- Email: [email protected]
The failure of a strain-softening material: II. The effect of geometrical and loading parameters
Theoretical and Applied Fracture Mechanics 11 (1989) 65-70
65
THE FAILURE OF A STRAIN-SOFTENING MATERIAL: II. THE EFFECT OF GEOMETRICAL AND LOADING PARAMETERS E. SMITH Joint Manchester University/UMIST, and Materials Science Centre, Grosvenor Street, Manchester M1 7HS, U.K.
This paper presents theoretical analyses of a variety of models which simulate crack growth in a strain-softening material, with attention being focused on the fully developed softening zone length and the value of the crack tip stress intensity associated with the attainment of such a state. Results from the models show that both these parameters can be very sensitive to both the initial crack configuration and the loading characteristics, and can differ appreciably from the values appropriate to a semi-infinite crack in a remotely loaded infinite solid. The present paper's results underline the view that the analytical results obtained in Part I, and other workers' numerical results for a specific material, are rather special. Part I analyzed the behaviour of a crack in a double cantilever beam specimen, and it was shown that the value of the crack tip stress intensity associated with a fully developed softening zone is essentially independent of the initial crack size and beam height, and is equivalent to the value for a semi-infinite crack in a remotely loaded infinite solid.
1. Introduction
As indicated in Part I [1], a wide range of materials exhibit a phenomenon known as strain softening, whereby if a material contains a crack and is stressed, the crack extends and its faces become bridged by unbroken ligaments whose average behaviour can be represented by a decreasing stress versus increasing crack face opening relationship. The crack extension is associated with an increasing crack resistance (i.e. increasing crack tip stress intensity K), an effect that has been observed in concretes, ceramics, and fibre cements. During this stable crack growth, a strain-softening zone gradually develops behind the crack tip, i.e. the leading edge of the softening zone, so that the crack length is equal to the initial crack length plus the length of the softening zone; this produces a stabilising influence on crack growth and leads to an increasing crack growth resistance ( K ) curve. Foote, Mai and CottereU [2] have calculated the K resistance curves for a specific fibre cement composite (linear softening law) in a double cantilever beam (DCB) geometry and showed that, while the fully developed softening zone size R is very dependent on the beam dimensions, the K value associated with a fully developed softening zone is essentially independent of these parameters and is approximately equivalent to the value appropriate for a semi-infinite crack in a remotely loaded infinite solid. 0167-8442/89/$3.50 © 1989, Elsevier Science Publishers B.V.
These numerical results have been underpinned by the analytical results obtained in Part I [1], where a simple Euler-Bernoulli representation of the DCB specimen was employed. The present paper also addresses the problem of crack growth in a strain-softening material, with the objective of studying other geometrical and loading configurations. With this objective in mind, we adopt a very simple description of the softening zone, and use the DBCS representation [3,4], where the stress within the softening zone in fact retains a constant value. This simple description should be adequate to give broad general conclusions regarding the effect of geometrical and loading parameters on crack growth resistance. Sections 2-6 apply the DBCS type representation to a variety of simple models, for which analytical results can be obtained, and show that the fully developed softening zone length R and the associated value K A of the crack tip stress intensity are very sensitive to both the initial crack configuration and the loading characteristics, and can differ appreciably from the values appropriate to a semi-infinite crack in a remotely loaded infinite solid. With regard to the crack tip stress intensity, these results contrast sharply with those obtained [1,2] for the double cantilever beam geometry. It would, therefore, seem that results for this specific geometry are special, and should not be used to draw general conclusions about crack growth in a strain-softening material.
E. Smith // The/allure of a strain-so[tening material: 11
66
2. The model of an isolated crack in a uniformly stressed infinite solid
Figure 1 shows the model of an isolated two-dimensional crack in an infinite solid subject to a uniform applied tensile stress normal to the crack plane; it is assumed that there is a fully developed softening zone at each crack tip. The initial crack size is 2a 0 and the crack size at the attainment of the fully developed zone is 2 a, the applied tensile stress being o; at this stage, the cohesion becomes zero at the initial crack tip. The tensile stress within each zone is Pc, while the opening displacement at the original crack tips is 8c; R - (a - a0) is the size of the fully developed zone at each crack tip. This model also approximately describes the behaviour of an edge crack of initial depth a o in the surface of a semi-infinite solid. Standard results for this crack model [3,4] are: 8(1 8c
v2)&ao
~rE
a0
- - = cos a
In sec( ~ra ] 2pc '
'I-able 1 The fully developed softening zone size R and the associated K:, value for an isolated crack in a uniformly stressed infinite solid (model in Fig. 1); both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid
R
0.1 0.5 1 5 10 oz
K~ = (2)
where E is Young's modulus and v is Poisson's ratio,, while the crack tip stress intensity K A associated with the development of the softening zone is:
52.11 1.95 1.39 1.07 1.03 1
KA
simplify to
~rEac
(6)
8(1 - v2)pc
and
(1)
,
2202 3.19 1.72 1.11 1.05 1
expressions for R and R~ -
K.I
V/ (1EPcSc - v 2)
(7)
which are the expressions for a semi-infinite crack in a remotely loaded infinite solid. With a 0 = ao/R~, R=R/R~, a n d K,A=KA/K~, it immediately follows f r o m relations (4)-(7) that do - exp
- 1,
(8)
(3)
KA =
and It immediately follows that -
X= R =exp
ao
---
-1,
8(1- v2)pcao
(4)
1
{exp[--~0]}"
~o-
lo-
3. The model of an isolated crack in an infinite solid, with the crack faces being subject to line loads
-
v2)pcao -
.
For the large-crack situation, i.e. a 0 ~ m, these
R 2
L
-1
Values of R and KA for various initial crack sizes are shown in Table 1. These results clearly show the extent to which decreasing the initial crack size increases b o t h the fully developed softening zone size and the associated K a value above the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
-
8(1 -
(5)
•
1
(9)
2pc a~0 ~/1 + )t sec 1 exp KA = ~ - ~
qoR
1
KA=~-a~0~exp(8olc°s
2ao
>
Fig. 1. The model of an isolated crack in an uniformly stressed infinite solid; there are fully developed softening zones at each crack tip.
Figure 2 shows the model of an infinite solid containing an isolated crack and subject to line
E. Smith / The failure of a strain-softening material." H
67
Table 2 The fully developed softening zone size R and the associated K A value for an isolated crack in an infinite solid with the crack faces being subject to line loads (model in Fig. 2); both parameters are expressed in terms of the corresponding values for a semi-infinite crack in an infinite solid
2ao 2a
Fig. 2. The model of an isolated crack in an infinite solid with
the centres of the crack faces being subject to line loads of magnitude P.
loads of magnitude P at the centres of the two faces. In every other respect, the situation is similar to that for the model in Fig. 1. Standard results for the model in Fig. 2 are [5]: 8(1 - v2)pca0 ~rE
~ao-
ao ao = ~
1000 100 10 1 0.1 0.01 0.001
0.0001 0.0015 0.0274 0.6106 9.2507 99.0099 1000
R R = _Ro _°
KA KA =
0.10 0.15 0.27 0.61 0.92 0.99
0.35 0.42 0.57 0.82 0.97 0.99
-
1.00
-
and KA=
~ 2 ~ 0 ~ / 1 + _-R cos -1
1
ao
ao
-- ~ 1 - -
(?)2}]
'
(lO)
~00COS-I(-~) 2pca 0p , P
(11) (12)
"
Normalizing the initial crack size a0, R = a - a 0, and K A with respect to the values for a semi-infinite crack in a remotely loaded infinite solid, as in the preceding section, it follows that the fully developed softening zone size R ( = R Roo) and the associated K A value ( = K A Koo) are given by the relations:
For this situation, so as to facilitate the computation, we select specific values of the ratio R/~0, and obtain the value of ~0 and thereby R from relation (13); h'A is then obtained from relation (14). Table 2 gives R and J~A in terms of the initial crack size. These results, which contrast sharply with those for the model in Fig. 1, clearly show the extent to which decreasing the initial crack size decreases both the fully developed softening zone size and the associated K A value below the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
4. The model of a solid containing two symmetrically situated deep cracks and with tension of the remaining ligament Fig. 3 shows the model of a solid containing two symmetrically situated deep cracks and with tension of the remaining ligament, due to the
1 °
X
(14)
1+
2pcao
X{cosh-I[~00]
KA=
1.00
11 (13)
tP R __ C
2L.
_J
R
2L
Cp Fig. 3. The model o f an infinite solid containing two symmetno
cally situated deep cracks and with tension of the remaining ligament.
E. Smith / Thefailure of a strain-softeningmaterial. 11
68
Table 3 The fully developed softening zone size R and the associated KA value for the model in Fig. 3; both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid L
0.2 0.4 0.6 0.8
R
KA
~ = _ R~
~ = _ R~
g - A = _K~
49.63 12.15 5.19 2.72
1.002 1.015 1.037 1.087
1.01 1.03 1.08 1.20
application of line loads P at a great distance from the ligament along the central axis which bisects the ligament. In every other respect, the situation at each crack tip is similar to that for the earlier models. Results for this model are [6]: 4 ( l - ),2)p~L [(1 + X) ln(1 + X) ,rrE
8c
+(1 -X)ln(1
-X)],
(15)
P X
2Lpc
Ka
V/'~-- (1 - 2-ff-~) ,
(16)
P ~gr~_, ,
~rE8~
(18)
4(1 - v2)p~ In 2 "
Normalizing L, R and K A with respect to the parameters associated with a semi-infinite crack in a remotely loaded infinite solid, i.e. Z = L/Ro~, = R / R ~ , and KA = KA/K~, it follows that the fully developed softening zone size and the associated K a value are given by the relations: = (I+x) KA=
~(1
2 ln(1 + X) + ( l - x )
--X2) t / ' '
Z - 1 - (1 - X 2 "
(19) ln(1 - X ) ' (20)
(21)
For this situation, so as to facilitate the computation, we select specific values of X_, o b t a i n L from relation (19), and thereby obtain R and K A from, respectively, relations (21) and (20); the results are shown in Table 3 in terms of the initial ligament size. The results show the extent to which decreasing the initial ligament size increases both the fully developed softening zone size and the associated K A value above the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
5. The model of an infinite solid containing a semi-infinite crack whose faces are subject to equal and opposite line loads
Figure 4 shows the model of an infinite solid containing a semi-infinite crack whose faces are subject to equal and opposite line loads of magnitude P. Standard results for this model are [5,7]:
(17)
where 2L is the initial ligament size and 2 L , is the ligament size associated with the attainment of a fully developed softening zone; the size R of such a zone is equal to ( L - L,). Expressions (15), (16), and (17) are valid provided that the crack tips do not coincide with each other, i.e. provided that X < 1, or R < L, the appropriate condition being (see relation (15)): 2L <
and
8(1 - vZ)pcR 8~=
,rrE x
--~--
In
a0+R
-1 (22)
P = 2 p c R ¢a ° [ R+ R ,
KA =
(23)
P V~ nr(a 02+ R ) "
(24)
Normalizing a 0 (the distance between the loading lines and the initial crack tip), R and K A with respect to the parameters associated with a semiinfinite crack in a remotely loaded infinite solid, P
)
inl)ll~lll~
~ ao R P Fig. 4. The model of an infinite solid containing a semi-infinite crack whose faces are subject to equal and opposite line loads.
E. Smith / Thefailureof a strain-softeningmaterial:H Table 4 The fully developed softening zone size R and the associated KA value for the model in Fig. 4; both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid
---
= a0
R
--
Standard results for this model are [5,8]:
~ESc 4(1 - v2)p¢
= [(ao + R) c°s-lt( aao° -+RR )
K A
ao
1000 100 10 1 0.1 0.01 0.001
0.00014 0.00198 0.03425 0.66983 9.3972 99.340 - 1000
0.14 0.20 0.34 0.67 0.94 0.99
0.37 0.45 0.58 0.82 0.97 0.99
- 1.00
- 1.00
× sin-1 7
i.e. a o = ao/R ~, R = R/R~, and KA = KA/Koo, it follows that the fully developed softening zone size and the associated K A values are given by the realations:
1
~ +lln
+1 +1
a/_~_==--+1
KA = v/R- .
-1
,
+ 2 af~oR]
R
ao+R
_ R~-aTocos_l(~ao+Ra o - R),
h
~=
69
(25)
1
(26)
For this situation, so as to facilitate the compuation, we select specific values of the ratio R___/60, and thus obtain ao and R via relation (25); K A is then obtained from relation (26). Table 4 gives R and K"A in terms of the initial crack size. These results, which are very similar to those for the model in Fig. 2 (see Table 1), show the extent to which decreasing a0, the distance between the loading lines and the initial crack tip, decreases both the fully developed softening zone size and the associated K A value below the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
6. The model of an infinite solid containing a semi-infinite crack whose faces are forced apart by a wedge of constant thickness
(27)
2(1--~2)Pc [(ao + R) cos-lt ao + R ] (28)
KA=
Eh
1
2(1 - ; )
72
(29)
(ao + R)
Normalizing, as in the preceding section, a0, R, and KA, it follows that the fully developed softening z o n e size R = R R ~ and the associated K A ( = K A g ~ ) value are given by the relations:
--~0 =
1[ 1+~oo
tan-1
g A = 7 1+
1+
~oo
ta"-'7 o
(31)
For this situation, so as to facilitate the computation, we select specific values fo the ratio ~ / a 0 , and thus obtain the values of a0 and R via relation (30); K A is then obtained from relation (31). Table 5 gives R and K--A in terms of the initial crack size. These results show the extent to which decreasing the initial crack size a 0 decreases both the fully developed softening zone
h I Ill/Ill/IliA ao
Figure 5 shows the model of an infinite solid containing a semi-infinite crack whose faces are forced apart by a wedge of constant thickness h.
(30)
'
R
Fig. 5. The model of an infinite solid containing a semi-infinite crack whose faces are forced apart by a wedge of constant thickness.
70
E. Smith / Thejailure of a strain-sq/~ening material: 11
Table 5 The fully developed softening zone size R and the associated K A value for the model in Fig. 5; both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid R
a0 1000 100 10 1 o. 1 0.01 0.001
ao= ~
R : R~
0.00042 0.00457 0.05685 0.81058 9.69122 99.67 - 1000
0.42 0.46 0.57 0.81 0.97 0.99 - 1.00
-KA KA= ~-~
0.51 0.53 0.61 0.82 0.97 0.99 - 1.00
size and the associated K A value b e l o w the corres p o n d i n g values for a semi-infinite crack in a remotely l o a d e d infinite solid. F o r this m o d e l , it should also be n o t e d (see eqs. (30) a n d (31)) that is always greater t h a n 4/¢r 2 a n d -KA is always greater than 1 / 2 , irrespective of the value of a 0.
7. Discussion T h e p a p e r has a d d r e s s e d the p r o b l e m of M o d e I plane strain crack growth in a strain-softening material, with a t t e n t i o n being focused on the fully d e v e l o p e d softening zone length a n d the value of the crack tip stress intensity that is a s s o c i a t e d with the a t t a i n m e n t of a fully d e v e l o p e d zone, i.e. when the cohesion vanishes at the original crack tip. A very simple d e s c r i p t i o n of the softening zone has been a d o p t e d , with the theoretical analyses being b a s e d on the D B C S representation, where the stress within the softening zone in fact retains a c o n s t a n t value t h r o u g h o u t the zone. This description ought to be a d e q u a t e to allow b r o a d general conclusions to be d r a w n c o n c e r n i n g the effect of geometrical a n d l o a d i n g p a r a m e t e r s on crack growth resistance. Sections 2 - 6 have used the D B C S r e p r e s e n t a t i o n for a range of simple m o d els, for which analytical results can be o b t a i n e d ,
a n d it has been shown that both the fully develo p e d softening zone size R a n d the associated crack tip stress intensity value K A are very sensitive to b o t h the initial crack size a n d the loading characteristics. The values of R a n d K A c a n differ a p p r e c i a b l y from the c o r r e s p o n d i n g values for a semi-infinite crack in a r e m o t e l y l o a d e d infinite solid, as the results in T a b l e s 1 5 clearly d e m o n strate. The softening zone size a n d critical K value can be less t h a n or greater than the values for a semi-infinite crack in a remotely l o a d e d infinite solid, even for the same initial crack configuration, when the l o a d i n g c o n d i t i o n s are different ( c o m p a r e the results in Tables 1 a n d 2). T h e results c o n t r a s t with those o b t a i n e d by F o o t e , M a i a n d Cotterell [2] and also those obt a i n e d in Part I [1], with r e g a r d to the crack tip stress intensity K A a s s o c i a t e d with the a t t a i n m e n t of a fully d e v e l o p e d softening zone in a d o u b l e cantilever b e a m specimen, where K A w a s shown to be essentially i n d e p e n d e n t of the b e a m d i m e n sions, a n d to be a p p r o x i m a t e l y equivalent to the value a p p r o p r i a t e for a semi-infinite crack in a r e m o t e l y l o a d e d infinite solid. It would, therefore, seem that results for the d o u b l e cantilever b e a m g e o m e t r y are special, a n d should not be used to d r a w general conclusions a b o u t crack growth in a s t r a i n - s o f t e n i n g material.
References [1] E. Smith, Theoret. Appl. Fracture Mech. 11, 59 (1989) (this issue). [2] R.M.L. Foote, Y.W. Mai and B. Cotterell, J. Mech. Pt~vs. Solids 34, 593 (1986). [3] D.S. Dugdale, J. Mech. Phys. Solids 8, 100 (1960). [4] B.A. Bilby, A.H. Cottrell and K.H. Swinden, Proc. Roy. Soc. A272, 304 (1963). [5] H. Tada, P.C. Paris and G.R. Irwin, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, PA (1973). [6] E. Smith, Int. J. Fracture 23, 213 (1983). [7] E. Smith, Mat. Sci. Eng. 62, 41 (1983). [8] E. Smith, Mat. Sci. Eng. 84, 127 (1986).
65
THE FAILURE OF A STRAIN-SOFTENING MATERIAL: II. THE EFFECT OF GEOMETRICAL AND LOADING PARAMETERS E. SMITH Joint Manchester University/UMIST, and Materials Science Centre, Grosvenor Street, Manchester M1 7HS, U.K.
This paper presents theoretical analyses of a variety of models which simulate crack growth in a strain-softening material, with attention being focused on the fully developed softening zone length and the value of the crack tip stress intensity associated with the attainment of such a state. Results from the models show that both these parameters can be very sensitive to both the initial crack configuration and the loading characteristics, and can differ appreciably from the values appropriate to a semi-infinite crack in a remotely loaded infinite solid. The present paper's results underline the view that the analytical results obtained in Part I, and other workers' numerical results for a specific material, are rather special. Part I analyzed the behaviour of a crack in a double cantilever beam specimen, and it was shown that the value of the crack tip stress intensity associated with a fully developed softening zone is essentially independent of the initial crack size and beam height, and is equivalent to the value for a semi-infinite crack in a remotely loaded infinite solid.
1. Introduction
As indicated in Part I [1], a wide range of materials exhibit a phenomenon known as strain softening, whereby if a material contains a crack and is stressed, the crack extends and its faces become bridged by unbroken ligaments whose average behaviour can be represented by a decreasing stress versus increasing crack face opening relationship. The crack extension is associated with an increasing crack resistance (i.e. increasing crack tip stress intensity K), an effect that has been observed in concretes, ceramics, and fibre cements. During this stable crack growth, a strain-softening zone gradually develops behind the crack tip, i.e. the leading edge of the softening zone, so that the crack length is equal to the initial crack length plus the length of the softening zone; this produces a stabilising influence on crack growth and leads to an increasing crack growth resistance ( K ) curve. Foote, Mai and CottereU [2] have calculated the K resistance curves for a specific fibre cement composite (linear softening law) in a double cantilever beam (DCB) geometry and showed that, while the fully developed softening zone size R is very dependent on the beam dimensions, the K value associated with a fully developed softening zone is essentially independent of these parameters and is approximately equivalent to the value appropriate for a semi-infinite crack in a remotely loaded infinite solid. 0167-8442/89/$3.50 © 1989, Elsevier Science Publishers B.V.
These numerical results have been underpinned by the analytical results obtained in Part I [1], where a simple Euler-Bernoulli representation of the DCB specimen was employed. The present paper also addresses the problem of crack growth in a strain-softening material, with the objective of studying other geometrical and loading configurations. With this objective in mind, we adopt a very simple description of the softening zone, and use the DBCS representation [3,4], where the stress within the softening zone in fact retains a constant value. This simple description should be adequate to give broad general conclusions regarding the effect of geometrical and loading parameters on crack growth resistance. Sections 2-6 apply the DBCS type representation to a variety of simple models, for which analytical results can be obtained, and show that the fully developed softening zone length R and the associated value K A of the crack tip stress intensity are very sensitive to both the initial crack configuration and the loading characteristics, and can differ appreciably from the values appropriate to a semi-infinite crack in a remotely loaded infinite solid. With regard to the crack tip stress intensity, these results contrast sharply with those obtained [1,2] for the double cantilever beam geometry. It would, therefore, seem that results for this specific geometry are special, and should not be used to draw general conclusions about crack growth in a strain-softening material.
E. Smith // The/allure of a strain-so[tening material: 11
66
2. The model of an isolated crack in a uniformly stressed infinite solid
Figure 1 shows the model of an isolated two-dimensional crack in an infinite solid subject to a uniform applied tensile stress normal to the crack plane; it is assumed that there is a fully developed softening zone at each crack tip. The initial crack size is 2a 0 and the crack size at the attainment of the fully developed zone is 2 a, the applied tensile stress being o; at this stage, the cohesion becomes zero at the initial crack tip. The tensile stress within each zone is Pc, while the opening displacement at the original crack tips is 8c; R - (a - a0) is the size of the fully developed zone at each crack tip. This model also approximately describes the behaviour of an edge crack of initial depth a o in the surface of a semi-infinite solid. Standard results for this crack model [3,4] are: 8(1 8c
v2)&ao
~rE
a0
- - = cos a
In sec( ~ra ] 2pc '
'I-able 1 The fully developed softening zone size R and the associated K:, value for an isolated crack in a uniformly stressed infinite solid (model in Fig. 1); both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid
R
0.1 0.5 1 5 10 oz
K~ = (2)
where E is Young's modulus and v is Poisson's ratio,, while the crack tip stress intensity K A associated with the development of the softening zone is:
52.11 1.95 1.39 1.07 1.03 1
KA
simplify to
~rEac
(6)
8(1 - v2)pc
and
(1)
,
2202 3.19 1.72 1.11 1.05 1
expressions for R and R~ -
K.I
V/ (1EPcSc - v 2)
(7)
which are the expressions for a semi-infinite crack in a remotely loaded infinite solid. With a 0 = ao/R~, R=R/R~, a n d K,A=KA/K~, it immediately follows f r o m relations (4)-(7) that do - exp
- 1,
(8)
(3)
KA =
and It immediately follows that -
X= R =exp
ao
---
-1,
8(1- v2)pcao
(4)
1
{exp[--~0]}"
~o-
lo-
3. The model of an isolated crack in an infinite solid, with the crack faces being subject to line loads
-
v2)pcao -
.
For the large-crack situation, i.e. a 0 ~ m, these
R 2
L
-1
Values of R and KA for various initial crack sizes are shown in Table 1. These results clearly show the extent to which decreasing the initial crack size increases b o t h the fully developed softening zone size and the associated K a value above the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
-
8(1 -
(5)
•
1
(9)
2pc a~0 ~/1 + )t sec 1 exp KA = ~ - ~
qoR
1
KA=~-a~0~exp(8olc°s
2ao
>
Fig. 1. The model of an isolated crack in an uniformly stressed infinite solid; there are fully developed softening zones at each crack tip.
Figure 2 shows the model of an infinite solid containing an isolated crack and subject to line
E. Smith / The failure of a strain-softening material." H
67
Table 2 The fully developed softening zone size R and the associated K A value for an isolated crack in an infinite solid with the crack faces being subject to line loads (model in Fig. 2); both parameters are expressed in terms of the corresponding values for a semi-infinite crack in an infinite solid
2ao 2a
Fig. 2. The model of an isolated crack in an infinite solid with
the centres of the crack faces being subject to line loads of magnitude P.
loads of magnitude P at the centres of the two faces. In every other respect, the situation is similar to that for the model in Fig. 1. Standard results for the model in Fig. 2 are [5]: 8(1 - v2)pca0 ~rE
~ao-
ao ao = ~
1000 100 10 1 0.1 0.01 0.001
0.0001 0.0015 0.0274 0.6106 9.2507 99.0099 1000
R R = _Ro _°
KA KA =
0.10 0.15 0.27 0.61 0.92 0.99
0.35 0.42 0.57 0.82 0.97 0.99
-
1.00
-
and KA=
~ 2 ~ 0 ~ / 1 + _-R cos -1
1
ao
ao
-- ~ 1 - -
(?)2}]
'
(lO)
~00COS-I(-~) 2pca 0p , P
(11) (12)
"
Normalizing the initial crack size a0, R = a - a 0, and K A with respect to the values for a semi-infinite crack in a remotely loaded infinite solid, as in the preceding section, it follows that the fully developed softening zone size R ( = R Roo) and the associated K A value ( = K A Koo) are given by the relations:
For this situation, so as to facilitate the computation, we select specific values of the ratio R/~0, and obtain the value of ~0 and thereby R from relation (13); h'A is then obtained from relation (14). Table 2 gives R and J~A in terms of the initial crack size. These results, which contrast sharply with those for the model in Fig. 1, clearly show the extent to which decreasing the initial crack size decreases both the fully developed softening zone size and the associated K A value below the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
4. The model of a solid containing two symmetrically situated deep cracks and with tension of the remaining ligament Fig. 3 shows the model of a solid containing two symmetrically situated deep cracks and with tension of the remaining ligament, due to the
1 °
X
(14)
1+
2pcao
X{cosh-I[~00]
KA=
1.00
11 (13)
tP R __ C
2L.
_J
R
2L
Cp Fig. 3. The model o f an infinite solid containing two symmetno
cally situated deep cracks and with tension of the remaining ligament.
E. Smith / Thefailure of a strain-softeningmaterial. 11
68
Table 3 The fully developed softening zone size R and the associated KA value for the model in Fig. 3; both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid L
0.2 0.4 0.6 0.8
R
KA
~ = _ R~
~ = _ R~
g - A = _K~
49.63 12.15 5.19 2.72
1.002 1.015 1.037 1.087
1.01 1.03 1.08 1.20
application of line loads P at a great distance from the ligament along the central axis which bisects the ligament. In every other respect, the situation at each crack tip is similar to that for the earlier models. Results for this model are [6]: 4 ( l - ),2)p~L [(1 + X) ln(1 + X) ,rrE
8c
+(1 -X)ln(1
-X)],
(15)
P X
2Lpc
Ka
V/'~-- (1 - 2-ff-~) ,
(16)
P ~gr~_, ,
~rE8~
(18)
4(1 - v2)p~ In 2 "
Normalizing L, R and K A with respect to the parameters associated with a semi-infinite crack in a remotely loaded infinite solid, i.e. Z = L/Ro~, = R / R ~ , and KA = KA/K~, it follows that the fully developed softening zone size and the associated K a value are given by the relations: = (I+x) KA=
~(1
2 ln(1 + X) + ( l - x )
--X2) t / ' '
Z - 1 - (1 - X 2 "
(19) ln(1 - X ) ' (20)
(21)
For this situation, so as to facilitate the computation, we select specific values of X_, o b t a i n L from relation (19), and thereby obtain R and K A from, respectively, relations (21) and (20); the results are shown in Table 3 in terms of the initial ligament size. The results show the extent to which decreasing the initial ligament size increases both the fully developed softening zone size and the associated K A value above the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
5. The model of an infinite solid containing a semi-infinite crack whose faces are subject to equal and opposite line loads
Figure 4 shows the model of an infinite solid containing a semi-infinite crack whose faces are subject to equal and opposite line loads of magnitude P. Standard results for this model are [5,7]:
(17)
where 2L is the initial ligament size and 2 L , is the ligament size associated with the attainment of a fully developed softening zone; the size R of such a zone is equal to ( L - L,). Expressions (15), (16), and (17) are valid provided that the crack tips do not coincide with each other, i.e. provided that X < 1, or R < L, the appropriate condition being (see relation (15)): 2L <
and
8(1 - vZ)pcR 8~=
,rrE x
--~--
In
a0+R
-1 (22)
P = 2 p c R ¢a ° [ R+ R ,
KA =
(23)
P V~ nr(a 02+ R ) "
(24)
Normalizing a 0 (the distance between the loading lines and the initial crack tip), R and K A with respect to the parameters associated with a semiinfinite crack in a remotely loaded infinite solid, P
)
inl)ll~lll~
~ ao R P Fig. 4. The model of an infinite solid containing a semi-infinite crack whose faces are subject to equal and opposite line loads.
E. Smith / Thefailureof a strain-softeningmaterial:H Table 4 The fully developed softening zone size R and the associated KA value for the model in Fig. 4; both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid
---
= a0
R
--
Standard results for this model are [5,8]:
~ESc 4(1 - v2)p¢
= [(ao + R) c°s-lt( aao° -+RR )
K A
ao
1000 100 10 1 0.1 0.01 0.001
0.00014 0.00198 0.03425 0.66983 9.3972 99.340 - 1000
0.14 0.20 0.34 0.67 0.94 0.99
0.37 0.45 0.58 0.82 0.97 0.99
- 1.00
- 1.00
× sin-1 7
i.e. a o = ao/R ~, R = R/R~, and KA = KA/Koo, it follows that the fully developed softening zone size and the associated K A values are given by the realations:
1
~ +lln
+1 +1
a/_~_==--+1
KA = v/R- .
-1
,
+ 2 af~oR]
R
ao+R
_ R~-aTocos_l(~ao+Ra o - R),
h
~=
69
(25)
1
(26)
For this situation, so as to facilitate the compuation, we select specific values of the ratio R___/60, and thus obtain ao and R via relation (25); K A is then obtained from relation (26). Table 4 gives R and K"A in terms of the initial crack size. These results, which are very similar to those for the model in Fig. 2 (see Table 1), show the extent to which decreasing a0, the distance between the loading lines and the initial crack tip, decreases both the fully developed softening zone size and the associated K A value below the corresponding values for a semi-infinite crack in a remotely loaded infinite solid.
6. The model of an infinite solid containing a semi-infinite crack whose faces are forced apart by a wedge of constant thickness
(27)
2(1--~2)Pc [(ao + R) cos-lt ao + R ] (28)
KA=
Eh
1
2(1 - ; )
72
(29)
(ao + R)
Normalizing, as in the preceding section, a0, R, and KA, it follows that the fully developed softening z o n e size R = R R ~ and the associated K A ( = K A g ~ ) value are given by the relations:
--~0 =
1[ 1+~oo
tan-1
g A = 7 1+
1+
~oo
ta"-'7 o
(31)
For this situation, so as to facilitate the computation, we select specific values fo the ratio ~ / a 0 , and thus obtain the values of a0 and R via relation (30); K A is then obtained from relation (31). Table 5 gives R and K--A in terms of the initial crack size. These results show the extent to which decreasing the initial crack size a 0 decreases both the fully developed softening zone
h I Ill/Ill/IliA ao
Figure 5 shows the model of an infinite solid containing a semi-infinite crack whose faces are forced apart by a wedge of constant thickness h.
(30)
'
R
Fig. 5. The model of an infinite solid containing a semi-infinite crack whose faces are forced apart by a wedge of constant thickness.
70
E. Smith / Thejailure of a strain-sq/~ening material: 11
Table 5 The fully developed softening zone size R and the associated K A value for the model in Fig. 5; both parameters are expressed in terms of the corresponding values for a semi-infinite crack in a remotely loaded infinite solid R
a0 1000 100 10 1 o. 1 0.01 0.001
ao= ~
R : R~
0.00042 0.00457 0.05685 0.81058 9.69122 99.67 - 1000
0.42 0.46 0.57 0.81 0.97 0.99 - 1.00
-KA KA= ~-~
0.51 0.53 0.61 0.82 0.97 0.99 - 1.00
size and the associated K A value b e l o w the corres p o n d i n g values for a semi-infinite crack in a remotely l o a d e d infinite solid. F o r this m o d e l , it should also be n o t e d (see eqs. (30) a n d (31)) that is always greater t h a n 4/¢r 2 a n d -KA is always greater than 1 / 2 , irrespective of the value of a 0.
7. Discussion T h e p a p e r has a d d r e s s e d the p r o b l e m of M o d e I plane strain crack growth in a strain-softening material, with a t t e n t i o n being focused on the fully d e v e l o p e d softening zone length a n d the value of the crack tip stress intensity that is a s s o c i a t e d with the a t t a i n m e n t of a fully d e v e l o p e d zone, i.e. when the cohesion vanishes at the original crack tip. A very simple d e s c r i p t i o n of the softening zone has been a d o p t e d , with the theoretical analyses being b a s e d on the D B C S representation, where the stress within the softening zone in fact retains a c o n s t a n t value t h r o u g h o u t the zone. This description ought to be a d e q u a t e to allow b r o a d general conclusions to be d r a w n c o n c e r n i n g the effect of geometrical a n d l o a d i n g p a r a m e t e r s on crack growth resistance. Sections 2 - 6 have used the D B C S r e p r e s e n t a t i o n for a range of simple m o d els, for which analytical results can be o b t a i n e d ,
a n d it has been shown that both the fully develo p e d softening zone size R a n d the associated crack tip stress intensity value K A are very sensitive to b o t h the initial crack size a n d the loading characteristics. The values of R a n d K A c a n differ a p p r e c i a b l y from the c o r r e s p o n d i n g values for a semi-infinite crack in a r e m o t e l y l o a d e d infinite solid, as the results in T a b l e s 1 5 clearly d e m o n strate. The softening zone size a n d critical K value can be less t h a n or greater than the values for a semi-infinite crack in a remotely l o a d e d infinite solid, even for the same initial crack configuration, when the l o a d i n g c o n d i t i o n s are different ( c o m p a r e the results in Tables 1 a n d 2). T h e results c o n t r a s t with those o b t a i n e d by F o o t e , M a i a n d Cotterell [2] and also those obt a i n e d in Part I [1], with r e g a r d to the crack tip stress intensity K A a s s o c i a t e d with the a t t a i n m e n t of a fully d e v e l o p e d softening zone in a d o u b l e cantilever b e a m specimen, where K A w a s shown to be essentially i n d e p e n d e n t of the b e a m d i m e n sions, a n d to be a p p r o x i m a t e l y equivalent to the value a p p r o p r i a t e for a semi-infinite crack in a r e m o t e l y l o a d e d infinite solid. It would, therefore, seem that results for the d o u b l e cantilever b e a m g e o m e t r y are special, a n d should not be used to d r a w general conclusions a b o u t crack growth in a s t r a i n - s o f t e n i n g material.
References [1] E. Smith, Theoret. Appl. Fracture Mech. 11, 59 (1989) (this issue). [2] R.M.L. Foote, Y.W. Mai and B. Cotterell, J. Mech. Pt~vs. Solids 34, 593 (1986). [3] D.S. Dugdale, J. Mech. Phys. Solids 8, 100 (1960). [4] B.A. Bilby, A.H. Cottrell and K.H. Swinden, Proc. Roy. Soc. A272, 304 (1963). [5] H. Tada, P.C. Paris and G.R. Irwin, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, PA (1973). [6] E. Smith, Int. J. Fracture 23, 213 (1983). [7] E. Smith, Mat. Sci. Eng. 62, 41 (1983). [8] E. Smith, Mat. Sci. Eng. 84, 127 (1986).