- Email: [email protected]
A solution for a dugdale crack subjected to a linearly varying tensile loading
Inr. J. Engng Sci. Vol. 8, pp. 85-95.
Pergamon Press 1970.
Printed in Great Britain
A SOLUTION FOR A DUGDALE CRACK SUBJECTED TO A LINEARLY VARYING TENSILE LOADING M. F. KANNINEN Advanced Solid Mechanics Division, Battelle Memorial Institute, Columbus, Ohio 43201, U.S.A. Abstract-The Dugdale crack model has been extended to include the effects of a linearly varying tensile loading. Under this type of loading the plastic zone lengths and the crack opening displacements at each end of a central crack are different. Expressions for these quantities in terms of the crack length, the parameters characterizing the loading, and the yield stress of the material are deduced by removing the singularities at the ends of the crack in accord with Dugdale’s postulate. It is found that the plastic zone lengthsare muchmore sensitive to a nonuniform loading of this kind than are the crack opening displacements. 1. INTRODUCTION IN DUCTILE fracture, crack growth takes place by the nucleation and coalesence of voids on the microscopic scale. This process is invariably accompanied by substantial plastic deformation and, when the dominant mode of this deformation is shear on 45” planes through-the-thickness, crack growth occurs under conditions which approximate plane stress [ 11. A useful model for this situation is that given by Dugdale [21 in which the plastic zone is taken to be simply an extension of the crack. It is, of course, recognized that a one dimensional plastic enclave represents a highly idealized situation. But, because the model supplies simple closed form relations, the results of which are found to be in good accord with experiment in many instances[3], the resulting simplicity often more than compensates for the lack of precision. Cracks in mild steel foil exhibit plastic zones like those of the Dugdale model. Consequently, this material provides an excellent experimental device for the study of ductile fracture under conditions approximating plane stress [4,5]. One difficulty which arises in testing center cracked foil coupons is that extreme care must be exercised to avoid a noticeable disparity in the lengths of the plastic zones at each end of the crack. It is of some importance, therefore, in interpreting the results of foil experiments, to be able to estimate the effects of a slightly misaligned loading. Of particular concern is the proclivity of such a loading to promote premature crack extension. Because ductile fracture is believed to be associated with a critical value of the crack opening displacement, expressions for this quantity are primarily required. In this paper an approximate solution is obtained for misaligned center cracked foil coupons by considering a Dugdale crack in the infinite plane subjected to a linearly varying tensile loading (Fig. 1). By removing the singularities in accord with Dugdale’s postulate, expressions for the plastic zone lengths are first obtained. Then, expressions for the crack opening displacements are obtained using the technique of Goodier and Field[6]. The results are given implicitly as nonlinear functions of the parameters characterizing the loading, the yield stress of the material and the crack length. More tractable relations are subsequently obtained by linearizing these results. 2. THE DUGDALE
CRACK
MODEL
In essence, the Dugdale crack model is obtained by superposing two elastic solutions. The first is that for a stress free crack of length 2a in an infinite plate under a 85
86
M. F. KANNINEN
Fig. 1. A Dugdale crack subjected to a linearly varying tensile loading. tension T. The second is for a crack loaded over intervals of length I= a - c at each of its ends by a tensile stress Y associated with the yield stress of the material. Dugdale recognized that the stress singularities in each solution not only occur at the same point but have exactly the same character. Hence, by adjusting the plastic zone lengths such that
uniform
?TT
C
a=cosZr the singularities can be made to exactly cancel leaving the stress field everywhere finite. A more direct way of deducing this equation was given by Goodier and Field[6]. Because the singular terms in the stress functions for each subproblem (of the KolosovMuskhelishvili complex variable type) differ by only a multiplicative constant, the singularities can be abolished by simply setting the coefficient of this term in the combined stress function to zero. In an equally expeditious manner, Goodier and Field were able to determine the normal displacements along the crack line for the Dugdale model. In particular, they found the crack opening displacement at the tip of the crack to be (K+
0c
=-Flog
1)
297
YC
(sec:f)
(2)
where K = (3 - v) / ( 1 •t u) for plane stress, 3-4~ for plane strain (V is Poisson’s ratio) and p is the shear modulus.
87
A solution for a Dugdale crack
These two results are quite important because much of the impetus for the growing popularity of the Dugdale model arises from experimental observations of the plastic zone length and the crack opening displacement. These observations are found to be in good agreement with equations (1) and (2) when the plastically deformed regions are narrow and wedge shaped [2-51. It is often incorrectly assumed that plastic deformation like that of the Dugdale model is always obtained if the material is thin enough. It should be emphasized, however, that the ‘thinness’ of the foil is not enough to assure this kind of deformation. Aluminum foil, for example, does not exhibit plastic enclaves of this kind. In addition to the specimen being thin, the material must work harden very little. Then, it will neck as soon as it yields giving throu~-the-thickness relaxation and, consequently, narrow elongated zones. 3. AN
EXTENSION
OF THE
DUGDALE MODEL VARYING LOADING
TO INCLUDE
A LINEARLY
In the present analysis an extension of the Dugdale model is made by including a linearly varying (or in-plane bending) tensile stress at infinity. Just as in previous work the approp~ate stress functions can be obtained by adapting results given by Muskhelishvili [7]. In place of the lone ‘singularity cancelling equation’ derived for the Dugdale model, however, two such equations are now obtained. The straightforward approach used by Goodier and Field to obtain the singularity cancelling equation and, hence, the plastic zone length cannot be used here. The terms in the stress functions which give rise to the singularities no longer all have the same functional dependence. Consequently, the stresses themselves must be evaluated. It is found that the stress singularities all have the same character (i.e. r-112,r being the distance from the crack tip) and they can be abolished by setting their combined coethcient to zero just as in the original problem. Formulation of the problem
Expressions relating the stresses and displacements to two analytic functions of a complex variable cpl(z) and Jll(z) are given by Muskhelishvili[8]. In applications of these relations to all but the most elementary regions it is necessary to use conformal mapping. The given region is represented on the 5 plane through the relation
whereupon ~(5) = (ol(z) and JI({) = JI, (2). When, as is the case here, the given region is mapped onto the infinite plane with a circular hole, it is natural to introduce polar coordinates by putting ([= pe w. The components of stress in the curvilinear coordinate system are then given [93as
(3)
where the prime notation indicates difIerentiation. These are related to the stress
88
M. F. KANNINEN
components
in the Cartesian system by the expressions
R-p^p+2iprij=-
jj:$#[o.-c~~+2i7~J.
The function which maps the infinite region outside the segment of x axis 1x1 < a onto the outside of the unit circle in the 5 plane is o(lJ =;
( > i+;
.
Hence, the singular points, which occur at 1x1= a, y = 0 in the z plane, are mapped onto the positions p = 1, 8 = 0,~ in the 4 plane. To remove the singularities it is sufficient to consider the stress components on the mapping of the prolongation of the crack line. These stresses can be obtained starting from the stress functions given by Muskhelishvili [7] for an elliptical hole in the infinite plane subjected to, (i) uniform tension at infinity, (ii) linearly varying tension at infinity and, (iii) internal pressure on a portion of its edge. By appropriately adapting these functions and then using equations (3) and (5), the & and p5 components of stress on the prolongation of the crack line are found to be as follows (p? = 0 in each case). First, for a uniform tensile loading uy = P acting at infinityt
[&I],=, = [di&_ = Pg (6)
ElI3=0 = [p^pltk,= -& Next, for the loading given by my = CLX at infinity
(7)
[phPle=o = - m1e=n = y--&. Finally, for an internal tension Y acting on the interval of length 1, on the right hand side of the slit and I, on the left hand side, with vanishing stress at infinity
tP will be related to T later.
89
A sdution for a Dugdale crack
and
sin &
+ tan-’
p+cosPB,
-tan-’
sin & P-cossz
I
(8)
where cos&
a-1, =---=; a
cl
(9)
a - I2_ cosp~=--a
c2
a’
The plastic zone lengths
After superposing the results given by equations (6), (7), and (8), the singular terms can be removed from each expression by evaluating the coefficient of the (p*- 1)-l term at p = 1 and setting it equal to zerot. The four equations so obtained reduce to ZP+cla-~[81+&+
sin&-sin&]
2P-aa-~DP,+Bz-sin&+sinS,1
= 0 =O.
Solving for & and p2 and using equations (9) then gives _Cl = a 2~
cos
I
Gy+
TP ~0s -_I 2Y
:y
sin-1
(
secEP
2Y >I
7rcxa 7TP sin-1 _secFf ( 4Y 11
(10)
which implicitly relate the plastic zone lengths to the parameters characterizing the applied stress. The applied stress has, for convenience, been taken with respect to a coordinate system centered on the interval of length 2a (i.e. the crack plus the plastic zones). In order to obtain a result in terms of parameters more relevant to a central crack in a plate of finite width, the origin must be relocated at the center of the free crack. This requires a translation of ( c2 - cl)/2 in the negative x direction. Then, referring to Fig. 1, T, the average applied stress for a central crack and 2c, the length of the free crack, can be related to P, a, cl and cZby the expressions
2c = Cl + cp. tit shouldperhapsbe noted that this procedure is valid only because here the singularity is a simple pole and, further, that the term does not vanish but takes on finite values at p = 1, as can be verified by a limiting procedure.
90
M. F. KANNINEN
Substituting from equation (10) then gives (11) and (12) Notice that in the special case where (Y= 0, P and T become identical whereupon equation (12) reduces to equation (l), Dugdale’s result. By examining the second of equations (lo), it can be seen that f,(= a - c2) will decrease as cxincreases until it vanishes for 2Y. (Y=--sm~-. 7ra
P (13)
Y
Thus, equation (13) apparently gives a relation for the maximum value of a for which these results are valid.7 The theory can be extended to higher values of a by introducing an additional postulate concerning the yield stress of the material in compression. In particular, it might be supposed that the yield stress in compression is also Y. Then, by rewriting equations (8) for a tension -Y acting in the interval of length l2 (the tension acting in 1, remains +Y) and proceeding as before, singularity cancelling equations can be derived for this case as well. The results are found to be identical to equations (IO). Hence, in the special case where the yield stresses in tension and compression are the same, equations (lo), (1 l), and (12), remain valid even for values of (Yin excess of that given by equation (13). However, because the theory developed here must be restricted to small values of o, as noted below, this result has little practical importance. The crack opening displacement
General expressions for the displacement field in the body can be obtained by the same kind of procedure that was followed to obtain the stress field. However, it is the normal displacement along the crack line that is of paramount importance and this quantity can be obtained much more conveniently following the technique of Goodier and Field[6]. For the point on the crack corresponding to a point 4 on the unit circle, they have shown that 2pv =-(K+
1) Re{@(c)}.
(14)
In particular, using the stress function for the Dugdale model they were able to deduce equation (2), via a limiting process, from the more general result given by equation (14). The same approach can be followed in the present work. Omitting the details, the result is found to be cos-’ % + cos-’
(a2-cf)1/2+
tThe argument of the inverse sine functions in equations (10) also imposes a requirement on a. But, it is not difficult to show that this requirement is less restrictive than equation (13).
A solution for a Dugdale crack
91
where uCl is the normal displacement at x = c,. By making use of equations (lo), equation (15) can be written in the alternative form
(;z;:r )I * ‘S-F ,ros*;; I rraasin
[
7rP+2sin1
vexa
nP
(
(161
In the special case where (II= 0, recalling that P then becomes equal to T, it can easily be seen that equation (16) reduces to equation (2), as it must. A similar expression for u,,, the crack opening displacement at the opposite end of the crack, can also be obtained. However, this quantity is not of interest because under the assumed loading v,, < U, < uCland crack extension will tend to be retarded at this end. It is the excess displacement (over that given by the uniform loading} that is important as this quantity will reflect the proclivity of the nonuniform loading to promote crack extension. 4. RESULTS
AND
DISCUSSION
For the calculation of the plastic zone lengths and the crack opening displacement with given values of the physically relevant parameters c, T, and OL,it is first necessary to solve the nonlinear equations (11) and (12) for the parameters a and P. Then, a - cz can be obtained from equations (10) and uC1from equation (16) 11=a-clandI,= Illustrative results obtained in this way are shown in Figs. 2 and 3. Because the applied stress (a,), = T +ax increases without limit as x increases, the yield stress will obviously be exceeded at positions other than at the a priori crack tip plastic zones. Hence, for this analysis to be valid, LY: must first of all be small enough so that monitions in the neighborhood of the crack tip are not si~fi~tly affected by this anomaly. This is expected to be the case provided that otc is small compared with both T and with the difference Y- T. In addition, because the finite boundaries of the component are ignored, the overall length of the crack plus the plastic zone must be small compared with these dimensions. It might be pointed out in this connection that by imposing boundary conditions appropriate to the way in which the foils are tested (i.e. displacement boundary conditions) significant differences are obtained between the finite plate and the infinite plate solutions for all crack lengths. This later point has been demonstrated by Hulbert, et ai.[4]_
With these restrictions in mind, the results shown in Figs. 2 and 3 can be used to estimate the effects of a slight misalignment in the loading on steel foil test coupons. However, it is difficult to apply these results directly because in an actual test the proper value of OLcannot be easily determined. A parameter which can be observed directly is the ratio of the plastic zone lengths. So, it may be more convenient to utilize this parameter as a measure of the misalignment. In this spirit the ratio ve,/vc (i.e. the
Fig. 2. Ratio of the plastic zone lengths for a linearly varying tensile loading.
i-1 -
@2-
Fig. 3. Crack opening displacement for a linearly varying tensile loading.
"C -!vc
k3-
A solution for a Dugdale crack
93
increase in the crack tip displacement over that which would be obtained if the same average loading were applied uniformly) is shown as a function of the ratio 1,/f, in Fig. 4. The dashed tine in Fig. 4 represents the hypothetical situation wherein the variation in the crack tip displacement is just equal to the variation in the plastic zone lengths.
Fig. 4. Crack opening displacement as a function of the plastic zone length ratio.
Because the actual results fall below this line, it can be seen that the crack tip displacement is always proportionately less affected by a skewed loading than are the plastic zone lengths. Furthermore, as the average loading is increased, this change becomes less and less. Thus, a misaligned loading always tends to manifest itself by elongating the plastic zones and only to a much less degree is the crack opening displacement increased. Finally, because the present theory is restricted to small values of cr, it is natural to seek more amenable relations by linearizing the foregoing results. In these circumstances, it is convenient to write uC1= v, + Au,: I, = I+ Al and l2 = I- Al. Then, retain-
94
M. F. KANNINEN
ing only terms linear in (Y,equations (10) can be combined to give
Al _“ZY 1
?rT ITT tan?,secTy. ~TQC l-cosz
(17)
L 2Y
Similarly, from equation (16) 7TT
hu,_rcYc --UC
tan??
(18)
4Y
Further simplification can be obtained by specializing the above to small values of T/Y. If this is done and the results combined to eliminate (Y,the simple relation (19) is obtained. Equation (19) is quite approximate (to what degree can be determined by checking it against the exact results shown in Fig. 4) but nonetheless very nicely points up the most important conclusion of the present work. This is, once again, that the discrepancy in the plastic zone lengths is a conservative indicator of the propensity of a small misalignment in the loading to lead to a premature fracture. Acknowledgments-The author is grateful to his colleagues Drs. G. T. Hahn and A. R. Rosenfield for many helpful discussions and to Mrs. Patricia Bare for her work on the manuscript. This work was supported by the Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio. REFERENCES [I] A. [2] D. 131 G. [4] L. [5] [6] [7] [8] [9]
R. ROSENFIELD, Metulf. Reo. 121,29 (1968). S. DUGDALE,J. Mech. Phys. Solids 8, 100 (1960). T. HAHN and A. R. ROSENFIELD, A.S.T.M.-STP432,5 (1968). E. HULBERT, G. T. HAHN, A. R. ROSENFIELD and M. F. KANNINEN, Proc. ofrhe 12th Int. Congress of/l&. Mech., Stanford, California (1968). (To be published by Springer 1970.) M. F. KANNINEN, A. K. MUKHERJEE, A. R. ROSENFIELD and G. T. HAHN, Mechanical Behaoior of Materials Under Dynamic Loads, edited by U. S. LINDHOLM, p. 96. Springer (1968). J. N. GOODIER and F. A. FIELD, Fracture of Solids, edited by D. C. DRUCKER and J. J. GILMAN, p. 103. Interscience (1963). N. I. MUSKHELISHVILI, Some Basic Problems of the Mathematical Theory of Elasticity, 4th Edn. pp. 343-353. Noordhoff,Griiningen(l963). N. I. MUSKHELISHVILI, Some Basic Problems of the Mathematical Theory of Elasticity, 4th Edn. pp. 112-l 14. Noordhoff, Grijningen (1963). N. I. MUSKHELISHVILI, Some Basic Problems of the Mathematical Theory of Elasticity, 4th Edn. p. 188. Noordhoff, Grijningen (1963). (Received
24 April 1969)
Resume-L’ttude suivant Dugdale, de la fissuration-type a et6 elargie pour tenir compte des effets de tensions lintairement variables. Sous une telle charge, les longueurs de la xone plastique et les dtplacements de l’ouverture de la fissure, a chacune des extremites d’une fissure cent&e, sont diff&ents. En supprimant les singularitCs aux extremites de la fissure, en accord avec le postulat de Dugdale, on obtient des expressions
A solutionfor a Dugdale cmck
95
pour ces valeurs en fonction de la longueur de la fissure, des param&res caract&isant la charge et de la charge de rupture.On trouve queles longueurs delazoneplastiquesontbeaucoupplussensiblesBunechargenon uniformedecegenrequenelesontlesdCplacementsdeI'ouverturedelafissure.
Zuammmnfasmmg-Das Dugdale Rissmodeg wurde erweitert urn die Wirkungen linear-veranderlicher Zugsbelastuug einzuschliessen. Unter dieser Art von Belastung sind die plastischen Zonenlangen und die Rissi%umgsverschiebungen an jedem Ende eines zentralen Risses verschieden. Ausdriicke fur diese Griissen in der Form der Risshge, der Parameter, die die Belastung beschreiben, und der Bruchspannung des Stoffes werden abgeleitet indem die Singularitaten an den Enden des Risses in Ubereinstimmung mit Dugdale’s Postulat entfemt werden. Es wird gefunden, dass die plastischen Zonenl&ngen weit mehr empfindlich fur ungleichmiissige Belastung dieser Art sind als die RissMungsverschiebungen. Sommario-Si b ampliato il modello d’incrinatura di Dugdale per comprendervi gli effetti di un carico di rottura variante linearmente. Con tale genere di carico le lunghezze delle zone plastichee glispostamenti dell'apertum dell'incrinatura a ciascuna estremita di un’incrinatura centrale sono differenti. Si deducono le espressioniper questi quantitativi in funzione della lunghezza dell’incrinatura, dei parametri the caratterizzano il carico, e della sollecitazionedi snervamento toghendo le singolarid alle estremita dell'incrinatura second0 il postulato di Dugdale. Si scopre the le hmghezze delle zone plastichesono molto pi6 sensibilia uncaricodisuguale diquestogenere diquantononlo A6cTparcT-PacmupaeTcs
sianoglispostamentidell'aperturadell'incrinatura.
MOLlenb TpeLUHHbl flarIl3gJlsHa C!lyYai?l303RekTBHs flHHetiH0U3MeHatOmeks Ilp~ HarpysKeTaKoro ponannwHbTnnacTwiecKnx30~ nnepe~emeHnnBbTxona Ha nOBepXHOCTb pa3ntiYHbl D,TflKa)KnOro KOHUa UeHTpa,TbHOg TpCU,HHbl. nyTeM yCTpaHeHBR OCO6eHHOCT& Ha KOHUaX TpelUWHbI,COrJlaCHO I-IOCTyJlOTy flarn3ttJln, BbIBOLIflTCaBb1pa~eHul-lLUls ynOM5lHyTbIX BenwrHH a @YHKUHL~ nnnHbr TpemwHbl, napaMeTpoB xapaKTepH3yromex Harpy3Ky H npenena TeKywcm MaTepHana. O6Hap)‘XWBaeTCfi, ST0 &TnHbI nnacTwiecKHx 30H ropa3no 6onee 'IyBCTBHTeJlbHblXK TaKOrO pODa HepaBHOMepHOg Harpy-JKe,9eM nepeMeU,eHHs BblXOllaTpeIUnHbl Ha nOBe,,XHOCTb.
pacrflruaatomeiI
riarpy3kn.
Pergamon Press 1970.
Printed in Great Britain
A SOLUTION FOR A DUGDALE CRACK SUBJECTED TO A LINEARLY VARYING TENSILE LOADING M. F. KANNINEN Advanced Solid Mechanics Division, Battelle Memorial Institute, Columbus, Ohio 43201, U.S.A. Abstract-The Dugdale crack model has been extended to include the effects of a linearly varying tensile loading. Under this type of loading the plastic zone lengths and the crack opening displacements at each end of a central crack are different. Expressions for these quantities in terms of the crack length, the parameters characterizing the loading, and the yield stress of the material are deduced by removing the singularities at the ends of the crack in accord with Dugdale’s postulate. It is found that the plastic zone lengthsare muchmore sensitive to a nonuniform loading of this kind than are the crack opening displacements. 1. INTRODUCTION IN DUCTILE fracture, crack growth takes place by the nucleation and coalesence of voids on the microscopic scale. This process is invariably accompanied by substantial plastic deformation and, when the dominant mode of this deformation is shear on 45” planes through-the-thickness, crack growth occurs under conditions which approximate plane stress [ 11. A useful model for this situation is that given by Dugdale [21 in which the plastic zone is taken to be simply an extension of the crack. It is, of course, recognized that a one dimensional plastic enclave represents a highly idealized situation. But, because the model supplies simple closed form relations, the results of which are found to be in good accord with experiment in many instances[3], the resulting simplicity often more than compensates for the lack of precision. Cracks in mild steel foil exhibit plastic zones like those of the Dugdale model. Consequently, this material provides an excellent experimental device for the study of ductile fracture under conditions approximating plane stress [4,5]. One difficulty which arises in testing center cracked foil coupons is that extreme care must be exercised to avoid a noticeable disparity in the lengths of the plastic zones at each end of the crack. It is of some importance, therefore, in interpreting the results of foil experiments, to be able to estimate the effects of a slightly misaligned loading. Of particular concern is the proclivity of such a loading to promote premature crack extension. Because ductile fracture is believed to be associated with a critical value of the crack opening displacement, expressions for this quantity are primarily required. In this paper an approximate solution is obtained for misaligned center cracked foil coupons by considering a Dugdale crack in the infinite plane subjected to a linearly varying tensile loading (Fig. 1). By removing the singularities in accord with Dugdale’s postulate, expressions for the plastic zone lengths are first obtained. Then, expressions for the crack opening displacements are obtained using the technique of Goodier and Field[6]. The results are given implicitly as nonlinear functions of the parameters characterizing the loading, the yield stress of the material and the crack length. More tractable relations are subsequently obtained by linearizing these results. 2. THE DUGDALE
CRACK
MODEL
In essence, the Dugdale crack model is obtained by superposing two elastic solutions. The first is that for a stress free crack of length 2a in an infinite plate under a 85
86
M. F. KANNINEN
Fig. 1. A Dugdale crack subjected to a linearly varying tensile loading. tension T. The second is for a crack loaded over intervals of length I= a - c at each of its ends by a tensile stress Y associated with the yield stress of the material. Dugdale recognized that the stress singularities in each solution not only occur at the same point but have exactly the same character. Hence, by adjusting the plastic zone lengths such that
uniform
?TT
C
a=cosZr the singularities can be made to exactly cancel leaving the stress field everywhere finite. A more direct way of deducing this equation was given by Goodier and Field[6]. Because the singular terms in the stress functions for each subproblem (of the KolosovMuskhelishvili complex variable type) differ by only a multiplicative constant, the singularities can be abolished by simply setting the coefficient of this term in the combined stress function to zero. In an equally expeditious manner, Goodier and Field were able to determine the normal displacements along the crack line for the Dugdale model. In particular, they found the crack opening displacement at the tip of the crack to be (K+
0c
=-Flog
1)
297
YC
(sec:f)
(2)
where K = (3 - v) / ( 1 •t u) for plane stress, 3-4~ for plane strain (V is Poisson’s ratio) and p is the shear modulus.
87
A solution for a Dugdale crack
These two results are quite important because much of the impetus for the growing popularity of the Dugdale model arises from experimental observations of the plastic zone length and the crack opening displacement. These observations are found to be in good agreement with equations (1) and (2) when the plastically deformed regions are narrow and wedge shaped [2-51. It is often incorrectly assumed that plastic deformation like that of the Dugdale model is always obtained if the material is thin enough. It should be emphasized, however, that the ‘thinness’ of the foil is not enough to assure this kind of deformation. Aluminum foil, for example, does not exhibit plastic enclaves of this kind. In addition to the specimen being thin, the material must work harden very little. Then, it will neck as soon as it yields giving throu~-the-thickness relaxation and, consequently, narrow elongated zones. 3. AN
EXTENSION
OF THE
DUGDALE MODEL VARYING LOADING
TO INCLUDE
A LINEARLY
In the present analysis an extension of the Dugdale model is made by including a linearly varying (or in-plane bending) tensile stress at infinity. Just as in previous work the approp~ate stress functions can be obtained by adapting results given by Muskhelishvili [7]. In place of the lone ‘singularity cancelling equation’ derived for the Dugdale model, however, two such equations are now obtained. The straightforward approach used by Goodier and Field to obtain the singularity cancelling equation and, hence, the plastic zone length cannot be used here. The terms in the stress functions which give rise to the singularities no longer all have the same functional dependence. Consequently, the stresses themselves must be evaluated. It is found that the stress singularities all have the same character (i.e. r-112,r being the distance from the crack tip) and they can be abolished by setting their combined coethcient to zero just as in the original problem. Formulation of the problem
Expressions relating the stresses and displacements to two analytic functions of a complex variable cpl(z) and Jll(z) are given by Muskhelishvili[8]. In applications of these relations to all but the most elementary regions it is necessary to use conformal mapping. The given region is represented on the 5 plane through the relation
whereupon ~(5) = (ol(z) and JI({) = JI, (2). When, as is the case here, the given region is mapped onto the infinite plane with a circular hole, it is natural to introduce polar coordinates by putting ([= pe w. The components of stress in the curvilinear coordinate system are then given [93as
(3)
where the prime notation indicates difIerentiation. These are related to the stress
88
M. F. KANNINEN
components
in the Cartesian system by the expressions
R-p^p+2iprij=-
jj:$#[o.-c~~+2i7~J.
The function which maps the infinite region outside the segment of x axis 1x1 < a onto the outside of the unit circle in the 5 plane is o(lJ =;
( > i+;
.
Hence, the singular points, which occur at 1x1= a, y = 0 in the z plane, are mapped onto the positions p = 1, 8 = 0,~ in the 4 plane. To remove the singularities it is sufficient to consider the stress components on the mapping of the prolongation of the crack line. These stresses can be obtained starting from the stress functions given by Muskhelishvili [7] for an elliptical hole in the infinite plane subjected to, (i) uniform tension at infinity, (ii) linearly varying tension at infinity and, (iii) internal pressure on a portion of its edge. By appropriately adapting these functions and then using equations (3) and (5), the & and p5 components of stress on the prolongation of the crack line are found to be as follows (p? = 0 in each case). First, for a uniform tensile loading uy = P acting at infinityt
[&I],=, = [di&_ = Pg (6)
ElI3=0 = [p^pltk,= -& Next, for the loading given by my = CLX at infinity
(7)
[phPle=o = - m1e=n = y--&. Finally, for an internal tension Y acting on the interval of length 1, on the right hand side of the slit and I, on the left hand side, with vanishing stress at infinity
tP will be related to T later.
89
A sdution for a Dugdale crack
and
sin &
+ tan-’
p+cosPB,
-tan-’
sin & P-cossz
I
(8)
where cos&
a-1, =---=; a
cl
(9)
a - I2_ cosp~=--a
c2
a’
The plastic zone lengths
After superposing the results given by equations (6), (7), and (8), the singular terms can be removed from each expression by evaluating the coefficient of the (p*- 1)-l term at p = 1 and setting it equal to zerot. The four equations so obtained reduce to ZP+cla-~[81+&+
sin&-sin&]
2P-aa-~DP,+Bz-sin&+sinS,1
= 0 =O.
Solving for & and p2 and using equations (9) then gives _Cl = a 2~
cos
I
Gy+
TP ~0s -_I 2Y
:y
sin-1
(
secEP
2Y >I
7rcxa 7TP sin-1 _secFf ( 4Y 11
(10)
which implicitly relate the plastic zone lengths to the parameters characterizing the applied stress. The applied stress has, for convenience, been taken with respect to a coordinate system centered on the interval of length 2a (i.e. the crack plus the plastic zones). In order to obtain a result in terms of parameters more relevant to a central crack in a plate of finite width, the origin must be relocated at the center of the free crack. This requires a translation of ( c2 - cl)/2 in the negative x direction. Then, referring to Fig. 1, T, the average applied stress for a central crack and 2c, the length of the free crack, can be related to P, a, cl and cZby the expressions
2c = Cl + cp. tit shouldperhapsbe noted that this procedure is valid only because here the singularity is a simple pole and, further, that the term does not vanish but takes on finite values at p = 1, as can be verified by a limiting procedure.
90
M. F. KANNINEN
Substituting from equation (10) then gives (11) and (12) Notice that in the special case where (Y= 0, P and T become identical whereupon equation (12) reduces to equation (l), Dugdale’s result. By examining the second of equations (lo), it can be seen that f,(= a - c2) will decrease as cxincreases until it vanishes for 2Y. (Y=--sm~-. 7ra
P (13)
Y
Thus, equation (13) apparently gives a relation for the maximum value of a for which these results are valid.7 The theory can be extended to higher values of a by introducing an additional postulate concerning the yield stress of the material in compression. In particular, it might be supposed that the yield stress in compression is also Y. Then, by rewriting equations (8) for a tension -Y acting in the interval of length l2 (the tension acting in 1, remains +Y) and proceeding as before, singularity cancelling equations can be derived for this case as well. The results are found to be identical to equations (IO). Hence, in the special case where the yield stresses in tension and compression are the same, equations (lo), (1 l), and (12), remain valid even for values of (Yin excess of that given by equation (13). However, because the theory developed here must be restricted to small values of o, as noted below, this result has little practical importance. The crack opening displacement
General expressions for the displacement field in the body can be obtained by the same kind of procedure that was followed to obtain the stress field. However, it is the normal displacement along the crack line that is of paramount importance and this quantity can be obtained much more conveniently following the technique of Goodier and Field[6]. For the point on the crack corresponding to a point 4 on the unit circle, they have shown that 2pv =-(K+
1) Re{@(c)}.
(14)
In particular, using the stress function for the Dugdale model they were able to deduce equation (2), via a limiting process, from the more general result given by equation (14). The same approach can be followed in the present work. Omitting the details, the result is found to be cos-’ % + cos-’
(a2-cf)1/2+
tThe argument of the inverse sine functions in equations (10) also imposes a requirement on a. But, it is not difficult to show that this requirement is less restrictive than equation (13).
A solution for a Dugdale crack
91
where uCl is the normal displacement at x = c,. By making use of equations (lo), equation (15) can be written in the alternative form
(;z;:r )I * ‘S-F ,ros*;; I rraasin
[
7rP+2sin1
vexa
nP
(
(161
In the special case where (II= 0, recalling that P then becomes equal to T, it can easily be seen that equation (16) reduces to equation (2), as it must. A similar expression for u,,, the crack opening displacement at the opposite end of the crack, can also be obtained. However, this quantity is not of interest because under the assumed loading v,, < U, < uCland crack extension will tend to be retarded at this end. It is the excess displacement (over that given by the uniform loading} that is important as this quantity will reflect the proclivity of the nonuniform loading to promote crack extension. 4. RESULTS
AND
DISCUSSION
For the calculation of the plastic zone lengths and the crack opening displacement with given values of the physically relevant parameters c, T, and OL,it is first necessary to solve the nonlinear equations (11) and (12) for the parameters a and P. Then, a - cz can be obtained from equations (10) and uC1from equation (16) 11=a-clandI,= Illustrative results obtained in this way are shown in Figs. 2 and 3. Because the applied stress (a,), = T +ax increases without limit as x increases, the yield stress will obviously be exceeded at positions other than at the a priori crack tip plastic zones. Hence, for this analysis to be valid, LY: must first of all be small enough so that monitions in the neighborhood of the crack tip are not si~fi~tly affected by this anomaly. This is expected to be the case provided that otc is small compared with both T and with the difference Y- T. In addition, because the finite boundaries of the component are ignored, the overall length of the crack plus the plastic zone must be small compared with these dimensions. It might be pointed out in this connection that by imposing boundary conditions appropriate to the way in which the foils are tested (i.e. displacement boundary conditions) significant differences are obtained between the finite plate and the infinite plate solutions for all crack lengths. This later point has been demonstrated by Hulbert, et ai.[4]_
With these restrictions in mind, the results shown in Figs. 2 and 3 can be used to estimate the effects of a slight misalignment in the loading on steel foil test coupons. However, it is difficult to apply these results directly because in an actual test the proper value of OLcannot be easily determined. A parameter which can be observed directly is the ratio of the plastic zone lengths. So, it may be more convenient to utilize this parameter as a measure of the misalignment. In this spirit the ratio ve,/vc (i.e. the
Fig. 2. Ratio of the plastic zone lengths for a linearly varying tensile loading.
i-1 -
@2-
Fig. 3. Crack opening displacement for a linearly varying tensile loading.
"C -!vc
k3-
A solution for a Dugdale crack
93
increase in the crack tip displacement over that which would be obtained if the same average loading were applied uniformly) is shown as a function of the ratio 1,/f, in Fig. 4. The dashed tine in Fig. 4 represents the hypothetical situation wherein the variation in the crack tip displacement is just equal to the variation in the plastic zone lengths.
Fig. 4. Crack opening displacement as a function of the plastic zone length ratio.
Because the actual results fall below this line, it can be seen that the crack tip displacement is always proportionately less affected by a skewed loading than are the plastic zone lengths. Furthermore, as the average loading is increased, this change becomes less and less. Thus, a misaligned loading always tends to manifest itself by elongating the plastic zones and only to a much less degree is the crack opening displacement increased. Finally, because the present theory is restricted to small values of cr, it is natural to seek more amenable relations by linearizing the foregoing results. In these circumstances, it is convenient to write uC1= v, + Au,: I, = I+ Al and l2 = I- Al. Then, retain-
94
M. F. KANNINEN
ing only terms linear in (Y,equations (10) can be combined to give
Al _“ZY 1
?rT ITT tan?,secTy. ~TQC l-cosz
(17)
L 2Y
Similarly, from equation (16) 7TT
hu,_rcYc --UC
tan??
(18)
4Y
Further simplification can be obtained by specializing the above to small values of T/Y. If this is done and the results combined to eliminate (Y,the simple relation (19) is obtained. Equation (19) is quite approximate (to what degree can be determined by checking it against the exact results shown in Fig. 4) but nonetheless very nicely points up the most important conclusion of the present work. This is, once again, that the discrepancy in the plastic zone lengths is a conservative indicator of the propensity of a small misalignment in the loading to lead to a premature fracture. Acknowledgments-The author is grateful to his colleagues Drs. G. T. Hahn and A. R. Rosenfield for many helpful discussions and to Mrs. Patricia Bare for her work on the manuscript. This work was supported by the Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio. REFERENCES [I] A. [2] D. 131 G. [4] L. [5] [6] [7] [8] [9]
R. ROSENFIELD, Metulf. Reo. 121,29 (1968). S. DUGDALE,J. Mech. Phys. Solids 8, 100 (1960). T. HAHN and A. R. ROSENFIELD, A.S.T.M.-STP432,5 (1968). E. HULBERT, G. T. HAHN, A. R. ROSENFIELD and M. F. KANNINEN, Proc. ofrhe 12th Int. Congress of/l&. Mech., Stanford, California (1968). (To be published by Springer 1970.) M. F. KANNINEN, A. K. MUKHERJEE, A. R. ROSENFIELD and G. T. HAHN, Mechanical Behaoior of Materials Under Dynamic Loads, edited by U. S. LINDHOLM, p. 96. Springer (1968). J. N. GOODIER and F. A. FIELD, Fracture of Solids, edited by D. C. DRUCKER and J. J. GILMAN, p. 103. Interscience (1963). N. I. MUSKHELISHVILI, Some Basic Problems of the Mathematical Theory of Elasticity, 4th Edn. pp. 343-353. Noordhoff,Griiningen(l963). N. I. MUSKHELISHVILI, Some Basic Problems of the Mathematical Theory of Elasticity, 4th Edn. pp. 112-l 14. Noordhoff, Grijningen (1963). N. I. MUSKHELISHVILI, Some Basic Problems of the Mathematical Theory of Elasticity, 4th Edn. p. 188. Noordhoff, Grijningen (1963). (Received
24 April 1969)
Resume-L’ttude suivant Dugdale, de la fissuration-type a et6 elargie pour tenir compte des effets de tensions lintairement variables. Sous une telle charge, les longueurs de la xone plastique et les dtplacements de l’ouverture de la fissure, a chacune des extremites d’une fissure cent&e, sont diff&ents. En supprimant les singularitCs aux extremites de la fissure, en accord avec le postulat de Dugdale, on obtient des expressions
A solutionfor a Dugdale cmck
95
pour ces valeurs en fonction de la longueur de la fissure, des param&res caract&isant la charge et de la charge de rupture.On trouve queles longueurs delazoneplastiquesontbeaucoupplussensiblesBunechargenon uniformedecegenrequenelesontlesdCplacementsdeI'ouverturedelafissure.
Zuammmnfasmmg-Das Dugdale Rissmodeg wurde erweitert urn die Wirkungen linear-veranderlicher Zugsbelastuug einzuschliessen. Unter dieser Art von Belastung sind die plastischen Zonenlangen und die Rissi%umgsverschiebungen an jedem Ende eines zentralen Risses verschieden. Ausdriicke fur diese Griissen in der Form der Risshge, der Parameter, die die Belastung beschreiben, und der Bruchspannung des Stoffes werden abgeleitet indem die Singularitaten an den Enden des Risses in Ubereinstimmung mit Dugdale’s Postulat entfemt werden. Es wird gefunden, dass die plastischen Zonenl&ngen weit mehr empfindlich fur ungleichmiissige Belastung dieser Art sind als die RissMungsverschiebungen. Sommario-Si b ampliato il modello d’incrinatura di Dugdale per comprendervi gli effetti di un carico di rottura variante linearmente. Con tale genere di carico le lunghezze delle zone plastichee glispostamenti dell'apertum dell'incrinatura a ciascuna estremita di un’incrinatura centrale sono differenti. Si deducono le espressioniper questi quantitativi in funzione della lunghezza dell’incrinatura, dei parametri the caratterizzano il carico, e della sollecitazionedi snervamento toghendo le singolarid alle estremita dell'incrinatura second0 il postulato di Dugdale. Si scopre the le hmghezze delle zone plastichesono molto pi6 sensibilia uncaricodisuguale diquestogenere diquantononlo A6cTparcT-PacmupaeTcs
sianoglispostamentidell'aperturadell'incrinatura.
MOLlenb TpeLUHHbl flarIl3gJlsHa C!lyYai?l303RekTBHs flHHetiH0U3MeHatOmeks Ilp~ HarpysKeTaKoro ponannwHbTnnacTwiecKnx30~ nnepe~emeHnnBbTxona Ha nOBepXHOCTb pa3ntiYHbl D,TflKa)KnOro KOHUa UeHTpa,TbHOg TpCU,HHbl. nyTeM yCTpaHeHBR OCO6eHHOCT& Ha KOHUaX TpelUWHbI,COrJlaCHO I-IOCTyJlOTy flarn3ttJln, BbIBOLIflTCaBb1pa~eHul-lLUls ynOM5lHyTbIX BenwrHH a @YHKUHL~ nnnHbr TpemwHbl, napaMeTpoB xapaKTepH3yromex Harpy3Ky H npenena TeKywcm MaTepHana. O6Hap)‘XWBaeTCfi, ST0 &TnHbI nnacTwiecKHx 30H ropa3no 6onee 'IyBCTBHTeJlbHblXK TaKOrO pODa HepaBHOMepHOg Harpy-JKe,9eM nepeMeU,eHHs BblXOllaTpeIUnHbl Ha nOBe,,XHOCTb.
pacrflruaatomeiI
riarpy3kn.