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Nonlinear mode II crack-tip fields for some Hookean materials
Theoretical and Applied Fracture Mechanics 6 (1986) 217-222 North-Holland
217
NONLINEAR MODE lI CRACK-TIP FIELDS FOR SOME HOOKEAN MATERIALS
C.L. C H O W Department of Mechanical Engineering, University of Hong Kong, Hong Kong C. OUYANG and X.Q. LU Department of Applied Mechanics, Fudan University, Shanghai, China
This paper presents a modified nonlinear Mode II crack model which is shown to satisfy the nonpenetrating crack surface boundary condition for homogeneous isotropic Hookean materials taking into account finite deformations. A recent investigation of the problem by Knowles [1] reveals apparent interpenetration of the crack surfaces Which is considered nonphysical and therefore invalid. This observation is confirmed when a general solution based on Knowles's perturbation boundary layer method to characterize the finite deformation effects on Mode II crack-tip fields for the materials is derived. By deducing complete 2nd order solutions of the problem, the Poynting nonlinear effect becomes self-evident at the crack tip and for Hookean materials with v < ~ there always exists the penetrating phenomenon between upper and lower crack surfaces.
1. Introduction
The linear elastic solution for Mode II cracks [2] has established that the normal displacements of upper and lower crack surfaces near the crack tip equal to zero, the original undeformed linear crack surfaces thus remain linear after deformation, and no interaction between these upper and lower crack surfaces occurs. In 1980, Stephenson [3] discussed the Mode II crack problems for a class of homogeneous, isotropic incompressible solids by finite deformation theory. He showed that when including the 2nd order terms of the deformation at the crack-tip, the deformation between the upper and lower crack surfaces could be open or penetrating. In 1981, Knowles [1] extended this analysis to compressible elastic materials and reached the same conclusions. Further investigation of the analysis by the authors revealed that Knowles's solutions fail to satisfy the stress-free crack surface boundary conditions imposed by physical considerations. This paper presents a modified model for nonlinear Mode II crack problems when finite deformations are taken into account for some
homogeneous, isotropic Hookean materials. The physical condition of free crack surface is achieved by superimposing appropriate compressive stresses along the crack surfaces near the crack tip. An attempt is also made in the investigation to identify the source of the problem association with Knowles's solution shown to be violating free crack surface condition. An analysis of the general nonlinear Mode II crack problem subject to small shear loading at infinity is performed using the perturbation scheme chosen earlier by Knowles [1]. A complete set of the 2nd order solutions of the problem is derived, revealing apparent interpenetration of the crack surfaces or the nonlinear Poynting effect. 2. Mode II crack in finite elasticity
Consider a plane strain problem with its section area ~ and a crack of length 2a shown in Fig. 1. Based on the Cartesian coordinates x 1, x2, the free crack surface boundary condition is expressed as 0"12=0"22 = 0
0167-8442/86/$3.50 (~) 1986, Elsevier Science Publishers B.V. (North-Holland)
,
for x 2 = O , - a < x l < a
(1)
218
C.L. Chow et al. / Nonlinear mode H crack-tip fields xl 2 .+ Xl~'
and 1, J are the strain invariants expressed as
~(I~
ul -- Kx2
u2=0 I'7
~.
i
/
I
--------I~1 t
I
1
/
(/
I
x ~
°)
/
x, I
t
i
"V
1
+ ~e,o%ru,,¢uo, ~ ,
(6)
(a,/3, p, "y = 1, 2)
/
W/, Wj in (5) are derivatives of W about I and J, 6q is Kronecker delta symbol, and e ~ is the
/ t
1
J= 1 + u
/
~ a-..~a~
I
l = 2 + 2u ..... + u,,t~u,,t~,
/
2-dimensional permutation symbol. To satisfy the one-to-one correspondence between the undeformed and deformed regions, J in (6) should be positive or
.../
I
J > O.
(7)
Fig. 1. Mode II crack.
From the transform between Cauchy and Piola stress tensor [4], we may express the constitutive equation for plane deformation by Cauchy stress tensor as follows: where % (i, j = 1, 2) are Piola stress tensor components. The boundary condition (1) may also be expressed by related Cauchy stress tensor, but then the deformed region ~ of ~ should be used. Since the crack length 2a is small compared to the characteristic length of ~, we may consider @ as an infinite region. Thus, the boundary condition at the outer boundary for the Model II crack becomes: u l--->kx 2,
k=tga,
u2---~ 0 ,
f o r x~ + x2---~ oo ,
2
(2)
where k is the shear loading at infinity as shown in Fig. 1, and will be required in a later section to be small. Using the Piola stress o~0, the equilibrium equations are: o'~,~=0,
a , / 3 = l , 2 , in ~
(3)
If the Cauchy stress is used, then these equations are
~-q,j=0,
2
(8)
+ ] W l ( u w + uj, i + u ~ , u j , p ) .
Assume that the body in undeformed state is free of stress, then we have from eq. (8) 2W1(2, 1) + Wj(2, 1) = 0.
Let the shear loading k be small and consider a perturbation expansion of displacements u. about the small parameter c as [1] l (t) u ~ = ~ ~,~ ,
a=l,2,
in ~
(4)
For a plane strain problem, the general constitutive equations for homogeneous, isotropic compressible solids expressed in terms of the Piola stress are given by o-~ = (2Wt + Wj)6~t ~ + 2 W , u~,~ + Wje~o %,up, , , (5)
where W= W ( I , J ) is the strain energy density,
(10)
/=1
where c =
.
(11)
Substituting (10) into (5), (6), we obtain the following: I = I o + cI I + c212 + o ( c 2 ) ,
i,j=l,2,
(9)
I 0 = 2, 12
11 = 2u~l)~ ,
_ ,~ ( 2 ) + -- ZUct,c t
. (1) . (1) Uct,flua,fl
,
J = Jo + cJ~ + c2J2 + 0 ( c 2 ) , Jo
=
1
(12)
,
J1
=
(13)
(14)
. (1)
IXot,o t ,
J2 = u(2) - u oO) or,or + ±e 2 ctA '~'By t , ~ u (1) A, y
(15)
(2) c5,t3 = co-(~) + c 2 _o~, 0 + 0(c2) ,
(16)
219
C.L. Chow et al. / Nonlinear mode H crack-tip fields
0.O) (1) ~t3 = {2(W1)(2) + ( W j ) ( z ) } 6 ~ + 2(WI) (,).",~,o I w ~(~)~
+ \ " J.I
-
u (1)
(17)
eap t~[3y p,y , +
a,8 Jr
2(Wl)(2)U(1) + (W+)%=p
+ 2(w,) O). (2) +
(1)
E~yUp,y , (18)
where
( W j ) (1) = ( W j ) o ,
( W I ) (1) = ( W l ) o
,
3. Small-scale nonlinear crack problem and second order solutions
Assume that small deformation simplification prevails in the whole physical area except the near crack-tip region. This is usually the case for many engineering materials. The so-called smallscale nonlinear crack problems may then be formulated. By a boundary layer approximation [1], we may confine ourselves within the near crack-tip region and thus reduce the problem to one with a semi-infinite crack in infinite plane as shown in Fig. 2. The free crack surface condition is given by
(Wj) (2) = ( W u ) o I 1 + ( W j j ) o J 1 , °'12 = 0"22= 0 (W,) <2) = ( W n ) o I 1 + ( W H ) o J 1 ,
for x 2- '- 0, -
GO<
x I¢ < 0 .
(25)
But the matching boundary condition becomes
(W+) (3) = (W,j)oI 2 + (W++)oJ2 1
,
2
(19)
2
"(1) ~ Ua'---~ C/.g~
for r = V ~ +
2 X2""~GO , ((3/ = 1 , 2 )
+ ~ [ ( W l l J ) o I 1 Jr- ( W j j j ) o J x ]
,
(26)
+ (WHj)oI1J 1 , ( W / ) (3) :
From the boundary layer matching condition (26), we may set
(Wl,)oI 2 + (Wlj)J 2
~. %=~%
+ I [ ( W H I ) o I2 + (Wljj)oJ211 + (WllJ)oliJ1 •
a13,0 = 0 .(1) _(1) = 0 12 = tJ22
in
(20)
for x 2 = 0 , - a < x
l
0 .(2)
a0,~
=0
(a=l,2).
u(2) tx = V~(0)ym,
0 ~< m e < I
(29) where A,/z, are Lam6 constants.
for x 2 = 0 , - a < x 1 < a
and the boundary condition at infinity is (2). Obviously, the 1st order solution u~1) is just that of the linear elastic Mode II crack solution given at the crack-tip as [2]
x; ~ + x;,~ ~ o~
1
till) = ~/T[ ( ; --'4P) sin ~ 0 + ~ sin (24) 4v-
cos~0+~sin
(28)
W = l ) t ( I ( I - 2)) 2 + ¼/x(I 2 -- 2 J 2 - 2I + 2 ) ,
(23)
u~1)=x/T
(27)
It is evident that eqs. (27) and (28) satisfy the outer boundary condition (26). Consequently, we should only discuss the 2rid order s o l , i o n s . For simplicity we confine the discussions to some Hookean materials. The strain energy density W ( I , J ) is expressed as
(22)
in
Or(2) _(2) = 0 12 = 0 2 2
+ c u2~ (2) + o ( c 2)
Here U~ • (1) is given by (24), and
The subscript 'o' means values at the initial state, I=2, J=l. The associated equations of equilibrium and boundary conditions may then be reduced as: 0.(0
(1)
Fig. 2. Nonlinear smaU-scale crack problem.
C.L. Chow et al. / Nonlinear mode 11 crack-tip fields
220
From (5) and (19), we have
(W,)o = ~ j , ,
__(2) cr22
~rT., x (1) (ll (4Wu + W J J ]~o u oe~2~ ,ex + Z~WzDU,,.tsu., ~
=
q- g W "1 ( " (1). \ "'jjlo\l'gl,lt42,2
(W,)o = - l * ,
(w,,)o = ~(½a + ~ ) ,
(w.)o
(W.)o = (w,,,)o = (w,.)o
(30)
= -~,
"1- ••.a[
= (w,,,)o = (w,,,)o
(1)
\ ' ' 1 1 ] o " 2 , 2 "q" ( W J j ) o U , , , ]
=0,
{2) + 2(W,)oU(2) {4(W/,)o + (W,,)o}U,,,,,~
for x~ = 0, - ~ < x ' < 0
With eqs. (31), (32) and (33), we can readily determine the displacements u~2) (a = 1, 2). Because this process is quite lengthy and involved, no detailed derivations will be given but the resulting solutions are: m e=0, U~2)--~
=--{[2(W,,)oU:{)bU(~)b
(33)
+ 2(Wz)oU~2~ + (Wj)oU(12~
=0.
Substituting (30), (24) into (18), we obtain or(2) expressed in terms of the displacements. The equilibrium equation is then used to deduce the basic equations on u~2) as
(1). (1),~ U2,1/~1,2)
--
a=1,2,
(34)
uI(O)
= a 0 + rh cos O + r/2 cos 20 + r13 cos 30
1
O) o) ] ,~
+ ~ (w,,)o~o,~,*o,~u,,~
+ [4(WH)oU(al,)uO )
E
"W
\
119 - 292v + 192v 2
= a0 +
32(v - 1)
+
9 - 18v + 8v 2 cos 20 16(v - I)
+
9 - 12v cos 3 0 , 32(v - 1)
(1) U(1) ] O//
cos 0
(35)
u~2) = bo + ¢o0 + ¢1 sin 0 + ¢2 sin 20 + ¢3 sin 30 Here a,/3, a, b, p, y, k each take the values 1, 2. The crack surface condition can also be expressed in terms of the displacements:
= b° +
tr(2) 12 = 2{(Wtt)oI1}u{1,~ - {(W,l)oJ,}u~'~ + 2(W,) oU{2,~-- t"W'j)oU2,1(2) =0,
for x~ = 0,
+
(1-2v)(5-av) 8(v - 1)
(1
32(v-- 1)
9 - 18v + 8v z
(32)
+
-~
16(v - 1)
8
0
sinO
sin 20 +
9 - 12v 3 2 ( , , - 1)
sin 3 0 . (36)
Table 1 v
~0
~1
~2
~:3
"01
"02
*13
Material
0.29 0.31
-0.2841 -0.2540
-0.9190 -0.9221
-0.3919 -0.3772
-0.2431 -0.2379
-2.221 -2.107
-0.3919 -0.3772
-0.2431 -0.2379
0.35
-0.2071
-0.9229
-0.3531
-0.2312
-1.935
-0.3531
-0.2312
0.33
-0.2332
-0.9221
-0.3680
-0.2350
-2.031
-0.3680
-0.2350
0.25
-0.3331
-0.9170
-0.4171
-0.2500
-2.417
-0.4171
-0.2500
0.2
-0.3938
-0.9141
-0.4469
-0.2578
-2.667
-0.4469
-0.2578
Steels Aluminum alloys Magnesium alloys Copper (hot rolled) Plastics Glass Concrete
C.L. Chow et al. / Nonlinear mode H crack-tip fields
Here v is the Poisson's ratio and the constant a0, b 0 are related to the rigid motion, and usually they could be taken as zero. For the purpose of illustration we present some numerical data of ~i, ~7i for different values of v related to some engineering Hookean material [5], see Table 1.
221
5. Poynting effect at the crack-tip and a physical Mode H crack model Let the related displacement between the upper and lower crack surfaces be 6 = u2('rr) - u2(-'tr ) .
(40)
Substituting eqs. (34) and (36) into eq. (40), we obtain 6 = 2"rrc2~0 .
4. Nonlinear Poynting effect
(41)
For Hookean materials, we have from eq. (29) We shall next examine the boundary condition along the crack surfaces from Knowles's solution [1]. For the Hookean materials, a and/3 expressed in eqs. (5) and (6) of [1] are reproduced as: ot
=
2(1 - 2v) gt
(2(WIi)o) = 2(1 -
v),
The solutions as given by [1] are
,[
2(2-3v)(1-v)
11
-7+T
13
_ 4v2]
J
1[(9)
2(1-v)
2 -g+4v
(l-v)
] 9,)
+i-6-12v-4v
1(
se2-
2(1- v)
~3 =
3 2(11v)(~(1-v)-3)
(38)
which indicates that for v < 1/2, the crack surface displacement is always 8 < 0,
(43)
and penetration between the upper and lower crack surface will occur. This is the so-called Poynting effect in finite elasticity. This means the free crack surface condition is no longer valid in finite fracture mechanics formulations even for small scale nonlinear crack problems. In order to nullify the nonphysical Poynting effect, a modified nonlinear Mode I! crack model is proposed. This is achieved by superimposing appropriate amount of compressive stresses along the crack surfaces, Fig. 3. 0 = ~oC2(A + 0"22
2 ,
v(1-v)-g+~
2tz)/r
,
v ,
•
for 0 = - + ~r
(39)
we find 2_(2)1 C or22 lO=---~r =
~ a k 2 ( 8 v - 3)(2v + 3)
16r(1 - 2v)
(44)
where ¢o is the parameter obtained in the original formulation of Mode II crack problems. All other equations and boundary condition at infinity are retained. For the modified model, we can obtain a valid solution which satisfies the compatibility con-
Substituting these results into the free crack surface condition "(2) 22 = 0
(42)
/3=0.
(37)
se°=-2(1-v)
6 = "rrc2(1 - 2v)(5 - 4v) 4(v - 1) '
#0.
The above verifies that Knowles's solution violates the physical boundary condition of free crack surface. Fig. 3. Modified Mode II crack model.
222
C.L. Chow et al. / Nonlinear mode H crack-tip fields
dition along the crack surfaces as 4
u~2) = ~'o0 + ~
~i sin i 0 ,
(45)
i=1
with SC'o= 0.
found to exhibit the nonlinear Poynting effect at the crack tip. The nonphysical phenomenon is verified by analyzing the finite deformation nonlinear problem subjected to small loading at infinity with a perturbation scheme.
(46)
The modified model does not therefore violate the mandatory nonpenetrating crack surface condition and provide one-to-one correspondence between the deformed and undeformed configuration.
6. Conclusions A compatible nonlinear Mode II crack model incorporating finite deformations is presented and shown to satisfy the physical requirement of nonpenetrating crack boundary surface. An earlier model for the problem is investigated and
References [1] J.K. Knowles, "A Nonlinear Effect in Mode II Crack Problems," Engrg. Fracture Mech. 15(3,4), 469-476 (1981). [2] J.R. Rice, "Mathematical Analysis in the Mechanics of Fracture," in: H. Liebowitz, ed., Fracture, Vol. II, Academic Press, New York (1968). [3] R.A. Stephenson, "The Equilibrium Field Near the Tip of a Crack for Finite Plane Strain of Incompressible Elastic Materials," Tech. Rep. No. 44, ONR Contract N00014-75-L-0196 (1980). [4] A.C. Eringen, Nonlinear Theory of Continuous Media, McGraw-Hill, New York (1962). [5] Y.C. Fung, Foundations of Solid Mechanics, PrenticeHall, Englewood Cliffs (1965).
217
NONLINEAR MODE lI CRACK-TIP FIELDS FOR SOME HOOKEAN MATERIALS
C.L. C H O W Department of Mechanical Engineering, University of Hong Kong, Hong Kong C. OUYANG and X.Q. LU Department of Applied Mechanics, Fudan University, Shanghai, China
This paper presents a modified nonlinear Mode II crack model which is shown to satisfy the nonpenetrating crack surface boundary condition for homogeneous isotropic Hookean materials taking into account finite deformations. A recent investigation of the problem by Knowles [1] reveals apparent interpenetration of the crack surfaces Which is considered nonphysical and therefore invalid. This observation is confirmed when a general solution based on Knowles's perturbation boundary layer method to characterize the finite deformation effects on Mode II crack-tip fields for the materials is derived. By deducing complete 2nd order solutions of the problem, the Poynting nonlinear effect becomes self-evident at the crack tip and for Hookean materials with v < ~ there always exists the penetrating phenomenon between upper and lower crack surfaces.
1. Introduction
The linear elastic solution for Mode II cracks [2] has established that the normal displacements of upper and lower crack surfaces near the crack tip equal to zero, the original undeformed linear crack surfaces thus remain linear after deformation, and no interaction between these upper and lower crack surfaces occurs. In 1980, Stephenson [3] discussed the Mode II crack problems for a class of homogeneous, isotropic incompressible solids by finite deformation theory. He showed that when including the 2nd order terms of the deformation at the crack-tip, the deformation between the upper and lower crack surfaces could be open or penetrating. In 1981, Knowles [1] extended this analysis to compressible elastic materials and reached the same conclusions. Further investigation of the analysis by the authors revealed that Knowles's solutions fail to satisfy the stress-free crack surface boundary conditions imposed by physical considerations. This paper presents a modified model for nonlinear Mode II crack problems when finite deformations are taken into account for some
homogeneous, isotropic Hookean materials. The physical condition of free crack surface is achieved by superimposing appropriate compressive stresses along the crack surfaces near the crack tip. An attempt is also made in the investigation to identify the source of the problem association with Knowles's solution shown to be violating free crack surface condition. An analysis of the general nonlinear Mode II crack problem subject to small shear loading at infinity is performed using the perturbation scheme chosen earlier by Knowles [1]. A complete set of the 2nd order solutions of the problem is derived, revealing apparent interpenetration of the crack surfaces or the nonlinear Poynting effect. 2. Mode II crack in finite elasticity
Consider a plane strain problem with its section area ~ and a crack of length 2a shown in Fig. 1. Based on the Cartesian coordinates x 1, x2, the free crack surface boundary condition is expressed as 0"12=0"22 = 0
0167-8442/86/$3.50 (~) 1986, Elsevier Science Publishers B.V. (North-Holland)
,
for x 2 = O , - a < x l < a
(1)
218
C.L. Chow et al. / Nonlinear mode H crack-tip fields xl 2 .+ Xl~'
and 1, J are the strain invariants expressed as
~(I~
ul -- Kx2
u2=0 I'7
~.
i
/
I
--------I~1 t
I
1
/
(/
I
x ~
°)
/
x, I
t
i
"V
1
+ ~e,o%ru,,¢uo, ~ ,
(6)
(a,/3, p, "y = 1, 2)
/
W/, Wj in (5) are derivatives of W about I and J, 6q is Kronecker delta symbol, and e ~ is the
/ t
1
J= 1 + u
/
~ a-..~a~
I
l = 2 + 2u ..... + u,,t~u,,t~,
/
2-dimensional permutation symbol. To satisfy the one-to-one correspondence between the undeformed and deformed regions, J in (6) should be positive or
.../
I
J > O.
(7)
Fig. 1. Mode II crack.
From the transform between Cauchy and Piola stress tensor [4], we may express the constitutive equation for plane deformation by Cauchy stress tensor as follows: where % (i, j = 1, 2) are Piola stress tensor components. The boundary condition (1) may also be expressed by related Cauchy stress tensor, but then the deformed region ~ of ~ should be used. Since the crack length 2a is small compared to the characteristic length of ~, we may consider @ as an infinite region. Thus, the boundary condition at the outer boundary for the Model II crack becomes: u l--->kx 2,
k=tga,
u2---~ 0 ,
f o r x~ + x2---~ oo ,
2
(2)
where k is the shear loading at infinity as shown in Fig. 1, and will be required in a later section to be small. Using the Piola stress o~0, the equilibrium equations are: o'~,~=0,
a , / 3 = l , 2 , in ~
(3)
If the Cauchy stress is used, then these equations are
~-q,j=0,
2
(8)
+ ] W l ( u w + uj, i + u ~ , u j , p ) .
Assume that the body in undeformed state is free of stress, then we have from eq. (8) 2W1(2, 1) + Wj(2, 1) = 0.
Let the shear loading k be small and consider a perturbation expansion of displacements u. about the small parameter c as [1] l (t) u ~ = ~ ~,~ ,
a=l,2,
in ~
(4)
For a plane strain problem, the general constitutive equations for homogeneous, isotropic compressible solids expressed in terms of the Piola stress are given by o-~ = (2Wt + Wj)6~t ~ + 2 W , u~,~ + Wje~o %,up, , , (5)
where W= W ( I , J ) is the strain energy density,
(10)
/=1
where c =
.
(11)
Substituting (10) into (5), (6), we obtain the following: I = I o + cI I + c212 + o ( c 2 ) ,
i,j=l,2,
(9)
I 0 = 2, 12
11 = 2u~l)~ ,
_ ,~ ( 2 ) + -- ZUct,c t
. (1) . (1) Uct,flua,fl
,
J = Jo + cJ~ + c2J2 + 0 ( c 2 ) , Jo
=
1
(12)
,
J1
=
(13)
(14)
. (1)
IXot,o t ,
J2 = u(2) - u oO) or,or + ±e 2 ctA '~'By t , ~ u (1) A, y
(15)
(2) c5,t3 = co-(~) + c 2 _o~, 0 + 0(c2) ,
(16)
219
C.L. Chow et al. / Nonlinear mode H crack-tip fields
0.O) (1) ~t3 = {2(W1)(2) + ( W j ) ( z ) } 6 ~ + 2(WI) (,).",~,o I w ~(~)~
+ \ " J.I
-
u (1)
(17)
eap t~[3y p,y , +
a,8 Jr
2(Wl)(2)U(1) + (W+)%=p
+ 2(w,) O). (2) +
(1)
E~yUp,y , (18)
where
( W j ) (1) = ( W j ) o ,
( W I ) (1) = ( W l ) o
,
3. Small-scale nonlinear crack problem and second order solutions
Assume that small deformation simplification prevails in the whole physical area except the near crack-tip region. This is usually the case for many engineering materials. The so-called smallscale nonlinear crack problems may then be formulated. By a boundary layer approximation [1], we may confine ourselves within the near crack-tip region and thus reduce the problem to one with a semi-infinite crack in infinite plane as shown in Fig. 2. The free crack surface condition is given by
(Wj) (2) = ( W u ) o I 1 + ( W j j ) o J 1 , °'12 = 0"22= 0 (W,) <2) = ( W n ) o I 1 + ( W H ) o J 1 ,
for x 2- '- 0, -
GO<
x I¢ < 0 .
(25)
But the matching boundary condition becomes
(W+) (3) = (W,j)oI 2 + (W++)oJ2 1
,
2
(19)
2
"(1) ~ Ua'---~ C/.g~
for r = V ~ +
2 X2""~GO , ((3/ = 1 , 2 )
+ ~ [ ( W l l J ) o I 1 Jr- ( W j j j ) o J x ]
,
(26)
+ (WHj)oI1J 1 , ( W / ) (3) :
From the boundary layer matching condition (26), we may set
(Wl,)oI 2 + (Wlj)J 2
~. %=~%
+ I [ ( W H I ) o I2 + (Wljj)oJ211 + (WllJ)oliJ1 •
a13,0 = 0 .(1) _(1) = 0 12 = tJ22
in
(20)
for x 2 = 0 , - a < x
l
0 .(2)
a0,~
=0
(a=l,2).
u(2) tx = V~(0)ym,
0 ~< m e < I
(29) where A,/z, are Lam6 constants.
for x 2 = 0 , - a < x 1 < a
and the boundary condition at infinity is (2). Obviously, the 1st order solution u~1) is just that of the linear elastic Mode II crack solution given at the crack-tip as [2]
x; ~ + x;,~ ~ o~
1
till) = ~/T[ ( ; --'4P) sin ~ 0 + ~ sin (24) 4v-
cos~0+~sin
(28)
W = l ) t ( I ( I - 2)) 2 + ¼/x(I 2 -- 2 J 2 - 2I + 2 ) ,
(23)
u~1)=x/T
(27)
It is evident that eqs. (27) and (28) satisfy the outer boundary condition (26). Consequently, we should only discuss the 2rid order s o l , i o n s . For simplicity we confine the discussions to some Hookean materials. The strain energy density W ( I , J ) is expressed as
(22)
in
Or(2) _(2) = 0 12 = 0 2 2
+ c u2~ (2) + o ( c 2)
Here U~ • (1) is given by (24), and
The subscript 'o' means values at the initial state, I=2, J=l. The associated equations of equilibrium and boundary conditions may then be reduced as: 0.(0
(1)
Fig. 2. Nonlinear smaU-scale crack problem.
C.L. Chow et al. / Nonlinear mode 11 crack-tip fields
220
From (5) and (19), we have
(W,)o = ~ j , ,
__(2) cr22
~rT., x (1) (ll (4Wu + W J J ]~o u oe~2~ ,ex + Z~WzDU,,.tsu., ~
=
q- g W "1 ( " (1). \ "'jjlo\l'gl,lt42,2
(W,)o = - l * ,
(w,,)o = ~(½a + ~ ) ,
(w.)o
(W.)o = (w,,,)o = (w,.)o
(30)
= -~,
"1- ••.a[
= (w,,,)o = (w,,,)o
(1)
\ ' ' 1 1 ] o " 2 , 2 "q" ( W J j ) o U , , , ]
=0,
{2) + 2(W,)oU(2) {4(W/,)o + (W,,)o}U,,,,,~
for x~ = 0, - ~ < x ' < 0
With eqs. (31), (32) and (33), we can readily determine the displacements u~2) (a = 1, 2). Because this process is quite lengthy and involved, no detailed derivations will be given but the resulting solutions are: m e=0, U~2)--~
=--{[2(W,,)oU:{)bU(~)b
(33)
+ 2(Wz)oU~2~ + (Wj)oU(12~
=0.
Substituting (30), (24) into (18), we obtain or(2) expressed in terms of the displacements. The equilibrium equation is then used to deduce the basic equations on u~2) as
(1). (1),~ U2,1/~1,2)
--
a=1,2,
(34)
uI(O)
= a 0 + rh cos O + r/2 cos 20 + r13 cos 30
1
O) o) ] ,~
+ ~ (w,,)o~o,~,*o,~u,,~
+ [4(WH)oU(al,)uO )
E
"W
\
119 - 292v + 192v 2
= a0 +
32(v - 1)
+
9 - 18v + 8v 2 cos 20 16(v - I)
+
9 - 12v cos 3 0 , 32(v - 1)
(1) U(1) ] O//
cos 0
(35)
u~2) = bo + ¢o0 + ¢1 sin 0 + ¢2 sin 20 + ¢3 sin 30 Here a,/3, a, b, p, y, k each take the values 1, 2. The crack surface condition can also be expressed in terms of the displacements:
= b° +
tr(2) 12 = 2{(Wtt)oI1}u{1,~ - {(W,l)oJ,}u~'~ + 2(W,) oU{2,~-- t"W'j)oU2,1(2) =0,
for x~ = 0,
+
(1-2v)(5-av) 8(v - 1)
(1
32(v-- 1)
9 - 18v + 8v z
(32)
+
-~
16(v - 1)
8
0
sinO
sin 20 +
9 - 12v 3 2 ( , , - 1)
sin 3 0 . (36)
Table 1 v
~0
~1
~2
~:3
"01
"02
*13
Material
0.29 0.31
-0.2841 -0.2540
-0.9190 -0.9221
-0.3919 -0.3772
-0.2431 -0.2379
-2.221 -2.107
-0.3919 -0.3772
-0.2431 -0.2379
0.35
-0.2071
-0.9229
-0.3531
-0.2312
-1.935
-0.3531
-0.2312
0.33
-0.2332
-0.9221
-0.3680
-0.2350
-2.031
-0.3680
-0.2350
0.25
-0.3331
-0.9170
-0.4171
-0.2500
-2.417
-0.4171
-0.2500
0.2
-0.3938
-0.9141
-0.4469
-0.2578
-2.667
-0.4469
-0.2578
Steels Aluminum alloys Magnesium alloys Copper (hot rolled) Plastics Glass Concrete
C.L. Chow et al. / Nonlinear mode H crack-tip fields
Here v is the Poisson's ratio and the constant a0, b 0 are related to the rigid motion, and usually they could be taken as zero. For the purpose of illustration we present some numerical data of ~i, ~7i for different values of v related to some engineering Hookean material [5], see Table 1.
221
5. Poynting effect at the crack-tip and a physical Mode H crack model Let the related displacement between the upper and lower crack surfaces be 6 = u2('rr) - u2(-'tr ) .
(40)
Substituting eqs. (34) and (36) into eq. (40), we obtain 6 = 2"rrc2~0 .
4. Nonlinear Poynting effect
(41)
For Hookean materials, we have from eq. (29) We shall next examine the boundary condition along the crack surfaces from Knowles's solution [1]. For the Hookean materials, a and/3 expressed in eqs. (5) and (6) of [1] are reproduced as: ot
=
2(1 - 2v) gt
(2(WIi)o) = 2(1 -
v),
The solutions as given by [1] are
,[
2(2-3v)(1-v)
11
-7+T
13
_ 4v2]
J
1[(9)
2(1-v)
2 -g+4v
(l-v)
] 9,)
+i-6-12v-4v
1(
se2-
2(1- v)
~3 =
3 2(11v)(~(1-v)-3)
(38)
which indicates that for v < 1/2, the crack surface displacement is always 8 < 0,
(43)
and penetration between the upper and lower crack surface will occur. This is the so-called Poynting effect in finite elasticity. This means the free crack surface condition is no longer valid in finite fracture mechanics formulations even for small scale nonlinear crack problems. In order to nullify the nonphysical Poynting effect, a modified nonlinear Mode I! crack model is proposed. This is achieved by superimposing appropriate amount of compressive stresses along the crack surfaces, Fig. 3. 0 = ~oC2(A + 0"22
2 ,
v(1-v)-g+~
2tz)/r
,
v ,
•
for 0 = - + ~r
(39)
we find 2_(2)1 C or22 lO=---~r =
~ a k 2 ( 8 v - 3)(2v + 3)
16r(1 - 2v)
(44)
where ¢o is the parameter obtained in the original formulation of Mode II crack problems. All other equations and boundary condition at infinity are retained. For the modified model, we can obtain a valid solution which satisfies the compatibility con-
Substituting these results into the free crack surface condition "(2) 22 = 0
(42)
/3=0.
(37)
se°=-2(1-v)
6 = "rrc2(1 - 2v)(5 - 4v) 4(v - 1) '
#0.
The above verifies that Knowles's solution violates the physical boundary condition of free crack surface. Fig. 3. Modified Mode II crack model.
222
C.L. Chow et al. / Nonlinear mode H crack-tip fields
dition along the crack surfaces as 4
u~2) = ~'o0 + ~
~i sin i 0 ,
(45)
i=1
with SC'o= 0.
found to exhibit the nonlinear Poynting effect at the crack tip. The nonphysical phenomenon is verified by analyzing the finite deformation nonlinear problem subjected to small loading at infinity with a perturbation scheme.
(46)
The modified model does not therefore violate the mandatory nonpenetrating crack surface condition and provide one-to-one correspondence between the deformed and undeformed configuration.
6. Conclusions A compatible nonlinear Mode II crack model incorporating finite deformations is presented and shown to satisfy the physical requirement of nonpenetrating crack boundary surface. An earlier model for the problem is investigated and
References [1] J.K. Knowles, "A Nonlinear Effect in Mode II Crack Problems," Engrg. Fracture Mech. 15(3,4), 469-476 (1981). [2] J.R. Rice, "Mathematical Analysis in the Mechanics of Fracture," in: H. Liebowitz, ed., Fracture, Vol. II, Academic Press, New York (1968). [3] R.A. Stephenson, "The Equilibrium Field Near the Tip of a Crack for Finite Plane Strain of Incompressible Elastic Materials," Tech. Rep. No. 44, ONR Contract N00014-75-L-0196 (1980). [4] A.C. Eringen, Nonlinear Theory of Continuous Media, McGraw-Hill, New York (1962). [5] Y.C. Fung, Foundations of Solid Mechanics, PrenticeHall, Englewood Cliffs (1965).