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The size of the fully developed softening zone associated with a crack in a strain-softening material—I. A semi-infinite crack in a remotely loaded infinite solid
hr. 1. Engng Sci. Vol. 27, No. 3, pp. 301-307, 1989 Printed in Great Britain.All rightsreserved
0020-7225/89
$3.OLl+0.00
Copyright@ 1989PergamonPressplc
THE SIZE OF THE FULLY DEVELOPED SOFTENING ZONE ASSOCIATED WITH A CRACK IN A STRAIN-SOFTENING MATERIAL-I. A SEMI-INFINITE CRACK IN A REMOTELY LOADED INFINITE SOLID E. SMITH Joint Manchester University/UMIST Department of Metallurgy and Materials Science, Grosvenor Street, Manchester Ml 7HS, U.K. (Communicated by B. A. BILBY) Abstract-The paper presents results from a theoretical analysis of a model which simulates Mode I plane strain crack extension in a strain-softening material. Attention is focussed on the fully developed softening xone size associated with a semi-infinite crack in a remotely loaded infinite solid. The results for a range of force laws show that this size is relatively insensitive to the details of the softening law, though being critically dependent on the maximum stress and displacement within the softening zone. The implications of the results to the crack sensitivity of strain-softening materials are briefly discussed.
1. INTRODUCTION
Materials such as for example concrete, soils and ceramics exhibit a phenomenon known as strain-softening, whereby the mechanical behaviour is characterised by a linear stress-strain curve until the attainment of the ultimate tensile strength; thereafter there is a reduction in the stress. If this behaviour pattern is carried over to a cracked solid, there exists a state of affairs where the faces of a crack are bridged by unbroken ligaments of material whose average behaviour can be characterised by a decreasing stress vs increasing crack face opening relationship. A simplistic description of this state of affairs is that there is a zone of material, coplanar with the crack, in which the material behaviour is nonlinear, the form of the nonlinearity being governed by the strain-softening characteristics of the material under consideration. When an initially cracked solid is progressively loaded, the crack extends and a strain softening zone 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, with the crack tip stress intensity K being measured at the leading edge of the softening zone. It was against this general background that Foote et al. [l] examined the fracture behaviour of a strain softening material. They calculated the K resistance curve for a fibre cement composite (linear softening law) having a double-cantilever-beam geometry (Mode I plain strain deformation), focussing particularly on the size R of the fully developed softening zone, the K value, K,,+ associated with the attainment of this state, and the way in which these parameters vary with the crack and beam dimensions. Prompted by this work, the author in studying the fracture resistance of a strain-softening material, has adopted a highly idealised approach [2] based on the use of the DBCS type force law [3,4] where the stress within the softening zone has a constant value pc, while the critical displacement at which cohesion vanishes is 6,. With the aid of this force-law, the effects of initial crack configuration and loading characteristics on the parameters R and KA were examined for a wide range of situations. This paper focusses on the determination of R for a semi-infinite crack in a remotely loaded infinite solid, for a wide range of idealised strain softening laws, with particular emphasis being given to the effects of the tail portion of the force law. It is shown that R, though being very dependent on pc and 6,, is relatively independent of the details of the softening law. The implications of the results to the general problem of the crack sensitivity of strain-softening materials are briefly discussed. 301
302
E. SMITH
2. BACKGROUND
THEORY
As indicated in the Introduction, and following Foote er af. [l], the crack tip is defined as the leading edge of the fully developed softening zone. The net stress intensity factor at the crack tip is numerically equal to the intrinsic fracture toughness Krc of the matrix material. For example, with fibre cements K rc is the toughness of the reinforced matrix material, in concretes it is that of the cement paste matrix, while in ceramics it is given by either the transgranular or intergranular toughness. In this analysis it is assumed that K rC = 0; this assumption simplifies the analysis and is unlikely to significantly affect the conclusions, bearing in mind the paper’s overall objectives, which are to examine the effect of the softening law on the size R of the fully developed softening zone. The softening zone associated with a semi-infinite crack in an infinite solid is shown in Fig. 1, with the co-ordinate system being as indicated. Assuming that the zone is fully developed in the sense that the crack face displacement at the trailing edge of the zone is such that there is then no cohesion between the crack faces, i.e. it is the displacement required to rupture the ligaments that bridge the crack faces, the governing relationship between the average tensile stress p(s) and the relative displacement V(S) within the softening zone is [5] du -= ds
4(1-
.‘)s”~ XE
R p(t) dt (1)
i () (s - t)t’”
where E is Young’s modulus, Y is Poisson’s ratio, and R is the size of the fully developed softening zone. If KA is the applied stress intensity due to the remote loadings applied to the solid, the crack tip condition is “P(S) I___ ds
K,$ i0 me
s1i2
(2)
-
singular integral equation (1) can be solved to give p(s) as (3)
whereupon straightfo~ard
manipulation
[2] of these equations gives
KA= (&%p
du)lt2>
(4)
showing that the value of I( (i.e. KA) associated with the attainment of a fully developed softening zone is directly related to the stress-displacement law within the softening zone; relation (4) can also be obtained directly by using the J path independent integral approach [6].
3. ANALYSIS
FOR
A POLYNOMIAL
DISPLACEMENT
Following earlier work [7] and with u = s/R, the displacement v = a Un+3i2~n+3i2 n
DISTRIBUTION
v is initially chosen as
9
R
Fig. 1. The general model of a softening zone associated with a ~mi-infinite crack in an infinite sotid; R is the size of the softening zone, which is assumed to be fully devetoped.
(5)
303
Softening zone size-1
where n is zero or a positive integer. simplification, P=
Substitution
in eqn (3) immediately
gives, after some
Ea,(n + 3/2)R”+ln (1 -C%(~), 4(1- vZ)n
(f-9
where J,(u) is given by the expression J,(u) =
(2r)! nlP$(r!)2(4u)'
(7)
r=O
for II = 0, 1, 2, 3 etc. To obtain a variety of stress-displacement laws of terms of the type defined by relations (5) and Thus, consider the simplest case where there are n = 1, when the displacement and stress are given
for the softening zone, one can use a series (6) and sum their respective contributions. only two terms corresponding to n = 0 and respectively by
u = a 0 R312u312 + aIR5/2$‘2, P =
(8)
8;,R::) (1 - u)‘nJo(u)+ 8z,R;) (1 - U)‘~J,(U).
(9)
Since u = 6, when u = 1 and assuming that p =pc at the leading edge of the softening zone, i.e. when u = 0, relations (8) and (9) give, with k = aIRlao,
and g = (1 - U)lQ[ l+r(U+;)]/(l+$).
C Furthermore
relations (8) and (9) give the size of the fully developed
R=
8(1-
softening zone as
v2)p,(l + k)’
while the value of KA associated with the development (10) and (11) as
of this zone is given by relations (4),
Fig. 2. Stress-displacement laws for the three values k = -2/S, 0 and +2/5.
304
E. SMITH Table 1. The fully developed softening zone size R for various softening laws, i.e. k values
-g
0.42
1
0.40
0
0.37
+$
0.36
+$
0.35
DBCS
0.39
Relations (lo)-(13) allow the size R of the fully developed softening zone and the associated KA value to be determined for a range of force laws which can be obtained by varying k. k must be less than 2/5 for the stress p to be less than pc, and k must be greater than -2/5 for p laws obtained from relations (10) and (11) for to be positive. The stress-displacement k = -2/5, k = 0 and k = +2/5 are shown in Fig. 2. Table 1 gives R for a range of k values between -2/5 and +2/5. These results are compared in Table 1 with the corresponding value obtained by assuming that the stress retains a constant value pc within the softening zone, this being a standard result for the DBCS model [3,4]. The KA value for any stress-displacement law can be determined directly using relation (4). Unfortunately there is no corresponding relation with regards to R. However the results in Table 1 clearly show that the value of R is not especially sensitive to the details of the force law, at least within the range k = -2/5 to k = +2/5; the critical parameters are the maximum stress pc and the maximum displacement 6,. Furthermore R is approximately given by the DBCS expression assuming a constant stress pc within the softening zone. The next section extends the considerations to a wider range of force laws, the analysis being based on the general theory developed in Section 2, and with emphasis being given to the effects of the tail portion of the force law.
4. ANALYSIS
FOR
A PIECE-WISE
DBCS
FORCE
LAW
This section considers the general force law shown in Fig. 3, the situation in the vicinity of the crack tip being as shown in Fig. 4; by varying the parameters q and A it is clearly possible to examine the effects of a wide range of softening laws. For this general force law, it follows from relation (1) and (4), after some manipulation, that JTEG,
8(1 - v2)pcR
=[4+(1-q)(l-;)1’2]+(1;;)Wln
1 ”
,” a 4
(14)
I-L
I 0
I
I
A
1
t
V/8,
Fig. 3. The idealised
stress-displacement
law analysed
in Section 4.
Softening zone size-1
Fig. 4. The situation in the vicinity of a crack tip for the force law in Fig. 3.
Writing l? = 8(1- Y~)~~R/~ES, and Ip = w/R allows relations (14) and (15) to be written respectively in the forms ;== [q + (1 - q)(l -
$ = (1 - t#yyq
+ (I-
#)‘n] + (I -*q)%l(; q)(l
- CjJy] - $ln(i
” I; 1 $J, ” ;;
1 ;;:::,,
relations which, in principle, allow the fully developed softening zone size R to be obtained for various vdues of the parameters q and h. The effect of a long tai1 in the force Iaw can be examined by considering the special case where q = 0, when relations (17) and (18) simplify to
# 1 + (1 - #)*‘* l=(l-#)l~+~In(l_ll_~)ln)J R g=(l-0). Table 2. The fully developed softening zone size R for the ideabed force lew shown in Fig. 3, with q = 4 and for variaw values of X
w
A
0.1
049
O-2
0.96
;::
0.5 0.6
X:3 O-9 1.0
0.91 0.84 0.75 064 O-51 0.36 0.19 0
O-05 0.10 0.15
147
0.21 0.27 0.34 0.43 0.54 0.69 1.00
0.52 o-43 O*38
0.33 0.39
E. SMITH
306
Ii cl”
u-
=e:
0.5
a
__1_
I
I
-----x
x
_
1
I
I 05
0
)
1.0
x
Fig. 5. The fully developed softening zone size R for the ideal force law shown in Fig. 3. The full curve shows the results (Table 2) for the case where q = 0, and the crosses (results from Table 3) are for the case where q = 0.5; the dotted line is the DBCS result for the case where q = 1.
Table 3. The fully developed softening zone size R for the idealised force law shown in Fig. 3, with q = 0.5, and for various values of A
0.2 0.4 0.6 0.8
0.64 044 0.28 0.14
0.36 0.36 0.38 0.43
By selecting appropriate values of #, relations (19) and (20) give R for various values of A, the results being shown in Table 2 and also in Fig. 5. For the special case where q = 0.5, relations (17) and (18) simplify to i = [l + (1 - #)““I + 3 ln(: T [: I ,“:J, %=(l-
$~)“~[l +(l-
Again by selecting appropriate A, the results
being
shown
$)“‘I -tln(:
(21)
T ii I $J.
values of #I, relations (21) and (22) give R for various 3 and also in Fig. 5.
(22) values
of
in Table
5. DISCUSSION
The clear conclusion that arises from inspecting the results in Fig. 5, particularly when they are viewed alongside the results in Table 1, is that for the case of a semi-infinite crack in a remotely loaded infinite solid, the fully developed softening zone size R is not especially sensitive to the details of the softening law, though it is dependent on pc (the maximum stress) and S, (the maximum displacement). R is approximately equal to the value for the DBCS model where the stress is constant within the softening zone, i.e. 0.39E36,/(1 - v2)pc. The magnitude of R can be approximately equated to the size D of crack in a strain-softening solid above which the semi-infinite crack-infinite solid relation for K [i.e. relation (4)] can be used to predict the failure of a stressed solid. The present paper’s results show that this critical size D has the DBCS value, i.e. 0.39B6,/(1 - v2)pc, but is essentially insensitive to the details of the force law. The failure stress will be less than that predicted using relation (4) if the crack size is less than D. As an example, for the case of a cellulose/asbestos fibre reinforced mortar, with the figures quoted by Foote et al. [l], i.e. E = 6OOOMPa, pc = 6MPa, 6, = O-8 mm, and ignoring Poisson’s ratio effects, D is approximately equal to 300 mm.
Softening zone size-1
REFERENCES [ 11 R. M. L. FOOTE, Y. W. MAI and B. COTIERELL, J. Me&. Phys. Solids 34,593 (1986). [2] E. SMITH, ht. J. Engng Sci. 27,309 (1989). [3] D. S. DUGDALE, J. Mech. Phys. Solids 8, 100 (1960). [4] B. A. BILBY, A. H, COTIRELL and K. H. SWINDEN, Proc. R. Sot. A272, 304 (1963). [5] J. R. WILLIS, J. Mech. Phys. Solids 15, 151 (1967). [6] J. R. RICE, In Fracture (Edited by H. Liebowitz), Vol. 2, p. 191. Academic Press, New York (1968). [7] E. SMITH, ht. J. Engng Fracture Me&. 6, 213 (1974). (Received 9 August 1988)
307
0020-7225/89
$3.OLl+0.00
Copyright@ 1989PergamonPressplc
THE SIZE OF THE FULLY DEVELOPED SOFTENING ZONE ASSOCIATED WITH A CRACK IN A STRAIN-SOFTENING MATERIAL-I. A SEMI-INFINITE CRACK IN A REMOTELY LOADED INFINITE SOLID E. SMITH Joint Manchester University/UMIST Department of Metallurgy and Materials Science, Grosvenor Street, Manchester Ml 7HS, U.K. (Communicated by B. A. BILBY) Abstract-The paper presents results from a theoretical analysis of a model which simulates Mode I plane strain crack extension in a strain-softening material. Attention is focussed on the fully developed softening xone size associated with a semi-infinite crack in a remotely loaded infinite solid. The results for a range of force laws show that this size is relatively insensitive to the details of the softening law, though being critically dependent on the maximum stress and displacement within the softening zone. The implications of the results to the crack sensitivity of strain-softening materials are briefly discussed.
1. INTRODUCTION
Materials such as for example concrete, soils and ceramics exhibit a phenomenon known as strain-softening, whereby the mechanical behaviour is characterised by a linear stress-strain curve until the attainment of the ultimate tensile strength; thereafter there is a reduction in the stress. If this behaviour pattern is carried over to a cracked solid, there exists a state of affairs where the faces of a crack are bridged by unbroken ligaments of material whose average behaviour can be characterised by a decreasing stress vs increasing crack face opening relationship. A simplistic description of this state of affairs is that there is a zone of material, coplanar with the crack, in which the material behaviour is nonlinear, the form of the nonlinearity being governed by the strain-softening characteristics of the material under consideration. When an initially cracked solid is progressively loaded, the crack extends and a strain softening zone 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, with the crack tip stress intensity K being measured at the leading edge of the softening zone. It was against this general background that Foote et al. [l] examined the fracture behaviour of a strain softening material. They calculated the K resistance curve for a fibre cement composite (linear softening law) having a double-cantilever-beam geometry (Mode I plain strain deformation), focussing particularly on the size R of the fully developed softening zone, the K value, K,,+ associated with the attainment of this state, and the way in which these parameters vary with the crack and beam dimensions. Prompted by this work, the author in studying the fracture resistance of a strain-softening material, has adopted a highly idealised approach [2] based on the use of the DBCS type force law [3,4] where the stress within the softening zone has a constant value pc, while the critical displacement at which cohesion vanishes is 6,. With the aid of this force-law, the effects of initial crack configuration and loading characteristics on the parameters R and KA were examined for a wide range of situations. This paper focusses on the determination of R for a semi-infinite crack in a remotely loaded infinite solid, for a wide range of idealised strain softening laws, with particular emphasis being given to the effects of the tail portion of the force law. It is shown that R, though being very dependent on pc and 6,, is relatively independent of the details of the softening law. The implications of the results to the general problem of the crack sensitivity of strain-softening materials are briefly discussed. 301
302
E. SMITH
2. BACKGROUND
THEORY
As indicated in the Introduction, and following Foote er af. [l], the crack tip is defined as the leading edge of the fully developed softening zone. The net stress intensity factor at the crack tip is numerically equal to the intrinsic fracture toughness Krc of the matrix material. For example, with fibre cements K rc is the toughness of the reinforced matrix material, in concretes it is that of the cement paste matrix, while in ceramics it is given by either the transgranular or intergranular toughness. In this analysis it is assumed that K rC = 0; this assumption simplifies the analysis and is unlikely to significantly affect the conclusions, bearing in mind the paper’s overall objectives, which are to examine the effect of the softening law on the size R of the fully developed softening zone. The softening zone associated with a semi-infinite crack in an infinite solid is shown in Fig. 1, with the co-ordinate system being as indicated. Assuming that the zone is fully developed in the sense that the crack face displacement at the trailing edge of the zone is such that there is then no cohesion between the crack faces, i.e. it is the displacement required to rupture the ligaments that bridge the crack faces, the governing relationship between the average tensile stress p(s) and the relative displacement V(S) within the softening zone is [5] du -= ds
4(1-
.‘)s”~ XE
R p(t) dt (1)
i () (s - t)t’”
where E is Young’s modulus, Y is Poisson’s ratio, and R is the size of the fully developed softening zone. If KA is the applied stress intensity due to the remote loadings applied to the solid, the crack tip condition is “P(S) I___ ds
K,$ i0 me
s1i2
(2)
-
singular integral equation (1) can be solved to give p(s) as (3)
whereupon straightfo~ard
manipulation
[2] of these equations gives
KA= (&%p
du)lt2>
(4)
showing that the value of I( (i.e. KA) associated with the attainment of a fully developed softening zone is directly related to the stress-displacement law within the softening zone; relation (4) can also be obtained directly by using the J path independent integral approach [6].
3. ANALYSIS
FOR
A POLYNOMIAL
DISPLACEMENT
Following earlier work [7] and with u = s/R, the displacement v = a Un+3i2~n+3i2 n
DISTRIBUTION
v is initially chosen as
9
R
Fig. 1. The general model of a softening zone associated with a ~mi-infinite crack in an infinite sotid; R is the size of the softening zone, which is assumed to be fully devetoped.
(5)
303
Softening zone size-1
where n is zero or a positive integer. simplification, P=
Substitution
in eqn (3) immediately
gives, after some
Ea,(n + 3/2)R”+ln (1 -C%(~), 4(1- vZ)n
(f-9
where J,(u) is given by the expression J,(u) =
(2r)! nlP$(r!)2(4u)'
(7)
r=O
for II = 0, 1, 2, 3 etc. To obtain a variety of stress-displacement laws of terms of the type defined by relations (5) and Thus, consider the simplest case where there are n = 1, when the displacement and stress are given
for the softening zone, one can use a series (6) and sum their respective contributions. only two terms corresponding to n = 0 and respectively by
u = a 0 R312u312 + aIR5/2$‘2, P =
(8)
8;,R::) (1 - u)‘nJo(u)+ 8z,R;) (1 - U)‘~J,(U).
(9)
Since u = 6, when u = 1 and assuming that p =pc at the leading edge of the softening zone, i.e. when u = 0, relations (8) and (9) give, with k = aIRlao,
and g = (1 - U)lQ[ l+r(U+;)]/(l+$).
C Furthermore
relations (8) and (9) give the size of the fully developed
R=
8(1-
softening zone as
v2)p,(l + k)’
while the value of KA associated with the development (10) and (11) as
of this zone is given by relations (4),
Fig. 2. Stress-displacement laws for the three values k = -2/S, 0 and +2/5.
304
E. SMITH Table 1. The fully developed softening zone size R for various softening laws, i.e. k values
-g
0.42
1
0.40
0
0.37
+$
0.36
+$
0.35
DBCS
0.39
Relations (lo)-(13) allow the size R of the fully developed softening zone and the associated KA value to be determined for a range of force laws which can be obtained by varying k. k must be less than 2/5 for the stress p to be less than pc, and k must be greater than -2/5 for p laws obtained from relations (10) and (11) for to be positive. The stress-displacement k = -2/5, k = 0 and k = +2/5 are shown in Fig. 2. Table 1 gives R for a range of k values between -2/5 and +2/5. These results are compared in Table 1 with the corresponding value obtained by assuming that the stress retains a constant value pc within the softening zone, this being a standard result for the DBCS model [3,4]. The KA value for any stress-displacement law can be determined directly using relation (4). Unfortunately there is no corresponding relation with regards to R. However the results in Table 1 clearly show that the value of R is not especially sensitive to the details of the force law, at least within the range k = -2/5 to k = +2/5; the critical parameters are the maximum stress pc and the maximum displacement 6,. Furthermore R is approximately given by the DBCS expression assuming a constant stress pc within the softening zone. The next section extends the considerations to a wider range of force laws, the analysis being based on the general theory developed in Section 2, and with emphasis being given to the effects of the tail portion of the force law.
4. ANALYSIS
FOR
A PIECE-WISE
DBCS
FORCE
LAW
This section considers the general force law shown in Fig. 3, the situation in the vicinity of the crack tip being as shown in Fig. 4; by varying the parameters q and A it is clearly possible to examine the effects of a wide range of softening laws. For this general force law, it follows from relation (1) and (4), after some manipulation, that JTEG,
8(1 - v2)pcR
=[4+(1-q)(l-;)1’2]+(1;;)Wln
1 ”
,” a 4
(14)
I-L
I 0
I
I
A
1
t
V/8,
Fig. 3. The idealised
stress-displacement
law analysed
in Section 4.
Softening zone size-1
Fig. 4. The situation in the vicinity of a crack tip for the force law in Fig. 3.
Writing l? = 8(1- Y~)~~R/~ES, and Ip = w/R allows relations (14) and (15) to be written respectively in the forms ;== [q + (1 - q)(l -
$ = (1 - t#yyq
+ (I-
#)‘n] + (I -*q)%l(; q)(l
- CjJy] - $ln(i
” I; 1 $J, ” ;;
1 ;;:::,,
relations which, in principle, allow the fully developed softening zone size R to be obtained for various vdues of the parameters q and h. The effect of a long tai1 in the force Iaw can be examined by considering the special case where q = 0, when relations (17) and (18) simplify to
# 1 + (1 - #)*‘* l=(l-#)l~+~In(l_ll_~)ln)J R g=(l-0). Table 2. The fully developed softening zone size R for the ideabed force lew shown in Fig. 3, with q = 4 and for variaw values of X
w
A
0.1
049
O-2
0.96
;::
0.5 0.6
X:3 O-9 1.0
0.91 0.84 0.75 064 O-51 0.36 0.19 0
O-05 0.10 0.15
147
0.21 0.27 0.34 0.43 0.54 0.69 1.00
0.52 o-43 O*38
0.33 0.39
E. SMITH
306
Ii cl”
u-
=e:
0.5
a
__1_
I
I
-----x
x
_
1
I
I 05
0
)
1.0
x
Fig. 5. The fully developed softening zone size R for the ideal force law shown in Fig. 3. The full curve shows the results (Table 2) for the case where q = 0, and the crosses (results from Table 3) are for the case where q = 0.5; the dotted line is the DBCS result for the case where q = 1.
Table 3. The fully developed softening zone size R for the idealised force law shown in Fig. 3, with q = 0.5, and for various values of A
0.2 0.4 0.6 0.8
0.64 044 0.28 0.14
0.36 0.36 0.38 0.43
By selecting appropriate values of #, relations (19) and (20) give R for various values of A, the results being shown in Table 2 and also in Fig. 5. For the special case where q = 0.5, relations (17) and (18) simplify to i = [l + (1 - #)““I + 3 ln(: T [: I ,“:J, %=(l-
$~)“~[l +(l-
Again by selecting appropriate A, the results
being
shown
$)“‘I -tln(:
(21)
T ii I $J.
values of #I, relations (21) and (22) give R for various 3 and also in Fig. 5.
(22) values
of
in Table
5. DISCUSSION
The clear conclusion that arises from inspecting the results in Fig. 5, particularly when they are viewed alongside the results in Table 1, is that for the case of a semi-infinite crack in a remotely loaded infinite solid, the fully developed softening zone size R is not especially sensitive to the details of the softening law, though it is dependent on pc (the maximum stress) and S, (the maximum displacement). R is approximately equal to the value for the DBCS model where the stress is constant within the softening zone, i.e. 0.39E36,/(1 - v2)pc. The magnitude of R can be approximately equated to the size D of crack in a strain-softening solid above which the semi-infinite crack-infinite solid relation for K [i.e. relation (4)] can be used to predict the failure of a stressed solid. The present paper’s results show that this critical size D has the DBCS value, i.e. 0.39B6,/(1 - v2)pc, but is essentially insensitive to the details of the force law. The failure stress will be less than that predicted using relation (4) if the crack size is less than D. As an example, for the case of a cellulose/asbestos fibre reinforced mortar, with the figures quoted by Foote et al. [l], i.e. E = 6OOOMPa, pc = 6MPa, 6, = O-8 mm, and ignoring Poisson’s ratio effects, D is approximately equal to 300 mm.
Softening zone size-1
REFERENCES [ 11 R. M. L. FOOTE, Y. W. MAI and B. COTIERELL, J. Me&. Phys. Solids 34,593 (1986). [2] E. SMITH, ht. J. Engng Sci. 27,309 (1989). [3] D. S. DUGDALE, J. Mech. Phys. Solids 8, 100 (1960). [4] B. A. BILBY, A. H, COTIRELL and K. H. SWINDEN, Proc. R. Sot. A272, 304 (1963). [5] J. R. WILLIS, J. Mech. Phys. Solids 15, 151 (1967). [6] J. R. RICE, In Fracture (Edited by H. Liebowitz), Vol. 2, p. 191. Academic Press, New York (1968). [7] E. SMITH, ht. J. Engng Fracture Me&. 6, 213 (1974). (Received 9 August 1988)
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