~
Engineering Fracture Mechanics Vol. 54. No. 3, pp. 405-422. 1996
Pergamon
0013-7944(95)00191-3
Copyright © 1996 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0013-7944/96 $15.00+0.00
HIGHER ORDER ASYMPTOTIC ELASTIC-PLASTIC
CRACK-TIP FIELDS UNDER ANTIPLANE SHEAR
S. YANG, F. G. YUAN and X. CAI
Department of Mechanical and Aerospace Engineering, North Carolina State University,
Raleigh NC 27695, U.S.A.
Abstract--The asymptotic stress and strain fields near the crack tip under antiplane shear are developed
in an elastic power-law hardening material. Using an asymptotic expansion and separation of variables
for the stress function, a series solution for all of the hardening exponents can be obtained. The stress
exponents for the higher order terms are analytically determined; the angular distributions which are
governed solely by plastic strains are also analytically obtained. Good agreement with the finite element
solutions confirms the proposed approach. It is further demonstrated that the first three terms, controlled
by two parameters, can be used to characterize the crack tip stress and strain fields with various hardening
exponents. Copyright © 1996 Elsevier Science Ltd
1. INTRODUCTION
THE ASYMPTOTICelastic-plastic crack-tip fields were analyzed by Hutchinson [1], and Rice and
Rosengren [2], henceforth, HRR, which described the crack-tip fields by a singular dominant term
in a power-law hardening material. The stress and strain fields are uniquely characterized by the
amplitude of the dominant term or the parameter J-integral introduced by Rice [3]. In some cases,
the single term characterization may not adequately describe the elastic-plastic fracture. Hence,
the effects of the higher order terms on the crack-tip stress fields have been under extensive
investigation. Li and Wang [4], and Sharma and Aravas [5] determined the second order term for
the Mode I plane strain condition. A more comprehensive analysis for higher order solutions which
include the second, third and fourth order terms and so on for both Mode I and Mode II under
either plane strain or plane stress condition was given by Yang et al. [6]. Higher order analysis for
Mode I plane strain was also investigated by Xia et al. [7].
For the Mode III crack problem, the leading term of the asymptotic elastic-plastic crack-tip
fields has been given by Reidel [8] and others [9, 10]. Aravas and Blazo [11] further investigated the
second order term and provided angular distribution of the second order term. In this paper, a
method for determining a series of the higher order terms is developed. Based on the small strain
deformation formulation and JE-deformation plasticity theory with power-law hardening, the
governing equations for the higher order fields, which include the second, third, fourth order terms
and so on, are derived. As previously presented in refs [4-6] for Mode I and Mode II loading, the
higher order solutions for the crack-tip fields including stress exponents and associated angular
distributions are e btained through numerical means. However, under antiplane shear, this paper
provides analytical solutions of stress exponents for higher order terms. Further, the angular
distributions for the higher order fields which are completely controlled by plastic strains can also
be given by analytical solutions under Mode III antiplane shear.
As an application, a comparison of the asymptotic solutions to the finite element solutions
is attempted. The results indicate that the three term expansion with two parameters can predict
or describe the stress, strain and displacement fields for all angles within a considerable distance
from the crack tip.
2. FUNDAMENTAL
EQUATIONS
Assume that a material is governed by a Ramberg-Osgood stress-strain relation in antiplane
shear in a dimensionless form:
= ~ + ~",
(1)
where u is a dimensionless material constant and n is a strain hardening exponent.
405
406
S. YANG et al.
The generalization of eq. (1) based on the ,/2 deformation theory of plasticity for Mode III
loading gives
(2)
7~ = za + o~3;'- lzp, fl = r,O ,
where 7a = 7a--, 3a = 3a.-, C~ = 3a3 a.
In this paper, the following normalized quantities defined by
3p = -Ya
-,
= --7~
30 YP 70
(3)
?
r = ~ , w = 70L
are used, where the barred quantities are the non-normalized field variables, L is a characteristic
length, 30 is the yield shear stress in pure shear, 7o = zo/G, G is the shear modulus, ~ is the stress
tensor, 7i is the engineering shear strain component and w is the displacement.
For small strain formulation, the strain-displacement relation is
cow
1 cow
7r= -~-r' 7 o - r coO "
(4)
The equilibrium equation is written as
c~r,
1 co30
cor+r~+7
3r = 0
(5)
•
The compatibility equation is given by
coTr
coO
co(reo)
cot
- 0.
(6)
By introducing a stress function q~(r,0) with
3r
1 CO~
r coO'
--
co~
T O ---~
--
cor '
(7)
- -
the equilibrium eq. (5) is automatically satisfied.
3. A S Y M P T O T I C ANALYSIS
Referring a polar coordinate system (r,0) with origin centered at the crack tip, an asymptotic
elastic-plastic crack-tip field is sought under condition of no unloading. In this paper, the problem
is formulated in terms of the stress function. An asymptotic expansion of the stress function 4~ is
attempted in the form of separation of variables:
=
- ZAkr~*+%(0)
k = 1,2,3,...
as r--*0,
(8)
k
where s, refer to stress exponents and s~ < s_, < s~ < .... It follows from eqs (7) and (8) that the stress
asymptotic expansion is
zt~ = Y'.Akr~k¢]~(O)
as r ~ O ,
(9)
k
where
e',=
-f~
~ ' , = (s~ + 1)A.
(lO)
Higher order asymptotic elastic-plastic crack-tip fields
407
The prime denotes the derivative with respect to 0. Using the binomial theorem to expand
zT-' and keeping terms up to at least the fourth order as r--.0, z 7- ~ can be written as
=A~'-~r (''-''''(~'')''-' { 1 + ( n -
A2ra'~"-A~r~'~z'3
1) [LA~
+ Z
+n
A' ra~4f'4
+ Z
+"" 1
f,-')-,]
A--:-~
-
+ (n - 3)e'2~ '31
where As, = sq - s,, f " = (~I')~I')) '/~
~)~)
Note that eq. (11) is valid for 1 < ~,. < x / ~ A : " , f " . The repeated subscript i in the expressions,
f " and f~q, is summed over the r and 0.
Substituting eq. (I 1) into the constitutive eq. (2), the expansion of shear strains is given by
?, = aA','r"',Pl': + a A T - ' A:""' + t"-Fl~' + aA','- ' A3r "~' + As3/~I3) "~- o~Ait- 'A4r"" + a~'/~l" + ""
+
..~,,I - ~zx2"
w,.,, + ~....~(,,
~zat
-11i + a A
.-3
A :3
,,. +3~,.:01~)
-.
+ ~ A ~ - 2 A 2 A 3 r '''
+ ~,.,
- + ~,,/~?)
-. +
...
+ A t r " ~ ~: + A2r~2~ 2) + A3r~fl ~) + ... ,
(12)
where the last three terms with factors ?~ are elastic strains; the rest of the terms are associated
with plastic strains and
t~l,, = (~,,),,-,~,,
~FI"' = (~")"-'[~'"' + (n -- 1)~'"'~"];
m = 2,3,4,..-
/~I') : ( f " ) " - ' ( n -- 1){~'~'?I-') + ½[e-'-'+ (n -- 3)(~'-')2]~,.''}
p}z)= ½(f,,),,- '(n -- l){[f-'-' + (n -- 3)(f'-')-']~ -~)
p13'= (f")"-~(n - 1){f'2¢13' + f~3~_,)+ [~_,3+ (n - 3)f'-'f'3]~"}.
(13)
Substituting these strain expansions into the compatibility equation, the strain compatibility
equation becomes.
aA','r"'Dl~;sO + ~A~'-IA:'-D2Q/qlf,.;s,_) + aA'l' -IA3r"~D3~:[~;s3)
+ etA'; - ~A4r"~D4(jqf4;s4) + ...
+ a A'~' - 2A ~_r"-+ a ' - D p l (.[i 0¢'~) + ~A~' - 3A ~r": + '-a'-'D,.,(/]~&)
+ ~A7 - "-A,.A3r"; + ~'-'D:0C~f.,d%) + ".
+A~r('--")"'D,,~([i) + A,_ra-")"'+~'-D:.(f,.) + A3r ''--")'' + a"'D,.3(/.~) + . . . .
0,
(14)
408
s. YANG et al.
where
D,(/i;s,) = (ffl")' - (ns, + 1)if},')
Dk(/i,/~;s~) = (/~')' -- (ns] + 1 + Ask)ffl#
Dp~,[~,...) = ~l.k') ' - (ns, + 1 + Ak)fil,k'
for k = 2,3,4,--.
for k = i,2,3,...
D,.~(A) = (fl?')' - (s~ + 1)~,k' for k = 1,2,3,...
Ai = 2As2, A2 = 3As,_, A3 = As,_ + As3,""
(15)
In the expansion of the compatibility eq. (14) all the terms which may become one of the first four
terms as r --, 0 are listed. The terms with D,,k are from the elastic strains, i.e. the first part of the
constitutive relation [eq. (2)]. The rest of the terms are from the plastic strains.
The traction free boundary conditions for antiplane shear loading are
rr=O
at0=0
r0=0
at0=Tr,
(16)
which can be expressed as
fi(0) =fi(Tr) = 0
k = 1,2,3,.-.
(17)
Equations (14) and (17) are general governing equations for the asymptotic fields.
Collecting the terms with the same order of r in eq. (14), the coefficients of each power should
be zero in order to satisfy the compatibility equation term-by-term in an asymptotic sense as
r ~ 0 . The first term in eq. (14) is the leading term of the expansion and compatibility equation
becomes D ~ ; s ] ) = 0, to the leading order. Asymptotic analyses indicate that higher order fields
are governed by either linear eigenvalue equations or linear nonhomogeneous equations.
Depending on the value of hardening exponent n, some higher order solutions (k/> 2) can be
obtained by citing D~.(fid~;&) equal to zero which is an eigenvalue problem. The resulting governing
equation D~(fd~.;s~)=0 with homogeneous boundary conditions [eq. (17)] forms a linear
homogeneous equation of second order in ft. In this case, the higher order solution includes an
independent parameter Ak. Since the terms Dr,,,~J;,...)~ 0 and D,.,,,(fi,,)~ 0 in the interval,
0 < 0 < n, (m i> 1), the compatibility equation implies that these terms must be matched with other
higher order terms such as D , ~ ; s k ) (k > m). This matching leads to a linear nonhomogeneous
governing equation. In the latter case, in general, the asymptotic solution does not contain an
independent parameter and the amplitude A, depends on the amplitudes of lower order terms.
4. GOVERNING EQUATIONS OF THE HIGHER ORDER FIELDS
Applying the approach in ref. [6] and comparing the order of the terms in eq. (14) as r--,0, the
governing equations can be obtained in succession. The leading term in eq. (14) is governed by
D~([i;s~) = O, 0 < 0 < rt
.1;(0) = ./i (re) = 0.
(18)
The solution of the above nonlinear eigenvalue equation has been given in refs [8, 12] in an
analytical form
I
sl = - - n+i
(19)
and
n + i / [ ~ / ( 1 -- k2sin20) + cosO][x/(l - k'-sin20) - k cosO] k k = n - I
n q
2(I-k) k
'
n+ i "
(20)
Higher order asymptotic elastic-plastic crack-tip fields
409
The amplitude A~ can be determined by applying the J-integral, i.e.
( 0LS ''+''
where In = rt(n + 1)/2n and J~ is normalized by setting ¢I]~(0) = 1.
It is a p p a r e n t from eqs (14) and (17), the higher-order fields depend on the value of hardening
exponent n. This will be discussed next.
4.1. Stress exponents Sk
Using compatibility eq. (14) and making order analysis similar to that used in ref. [6], the stress
exponents sk can be determined in a sequential manner, and hence are written in the following forms
S I ~
Sit s ~ _
1
n+l
s2 = min {s~,(2 -- n)sl}
s2>s L
s3 = min {s~',s~,(2 - n)st,s2 + As2,(2 - n)s~ + As2}
s3>s 2
s, = min{s~,s~',sg,(2 - n)s~,s2 + As2,(2 - n)s~ + As2
s4>x~
s2 + 2As2,ss + As2,(2 - n)s~ + As3}
•"
(21)
where s~' < s~ < s~ < s~ < ... s~! (k = 2,3,4,...) are the eigenvalues determined by the linear eigenvalue
equations with h o m o g e n e o u s b o u n d a r y conditions
I h
Dk~oq;&)
= 0, 0 < 0 < rt
~!)'(0)
(22)
= f~!(r~) = 0 .
(23)
Equation (22) with k n o w n J] obtained from eq. (20) are second-order linear h o m o g e n e o u s
differential equations inJ~ (k > 1). The superscript h in s~ andJ~ designates that they are solutions
for the higher order fields governing by h o m o g e n e o u s equations. The eigenvalues sg are indeed the
stress exponents of higher order terms with an arbitrarily independent amplitude for pure power
law hardening material. The analytical formula for s~' can be derived from s~' = - (n6j + 2)/(n + l)
in ref. [12] and is written as
~j,= x/(n - 1) 2 + 4(2j - 1 ) 2 n - n - 3
2(n + 1)
' j = 1,2,3,...
(24)
Numerical solutions to eqs (22) and (23) are in agreement for both eigenvalues sg and eigenvectors
~ . F o r a given n (n > 1), it is easy to obtain the stress exponents sk from eq. (21). The dependence
o f the stress exponents sk (k = 1,2,3,4) on the hardening exponent n is shown in Fig. 1. Table 1
provides the analytical expressions of s, (k = 1,2,3,4) in terms of s; or functions of n for various
hardening exponents n. ~ are the stress exponents for pure power law hardening materials under
M o d e I I I loading ]112] without consideration of the dependent terms D,.k(J~) in the compatibility
eq. (14), i.e.
- s~',
1
s¢=
n+
1
s~ = 2s~ - s¢ = 2s~ - s~',
s~ = x/(n - 1)2 + 3 6 n - n - 3
2 ( n + 1)
s~ = x/(n - 1)2 + 1 0 0 n - n - 3
= s~'
-'
(25)
= s~',
2(n + 1)
= 3s~ - 2s~ = 3s~ - 2s~',"',
where sf < s~ < s~ < s~.... It can be seen from eq. (25) that the third and fifth order terms do not
contain independent p a r a m e t e r s in pure power law hardening materials.
410
S. Y A N G
1.5
I
et al.
I
I
I
%.
1.0
0.5
g
g
0.0
°-°
°'°
~
'-"
.....
S1
. . . . .
S4
-0.5
-1.0
,
J
I
i
~
4
~
I
i
i
r
8
I
i
~
12
~
I
,
~
16
20
n
F i g . 1. V a r i a t i o n
o f s t r e s s e x p o n e n t s s~ (k = 1,2,3,4) o n t h e h a r d e n i n g
exponent
n.
T h e stress e x p o n e n t s c a n be d e t e r m i n e d a l t e r n a t i v e l y b y
s,=s~'=
-
n+
1
1
(26)
s2 = min{s~,(2 - n)s~}
v2 > s I
s3 = min{s~,s~,(2 - n)s,,(2 - n)s, + A&}
s 3 > ~2
p
1
P
P
s4 = mln{s4,s>s,_,(2 - n)&,(2 - n)s, + As_,,(2 - n)s, + As~}
'~4 > ~3
where s~' < s; < s~ < s; < ....
4.2. Governing equations for the higher order field
A f t e r d e t e r m i n i n g stress e x p o n e n t &, collecting coefficients o f all the t e r m s with p o w e r sk in
r, a n d e q u a t i n g coefficients o f this p o w e r o f r to zero, the c o r r e s p o n d i n g g o v e r n i n g e q u a t i o n for
the k t h o r d e r field c a n be o b t a i n e d . T o describe this p r o c e d u r e , we c o n s i d e r the f o l l o w i n g cases
with n = 3 , 1 0 a n d n > 13.
4.2.1. Governing equations .['or n = 3. Since
sl =
- 0.25,
s~' = 0.5729,
s~' = 1.4294,
s~ = 2.2914,
we have, f r o m eq. (21) o r (26),
s,=
--0.25,
s.,=(2-n)s,=0.25,
s3 = s~ = s~ = 0.5729,
s4 = s_, + As_, = (2 - n)s, + As: = (3 - 2n)s, = 0.75
ss = s~ + As2 = (2 -- n)s, + As3 = (1 - n)s, + s~ = 1.0729.
T h e g o v e r n i n g e q u a t i o n s o f h i g h e r o r d e r terms are as follows:
(I) T h e s e c o n d o r d e r term
~A','- ' A2D,_(I;,I;;s,_) + A,D,.,(I;) = O.
(27)
Since s_, = (2 - n)s, ¢ s~!, (k = 1,2,...), i.e.s., is n o t a n e i g e n v a l u e o f the h o m o g e n e o u s p r o b l e m
g o v e r n e d b y D2(/',,[~;s2) = 0. A c c o r d i n g to the t h e o r y o f o r d i n a r y differential e q u a t i o n s , the
1.9041
A , = A~(J)
5 < n < 11.3627
A2
A_, = ct - ~A~ "
s2=
(n - 2)
(n+l)'
s2,A,_
s4 = s ~ , A 4
s~ =
-
s,
'
((nn +- 2)
1)'
s4 = 2s,
-
(n +
2(n+ +36n
1) - n - 3 , s{ = 2s¢ - s~' =
I)-'
-
(3n-4)
(n+ 1)'
) ' s¢- = x / ( n -
(n-2)
(2n-3)
(n+ 1)' (n+ 1)'
-
x/(n
1)2 + 36n
(n + 1)
n - 2
then the corresponding higher order terms are controlled by both elastic and plastic strains.
Ifs~=
i.e., g
' s4e =
x//(n
-
-'A~-2"
~ ~A4 3,,
-
A4 = ~ - I A ~ - "
A4 = A~/AI
-
(2n--3)
~ =~
(n + 1) ' ,~4
(n - 2) , A3 = Qt ' A ~ - "
s3 = (n + 1 ) '
s3 = 2s2 -- sl , A3 = A 2 / A ,
$4,A4
(3n - 4)
( n + 1) ' A 4 =
s4 = s_g, A4
s4=
s4 =
(2n - 3)
( n + 1) ' A 3 = c t - ' - A ~ - ' - "
s3 = s.~, A3
s3=
s3,A3
1) 2 + 100n - n
2(n + 1)
_
3
sf a r e stress e x p o n e n t s o f t h e t e r m s w h i c h a r e d e c i d e d solely b y p l a s t i c s t r a i n s o r stress e x p o n e n t s f o r p u r e p o w e r - l a w m a t e r i a l ,
13 < n
11.3627 < n < 13
s, = s~
si,Ai
2.5352 < n < 5
1.9041 < n < 2 . 5 3 5 2
1 <n<
n
T a b l e I. E x p r e s s i o n s o f s t r e s s e x p o n e n t s s~ a n d a m p l i t u d e s A~ in a s y m p t o t i c e x p a n s i o n
which control
J , A_,
J , A3
AI o r J
Ai o r J
three-term expansion
Parameters
o~
i
¢'0
J
Q
O
S. Y A N G e t al.
412
homogeneous problem only yields a trivial h o m o g e n e o u s solution. Hence, the linear
n o n h o m o g e n e o u s problem governed by eq. (27) has a unique particular solution for a known./'~.
Solving eq. (27) leads to the solution A,_ = A~-"/~ and)C_, is the solution o f the following equation:
D2(l~f2;s2) + D,,,~) = 0
s, = (2 -- n)s,, A,_ = A~ "/~,
(28)
with . ~ ( 0 ) = f _ , ( n ) = 0. It is noted that similar situations will occur in other n o n h o m o g e n e o u s
governing equations in the latter part o f the paper and similar choices o f amplitude will be made.
(2) The third order term
(29)
D3@~;s3) = 0, s~ = s~.
An amplitude At is an independent parameter associated with the solution.
(3) The fourth order term
c~AI'- ' A 4 D , ~ f4;s4) + c~A':- ZA~Dp~(Jl f2) + A2D,.z(f,_) = 0
O , 0 q & ; s 4 ) + O~,0q05) + D,,.,05) = 0
(30)
or
s4 = (3 - 2n)s~, A4 = A~-2"/c~ 2 .
(4) The fifth order term
~A'~'- ' AsDsOq~oCs;ss) + ~A]'- 2A~A3Dp3(JI~f_~0¢3) + A3D,.30~3) = 0
Os0q,/~;ss) + Op3(Jqlg(2~) "[- D,.3(/~) = 0
(31)
or
p
s5 = (1 - n)s, + s2, A5
~
AI
- - II
A3/O~ .
F r o m the governing equations, it can be seen that the second, fourth and fifth order fields are
controlled by both elastic and plastic strains, while the first and third order fields are controlled
solely by plastic strains. In addition, the elastic strains enter the second term o f the asymptotic
solution. The asymptotic expansion up to the fifth order depends on the two independent
parameters A~ (or J) and A3. Solving the governing equations, the numerical solutions for the higher
order fields can be obtained. Note that the analytical solution for the third order field can be
obtained from ref. [12].
4.2.2. Governing equations f o r n = 10. Since
s~ = - 0.0909, s~ = 0.3636, s~' = 0.9036, s~ = 1.4626,
we have, from eq. (21) or (26),
sl = - 0.0909, s2 = s~ = st = 0.3636, s3 = (2 - n)s~ = 0.7273,
s4 = 2s_, - sl = 2s¢ - sl = 0.8182, s5 = s~ = s~ = 0.9036.
The governing equations are written as follows:
(I) The second and fifth order terms
D~(J',,/2;s~.) = O. k = 2.5
(32)
s: = sL s5 = st.
(2) The third order term
~AI'-'A,D,(/I,/;;s,) + , 4 , 0 , 4 / i ) = 0
D~(/',,/i;s3) + D,,,(/I) = 0
or
A3 = A~-"/~, s3 = (2 - n)s,.
(33)
Higher order asymptotic elastic-plastic crack-tip fields
413
(3) The fourth order term
+
= 0
(34)
D4q;0¢4;s4) q- D p , ( j q l t f 2 ) = 0
or
A, = A~/AI, s, = s~ .
It is obvious that the effect o f elastic strains enters the third order term. The asymptotic solution
up to the fourth order term depends on two independent parameters A~ (or J) and A.,. N o t e that
analytical solutions for the second, fourth and fifth order fields which are completely controlled
by plastic strains can be found in ref. [12].
4.2.3. Governing equations f o r n >
13. In this case, we have the following exponents
s, = s~, k = 1,2,3,4
and the governing equations for higher order fields are:
(1) The second and fourth order terms
(35)
D k ~ f k ; S k ) = 0, k = 2 and 4
s2 = st, s4 = s~
(2) The third order term
D30q0q;s3) + Dp~0qd~) = 0
s3 = 2s~ - sl, A3
=
(36)
A~/AI.
F o r the asymptotic solution up to the fourth order term, all the terms are attributed solely by the
plastic strains. The: three-term asymptotic solution depends on two independent parameters A~ (or
J) and A2. N o t e that analytical solutions for the second to the fourth order terms are given by
ref. [12].
In analytically determining the stress exponents, Sk are selected from the smallest exponent
in the brackets in eq. (26) for a given hardening exponent n. F o r some particular values
o f n, we can find that exponents from plastic strains and elastic strains become equal,
s, = s~ = (2 - n)sl + As,,, (k >>,j , m ) . These values can be extracted graphically from the junctions
o f any two continuously differentiable curves for a given sk in Fig. 1. These few special cases for
some higher-order fields J~(k > 1) with the particular values o f n requires special attention.
Considering the first four terms in Table 1, the cases can be summarized as follows:
13 + x / ~ ]
n
-
n -
12
4+,/53
~
s4 = s~ = (2 - n)s~ + As3,
~ 1 9041
=
"
= 2.5352,
n = 5,
n -
17 +
n = 13,
3
4th order term
'
~ 11.3627,
s3 = s4 = s¢ = (2 - n)st + As_,, 3rd and 4th order terms
s2 = s3 = s~ = (2 - n)sl,
2nd and 3rd order terms
s4 = s~ = (2 - n)st + As,_,
4th order term
s~ = s4 = sf = (2 - n)s~,
3rd and 4th order terms
s4 = s~ = (2 - n)s~,
4th order term.
(37)
In these discrete cases, although the higher-order fields are governed by a linear n o n h o m o g e n e o u s
equation, the solutions m a y or m a y not contain an arbitrarily independent parameter.
F o r example, n = 5, according to the preceding analysis, there are two cases needed to be
considered.
414
S. YANG et al.
4.2.3. I. The second and third order terms. These two orders merge into a single term when
s2 = s~ = s~( = s~) = (2 - n)s, = 0.5
and the governing equation for the new second order field with the boundary conditions, identical
to eq. (28), are
D 2 ~ ; s 2 ) + D,,,(fi) = 0
(38)
rE(0) =f2(n) = 0, A2 = A~-"/~.
Since s_, = s~, s2 is an eigenvalue of the homogeneous problem, D20q,f_,;s2) = 0. Hence, the
homogeneous problem has a nontrivial solution defined as f2' which adds an additional independent
parameter. The analytical form o f ~ can be obtained in ref. [12]. The complete solution of eq. (38)
can be written as
f-, =f2' + ~ ,
(39)
where ~ is a particular solution of eq. (38). ~ can be determined numerically.
4.2.3.2. The fourth order term
s4 = s~( ~ s~) = (2 - n)s, + As2 _~ 1.1667.
The governing equation for the new third order field with the boundary conditions is
~A~ - 'A,D4Qq~;s4) + a A f - :A~D,,(ft&) + AED,2(J~) = 0
(40)
j~(o) = f i ( ~ ) = o.
Since s4 ~ s~, s4 is not an eigenvalue for the homogeneous problem, O40']d'¢4;S4)=0. Hence, the
homogeneous problem has only a trivial solution. The solution of eq. (40) consists of particular
solutions only, one from Dp,~f_,) and the other from D,,,~). Therefore the solution does not
contain an independent parameter.
In a similar manner, the governing equations of other particular n in eq. (37) can be obtained.
Summarizing the analysis, we can reach the following conclusions:
i. when s, = s~' and s, ~ ( 2 - n)s, + As .... the higher order field is governed by the linear
homogeneous equation, controlled by plasticity only, and described by an independent parameter;
2. when s, = s~' and s~ = ( 2 - n)s~ + As,,,, the higher order field is governed by the linear
nonhomogeneous equation, controlled by both elasticity and plasticity, and the solution includes
an independent parameter;
3. when s~ = sf, sk ¢ s~' and s, 4:(2 - n)s, + As,,,, the higher order field is governed by a linear
nonhomogeneous equation, controlled by plasticity only, and the solution does not include an
independent parameter;
4. when s, = s~', s, ~ s~' and s~ = (2 - n)s, + As,,,, the higher order field is governed by a linear
nonhomogeneous equation, controlled by both elasticity and plasticity, and the solution does not
include an independent parameter;
5. when s, ¢= sp, thus s, = (2 - n)s, + As,,,, the governing equation is nonhomogeneous, the
higher order field is controlled by both elasticity and plasticity and described by the parameters
belonging to the lower order terms. In other words, the solution does not include an independent
parameter;
6. when s, = s; and s, 4:(2 - n)s, + As,,,, analytical solution can be obtained from ref. [12],
i.e../~ =.Le..ff is the angular function of the jth order field for pure power law hardening materials
with the same value of n.
The same procedure can be continued for other higher-order fields.
Higher order asymptotic elastic-plastic crack-tip fields
415
5. FORMS OF THE ASYMPTOTIC CRACK-TIP FIELDS
After solving the governing equations for.A and s~, the stress expansion can be expressed in
the form:
(41)
zi = A~r~'~ u + A2r"-~ii 2~ + A3r~'~i 3~ + "", i = r,O ,
where ~*> are given by eq. (10). Depending on n, the amplitude A, (k >~ 2) discussed in the preceding
section may be undetermined or determined by A~..... A,_ ~ and given in Table I.
Collecting the ~:erms having like powers of r in eq. (12) and rewriting the equation, we obtain
the asymptotic expansion for strains
~ = aA'~'r"~i" + ~ A I ' - IA2r"~' + ~,~,2~ + ctA~'- ' A3r '~' + ~,~,~3~ + ...
(42)
where ~*~ (1 ~< k ~< 4), depending on the form of the governing equation for f , , can be written in
the form:
/~I~
~/ k)
when D~ = 0
/~I~' +/~I"
when D~ + D~, = 0
/~I~ + ~')
when Da + D,,I = 0
(43)
ffl,~+/~I~, + ~2~ when D , + Dp, + D,.2 = 0
ffl ~' + ffl"' + ffl3' when D, + Oe~ + Dp3 = 0
-Pl*' + ffl-" + Pl ~' + ~3,
when D~. + Dr,. + D~ + D,.3 = 0,
where expressions of/~I *~, Pl*' have been given by eq. (13).
Integrating the relation 7,. = 8 w / t 3 r , the displacement asymptotic expansion can be expressed
as
w = w ~ + ~tAi'r ''~' + ~1~ + orAl'-~A~_r"" + 1+ ~,_~12>+ o~A]'-~A3r "~' + ~+ a,~,o~ + ... ,
(44)
where w R is the rigid body displacement and
~l*, _
~,k~
ns~ + I + A s ,
(45)
"
Multi-term expansion can be used to describe the elastic-plastic crack-tip fields. F r o m Table 1,
the four-term expansions for the stress field are given by
Atr'~,~ u + a-~(A~r'~,)'--"~'-~+ ct-'-(A~r'~')3-2"~ 3~ + ~t-3(A~r',)4-3"f14~ + ... for 1 < n < 1.9041
Air~'~i u + ~-~(A~r"')2-"~ ~-~+ ct--*(Air~t)3-2"~,.3~-F A4r~4~l4~ + . " for 1.9041 < n < 2.5352
Air"'fi u + ct-I(A~r")'--"~ 2~ + A 3 r ' ~ 3~ + 0~-'*(Ajr~')3-2"~i4> 4- "" for 2.5352 < n < 5
A~r"fl" + A2r"-~"~ + ~-~(Atr")'--"~3~ + A ~ A ( 'r2~'--"'~4~ + "'" for 5 < n < 11.33627
Ajr"i~) + A,r"'-~ 2J + A~_Ai-Ir'-"--"~i 3~ + ct-I(Air"~)'--"~4~-I- "" for 11.3627 < n < 13
Atr"'~u+
A2r'-~'-~+ A ] A ( 1r-%-~'~3~ + A 4 r ' 4 ~ i 4 1 - F
""
for 13 < n.
(46)
It is clear from Table 1 that the three-term expansion is characterized by one or, at most, two
parameters. Similarly, these multi-term expansions for strain and displacement fields can be
obtained from eqs (43) and (45).
416
s YANG et al.
For the special cases of n in eq. (37), the multi-term expansions of stress fields can be
modified as
A,r,,fl~) + ct ,(A,r,,)2-,,~,2) + ~-2(A,r,,)3
~_,,~3)+ ~ 3(A,r,,)4- 3,,(A4f(a)h + ~4)/,) +
for n ~_ 1.9041
Air~,...glt)_.k ~-i(Air.,,)2-,,~.2) + ~-2(A ~~.,,~3
.~(3)/,+ .?13)/,)+ .. • f o r n ~ 2 . 5 3 5 2
s 2,,tA
~3~i
z~ =-~
A,r"'~l t) + o~-'(A~r")2-"(A2fl ')h + fl 2)') + "" for n = 5
A,r"~i '~ + A,r"-'?l q- r 12-")"(0~
Air~,~ 11 + A,r"-.?l
2) +
_
A ;'_
A_,
-
'a",11-,,.r-(3),,i + A~_At- 'f1312) + "" for n = 11.3627
-, r .,,,_
. . n
. . . . . ~I ~ + ~ ~(A,r") ~2 "~(A,g "~' + ~I"~) + . . for
13,
(47)
where the superscripts h and p on the angular distributions denote the homogeneous and particular
solutions, respectively. ~(3~ and ~3~_,are given by f~ and j~, respectively, while J~ and f i are the
solutions of the following equations:
D3(/q~ofi;s3) + D,.,0q) = 0, A3 = ~- IAf
D3(ff;OC3,s3)-t- D,,,(Jq.f_,) = 0,
A3 =
"
(48)
A~_A,-i,
respectively.
6. N U M E R I C A L
RESULTS
AND
DISCUSSION
The governing equations (eigenvalue problems or nonhomogeneous equations) are solved by
using a shooting technique• This technique integrates the ordinary differential equations utilizing
the Runge-Kutta method with adaptive step sizes to ensure the accuracy. The initial guess for the
initial values is adjusted systematically by the multi-dimensional Newton-Raphson method, and
the procedure repeated until the boundary conditions at the end of the integration and some
predetermined convergence criterion are satisfied•
Figures 2 and 3 illustrate the first four terms of stress angular distribution for n = 3 and n = 10
In the figures, ~*~, k = 1,2,3,4 are normalized (fl~'(0) = 1) if the kth order term is governed by a
homogeneous equation• For the higher order terms governed by non-homogeneous equations,
0.0~
1.2
0
n=3
1 . . . . . . . . . . . . . . ~.•.. .......
-0.05
0.8
-0.1
0.6
43.15
0.4
-0.2
0.2
-0.25
0
45
e
(Deg0eree)
................................................. .~.....
-0.3
135
........
0
leo
' ........
45
' ........
e
90
(~er~)
...."
' ........
135
180
0.~
2
O,O6
I
0.04
!
0
0.02
•
-1
"''
'..
0
-2
-0.02
-3
-0,04
0
45
O
(l~9011re~)
t3s
18o
......... ;~4)
0
45
0
~elp'ee)
135
F i g 2. The stress angular distribution function of the first four terms for n = 3.
t~
Higher order asymptotic elastic-plastic crack-tip fields
1.2
4
I
2
417
n=10
I
0.8
0
!
0.6
-2
0.4
-4
0.2
.........+
-6
0
45
90
135
90
135
180
e (Degree)
e (i~)
200
0.2
a=lO
__
~3)
I{]0
0.1
!
........................
.-..._t
45
180
0
!
0
"'" ..........
-O.I
-I{]0
-O.2
-200
45
e (~gn~)
13s
-300
0
iso
" . ".
~
.......
. . i"
",,....,,
........
....
45
e ~g.~)
Xk
i"~': : ",'i"i ....
135
:/"
'
,
iso
Fig 3 The stress angular distribution function of the first four terms for n = 10
normalization is unnecessary. It is noticeable that the first term for each case is well-known from
eq. (20). For n = 3, the second and fourth terms are elastic-plastic terms with the corresponding
amplitude dependent on the J value. The third order term is the pure plastic term which is exactly
the second order term in the power-law material solution [12]. For the lower hardening material
n = 10, the plastic effect is stronger than that o f n = 3 and the elastic effect b e c o m e s weaker. I n
this case the second and fourth order terms b e c o m e the pure plastic term and the third term is
the elastic-plastic term. The second and fourth order terms can also be determined by analytical
expression in ref. [12].
To validate the asymptotic expansion solution, an infinite strip o f material shown
in Fig. 4 occupying the region, - o o < ~ < o o , - h < f i < h, with a semi-infinite crack
( - oo < ~ < 0, fi = 0) is studied in detail. Antiplane shear is applied on the upper and lower
boundaries o f the strip, with • I,~=,, = w0 and • I,~= _ h = - w.. All the barred quantities are defined
in the non-normalized sense. To normalize all the quantities for the problem, we define L = h,
yo = wo/h in eq. (3). In the following figures, all the variables are referred to normalized quantities.
A nonlinear finite element analysis was carried out using A B A Q U S with specially designed user
W=Woet~--o)
~=0
.......
ili
~=0
0
zoLv.<o! .............. 2h
x
w = -w0 (Txffi0)
Fig. 4. A semi-infinite crack in an infinite strip under constant displacements along the edges.
418
S. Y A N G et al.
1.8
O.S
n=3, a = l
O.7
n=3, a = l
8=90"
1.6
0=45"
IA
0.6
o
~f 0.5
",
^
.
. . . . . . . . Leading T a m Solution
_ Three Term Solution
1
0.4
0.8
~
o
o
o o
o o
0.3
0.6
i
0.2
i
,
i
I
0
. . . .
0.05
i
. . . .
0.1
r
i
. . . .
0.4
0.15
. . . .
i
0.2
. . . .
0.05
l
. . . .
0.1
r
I
. . . .
0.15
0.2
2.5
n=3,
n=3, a = l
8=180"
a=l
2.5
e=135"
2
,..
1.5
x,
.....*-...~t,~d,,Tn Sol.~,
1.5
~.....
-..~....~L~.. ~.~ Solu~oo
" ° ° ° ° " " ' ' " ' ' ~ ...............................
I
I
. . . .
0.5
i
. . . .
0.05
i
. . . .
0.1
f
i
. . . .
0.5
0.15
. . . .
i
0.2
. . . .
0.05
i
. . . .
0.1
r
i
i
i
i
i
0.15
0.2
Fig. 5. The radial distribution of stress component z~ for n = 3, • = 1.
defined element to account for the material constitutive relation. Convergence of the mesh is studied
by comparing the results from meshes with different sizes. The analysis uses a mesh with the size
of four-node isoparametric element at the crack tip being 0.02% of the strip height. In addition,
4
4
n=3, a = l
3.5
3
3
2.5
xe
n=3, a = l
8=45"
33
' ~ . . . . I1=0°
2.5
o
~'.
2
F~
. . . . . . . Leading Tern1 So!ution
~e
1.5
2
o__l
\
13
1
1
. . . .
0.5
i
0
. . . .
0.05
i
. . . .
0.1
r
i
. . . .
0.15
03
0.2
0.05
3
0.1
f
0.15
0.2
1.6
n=3,
2.5
ix=l
1.4,
0=90"
n=3,
a=l
0=135"
1.2
2
13
1
xo
~.....
0.8
....°...~,,__ Sol,~o
~".... .... ,
.....*...~m~,a
Solu,~
0.6
1
0.4
O.5
0.2
0.05
0.1
r
0.15
0.2
0.05
0.1
r
Fig. 6. The radial distribution of stress component ~. for it = 3, • = I.
0.15
0.2
H i g h e r o r d e r a s y m p t o t i c elastic-plastic c r a c k - t i p fields
419
6
n=3,
a--I
n=3,
a--I
5
0=90"
0=45"
4
~r
3
"!
o
~
Three T e r m Solution
2
1
" ..%..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
0
i
.
i
i
i
0.05
0
,
.
.
.
.
,~
. . . .
i
0.1
r
0.15
,
-
•
i
. . . .
,
0.05
0.2
. . . .
0.1
r
A . . . .
0.15
0.2
I0
a=l
n=3,
n=3,
.~.. O=I3F
~
¥f
a=l
0=180"
Yr
i
•
-
-
i
.
"':,.~
........ L e a d i n g T e r m Solution
~
T a m Solution
"**.**'~___________~
"-....~,.~
.
.
.
i
0.05
.
.
.
.
i
0.I
r
._.°_ F~nlI:
. . . .
. . . .
0.15
i
0.2
. . . .
i
0.05
T e m t Solution
. . . .
0.I
r
i
. . . .
0.2
0.15
Fig. 7. The radial d i s t r i b u t i o n of strain c o m p o n e n t ~;,, for n = 3 and ~ = I.
the leading solution was calculated by evaluating the J-integral around the crack tip. The J-integral
for this example can also be obtained analytically:
J
=
2~n ~,,+ 1)
r0y0L ~2 + ~
(49)
,
20
14
n=3,
a=l
12
0=0"
10
7e 8
6
10 1o
°__
......... L e a d i n g T e r m Solution
.....o....
4
5
2
. . . .
i
.
i
i
•
0.05
i
.
.
.
i
0.1
r
!
i
i
,
0.15
0
i
0.2
. . . .
i
0
. . . .
i
0.05
. . . .
0.I
r
i
. . . .
0.15
0.2
I0
."=1
8
n=3,
.....~.e
a=l
= 135"
6
Ye
4
"
-
o FE~
-........
I.h~|
.
~
~
Term ~lut~n
.....
2
. . . .
0
0
i
0.0~
. . . .
i
0.I
r
.
.
.
i
J
0.15
. . . .
•
0.2
i
•
-
i
0.05
.
.
.
i
P
.
.
.
0.1
r
Fig. 8. The radial d i s t r i b u t i o n o f strain c o m p o n e n t 7,, for n = 3 a n d • = I.
EFM 5413 E
.
i
0.1J
i
i
i
i
0.2
420
S. Y A N G
et al.
1.4
1.2
n=3,
1
a=l
~.~"........
r=0.1
nffi3, ¢gffil
"'....
. . . / " ' " .... -.-...o.'". .-"
0.8
+"~........ ""~
1.2
r=O.1
1
"'.-.~
...o
%..
0.8
'~r 0.6
0.6
0.4
........ Leading Term Solution
0.4
0.2
~ , ~ . "..,
. . . . . Two Tram ~Intiou
~
:'...
0.2
.
0
0
4.5
0
90
135
e (Degree)
180
45
90
135
e (~e~ee)
180
1.2
............................
0.8
1
rffi0.2
nffi3, (xffil
....
.~.,,,...,""-...,... ""-..
P°°Ooo
r'+0.2
0.8
0.6
"-.%..
~e
"...%
0.6
0.4
0.4
,~./-
0.2
.........~ : , , ~
o,,~_,,"~
--...-Two
T-am Solution
0.2
0
0
45
0
90
135
e (~gme)
o +
........ ~
Term Solutiou
. . . . . Two Ta'm Solution
Tctm Solution
180
F i g . 9. T h e a n g u l a r d i s t r i b u t i o n
0
45
~%~'-.. "..
" ~ . . _ "..
-'~...
90
135
e (~ee)
180
o f s t r e s s f o r n = 3, c< = I.
with + satisfying
+ -t- c++"= 1.
The numerically determined J value agrees almost identically with the analytical expression.
3.5
nffi3, a l l
r=0,1
1.5
3
.~.- . . . . ". . . . . . .
..-~"'" o o o o o o o o o
oo
Yf
........ °-
o o
°
o o
.
.
.
.
.
.
.
.
_
n=3,
otfl
2
Te
..
°.,
,
1.5
....... I..~ding Term Solmion
. . . . . Two Tram Solution
" I ' ~ Twin Solution
1
- .... ~
T a m Solution
. . . . . Two Tram Solution
.
45
.......... ~.......
o~
_~ "j"
~...%."
~
2.5
.,..~.~
~.J..<o.1--"
0.5
!
O.5
.
0
90
135
o (~g~e)
180
.
.
.
.
0
.
.
I
.
.
.
.
.
45
.
.
I
.
"'~R~...
~l~,..'~.
" " ~
.
I
.
.
90
o ( ~ s , ee)
.
.
.
I
.
.
.
.
.
.
.
135
180
2-~
1.2
nffi3, C(ml
1
. ~ """" """ ~ ....
ra0.2
....?
0.8
oo
0.6
~
0t=l
" g ' o. ~ . , .
~ 6
~
.... ~°
....
•
° ° ~
oo
n=3,
2
..~'"
o
.
.....
1.5
°o
:.---i"----.?-..'
.........
~i .....
Ye
.......
1
0.4
o
..... ....
o~-"
..-- ~
:/.:.~/"
T.~.~.
0.5
0.2
. . . . . . . . . . . . . . . .
0
0
45
9O
e (Desree)
135
180
F i g . 10. T h e a n g u l a r d i s t r i b u t i o n
0
~
~
.
_
. . . . . Two Tm'mSolution
Tlzw Tmu Selutlm
........
a ........
a ........
45
90
o (~sree)
o f s t r a i n f o r n = 3 a n d ct = I.
a . . *-. * ~*1 ~ -
135
180
Higher order asymptotic elastic-plastic crack-tip fields
421
The material parameters n = 3 and 0~= 1 are selected for the study. A truncated three-term
expansion solution with two independent parameters, J and A3, is presented. The value of A3 is
determined by a point matching technique with the stress value obtained from the FEM. The stress
Zo at a chosen point 0 = 90 ° and r = 0.05 is set equal to the three-term asymptotic expansion result
to determine the value of A3. The A3 in this example has the value of 0.11 for normalized stress
distribution. Since the truncated asymptotic solution has been used to match the complete solution
from the FEM, the A3 value only approximates the true A3 in the asymptotic expansion.
The stress and strain distributions at four different angles are plotted in Figs 5-8 with the
coordinate r being the normalized radial distance. It is obvious that all the three solutions match
quite well in a region near the crack tip, and the three-term solution agrees well with FEM results
over a wider region. Finally, the effect of multi-term expansion and independent parameters on
the stress and strain distributions is considered. The stress and strain components using two term
expansion controlled by a single parameter J, three and four term expansions determined by two
parameters J and A3 are shown in Figs 9 and 10. The results from FEM and the leading term
solutions are also shown in the figures. In these figures, the angular distributions at r = 0.1 and 0.2
are demonstrated. It is clearly seen that the two term expansion with a single controlled parameter
J improves somewhat better than the leading term solution; the distribution of stresses and strains
still departs from the FEM results. Comparing the solution controlled by two parameters between
three term and four term asymptotic expansions, they agree well with each other and match closely
with the FEM results. In general, the prediction of the stress distribution from multi-term
asymptotic expansion fits more accurately than that of strain distribution.
7. CONCLUSIONS
A series solution with assumed separation of variables form for the stress and strain fields near
the crack tip in ar~ elastic power-law hardening material under antiplane shear has been developed.
The leading term is analytically obtained by solving a nonlinear eigenvalue problem. The higher
order fields are governed by either linear homogeneous eigenvalue equations or linear
nonhomogeneous governing equations. The stress exponents of higher order field for any hardening
exponent are analytically determined. The governing equations for higher order terms which are
controlled solely by the plastic strains can also be obtained analytically. However, the governing
equations governed by elastic and plastic strains need to be solved numerically. With the
analytically determined stress exponents, distinct regions resulting from different strain hardening
exponents where the higher order terms up to the fourth order attributed to the plastic strains or
elastic and plastic strains can be identified and are shown in the Table 1. It has been demonstrated
that a truncated three term solution with two parameters accurately characterizes the crack tip
stress and strain fields.
Acknowledgements--This research is supported by the National Science Foundation Grant No. MSS-9202223 and the U.S.
Air Force Office of Scientific Research under Grant F49620-95-1-0256.
REFERENCES
[I] J. W. Hutchinson, Singular behavior at the end of a tensile crack in hardening materials. J. Mech. Phys. Solids 16,
13-31 (1968).
[2] J. R. Rice and G. F. Rosengren, Plastic stress and strain fields at a crack tip. J. Mech. Phys. Solids 16, 1-13 (1968).
[3] J. R. Rice, A p~.th independent integral and approximate analysis of strain concentration by notches and cracks.
ASME, J. appl. Mech. 35, 379-386 (1968).
[4] Y. Li and Z. Wang, Higher order asymptotic field of tensile plane strain nonlinear crack problem. Scientia Sinica
(Series A) 29, 942-955 (1986).
[5] S. M. Sharma and N. Aravas, Determination of higher-order terms in asymptotic elastic-plastic crack tip solutions.
J. Mech. Phys. "~olids 39, 1043-1072 (1991).
[6] S. Yang, Y. J. Chao and M. A. Sutton, Higher order asymptotic crack tip fields in a power-law hardening material.
Engng Fracture Mech. 45, 1-20 (1993).
[7] L. Xia, T. C. Wang and C. F. Shih, Higher order analysis of crack tip fields in elastic-plastic power-law hardening
materials. J. Mech. Phys. Solids 41, 665~587 (1993).
[8] H. Reidel, Cracks loaded in anti-plane shear under creep conditions. Z. Metallkunde 69, 755-760 (1978).
[9] J. R. Rice, Stresses due to a sharp notch in a work-hardening elastic-plastic material loaded by longitudinal shear.
ASME, J. appl. Mech. 34, 287-298 (1967).
422
S. YANG et al.
[10] C. Atkinson and C. R. Champion, Some boundary-value problems for the equation V-([ V~ J"V4~)= O. Q. i . Mech.
appl. Math. 37, 401-419 (1984).
[11] N. Aravas and D. A. Blazo, Higher order terms in asymptotic elastoplastic mode-Ill crack tip solutions. Aeta Mech.
90, 139-153 (1991).
[12] F. G. Yuan and S. Yang, Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear. Int.
J. Fracture 69, 1-26 (1995).
(Received 9 September 1994)