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Effect of yield surface vertex on crack-tip fields in mode III
J. Mech. Ph_vs. Solids Vol. 29, No. 2, pp. 143-152, Printedin Great Britain.
0022-5096/8l/O20143-IO $02.0010 f? 1981 Pergamon Press Ltd.
1981
EFFECT OF YIELD SURFACE VERTEX CRACK-TIP FIELDS IN MODE III K. K. Lo and D. Division of Engineering, Brown
University,
ON
PEIRCE
Providence,
RI 02912, U.S.A.
(Received 7 February 1980)
ABSTRACT AN ASYMPTOTICcrack-tip
analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic-plastic solid. as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the Jz flow theory of plasticity.
1. IN STABLE
IN-~R~DUC-~I~N
crack growth, a material element near the crack tip typically experiences a strong nonproportional loading history as the crack tip advances through the elasticplastic solid. It has been found in plastic buckling problems (HUTCHINSON, 1974) that classical flow theory with a smooth yield surface normally overestimates the incremental stiffness of the material for nonproportional stress paths. For nearly proportional loading, the material is modelled more closely by the deformation theory of plasticity. However, when loading is highly nonproportional, deformation theory does not apply because it does not account for the unloading behavior. For a stationary crack under a monotonically increasing load, predictions by flow and deformation theory coincide because the loading is proportional everywhere. Recently, CHRISTOFFERSENand HUTCHINSON (1979) constructed a flow theory of plasticity which develops a corner on the yield surface. This theory gives the same predictions as deform ition theory for nearly proportional loading, while it provides a smooth transition for the strain-rate between the loading and unloading regimes. In this paper, the asymptotic crack-tip fields for quasi-static steady crack growth in antiplane shear (Mode III) are determined with the use of the above-mentioned corner theory. Similar analyses based on the J, flow theory and deformation theory were carried out by AMAZIGO and HUTCHINSON (1977) and SLEPYAN(1973). Our main objective here is to compare the corner theory results with those predicted by the J, flow theory and to assess the extent of the differences in the results predicted by the two different plasticity theories. 143
K. K. Lo and D.
144
2.
PEIR(.E
FORMULATIONAND ANALYSIS
Following the notation of CHRISTOFFERSEN and HUTCHINSON(1979), let the shear stresses and strains be r, E 03% and ;I, s 2~~~ (9 = 1,2). Cartesian coordinates X, moving with the crack tip will be used (see Fig. 1). As in AMAZIGOand HUTCHINSON (1977), we assume a bilineur hardening stress-strain curve in simple shear. As mentioned in the Introduction, the J, corner theory of CHRISTOFFERSEN and HUTCHINSON(1979) is used here to account for the strongly nonproportional loading histories near the crack tip. The basic idea of the theory is the existence of a plastic strain-rate potential Wp(6) which is homogeneous of degree two in the stress-rate. i.e. WP(cr) = ; /I($) c 6 ir,+
(1)
where h($) is a suitable “transition” function (!I($) corresponds to CHRISTOFFERSEN and HUTCHINSON’S (1979) f(O)), C is the plastic compliance tensor for nearly proportional loading, and ir is the stress-rate. Corner theory differs from the classical flow theories of plasticity by virtue of the transition range which bridges the gap between the usual loading and unloading regimes. The transition angle cl/, which is homogeneous of degree zero in the stress-rate, is defined implicitly by the direction of the loading increment as cos $ = CIir/(C66)!.
(2)
where the tensor 1, is directed along the axis of the yield cone in stress-rate space. k is normalized so that Chl = 1 ; hence $ is inherently positive and lies in the range 0 < II/ < Z. Note that C1 is directed along the axis of the yield cone in plastic strain rate space. Given Wp, then, we may obtain the plastic strain rate from the relation tP = ?Wp/c%. Complete details of the theory are furnished by CHRISTOFFERSENand HLJTCHIMON (1979). In antiplane shear the constitutive relation derived from (1) becomes that given by DEAN and HWCHINSON (1979), viz. G $, = (Q + $ Q* cotan [j) 2, - 4 Q* (sinfl cosB)- ’ s,i/s.
(3)
is the effective shear stress, and 0 Here. G is the shear modulus, T = (7:+7J ’ “’ denotes the material derivative of (. ). The angle 8. which turns out to be a more convenient transition angle. is related to the angle $ by tanlC/ = (G/G,-
1 )I” (G/G, - 1) ’ ’ tan/j.
(4)
RANSITION
FIN;.1. Crack-tip region showing co-ordinate according
systems and plastic loading and elastic unloading to (a) _I2 flow theory and (b) .JL corner theory.
+ c 6 ir = c,,,, ti,, kr-,. in the usual notation.
sectors
Effect of yield surface
vertex on crack-tip
where G, and G, are the tangent and secant moduli $, lies in 0 d /? < rr. Q may be written as
fields in
Mode III
145
in simple shear. Note that /3, like
Q(B) = 1 + [(G/G, - 1) cos*B + (G/G, - 1) sin*/llh($).
(5)
provided that we consider $ as a function of p. In (3) Q* = dQ/dp. As h(ll/) is taken to be unity for nearly proportional loading (say, $ < $o), equation (3) is identical to that given by the J2 deformation theory in that range. Elastic unloading occurs when, say, $ 3 $,, where h($) is taken to vanish. In the J, corner theory, $, is specified by [see CHRISTOFFERSENand HUTCHINSON (1979, equation (3.18))] tan+,
= -~,(T*-T~))~~*(G/G,--~)~~*(G/G~-~)-~’*,
(6)
where T~(> 0) is the yield stress in shear. Assuming that the stresses are singular at the crack tip, we can simplify the above description by noting that G, and G, become equal asymptotically. At the tip, (4) implies that $ = /?, and (6) tells us that tiE = j3, = n. (This corresponds to an infinitely sharp spike on the yield surface. While such a spike may not model material behavior accurately, it does provide a low stiffness limit. The angle 8, could equally well be chosen as less than n, especially if one seeks to preserve a reasonable corner in the presence of singular stresses.) The parameter $. = &( < B,) which bounds the range of nearly proportional loading may be chosen arbitrarily, but it turns out that the results are quite insensitive to any reasonable choice of &,. Hereafter, we use 8, &, and p, for $, $. and $,. Thus, with the above simplifications, from (5) we obtain
Q(B) = 1+ (G/G, - 1P(B) for the asymptotic analysis. The function h(B) used here is defined CHRISTOFFERSEN and HUTCHINSON’S (1979, equation (2.46)) g(4), Referring to their paper, we find that
(7) implicitly by with m = 3.
(8) The additional
relations
tar@-4) = -Ss’(d)/s(4)= 44)
(9)
WM4J) = Cl +12(4)1-1
(10)
and
provide all the required information about h(b) and its derivatives. The relation (8), with /?,, = 0, was used by DEAN and HUTCHINSON (1979) to characterize the corner theory in their analysis. For the Mode III crack problem discussed here, (3) together with (7)-(10) provide the full constitutive description. By way of contrast, the J, flow theory used by AMAZIGO and HUTCHINSON (1977) leads to Gj, = ?, + (G/G, - 1 )T,+/T for plastic
loading.
(11)
K. K. Lo and D.
146
PEIRCE
From previous studies (CHITALEY and MCCLINTOCK, 1971; SLEPYAN, 1973; AMAZIGO and HUTCHINSON, 1977) based on other plasticity theories, it seems plausible that the different loading sectors near the crack tip when corner theory is used are as shown in Fig. 1. Total loading (p d &,) takes place directly ahead of the crack tip, so that for the polar angle 0 (with the crack tip as the origin) in [0, O,] the response is governed by deformation theory. To develop the solution in this sector, we use the equilibrium and compatibility conditions 5, iI = 0,’ 71.2
-
(12)
Y2.1 .
=
0, I
the latter becoming (f&s).2 when deformation such that
- (f&s).,
theory applies. The introduction
(13)
= 0 of a stress-rate
function
0(x,,
x2)
(14) ensures equilibrium, and, with the bilinear compatibility equation implies that
effective stress-strain
curve used here, the
v20 = 0,
(15)
provided the stresses are singular at the crack tip. Following HUTCHINSON (1977), we seek a separable solution of the form
AMAZIGO and
(16)
0 = QV(@
where s (referred to as the singularity) and f(0) are to be found, and the amplitude K, relating the intensity of the near and far fields, cannot be determined in an asymptotic analysis of a stably growing crack. Symmetry and a convenient normalization give the boundary conditions f’(0) = 0, Equations
(15)-(17)
at this point
t The usual summation
(17)
oxeye,.
(18)
yield the solution f(e) = cos(s@,
It is convenient notation, viz.
f(0) = 1.
convention
to introduce
applies.
AMAZIGO and
HUTCHINSON’S (1977)
Effect of yield surface vertex on crack-tip
fields in Mode III
147
from which it follows that t, = -s cosOf+sin0f’,
t, = s sin0f+cos0f’, t* = - SJ
fr =f’,
T, = f.
sin0 T; = s co& T, + t 1, T = (Tf+T;)‘,‘=
(20)
(T,2+7;f)‘!2,
t = Tm1(T,t,+T,t2)=
I
T-‘(T,t,+Tots),
where t,, t, and T,, To are the radial and circumferential respectively. From the relations (20). we may write deformation theory solution using (18); in particular,
T2 = cos(sO),
T, = sin(s0), T, = sin((s+
components of t and T, down full details of the
1)0),
& = cos((s+
The deformation theory solution holds until the transition (2), one can show that this occurs when
(21) 1)O).
I angle p reaches /&. Using
(22)
W,t/J)~ = cos PO,
or when co& = co&,. It follows, interestingly, that the polar angle O0 bounding the total loading region is exactly equal to the transition angle &,. As shown in Fig. 1, transition behavior is anticipated in the range 8, < 0 d 0, (corresponding to /I0 d fi 6 PC), so the full constitutive relation (3) of the corner theory comes into play. Equilibrium is still satisfied with the introduction of the stress-rate function @. The requirement of compatibility, as we again use (16), leads to f”[T2N+T,2M+P(f’/t,t,-T,ItT)]+MtT2(T,T-’) +MT,T[(T,T-‘)‘f’+(T-‘7&J] + P{ T - ’ T’ + [s( r,f)’
- rf’]/Tt
+ s2ff’/t,t,)
+s2NT2f - sMtTT, = 0,
(23)
where P = cotan fl(T’f’N* N = N(B) =
+ TtT,M*),
QW’)+_1Q*(P) cotat@,
M = M(B) = -fQ*(fi)/sir$
co@,
Again following AMAZIGO and HUTCHINSON(1977), we use sin 0T; = s cos t3T,+ t, from (20) together with (23) to cast the problem as a system of three first-order nonlinear ordinary differential equations for the dependent variables f, f’, T, as functions of 0. The integration is begun with the three boundary-values found at the edge of the total loading sector (0 < 8 < e,,), i.e. f(&)
= cos(s&),
f’(0,)
= - sin(&),
q(0,)
= sin(&).
(24)
K. K. Lo and D. PEIIRCL
I48
For a particular case, the material is characterized by constant values of c( = G,/G and BO, and the integration is carried out with a trial value for s until the elastic unloading region is reached (at 0 = 0,) as shown in Fig. 1. Since the polar angle 0 = eP corresponds to the transition angle 8, = n, the integration through the transition region stops when t = -/z. At this stage, the continuity AMAZIGO and HUTCHINSON 1977, equation (2.24)) ,f”(fI,)+s
cotan(s(rr-B,))f(f?,)
condition
= 0
(see
(25)
is checked. This condition corresponds to continuity of traction-rate and displacement-rate across the elastic-plastic boundary (0 = 0J. If condition (25) is not satisfied sufficiently, then a new iterate for s is found and integration is started again from (24) until adequate convergence of s is obtained. For 0 2 (jr,, the elastic solution holds [cf. AMAZIGO and HUTCHINSON (1977, equation (2.23))] : T, =f=f(fI,)sin(s(lr-fl))/sin(s(~-0,)). Tl =
A,(sin
~I)S+~‘(O~)COS(S(~-(O))/~~~(~(~~-(I,)),
(fIP d 0 d rr),
where
(26) 49 = CTl(~[)-.f(B;)cot
Mn-B,))l/(sin
0,)s.
:
and it should be noted that we neglect any reversed loading zone at the crack flanks. For different values of r = G,/G, and with B0 = 0.01, the values of s and 8, are given in Table 1. For comparison purposes, the corresponding values from AMAZIGO and HUTCHINSON(1977) which are based on the J, flow theory, are also listed. In the particular case when r = 0.5. s and 0, are given in Table 2 for different values of /IO.
TABLE 1. Values
of s and
tip for
various
values
J, flow theory. Results from AMAZIGOand HUTCHINSON(1977)
0.1 0.2 0.3 0.5 0.7 0.8
-
0.207 0.277 0.325 0.394 0.444
1.221 1.329 1.393 1.473 1.523
ofcl =
G,/G.
8, = 0.01 rad
J, corner theory
-0.193 - 0.258 - 0.301 - 0.377 -0.433 - 0.457
2.39 2.61 2.70 2.95 3.05 3.09
Effect of yield surface vertex on crack-tip fields in Mode III
149
TABLE 2. Values of s and 8, for various values of 8,. u = G,/G = 0.5
fMrad)
s
e&ad)
0.005 0.01
- 0.377 - 0.377 - 0.375 - 0.373 -0.371 -0.371
2.95 2.95 2.92 2.91 2.86 2.85
0.2 0.4 0.8
1.0
3.
DISCUSSIONOF THE RESULTS
Values of the singularity s are calculated only for the range c1= 0.1 to c( = 0.8 since corner theory is not meaningful with perfectly-plastic (c( = 0) or elastic (a = 1) materials. In the case of perfect plasticity, the initially smooth yield surface remains fixed throughout the deformation ; J, corner theory, however, predicts a sharp spike on the yield surface in the presence of singular stresses no matter how small (x becomes. Clearly, the corner theory does not admit passage to the perfectly-plastic (a = 0) limit. Turning to elasticity, we find that the constitutive relation (3) reduces to the linear elastic law if a = 1. Calculations for CYnear unity indicate that the elastic singularity and stress fields are in fact approached. As can be seen from Table 1, however, 0, is nearing 7~ rather than fn (AMAZIGO and HUTCHINSON, 1977) as a approaches unity. Again, this is readily understood by considering the shape of the yield surfaces in both the J, corner and J, flow theories. In the former case, it is easily seen that u = 1 implies f3p= fi,; and since 8, = 7casymptotically, we must have that f3p approaches n: as LYnears unity. The J, flow theory, by contrast, has an effective p, of f~ because the smooth yield surface expands isotropically, and hence AMAZIGO and HUTCHINSON (1977) found that 8, approaches 37~. For the range of c1considered, then, we find that the extent of the plastic zone for the J, corner theory, as indicated by 8,, is much greater than that predicted by the J, flow theory. This appears to be consistent, at least qualitatively, with DEAN and HUTCHINSON’S(1979) finite element results, although singular stress fields are not incorporated in their calculations. For cases that do not border on elasticity, we may understand the larger plastic zone in the corner theory as arising because transition behavior is accounted for. Whereas the J, flow theory admits plasticity only as total loading, the J, corner theory accounts for plasticity outside the total loading regime, and thus can describe more material as plastically deforming, as it does here. (Physically, the transition regime is meant to correspond to deactivation of some, but not all, of the slip systems of the deforming material.) The total loading boundary flo( = 0,) is chosen arbitrarily in the present analysis. We use the value &, = 0.01 to approximate the more realistic “thoroughly nonlinear” characterization (CHRISTOFFERSENand HUTCHINSON, 1979), but, as seen from Table 2, for moderate hardening (a = 0..5), the singularity s is quite insensitive to any
150
K. K. Lo and D. PEIRCE
reasonable choice of &. (This indicates that the incremental stiffness through much of the transition region corresponds closely to that of deformation theory.) It must also be mentioned that the present work neglects any rrvcrsed yielding zone at the crack flanks. Such a zone, which subtends an angle of less than $ according to the perfectly-plastic analysis (CHITALEY and MCCLINTOCK. 1971), should have negligible effect on the remainder of the field. As for the crack-tip singularity s, we find that SLEPYAN’S(1973) results, based on a pure deformation theory analysis, correspond mow closely to the J2 flow theory singularities (AMAZIGO and HUTCHINSON, 1977) than do the results of our J, corner theory analysis. While we might normally expect the corner theory singularities to fall in between those from the deformation and flow theories, we cannot anticipate this here because of the inherent discontinuity in strain-rate that is present in Slepyan’s solution. As there is a definite transition from plastic loading to elastic unloading in the present Mode III problem, it is questionable whether deformation theory has application here. Be that as it may. there remains a degree of arbitrariness in the way Slepyan chooses his singularity; although Qp and s are uniquely related. the choice of either one of these parameters is indeterminate within a certain range. It is not disturbing, therefore, that Slepyan’s results should lie nearer to those of Amazigo and Hutchinson than do our present results. Remembering that corner theory involves no strain-rate discontinuity, then, we remark that the present analysis affords a more rigorous means of reducing the high incremental stiffness associated with the J2 flow theory. For the asymptotic problem discussed here, however, the J, corner theory probably reduces the incremental stiffness too much, since the corner on the yield surface is collapsed to such a sharp point in the presence of singular stresses. We do suggest, though, that the present calculations represent rigorous /O\Vstiffnes.s bounds on the actual stress fields and singularities for crack growth in antiplane shear, while Amazigo and Hutchinson’s Mode III results furnish the bounds for high stif’wss. In Fig. 2, the shear stress Q-factors T,, T, from the J, corner theory and the J, flow theory are compared within their respective plastic zones (0 d 0 6 8,) for the case a = 0.1. While the fields are substantially different. it is nevertheless interesting to
0.0 FIG.
2. Variation
0.2
0.4
06 0.6 0 (rod.)+
I .o
I.2
with polar angle 0 of the shear stress O-factors T., T, for J, corner and HUTCHINSON’S (1977) J2 flow theory.
theory and AMAZIGO
Effect of yield surface vertex on crack-tip fields in Mode III
151
note that the J, corner theory stresses exhibit the reduced resistance to nonproportional loading that one would expect to see. This reduced resistance is evident in Fig. 3, where the variations of T,/T*at a fixed material point are explicitly plotted. Still, although the shear stresses themselves differ, the effective shear stress T is found to be virtually the same for both the J, flow and corner theories. Although the effect of nonproportional loading near the crack tip as modelled by the corner theory is not too drastic here, it remains unclear whether such a conclusion can be drawn in plane strain (Mode I). Further study comparing the J, corner theory for the Mode I problem would help to assess the importance of nonproportional loading in plane-strain crack growth. A Mode I analysis following that of AMAZIGO and HUTCHINSON (1977), but using the J, corner theory instead of the J, flow theory, seems possible ; such a study, however, would need to consider a reversed plastic loading zone at the crack flanks, since a perfectly-plastic material in steady Mode I crack growth exhibits a reversed loading region that is not negligible (RICE, DRUGAN and SHAM, 1979). While the plane-strain analysis would certainly be desirable, it would also be substantially more involved than the present study.
PATH OF MATERIAL POINT CONSIDERED
;
1.0 CRACK
(X,
FIG.3. Variation of shear stress ratio
?,/re
= CONSTANT)
for a fixed material point.
ACKNOWLEDGEMENT This study is supported by the NSF Materials Research Laboratory at Brown University.
I52
K. K. Lo and D. PEIRCE
REFERENCES 1977
J. Mrch. Phys. Solids 25, 81
CHITALLY,A. D. and MCCLINTOCK,F. A. CHRISTOFFERSEN, J. and HUTCHINSON,J. W. DEAN, R. and HUTCHINSON,J. W
1971
Ibid. 19, 147.
1979
Ibid. 27, 465
1979
HUTCHINSON,J. W. RICE, J. R., DRUGAN, W. J. and SHAM. T. L.
1974 1979
SL~PYAN,L. I.
1973
Quasi-Static Steady Crack Growth in Small Scale Yielding. Harvard University, Division of Applied Sciences, Tech. Report MECH-1 I. ,4dv. appl. Mech. 14, 67. Ela.stic-mP/a.stic Anulysis ofGrowing Cracks. Brown University, Division of Engineering, Tech. Report No. 65. IZY. Akud. Nauk SSSR, Mekh. Tverd. Tela 8, 139. (Translated from the Russian)
AMAZIGO,J. C. and HUTCHINSON,J. W.
0022-5096/8l/O20143-IO $02.0010 f? 1981 Pergamon Press Ltd.
1981
EFFECT OF YIELD SURFACE VERTEX CRACK-TIP FIELDS IN MODE III K. K. Lo and D. Division of Engineering, Brown
University,
ON
PEIRCE
Providence,
RI 02912, U.S.A.
(Received 7 February 1980)
ABSTRACT AN ASYMPTOTICcrack-tip
analysis of stress and strain fields is carried out for an antiplane shear crack (Mode III) based on a corner theory of plasticity. Because of the nonproportional loading history experienced by a material element near the crack tip in stable crack growth, classical flow theory may predict an overly stiff response of the elastic-plastic solid. as is the case in plastic buckling problems. The corner theory used here accounts for this anomalous behavior. The results are compared with those of a similar analysis based on the Jz flow theory of plasticity.
1. IN STABLE
IN-~R~DUC-~I~N
crack growth, a material element near the crack tip typically experiences a strong nonproportional loading history as the crack tip advances through the elasticplastic solid. It has been found in plastic buckling problems (HUTCHINSON, 1974) that classical flow theory with a smooth yield surface normally overestimates the incremental stiffness of the material for nonproportional stress paths. For nearly proportional loading, the material is modelled more closely by the deformation theory of plasticity. However, when loading is highly nonproportional, deformation theory does not apply because it does not account for the unloading behavior. For a stationary crack under a monotonically increasing load, predictions by flow and deformation theory coincide because the loading is proportional everywhere. Recently, CHRISTOFFERSENand HUTCHINSON (1979) constructed a flow theory of plasticity which develops a corner on the yield surface. This theory gives the same predictions as deform ition theory for nearly proportional loading, while it provides a smooth transition for the strain-rate between the loading and unloading regimes. In this paper, the asymptotic crack-tip fields for quasi-static steady crack growth in antiplane shear (Mode III) are determined with the use of the above-mentioned corner theory. Similar analyses based on the J, flow theory and deformation theory were carried out by AMAZIGO and HUTCHINSON (1977) and SLEPYAN(1973). Our main objective here is to compare the corner theory results with those predicted by the J, flow theory and to assess the extent of the differences in the results predicted by the two different plasticity theories. 143
K. K. Lo and D.
144
2.
PEIR(.E
FORMULATIONAND ANALYSIS
Following the notation of CHRISTOFFERSEN and HUTCHINSON(1979), let the shear stresses and strains be r, E 03% and ;I, s 2~~~ (9 = 1,2). Cartesian coordinates X, moving with the crack tip will be used (see Fig. 1). As in AMAZIGOand HUTCHINSON (1977), we assume a bilineur hardening stress-strain curve in simple shear. As mentioned in the Introduction, the J, corner theory of CHRISTOFFERSEN and HUTCHINSON(1979) is used here to account for the strongly nonproportional loading histories near the crack tip. The basic idea of the theory is the existence of a plastic strain-rate potential Wp(6) which is homogeneous of degree two in the stress-rate. i.e. WP(cr) = ; /I($) c 6 ir,+
(1)
where h($) is a suitable “transition” function (!I($) corresponds to CHRISTOFFERSEN and HUTCHINSON’S (1979) f(O)), C is the plastic compliance tensor for nearly proportional loading, and ir is the stress-rate. Corner theory differs from the classical flow theories of plasticity by virtue of the transition range which bridges the gap between the usual loading and unloading regimes. The transition angle cl/, which is homogeneous of degree zero in the stress-rate, is defined implicitly by the direction of the loading increment as cos $ = CIir/(C66)!.
(2)
where the tensor 1, is directed along the axis of the yield cone in stress-rate space. k is normalized so that Chl = 1 ; hence $ is inherently positive and lies in the range 0 < II/ < Z. Note that C1 is directed along the axis of the yield cone in plastic strain rate space. Given Wp, then, we may obtain the plastic strain rate from the relation tP = ?Wp/c%. Complete details of the theory are furnished by CHRISTOFFERSENand HLJTCHIMON (1979). In antiplane shear the constitutive relation derived from (1) becomes that given by DEAN and HWCHINSON (1979), viz. G $, = (Q + $ Q* cotan [j) 2, - 4 Q* (sinfl cosB)- ’ s,i/s.
(3)
is the effective shear stress, and 0 Here. G is the shear modulus, T = (7:+7J ’ “’ denotes the material derivative of (. ). The angle 8. which turns out to be a more convenient transition angle. is related to the angle $ by tanlC/ = (G/G,-
1 )I” (G/G, - 1) ’ ’ tan/j.
(4)
RANSITION
FIN;.1. Crack-tip region showing co-ordinate according
systems and plastic loading and elastic unloading to (a) _I2 flow theory and (b) .JL corner theory.
+ c 6 ir = c,,,, ti,, kr-,. in the usual notation.
sectors
Effect of yield surface
vertex on crack-tip
where G, and G, are the tangent and secant moduli $, lies in 0 d /? < rr. Q may be written as
fields in
Mode III
145
in simple shear. Note that /3, like
Q(B) = 1 + [(G/G, - 1) cos*B + (G/G, - 1) sin*/llh($).
(5)
provided that we consider $ as a function of p. In (3) Q* = dQ/dp. As h(ll/) is taken to be unity for nearly proportional loading (say, $ < $o), equation (3) is identical to that given by the J2 deformation theory in that range. Elastic unloading occurs when, say, $ 3 $,, where h($) is taken to vanish. In the J, corner theory, $, is specified by [see CHRISTOFFERSENand HUTCHINSON (1979, equation (3.18))] tan+,
= -~,(T*-T~))~~*(G/G,--~)~~*(G/G~-~)-~’*,
(6)
where T~(> 0) is the yield stress in shear. Assuming that the stresses are singular at the crack tip, we can simplify the above description by noting that G, and G, become equal asymptotically. At the tip, (4) implies that $ = /?, and (6) tells us that tiE = j3, = n. (This corresponds to an infinitely sharp spike on the yield surface. While such a spike may not model material behavior accurately, it does provide a low stiffness limit. The angle 8, could equally well be chosen as less than n, especially if one seeks to preserve a reasonable corner in the presence of singular stresses.) The parameter $. = &( < B,) which bounds the range of nearly proportional loading may be chosen arbitrarily, but it turns out that the results are quite insensitive to any reasonable choice of &,. Hereafter, we use 8, &, and p, for $, $. and $,. Thus, with the above simplifications, from (5) we obtain
Q(B) = 1+ (G/G, - 1P(B) for the asymptotic analysis. The function h(B) used here is defined CHRISTOFFERSEN and HUTCHINSON’S (1979, equation (2.46)) g(4), Referring to their paper, we find that
(7) implicitly by with m = 3.
(8) The additional
relations
tar@-4) = -Ss’(d)/s(4)= 44)
(9)
WM4J) = Cl +12(4)1-1
(10)
and
provide all the required information about h(b) and its derivatives. The relation (8), with /?,, = 0, was used by DEAN and HUTCHINSON (1979) to characterize the corner theory in their analysis. For the Mode III crack problem discussed here, (3) together with (7)-(10) provide the full constitutive description. By way of contrast, the J, flow theory used by AMAZIGO and HUTCHINSON (1977) leads to Gj, = ?, + (G/G, - 1 )T,+/T for plastic
loading.
(11)
K. K. Lo and D.
146
PEIRCE
From previous studies (CHITALEY and MCCLINTOCK, 1971; SLEPYAN, 1973; AMAZIGO and HUTCHINSON, 1977) based on other plasticity theories, it seems plausible that the different loading sectors near the crack tip when corner theory is used are as shown in Fig. 1. Total loading (p d &,) takes place directly ahead of the crack tip, so that for the polar angle 0 (with the crack tip as the origin) in [0, O,] the response is governed by deformation theory. To develop the solution in this sector, we use the equilibrium and compatibility conditions 5, iI = 0,’ 71.2
-
(12)
Y2.1 .
=
0, I
the latter becoming (f&s).2 when deformation such that
- (f&s).,
theory applies. The introduction
(13)
= 0 of a stress-rate
function
0(x,,
x2)
(14) ensures equilibrium, and, with the bilinear compatibility equation implies that
effective stress-strain
curve used here, the
v20 = 0,
(15)
provided the stresses are singular at the crack tip. Following HUTCHINSON (1977), we seek a separable solution of the form
AMAZIGO and
(16)
0 = QV(@
where s (referred to as the singularity) and f(0) are to be found, and the amplitude K, relating the intensity of the near and far fields, cannot be determined in an asymptotic analysis of a stably growing crack. Symmetry and a convenient normalization give the boundary conditions f’(0) = 0, Equations
(15)-(17)
at this point
t The usual summation
(17)
oxeye,.
(18)
yield the solution f(e) = cos(s@,
It is convenient notation, viz.
f(0) = 1.
convention
to introduce
applies.
AMAZIGO and
HUTCHINSON’S (1977)
Effect of yield surface vertex on crack-tip
fields in Mode III
147
from which it follows that t, = -s cosOf+sin0f’,
t, = s sin0f+cos0f’, t* = - SJ
fr =f’,
T, = f.
sin0 T; = s co& T, + t 1, T = (Tf+T;)‘,‘=
(20)
(T,2+7;f)‘!2,
t = Tm1(T,t,+T,t2)=
I
T-‘(T,t,+Tots),
where t,, t, and T,, To are the radial and circumferential respectively. From the relations (20). we may write deformation theory solution using (18); in particular,
T2 = cos(sO),
T, = sin(s0), T, = sin((s+
components of t and T, down full details of the
1)0),
& = cos((s+
The deformation theory solution holds until the transition (2), one can show that this occurs when
(21) 1)O).
I angle p reaches /&. Using
(22)
W,t/J)~ = cos PO,
or when co& = co&,. It follows, interestingly, that the polar angle O0 bounding the total loading region is exactly equal to the transition angle &,. As shown in Fig. 1, transition behavior is anticipated in the range 8, < 0 d 0, (corresponding to /I0 d fi 6 PC), so the full constitutive relation (3) of the corner theory comes into play. Equilibrium is still satisfied with the introduction of the stress-rate function @. The requirement of compatibility, as we again use (16), leads to f”[T2N+T,2M+P(f’/t,t,-T,ItT)]+MtT2(T,T-’) +MT,T[(T,T-‘)‘f’+(T-‘7&J] + P{ T - ’ T’ + [s( r,f)’
- rf’]/Tt
+ s2ff’/t,t,)
+s2NT2f - sMtTT, = 0,
(23)
where P = cotan fl(T’f’N* N = N(B) =
+ TtT,M*),
QW’)+_1Q*(P) cotat@,
M = M(B) = -fQ*(fi)/sir$
co@,
Again following AMAZIGO and HUTCHINSON(1977), we use sin 0T; = s cos t3T,+ t, from (20) together with (23) to cast the problem as a system of three first-order nonlinear ordinary differential equations for the dependent variables f, f’, T, as functions of 0. The integration is begun with the three boundary-values found at the edge of the total loading sector (0 < 8 < e,,), i.e. f(&)
= cos(s&),
f’(0,)
= - sin(&),
q(0,)
= sin(&).
(24)
K. K. Lo and D. PEIIRCL
I48
For a particular case, the material is characterized by constant values of c( = G,/G and BO, and the integration is carried out with a trial value for s until the elastic unloading region is reached (at 0 = 0,) as shown in Fig. 1. Since the polar angle 0 = eP corresponds to the transition angle 8, = n, the integration through the transition region stops when t = -/z. At this stage, the continuity AMAZIGO and HUTCHINSON 1977, equation (2.24)) ,f”(fI,)+s
cotan(s(rr-B,))f(f?,)
condition
= 0
(see
(25)
is checked. This condition corresponds to continuity of traction-rate and displacement-rate across the elastic-plastic boundary (0 = 0J. If condition (25) is not satisfied sufficiently, then a new iterate for s is found and integration is started again from (24) until adequate convergence of s is obtained. For 0 2 (jr,, the elastic solution holds [cf. AMAZIGO and HUTCHINSON (1977, equation (2.23))] : T, =f=f(fI,)sin(s(lr-fl))/sin(s(~-0,)). Tl =
A,(sin
~I)S+~‘(O~)COS(S(~-(O))/~~~(~(~~-(I,)),
(fIP d 0 d rr),
where
(26) 49 = CTl(~[)-.f(B;)cot
Mn-B,))l/(sin
0,)s.
:
and it should be noted that we neglect any reversed loading zone at the crack flanks. For different values of r = G,/G, and with B0 = 0.01, the values of s and 8, are given in Table 1. For comparison purposes, the corresponding values from AMAZIGO and HUTCHINSON(1977) which are based on the J, flow theory, are also listed. In the particular case when r = 0.5. s and 0, are given in Table 2 for different values of /IO.
TABLE 1. Values
of s and
tip for
various
values
J, flow theory. Results from AMAZIGOand HUTCHINSON(1977)
0.1 0.2 0.3 0.5 0.7 0.8
-
0.207 0.277 0.325 0.394 0.444
1.221 1.329 1.393 1.473 1.523
ofcl =
G,/G.
8, = 0.01 rad
J, corner theory
-0.193 - 0.258 - 0.301 - 0.377 -0.433 - 0.457
2.39 2.61 2.70 2.95 3.05 3.09
Effect of yield surface vertex on crack-tip fields in Mode III
149
TABLE 2. Values of s and 8, for various values of 8,. u = G,/G = 0.5
fMrad)
s
e&ad)
0.005 0.01
- 0.377 - 0.377 - 0.375 - 0.373 -0.371 -0.371
2.95 2.95 2.92 2.91 2.86 2.85
0.2 0.4 0.8
1.0
3.
DISCUSSIONOF THE RESULTS
Values of the singularity s are calculated only for the range c1= 0.1 to c( = 0.8 since corner theory is not meaningful with perfectly-plastic (c( = 0) or elastic (a = 1) materials. In the case of perfect plasticity, the initially smooth yield surface remains fixed throughout the deformation ; J, corner theory, however, predicts a sharp spike on the yield surface in the presence of singular stresses no matter how small (x becomes. Clearly, the corner theory does not admit passage to the perfectly-plastic (a = 0) limit. Turning to elasticity, we find that the constitutive relation (3) reduces to the linear elastic law if a = 1. Calculations for CYnear unity indicate that the elastic singularity and stress fields are in fact approached. As can be seen from Table 1, however, 0, is nearing 7~ rather than fn (AMAZIGO and HUTCHINSON, 1977) as a approaches unity. Again, this is readily understood by considering the shape of the yield surfaces in both the J, corner and J, flow theories. In the former case, it is easily seen that u = 1 implies f3p= fi,; and since 8, = 7casymptotically, we must have that f3p approaches n: as LYnears unity. The J, flow theory, by contrast, has an effective p, of f~ because the smooth yield surface expands isotropically, and hence AMAZIGO and HUTCHINSON (1977) found that 8, approaches 37~. For the range of c1considered, then, we find that the extent of the plastic zone for the J, corner theory, as indicated by 8,, is much greater than that predicted by the J, flow theory. This appears to be consistent, at least qualitatively, with DEAN and HUTCHINSON’S(1979) finite element results, although singular stress fields are not incorporated in their calculations. For cases that do not border on elasticity, we may understand the larger plastic zone in the corner theory as arising because transition behavior is accounted for. Whereas the J, flow theory admits plasticity only as total loading, the J, corner theory accounts for plasticity outside the total loading regime, and thus can describe more material as plastically deforming, as it does here. (Physically, the transition regime is meant to correspond to deactivation of some, but not all, of the slip systems of the deforming material.) The total loading boundary flo( = 0,) is chosen arbitrarily in the present analysis. We use the value &, = 0.01 to approximate the more realistic “thoroughly nonlinear” characterization (CHRISTOFFERSENand HUTCHINSON, 1979), but, as seen from Table 2, for moderate hardening (a = 0..5), the singularity s is quite insensitive to any
150
K. K. Lo and D. PEIRCE
reasonable choice of &. (This indicates that the incremental stiffness through much of the transition region corresponds closely to that of deformation theory.) It must also be mentioned that the present work neglects any rrvcrsed yielding zone at the crack flanks. Such a zone, which subtends an angle of less than $ according to the perfectly-plastic analysis (CHITALEY and MCCLINTOCK. 1971), should have negligible effect on the remainder of the field. As for the crack-tip singularity s, we find that SLEPYAN’S(1973) results, based on a pure deformation theory analysis, correspond mow closely to the J2 flow theory singularities (AMAZIGO and HUTCHINSON, 1977) than do the results of our J, corner theory analysis. While we might normally expect the corner theory singularities to fall in between those from the deformation and flow theories, we cannot anticipate this here because of the inherent discontinuity in strain-rate that is present in Slepyan’s solution. As there is a definite transition from plastic loading to elastic unloading in the present Mode III problem, it is questionable whether deformation theory has application here. Be that as it may. there remains a degree of arbitrariness in the way Slepyan chooses his singularity; although Qp and s are uniquely related. the choice of either one of these parameters is indeterminate within a certain range. It is not disturbing, therefore, that Slepyan’s results should lie nearer to those of Amazigo and Hutchinson than do our present results. Remembering that corner theory involves no strain-rate discontinuity, then, we remark that the present analysis affords a more rigorous means of reducing the high incremental stiffness associated with the J2 flow theory. For the asymptotic problem discussed here, however, the J, corner theory probably reduces the incremental stiffness too much, since the corner on the yield surface is collapsed to such a sharp point in the presence of singular stresses. We do suggest, though, that the present calculations represent rigorous /O\Vstiffnes.s bounds on the actual stress fields and singularities for crack growth in antiplane shear, while Amazigo and Hutchinson’s Mode III results furnish the bounds for high stif’wss. In Fig. 2, the shear stress Q-factors T,, T, from the J, corner theory and the J, flow theory are compared within their respective plastic zones (0 d 0 6 8,) for the case a = 0.1. While the fields are substantially different. it is nevertheless interesting to
0.0 FIG.
2. Variation
0.2
0.4
06 0.6 0 (rod.)+
I .o
I.2
with polar angle 0 of the shear stress O-factors T., T, for J, corner and HUTCHINSON’S (1977) J2 flow theory.
theory and AMAZIGO
Effect of yield surface vertex on crack-tip fields in Mode III
151
note that the J, corner theory stresses exhibit the reduced resistance to nonproportional loading that one would expect to see. This reduced resistance is evident in Fig. 3, where the variations of T,/T*at a fixed material point are explicitly plotted. Still, although the shear stresses themselves differ, the effective shear stress T is found to be virtually the same for both the J, flow and corner theories. Although the effect of nonproportional loading near the crack tip as modelled by the corner theory is not too drastic here, it remains unclear whether such a conclusion can be drawn in plane strain (Mode I). Further study comparing the J, corner theory for the Mode I problem would help to assess the importance of nonproportional loading in plane-strain crack growth. A Mode I analysis following that of AMAZIGO and HUTCHINSON (1977), but using the J, corner theory instead of the J, flow theory, seems possible ; such a study, however, would need to consider a reversed plastic loading zone at the crack flanks, since a perfectly-plastic material in steady Mode I crack growth exhibits a reversed loading region that is not negligible (RICE, DRUGAN and SHAM, 1979). While the plane-strain analysis would certainly be desirable, it would also be substantially more involved than the present study.
PATH OF MATERIAL POINT CONSIDERED
;
1.0 CRACK
(X,
FIG.3. Variation of shear stress ratio
?,/re
= CONSTANT)
for a fixed material point.
ACKNOWLEDGEMENT This study is supported by the NSF Materials Research Laboratory at Brown University.
I52
K. K. Lo and D. PEIRCE
REFERENCES 1977
J. Mrch. Phys. Solids 25, 81
CHITALLY,A. D. and MCCLINTOCK,F. A. CHRISTOFFERSEN, J. and HUTCHINSON,J. W. DEAN, R. and HUTCHINSON,J. W
1971
Ibid. 19, 147.
1979
Ibid. 27, 465
1979
HUTCHINSON,J. W. RICE, J. R., DRUGAN, W. J. and SHAM. T. L.
1974 1979
SL~PYAN,L. I.
1973
Quasi-Static Steady Crack Growth in Small Scale Yielding. Harvard University, Division of Applied Sciences, Tech. Report MECH-1 I. ,4dv. appl. Mech. 14, 67. Ela.stic-mP/a.stic Anulysis ofGrowing Cracks. Brown University, Division of Engineering, Tech. Report No. 65. IZY. Akud. Nauk SSSR, Mekh. Tverd. Tela 8, 139. (Translated from the Russian)
AMAZIGO,J. C. and HUTCHINSON,J. W.