtch the fracture on the surface will take place has also been denved, and st has been shown that the cnttcal veloctty at which the fracture will take place IS also a function of initral stresses present m the body. The numerical values for the velocittes of moving toad for gene~ti~f~ctu~ have been calculated for different prestressed parameters rn case of ice
INTRODUCTION
The response of moving normal load over a surface is a subject of investigation because of its possible application in determining the strength of a structure. The steady-state solution of the problem of moving normal load over an elastic half-space was given by Cole and Huth[ I] and Craggs[Z], who derived a relatively simple closed form solution, exhibiting a resonance effect at a critical load velocity, which in this case equals the velocity of Rayleigh waves. The problem considered by Cole and Huth[l] had been discussed previously by SneddonU] by a somewhat different method. However, Sneddon[3] treats only a particular case (the subsonic case). Stresses developed in a transversely isotropic elastic half-space due to normal moving load over a rough surface have been determined by Mukhetjee[rl]. The problem of moving load on a plate resting on an elastic half-space has been solved by Sackman[S] and Miles[6]. The relevant systematic approach towards the problem of moving load on a plate resting on an elastic half-space has been given by Achenbach, Keshava and He~ann[7]. Mukhopadhyay~~] has studied the problem of normal moving load over a transversefy isotropic layer lying on a rigid foundation. The earth is an initially stressed medium. Due to slow process of creep, atmospheric pressure, gravity, difference of temperature, etc. stresses of large magmtude, called initial stresses or prestresses, are present inside the earth. It is of interest to study the effect of the initial stresses while studying the problems of earth model. The present paper is concerned with the effect of the prestresses on the generation of incremental stresses leading to fracture due to a normal moving load on the rough surface of the transversely isotropic elastic half-space. The stresses have been obtained in closed form, and numerically the values have been calculated for different magnitudes of the prestressed parameters in the case of ice, which is considered as transversely isotropic in nature (Hearmon[9]). The critical velocity of resonance causing fracture has also been evaluated for different prestressed parameters. In the absence of initial stresses the results have been shown to coincide with the result obtained by Mukhejee141. Further, by making the medium to be isotropic, the surface to be smooth and removing the initial stresses, the results have been shown to tally with those of Cole and Huth[ I]. The static resufts derived here for the isotropic elastic half-space without initial stresses also agree with the result given by Timoshenko[lO]. 283
284
S.DEY CI (II FORMULATION
Constder a transversely isotropic elastic compressive stresses SI I and SIT acting along normal load F moves with a constant speed ofx. The dynamic equations of motion in an Biot[ I I]:
half-space z 3 0 under uniform mitral the x- and z-directions, respectively. A u over the surface along the du-ection initially stressed medtum are given by
where P = S3,- SII,and the incremental stress components s,, in terms of displacement components u and w may be written as
au $11 =
(Cl1
f
$33 =
c33 a” az
Sl3 =
s3i
=
P)-- &_ +
k13
+
au Cl3 ax,
c44
z az
+
+
aw z
9
(2) aw ax >
(
PI
’
c,, being the elastic constants of the medium considered. The stress equations of motion (I) with the help of (2) become
(Cl1+ PI
(
t$+ (c&+9$+ (c,,+c,+$g=p$. P
C&
-
-2
1
a2w s+c33az’+
P
a2w
a2u Gz=
c13+c44+5 (
(3)
a2w PS’
)
The boundary conditions may be written as
Af: = - F6tx - zd = -
cos k(x - d) dk,
(4)
where Af,, Af, are the incremental normal and shear boundary forces, respectively, per unit initial area and are given bytt 11
(3
Further, S(x) is the Dirac delta function of argument x, and R is the frictional coefficient.
285
Dynamic response of normal movmg load SOLUTION
The steady-state solutions of equations of motton (3) may be assumed to be If
I,I
=
W=
I
emhY:
A
sin k(x - VI) dk,
cos k(x - VI) dk +
,,%C eWh*:cos k(x - z,t) dk +
I,,I
D e-
(6)
‘ye sin k(x - zu) dk.
Inserting (6) into (3) the following equations for the constants A, B, C, D may be obtained: [PVZ- (c,, + P) + (c.a + P/2) q2]A - g[c,3 + CM + P/2]D = 0, q[clJ + cti + P/2]A + [PI,* - (CM - P/2) + aaq*lD
= 0,
(7)
[p$ - (c,, + P) + ((.a + P/2)q*]B + Q[C,S + cu f P/2lC = 0, q[c,3 + CM + P/2]B - [PI,* - (CU - P/2) + cu$]C
= 0.
Equations (7) are consistent if q4
+
-
pv*
cc44
-
P/2) +
pu*
c33
[
-
(c,,
cqq
+
+ P) + (Cl3 + c44 + P/2)*
PI2
c&44
+ [PZj2 -
+
(Cl,
+ P/2)
I q2
P)l[pv2 - (c44 - P/2)1
c33tc.44 +
= 0.
(8)
pm
Let qf and qj be the two roots of eqn (8). Both q, and q2 will be real if qf. q$ are positive, i.e. if [pz? - (c,, + P)] [pz? - (cu - P/2)1 is positive. Hence, 1, must be either greater than or less than both [(c,, + P)lp],‘* and [(c.,., + P/2)/p],‘*. But if v is greater than both [(c,, + P)lpl”* and [(CM - P/2)lpl,n. the coefficient of 4’ and the constant term in eqn (8) become positive; therefore both qf and q$ become negative. Therefore, q, and q2 will be both positive when and only when v is less than both [(CII + P)lp]‘n and [(CM - P/2)lp],n. Then the solution (6) may be written in explicit form as
If
=
I(,I
Ale-&I:
cos k(.r - vt) dk +
/
,,%A2e-h2.‘ cos k(r - IV) dh
sin k(x - r~) dk + 1%’ =
I
x
,~
sin k(x - TV) dk. -z
m&e -‘Wecos k(.r - VI) dk +
m2B2e-hqz’ cos k(x - vt) dk
X -I
I,
sin k(x - rd) dk -
m,A,e-hYhr
m2ALle-h1~zrsin k(x - vr) dh,
where
(c,, ml.2
=
P/2) yf.2 ((.I? + (‘44+ P/2) 4, 2 *
+ P) - PI’:! -
(C&j
+
286
S DEY et (II
The boundary conditions (4) with (5) and (9) yield AtH2cti
-
S33 -
Slibl
+ a44
+ Pl9iJ
+ &L(2C44 - s33 - Sl lb2 -
Ailmi91~3
BiK2C44 - s33 - S,ihl
(cl\
+ &,)I
+
+
2FR
+ m21
A2jmtq2c33
-
- x
((.I3
+
= 0,
&*)I
=
0.
&j21
=
0.
+
=
0,
‘(to)
+ (2c44 + P)q,l + BJ(2c44 -
BlK1;‘i3
+ m44
S33)
ml9IL.331 +
-
s33
-
B2ki3
Sidm2
+
s33)
+
-
(2~44
f
m292(.33]
5
from which the constants are obtained as 21%
A
’A2=
m2w33
mtqic33
F
1
-
6~13 + S3t)
A*
[
B,=--
9
A*
2FR
-x
(~13+ S33)
-
[
Irk
(2~44 - S,s -
nh
St&b
1I
+ (2~44 +
Pk2
A*
[ S33
-
Silh
+
.
I (2~44
+
P)9i
A*
1
,
where A* = [(2c4 - S3j -
K2c44 -
Sillrni
$33
-
+
Siilm2
(2~44 + P)qtl [m2q2c33
-
ki3
(2c44 + P)92l Imiqic33 -
+
+
ki3
S341 +
S3dl.
Thus, the incremental stresses in the medium, due to the moving load over the surface of an initially stressed elastic half-space, are [hi
-
-t
91) {t2c44 -
-
s3.l
Sil)m2
+
(2ca + P)92fe-‘y”
(mz + 92) ((2~44- St.1 - S~i)mi + (2~4~ f P)qi}e-“f”]
+ 2R
m2
+
-
y2)
(mi
(m19i~33 +
(~13
+
Sin
kfx -
IV)
dtl
S3d}ewAuz;
1
91) {mz92(*33- (ci3 + S33)}e-A“‘;]cos A(.u- IV) dX
and l((‘l3 -
ti I3
m2w33)
((2~~ - SU - S,,)m, + (2C.44+ P)91}e-A””
- rn,9l~33) ((2r44 - St - Sli)W
+ (2~~ + P)91}e-AC’1z] cos A(.Y- vt) dk + - k13 + W}e- ktr -
(rtlly,c33
-
I
(,%2Rl(m,qi( 31 - ( 14 &y,c*xt
(.,3) {Miq,(.tt -
(tit
+ S3t)}e-“f”r]
sin A(.r -
rjt)
1.
dX
287
Dynamic response of normal movmg load
It may be noted that the integrals in the expressions for s13 and sv converge for positive values of 91, 42, i.e. for values of II less than both [(cl, + P)/p]‘” and NC44- P/2)/p] IR. Evaluating the integrals, the expressions for the stresses may be written as
Sl3
=
FCq4 A*
-
_
K
+ 91)
+
[(2c44
bl
h2
92)
l(2c44 - s33 - Sl lb2 (x - vf)2 + 9:22 -
b2
Sll)rnl
+
m44
f79ll
+
zrr)2+ 9122
(x +2R
-
s33
(2c44 + Pkj21
+
+ 92) b191C33
-
(Cl3 +
uf)
s33)192z
9:z2
Vf)2 +
(x -
(x -
MI + 91)
bn292C33 -
k13
Vf)2 +
(x -
+ S331191z
9:z2
I
1
(11)
and C44 -
s33 -
Sllbl
+
(2CU
N2CU
-
s33 -
Sll)M2
+
(2CU
+
2R
(mlqiC33
-
~13) b292C33 uf)2 +
(x -
I _
(m292C33 -
(Cl3 -
+ P)921 (Cl3 -
vf)2 +
(x -
p)9l1
m292C33)92z
ut)2 + 9jz2
(x _
+
-
ml9lCl3)9lz
9:z2 (Cl3 + S33)l
9:z2
~13) klqlC33
-
k13
+
S3311
ur)2 + 9tz2
(x -
fx
I
- Id) 1.
(12)
Thus, it is seen from expressions (1 I) and (12) that the stresses generated due to the moving load are functions of the inherent initial stresses (or prestresses) existing in the elastic solid, irrespective of the anisotropic nature of the solid. In the absence of initial stresses results (II) and (12) coincide with the result obtained by Mukherjee[4]. Further, if we reduce the half-space from an anisotropic to an isotropic medium, i.e. cU + )I, c13 + A, cj3 --+ (A + 2p), the developed incremental stresses may be written in nondimensionalized form as F
Sl3 = -
(ml
7rA* _
[I
+
91) [(I
t; -
<‘)m2 + (1 + 5 -
5’)921
(x - ZJf)2+ 9:z2
(m2 + 92) [(I
-
5 -
(x + R
-
(m2 +
11’)mt +
92) bl91
(x -
I _
MI
+
(1 +
g -
1;‘)9,1
rV)2 + 9lz2 1) b2b2)
+ 2(1 -
-
Vf)
6)192Z
zu)2 + 9lz2
91) b292
-
1) b2/P2)
(x - rQ)2 +
9:z2
+
II
2(1 - 5)191z
1
(13)
288
S 5 - 5’hI
DEY CI ~1
+ (1 + 5 - 5’)41} {(I - m292) (a*@‘)
2)yg
-
Lx - rtry + 9:z* _
+
_
it1 - 5 - 5’)W + (1 +
5 - [‘I@} {(I - mlqI) (a2/p2) - 2}9,z (x - ItI)2 + 9:z’
R
Km191- I) (a’lf32)
+
I
b2y2
-
I) (a'@')
+
21 [(m292 - 1) (a2/P2) + 2(l -
Lx -
-
at)2 +
I) (azIp
13
+ 2(1 - 511 (x _ z,,)
qfz2
I
(14) where A*
=
ItI -
-
5 -
[(I -
5 -
a2/f3’ ml.2
5'h
+
5')m2
+
5 -
(1 +
(')q,l
5 -
5'1421
+
h,q,
-
[
31
I)$
-
5)
+2(1-o
1 1 ,
- (I + 5 - {‘)9:,2 2(5+ 5')- 2J2/P2 b2/B2 + 5 + 5' - 1)ql.z
=
and 91.2 =
(1 +
1*
-b&d=*
‘I2
2
in which b
_
c =
7J2/p2 + a2/p2 5 {zJ~I~~~ -
5'
a2/p2
5 {' 5’) 1)'’ + 5') + (a2/p2 (a2/P2) + (I 5 + 1 + zJ21fPI+&-5 a2/p2 2(5
2(5 - r’,} {V21P2+ 5 - 5’ - I} (a2/P2) (1 + 5 - 5’) -
Again, if we remove the inttlal stresses, i.e. SII = Sj3 --) 0, the above results reduce for smooth surface (i.e. R = 0)as below: -=F St3
31 +A*9392 [ (x -
I#)' +
1
1
9;z2 - (.r - 7,#I + 9Tz2 (x - 1”)
(15)
and 49:9zz (x -
742
+
where A* = (1 + 9% - 49,92. yi = (I -
7r2/p2).
93 = (I -
a2
2pJ/p,
fit = CL/p.
=
(A
+
7j2/a2),
qTz'
1 '
(16)
289
Dyniimlc response of normal movmp ludd
Using the notations X = s - ttt, CL = Q, Cl = 8, ML = 1,/a, MI = t@, Pi = (I f3i = (1 - Mtr), 4. = CP + P5z2), rF = CP + Pfz% 9~ = tan-‘&zt’x), 07 = tan-‘(&ztfx), results (13) and (14) are seen to coincide with the results obtained by Cole and Huth([l], page 434). From (11) and ( 12). the stresses at x = vt, i.e. at the point directly below the load, may be obtained as
M),
_ (ml + 41)
fm2q2N
(1 + 2qs)l
-
1
41
$33
=
-
F
nzA.*
(f-
m2cltN)
[(I
-
q;
-
q,)m,
+
(1
+
q;
-
1
(17)
q,)q,]
q2
C
-
(I -
wq1~) [(I - q; - q1h
f (1 + q; - qr)q21
4r
.
(18)
1
where
A* = hqz~
- (1 + 2qd t(1 - q; - q~)m, + (1 + 7; -
in which N = Sratic
c33h3.
t@‘bqiN r)3
=
(1 + b)l
s33!2C,3,
7.j;
[(I =
-
qr; -
q)m2
q,)q,] +
(1
+
q;
-
q,)qz].
and qi = 5,,/2c’44.
s33/k.~
L’usc
For the static case we consider zj = 0, By making the medium isotropic and removing the initial stresses, we get the following static results: 2F $13
=
-
TZ
x (x2
f
2’ z2)“2
(x2
+
z2)3’2
and $33
=
2F
--
TZ
z” (x2
+
z2)2
*
The above results tally with the well-known static result obtained by Timoshe~o(flOJ, page 84).
NUMERICAL
RESULTS
The stresses as given in ( 17) and ( 18) have been calculated in case of ice, Following Hea~on[9], the elastic constants for ice at - 16°C are CII
= 1.36
c33
=
x
IO” dyn/cm2,
1.46 x IO” dyn/cm2,
cu = 0.32 x 10” dynlcm’, Cl2
=
0.67 x 10” dyn/cm2,
Cl3
=
0.52 x 10” dyn/cm2,
(19)
290
S DEY et 01. Table I. v = 100 mkcc
z=lm
rl3
51:
-%
i.: -0.2 0.0 02 05 0.0
08 0.3 -0.3 8;
is -0.5 -;;
0:8 0.0
0:o 0.0
(~336
-a
-1 112 -5.732 -7.219 -5.133 -t.i60 -6.057 -7.289
x x x x x x
10-2 lO-3 lO-3 lo-’ 10-2 10-z IO-’
-1.147
x lo-’
fs13iFRLt I529 1.779 -0.915 1.899 I%7 4.431 0.132
x x x x x x
IO-’ 10-j lo-4 IO-’ 10-3 10-l 10 ’
z=Sm 0.3
3.558 x IO-’ x -03.055 382 x lo-’
-0:2 :*:
-0.3 08
-0:s ii+:
-1.444 -2.224
x 1O-3 10-3
8::
0.3 0.0
-0.5 0.5
-2.321 -1.027
x IO-j IO-’
2.762 3.800 x lo-* IO-’
8.:
83
X:l
-1.211 -1.458
x lo-’
0.255 x lo-’ 0.887 lo-5
Table 2. v = 200 mkcc L=
lm
l-l3 0.2 0.5 -0.2
d ifi -0’3
?I 0.5 i:: -:::
(~33IFh-w
h3IFfLv1
-5.748x 10-9 -1.124 x lO-2 1.408 x 10-2
1865 x lo-’ I.544 x 10-3 I762 x 1O-2
-4.657 -5.147
x 10-3 10-j
1901 1974 x lo-” 10-Z
8;
x3
8::
0.0 0’8
i:8
-7.306 -1.262
x lo-* 10-3
0.147 1242 x IO-’ 10-4
0.2 0.5 -0.2
0.3
0.5 0.2 0.5
-1 149 x 10-3 -2.240 x 10-9 2.817 x lO-3
0.373 x 10-3 0.248 x lO-3 3.523 x lO-3
z=5m
ii:: 0:s 0.0
_g ii.! 0:8 0.0
-0.5 0.5 0.0 0.0
-1.029 -3.948 -2,524 -1.461
x lO-3 lo-” x lO-3 x 10-3
-0.893 3 147 x 10-4 lo-’ 2484 x 10-j 0.293 x IO-‘
P = 0.917 g-cm3. The numerical values of s&FR and s&F at points directly below the load are calculated for various prestressed parameters and for different z, The values of 11taken are 100 mlsec and 200 mlsec. The numerical results are in Tables 1 and 2. DISCUSSION It isobserved that A* occurs in the denominator of the expressions for the stress com~nents. Hence, in case A* = 0 there wifl be‘a hi stress con~ent~tion which will lead the material towards fracture. Thus A* = 0 will determine the velocity v of the moving load F at which the fracture will take place. The presence of ql, q~ and q; in the equation A* = 0 shows the influence of prestresses in the medium on the formation of fracture. From A* = 0 the values of rJm have been calculated for tee usmg the same elastic constants as given in (19) for different sets of prestress parameters -q,, r(3 and q$, and the following results have been obtained.
Dynamic
?I 0 0 -0.2 -02 02
response
of normal rll
rli 0 0 -0.3 -0 3 0.3
0 05 0 -05 -05
movmg
hd
291
+ 1:1GTp 0 9s84 I 2125 0 9114 0 6365 0.3816
The first result is for the maternal free from initial stresses, and in this case the velocity at which the fracture will take place is very close to the velocity of Rayleigh wave propagation in the transversely isotropic half-space. This was expected too. The result shows that if the tensile prestress in the x-direction is increased, the fracture occurs at higher velocity of the moving load, whereas if the prestress in the x-direction is compressible, then lower velocity can create fracture. Further, it has been observed during calculation that if the tensile initial stress in the z-direction is higher, then there is less possibility of fracture. From the numerical results, specifically, it may be said that the development of the incremental normal and the shearing stresses induced by the initial tensile stresses along the z-direction increases. However, the normalized initial tensile stress along the x-direction increases the development of shearing stress but decreases the normal stress. The presence of either the tensile initial stress or compressive initial stress in both directtons increases the shearing stress but decreases the normal stress. Also, it may be noted that the development of shearing stress in the case of initial compressive stress is comparatively less than the tensile case, but for normal stress the behaviour is just the opposite. REFERENCES I
: 4 5. 6. 7 8. 9. 10 II.
J Cole and J Huth, Stresses produced in a half plane by moving loads. / Appl. Merh. 25, 433-436 (1958). J W. Craggs. On two-drmenaonal waves in an clastlc half plane. Proc. Curnb. fIri/. Sot. 56,269 (1960) I. N. Sneddon. Stress produced by a pulse of pressure moving along the surface of a semi-mfimte solid. Rend. Circ . MU. Palcrtnn 2, 57-62 (1952). S MukherJee, Stresses produced by a load movmg over a rough boundary of a semi-mfinue transversely isotropic sohd. Pure Appl. Geophys. 72, 45-50 (1%9). J L Sackman, Uniformly moving load on a layered half-plane. J Engng Mech. DIV. Pror. AXE 87 EM,, 75-89 (l%l). I. W Miles, Response of a layered half-space to a movmg load. J. Appl Mech. 33,680-681 (1966). J D. Achenbach, S. P. Keshawa and G. Henmann, Moving load on a plate restmg on an elastm halfspace. 1. Appl. Mech. 24, 910-914 (1967). A. Mukhopadhyay, Stresses produced by a normal load moving over a transversely isotropic layer of se lymg on a rigid foundation. Pure Appl. Geophys. 60, 29 (1965). R. F. S. Hearmon, An Introduction IO Applied Anisotropic Ehticuy, p. 38. Oxford Umverslty Press, Oxford (l%l). S. Timoshenko, The Theory of Elasriciry, pp. 82-84. McGraw-Hill, New York (1934). M. A. Biot, Mechanics of Incremental Defirmarions. John Wiley, New York (l%S).
fnr J SohJsS~rucr,rrrsVol 22. No 3, PP 283-291. 1986 Prmed I” Great Bntam
DYNAMIC RESPONSE OF NORMAL MOVING LOAD IN AN INITIALLY STRESSED TRANSVERSELY ISOTROPIC HALF-SPACE S. DEY, S. IL CHAKRABORTV and M. Departmenl
of Physics and Mathematws,
CHAKRABORTY
lndlan School of Mines, Dhanbad-826004,
India
Abstract-In thts paper the stresses developed m a transversely ~sotroptc half-space under compressive prestressed condittons have been ohtamed due to a movrng normal load on the rough surface. It has been shown that the exrsttng prestresses have much effect on the development of mcremental stresses due to normal load. The numerical calculations for developed stresses have been done for the case of ice for different values of prestressed parameters and have been presented m tabuiar form The condition under whtch the fracture on the surface will take place has also been denved, and st has been shown that the cnttcal veloctty at which the fracture will take place IS also a function of initral stresses present m the body. The numerical values for the velocittes of moving toad for gene~ti~f~ctu~ have been calculated for different prestressed parameters rn case of ice
INTRODUCTION
The response of moving normal load over a surface is a subject of investigation because of its possible application in determining the strength of a structure. The steady-state solution of the problem of moving normal load over an elastic half-space was given by Cole and Huth[ I] and Craggs[Z], who derived a relatively simple closed form solution, exhibiting a resonance effect at a critical load velocity, which in this case equals the velocity of Rayleigh waves. The problem considered by Cole and Huth[l] had been discussed previously by SneddonU] by a somewhat different method. However, Sneddon[3] treats only a particular case (the subsonic case). Stresses developed in a transversely isotropic elastic half-space due to normal moving load over a rough surface have been determined by Mukhetjee[rl]. The problem of moving load on a plate resting on an elastic half-space has been solved by Sackman[S] and Miles[6]. The relevant systematic approach towards the problem of moving load on a plate resting on an elastic half-space has been given by Achenbach, Keshava and He~ann[7]. Mukhopadhyay~~] has studied the problem of normal moving load over a transversefy isotropic layer lying on a rigid foundation. The earth is an initially stressed medium. Due to slow process of creep, atmospheric pressure, gravity, difference of temperature, etc. stresses of large magmtude, called initial stresses or prestresses, are present inside the earth. It is of interest to study the effect of the initial stresses while studying the problems of earth model. The present paper is concerned with the effect of the prestresses on the generation of incremental stresses leading to fracture due to a normal moving load on the rough surface of the transversely isotropic elastic half-space. The stresses have been obtained in closed form, and numerically the values have been calculated for different magnitudes of the prestressed parameters in the case of ice, which is considered as transversely isotropic in nature (Hearmon[9]). The critical velocity of resonance causing fracture has also been evaluated for different prestressed parameters. In the absence of initial stresses the results have been shown to coincide with the result obtained by Mukhejee141. Further, by making the medium to be isotropic, the surface to be smooth and removing the initial stresses, the results have been shown to tally with those of Cole and Huth[ I]. The static resufts derived here for the isotropic elastic half-space without initial stresses also agree with the result given by Timoshenko[lO]. 283
284
S.DEY CI (II FORMULATION
Constder a transversely isotropic elastic compressive stresses SI I and SIT acting along normal load F moves with a constant speed ofx. The dynamic equations of motion in an Biot[ I I]:
half-space z 3 0 under uniform mitral the x- and z-directions, respectively. A u over the surface along the du-ection initially stressed medtum are given by
where P = S3,- SII,and the incremental stress components s,, in terms of displacement components u and w may be written as
au $11 =
(Cl1
f
$33 =
c33 a” az
Sl3 =
s3i
=
P)-- &_ +
k13
+
au Cl3 ax,
c44
z az
+
+
aw z
9
(2) aw ax >
(
PI
’
c,, being the elastic constants of the medium considered. The stress equations of motion (I) with the help of (2) become
(Cl1+ PI
(
t$+ (c&+9$+ (c,,+c,+$g=p$. P
C&
-
-2
1
a2w s+c33az’+
P
a2w
a2u Gz=
c13+c44+5 (
(3)
a2w PS’
)
The boundary conditions may be written as
Af: = - F6tx - zd = -
cos k(x - d) dk,
(4)
where Af,, Af, are the incremental normal and shear boundary forces, respectively, per unit initial area and are given bytt 11
(3
Further, S(x) is the Dirac delta function of argument x, and R is the frictional coefficient.
285
Dynamic response of normal movmg load SOLUTION
The steady-state solutions of equations of motton (3) may be assumed to be If
I,I
=
W=
I
emhY:
A
sin k(x - VI) dk,
cos k(x - VI) dk +
,,%C eWh*:cos k(x - z,t) dk +
I,,I
D e-
(6)
‘ye sin k(x - zu) dk.
Inserting (6) into (3) the following equations for the constants A, B, C, D may be obtained: [PVZ- (c,, + P) + (c.a + P/2) q2]A - g[c,3 + CM + P/2]D = 0, q[clJ + cti + P/2]A + [PI,* - (CM - P/2) + aaq*lD
= 0,
(7)
[p$ - (c,, + P) + ((.a + P/2)q*]B + Q[C,S + cu f P/2lC = 0, q[c,3 + CM + P/2]B - [PI,* - (CU - P/2) + cu$]C
= 0.
Equations (7) are consistent if q4
+
-
pv*
cc44
-
P/2) +
pu*
c33
[
-
(c,,
cqq
+
+ P) + (Cl3 + c44 + P/2)*
PI2
c&44
+ [PZj2 -
+
(Cl,
+ P/2)
I q2
P)l[pv2 - (c44 - P/2)1
c33tc.44 +
= 0.
(8)
pm
Let qf and qj be the two roots of eqn (8). Both q, and q2 will be real if qf. q$ are positive, i.e. if [pz? - (c,, + P)] [pz? - (cu - P/2)1 is positive. Hence, 1, must be either greater than or less than both [(c,, + P)lp],‘* and [(c.,., + P/2)/p],‘*. But if v is greater than both [(c,, + P)lpl”* and [(CM - P/2)lpl,n. the coefficient of 4’ and the constant term in eqn (8) become positive; therefore both qf and q$ become negative. Therefore, q, and q2 will be both positive when and only when v is less than both [(CII + P)lp]‘n and [(CM - P/2)lp],n. Then the solution (6) may be written in explicit form as
If
=
I(,I
Ale-&I:
cos k(.r - vt) dk +
/
,,%A2e-h2.‘ cos k(r - IV) dh
sin k(x - r~) dk + 1%’ =
I
x
,~
sin k(x - TV) dk. -z
m&e -‘Wecos k(.r - VI) dk +
m2B2e-hqz’ cos k(x - vt) dk
X -I
I,
sin k(x - rd) dk -
m,A,e-hYhr
m2ALle-h1~zrsin k(x - vr) dh,
where
(c,, ml.2
=
P/2) yf.2 ((.I? + (‘44+ P/2) 4, 2 *
+ P) - PI’:! -
(C&j
+
286
S DEY et (II
The boundary conditions (4) with (5) and (9) yield AtH2cti
-
S33 -
Slibl
+ a44
+ Pl9iJ
+ &L(2C44 - s33 - Sl lb2 -
Ailmi91~3
BiK2C44 - s33 - S,ihl
(cl\
+ &,)I
+
+
2FR
+ m21
A2jmtq2c33
-
- x
((.I3
+
= 0,
&*)I
=
0.
&j21
=
0.
+
=
0,
‘(to)
+ (2c44 + P)q,l + BJ(2c44 -
BlK1;‘i3
+ m44
S33)
ml9IL.331 +
-
s33
-
B2ki3
Sidm2
+
s33)
+
-
(2~44
f
m292(.33]
5
from which the constants are obtained as 21%
A
’A2=
m2w33
mtqic33
F
1
-
6~13 + S3t)
A*
[
B,=--
9
A*
2FR
-x
(~13+ S33)
-
[
Irk
(2~44 - S,s -
nh
St&b
1I
+ (2~44 +
Pk2
A*
[ S33
-
Silh
+
.
I (2~44
+
P)9i
A*
1
,
where A* = [(2c4 - S3j -
K2c44 -
Sillrni
$33
-
+
Siilm2
(2~44 + P)qtl [m2q2c33
-
ki3
(2c44 + P)92l Imiqic33 -
+
+
ki3
S341 +
S3dl.
Thus, the incremental stresses in the medium, due to the moving load over the surface of an initially stressed elastic half-space, are [hi
-
-t
91) {t2c44 -
-
s3.l
Sil)m2
+
(2ca + P)92fe-‘y”
(mz + 92) ((2~44- St.1 - S~i)mi + (2~4~ f P)qi}e-“f”]
+ 2R
m2
+
-
y2)
(mi
(m19i~33 +
(~13
+
Sin
kfx -
IV)
dtl
S3d}ewAuz;
1
91) {mz92(*33- (ci3 + S33)}e-A“‘;]cos A(.u- IV) dX
and l((‘l3 -
ti I3
m2w33)
((2~~ - SU - S,,)m, + (2C.44+ P)91}e-A””
- rn,9l~33) ((2r44 - St - Sli)W
+ (2~~ + P)91}e-AC’1z] cos A(.Y- vt) dk + - k13 + W}e- ktr -
(rtlly,c33
-
I
(,%2Rl(m,qi( 31 - ( 14 &y,c*xt
(.,3) {Miq,(.tt -
(tit
+ S3t)}e-“f”r]
sin A(.r -
rjt)
1.
dX
287
Dynamic response of normal movmg load
It may be noted that the integrals in the expressions for s13 and sv converge for positive values of 91, 42, i.e. for values of II less than both [(cl, + P)/p]‘” and NC44- P/2)/p] IR. Evaluating the integrals, the expressions for the stresses may be written as
Sl3
=
FCq4 A*
-
_
K
+ 91)
+
[(2c44
bl
h2
92)
l(2c44 - s33 - Sl lb2 (x - vf)2 + 9:22 -
b2
Sll)rnl
+
m44
f79ll
+
zrr)2+ 9122
(x +2R
-
s33
(2c44 + Pkj21
+
+ 92) b191C33
-
(Cl3 +
uf)
s33)192z
9:z2
Vf)2 +
(x -
(x -
MI + 91)
bn292C33 -
k13
Vf)2 +
(x -
+ S331191z
9:z2
I
1
(11)
and C44 -
s33 -
Sllbl
+
(2CU
N2CU
-
s33 -
Sll)M2
+
(2CU
+
2R
(mlqiC33
-
~13) b292C33 uf)2 +
(x -
I _
(m292C33 -
(Cl3 -
+ P)921 (Cl3 -
vf)2 +
(x -
p)9l1
m292C33)92z
ut)2 + 9jz2
(x _
+
-
ml9lCl3)9lz
9:z2 (Cl3 + S33)l
9:z2
~13) klqlC33
-
k13
+
S3311
ur)2 + 9tz2
(x -
fx
I
- Id) 1.
(12)
Thus, it is seen from expressions (1 I) and (12) that the stresses generated due to the moving load are functions of the inherent initial stresses (or prestresses) existing in the elastic solid, irrespective of the anisotropic nature of the solid. In the absence of initial stresses results (II) and (12) coincide with the result obtained by Mukherjee[4]. Further, if we reduce the half-space from an anisotropic to an isotropic medium, i.e. cU + )I, c13 + A, cj3 --+ (A + 2p), the developed incremental stresses may be written in nondimensionalized form as F
Sl3 = -
(ml
7rA* _
[I
+
91) [(I
t; -
<‘)m2 + (1 + 5 -
5’)921
(x - ZJf)2+ 9:z2
(m2 + 92) [(I
-
5 -
(x + R
-
(m2 +
11’)mt +
92) bl91
(x -
I _
MI
+
(1 +
g -
1;‘)9,1
rV)2 + 9lz2 1) b2b2)
+ 2(1 -
-
Vf)
6)192Z
zu)2 + 9lz2
91) b292
-
1) b2/P2)
(x - rQ)2 +
9:z2
+
II
2(1 - 5)191z
1
(13)
288
S 5 - 5’hI
DEY CI ~1
+ (1 + 5 - 5’)41} {(I - m292) (a*@‘)
2)yg
-
Lx - rtry + 9:z* _
+
_
it1 - 5 - 5’)W + (1 +
5 - [‘I@} {(I - mlqI) (a2/p2) - 2}9,z (x - ItI)2 + 9:z’
R
Km191- I) (a’lf32)
+
I
b2y2
-
I) (a'@')
+
21 [(m292 - 1) (a2/P2) + 2(l -
Lx -
-
at)2 +
I) (azIp
13
+ 2(1 - 511 (x _ z,,)
qfz2
I
(14) where A*
=
ItI -
-
5 -
[(I -
5 -
a2/f3’ ml.2
5'h
+
5')m2
+
5 -
(1 +
(')q,l
5 -
5'1421
+
h,q,
-
[
31
I)$
-
5)
+2(1-o
1 1 ,
- (I + 5 - {‘)9:,2 2(5+ 5')- 2J2/P2 b2/B2 + 5 + 5' - 1)ql.z
=
and 91.2 =
(1 +
1*
-b&d=*
‘I2
2
in which b
_
c =
7J2/p2 + a2/p2 5 {zJ~I~~~ -
5'
a2/p2
5 {' 5’) 1)'’ + 5') + (a2/p2 (a2/P2) + (I 5 + 1 + zJ21fPI+&-5 a2/p2 2(5
2(5 - r’,} {V21P2+ 5 - 5’ - I} (a2/P2) (1 + 5 - 5’) -
Again, if we remove the inttlal stresses, i.e. SII = Sj3 --) 0, the above results reduce for smooth surface (i.e. R = 0)as below: -=F St3
31 +A*9392 [ (x -
I#)' +
1
1
9;z2 - (.r - 7,#I + 9Tz2 (x - 1”)
(15)
and 49:9zz (x -
742
+
where A* = (1 + 9% - 49,92. yi = (I -
7r2/p2).
93 = (I -
a2
2pJ/p,
fit = CL/p.
=
(A
+
7j2/a2),
qTz'
1 '
(16)
289
Dyniimlc response of normal movmp ludd
Using the notations X = s - ttt, CL = Q, Cl = 8, ML = 1,/a, MI = t@, Pi = (I f3i = (1 - Mtr), 4. = CP + P5z2), rF = CP + Pfz% 9~ = tan-‘&zt’x), 07 = tan-‘(&ztfx), results (13) and (14) are seen to coincide with the results obtained by Cole and Huth([l], page 434). From (11) and ( 12). the stresses at x = vt, i.e. at the point directly below the load, may be obtained as
M),
_ (ml + 41)
fm2q2N
(1 + 2qs)l
-
1
41
$33
=
-
F
nzA.*
(f-
m2cltN)
[(I
-
q;
-
q,)m,
+
(1
+
q;
-
1
(17)
q,)q,]
q2
C
-
(I -
wq1~) [(I - q; - q1h
f (1 + q; - qr)q21
4r
.
(18)
1
where
A* = hqz~
- (1 + 2qd t(1 - q; - q~)m, + (1 + 7; -
in which N = Sratic
c33h3.
t@‘bqiN r)3
=
(1 + b)l
s33!2C,3,
7.j;
[(I =
-
qr; -
q)m2
q,)q,] +
(1
+
q;
-
q,)qz].
and qi = 5,,/2c’44.
s33/k.~
L’usc
For the static case we consider zj = 0, By making the medium isotropic and removing the initial stresses, we get the following static results: 2F $13
=
-
TZ
x (x2
f
2’ z2)“2
(x2
+
z2)3’2
and $33
=
2F
--
TZ
z” (x2
+
z2)2
*
The above results tally with the well-known static result obtained by Timoshe~o(flOJ, page 84).
NUMERICAL
RESULTS
The stresses as given in ( 17) and ( 18) have been calculated in case of ice, Following Hea~on[9], the elastic constants for ice at - 16°C are CII
= 1.36
c33
=
x
IO” dyn/cm2,
1.46 x IO” dyn/cm2,
cu = 0.32 x 10” dynlcm’, Cl2
=
0.67 x 10” dyn/cm2,
Cl3
=
0.52 x 10” dyn/cm2,
(19)
290
S DEY et 01. Table I. v = 100 mkcc
z=lm
rl3
51:
-%
i.: -0.2 0.0 02 05 0.0
08 0.3 -0.3 8;
is -0.5 -;;
0:8 0.0
0:o 0.0
(~336
-a
-1 112 -5.732 -7.219 -5.133 -t.i60 -6.057 -7.289
x x x x x x
10-2 lO-3 lO-3 lo-’ 10-2 10-z IO-’
-1.147
x lo-’
fs13iFRLt I529 1.779 -0.915 1.899 I%7 4.431 0.132
x x x x x x
IO-’ 10-j lo-4 IO-’ 10-3 10-l 10 ’
z=Sm 0.3
3.558 x IO-’ x -03.055 382 x lo-’
-0:2 :*:
-0.3 08
-0:s ii+:
-1.444 -2.224
x 1O-3 10-3
8::
0.3 0.0
-0.5 0.5
-2.321 -1.027
x IO-j IO-’
2.762 3.800 x lo-* IO-’
8.:
83
X:l
-1.211 -1.458
x lo-’
0.255 x lo-’ 0.887 lo-5
Table 2. v = 200 mkcc L=
lm
l-l3 0.2 0.5 -0.2
d ifi -0’3
?I 0.5 i:: -:::
(~33IFh-w
h3IFfLv1
-5.748x 10-9 -1.124 x lO-2 1.408 x 10-2
1865 x lo-’ I.544 x 10-3 I762 x 1O-2
-4.657 -5.147
x 10-3 10-j
1901 1974 x lo-” 10-Z
8;
x3
8::
0.0 0’8
i:8
-7.306 -1.262
x lo-* 10-3
0.147 1242 x IO-’ 10-4
0.2 0.5 -0.2
0.3
0.5 0.2 0.5
-1 149 x 10-3 -2.240 x 10-9 2.817 x lO-3
0.373 x 10-3 0.248 x lO-3 3.523 x lO-3
z=5m
ii:: 0:s 0.0
_g ii.! 0:8 0.0
-0.5 0.5 0.0 0.0
-1.029 -3.948 -2,524 -1.461
x lO-3 lo-” x lO-3 x 10-3
-0.893 3 147 x 10-4 lo-’ 2484 x 10-j 0.293 x IO-‘
P = 0.917 g-cm3. The numerical values of s&FR and s&F at points directly below the load are calculated for various prestressed parameters and for different z, The values of 11taken are 100 mlsec and 200 mlsec. The numerical results are in Tables 1 and 2. DISCUSSION It isobserved that A* occurs in the denominator of the expressions for the stress com~nents. Hence, in case A* = 0 there wifl be‘a hi stress con~ent~tion which will lead the material towards fracture. Thus A* = 0 will determine the velocity v of the moving load F at which the fracture will take place. The presence of ql, q~ and q; in the equation A* = 0 shows the influence of prestresses in the medium on the formation of fracture. From A* = 0 the values of rJm have been calculated for tee usmg the same elastic constants as given in (19) for different sets of prestress parameters -q,, r(3 and q$, and the following results have been obtained.
Dynamic
?I 0 0 -0.2 -02 02
response
of normal rll
rli 0 0 -0.3 -0 3 0.3
0 05 0 -05 -05
movmg
hd
291
+ 1:1GTp 0 9s84 I 2125 0 9114 0 6365 0.3816
The first result is for the maternal free from initial stresses, and in this case the velocity at which the fracture will take place is very close to the velocity of Rayleigh wave propagation in the transversely isotropic half-space. This was expected too. The result shows that if the tensile prestress in the x-direction is increased, the fracture occurs at higher velocity of the moving load, whereas if the prestress in the x-direction is compressible, then lower velocity can create fracture. Further, it has been observed during calculation that if the tensile initial stress in the z-direction is higher, then there is less possibility of fracture. From the numerical results, specifically, it may be said that the development of the incremental normal and the shearing stresses induced by the initial tensile stresses along the z-direction increases. However, the normalized initial tensile stress along the x-direction increases the development of shearing stress but decreases the normal stress. The presence of either the tensile initial stress or compressive initial stress in both directtons increases the shearing stress but decreases the normal stress. Also, it may be noted that the development of shearing stress in the case of initial compressive stress is comparatively less than the tensile case, but for normal stress the behaviour is just the opposite. REFERENCES I
: 4 5. 6. 7 8. 9. 10 II.
J Cole and J Huth, Stresses produced in a half plane by moving loads. / Appl. Merh. 25, 433-436 (1958). J W. Craggs. On two-drmenaonal waves in an clastlc half plane. Proc. Curnb. fIri/. Sot. 56,269 (1960) I. N. Sneddon. Stress produced by a pulse of pressure moving along the surface of a semi-mfimte solid. Rend. Circ . MU. Palcrtnn 2, 57-62 (1952). S MukherJee, Stresses produced by a load movmg over a rough boundary of a semi-mfinue transversely isotropic sohd. Pure Appl. Geophys. 72, 45-50 (1%9). J L Sackman, Uniformly moving load on a layered half-plane. J Engng Mech. DIV. Pror. AXE 87 EM,, 75-89 (l%l). I. W Miles, Response of a layered half-space to a movmg load. J. Appl Mech. 33,680-681 (1966). J D. Achenbach, S. P. Keshawa and G. Henmann, Moving load on a plate restmg on an elastm halfspace. 1. Appl. Mech. 24, 910-914 (1967). A. Mukhopadhyay, Stresses produced by a normal load moving over a transversely isotropic layer of se lymg on a rigid foundation. Pure Appl. Geophys. 60, 29 (1965). R. F. S. Hearmon, An Introduction IO Applied Anisotropic Ehticuy, p. 38. Oxford Umverslty Press, Oxford (l%l). S. Timoshenko, The Theory of Elasriciry, pp. 82-84. McGraw-Hill, New York (1934). M. A. Biot, Mechanics of Incremental Defirmarions. John Wiley, New York (l%S).