J-Q Characterization of Propagating Cracks.1023_A-1007558400880

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International Journal of Fracture 94: 357369, 1998. 1998 Kluwer Academic Publishers. Printed in the Netherlands.

J -Q characterization of propagating cracks

A. TRDEGRD, F. NILSSON and S. STLUND Department of Solid Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden;e-mail: [email protected]

Received 15 December 1997; accepted in revised form 12 September 1998

Abstract. An investigation is performed to determine to what extent the state at a growing crack tip vicinity canbe characterised by J and Q calculated from FE analyses of successively stationary crack tip positions. FE modelsin two dimensions of single edge notch bend and double edge cracked panel specimens with several different cracklengths are used to cover a range of load and constraint levels. The stress and strain elds are compared betweendifferent specimens keeping J - and Q -values equal. A remeshing technique in the commercial FE-code ABAQUSis used to enhance the efciencyof the analysis. The results show that theJ -Q -theory provides reasonably accuratecrack tip characterization also for growing cracks. This leads to the conclusion that FE analyses of successivestationary cracks rather than full FE propagation analyses are sufcient. The limit of validity for propagation issimilar to the validation limit for the stationary case, although somewhat more restrictive.

Keywords: Fracture, elastic-plastic fracture, constraint, two-parameter characterization, stable crack growth, niteelement method, remeshing.

1. Introduction

In recent years much effort has been spent on the question of how to characterize the stateat stationary crack tips in elastic-plastic materials when the loading is so elevated that aone-parameter characterization is no longer possible. The characterization may be made forinstance by the J -integral. Some consensus has been reached that, at least approximately,considerable improvement in the characterization capacity can be obtained by the so calledQ -parameter (cf. ODowd and Shih, 1993), which provides a way to quantify the loss of constraint at a crack tip, which in turn is caused by increasing plastic ow at the crack tip. Thecorresponding characterization problem for a propagating crack has, on the contrary, not beenvery much studied. The crack growth considered here is stable and quasi-static.

It is well known from theoretical and numerical analyses that the stress and strain eldsat the growing tip are considerably different from those of a stationary crack tip. Thus it isfor instance well known (cf. Nilsson (1992) and others) that the J -integral becomes path-dependent for moving cracks and its near tip value approaches zero. The results in (Nilsson,1992) and other investigations indicate that by taking the integration path sufciently remote

from the crack tip a value is obtained that usually coincides with the so called deformationtype J . This quantity is dened as the J -value that would have been obtained for a stationarycrack tip at the current position of the moving crack tip and subjected to the same remoteloading. In the following the term J should be understood in this sense.

The question of whether J can characterize the state of moving crack tips was also studiedfor some cases in (Nilsson, 1992). The conclusion was that J provides a poor characterizationif the load is elevated much above the ASTM E-399 limit for linearly elastic fracture mech-

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J -Q characterization of propagating cracks 359

2. Denition of crack tip quantities and problem formulation

Numerous investigators have discussed the two-dimensional state at a stationary crack tip inan elastic-plastic material. Following ODowd and Shih (1993) it is assumed that the statein the vicinity of a stationary crack tip under plane strain conditions can be represented to asufcient degree of accuracy by

ij = ( ij )SSY + ij = ( ij )SSY + Q 0ij . (1)

The term ( ij )SSY is the stress eld obtained from a standard two-dimensional plane strainsmall scale yielding (SSY) analysis, and thus directly connected to the value of J . The yieldstress is denoted by 0 and Q is the deviation in hydrostatic stress from the SSY-solution,scaled by 0. It has to be realized that the adopted form of the crack tip eld is not a math-ematically exact statement. The non-singular part is in fact not a constant but rather a slowlyvarying eld. This requires a predened way of evaluating Q and here this will be taken as

Q = ( )SSY

0

, at = 0, r = 2J / 0, (2)

where r and denote polar coordinates centered at the crack tip in deformed state. The stress is subsequently circumferential stress.

The description (1) breaks down when the loading becomes sufciently high. In order to judge when the description is applicable it has been suggested by for example ODowd andShih (1993) that the value of must not vary too much over a certain distance ahead of the crack tip. Dene a normalized radius as r = r/(J/ 0). If the mean gradient of over1 < r < 5 is

= ( r = 5) ( r = 1)

4 , (3)

then 0

< 0.1, (4)

provides a reasonable limit for the applicability of the Q -characterization.As remarked earlier a characterization of the state of a moving crack tip analogous to (1)

is not known and there is in fact no reason to assume that this is even possible. In order toinvestigate the use of analyses of stationary cracks to describe the state of propagating cracks,stresses, strains and crack opening angle COA are here compared between corresponding FE-models of propagating cracks. As already mentioned in the introduction, the strategy is tocompare the states at a propagating crack tip between different geometries at the same valueof J and Q dened in the way described above. It should be noted that during the propagationphase up to the point of comparison the values of Q do not necessarily coincide. In order toobtain reasonably realistic crack growth histories, a crack growth law of the following form isassumed

J J 0

= + ad

, (5)

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360 A. Trdegrd et al.

Figure 1. Geometries considered in the analyses including FE meshes.

where

J 0 = (d 20 )/ 5E. (6)

With the choice of J /J 0 = 1 the load level corresponds approximately to the ASTM E-399-limit for linear elastic fracture mechanics, d is a characteristic length of the specimengeometry as dened in Figure 1. When the crack length (and ligament length) is d/ 2 theASTM E-399-limit is exactly fullled at J /J 0 = 1. The Youngs modulus of the material isdenoted E . The dimensionless constants and have been assigned different values in thisstudy. In all analyses equals 1.25 while is chosen as 62.5 or 353 depending on the targetvalue of the loading. These values may be compared to the condition suggested as a limit forthe J -characterization by Hutchinson and Paris (1979)

=(d/ 2)

J dJ da 1,

(7)

where 1 10 is considered sufcient. Strict adherence to this condition is however notcritical in the present context since the inclusion of the Q -parameter is intended to improvethe characterization capacity. It should be noted that the resistance curve usually is affectedby the constraint level, see for instance Hancock et al. (1993) and Faleskog (1995).

Consider now two arbitrary different geometries (1 and 2) for which J 1(P,a) ,Q 1(P,a)and J 2(P,a) ,Q 2(P,a) , respectively, are known either from numerical analyses of successive

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stationary crack tip positions or from references such as ODowd and Shih (1993). P heredenotes the value of the chosen load parameter. In the calculations performed here, the loadingwas imposed by prescribing the displacement at the boundary. A suitable pair of targetvalues for J and Q is chosen so that J 1 = J 2 and Q 1 = Q 2 and from this the crack length atwhich the comparison is to be performed can be determined. In general this crack length willbe different for the two geometries. By the assumed propagation law (5), the crack growthincrement a is then also given. What remains to be determined are the loading historiesduring the propagation phase of the two geometries and these are obtained from the J (P,a)relation for each geometry so that the propagation law (5) is satised.

The material assumptions are those of an elastic-plastic material obeying von Mises owcriterion with an associated ow rule and isotropic power law hardening. The calculationshave been performed under nite strain assumptions and the constitutive law can be written inthe form

ij = E ijkl

E1 +

s ij skl

(1+ )E

2 e3

2h + 23

2e

kl , (8)

where ij is the Jaumann stress rate and kl the rate of deformation tensor. Theelastic modulustensor components are denoted by E ijkl , the stress deviator components by s ij , the effectivestress by e , pe the effective plastic strain and h = d f / d pe is the plastic hardening modulus.Plastic loading yields = 1 whereas for elastic loading or any unloading = 0. The owstress, and thus the hardening is given by the relation

pe = 0E

f 0

n

f E

for e > 0. (9)

In this study a value of the hardening exponent n = 5 has been assumed, also E / 0 = 500and = 0.3.

The geometries considered are those shown in Figure 1, that is a single edge notch bend(SEN(B)) specimen and a double-edge cracked panel (DECP). In addition, comparisons witha so called modied boundary layer (MBL) model were also made. This is a model where theouter region remains elastic and very large compared to the size of the plastic zone around thecrack with boundary conditions according to rst and second terms of the series expansion of stresses for a crack tip in a linearly elastic material

3. Analysis

In the present computations of propagating cracks, the load is rst applied to the structure withthe crack tip being stationary. When J reaches the value for initiation, the crack is advanced

by node release through a small part of the structure. This part of the structure has a meshspecially designed for propagation as shown in Figure 2b. All FE analyses in this study aremade assuming two-dimensional plane strain conditions and nite strain effects are accountedfor.

As the loading is applied to the FE mesh with the initially stationary crack the tip, bluntingcauses the mesh to distort due to large strains. When the FE mesh of the blunted crack isconsidered to be too distorted to provide reliable results or when the tip of a propagating

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(a) (b)

Figure 2. Crack tip meshes. (a) Initial mesh for stationary analysis. (b) Mesh after some amount of crack growthusing remeshing.

Figure 3. Remeshing of crack tip, stationary crack. The meshes shown are: (1) Initial mesh. (2) Mesh becomesdistorted due to loading. (3) New mesh with the same outer shape as previous mesh. (4) Mesh deformed due tofurther loading.

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The comparison with the highest load, comparison 3, was made with a/d = 0.5 for bothDECP and SEN(B). This load is far above the ASTM E-399-limit for linearly elastic fracturemechanics. The relation to the limit for non linear fracture mechanics according to ASTME-813 is here 1.8 times to high for the bend specimen SEN(B). The corresponding value forthe tensile geometry DECP is 14.

The MBL model was subjected to the same loading in terms of J and Q as the nite geo-metries. The crack tip was also forced to propagate a corresponding distance a . The reasonfor performing these analyses was to see whether a large plastic zone in a nite geometry hasany effect on the results. In an MBL analysis the plastic zone is always small compared to theouter radius.

Originally, the constants and in the propagation law (5) were chosen in accordancewith Nilsson (1992) to have the values 1.25 and 62.5, respectively. But as the load was veryhigh in comparison 3 this choice would have caused the crack to propagate through the wholeligament before reaching the desired load level. The slope (i.e. ) of the propagation lawwas therefore increased to = 353. This gives a crack propagation a of 25 percent of thespecimen width. The propagation takes place over almost 300 elements.

The dimensionless constants and have, as mentioned above, been assigned different

values in the different comparisons. This should have no great effect on the solution as longas it is observed that in all the comparisons the crack tip has propagated well through thearea close to the zone affected by the blunting of the original crack tip. If the crack were tostop at a relatively small distance after initiation, for instance at about one crack tip openingdisplacement, the results might have been affected by the severe plastic deformation causedby the blunting. With these values in equation (7) ranges from 141 to 1.97 in comparison 3and 25 to 12.5 in comparison 1. The effect of changing was to studied to some extent in theMBL model.

The analyses were performed following the same procedure of remeshing and node relax-ation as in the previously discussed performance study Trdegrd et al. (1998). The numberof elements in the analyses of the stationary crack for comparisons 1, 2 and 3 are 2407, 2007and 1815 elements respectively. In the analysis of the growing crack, the number of elementswere 2969, 2664 and 2286 elements for the comparisons 1, 2 and 3, respectively. The ratioof specimen width d and minimum element length l0 was also varied. In the analysis of thestationary crack this ratio falls in the range of 2 103 7 104 for comparison 3, 2 and 1 withthe coarsest mesh in comparison 3. The reason why the ratios differ so much in the analysisof stationary cracks is that the minimum element length is related to the crack tip openingdisplacement and thereby to the load level. In the analysis of the growing crack the ratio of (d/l 0) is about 1.5 104 for comparison 1 and around 1 103 and for both comparison 2 and3. The reason why the ratios are almost the same in the last two comparisons is that the crackgrowth increment a is almost the same, and the number of elements over which growthoccurs is also the same. The number of elements used in the MBL analyses was 3904 andthe crack growth proceeded over 90 elements. This is less than in the analyses of the nite

geometries, but the accuracy is still considered to be sufcient.

4. Results

For each of the comparisons the normal stress perpendicular to and along the prospectivecrack plane is shown for the case where the crack tip is stationary all the time at the chosen

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(a) (b)

(c)

Figure 4. Comparison 1 (lowest load level) J /J 0 = 2.5 and Q = 0.19. (a) stresses ahead of stationary cracktip, (b) stresses ahead of propagated crack tip, (c) total strains ahead of crack tip.

comparison position and subjected to the nal loading (Figure 4a, Figure 5a and Figure 6a).For the cracks that have propagated to this position the same stresses are shown (Figure 4b,Figure 5b and Figure 6b) together with the corresponding total strains (Figure 4c, Figure 5cand Figure 6c). In all these gures the results from the nite geometries are shown whilefor the MBL analyses results are shown only for the stress at propagation for the lowest andhighest load, respectively.

Considering rst the results for stationary cracks (Figure 4a, Figure 5a and Figure 6a) it isseen that the J -Q description is adequate for the DECP geometry for all three loading levelsand condition (4) is certainly satised. For the SEN(B) geometry this is also the case for thetwo lowest load levels. The condition (4) is for instance satised since here | / 0| = 0.04.

A large discrepancy is however evident for the highest load level. It is clearly seen in Figure 6athat the stress in the SEN(B) specimen does not satisfy the condition (4) for the differencestress since | / 0| = 0.5. What is seen in Figure 6a is actually the global bending stressof the SEN(B) specimen and thus it is not expected that the stresses of the propagating crackshould agree at this load level. The strains show even less agreement, as is evident in Figure 6c.The variations Q vs. crack tip radius for the geometries DECP and SEN(B) are described ingreater detail by ODowd and Shih (1993).

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(a) (b)

(c)

Figure 5. Comparison 2 (intermediate load level), J /J 0 = 20 and Q = 0.29. (a) stresses ahead of stationarycrack tip, (b) stresses ahead of propagated crack tip, (c) total strains ahead of crack tip.

For the propagating cracks, the comparisons 1 and 2 at lower loads are encouraging (Fig-ure 4b and Figure 5b) since the stress agrees well for the SEN(B), the DECP geometry andthe MBL results. For instance, at a normalized radius r of 2 the stresses differ 1.5 percent forthe lowest load. The strains shown in Figure 4c also show good agreement. The difference instrains that can be seen in Figure 4c at normalized radius r = 1 emanates from the natureof the nite element mesh used in the propagation analyses and has no physical background.The reason is a cluster of elements by the end of the rened zone as can be seen in the meshFigure 2b.

In comparison 2, the stresses of the propagating cracks (Figure 5b) are not identical butshow good agreement. At a normalized radius r of 2 the stresses are almost identical. The

strains shown in Figure 5c also show good agreement.In comparison 3, at the highest load level certain deviations are however evident. That theresult for the SEN(B) differs considerably from the other two geometries is not surprisingin view of the substantial disagreement for the stationary crack. More interesting is that theDECP results also deviate from the MBL results even though the difference is moderate. It isthus found that for this case the J -Q -description is not satisfactory, since a deviation occurseven though the results for a stationary crack are almost identical. Thus the J -Q -description

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(a) (b)

(c)

Figure 6. Comparison 3 (highest load level), J /J 0 90 and Q = 0.88. (a) stresses ahead of stationary cracktip, (b) stresses ahead of propagated crack tip, (c) total strains ahead of crack tip.

Table 2. Crack opening angle for different comparisons.

COA/[rad] DECP SEN(B) MBL

Comparison 1 0.0326 0.0314 0.0519Comparison 2 0.0274 0.0300 0.0328Comparison 3 0.0278 0.0200 0.0488

probably has a somewhat smaller region of applicability in the case of a growing crack thanfor the case of stationary cracks.

In summary it appears as the agreement between results for the two geometries and the

MBL results is good as long as the yielding is not too high.The crack opening angle (COA) has also been evaluated and a comparison is shown inTable 2 below. The crack opening angle varies slightly with the distance from the cracktip. The angle COA is here dened as the angle between half crack opening measured atnormalized radius r = 2 and the distance from the crack tip, i.e. COA = arctan(u/r) where udenotes the crack surface displacement. As can be seen in Table 2 there is no tendency for ahigher value of COA of the SEN(B) specimen compared to the DECP specimen or vice versa.

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5. Conclusions

The results of the present investigation do not contradict the hypothesis that the J -Q -theorycan be used to characterize the state at a propagating crack, also at load levels far above theASTME-399-limit. The limit of validity for propagation is most likely similar to the validationlimit for the stationary case although our results for the highest load indicate that the range of validity may be somewhat more limited. This points to the conclusion for practical purposesit is sufcient with FE analyses of successive stationary cracks, not full FE analyses of thepropagating crack, to characterize the state at a propagating crack. Of course an extensiveanalysis including many cases is needed to verify the hypothesis more thoroughly but thepresented results are encouraging.

For a bend specimen, the global bending stress at load levels above the limit load becomessignicant and destroys the Q -characterization as the stresses close to the crack tip differ toomuch from the SSY-solution. This effect is not as apparent in a tensile geometry, since it ismuch further away from general yield in the examples considered.

Even though the stresses in comparison 3, which do not all full the condition (4), differconsiderably between the specimen DECP and SEN(B) also at propagation, they show coher-

ence at small radii, close to the crack tip. Over which distances the states have to coincidecannot be answered by this type of analysis but depends on the micro-structural distanceinherent in the real problem under consideration.

The corresponding MBL analysis shows, however, that if the size of the plastic zone be-comes of the order of the specimen size, it will also have an effect on the stress state. In such acase J and Q alone cannot be regarded as describing the state at the crack tip. This differenceis somewhat more pronounced for growing cracks than for stationary cracks.

References

ABAQUS (1993). Users and theory manuals, version 5.3. Hibbitt, Karlsson and Sorensen, Inc., (HKS),Pawtucket, RI, USA.

Dodds, R.H., Tang, M. and Anderson, T.L. (1995). Numerical modelling of ductile tearing effects on cleavagefracture toughness. ASTM STP 1244 (Edited by M. Kirk and A. Bakker), American Society for Testing andMaterials, West Conshohoken, Pa, 100133.

Faleskog, J. (1995). Effectsof local constraint along three-dimensional crack fronts - anumerical and experimentalinvestigation. Journal of the Mechanics and Physics of Solids 43, 447493.

Faleskog, J. (1994). An experimental and numerical investigation of ductile crack growth characteristics in surfacecracked specimens under combined loading. International Journal of Fracture 68, 99126.

Hancock, J.W., Walter, G.R. and Parks, D.M. (1993). Constraint and toughness parameterized by T . Constraint Effects in Fracture, ASTM STP 1171 (Edited by E.M. Hackett, K.-H. Schwalbe, and R.H. Dodds). AmericanSociety for Testing and Materials, West Conshohoken, Pa, 2140.

Hutchinson, J.W. and Paris,P.C. (1979). Stabilityanalysis of J -controlled crack growth. ASTM STP 668 (Edited byJ. D. Landes, J. A. Begley and C. A. Clarke). American Society of Testing and Evaluation, West Conshohoken,Pa, 3764.

Nilsson, F. (1992). Numerical investigation of J -characterization of growing crack tips. Nuclear Engineering and Design 133 , 457463.

ODowd, N.P. and Shih, C.F. (1993). Two-parameter fracture mechanics: Theory and applications. NUREG/CR-5958 , CDNSWC/SME-CR-16-92, United States Nuclear Regulatory Commission.

Trdegrd, A., Nilsson, F. and stlund, S. (1998). FEM-remeshing technique applied to crack growth problems.Computer Methods in Applied Mechanics and Engineering 160(12), 115131.

Varias, A.G. and Shih, C.F. (1993). Quasi-static crack advance under a range of constraints steady state eldsbased on a characteristic length. Journal of the Mechanics and Physics of Solids 41, 835861.

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J-Q Characterization of Propagating Cracks.1023_A-1007558400880