FINITE ELEMENT ANALYSIS OF VOID GROWTH NEAR A …FINITE ELEMENT ANALYSIS OF VOID GROWTH NEAR A...

J. Mech. Phys. Solids Vol. 33, No. 1, Pp. 25-49, 1985. Printed in Great Britain.

0022-5096/85 $3.00 + 00.00 0 1985 Pergamon Press Ltd.

FINITE ELEMENT ANALYSIS OF VOID GROWTH NEAR A BLUNTING CRACK TIP

N. ARAVAS and R. M. MCMEEKING

Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.

(Received 27 March 1984)

ABSTRACT

LARGE deformation finite element analysis has been used to study the near crack tip growth of long cylindrical holes aligned parallel to the plane of a mode I plane strain crack. The near crack tip stress and deformation fields are analyzed. The results show that the holes are pulled towards the crack tip and change their shape to approximately elliptical with the major axis radial to the crack. They also grow faster directly ahead of the crack than at an angle to the crack plane. Several crack-hole coalescence criteria are discussed and estimates for the conditions for fracture initiation are given and compared with experimental results. The range of estimates now available from finite element calculations coincides quite well with the range of experimental data for materials containing inclusions which are only loosely bonded to the matrix.

1. IN-I-R~DuC~~N

IN this paper we will be concerned with theoretical calculations of void growth near a blunting crack tip in an elastic-plastic material and with its consequences for crack propagation. In ductile metals, failure by coalescence of microscopic voids is an important fracture mechanism both in nominally uniform stress fields (ROGERS, 1960;

GURLAND and PLATEAU, 1963 ; BLUHM and MORRISEY, 1966 ; Cox and Low, 1974) and ahead of an existing crack (BEACHEM, 1963 ; BEACHEM and YODER, 1973 ; CLAYTON and KNOTT, 1976). The voids are nucleated by fracture or interfacial decohesion of inclusions and grow due to plastic straining of the surrounding matrix material. The void growth can be interrupted by the formation of void sheets between larger voids or between large voids and the crack tip. These sheets are composed of smaller voids nucleated at precipitate particles or carbides and lead to final coalescence of large voids with each other or with the crack tip.

Although void nucleation is the subject of a great deal of research, the results are not yet so clear that conditions for nucleation can be used in void growth calculations. A critical normal stress at the matrix-particle interface (ARGON and IM, 1975 ; BEREMIN, 1981) or a critical strain of the matrix material (GURLAND and PLATEAU, 1963 ; BROWN

and EMBURY, 1973 ; GODS and BROWN, 1979) have been proposed as void nucleation criteria.

On the other hand, void growth has been studied by several researchers. BERG (1962)

studied the growth of an elliptical hole in a remotely uniform stress field in a linear

25

26 N. ARAVAS and R. M. MCM~KING

viscous material under either plane strain or plane stress conditions. MCCLINTOCK (1968) analyzed the growth of a long cylindrical hole of elliptical cross section in a remotely uniform stress field in a rigid perfectly-plastic material under conditions of generalized plane strain. Following this, RICE and TRACEY (1969) analyzed the growth of isolated spherical and cylindrical voids in rigid perfectly-plastic materials. The growth of a spherical void ahead of a blunting crack was analyzed in an approximate way by RICE and JOLSON (1970), who used the results of Rrcz and TRACEY (1969). RICE

and JOHNSON (1970) identified the remote field of Rrcf! and TRACEY’S (1969) analysis with a slip-line solution for the local stress and defo~ation fields of a blunting crack computed as if no void were present. In similar work, MCMEEKING (1977) studied the growth of spherical voids ahead of the crack tip and at 45” to it using his results from detailed finite element analyses of a blunting crack. HELLAN (1975) presented an approximate study for the growth of isolated spherical voids by plasticity or creep. Following this, BUDIANSKY, HUTCHINSON and SLUT~KY (1982) studied the growth of spheroidal voids in non-linear viscous materials and obtained an improvement to the result of RICE and- TRACXY (1969). In all the above analyses the nearest neighbor interaction or the crack-void interaction has not been taken into account. TRACEY

(1971) analyzed the growth of long cy~nd~~al holes in a rigid-plastic, strain hardening material, taking hole interaction into account. N~~L~~~ (1972) studied nume~~ally the growth of long cylindrical holes in cubic arrays in an elastic-plastic material under plane strain conditions of macroscopically uniform strain, taking hole interaction into account. BURKE and NIX (1979) solved the same problem numerically for a creeping material. In an effort to obtain estimates for the yield surface for porous ductile media, GUR~~N (1977a, b) studied the growth of cylindrical holes in cylindrical cells and spherical voids in spherical cells and developed pressure sensitive dilatant plasticity laws. ANDERSSON (1977) presented a simplified model for the growth of spherical holes ahead of a crack tip. TVERGAARD (198 1,1982a, b) using GUR~ON’S (1977a, b) constitutive law obtained detailed finite element results for shear band instab~ities taking hole interaction into account.

Experimental results for ductile fracture show that the fracture toughness of metals depends on the size and spacing of void nucleating second phase particles (EzLLISSIzR, 1968 ; Cox and Low, 1974 ; VAN STONE, MERCHANT and Low, 1974 ; GREEN and KNOTT, 1976 ; RAWAL and GURLAND, 1976). A summary of the experimental results of other researchers for the effect of the spacing to size ratio of void nucleating second phase particles on the crack tip opening displacement (COD) for fracture initiation has been given by MCMEEKING (1977). Based on his hole growth calculations and following RICE and JOHNSON (1970), he also gave estimates for the crack tip opening displacement for fracture initiation which are in fairly good agreement with some ex~~mental results. In addition, experiments show that the measured fracture toughness for rolled steel depends on the orientation of the cut specimens to the rolling direction (SMITH and KNOTT, 197 1; GUN and KNOTT, 1976 ; HANCOCK and MACKENZIE, 1976). Specimens cut in the long transverse direction (Fig. 1) and notched parallel to the rolling direction give a planar distribution of roughly circular inclusions, where hole nucleation takes place after plastic straining of the matrix material. These specimens also show a smaller fracture toughness than those cut parallel to the rolling direction, mainly it seems, because of the different orientation of the inclusions to the crack plane.

Void growth near a blunting crack tip

Standard transverse testpiece

FIG. 1. Test piece orientations.

27

In this paper we try to model the near crack tip hole growth for a specimen cut in the long transverse direction and to estimate the conditions for fracture initiation. Using large deformation finite element analysis, we study the growth of long cylindrical holes in an elastic-plastic material near a mode I plane strain blunt crack. The results of the calculations provide a reasonable model for the behavior of holes generated by long stringers parallel to the crack, like those in specimens cut in the long transverse direction (SMITH and KNOTT, 1971; GREEN and KNOTT, 1976). Initially, an elastic perfectly-plastic material was used and two different configurations were analyzed : one with a single hole ahead of the crack tip and one with two holes placed symmetrically at 30” to the crack line. In the 30” case the distance between the holes and the crack was 10 times the hole diameter. In this way we model a material with 10 to 1 ratio of average inclusion spacing to average inclusion diameter. In the case where the hole is directly ahead of the crack, three solutions were obtained with the distance between the crack tip and the hole being 6.7,lO and 20 times the hole diameter respectively. In this way we examine the effect of spacing to size ratio of the inclusions on the hole growth and fracture initiation, Finally, three more solutions were obtained for a strain hardening material with the hole being ahead of the crack tip at a distance 5,10 and 15 times the hole diameter respectively.

In our finite element model the holes already exist in the unstressed material, while during the actual fracture process the holes are nucleated at inclusions after some plastic straining of the matrix. Also the initial cross-section shape of the holes is assumed to be circular, which is not always the case. But if one takes into account that holes could preexist due to fatigue cracking, rolling or prestraining of the specimen and that in some alloys holes nucleate .at relatively low strains at elongated particles, our model could be considered to be a good approximation to actual fracture processes for those particular situations.

The near crack tip stress and deformation fields as well as the hole growth are examined as functions of the applied load. Comparisons with MCMEEKING’S (1977) estimates for the near crack tip growth of spherical voids and with MCCLINTOCK’S (1968) theoretical model for the growth of long cylindrical holes in remotely uniform stress fields are made. The effect of strain hardening on hole growth and fracture initiation is also studied. Several hole-crack coalescence criteria are discussed and

28 N. AUAVAS and R. M. MC~~EBKING

applied, and estimates for the conditions for fracture initiation are given and compared with experimental results.

2. FORMULATION OF THB BOUNDARY VALUE PROBLEM

Assuming that the plastic zone size is negligible compared to any geometric dimension of the specimen, the small scale yielding solution for the growth of holes near a blunting crack can be achieved by solving the following el~ti~-pl~tic boundary value problem. Traction free conditions are applied on the crack face and displacement boundary conditions remote from the tip are applied incrementally to impose an asymptotic dependence on the mode I elastic crack tip singular stress field of IRW (1960). This asymptotic displacement field is of the form

(1)

where ui is the applied displacement field, I and 6 are polar coordinates with origin at the crack tip, K, is the elastic mode I stress intensity factor of IRWIN (19~), G is the elastic shear modulus, v is Poisson’s ratio and & are dime~io~ess universal functions. In this way, crack tip plasticity is accounted for in the manner of boundary layer formulation as discussed by RICE (1967,1968).

The constitutive law used represents the Jz flow theory and accounts for rotation of the principal deformation axes; its form is

*zii = 2G Sik Sjr f ~ 6, S,, - - -

for plastic loading, and

for elastic loading or any unloading, where z is the Kirchhoff stress defined by

z = Ja,

where Q is the true stress, J is the ratio of volume in the current state to volume in the undefo~ed state, D is the defo~ation rate tensor defined as the symmetric part of the spatial velocity gradient, 5, is Kronecker’s delta,

l$j = ~ij-~6ij~&k, f2 = *$jT;,

h is the slope of the uniaxial Kirchoff stress vs logarithmic plastic strain curve, and the superposed * denotes the Jaumann or co-rotational stress-rate. The form of the above constitutive law is discussed by MCMBKING and RICE (1974). The governing equations of equilibrium, including the effects of volume change, are enforced through a variational principle also discussed by MCMEEKING and RICE (1974).


3. THE FINITE ELEMENT FORMULATION

29

The finite element method was used to solve the boundary value problem formulated in the previous section. The ABAQUS (1982) general purpose finite element program developed by Hibbitt, Karlsson and Sorensen Inc., Providence R.I., was used for the computations. In addition, some cases were computed using a separate but similar program and identical results were obtained. The analysis was done incrementally using the updated-Lagrangian formulation of MCMEEKING and RICE (1974) for large elastic-plastic deformation. Equilibrium correction was used at the end of each increment.

The near crack tip mesh used for the case of a single hole ahead of the crack is shown in Fig. 2(a) in its undeformed configuration. This mesh was surrounded by that shown in Fig. 2(b). A total of 463 nodes and 402 plane strain quadrilateral isoparametric

FIG. 2. (a) Finite element mesh used in the near crack tip region for the case of a hole ahead of the crack. (b) Finite element mesh surrounding the one shown in Fig. 2(a).

30 N. ARAVAS and R. M. MCMEEKING

FIG. 3. Finite element mesh used in thenear crack tip region for the case of the-holes at 30” to thecrack plane.

elements with 4 stations for the integration of the stiffness were used (ZIENKIEWICZ,

1977). Figure 3 shows the near crack tip mesh for the case with the two holes at 30” to the crack line. This mesh was surrounded by one similar to that shown in Fig. 2(b). For this case 626 nodes and 547 elements were used. For the meshes shown in Figs. 2 and 3, the distance between the crack tip and the hole is 10 times the hole diameter. The very fine mesh near the crack tip and around the holes was required for accurate modelling of the tip blunting and hole growth. Also the outer semicircular perimeter must be remote from the crack tip plastic zone because the boundary conditions used to enforce small scale yielding conditions were applied on this perimeter as if the perturbation caused by the plastic zone was negligible. For all cases analyzed, the radius of the outer semicircular perimeter was 3000 times the initial radius of the notch root.

The elements used had an independent interpolation for the dilatation in order to avoid artificial constraints on incompressible modes (NAGTEGAAL, PARKS and RICE,

1974). This is precisely equivalent to using reduced (i.e. 1 point) integration for the dilatational contribution to the element stiffness matrix (RICE, MCMEEKING, PARKS

and SORENSEN, 1979). The symmetry condition was applied on the axis of symmetry. The small scale

yielding solution was achieved by generating increments from (1) and imposing these at the outer radius of the finite element mesh in Fig. 2(b). The displacement increments were generated by setting r equal to the outer radius and increasing K, from zero in increments. As long as the plastic zone size is small compared to the outer radius of the mesh, the solution will be an accurate boundary layer formulation of the blunting of a notch with near tip holes under small scale yielding conditions.

4. REXJLTS OF PLANE STRAIN NEAR TIP HOLE GROWTH BY FINITE ELEMENTS

Calculations were carried out for a material with a,,/E = l/300, v = 0.3 and a non- hardening (iV = 0) uniaxial Kirchhoff stress/logarithmic tensile strain curve, where rro

Void growth near a blunting crack tip 31

is the tensile yield stress. In addition, calculations for a power-law hardening material were carried out with N = 0.2 in a uniaxial stress-strain law of the form

where d is the equivalent plastic strain defined as

where DP is the plastic part of the deformation rate. In this section we present results for the following three cases : (i) a non-hardening

material (N = 0) with a hole ahead of the crack, (ii) a non-hardening material (N = 0) with holes at 30”, and (iii) a hardening material (N = 0.2) with a hole ahead of the crack. For all three cases the distance between the hole and the crack tip was 10 times the hole diameter. The results for other spacings between the hole and the crack tip are similar and are not discussed at length.

4.1. Deformed conjigurations

Figures 4-6 show the near tip deformed finite element meshes superimposed on the undeformed ones at different J/o,ay levels for the different configurations and materials analyzed, where for small scale yielding and plane strain conditions

l-v2 J=-K:

is RICE’S (1968) J-integral, E is the Young’s modulus and a’: is the initial hole diameter. It is clear that, for all cases, the holes are actually pulled towards the crack tip and

change their shape from circular to approximately elliptical with the major axes radial to the crack. For the case of the two holes at 30” to the crack line, the results show that there is a region between the two holes which has never deformed plastically. This region is shown by the A-A line on Fig. 5. This elastic region is perhaps artificial

FIG. 4. Deformed fmite element mesh in the near crack tip region superimposed on the undeformed mesh (dashed line) for a material with E/c,, = 300, v = 0.3 and N = 0, at a load level J/c& = 4.

32 N. ARAVAS and R. M. MCMEIXING

A A

FIG. 5. Deformed finite element mesh in the near crack tip region superimposed on the undeformed one for a material with E/u, = 300, v = 0.3 and N = 0, at a load level J/a,& = 6.8.

because, as mentioned in the Introduction, the holes, when nucleated, would already be surrounded by material deforming plastically. The results suggest that, in the geometry analyzed, after the holes are nucleated, the region between the two holes is only moderately stressed, while the region between the crack tip and the holes is intensively loaded, i.e. the crack-hole interaction is much stronger than the interaction between the two holes.

FIG. 6. Deformed finite element mesh in the near crack tip region superimposed on the undeformed one for a material with E/u, = 300, v = 0.3 and N = 0.2, at a load level J/aoay = 19.1


4.2. Hole growth and crack tip opening displacement results

33

Figures 7-9 show the hole growth, the COD and the size of the ligament between the crack tip and the holes, as functions of the applied load for the different configurations and materials analyzed. The symbols are explained in the corresponding insets.

In all cases the holes expand in all directions but the growth towards the crack tip is much faster than the growth perpendicular to this direction. This shows that the effect of the interaction of the neighboring free surfaces on the hole growth is stronger than the effect of the mainly tensile stress field ahead of the crack tip. We also mention that BUDIANSKY et al. (1982) have found that spherical voids in rigid-perfectly plastic materials will tend to grow oblate when triaxiality is high. In our problem though, the interaction between the holes and the crack seems to be the dominant reason for the subsequent shapes of the cylindrical holes. In addition, comparison of Figs. 7 and 8 shows that the cylindrical hole ahead of the crack grows faster than those at 30”.

The above results show that the growth of long cylindrical holes near cracks is rather different from what has so far been inferred for the growth of spherical holes.

5

4

3

2

1

0 %

_- i-- 0

0 1 2 3 4 5 6 J

a0 3 0

FIG. 7. Plot of the COD, the dimensions of the hole and the size of the ligament between the crack tip and the hole vs load for a material with E/u0 = 300, v = 0.3 and N = 0. In the undeformed configuration the distance of the crack tip from the hole was 10 times its diameter. The superscript 0 indicates the value of the corresponding quantity in the undeformed configuration. The dashed lines are based on MCCLINMCK’S

(1966) formula.

I I I I I b-b0 i

FIG. 8. Plot of the COD, the dimensions of the hole and the size of the liment between the crack tip and the hole vs load for a material with E/g,, = 300, Y = 0.3 and N = 0. In the undeformed c&&ration, the distance

of the crack tip from the hole was 10 times its diameter.

MCMEEKING (1977) found that spherical holes grow faster normal to the tip-hole direction, and that spherical holes at 45” grow, initially, faster than those ahead of the crack tip. In addition, comparison with MCMBBICING’S (1977) estimates for the growth of spherical holes shows that a cylindrical hole ahead of a crack grows about six times faster than a spherical one.

The COD plotted in Figs. 7-9 was de&red as the opening distance between the intercept of two 45”-lines drawn back from the tip with the deformed crack profile. Comparison of Fig. 7 (hole at O”) with Fig. 8 (holes at 30”) shows that, for the perfectly- plastic material, the COD is the same for both cases at low load levels while a difference appears at higher loads. The reason is that the growth of the 30”-holes starts to affect the deformation of the blunting crack at the place where the COD is measured, and this results in a higher COD. For the case of the hole ahead of the crack, the COD is closer to the one MCMEEKING (1977) found in the absence of a hole, because the influence of the hole growth on the defo~ation of the point where the COD is measured is small.

In addition, for the case of the hole ahead of the crack, Figs. 7 and 9 show that, for both the erectly-plastic and the power-law hardening material, the rate of increase of ai (see Fig. 7) is actually the same as the rate of increase of the COD as deformation proceeds. It appears that the hole behaves as if it were the tip of a crack extending beyond the true tip. The material between the true crack tip and the hole deforms as a ligament controlled by the motion of the surfaces of the fictitious extended crack.


b - bU

N = 0.2

2 4 6 6 10 12 14 16 18

FIG. 9. Plot of the COD, the dimensions of the hole and the size of the ligament between the crack tip and the hole vs load for a material with E/CT, = 300, v = 0.3 and N = 0.2. In the undeformed configuration, the

distance of the crack tip from the hole was 10 times its diameter.

The dashed lines in Fig. 7 show the predicted growth of a cylindrical hole ahead of a crack line using MCCLINTOCK’S (1968) equations for the growth of cylindrical holes in remotely uniform stress fields. These calculations were made using an incremental form of MCCLINTOCK’S (1968) equations with the remote field of his analysis identified with the local stress and deformation fields of MCMEEKING’S (1977) calculations at the current void site, computed as if no void were present. In this way the strong interaction between the neighboring free surfaces is neglected. MCCLINTOCK’S (1968) equations were used in an incremental form in order to take into account the change in stress and deformation fields at a fixed material point as the crack blunts. The results underestimate the hole growth by a factor of 6. They also predict that the hole will grow faster in the direction normal to the crack line in contradiction to the finite element analysis results. This shows again that the interaction between the crack tip and the hole surface in the two-dimensional plane strain case is indeed very strong and must be taken into account.


4.3. Stress and plastic strain distributions in the near tipJield

In Figs. 10-12 the true stress gee (see insets), at different load levels, is plotted against the distance from the deformed crack tip for the different cases analyzed. The distance is normalized by the initial hole diameter a:. The arrows on the R/a: axis indicate the position of the left end of the hole (point A in the insets) for the corresponding load levels. Figures 13-18 show equivalent plastic strain and hydrostatic stress contours in the near crack tip regions.

As is shown in Figs. 13-15 the regions between the crack and the holes are regions of very high plastic strains. So, one can expect high triaxial stresses there, as occurs in general ahead of crack tips under plane strain conditions. However, Figs. 10 and 11 show that this is not the case for the perfectly-plastic material. The reason is that the absolute values of the deviatoric stress components are bounded by the yield stress of the material and high hydrostatic stresses cannot be maintained on the blunted crack tip and on the hole surface. As a result, there is a maximum (or maxima) for a,, in the ligaments between the crack tip and the holes, coinciding with the maximum (or maxima) for the hydrostatic stress. Also for the perfectly-plastic material, Figs. 10 and

5 I I I I I I I I I I I I I

I I I I

0 I I I I I J.1 J-l 41 11 0 1 2 3 4 5 6 7 8 9 10

FIG. 10. Plot of stress a,.,&, along the ligament between the crack tip and the hole at different load levels for a material with E/u, = 300, v = 0.3 and N = 0. In the undeformed configuration, the distance of the crack tip from the hole was 10 times its diameter. Note that (r. is the yield stress in tension and that the arrows on the

horizontal axis show the position of point A (see inset).


5

4 N=O

3

Oee -

00

2 /r - . 0 I I

Uo a1 ! I I 1 I

I I I I I I

I I I I I I

I I I I I I It I I t I It il 1 2 3 4 5 6 7 6 9

R 0 9

I

FIG. 11. Plot of the stress u.&~ along the ligament between the crack tip and the hole at different load levels for a material with E/u, = 300, Y = 0.3 and N = 0. In the undeformed configuration, the distance of the crack

tip from the hole was 10 times its diameter.

11 show that as the load increases, triaxiality decreases in the ligaments between the crack tip and the holes. This happens because the size of the ligament decreases with increasing load and hence there is not enough distance for high triaxiality to develop, since hydrostatic stress must be low at both ends of the ligament. Also, the hydrostatic stress in the ligament decreases faster with increasing load for the case of the hole ahead of the crack than for the holes at 30”. This is because, as mentioned in the previous section, increasing load results in a faster hole growth ahead of the crack than at 30” to it, which means that the ligament size decreases faster ahead of the crack.

On the other hand, for the power-law hardening material, the deviatoric stress components can increase well above the yield stress with continuing plastic straining. Therefore, as shown in Fig. 12, high triaxial stresses appear between the crack tip and the hole and they increase with increasing load. Another hardening effect is the upturn in stress close to the crack tip, which arises from the elevation of flow stress by the large plastic strains in this area.

Finally, we mention that the numerical calculation breaks down before the hole coalesces with the crack tip. This occurs because of the very high distortion of the elements in the ligament between the crack tip and the hole. The elements which are


7

6

%

oo5

4

3

0

3-J-Q CRACK

0 12 3 4 5 6 7 8 9 10

R 0 5

FIG. 12. Plot of the stress U&T,, along the ligament between the crack tip and the hole at different load levels for a material with E/u, = 300, v = 0.3 and N = 0.2. In the undeformed configuration the distance of the hole

from the crack tip was 10 times its diameter.

close to the crack tip or the hole become very slender because of the high local plastic strains. Eventually, the elements distort to such an extent that a relatively small increment of deformation causes a node to pass through the opposite edge. This element becomes singular. The problem can be avoided by reducing the step size or remeshing. However, we considered the extent of the calculations to be adequate and we simply terminated them. The deformed finite element meshes shown in Figs. 4-6 and the corresponding contours (Figs. 13-18) are for load levels close to those at which the calculations were stopped. Also, all the results presented in this section are for the case where the distance between the hole and the crack tip is 10 times the hole diameter. As mentioned in the Introduction, more cases with different values of this ratio were analyzed for the con&uration with the hole ahead of the crack. The results are similar to those shown in this section, as far as stress and deformation fields are concerned. Of


FIG. 13. Contour plots of Ep in the near crack tip region for the case of a hole ahead of the crack for a material with E/u, = 300, v = 0.3 and N = 0, at a load level J/u,r$ = 4. In the undeformed configuration, the


course the hole coalesces with the crack tip at different load levels, depending on the initial spacing, and this is discussed in the following section.

5. COALEWENCE OF THE HOLJB WITH THE CRACK TIP

As mentioned in the Introduction, in high strength steels an important process leading to ductile failure is the linking of voids formed from inclusions with voids

FIG. 14. Contour plots of Ep in the near crack tip region for the case of the holes at 30” to the crack plane for a material with E/u, = 300, v = 0.3 and N = 0, at aload level J/u,o~ = 6.8. In the undeformed conftguration,

the distance of the crack tip from the hole was 10 times its diameter.

40 N. ARAVM and R. M. MCMPSKING

FIG. 15. Contour plots of gp in the near crack tip region for the case of a hole ahead of the crack for a material with E/c,, = 300, v = 0.3 and N = 0.2, at a load level J/u& = 19.1. In the undeformed um@tration, the


formed from precipitates growing in localized shear bands. Our finite element model took no account of the presence of such precipitates; so, shear bands induced by the precipitates would not develop in the calculations. However, shear banding could be independent of the precipitates, at least in perfectly-plastic materials, and in that case localization could occur in the calculations. In the finite element results no tendency for localization of plastic flow in the ligament between the crack tip and the hole was observed, in the sense that no accelerated shearing in the ligament appeared. As mentioned in the previous section the numerical analysis breaks down before the hole coalesces with the crack tip because of the very high distortion of the elements in the ligament. But, as can be seen in Figs. 7-9, the size of the ligament is close to being a linear function of J and an extrapolation to zero size could be used to get an estimate for the load for fracture initiation due to complete thinning and chisel point fracture of the ligament. The crack growth predictions that result could be considered to be the

N=O

Fs. 16. Hydrostatic stress (oU/3a0) contours in the near crack tip region for the case of a hole ahead of the crack for a material with E/u, = 300, v = 0.3 and N = 0, at a load level J/u,at = 4. In the undeformed

configuration, the distance between the crack tip and the hole was 10 times its diameter.


N=O

FIG. 17. Hydrostatic stress (uU/3co) contours in the near crack tip region for the case with the holes at 30” to the crack plane for a material with E/u, = 300, v = 0.3 and N = 0, at a load level J/o,a~ = 6.8. In the

undeformed configuration, the distance between the crack tip and the hole was 10 times its diameter.

extreme result for unbonded inclusions. If localization or some other mechanism intervenes to break the ligament between the hole and the crack tip, then, obviously, crack propagation will occur sooner than in the chisel point case. There are several criteria which can be used to represent the effect of early localization. We have used these to obtain estimates for the conditions under which crack propagation would take

FIG. 18. Hydrostatic stress (uLJ3a,J contours in thenear crack tip region for the case of a hole ahead of the crack for a material with E/u0 = 300, v = 0.3 and N = 0.2, at a load level J/u04 = 19.1. In the undeformed

contiguration, the distance between the crack tip and the hole was 10 times the hole diameter.

42 N. ARAVAS and R. M. MCM~KING

place. When the criterion for coalescence is satisfied, the hole growth is interrupted and crack extension is assumed to occur. At this point the values of .I and the COD for fracture initiation can be obtained. Any further increase of the COD due to coalescence is known to be small (GREEN and KNOTT, 1976). Before applying any of the coalescence criteria, a review of them is given and their applicability to our problem is discussed.

Several researchers (MCCLMTOCK, KAPLAN and BERG, 1966 ; ROSENFIELD and HAHN, 1966; THOMASON, 1968 ; MELANDER and STBLHLBERG, 1980), trying to explain the formation of shear bands, developed conditions for hole coalescence in terms of remotely applied uniform stress or strain fields. But the stress and strain fields in the near crack tip region are not at all uniform and these criteria cannot be used directly for our problem. KFUFFT (1964) proposed that ductile fracture by void-crack coalescence will occur when the instability strain for a uniaxial tensile specimen is achieved at the crack tip over a distance equal to the spacing of void nucleating particles. CLAUSING (1970) suggested that the instability strain for a plane strain specimen should be used in the model. But these criteria can only be applied to strain hardening materials and not to the idealized perfectly-plastic material used in most of our analyses.

On the other hand, several coalescence criteria which involve the comparison of some void dimension with the size of the ligament between the crack tip and the hole have been suggested. This kind of criterion can be used in our analysis. RICE and JOHNSON (1970) proposed that coalescence occurs when the size of the ligament between the crack tip and the hole becomes equal to the vertical diameter of the void. BROVVN and EMTNRY (1973) proposed that as soon as the spacing of neighboring voids becomes equal to their length, a slip plane can be drawn between the voids and the plastic constraint preventing local deformation is lost. They assumed that once this condition is achieved, any further plastic flow is localized in one plane and ductile fracture ensues immediately. TAIT and TAPLIN (1979), studying the growth of small drilled holes in sheet tensile specimens, suggested that hole coalescence occurs when the ratio of the hole major axis to the hole spacing reaches a critical value. They found this critical value to be 1.42 for the commercial Al-Mg alloy (grade 5083) they used in their experiments. This criterion was not applied to our problem because values for other materials were not given. Also, LE ROY, EMBURY, EDWARD and ASHBY (1981) proposed that void linkage occurs when the longest axis 2R{ of the void is of the order of magnitude of the mean planar neighbor spacing ;i, when 2R$ = 41, where 4 is considered to be a material constant. Experimental data revealed average values for 4 from 1.23, for elongated particles that nucleate voids at very low strains, to 0.83, for spherical particles that nucleate voids at much larger strains.

The above mentioned geometric criteria were applied to our analysis in the forms discussed below. RICE and JOHNSON’S (1970) criterion was used in the form d = a, (see insets in Figs. 7-9). A geometric interpretation of BROWN and EMJHJRY’S (1973)

criterion, as we applied it, is given in Fig. 19, or equivalently d = (,/(2)- l)(R +I), where R = b/2 and I = a,/2 were used. The criterion of LE ROY et al. (198 1) was used as a2 = c$d, where 4 = 1.23 is the value for long cylindrical holes.

The results from the application of the above criteria as well as from extrapolation to zero ligament size are summarized in Table 1, where b, is the value of the COD for fracture initiation and D is the distance between the center of the hole and the center of the blunt crack tip in the undeformed configuration. The results show that the criterion


CRACK

FIG. 19. Our geometric interpretation of the criterion of BROWN and EMBURY (1973).

of LE ROY et al. (1981) &es the smallest value for b, and extrapolation to zero ligament size gives the largest value, since localization is ignored.

In addition to the above discussed and applied coalescence criteria, which are mostly based on experimental observations, theoretical work on the conditions for localization of flow has been done. RUDNICKI and RICE (1975) considered the localization as an instability in the macroscopic constitutive description of the inelastic deformation of the material. Their results show that the conditions for localization are very sensitive to the kind of constituve law used. Also, YAMAMO’IQ (1979), using GURSON’S (1977a, b) law, studied the conditions for localization in ductile fracture of void containing materials. But before trying to apply either of the above approaches to the final fracture process, one has to decide whether voids formed on precipitates cause shear band localization or merely appear as a result of it. Experiments have been inconclusive as to this point.

Finally, we mention that an alternative to our approach would be to use GIJRSON’S (1977a, b) law in the finite element calculations, to describe the constitutive behavior in the ligament between the hole and the crack tip. In this way the formation of smaller voids at precipitates in the ligament is taken into account and a better picture for the final fracture process could be obtained. This is left for future investigations.

6. COMPARISON WITH EXPFMMENTAL RESULTS AND DEXUSSION

As mentioned in Section 5, Table 1 shows the ratio of b,/D for different values of D/a? when a hole coalesces with thecrack tip. In the Table, b, is the COD for fracture vitiation, L) is the undeformed spacing between the hole and the crack tip and cy is the initial diameter of the hole cross-section. The angle 8 is the orientation of the hole to the crack plane.

The results of Table 1 for 8 = 0” show bf/D to be insensitive to changes of the D/a: ratio. This is seen to be the case for both the perfectly-plastic and the power-law hardening material. It suggests that for a preexisting long cylindrical hole ahead of the crack, bf depends mainly on the distance of the hole from the crack tip and less on the hole size. In addition Table 1 shows that the case with the hole at 30” to the crack plane gives values for b, about WA higher than those with the hole ahead of the crack. This happens because the hole grows slower at 30”. It indicates that, for a given hole size and

N. ARAVAS and R. M. MCMREKING

Void growth near a blunting crack tip 4.5

DATA FROM FRACTURE TOUGHNESS TESTS High strength steel (MnS inclurionsl, Rice and

Johnson (19701, Pellissier (19991

AISI 4340 steel (MnS inclusions)

19-Ni, 200 maraging steel cox & Low (1974)

(Ti(C, Nl inclusions)

Mild steel (Spheroidized FeSC inclusionsI, Rawal &

Gurland (1976)

DATA FROM CRACK GROWTH INITIATION TESTS

ENlA mild steel

Prestrained ENlA ENlA (Longitudinal specimen)

Green 81 Knott (1976)

c I Mn steel

FRACTURE MODELS

b 0 - Small scale yielding, Rice &Johnson (1970). McMeeking

(19771

0 I I I Finite element results

0 5 10 15

D/a:

FIG. 20. Data for COD at initiation of crack growth or fracture, related to inclusion spacing D and particle size a:. Also, results for fracture from RICE and JOHNSON (1970) and MCMEEKING (1977) and results from 6ur

finite element analysis.

distance from the crack tip, b, depends strongly on the position of the hole with respect to the crack plane. However, critical void coalescence conditions will be met sooner for the void at 0” compared to other voids equally distant from the crack tip. In the following, we will use conditions met for holes at 0” as the critical ones for crack growth initiation caused by coalescence of holes with the crack tip.

If we identify D with the average two-dimensional spacing between inclusions which nucleate holes and a: with the average hole nucleating inclusion size, we can compare our results with experimental data. Figure 20 shows experimental results of several researchers together with the theoretical calculations of RICE and JOHNSON (1970) and MCMEEKING (1977), and our finite element results for 6 = 0” and N = 0 and 0.2. Since our results for bf/D depend on the coalescence criterion used, we give in Fig. 20 the range of the values obtained after the application of the several, different coalescence criteria.

Most of the experimental points shown in Fig. 20 are for materials containing loosely bonded and approximately equiaxed second phase particles. Only the material used by GREEN and KNOTT (1976) contained elongated inclusions. They used EnlA mild steel and specimens cut both in the longitudinal and the long-transverse direction; the hardening exponent, N, for this material is about 0.25. So, it seems appropriate to compare our finite element results with their experimental data for the specimens cut in the long transverse direction. Those points are connected with the chained line in Fig. 20. Such a comparison shows that the finite element results underestimate b,. In addition, the experimental results show a strong dependence of b,JD on D/a: which does not appear in the finite element results. One reason for this could be that in the finite element model the holes are assumed to preexist and grow immediately after the application of the load. In this way the stress or strain (and the corresponding COD) to

46 N. ARAVM and R. M. MCMEIXING

nucleate the holes is ignored. However, the hole nucleation conditions depend on the inclusion size a’: ; actually, the larger the inclusion the easier the nucleation (LE ROY et al., 1981). As a result, large holes (which for fixed D means smaller D/a:) will nucleate earlier and coalesce sooner with the crack tip than smaller ones ; so, for fixed D, b,/D is expected to increase with D/a:. There may be other reasons for the finite element results underestimating the experimental data. We mention again that in our finite element analysis the holes were assumed to extend through the whole thickness of the specimen. However, during the actual fracture process of specimens cut in the long transverse direction the nucleated holes at elongated inclusions have a length which is certainly smaller than the thickness of the specimen. This will constrain the rate of growth of these holes compared to the purely two-dimensional growth rates of the pure cylinders used in the finite element calculations. In addition, in practice the holes are placed randomly about the crack tip and thus the hole nearest to the tip will not be at 0 everywhere along the crack front. Where the nearest hole is not at 0”, the initiation of crack propagation will occur later than where the nearest void is at 0”. This will delay crack growth as a whole and lead to higher values for b, than those predicted from 0 holes alone.

On the other hand our perfectly-plastic calculations seem to agree well with the experimental results for prestrained EnlA. The effect of prestraining is to reduce the hardening capacity of the material and to nucleate holes, if the critical conditions for nucleation are reached. For the EnlA mild steel used by GREEN and KNOTT (1976) the nucleation strain is of the order of the yield strain and the holes are nucleated relatively easily. Also GREEN and KNOTT (1976) report that the extent of prestrain was sufficient to exhaust the hardening capacity of the material altogether. This has the effect of making the material and inclusion conditions almost exactly the same as for our perfectly- plastic finite element calculations. Thus, it is possible that the finite element calculations and the models for final coalescence predict the material behavior quite well in this particular case. Consequently, it may not be necessary to invoke explanations of extensive early localization as do GREEN and KNOTT (1976) to explain the low toughness of this case.

Figure 20 also shows the estimates of RICE and JOHNSON (1970) and MCMEEKING (1977) for a material containing spherical inclusions. As mentioned in the Introduction, they studied the near crack tip growth of spherical voids. In their calculations the remote field of the analysis of RICE and TRACEY (1969) was identified with the near crack tip stress and deformation fields of a blunting crack computed as if no voids were present. They assumed that coalescence will occur when the ligament between the void and the crack is the same size as the maximum dimension of the coalescing void. Their estimates for b,/D are insensitive to the value of the hardening exponent, IV, and the orientation, 0, of the void with respect to the crack plane. Figure 20 shows that their results overestimate b,/D. This may be so, because spherical voids grow slower than elongated ones and because the interaction between free surfaces of the blunting crack and the void is ignored in their calculations.

As shown in Fig. 20, our results for the cylindrical inclusion case and the estimates for the spherical inclusion of RICE and JOHNSON (1970) and MCMEEKING (1977) span the range of experimental results for fairly loosely bonded inclusions in appropriate orientations (the Fe& inclusions in the mild steel used by RAWAL and GURLAND (1976)


are well bonded with significant nucleation strains). This seems to explain the range of the measured toughnesses in experiments, in terms of the inclusion size and spacing and the orientation of non-equiaxed particles with respect to the crack plane.

We have also recently noted a paper by AOKI, KISHIMOTO, TAKEYA and SAKATA (1984) which includes calculations of hole growth near a blunting crack tip. However, our results are more extensive and detailed than those of AOKI et al. (1984) who are more concerned with the near crack tip stress and deformation fields due to dilatant plasticity.

Finally, we mention that, for the case of spherical voids, more accurate estimates for b,/D could be obtained if the finite element method was used to study the near crack tip behavior of this kind of void. Such a problem will necessarily be three-dimensional ; this, together with the fact that a fine mesh will have to be used in the near crack tip region, will make the computation extensive.

7. CLOSURE

Using large deformation finite element analysis we have studied the near crack tip growth of long cylindrical holes parallel to the plane of a mode I plane strain crack. It is found that the holes are actually pulled towards the crack tip and change their shape to approximately elliptical with the major axis radial to the crack. The results also show that the holes grow faster directly ahead of the crack than at an angle to the crack plane. Comparison with estimates for the near crack tip growth of spherical voids indicate that cylindrical holes grow about six times faster. Using several crack-hole coalescence criteria, we take into account, in an approximate way, the localization of flow in the ligament between the crack tip and the hole, caused perhaps by the formation of smaller voids nucleated at precipitate particles or carbides. Estimates for the condition for fracture initiation are given and compared with experimental results. The finite element results for a perfectly-plastic material seem to agree well with the experimental results for prestrained EnlA. Our results for the power-law hardening material underestimate the experimental data for reasons which are probably associated with the idealizations utilized in the finite element model. However, the results obtained here and previous estimates involving initially spherical voids span the range of experimental data for ductile crack propagation involving voids nucleated from loosely bonded inclusions.

ACKNOWLEDGEMENT

This work was carried out while both authors were supported by Grant MEA 82-l 1018 from the National Science Foundation. We are also grateful to Hibbitt, Karlsson and Sorensen, Inc. for provision of the ABAQUS finite element program and to Professor D. M. Parks for many helpful discussions.

ABAQUS 1982 User’s Manual, Version 4, Hibbitt, Karlsson and Sorensen, Inc., Providence, RI 02906.

48 N. ARAVAS and R. M. MCMEKING

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