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Three-dimensional transient wave response in a cracked elastic solid
agog
Pergamon
Fracture ~e~h~nirs Vol. 48, No. 4, pp. 545-5-552,1994 Copyright @ 1994 Ekemr Science Ltd. Printed in Great Britain. All rights reserved 0013-7P44(P4)EOO22-9 0013-7944/94 $7.00 + 0.00
THREE-DIMENSIONAL TRANSIENT WAVE RESPONSE IN A CRACKED ELASTIC SOLID LI XIANG-PING and LIU CHUN-TU Institute of Mechanics, Academic Sinica. Beijing 100080, P.R.China Abstract-The three-dimensional transient wave response problem is presented for an infinite elastic medium weakened by a plane crack of infinite length and finite width. Tractions are applied suddenly to the crack, which simulates the case of impact loading. The integral transforms are utilized to reduce the problem to a standard Fredholm integral equation in the Laplace transform variable and sequentially invert the Laplace transforms of the stress components by numerical inversion method. The dynamic mode I stress intensity factors at the crack tip are obtained and some numerical results are presented in graphical form.
1. INTRODUCTION
importance in structural analysis is the transient response of a flaw to a time dependent stress field. In fracture m~hani~s, it is important to reveal the behavior of the dynamic stresses in the vicinity of the crack end. Dynamic response of cracks to elastic waves in the two-dimensional case has been investigated by many authors, including Matte [I], Sih and Embley [2], Achenbach and Gautesen [3], Zhang and Achenbach [4], and Chen Bao-Xing and Zhang Xiang Zhou [5]. From the published literature it is seen that three-dimensional phenomena have received relatively little attention in dynamic fracture mechanics, especially in transient wave response. In 1977, Achenbach and Gautesen investigated the three-dimensional steady-state elastodynamic response of an unbounded solid containing a semi-infinite crack [6]; Itou investigated the three-dimensional steady-state response of an infinite elastic medium weakened by a plane crack of infinite length and finite width [7]. Freund proposed a general procedure for determining the stress intensity factor histories for a class of three-dimensional elastodynamic crack problems, and extensions of the procedure are considered by other authors [8-IO]. In the present paper a three-dimensional elastodynamic transient problem is considered. The tractions are applied suddenly to the crack which simulates the case of impact loading. The integral transformation reduces the problem to that of solving a pair of dual integral equations. These equations are usually convertible to Fredholm equations of the second kind which can be solved nume~cally. Some numerical results are presented for particular load situations. OF CONSIDERABLE
2. FUNDAMENTAL Let x, , x2, x3 be Cartesian coordinates. is located along the x,-axis from - 1 to + crack, as shown in Fig. 1. The coordinates the half width of the crack. In vector notation the Navier equation elastic solid is written as
EQUATION
An open crack of infinite length but of finite width 1 and the x,-axis coincides with the center line of the are regarded as dimensionless quantities referring to governing the displacement vector u for an isotropic
ii=c:v(v*u)-c;vx(vxu),
(1)
where C, and C, are the dilatational and shear wave speeds, respectively. A standard approach when solving eq. (1) is to introduce the displacement potentials cp and tj through the Helmholtz decomposition of the displacement vector, i.e.: u=v~+vxY
V*Y=O. 545
(2)
546
LI XIANG-PING
and LIU CHUN-TU
Fig. 1. Geometry and coordinate system.
The scalar dilatational wave potential cpand the vector shear wave potential Y = (‘y,, Y,,, Yz) satisfy the uncoupled wave equations
The two potentials are coupled through the boundary conditions that characterize the problem to be described. The stress and displacement field due to sudden application of normal tractions of the crack
Fig. 2. Mode I stress intensity factor for the case where b = 0.1, P,, = 1.0.
547
3D transient wave response in cracked elastic solid
2.53
Fig. 3. Mode I stress intensity factor for the case where b = 0.8,
PO = 1.0.
surface may be found by solving the preceding field equations (3) subject to zero initial conditions, and the following boundary conditions at x2 = 0: c,,(x, 0, r, 0 = - PI&(x, z)f(t) u,(x, 0, z, t) = 0 CJ = 0 xy= crYZ
1x1< 1 for lx] > 1 for -coooxxco
(4)
for all times t and z. In addition, the condition on displacement at infinity is lim [u,, au, a,] = 0. x2+jJ-W
(5)
The parameter PO is a constant with the dimension of stress and g,(x, z) is restricted to functions that are even in x, i.e. gc( - x, r) = g,(x, r).
3. DUAL INTEGRAL
(6)
EQUATIONS
Transform techniques are now used to obtain the dual integral equations. First, a one-sided Laplace transform over time is introduced: m cp(x, y, z, t)e -” dt (7) cp*(x9 Y, r, s) = I0 then the dependence on x, z is suppressed by taking the two-sided Laplace transform: WV, Y, (9 3) =
cp* . eKm+tz)& &.
(8)
Ll XIANG-PING and LIU CHUN-TU
548
The shear wave potential may be treated in a similar manner. General solutions in the transformed domain, which are bounded as y + + co, can be written as @ =: A *e-“J’ Y = (B,, B,, B,)r *e+,
(9)
where u = &s/c:
+ r+ + t2)
B = J(S2lC3 + rt2 f r2).
(10)
The complex plane is cut in order for Re(g) 2 0 and Re@) 3 0. Thus the expressions for the transform ~spIa~ment components are u,=
-iqA
e-“y+ic
*Bre-@Y-@ .BZ*e-”
U,, = - uA evay- i(B, emfly+ iqB, eMBy - it . A eTay- iqB, e-fly+ @B, e-%Y* ,
u,= the transformed
stress components
(11)
are
oy,, = k * [(/?’ + q2 + <*)A emaY+ 2ie - flBxemBY - 2i@ - B, evBJj Dxy= p - [2iaA . q . e-‘y - 5 - qB, eeBY- i# - Bye-@’+ (/I2+ q2)euBJ’j o,,=p.[2icttA
.e-“-(~z+~2)B~e-8y+ig~*Bye-BY+T~B,e-3y]
fJxx= p . [(S’,K’z - 2#12+ 2t2)A e-‘Y + 2iq#I - B, edpy- 25 * q - Bye-@J. The transformation of the condition equations; we easily obtain:
V. Y = 0 and the boundary
(12)
conditions provide five
A = a-’ . (1 - 2f12* C~/S2)U_(~, 5) B, = - 2% . CZ/S2U_(q, 5) By = 0 B, = 2i@ZZ/S2U_(q, 5)
(13)
where
U-h
e>=
m
Co
SI
--oD-co
ii_(x, z, s)e’-‘”
dx dz.
(14)
The minus subscript is used to denote a function that is nonvanishing in the range /xl< 1, while it vanishes out of this range. Consequently, the transformed components of stress and displacement can be represented by a single unknown coefficient u - (Q {). Upon substitution of these results into the remaining boundary conditions, and taking the inverse Fourier transform, we find that 1 * -=“dq=O 2;; s _;ou-e
/xl> 1
where W?, 5) = (l/k2 + 2W + ?>I2 with the parameter k defined as
W + T2M?Y W(% 0
549
3D transient wave response in cracked eIastic solid
By asymptotic analysis, we know that
x1 - c:> ~+Wltl)=x~+O(l/~) kZ
wrt,C)
m+
asq+fao.
(17)
Noting that g,(x, <) is even of x, we have m
U_ cos(xq) dl = 0
x &1
Pa)
s0 ow ~
U-
cm(q)
dtt
(W
,
I
For the eq. (18b), we have
c w
Jo
$Y_cos(xq)dtj
+~o~($-&+_cos(xrj)d~
=$$F(S)
Odx < 1.
(19)
Now, setting
substitution of eq. (20) into eqs (18a) and (19), followed by an interchange integration, yields
d
-. dx
’ Mbit) _ p2)1/2 s 0 t x2
npoge dp
=
-~~Pq(~ic)~ow
G(tl)Jot~rl)cos(w)drt
+7
F(S)
of the order of
o
<
1,
& (21b)
where we have noted the identities
aJo(~s)Wxtt) dtl dp =
&
(22)
and *
Jo~tt)tcostxtt)+ i WvN drl =
2
$ TX{)!, 2
‘j2
Ixl>p
I4
(234 Wb)
The imaginary part of eq. (23) is substituted into eq. (22) to arrive at the left side of eq. (21b). The real part of eq. (23b) implies that eq. (21b) is automatically satisfied since x >, 1 and 0 < q < 1. If
f(x) = ;;
I
; (x2- q2)-“2qg(q)dv,
then the function g(z) can be expressed as g(z)=
$2 - x2)-“2j(x) dx s0
and we also have i
o~,,2_,2)-I/2
cosfkx) dx = Jo(kz).
s
Hence, by multiplying eq. (21b) by 2n-‘(~~ - xr)-“* and integrating x from 0 to t, we
qtr,t)+
(24)
LI XIANG-PING
550
and LIU CHUN-TU
where
(25) Equation (25) is a standard Fredholm integral equation of the second kind, which can be solved by numerical methods, whose kernel k(t, p) is symmetric in p and r. For rapid convergence of the infinite integral, define a function d(q) as d(q) = W~)/tt + H/W + E2),
(26)
where, in order that d(q) be of the order of q -6 for large q, H and E are chosen as
(27) If we set 5 = 0, we easily obtain
xHE2 = -&c~-6c:+zc:+
which is similar to the two-dimensional W, P) =
I),
result in M’s paper [2]. Thus
m d(V)- ------j H q2+E s(0
>
ttJo(~rl)Jo(~) dtl.
(28)
Assume that p < T, since
I and the symmetry relation
om~~~o(~~)~ot~~)
dl =
~~(~~)~o(~?~
(29)
k(r, P) = W, r); the expression for the kernel-in-&e-Fmdholm W, P) = where IO and k0 are the zero-order respectively.
~r,(&)ko(Ei:)
(30)
integral equation becomes m +
rld(rl>J&)
I0 modified Bessel functions
dtl,
(30
of the first and second kind,
3D ~nsien~ wave response in cracked elastic solid
Introducing
the non-dimensional
551
variable
eq. (24) becomes &r,5)+
‘pQ(p,<)k(r,p)dp =f r(r2-X2)-112gpdX. (32) s0 s0 In order to invert eq. (32), we approximate the function Q(r, 5) by a series of step functions N
(33) where o r’(,&i,&,) and P”=n/N n=0,1,2 f..., N. %= 1 1 rE(P,-l,Pn) The coefficient q,, is assumed to be the discrete value &, r), where t,=(2n-1)(2iv)-’
(n=1,2,...,N)
is the midpoint of the interval (p,_ , , p,,). Substitution of eq. (33) into eq. (32) yields the matrix equation: n=l,2,...,N
5 A,q,=R,, ##=I
Pnt A,=d,+
s Pm-1
pk(r,> p) dp.
4. THE DYNAMIC STRESS INTENSITY
(34)
FACTOR
Once the coefficient qn is known from eq. (34), the entire stress field is obtainable in the Laplace transform variable. Here the significant quantity to be calculated is the dynamic stress intensity factor. By asymptotic analysis, from eq. (12) we have 1
G CYY=
G
(1 - Ci) -&
’ pq dp
I0
m (1 - ~~)Jo@~)sin(x~) dq s0 (35)
therefore
k,k s)=
2
2(1 - Ci)p +,--n -F(S)
&
n&F(S) Xk2
som
s
O” o q(l, S)cos(&)d<
q(L t)cos(
(36)
Applying Laplace inverse transform (36) then yields k, (z, s)&’ . ds. s Br In this paper, numerical inversion of Laplace transform has been carried out by use of Jacobi polynomials. k,(z, t) is written in the following form: W,
t) = &
N
C,p~“~fl(2e-“’ - l),
k,(z,t)x~* d-=71
n=O
(37)
552
LI XIANG-PING
and LIU CHUN-TU
where P~“,~‘(x) is a Jacobi polynomial of degree n and C, can be determined from the following equations:
k
(k - l)(k - 2). . *(k -(m - 2))
c
~~,(k+B)(k+B+l)...(k+B+m)
=
@[(B
+
k)d]
m - C,
= a~[(/? + 1)6]
B+1
k
>
,
2
k = 1,
with /_land 6 being parameters used in the numerical computation. 5. NUMERICAL RESULTS AND CONCLUSIONS For a numerical calculation, the function g,(x, z) is assumed as g,(x,z)=
cosx 1 + (bz)Z’
where b is the parameter which governs the distribution of the applied load along the z-direction. f(t) is the Heaviside step function Z-Z(t).The only material parameter co = c,/c, is taken as 0.542. In numerical solution of the Fredholm equation, N = 20. The value of 4( 1, 5) can be obtained from a quadratic extrapolation of d(r, 5). Numerical inversion parameters p = 0.024, 6 = 0.05-0.3 and n = 5-8 terms are chosen. The semi-infinite integral is divided into two parts, then evaluated by 12-point Gaussian quadrature. Curves of the variation of the dynamic mode I stress intensity factor with time (T = c,t) and position z are plotted in Figs 2 and 3, when b = 0.1 and 0.8, respectively. PO is taken as 1.0. These curves show a general feature in three-dimensional graphical form. For small values of t, the dynamic stress intensity factor is zero and then increases rapidly up to a maximum at T = 2.03 (T = c,t), i.e when the Rayleigh wave is first observed at the crack edge. After reaching a maximum, the dynamic stress intensity factors begin to decrease and then oscillate around the corresponding steady-state values. REFERENCES
111A. W. Matte, Die Beugung elastischer Wellen an der Halbebene. 2. angew. Math. Mech. 33, l-10 (1953). PI G. C. Sih, G. T. Embley and R. S. Ravera, Impact response of a finite crack in plane extension. Int. J. Solids Structures
8, 977-993 (1972). [31 J. D. Achenbach and A. K. Gautesen, A ray-theory for elastodynamic stress-intensity factors. J. appl. Mech. 45, 803-806 (1978).
141 Ch. Zhang and J. D. Achenbach, Scattering by multiple crack configurations. J. appl. Mech. 55, 104410 (1988). [51 Chen Bao Xing and Zhang Xiang Zhou, Dynamic mode II stress intensity factors of an infinite cracked medium subjected to transient concentrated forces. Znt. J. Fracture 57, 1833198 (1992). WI J. D. Achenbach and A. K. Gautesen, Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D loading. J. appl. Mech. 44, 243-249 (1977). 171 S. Itou, Three-dimensional wave propagation in a cracked elastic solid. J. appl. Mech. 45, 807-811 (1978). PI L. B. Freund, The stress intensity factor due to three dimensional transient loading of the faces of a crack. J. Mech. Phys. Soliak 35, 61-72 (1987). PI J. C. Ramirez, The three dimensional stress intensity factor due to the motion of a load on the faces of a crack. Q. appl. Math. XLV, 361-376 (1987). VOI C. R. Champion, The stress intensity factor history for an advancing crack under three-dimensional loading. Znt. J. Solids Structures 24, 285-300 (1988). [111 J. D. Achenbach, A. K. Gautesen and H. McMaken, Rays Methodr for Waves m Elastic Solids. F’itman, Boston, MA (1982). (Received 16 June 1993)
Pergamon
Fracture ~e~h~nirs Vol. 48, No. 4, pp. 545-5-552,1994 Copyright @ 1994 Ekemr Science Ltd. Printed in Great Britain. All rights reserved 0013-7P44(P4)EOO22-9 0013-7944/94 $7.00 + 0.00
THREE-DIMENSIONAL TRANSIENT WAVE RESPONSE IN A CRACKED ELASTIC SOLID LI XIANG-PING and LIU CHUN-TU Institute of Mechanics, Academic Sinica. Beijing 100080, P.R.China Abstract-The three-dimensional transient wave response problem is presented for an infinite elastic medium weakened by a plane crack of infinite length and finite width. Tractions are applied suddenly to the crack, which simulates the case of impact loading. The integral transforms are utilized to reduce the problem to a standard Fredholm integral equation in the Laplace transform variable and sequentially invert the Laplace transforms of the stress components by numerical inversion method. The dynamic mode I stress intensity factors at the crack tip are obtained and some numerical results are presented in graphical form.
1. INTRODUCTION
importance in structural analysis is the transient response of a flaw to a time dependent stress field. In fracture m~hani~s, it is important to reveal the behavior of the dynamic stresses in the vicinity of the crack end. Dynamic response of cracks to elastic waves in the two-dimensional case has been investigated by many authors, including Matte [I], Sih and Embley [2], Achenbach and Gautesen [3], Zhang and Achenbach [4], and Chen Bao-Xing and Zhang Xiang Zhou [5]. From the published literature it is seen that three-dimensional phenomena have received relatively little attention in dynamic fracture mechanics, especially in transient wave response. In 1977, Achenbach and Gautesen investigated the three-dimensional steady-state elastodynamic response of an unbounded solid containing a semi-infinite crack [6]; Itou investigated the three-dimensional steady-state response of an infinite elastic medium weakened by a plane crack of infinite length and finite width [7]. Freund proposed a general procedure for determining the stress intensity factor histories for a class of three-dimensional elastodynamic crack problems, and extensions of the procedure are considered by other authors [8-IO]. In the present paper a three-dimensional elastodynamic transient problem is considered. The tractions are applied suddenly to the crack which simulates the case of impact loading. The integral transformation reduces the problem to that of solving a pair of dual integral equations. These equations are usually convertible to Fredholm equations of the second kind which can be solved nume~cally. Some numerical results are presented for particular load situations. OF CONSIDERABLE
2. FUNDAMENTAL Let x, , x2, x3 be Cartesian coordinates. is located along the x,-axis from - 1 to + crack, as shown in Fig. 1. The coordinates the half width of the crack. In vector notation the Navier equation elastic solid is written as
EQUATION
An open crack of infinite length but of finite width 1 and the x,-axis coincides with the center line of the are regarded as dimensionless quantities referring to governing the displacement vector u for an isotropic
ii=c:v(v*u)-c;vx(vxu),
(1)
where C, and C, are the dilatational and shear wave speeds, respectively. A standard approach when solving eq. (1) is to introduce the displacement potentials cp and tj through the Helmholtz decomposition of the displacement vector, i.e.: u=v~+vxY
V*Y=O. 545
(2)
546
LI XIANG-PING
and LIU CHUN-TU
Fig. 1. Geometry and coordinate system.
The scalar dilatational wave potential cpand the vector shear wave potential Y = (‘y,, Y,,, Yz) satisfy the uncoupled wave equations
The two potentials are coupled through the boundary conditions that characterize the problem to be described. The stress and displacement field due to sudden application of normal tractions of the crack
Fig. 2. Mode I stress intensity factor for the case where b = 0.1, P,, = 1.0.
547
3D transient wave response in cracked elastic solid
2.53
Fig. 3. Mode I stress intensity factor for the case where b = 0.8,
PO = 1.0.
surface may be found by solving the preceding field equations (3) subject to zero initial conditions, and the following boundary conditions at x2 = 0: c,,(x, 0, r, 0 = - PI&(x, z)f(t) u,(x, 0, z, t) = 0 CJ = 0 xy= crYZ
1x1< 1 for lx] > 1 for -coooxxco
(4)
for all times t and z. In addition, the condition on displacement at infinity is lim [u,, au, a,] = 0. x2+jJ-W
(5)
The parameter PO is a constant with the dimension of stress and g,(x, z) is restricted to functions that are even in x, i.e. gc( - x, r) = g,(x, r).
3. DUAL INTEGRAL
(6)
EQUATIONS
Transform techniques are now used to obtain the dual integral equations. First, a one-sided Laplace transform over time is introduced: m cp(x, y, z, t)e -” dt (7) cp*(x9 Y, r, s) = I0 then the dependence on x, z is suppressed by taking the two-sided Laplace transform: WV, Y, (9 3) =
cp* . eKm+tz)& &.
(8)
Ll XIANG-PING and LIU CHUN-TU
548
The shear wave potential may be treated in a similar manner. General solutions in the transformed domain, which are bounded as y + + co, can be written as @ =: A *e-“J’ Y = (B,, B,, B,)r *e+,
(9)
where u = &s/c:
+ r+ + t2)
B = J(S2lC3 + rt2 f r2).
(10)
The complex plane is cut in order for Re(g) 2 0 and Re@) 3 0. Thus the expressions for the transform ~spIa~ment components are u,=
-iqA
e-“y+ic
*Bre-@Y-@ .BZ*e-”
U,, = - uA evay- i(B, emfly+ iqB, eMBy - it . A eTay- iqB, e-fly+ @B, e-%Y* ,
u,= the transformed
stress components
(11)
are
oy,, = k * [(/?’ + q2 + <*)A emaY+ 2ie - flBxemBY - 2i@ - B, evBJj Dxy= p - [2iaA . q . e-‘y - 5 - qB, eeBY- i# - Bye-@’+ (/I2+ q2)euBJ’j o,,=p.[2icttA
.e-“-(~z+~2)B~e-8y+ig~*Bye-BY+T~B,e-3y]
fJxx= p . [(S’,K’z - 2#12+ 2t2)A e-‘Y + 2iq#I - B, edpy- 25 * q - Bye-@J. The transformation of the condition equations; we easily obtain:
V. Y = 0 and the boundary
(12)
conditions provide five
A = a-’ . (1 - 2f12* C~/S2)U_(~, 5) B, = - 2% . CZ/S2U_(q, 5) By = 0 B, = 2i@ZZ/S2U_(q, 5)
(13)
where
U-h
e>=
m
Co
SI
--oD-co
ii_(x, z, s)e’-‘”
dx dz.
(14)
The minus subscript is used to denote a function that is nonvanishing in the range /xl< 1, while it vanishes out of this range. Consequently, the transformed components of stress and displacement can be represented by a single unknown coefficient u - (Q {). Upon substitution of these results into the remaining boundary conditions, and taking the inverse Fourier transform, we find that 1 * -=“dq=O 2;; s _;ou-e
/xl> 1
where W?, 5) = (l/k2 + 2W + ?>I2 with the parameter k defined as
W + T2M?Y W(% 0
549
3D transient wave response in cracked eIastic solid
By asymptotic analysis, we know that
x1 - c:> ~+Wltl)=x~+O(l/~) kZ
wrt,C)
m+
asq+fao.
(17)
Noting that g,(x, <) is even of x, we have m
U_ cos(xq) dl = 0
x &1
Pa)
s0 ow ~
U-
cm(q)
dtt
(W
,
I
For the eq. (18b), we have
c w
Jo
$Y_cos(xq)dtj
+~o~($-&+_cos(xrj)d~
=$$F(S)
Odx < 1.
(19)
Now, setting
substitution of eq. (20) into eqs (18a) and (19), followed by an interchange integration, yields
d
-. dx
’ Mbit) _ p2)1/2 s 0 t x2
npoge dp
=
-~~Pq(~ic)~ow
G(tl)Jot~rl)cos(w)drt
+7
F(S)
of the order of
o
<
1,
& (21b)
where we have noted the identities
aJo(~s)Wxtt) dtl dp =
&
(22)
and *
Jo~tt)tcostxtt)+ i WvN drl =
2
$ TX{)!, 2
‘j2
Ixl>p
I4
(234 Wb)
The imaginary part of eq. (23) is substituted into eq. (22) to arrive at the left side of eq. (21b). The real part of eq. (23b) implies that eq. (21b) is automatically satisfied since x >, 1 and 0 < q < 1. If
f(x) = ;;
I
; (x2- q2)-“2qg(q)dv,
then the function g(z) can be expressed as g(z)=
$2 - x2)-“2j(x) dx s0
and we also have i
o~,,2_,2)-I/2
cosfkx) dx = Jo(kz).
s
Hence, by multiplying eq. (21b) by 2n-‘(~~ - xr)-“* and integrating x from 0 to t, we
qtr,t)+
(24)
LI XIANG-PING
550
and LIU CHUN-TU
where
(25) Equation (25) is a standard Fredholm integral equation of the second kind, which can be solved by numerical methods, whose kernel k(t, p) is symmetric in p and r. For rapid convergence of the infinite integral, define a function d(q) as d(q) = W~)/tt + H/W + E2),
(26)
where, in order that d(q) be of the order of q -6 for large q, H and E are chosen as
(27) If we set 5 = 0, we easily obtain
xHE2 = -&c~-6c:+zc:+
which is similar to the two-dimensional W, P) =
I),
result in M’s paper [2]. Thus
m d(V)- ------j H q2+E s(0
>
ttJo(~rl)Jo(~) dtl.
(28)
Assume that p < T, since
I and the symmetry relation
om~~~o(~~)~ot~~)
dl =
~~(~~)~o(~?~
(29)
k(r, P) = W, r); the expression for the kernel-in-&e-Fmdholm W, P) = where IO and k0 are the zero-order respectively.
~r,(&)ko(Ei:)
(30)
integral equation becomes m +
rld(rl>J&)
I0 modified Bessel functions
dtl,
(30
of the first and second kind,
3D ~nsien~ wave response in cracked elastic solid
Introducing
the non-dimensional
551
variable
eq. (24) becomes &r,5)+
‘pQ(p,<)k(r,p)dp =f r(r2-X2)-112gpdX. (32) s0 s0 In order to invert eq. (32), we approximate the function Q(r, 5) by a series of step functions N
(33) where o r’(,&i,&,) and P”=n/N n=0,1,2 f..., N. %= 1 1 rE(P,-l,Pn) The coefficient q,, is assumed to be the discrete value &, r), where t,=(2n-1)(2iv)-’
(n=1,2,...,N)
is the midpoint of the interval (p,_ , , p,,). Substitution of eq. (33) into eq. (32) yields the matrix equation: n=l,2,...,N
5 A,q,=R,, ##=I
Pnt A,=d,+
s Pm-1
pk(r,> p) dp.
4. THE DYNAMIC STRESS INTENSITY
(34)
FACTOR
Once the coefficient qn is known from eq. (34), the entire stress field is obtainable in the Laplace transform variable. Here the significant quantity to be calculated is the dynamic stress intensity factor. By asymptotic analysis, from eq. (12) we have 1
G CYY=
G
(1 - Ci) -&
’ pq dp
I0
m (1 - ~~)Jo@~)sin(x~) dq s0 (35)
therefore
k,k s)=
2
2(1 - Ci)p +,--n -F(S)
&
n&F(S) Xk2
som
s
O” o q(l, S)cos(&)d<
q(L t)cos(
(36)
Applying Laplace inverse transform (36) then yields k, (z, s)&’ . ds. s Br In this paper, numerical inversion of Laplace transform has been carried out by use of Jacobi polynomials. k,(z, t) is written in the following form: W,
t) = &
N
C,p~“~fl(2e-“’ - l),
k,(z,t)x~* d-=71
n=O
(37)
552
LI XIANG-PING
and LIU CHUN-TU
where P~“,~‘(x) is a Jacobi polynomial of degree n and C, can be determined from the following equations:
k
(k - l)(k - 2). . *(k -(m - 2))
c
~~,(k+B)(k+B+l)...(k+B+m)
=
@[(B
+
k)d]
m - C,
= a~[(/? + 1)6]
B+1
k
>
,
2
k = 1,
with /_land 6 being parameters used in the numerical computation. 5. NUMERICAL RESULTS AND CONCLUSIONS For a numerical calculation, the function g,(x, z) is assumed as g,(x,z)=
cosx 1 + (bz)Z’
where b is the parameter which governs the distribution of the applied load along the z-direction. f(t) is the Heaviside step function Z-Z(t).The only material parameter co = c,/c, is taken as 0.542. In numerical solution of the Fredholm equation, N = 20. The value of 4( 1, 5) can be obtained from a quadratic extrapolation of d(r, 5). Numerical inversion parameters p = 0.024, 6 = 0.05-0.3 and n = 5-8 terms are chosen. The semi-infinite integral is divided into two parts, then evaluated by 12-point Gaussian quadrature. Curves of the variation of the dynamic mode I stress intensity factor with time (T = c,t) and position z are plotted in Figs 2 and 3, when b = 0.1 and 0.8, respectively. PO is taken as 1.0. These curves show a general feature in three-dimensional graphical form. For small values of t, the dynamic stress intensity factor is zero and then increases rapidly up to a maximum at T = 2.03 (T = c,t), i.e when the Rayleigh wave is first observed at the crack edge. After reaching a maximum, the dynamic stress intensity factors begin to decrease and then oscillate around the corresponding steady-state values. REFERENCES
111A. W. Matte, Die Beugung elastischer Wellen an der Halbebene. 2. angew. Math. Mech. 33, l-10 (1953). PI G. C. Sih, G. T. Embley and R. S. Ravera, Impact response of a finite crack in plane extension. Int. J. Solids Structures
8, 977-993 (1972). [31 J. D. Achenbach and A. K. Gautesen, A ray-theory for elastodynamic stress-intensity factors. J. appl. Mech. 45, 803-806 (1978).
141 Ch. Zhang and J. D. Achenbach, Scattering by multiple crack configurations. J. appl. Mech. 55, 104410 (1988). [51 Chen Bao Xing and Zhang Xiang Zhou, Dynamic mode II stress intensity factors of an infinite cracked medium subjected to transient concentrated forces. Znt. J. Fracture 57, 1833198 (1992). WI J. D. Achenbach and A. K. Gautesen, Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D loading. J. appl. Mech. 44, 243-249 (1977). 171 S. Itou, Three-dimensional wave propagation in a cracked elastic solid. J. appl. Mech. 45, 807-811 (1978). PI L. B. Freund, The stress intensity factor due to three dimensional transient loading of the faces of a crack. J. Mech. Phys. Soliak 35, 61-72 (1987). PI J. C. Ramirez, The three dimensional stress intensity factor due to the motion of a load on the faces of a crack. Q. appl. Math. XLV, 361-376 (1987). VOI C. R. Champion, The stress intensity factor history for an advancing crack under three-dimensional loading. Znt. J. Solids Structures 24, 285-300 (1988). [111 J. D. Achenbach, A. K. Gautesen and H. McMaken, Rays Methodr for Waves m Elastic Solids. F’itman, Boston, MA (1982). (Received 16 June 1993)