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Line-spring model for surface cracks in a reissner plate
hf. 1. Engrrg Sri. Vol. 19, pp. 1331~1340, 1981 Prinkd in Great Briiain.
LINE-SPRING MODEL FOR SURFACE CRACKS IN A REISSNER PLATE? F. DELALEand F. ERJJOGAN LehighUniversity, Bethlehem, PA 18015,U.S.A. Abstract-In this paper the ljne-spring model developed by Rice and Levy for a surface crack in elastic plates is reconsidered. The problem is formulate by using Reissner’s plate bending tbecrry. For the plane strain problem of a strip containing an edge crack and subjected ta tension and bending new expressions far stress intensity factors are used which are valid up to a depth-to-thickness ratio of 0.8. The stress intensity factors for a semi-elliptic and a rectangular crack are calculated. Considering the simplicity of the technique and the severity of the underlying assumptions, the results compare rather well with the existing finite element solutions.
1. INTRODUCTION IN THIS paper the line-spring model developed in[lJ for an approximate solution of a plate containing a part-through crack is reconsidered. The main purpose of the study is to see if one could improve upon the results obtained in[l] by taking into account the transverse shear effects in formulating the bending of the cracked plate. At the present time, to obtain a reasonably accurate solution of the surface crack problem, rather extensive numerical work is needed. Such solutions are obtained by using the finite element methodI2-41, the alternating method[5,6], or the boundary integral equation method[7]. If the technique based on the line-spring model can be shown to give results within an acceptable degree of accuracy, not only would it save large amounts of computer time in solving the part-through crack problems, as it was shown in[8,9], it could also be used to study certain aspects of the plasticity prob1em.S In this study the bending part of the problem is formulated by using Reissner’s piate theory fl 1,12J. Recent results given inff3f are used to incorporate the edge crack solution of a strip subjected to tension or bending into the fomulation of the problem. The formulation is somewhat different from that given in[lJ in that the derivatives of the crack surface displacement and rotation are used in deriving the integral equations of the problem,
2. THE COMPLIANCE
COEFFICIENTS
The desaiption of the line-spring model has been given in[l,8]. In a plate containing a part-through surface crack under membrane loading and/or bending the stresses in the net ligament would have a constraining effect of the crack surface displacements (Fig. 1). Then the basic idea underlying the model is that by representing the stresses in the net ligament by a membrane load N and a bending moment M and the crack surface displacements due to N and M by an opening 6 and a rotation 8, all referred to the midplane of the plate and distributed along the length of the crack, and by assuming that the relationship between IN, M) and (S,8) may be approximated by that of the plane strain results obtained from the solution of an edge-cracked strip, from the boundary and continuity conditions of the plate expressed for the plate containing the crack one could obtain a pair of integral equations for the functions N and M or 6 and 8.
tThis work was supported by the Department of Transportation under the contract DOT-RC-82007,by NSF under the
GrantENG.78-09737, and by NASA-Langley under the Grant NGR39-007-011. tSee also[lOJ for the application of the line-spring concept to cylindrical shells containing a part-through crack with a faiiy-yietded net ligament. 1331
F. DELALE and F. ERDOGAN
1332
A%(Y)
/1
1
NC0
KM,
/
o7’IV Lo
_% x,(x)
h
(of
Fig. 1. Geometry of the plate with a part-through surface crack.
Referring to[l] and Fig. l(b), for a symmetrically loaded infinite strip having an edge crack let h8 = A,p + Atbm,
(1)
(2) where (3) Also let the stress intensity factor for the strip be (4) where g, and gb are functions of L/h. From the expressions of the rate of change of potential energy in terms of crack closure energy and change of compliance as given by [I, 143
one may write 6 =
*cl- v2jh (a E
tt
@bt@
(I;
+
+
a
tbm)
abbm)
7
(6) (7)
Line-spring model for surface cracks in a Reissner plate
1333
1 L gigi dL, (i, j = b, t). “G =i; I 0
(8)
where
The functions g, and gb are obtained from the recent results given in[13] which are valid for 0 < L/h 5 0.8 and may be expressed as gc(t) = ~(1r~)(1.1216+6.5200~2- 12.3877~+89.0554p - 188.6080r8+ 207.3870r’”- 32.0524t1*),
(9)
g,,([) = ~(7r~)(1.1202- 1.88725+ 18.0143&87.3851t3 + 241.91249 - 319.9402r5+ 168.0105&
(10)
where 5 = L(x,)/h. Substituting from (9) and (10) into (8) the compliance coefficients are found to be
where C$‘) are given in Table 1. 2. THE INTEGRAL EQUATIONS
an “infinite” plate containing a part-through surface crack of length 2a and depth L(x2) be subjected to a uniform membrane load and a bending moment away from the crack region given by (Fig. l(a)) Let
N,, = Nm, M,,= Mm.
(14)
Table 1. Coefficients CC,)in eqns (1l)-(13) -r
n
C$
0
1.9761
1.9735
1:9710
1
il.4870
-2.2166
-4.4277
2
7.7086
21.6051
34.4952
3 4 5
15.0143
-69.3133
-165.7321
280.1207
196.3000
626.3926
-1099.7200
-406.2608
-2144.4651
6 7
3418.9795
644.9350
7043.4169
-7686.9237
-408.9569
-19003.2199
a
12794.1279
-159.6927
37853.3028
9
-13185.0403
-988.9879
-52595.4681
10
7868.2682
4266.5487
48079.2948
11
-1740.2463
-2997.1408
-25980.1559
12
124.1360
-6050.7849
6334.2425
13
8855.3615
14
3515.4345
15
-11744.1116
16
4727.9784
17
1695.6087
ia
-845.8958
1334
F. DELALE and F. ER~~GAN
Referring, for example to[l5] for the general formulation, the basic equations of the pIate problem may be expressed as follows
v4#J= 0,
(15)
v4w= 0,
(16)
uv2J,-$-w=o,
(17)
(21) (22)
(23) (24)
The dimensionless quantities which appear in (IS)-(25) are defined in Appendix A in terms of the standard variables, constants and field parameters, Assuming that xl = 0 and x2 = 0 are planes of symmetry with respect to loading and geometry and that the problem has been reduced to a ~ert~rbatiun problem in which the crack surface stress and moment resultants are the only nonzero external loads, by expressing
& j-1fdx, a) epiay da, w(x, y) = $ -1 f&.x, a) eeioy dar, I 44~ y) =
0(r, y ) = & j_L f&x, (2)e -iay da, $(x, Y)= & f__If4(x, a) Py
da,
f26a-d)
it may be shown that[lS]
(27a-d)
where x > 0
Line-spring model for surface cracks in a Reissner plate
1335
and the unknown functions A,, . . . , A5 are to be determined from the boundary conditions at x =0+ . From (19), (21), (23) and (24) the relevant components of stress, moment, and transverse shear resultants may be expressed as
Mm= 2
_/;((1- 4d(244 - x)Ada) - &(a)1
+ Z(ajA,(c~)]eVialxeWiyada
arA5(o)e+ esiyada,
Mxy= - y( 1 - Y)& \;
(2%
a[(x]~~j- 2/oz2- l)A,(a).
+ la\Aj(cu)]e-lal’e-@ da -y(l-
~)‘~~_~((l’+?)AS(~)e-*
-z (I- Y)$ I_=aAS *
Nxy =
(30)
; a2A4(a)e-‘alxemiya da I r
v, = -;
N,., = -
emiYada,
&
e-” eeiyada,
(31)
a’[A,(a) + xA,(a)] ewiatx eeiyada >
T&/_= a{- In/A,(a) t %
(1 - xlaI)A2(a)) ewinlxeciyada,
h -! = 12(1- r+2’
(32) (33)
(34)
For the symmetric problem under consideration, the boundary conditions may be expressed as Nxy(O, Y) = 0, W,(O, Y) = 0, V,(O, y) = 0, --CD < y &W,y)=~(-c~+~4,
-
(35a-c)
l
40, Y) = 0, lYI > 1,
1 M,(O + , y) = 6~ ( - mm + ml, -l
MO, Y) = 0, IYI > 1
where
U%= N,lh, m, = 6MJh*,
(38)
(T(Y)and m(y) are given in terms of the net ligament stress and moment resultants N and M by (3), i.e. @(y) = N(x2)lh = N(ay)lh, m(y) = 6M(x2)/h2 = 6M(ay)/@
(39)
and the remaining dimensionless quantities are defined in Appendix A. Three of the unknown functions A ,, . . . , A5 may be eliminated by using the homogeneous conditions (35) and the mixed boundary conditions (36) and (37) determine the remaining two. Defining now G&(0+
9Y)= G,(Y),;
u(O+, y)= G*(y),
(4Oa,b)
I336
F. DELALE
and F. ERDOGAN
after some relatively straightforward manipulations (36) and (37) may be reduced to[15] ---
4K(1-V) 1 l+v (t-y)3
---~~ +&&K(PI’-yl)]
GW=$,lukl:
(41) (42)
,8 = [2/K(1 - Y)]1’2,
(43)
where Kz is the modified Bessel function of the second kind. Noting that the crack opening displacement S and the rotation 8 are related to G, and G2 by 8=2p,(ot,y)=2
’ G,(t)dt, I -1
6=2au(Ot,y)=2a
(44)
1’ G,(t)dt,
(45)
-I
substituting from (6) (7), (44) and (45) into (41) and (42) we obtain
ybb(Y)
3tv 1 ’ G,(t)dt -$$$ j-l [----l+vt-y -,
4K(1-V) 1 ltv (t-y)3
+&,++(Blt-Y])]
G,(r)dt+n,(y)/’
MY) /yG,(f)df+y,(~)j-y -1
-1
-1
G,(t)dt=~,lyl
G,(t)dt~~_~~d’=~,lyl
(46) (47)
wheret 4~)
= E(yrr(yb
m(Y)
=
+ rrd~)Pxl, 6Ehh’)u + Ybb(y)&l,
Ytt= &y?
Ybb
=
1 36(1 _
-
&.
v2)
Ma, b)
2,
1 yrb-6(1-;ij~‘Yb,=-6h(llIw2)~~’ b
=
(Y&bb
(49)
The integral eqns (46) and (47) must be solved under the following single-valuedness conditions
I
I
-1
G,(t) dt = 0,
I
1
G,(t) dt = 0.
-1
(5% b)
From the following asymptotic behavior of K,(z) for small values of z
it may be shown that (46) has only a Cauchy type singularity. Hence, the solution of the system of singular integral equations is quite straightforward and may be obtained by using the numerical method described, for example, in[16].
TNotethat%(f) = Qj(UxJh) = aij(QOY)lh) = a,iU(y)lh), andhence
(i, i) = (t. b).
once I(y) is specified, yij are functions of y only,
1337
Line-spring model for surface cracks in a Reissner plate 4. SOLUTION AND RESULTS
The solution of the integral eqns (46) and (47) is of the following form
G(t) =
(1
F1(t) , G*(f) =(lf$;l,2, _
tz)l
W&b)
2,
and conditions (50) and symmetry considerations imply that F,(r) = - Fi( - t), (i = 1,2). After solving for F, and F2, the stress and moment resultants g(y) and m(y) may be determined from (48) by using (44), (45) and (49). The stress intensity factor K(x,) = K(y) along the crack front is then determined from (4) by using (9) and (10). It should be noted that the results of the bending problem are dependent on the Poisson’s ratio V.In the examples considered in this paper, u is assumed to be 0.3. The numerical results are obtained for two different crack profiles, namely a semi-elliptic and an idealized rectangular cut with various length to depth ratios al&,, Lo being the crack depth at x2 = 0. For these two cases the function I(y) is given by L(x*) =
L,V/(1 - (x*/a)*) = I(y) = L,V\/(1 -
y2),
(53)
for the semi-elliptic crack, and L(x*)=LqJ=I(y). -l
(54)
for the rectangular crack. Figures 2 and 3 show the stress intensity factor at the deepest penetration point of a semi-elliptic surface crack in an infinite plate under uniform tension N, and bending moment M,, respectively. In these figures as well as in Figs. 4 and 5 the stress intensity factor is normalized with respect to K, which is the corresponding value for an edge-cracked strip under plane strain conditions with the same Lo/h ratio (Fig. 1). Figures 4 and 5 show the results for a “rectangular” surface crack. It may be seen that the K values for the rectangular crack are consistently higher than the corresponding values for the semi-elliptic crack. Figures 6-8 show the comparison of the results obtained from the line-spring model with that found from the finite element solution& 3,171 for a plate containing a semi-elliptic surface crack. For crack depth (I;,,) to half crack length (a) ratios of 0.2 and 0.6 the comparison of the
0
0.25
I
4
16
64
a/h
Fig. 2. The stress intensity factor at the maximumpenetration point of a semi-elliptic surface crack inan infinite plate under uniform tension (v = 0.3).
0
0.25
Fig. 3.
I
a/h
4
64
0.25
I
Fig. 4.
I
I a/h
4
I
I 16
I
surface crack in an infinite plate under
surface crack in an
Fig. 5. The stress intensity factor at the midpoint of a rectangular surface crack in an infinite plate under pure bending (u = 0.3).
Fig. 4. The stress intensity factor at the midpoint of a rectangular uniform tension (v = 0.3).
I 64
Fig. 3. The stress intensity factor at the maximum penetration point of a semi-elliptic infinite plate under pure bending (u = 0.3).
16
0.25
Fig. 5.
o/h
4
I
I 16
I
I 64
Line-spring model for surface cracks in a Reissner plate.
0 _:
(D
d
w
d
t
B OJ
t d
N
0
d Il.
a
0
1339
1340
F. DELALE and F. ERDOGAN
stress intensity factors at the deepest penetration point (x2 = 0 = y) of the crack is shown in Fig. 6. In Figs. 6-8 the normalized stress intensity factor F is defined by[Z, 3,171
where S = (T, = N,/h for tension, S = m3 = 6M,/h* for bending and Q represents the complete etliptic integral of the second kind and may be approximated by [2,3]
For L&a = 0.2, Figs. 7 and 8 show the comparison of the stress intensity factors along the crack front. From Fig. 6 as well as from Figs. 7 and 8 it may be observed that the results obtained from the line-spring model are quite acceptable.
REFERENCES [i] J. R. RICE and N. LEVY, J. Appi. Mech. 39, 185(1972). ]2] I. S. RAJU and J. C. NEWMAN Jr., J. Eugug Frucl. Me& 11,817 (1979). [3] J. C. NEWMAN Jr., A Relies and Assessment of the S&Tess-InfeusifyFactors for Surface Cracks. NASA, Technieai Memorandum 78803(1978). [4] S. N. ATLURI, K. KATHIRESAN, A. S. KOBAYASHI and M. NAKAGAKI, In Proc. 3rd Int. C&f. Pressure Vessel Technology, Part III, pp. 527-533.ASME, New York (1977). [5] F. W. SMITH and D. R. SORENSEN, Iat. J. Frucf. Mech. 12,47 (1976). [6] R. C. SHAH and A. S. KOBAYASHI, In The Surface Crack: Physical Problems and Computational Solutions (Edited by J. L. Swedlow), pp. 79-124 (1972). [7] J. HELIOT, R. C. LABBENS and A. PELLISIER-TANON, Fracture Mechanics ASTM, STP 677, pp. 341-364 (1979). [S] 1. R. RICE, In The Surface Crack: Physical Problems and Computafional Solutions (Edited by J. L. Swedlow), pp. 171-185,ASME, New York (1972). [P] D. M. PARKS, The Inelastic Line-Spring: Estimates of Elastic-Plastic Fracture Mechanics Parameters for SurfaceCracked Plates and Shells. Paper 8O-C2/PVP-109,ASME (1980). IO] F. ERDOGAN and hf. RATWANI, Iut. 1. Fract. Mech. 8,413 (1972). 111E. REISSNER, 3. Awl. Mech. 12, Trans. ASME PD.A69-A77 (1945). 12] E. REISSNER, Q. i &~l. Math. 5.55 (1948). . ‘131A. C. KAYA and F. ERD~AN. iut. J. &act. 16. 17111980). j14] G. R. IRWIN, StrucfuratMe~h~~~~s (Edited by J. N. Goddier and N. J. Hoff), p, 557. Pergamon Press, Oxford (1960). IS] F. DELALE and F. ERDOGAN, Q. J. Appl. Moth. 37,239 (1979). ‘161F. ERDOGAN, Mechanics Today, (Edited by S. Nemat-Nasser), Vol. 4, pp. i-86. Pergamon Press Oxford (1978). 171J. C. NEWMAN Jr. and I. S. RAJU, Analysis of Surface Cracks in Finite Plates Under Tension or Bending Loads. NASA Technical Paper No. 1578(1979).
(Received 21 Nouember 1980) APPENDIX x = x,/a, y = x,/a, z = x,/a, u = &a, 0 = &a, w = &a, e=&
.a=a,>s,=P%
o, = o, ii& of,. = Q&E>uxY= o,JE, KII = 2, My8= $,
(a, 8) =(x, yf,(i.j) = (1.2).
V, = VJhB, V, = V,/hB, &--!!___ K = 5, 62(l+u)’
A4= 12(1- v2)a2/h2.
(AU 642) (A3) (A4) (AS) (A6) (A7)
LINE-SPRING MODEL FOR SURFACE CRACKS IN A REISSNER PLATE? F. DELALEand F. ERJJOGAN LehighUniversity, Bethlehem, PA 18015,U.S.A. Abstract-In this paper the ljne-spring model developed by Rice and Levy for a surface crack in elastic plates is reconsidered. The problem is formulate by using Reissner’s plate bending tbecrry. For the plane strain problem of a strip containing an edge crack and subjected ta tension and bending new expressions far stress intensity factors are used which are valid up to a depth-to-thickness ratio of 0.8. The stress intensity factors for a semi-elliptic and a rectangular crack are calculated. Considering the simplicity of the technique and the severity of the underlying assumptions, the results compare rather well with the existing finite element solutions.
1. INTRODUCTION IN THIS paper the line-spring model developed in[lJ for an approximate solution of a plate containing a part-through crack is reconsidered. The main purpose of the study is to see if one could improve upon the results obtained in[l] by taking into account the transverse shear effects in formulating the bending of the cracked plate. At the present time, to obtain a reasonably accurate solution of the surface crack problem, rather extensive numerical work is needed. Such solutions are obtained by using the finite element methodI2-41, the alternating method[5,6], or the boundary integral equation method[7]. If the technique based on the line-spring model can be shown to give results within an acceptable degree of accuracy, not only would it save large amounts of computer time in solving the part-through crack problems, as it was shown in[8,9], it could also be used to study certain aspects of the plasticity prob1em.S In this study the bending part of the problem is formulated by using Reissner’s piate theory fl 1,12J. Recent results given inff3f are used to incorporate the edge crack solution of a strip subjected to tension or bending into the fomulation of the problem. The formulation is somewhat different from that given in[lJ in that the derivatives of the crack surface displacement and rotation are used in deriving the integral equations of the problem,
2. THE COMPLIANCE
COEFFICIENTS
The desaiption of the line-spring model has been given in[l,8]. In a plate containing a part-through surface crack under membrane loading and/or bending the stresses in the net ligament would have a constraining effect of the crack surface displacements (Fig. 1). Then the basic idea underlying the model is that by representing the stresses in the net ligament by a membrane load N and a bending moment M and the crack surface displacements due to N and M by an opening 6 and a rotation 8, all referred to the midplane of the plate and distributed along the length of the crack, and by assuming that the relationship between IN, M) and (S,8) may be approximated by that of the plane strain results obtained from the solution of an edge-cracked strip, from the boundary and continuity conditions of the plate expressed for the plate containing the crack one could obtain a pair of integral equations for the functions N and M or 6 and 8.
tThis work was supported by the Department of Transportation under the contract DOT-RC-82007,by NSF under the
GrantENG.78-09737, and by NASA-Langley under the Grant NGR39-007-011. tSee also[lOJ for the application of the line-spring concept to cylindrical shells containing a part-through crack with a faiiy-yietded net ligament. 1331
F. DELALE and F. ERDOGAN
1332
A%(Y)
/1
1
NC0
KM,
/
o7’IV Lo
_% x,(x)
h
(of
Fig. 1. Geometry of the plate with a part-through surface crack.
Referring to[l] and Fig. l(b), for a symmetrically loaded infinite strip having an edge crack let h8 = A,p + Atbm,
(1)
(2) where (3) Also let the stress intensity factor for the strip be (4) where g, and gb are functions of L/h. From the expressions of the rate of change of potential energy in terms of crack closure energy and change of compliance as given by [I, 143
one may write 6 =
*cl- v2jh (a E
tt
@bt@
(I;
+
+
a
tbm)
abbm)
7
(6) (7)
Line-spring model for surface cracks in a Reissner plate
1333
1 L gigi dL, (i, j = b, t). “G =i; I 0
(8)
where
The functions g, and gb are obtained from the recent results given in[13] which are valid for 0 < L/h 5 0.8 and may be expressed as gc(t) = ~(1r~)(1.1216+6.5200~2- 12.3877~+89.0554p - 188.6080r8+ 207.3870r’”- 32.0524t1*),
(9)
g,,([) = ~(7r~)(1.1202- 1.88725+ 18.0143&87.3851t3 + 241.91249 - 319.9402r5+ 168.0105&
(10)
where 5 = L(x,)/h. Substituting from (9) and (10) into (8) the compliance coefficients are found to be
where C$‘) are given in Table 1. 2. THE INTEGRAL EQUATIONS
an “infinite” plate containing a part-through surface crack of length 2a and depth L(x2) be subjected to a uniform membrane load and a bending moment away from the crack region given by (Fig. l(a)) Let
N,, = Nm, M,,= Mm.
(14)
Table 1. Coefficients CC,)in eqns (1l)-(13) -r
n
C$
0
1.9761
1.9735
1:9710
1
il.4870
-2.2166
-4.4277
2
7.7086
21.6051
34.4952
3 4 5
15.0143
-69.3133
-165.7321
280.1207
196.3000
626.3926
-1099.7200
-406.2608
-2144.4651
6 7
3418.9795
644.9350
7043.4169
-7686.9237
-408.9569
-19003.2199
a
12794.1279
-159.6927
37853.3028
9
-13185.0403
-988.9879
-52595.4681
10
7868.2682
4266.5487
48079.2948
11
-1740.2463
-2997.1408
-25980.1559
12
124.1360
-6050.7849
6334.2425
13
8855.3615
14
3515.4345
15
-11744.1116
16
4727.9784
17
1695.6087
ia
-845.8958
1334
F. DELALE and F. ER~~GAN
Referring, for example to[l5] for the general formulation, the basic equations of the pIate problem may be expressed as follows
v4#J= 0,
(15)
v4w= 0,
(16)
uv2J,-$-w=o,
(17)
(21) (22)
(23) (24)
The dimensionless quantities which appear in (IS)-(25) are defined in Appendix A in terms of the standard variables, constants and field parameters, Assuming that xl = 0 and x2 = 0 are planes of symmetry with respect to loading and geometry and that the problem has been reduced to a ~ert~rbatiun problem in which the crack surface stress and moment resultants are the only nonzero external loads, by expressing
& j-1fdx, a) epiay da, w(x, y) = $ -1 f&.x, a) eeioy dar, I 44~ y) =
0(r, y ) = & j_L f&x, (2)e -iay da, $(x, Y)= & f__If4(x, a) Py
da,
f26a-d)
it may be shown that[lS]
(27a-d)
where x > 0
Line-spring model for surface cracks in a Reissner plate
1335
and the unknown functions A,, . . . , A5 are to be determined from the boundary conditions at x =0+ . From (19), (21), (23) and (24) the relevant components of stress, moment, and transverse shear resultants may be expressed as
Mm= 2
_/;((1- 4d(244 - x)Ada) - &(a)1
+ Z(ajA,(c~)]eVialxeWiyada
arA5(o)e+ esiyada,
Mxy= - y( 1 - Y)& \;
(2%
a[(x]~~j- 2/oz2- l)A,(a).
+ la\Aj(cu)]e-lal’e-@ da -y(l-
~)‘~~_~((l’+?)AS(~)e-*
-z (I- Y)$ I_=aAS *
Nxy =
(30)
; a2A4(a)e-‘alxemiya da I r
v, = -;
N,., = -
emiYada,
&
e-” eeiyada,
(31)
a’[A,(a) + xA,(a)] ewiatx eeiyada >
T&/_= a{- In/A,(a) t %
(1 - xlaI)A2(a)) ewinlxeciyada,
h -! = 12(1- r+2’
(32) (33)
(34)
For the symmetric problem under consideration, the boundary conditions may be expressed as Nxy(O, Y) = 0, W,(O, Y) = 0, V,(O, y) = 0, --CD < y &W,y)=~(-c~+~4,
-
(35a-c)
l
40, Y) = 0, lYI > 1,
1 M,(O + , y) = 6~ ( - mm + ml, -l
MO, Y) = 0, IYI > 1
where
U%= N,lh, m, = 6MJh*,
(38)
(T(Y)and m(y) are given in terms of the net ligament stress and moment resultants N and M by (3), i.e. @(y) = N(x2)lh = N(ay)lh, m(y) = 6M(x2)/h2 = 6M(ay)/@
(39)
and the remaining dimensionless quantities are defined in Appendix A. Three of the unknown functions A ,, . . . , A5 may be eliminated by using the homogeneous conditions (35) and the mixed boundary conditions (36) and (37) determine the remaining two. Defining now G&(0+
9Y)= G,(Y),;
u(O+, y)= G*(y),
(4Oa,b)
I336
F. DELALE
and F. ERDOGAN
after some relatively straightforward manipulations (36) and (37) may be reduced to[15] ---
4K(1-V) 1 l+v (t-y)3
---~~ +&&K(PI’-yl)]
GW=$,lukl:
(41) (42)
,8 = [2/K(1 - Y)]1’2,
(43)
where Kz is the modified Bessel function of the second kind. Noting that the crack opening displacement S and the rotation 8 are related to G, and G2 by 8=2p,(ot,y)=2
’ G,(t)dt, I -1
6=2au(Ot,y)=2a
(44)
1’ G,(t)dt,
(45)
-I
substituting from (6) (7), (44) and (45) into (41) and (42) we obtain
ybb(Y)
3tv 1 ’ G,(t)dt -$$$ j-l [----l+vt-y -,
4K(1-V) 1 ltv (t-y)3
+&,++(Blt-Y])]
G,(r)dt+n,(y)/’
MY) /yG,(f)df+y,(~)j-y -1
-1
-1
G,(t)dt=~,lyl
G,(t)dt~~_~~d’=~,lyl
(46) (47)
wheret 4~)
= E(yrr(yb
m(Y)
=
+ rrd~)Pxl, 6Ehh’)u + Ybb(y)&l,
Ytt= &y?
Ybb
=
1 36(1 _
-
&.
v2)
Ma, b)
2,
1 yrb-6(1-;ij~‘Yb,=-6h(llIw2)~~’ b
=
(Y&bb
(49)
The integral eqns (46) and (47) must be solved under the following single-valuedness conditions
I
I
-1
G,(t) dt = 0,
I
1
G,(t) dt = 0.
-1
(5% b)
From the following asymptotic behavior of K,(z) for small values of z
it may be shown that (46) has only a Cauchy type singularity. Hence, the solution of the system of singular integral equations is quite straightforward and may be obtained by using the numerical method described, for example, in[16].
TNotethat%(f) = Qj(UxJh) = aij(QOY)lh) = a,iU(y)lh), andhence
(i, i) = (t. b).
once I(y) is specified, yij are functions of y only,
1337
Line-spring model for surface cracks in a Reissner plate 4. SOLUTION AND RESULTS
The solution of the integral eqns (46) and (47) is of the following form
G(t) =
(1
F1(t) , G*(f) =(lf$;l,2, _
tz)l
W&b)
2,
and conditions (50) and symmetry considerations imply that F,(r) = - Fi( - t), (i = 1,2). After solving for F, and F2, the stress and moment resultants g(y) and m(y) may be determined from (48) by using (44), (45) and (49). The stress intensity factor K(x,) = K(y) along the crack front is then determined from (4) by using (9) and (10). It should be noted that the results of the bending problem are dependent on the Poisson’s ratio V.In the examples considered in this paper, u is assumed to be 0.3. The numerical results are obtained for two different crack profiles, namely a semi-elliptic and an idealized rectangular cut with various length to depth ratios al&,, Lo being the crack depth at x2 = 0. For these two cases the function I(y) is given by L(x*) =
L,V/(1 - (x*/a)*) = I(y) = L,V\/(1 -
y2),
(53)
for the semi-elliptic crack, and L(x*)=LqJ=I(y). -l
(54)
for the rectangular crack. Figures 2 and 3 show the stress intensity factor at the deepest penetration point of a semi-elliptic surface crack in an infinite plate under uniform tension N, and bending moment M,, respectively. In these figures as well as in Figs. 4 and 5 the stress intensity factor is normalized with respect to K, which is the corresponding value for an edge-cracked strip under plane strain conditions with the same Lo/h ratio (Fig. 1). Figures 4 and 5 show the results for a “rectangular” surface crack. It may be seen that the K values for the rectangular crack are consistently higher than the corresponding values for the semi-elliptic crack. Figures 6-8 show the comparison of the results obtained from the line-spring model with that found from the finite element solution& 3,171 for a plate containing a semi-elliptic surface crack. For crack depth (I;,,) to half crack length (a) ratios of 0.2 and 0.6 the comparison of the
0
0.25
I
4
16
64
a/h
Fig. 2. The stress intensity factor at the maximumpenetration point of a semi-elliptic surface crack inan infinite plate under uniform tension (v = 0.3).
0
0.25
Fig. 3.
I
a/h
4
64
0.25
I
Fig. 4.
I
I a/h
4
I
I 16
I
surface crack in an infinite plate under
surface crack in an
Fig. 5. The stress intensity factor at the midpoint of a rectangular surface crack in an infinite plate under pure bending (u = 0.3).
Fig. 4. The stress intensity factor at the midpoint of a rectangular uniform tension (v = 0.3).
I 64
Fig. 3. The stress intensity factor at the maximum penetration point of a semi-elliptic infinite plate under pure bending (u = 0.3).
16
0.25
Fig. 5.
o/h
4
I
I 16
I
I 64
Line-spring model for surface cracks in a Reissner plate.
0 _:
(D
d
w
d
t
B OJ
t d
N
0
d Il.
a
0
1339
1340
F. DELALE and F. ERDOGAN
stress intensity factors at the deepest penetration point (x2 = 0 = y) of the crack is shown in Fig. 6. In Figs. 6-8 the normalized stress intensity factor F is defined by[Z, 3,171
where S = (T, = N,/h for tension, S = m3 = 6M,/h* for bending and Q represents the complete etliptic integral of the second kind and may be approximated by [2,3]
For L&a = 0.2, Figs. 7 and 8 show the comparison of the stress intensity factors along the crack front. From Fig. 6 as well as from Figs. 7 and 8 it may be observed that the results obtained from the line-spring model are quite acceptable.
REFERENCES [i] J. R. RICE and N. LEVY, J. Appi. Mech. 39, 185(1972). ]2] I. S. RAJU and J. C. NEWMAN Jr., J. Eugug Frucl. Me& 11,817 (1979). [3] J. C. NEWMAN Jr., A Relies and Assessment of the S&Tess-InfeusifyFactors for Surface Cracks. NASA, Technieai Memorandum 78803(1978). [4] S. N. ATLURI, K. KATHIRESAN, A. S. KOBAYASHI and M. NAKAGAKI, In Proc. 3rd Int. C&f. Pressure Vessel Technology, Part III, pp. 527-533.ASME, New York (1977). [5] F. W. SMITH and D. R. SORENSEN, Iat. J. Frucf. Mech. 12,47 (1976). [6] R. C. SHAH and A. S. KOBAYASHI, In The Surface Crack: Physical Problems and Computational Solutions (Edited by J. L. Swedlow), pp. 79-124 (1972). [7] J. HELIOT, R. C. LABBENS and A. PELLISIER-TANON, Fracture Mechanics ASTM, STP 677, pp. 341-364 (1979). [S] 1. R. RICE, In The Surface Crack: Physical Problems and Computafional Solutions (Edited by J. L. Swedlow), pp. 171-185,ASME, New York (1972). [P] D. M. PARKS, The Inelastic Line-Spring: Estimates of Elastic-Plastic Fracture Mechanics Parameters for SurfaceCracked Plates and Shells. Paper 8O-C2/PVP-109,ASME (1980). IO] F. ERDOGAN and hf. RATWANI, Iut. 1. Fract. Mech. 8,413 (1972). 111E. REISSNER, 3. Awl. Mech. 12, Trans. ASME PD.A69-A77 (1945). 12] E. REISSNER, Q. i &~l. Math. 5.55 (1948). . ‘131A. C. KAYA and F. ERD~AN. iut. J. &act. 16. 17111980). j14] G. R. IRWIN, StrucfuratMe~h~~~~s (Edited by J. N. Goddier and N. J. Hoff), p, 557. Pergamon Press, Oxford (1960). IS] F. DELALE and F. ERDOGAN, Q. J. Appl. Moth. 37,239 (1979). ‘161F. ERDOGAN, Mechanics Today, (Edited by S. Nemat-Nasser), Vol. 4, pp. i-86. Pergamon Press Oxford (1978). 171J. C. NEWMAN Jr. and I. S. RAJU, Analysis of Surface Cracks in Finite Plates Under Tension or Bending Loads. NASA Technical Paper No. 1578(1979).
(Received 21 Nouember 1980) APPENDIX x = x,/a, y = x,/a, z = x,/a, u = &a, 0 = &a, w = &a, e=&
.a=a,>s,=P%
o, = o, ii& of,. = Q&E>uxY= o,JE, KII = 2, My8= $,
(a, 8) =(x, yf,(i.j) = (1.2).
V, = VJhB, V, = V,/hB, &--!!___ K = 5, 62(l+u)’
A4= 12(1- v2)a2/h2.
(AU 642) (A3) (A4) (AS) (A6) (A7)