Large crack tip opening in thin elastic-plastic sheets

International Journal of Fracture 45: 103-122, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands. 103

Large crack tip opening in thin elastic-plastic sheets

CRAIG L. HOM* and ROBERT M. McMEEKING Department of Materials and Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA

Received 10 January 1988; accepted in revised form 15 January 1989

Abstract. Large deformation, three-dimensional finite element analysis has been used to study the blunting of a mode I crack tip in a thin elastic-plastic sheet. The near tip stress and deformation fields were analyzed, and the results compared with two-dimensional plane stress solutions. A double shear band was found ahead of the crack tip which created a three-dimensional zone extending several sheet thicknesses before the crack. This mechanism leads to an irregular blunt tip shape at the midplane of the sheet. Distinct differences in crack tip shape and the deformation fields were found between the perfect plasticity solution and the strain hardening solution. The analysis was also compared with experimental results obtained by other investigators.

1. Introduction

The analysis of a crack in a thin ductile sheet can be important for the interpretation of fracture data which are typically obtained from precracked plate specimens. In this context, thin is taken to mean that the plate thickness is much smaller than all other specimen dimensions including crack length. Several theoretical and experimental investigators have examined the plastic behavior near the tip of a straight through thickness crack. Dugdale [1] observed that the plastic zone before a crack tip in a thin sheet of mild steel involves slip on two planes at 45 deg to the surface of the sheet. By assuming that this type of plastic zone develops as a narrow strip ahead of the crack tip, Dugdale modelled the thin sheet with a crack as an elastic plane stress problem and obtained an estimate of the plastic zone length. Hahn and Rosenfield [2] determined experimentally that a plane stress state exists at a crack tip when the size of the plastic zone is approximately equal to the thickness of the plate. Later, Hutchinson [3, 4] determined the near crack tip fields for plane stress with small deformations and small scale yielding. The resulting stresses and strains are singular at the crack tip for a strain hardening material, since no allowance is made for blunting in the analysis.

Since they are two-dimensional formulations, the Dugdale and Hutchinson models do not take into account the effect that finite sheet thickness has on the deformation and stress fields near a crack tip in a thin elastic-plastic sheet. Due to finite sheet thickness, the stress field will be three-dimensional in nature in the near tip region. While a plane stress state exists at the surface of the plate, higher stress triaxiality can build up at the midplane near the tip of the crack. Indeed, if the plate is sufficiently thick compared with the crack tip opening, a plane strain constraint will be present at the midplane of the sheet near the tip. Beyond the three-dimensional near tip region, the stresses and strains in thin sheets conform to plane

* Formerly Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA.

104 Craig L. Hom and Robert M. McMeeking

stress. If the plastic zone extent is smaller or comparable to the original sheet thickness then small scale yielding will prevail and the material in plane stress surrounding the three-dimensional region will be characterized by the standard square root singularity for stress. However, if the plastic zone is much larger than the plate thickness, then the material surrounding the three-dimensional region will develop a stress field similar to Hutchinson's [3, 4] plane stress solution.

Zehnder, Rosakis and Narasimhan [5] have investigated experimentally the effect of thickness on the near tip crack fields in thin sheets using the method of caustics. By comparing their experimental data with Hutchinson's solution, they found a region near the tip of the crack where the deformation field is not plane stress, but three-dimensional in nature. The extent of this region for plastically deforming materials is 0.6t0, where to is the specimen's original thickness. Using laser speckle, moir6 and holographic speckle interfero- metry techniques, Wu and Chiang [6], and Chiang and Hareesh [7] also found that three- dimensional effects were limited in the sheet to a distance of 0.8 to 1.5t 0 from the crack tip.

Levy and Marcal [8], and Sih, Hilton, Hartranft and Kiefer [9] have used three-dimensional finite element analysis to study cracks in panels of finite thickness. This type of analysis was also carried out by Wilkins [10] and Ayres [11] with finite difference methods and by Malik and Fu [12] with the method of lines. All of these investigators used a small deformation formulation and obtained plastic zone shapes and stress distributions ahead of the crack. However in all the analyses the plate thickness was comparable with the ligament size between the crack and the free edge of the panel. During loading the plastic zone shapes remained distinctively plane strain until the panels were in large scale yielding. In view of this, these analyses just mentioned addressed one aspect of three-dimensional behavior in finite thickness sheets, namely the interaction between sheet thickness and ligament size in specimens behaving initially with a plane strain constraint. The aspect of the three- dimensionality that pertains to the interaction of the crack tip region with a plane stress field in the surrounding material was not involved. As noted above, this behavior occurs in specimens in which the thickness is much less than other specimen dimensions.

In addition to finite thickness, large geometry changes in the crack tip region will modify the stress and displacement fields of Hutchinson's [4] singular solution. For plane strain, Rice and Johnson [13] used a slip line solution to determine the plastic crack tip stress and strain states due to blunting when there is perfect plasticity. They found a stress field with lower triaxiality ahead of the blunt crack than predicted by the singular small deformation solution [3]. Also, Rice and Johnson observed that the high strains directly ahead of the crack tip inducing void growth and fracture are only predicted with the large deformation blunting analysis. Later, McMeeking [14] included large deformation plasticity in a finite element analysis of plane strain crack tip opening for perfectly plastic and strain hardening materials, confirming the general conclusions of Rice and Johnson.

The blunting crack problem in plane stress was first examined by Nishimura and Achenbach [15]. They used the solution for an expanding circular hole in plane stress as the near tip field. An approximate solution for the deformation gradient directly ahead of a blunt crack tip was obtained by matching this field to a solution of Thomason [16] which is similar to Hutchinson's. Results obtained for an elastic-perfectly plastic material predicted strains higher than Hutchinson's solution near the crack tip. However it should be noted that plane stress theory was used by Nishimura and Achenbach [15] and they did not take account of differential thinning of the plate.

Large crack tip opening in thin elastic-plastic sheets 105

In this paper, we present the results of finite element analysis for the blunting of a through thickness crack tip in a thin sheet. The numerical analysis is based on a large deformation theory so that crack tip blunting was fully accounted for. The initial configuration had a semicircular tip shape. Three-dimensional finite elements were used to allow differential thinning near the crack tip. A mode I loading was applied to the sheet, and conditions of small scale yielding and plane stress far away from the crack tip were assumed. The initial thickness to initial notch width ratio was chosen to be 5 in one set of calculations and 10 in another. Results were obtained for both perfect plasticity and strain hardening. The stress and strain states in the near tip region and the crack tip opening displacements are presented. Comparisons with the approximate stress fields of Hutchinson [4], and of Nishimura and Achenbach [15] are made to illustrate the effect of finite sheet thickness on the near tip fields.

2. Problem formulation

The small scale yielding solution for a crack in an elastic-plastic thin sheet can be obtained by assuming that the plastic zone size is small compared with any planar dimension of the sheet. The crack is a straight through thickness crack, and traction free boundary conditions must be applied to the crack surface and the faces of the sheet. An asymptotic dependence on the mode I elastic crack tip singular plane stress field of Irwin [17] is obtained by incrementally applying displacement boundary conditions remote from the crack tip. This asymptotic displacement field is of the form

u i ~ fti(v,O) a s r ~ oo (1) 2G

where ui is the applied displacement field, r and 0 are polar coordinates with the origin at the crack tip, K~ is the elastic mode I stress intensity factor, G is the shear modulus, v is Poisson's ratio and fii are dimensionless functions given by Irwin [17].

The constitutive law used was J2 flow theory (Von Mises criterion with an associated flow law) with an adjustment to the elasticity for the rotation of the principal deformation axes. This form of the constitutive law is discussed by McMeeking and Rice [18]. The governing equations of equilibrium, including the effects of volume change, are enforced through a variational approach also discussed by McMeeking and Rice.

The sheet has an initial thickness of to and the crack an initial notch width of b 0 . The ratio to/b o is the only length ratio of significance to the problem, and it is chosen so that the plastic zone can become eventually large compared with the sheet thickness. In this manner, the three-dimensional zone near the crack tip is eventually surrounded by yielded material with a stress field similar to Hutchinson's [4] plane stress solution (see Fig. 1). Ratios of 5 and 10 for to/b o will be examined in this paper.

The coordinate system used in this paper is chosen so that the sheet's out-of-plane dimension is the Z-axis. The origin is chosen so that the midplane of the plate is Z = 0 and Z = to~2 is one of the plate's free edges, the other being at -to~2. The X-axis lies at the intersection of the crack plane the the midplane of the sheet. The problem being solved and the coordinate system are shown schematically in Fig. 1.


Uc~ = - ~ ~ (O)

f 3

......

S

f Plastic Zone

Three-Dimensional Zone

/ I \

Fig. 1. The boundary value problem of a mode I crack in a sheet of finite thickness.

3. The finite element formulation

The finite element method was used to solve the boundary value problem formulated in the previous section. The general finite element program ABAQUS [19] was used to perform the computations. This code implements a version of the updated-Lagrangian formulation of McMeeking and Rice [18] for large elastic-plastic deformation. Load was applied incrementally and a backward Euler algorithm is used to ensure satisfactory convergence of the solution [19].

The three-dimensional finite element mesh used to solve the problem of a crack in a thin sheet is shown in Fig. 2(a, b, c). Due to the symmetry of the loading, only one fourth of the sheet was modelled. That is, the mesh represents the portion of the specimen for which Z ~> 0 and Y >~ 0 and boundary conditions were used to impose symmetry conditions on the planes Z = 0 and Y = 0. The mesh in Fig. 2(a) is one element deep and surrounds the mesh in Fig. 2(b). The mesh in Fig. 2(b) is two elements deep and surrounds the mesh in Fig. 2(c). Finally the mesh in Fig. 2(c) contains the crack tip and is four elements deep. The outer semicircular perimeter of the mesh is 3000 times the initial radius of the crack's tip. A total of 3344 nodes and 440 twenty noded brick elements with 8 integration stations for the stiffness were used (Zienkiewicz [20]). The brick elements had an independent interpolation of the dilatation in order to avoid artificial constraints on incompressible modes (Nagtegaal, Parks and Rice [21]).

A small scale yielding solution was obtained by incrementally applying the displacement field of (I) to the outer perimeter of the finite element mesh. As long as the plastic zone size was small compared with the outer radius of the mesh and large compared with the sheet thickness, the analysis was a valid representation of small scale yielding for a plane stress crack.

(A)

(B)

(c)

b o /2

o


210 b o

Fig. 2. The three-dimensional finite element mesh used to solve the boundary value problem.

The finite element computat ions were carried out on a Convex C1-XP2 at the University of California, Santa Barbara. A typical run of 70 increments with 4 iterations per increment required 4000 min of computer time to blunt the crack tip to three times its initial width.

4. R e s u l t s o f the f in i te e l e m e n t c a l c u l a t i o n

The calculations were carried out for a material with ao/E = 1/100 and v = 0.3. Both a non-hardening and hardening material were examined. The uniaxial Kirchhoff stress- logarithmic strain law used was of the form

- + - - ~p (7 0 (7 0

where ~P is the equivalent plastic strain defined as

~p f 2 p = (~diPdiP.) ~/2 dt


MIDPLANE . . / ' U - - ] - - - 7 ...... .~ \ i / ~ ",,

QUARTE~

FREE EDGE

L - - ~ ' _ _ . [ _ _ ± _ _ k _ _ i 2

Fig. 3. The deformed mesh at the sheet's midplane, quarter plane and free surface for N = 0 and to/b o = 5 at a load level J/aobo = 2.0. Dashed lines indicate the undeformed mesh.

where d p is the plastic part of the deformat ion rate. For the strain hardening cases the exponent N = 0.1. The parameter G is the shear modulus and a 0 is the yield stress. The Kirchhoff stress t is equal to Ja , where a is the Cauchy stress and J is the ratio o f

volume in the deformed state to volume in the undeformed state. This choice of hardening law was made so that the analysis corresponds with previous calculations for plane strain

[14].

4.1. Deformed crack tip configurations

The deformed crack tip shapes with to/bo = 5 for the sheet's midplane (Z = 0), quarter plane (Z = t0/4) and free surface (Z = t0/2) are shown in Fig. 3 for the non-hardening (N = 0) case at load level J/crob o = 2.0 and Fig. 4 for the hardening (N = 0.1) case at load level J/c~ob o = 2.7. (Note that Z is the position on the Z-axis of material points in the undeformed state.) The dashed lines denote the undeformed mesh while the solid lines indicate the deformed mesh. The J-integral of Rice [22] is denoted as J, and for small scale

yielding and plane stress conditions

j - E


MIDPLANE

109

QUARTERPLANE

. / / / ~

/ ,"

/ ' / " x

FREE EDGE

~ - - - q ¸ - - .

Fig. 4. The deformed mesh at the sheet's midplane, quarter plane and free surface for N = 0.1 and to/bo = 5 at a load level J/aobo = 2.7. Dashed lines indicate the undeformed mesh.

where E is the Young's modulus. Both Figs. 3 and 4 indicate that the deformation near the tip of the crack is three-dimensional in nature. In the midplane of the sheet the deformation is concentrated directly ahead of the crack tip, and elements at the crack tip along the plane of symmetry are stretched by large amounts in the tensile direction. The deformation at the free edge and the quarter plane is more evenly distributed about the crack tip. The deformed meshes for a perfectly plastic material and a strain hardening one are generally similar. However the tip elements in the hardening case are slightly less distorted at the midplane due to the strain inhibiting effect of the hardening.

Figures 5 and 6 show plan views of the deformed crack plane for the non-hardening case at load level J/~obo = 2.0 and the strain hardening case at load level J/tyob o = 2.7 respectively both for to/b o = 5. The results show the crack advances more rapidly at the midplane of the sheet than at the free surface, and the shape of the crack front creates a lip at the edge of the plate. The figures also indicate that large out-of-plane displacements make the sheet very thin in the region near the crack tip.

Figure 7 is a plot of the current notch width measured at the midplane versus load for the non-hardening and hardening cases with to/bo = 5 and 10. The definition of notch width b is based on point A' shown in the inset of Fig. 7. The crack tip response to load is stiffer for materials with strain hardening, and to a smaller degree stiffer with increasing sheet thickness. The finite element results show there is a linear relationship between the crack tip

1 l0 Craig L. Hom and Robert M. M c M e e k i n g

Midplane C r a c k i ~ F r l n t ~ -+ -- -- ~ ~ 1

Free Surface Fig. 5. The deformed mesh in the crack plane for N = 0 and to/b o = 5 at a load level J/aob o = 2.0 superimposed on the undeformed mesh.

Midplane Crack Front 1

Free Surface Fig. 6. The deformed mesh in the crack plane for N = 0.1 and to/bo = 5 at a load level J/~o b0 = 2.7 superimposed on the undeformed mesh.

opening displacement and the J-integral as expected f rom previous work. The coefficient n for the relationship b = nJ[cr o is given in Table 1 where the slope of the lines in Fig. 7 are used to determine this result for a crack with b0 = 0. The behavior of a plane stress crack (Dugdale [1]) and a plane strain crack (McMeeking [14]) for N = 0 are also plotted in Fig. 7 for comparison. The finite element case with the smaller sheet thickness and perfect plasticity agrees well with the plane stress behavior. For the larger sheet thickness, the finite element results at the midplane are a little closer to the behavior of a plane strain crack. With increasing sheet thickness, the crack tip opening displacement at the midplane should approach the plane strain behavior. However, a crude extrapolat ion of the finite element data shows that the thickness must approach to/bo = 80 for plane-strain-like behavior to


4 I I I I

o

..o 2 -(3

N=O.O N=0.1 PLane Stress ,/,'" z - " ' ~

,,,",,,/,-'" ~ ane Strain

bo I ~ A'

I I 1 2

J/%bo

. . . . to/bo=lO - - to/bo= 5

I I 3 4

Fig. 7. A plot of the notch width versus the applied load for the non-hardening and hardening materials.

0 s

01 I I 1 2

J/%bo

- - to/bo= 10 - - to/bo= 5

I I 3 4

Fig. 8. A plot of the thickness at the crack tip versus the applied load for the non-hardening and hardening materials.

Table 1. Value of n = b/(J/a o) from the finite element calculations

E/~ o to/b o N n

100 10 0 0.96 100 10 0.1 0.73 100 5 0 0.98 lO0 5 O. 1 0.77

o c c u r at the m i d p l a n e . A s in the p l a n e s t r a in case, the c r a c k t ip o p e n i n g is less w h e n the re

is s t r a in h a r d e n i n g .

F i g u r e 8 is a p l o t o f the c u r r e n t t h i ckness t a t the c r a c k t ip versus the J - in t eg ra l . F o r the

t h i n n e r case to /b o = 5, the re is m o r e re la t ive o u t - o f - p l a n e d i s p l a c e m e n t t h a n the t h i ck case

to/bo -- 10. A l so , the shee t th ins m o r e in the nea r c r a c k t ip r eg ion fo r a pe r fec t ly p l a s t i c

112 Craig L. Horn and Robert M. McMeeking

.56

Fig. 9. A contour plot of the differential thickness t / t o for the case N = 0.1 and to/bo = 5 at a load level J /% = 2.0.

.96

" S 8

.64

Fig. 10. A contour plot of the differential thickness t / t o for the case N = 0.1 and to/b o = 5 at a load level

J / % b o = 2.7.

material than for a strain hardening material. These two results confirm that the thinner the plate the more predominant is the plane stress effect and that low strain hardening allows concentration of strain to take place. The last point is further illustrated in Figs. 9 and 10, contour plots of the differential thickness for the non-hardening and strain hardening cases with to/bo = 5. For perfect plasticity, the out-of-plane displacement is concentrated in a region directly ahead of the crack tip. However for the strain hardening case the out-of-plane displacement is more diffuse.

4.2. Stress and strain distributions in the near tip field

In Figs. 11, 12 and 13 the near tip stress a00 versus distance to the crack tip is plotted for the case to/bo = 5 and N = 0 at load level J/aobo = 2.0 and angles 0 = 0 °, 30 ° and 60 °. The distance is normalized with respect to b the notch width at the midplane. Directly ahead of the crack tip, the stresses at the midplane are higher than the stresses at the free edge. This reflects the high triaxiality of the near tip stress field at the midplane as illustrated in Fig. 14, a contour plot of the hydrostatic stress in the crack plane for to/bo = 5 and N = 0 at load level J/aobo = 2.0. Since the hydrostatic stress along the edge of the sheet is constrained by the free surface, o-00 is lower there and roughly uniform as reflected in the plots in Figs. 11, 12 and 13 for the free edge. At the midplane, o-00 is higher since there is no free surface to impose a constraint on the hydrostatic stress and triaxiality builds up in the standard


L3

2.5

2.0

1.5

1.0

0.5

I I I I

~ / M i d p l a n e (z=0) r ,~eol

Quarterplane (z =t/4) S - " ]

_ ~ _ _ _ _ . _ . ~ - _ - _ .

"Hutchinson ..... \ I . . . .

Nis imura and Achenbach

0 I I I I 0 2 4 6 8 10

r /b

Fig. 11. The n e a r t ip s t ress fields for the case N = 0, to/b o = 5 a n d 0 = 0 ° a t a l o a d level J/aob o = 2.0.

2 .5 I I I I

2 . 0 - r ~ e -

bo 1.5 - Free edge (z=t/2) e= 30 °

b 1.0 - - ~ Hutchinson

o.s-f f V v ' Midplane (z=0)

0 I I I I 0 2 4 6 8 10

r /b

Fig. 12. The n e a r t ip s t ress f ields for the case N = 0, to/b o = 5 a n d 0 = 30 ° a t a l o a d level J/~obo = 2.0.

2'~ I - ' ' ' ~oo 1

s i 1.5 Free edge (z =t/2) 8 = 60 °

b ] 0 Quar t e rp lane ( z = t / 4 )

0 .5 - ~ ~ Hutchinson

J / - Midplane (z =0)

0 I I I I 0 2 4 6 8 10

r /b

Fig. 13. The n e a r t ip s t ress f ields for the case N = 0, to/b o = 5 a n d 0 = 60 ° a t a l o a d level J/aobo = 2.0.


<@ 1.2

1° j ~ "

0.4

Fig. 14. A contour plot of hydrostatic stress in the crack plane for the case N = 0, to/b o = 5 at a load level J/aobo = 2.0.

fashion. However, the hydrostatic stress along the midplane cannot be maintained on the surface of the crack tip, and therefore a maximum in o-00 exists corresponding to the maximum for hydrostatic stress, some distance from the notch tip. The stress maximum occurs directly ahead of the crack (0 -- 0) and its value is approximately 2o- 0. At angles of 30 ° and 60 ° to the crack tip, the stresses in the near tip region are highest at the free edge of the plate.

Figures 11, 12 and 13 indicate also the extent of the three-dimensional stress state near the crack tip. Adjacent to the tip, the stresses depend on the position of the material point through the thickness. However, further away from the crack tip, the stresses become independent of that coordinate. Thus plane stress conditions prevail there. The size of the three-dimensional zone is taken to be the distance where the stress field satisfies

o-~dplane __ O00,free surface

free surface O-00

= 5% (2)

At 0 = 0, the three-dimensional region extends out to 5.7 to, whereas at 0 = 30 deg it spreads only as far as 3.0 t 0. At 60 deg, the region is smaller still at 1.8 to. At the stage represented by Figs. 11-14, the crack tip opening is about 60 percent of the original thickness. At larger relative openings one would expect some changes to the extent of the three-dimensional stress region.

The solutions of Hutchinson [3, 4] for a sharp plane stress crack tip and Nishimura and Achenbach [15] for a blunting plane stress crack tip are also plotted in Figs. 11, 12 and 13. As expected, the plane stress solutions agree with the finite element results ahead of the crack at the free surface of the sheet and underestimate the stresses at the midplane. The stresses at all three sections approach the plane stress solution far away from the crack tip. On the other hand, the stresses at the midplane near the crack tip are not as high as the maximum stresses predicted by Rice and Johnson [13] and McMeeking [14]. In that case, o-00 reaches a maximum value of 3O-o, and therefore a plane strain field does not develop even at the midplane near the crack tip in the thin section results.

In Figs. 15, 16 and 17 the corresponding near tip effective plastic strain fields are plotted for the case to/bo = 5 and N = 0 at load level J/o-obo = 2.0 and 0 = 0 °, 30 ° and 60 °. At


2.0

1.5

~Q- 1.0

0.5

0 0

I I I I

Midplane (z=O) 0 = 0 °

Quarterplane (z =t/4)

I I I I 2 4 6 8 10

r/b

Fig. 15. T h e n e a r t ip s t r a in fields for the case N = 0, to/b o = 5 a n d 0 = 0 ° a t a l o a d level J/aob o = 2.0.

2.0

1.5

o_ 1.0

0.5

0 0

I I I I

~ Free edge (z=t/2)

~ //Quarterplane (z =t/4)

~.~idplan..._L e (z,= O)

2 4 r/b

0 = 30 °

I I 6 8 10

Fig. 16. T h e n e a r t ip s t r a in fields for the case N = 0, to/b o = 5 a n d 0 = 30 ° a t a l o a d level J/aob o = 2.0.

2.0

1.5

a. 1.0

I I I I

~ Free edge (z=t/2)

0 = 60 °

0.5 / Quarterplane (z = t/4) /

0 ~ . ~ M/dplane (z =t0) I I

0 2 4 6 8 10 r/b

Fig. 17. T h e n e a r t ip s t r a in fields fo r the case N = 0, to/b o = 5 a n d 0 = 60 ° a t a l oad level J/crob o = 2.0.


0.1

'05

02

Fig. 18. A contour plot of effective plastic strain for a cross section ahead of the crack tip for the case N = 0,

to/bo = 5 at a load level J/aobo = 2.0.

[ I I ]

3 [ ~ Midplane (z=O)

.~ ~ Quarterplane (z =t/4)

o 2 .\\ ~ Free edge (z=t/2)

1 Hutchinson ~ x ~YY

0 I [ I [ 0 1 2 3 4 5

x/b Fig. 19. The near tip stress fields directly ahead of the crack for the case N = 0.1 and to/b o = 5 at a load level

J]Gobo = 2.7.

0 = 0 °, the plastic strains are highest at the midplane, while at 0 = 30 ° and 60 °, they are highest at the free edge. These last two plots reflect the concentration of strain into narrow crossed bands in front of the crack. This feature is seen most clearly in the contour plot of the effective plastic strain shown in Fig. 18 for a cross section of the sheet at a distance of 4.25 b0 ahead of the crack tip. Most of the plastic strain is concentrated at the crack tip in the midplane of the sheet. Two shear bands similar to the cross slip observed by Dugdale exist directly ahead of the crack tip on planes approximately 45 ° to the plane of the sheet. This double shear band accounts for the large deformation seen at the midplane directly ahead of the crack tip and at the free edge for 0 = 30 ° and 60 °.

Figure 19 shows the stress field directly ahead of the crack for the case to/bo = 5 and N = 0.1 at load level J/crob o = 2.7. With to/bo = 10, Figs. 20 and 21 show the near tip stress


I } l I

3 ~~oyy / Midplane (z =0)

~ i 2 //~,.,, " ~ / / ~ Quarterplane (z=t/4)

Free edge (z=t/2) Achenbach 1 |

0 1 2 3 4 5 x/b

Fig. 20. The near tip stress fields directly ahead o f the crack for the case N = 0 and to/b o = 5 a t a load level

J/aob o = 2.1.

~ 0 2

0

I I I

Midplane (z=0)

Quarterplane (z =t/4)

tchinson

\ Free edge (z =t/2)

I J 1 2

x/b

I _ _

(Syy !

I I 3 4 5

Fig. 21. The near tip stress fields directly ahead o f the crack for the case N = 0 . i and to/bo = 10 a t a load level

J/aob o = 2,8,

field ahead of the crack for N = 0 at load level J/aobo = 2.1 and N = 0.1 at load level J/aob o = 2.8 respectively. The difference in thickness appears to have little effect on the stress field directly ahead of the crack. It seems that only for much larger values of to/bo will the stresses in the interior near the tip be higher and closer to the plane strain state. As expected, strain hardening increases the level of stress ahead of the crack tip. The maximum stress at the midplane is 2.7a 0 for a strain hardening material compared with 2.0~r 0 for a perfectly plastic material. Using (2) a comparison of the stresses at the midplane and free surface in Fig. 19 indicates the three-dimensional zone extends 1.8 to ahead of the crack tip for a strain hardening material. Also, plotted with Figs. 19-21 are the plane stress crack solutions of Hutchinson, and Nishimura and Achenbach. The finite element results at the free edge agree well with the plane stress solutions; and far away from the crack tip the stresses predicted by the finite element analysis at all sections approach the plane stress solutions. However, near the tip, the blunting limits the elevation of the stresses seen in the singular solution of Hutchinson [4].


Y / b o

50

25

-21 -

- 5 0

1.1 1.3 1.8 2.0 = J / a o b o

i_

rp

I I I I 25 50 75 100

X /bo

Fig. 22. Pla s t i c z o n e s f o r N = 0 a t v a r i o u s l o a d levels , J/aob o.

4.3. Plastic zone shapes

For most of the load history in the calculations, the thickness of the plate was small compared with the plastic zone and the solution away from the crack tip was plane-stress- like. Therefore, the plastic zones were continuous through the thickness of the sheet. The plastic zones of the non-hardening and strain hardening materials are very different in shape. Figure 22 shows that the plastic zone is not self similar at different load levels J/aob o for the non-hardening material. The plastic zone does not spread transverse to the crack but extends ahead with increasing load. For the non-hardening case, the finite element results relate the length of the plastic zone rp to the J-integral by

EJ = m ag (3)

where m = 0.28. Figure 23 shows the plastic zone shapes for the strain hardening case at different load

levels. Strain hardening makes the plastic zone more diffuse than the perfectly plastic solution and it has now a self similar appearance as the load increases. For the strain hardening case, the finite element results relate the length of the plastic zone to the J-integral by (3) with m = 0.23.

For perfect plasticity, the plastic zone size of the finite element calculation is smaller than that predicted by the small strain theories. The parameter m -- 0.393 according to the model of Dugdale and m = 0.300 from the analysis of Achenbach and Dunayevsky [23].

4.4. Comparison with experimental results

Finally, a comparison can be made between the finite element results for the case N = 0.1 and to/b o = 10 and the displacement field measured by Chiang and Hareesh [7] for thin


I 25

Y/bo

1.4 1.9 2.3 2.8 = J / % b o

rp

-50 t t t I 25 50 75 100

X/bo

Fig. 23. Plastic zones for N = 0.1 at var ious load levels, J/aob o.

119

% "-1

I

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

I I I I I I |

- - Finite Element Results

• Experimental Data

x 2

I I I I I I 0.5 1.0 1.5 2.0 2.5 3.0 3.5

r/to

Fig. 24. Compar i son of the out-of-plane displacements predicted by the finite element analysis and measured experimental ly by Chiang and Hareesh [7] at 0 = 0 °.

sheets of A1 6061-T6 using a combined laser speckle-moir6 method. The material and geometric parameters for the experiment were N = 0.054, E/ao = 246 and to/b o = 8. Thus, there is some discrepancy between the parameters involved in the experiments and the calculations. Figure 24 shows the out-of-plane displacement directly ahead of the crack tip obtained by Chiang and Hareesh as well as that arising in the finite element calculation. The out-of-plane displacements at 0 = 30 deg are also shown in Fig. 25. Generally the agreement is good between the two results. However, the finite element computation predicts slightly lower displacements ahead of the crack than observed experimentally. One reason for this difference is that the specimen used in the experiment was thinner and made of a lower hardening material than the specimen modelled in the finite element calculation. As we have observed earlier, thin sheets made of non-hardening materials deform out of plane more than thicker sheets of hardening material. The displacements at 30 deg to the crack plane predicted by the finite element analysis are closer to the experimental results. Therefore the finite

120 Craig L. Hom and Robert M. MeMeeking

I

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

I I I I I I

- - Finite Element Results

• Experimental Data

x 2

- ~ x 1

I I I I I 0.5 1.0 1.5 2.0 2.5

r/to

I I 3.0 3.5

Fig. 25. Comparison of the out-of-plane displacements predicted by the finite element analysis and measured experimentally by Chiang and Hareesh [7] at 0 = 30 °.

element solution's out-of-plane displacements are more diffuse and less concentrated than the displacement field observed experimentally.

5. Discussion

The finite element results shows that a three-dimensional zone exists in the near tip region for a blunting crack in a sheet of finite thickness as a result of differential thinning. Due to a buildup of hydrostatic stress, the stresses in the near tip region at the midplane of the sheet are higher than predicted by plane stress solutions. However this hydrostatic stress cannot be maintained at the free surface of the sheet, and the stress field at the edge is closer to the plane stress solutions. Some of these complexities are present in three-dimensional calculations for sharp cracks (no blunting allowed for) in thin sheets. However, the blunting introduces further complications of the stress field compared with solutions based on sharp cracks. The three-dimensional features of the stress field extend six sheet thicknesses ahead of the crack tip for perfect plasticity and two sheet thicknesses ahead of the crack tip for strain hardening. As mentioned earlier, researchers have measured experimentally the extent of this three-dimensional zone and found it to be approximately one sheet thickness around the crack. A possible reason for the difference in size between the experimentally measured three-dimensional zone and the zone predicted by the finite element method is the method of measurement. Experimental estimates of the zone size are based on a comparison of the measured out-of-plane displacements and Hutchinson's [4] plane stress solution. We believe a better method for defining the three-dimensional zone in the calculations is by comparison of the stress and strain fields at the edge and midplane of the sheet as we have done for our analysis. That is, the three-dimensional zone is defined as that region where the free edge and the midplane stress differ. Reasonable agreement between the numerical calculations and the experimental displacements (Figs. 24 and 25) indicate that the calculations can be used to interpret the experiments and that the three-dimensional zone does indeed spread over larger distances in thin sheets than previously thought.


The extent of the three-dimensional zone has been phrased in terms of the sheet thickness in these discussions. Of course, there is another length scale involved, namely the crack tip opening. We must ask whether it is this that sets the length scale for the three-dimensional zone around the tip rather than the sheet thickness. In some of the results portrayed in the figures, the crack tip opening is as large as 2/3 of the original sheet thickness. Inspection of the stress and strain profiles in Figs. 11-13, 15-17, 20 and 21 reveal that the blunt crack tip seems to influence the solution in much the same way as is the case in plane strain. That is, the concentration of deformation ahead of the crack and the constraint limiting effect of the blunt tip spreads over only a few crack tip opening displacements. Indeed, from the stress plots, the extent seems to be only about one current crack tip opening. Thus the effect of the bluntness dies off well before the range of 10 or so crack tip openings around the tip that contains the three-dimensional zone. Thus we can deduce that in the solutions we have obtained, the plate thickness rather than the crack tip opening determines the scale of the three-dimensional zone. However, it may be possible that the effects observed in the solutions arise from an interaction of the plate thickness and the crack tip opening which depends on the existence of some bluntness of the crack involved.

The finite element results indicate also how fracture initiates in thin sheets with cracks. The ductile failure of metals occurs when microvoids nucleate from second phase particles, grow and coalesce with other voids or with a crack tip. Void initiation and growth generally require high plastic strains and probably high triaxial stresses. Therefore in the case of a crack in a thin sheet, voids will be concentrated in the double shear band ahead of the crack tip and failure probably initiates at the intersection of the two bands. The crack will follow a path through the highest void concentrations, so the fracture surface will develop slanted 45 deg to the plane of the sheet. This interpretation of the finite element results agrees with experiments by Brock [24], who observed slanted fracture surfaces in thin sheets made of metal alloys. Brock also found the fracture surface was not slanted but square for thicker plates with cracks under plane strain conditions. However, he observed the fracture surface was slanted at the edge of the plate where there is always a low constraint.

Finally, the analysis presented in this paper concerns very thin sheets. Plane stress behavior occurs very close to the crack tip, so the small scale yielding plastic zone beyond the three-dimensional region is plane stress in character at the midplane of the sheet. In thicker plates, the entire plastic zone could be three-dimensional. While the free surface of the plate is in a plane-stress-like condition, the midplane could be in plane strain. Analysis of this problem would provide a better understanding of the transition of a plane stress crack to a plane strain crack with increasing plate thickness.

Acknowledgments

This work was carried out while both authors were supported by Grant MSM85-04411 from the National Science Foundation. The provision of the ABAQUS program by Hibbitt, Karlsson and Sorensen Inc. of Providence, RI is gratefully acknowledged.

References

1. D.S. Dugdale, Journal of the Mechanics and Physics of Solids 8 (1960) 100-104. 2. G.T. Hahn and A.R. Rosenfield, Acta Metallurgica 13 (1965) 293-306.

122 Craig L. Hom and Robert M. McMeek ing

3. J.W. Hutchinson, Journal of the Mechanics and Physics of Solids 16 (1968) 13-31. 4. J.W. Hutchinson, Journal of the Mechanics and Physics of Solids 16 (1968) 337±347. 5. A.T. Zehnder, A.J. Rosakis and R. Narasimhan, "Measurement of the J Integral with Caustics: an

Experimental and Numerical Investigation", SM Report 86-8, California Institute of Technology (1986). 6. X.P. Wu and F.P. Chiang, "Three Dimensional Crack-Tip Deformation in a Plastically Deformed Three-

Point Bend Specimen", Tech. Report No. 475, State University of New York at Stony Brook (1986). 7. F.P. Chiang and T.V. Hareesh, "Three Dimensional Crack Tip Deformation: An Experimental Study and

Comparison to HRR Field", Tech. Report No. 481, State University of New York at Stony Brook (1986). 8. N. Levy and P.V. Marcal, "Three-Dimensional Elastic-Plastic Stress and Strain Analysis for Fracture

Mechanics, Phase I: Simple Flawed Specimens, Phase II: Advanced Flawed Specimens", USAEC Report HSST-TR-12 (1970).

9. G.C. Sih, P.D. Hilton, R.J. Hartranft and B.V. Kiefer, "Three-Dimensional Stress Analysis of a Finite Slab Containing a Transverse Central Crack", Tech. Report FSM-75-69, Lehigh University (1975).

10. M.L. Wilkins, "Calculation of Elastic-Plastic Flow", Rept. TID-4500, UC 34, Lawrence Livermore Labora- tory (1969).

11. D.J. Ayres, Engineering Fracture Mechanics 21 (1970) 87-106. 12. S.N. Malik and L.S. Fu, International Journal of Fracture 18 (1982) 45-63. 13. J.R. Rice and M.A. Johnson, Inelastic Behavior of Solids, M.F. Kanninen et al. (eds.), McGraw-Hill, New

York (1970) 641-672. 14. R.M. McMeeking, Journal of the Mechanics and Physics of Solids 25 (1977) 357-381. 15. N. Nishimura and J.D. Achenbach, Journal of the Mechanics and Physics of Solids 34 (1986) 147-165. 16. P.F. Thomason, in Fracture Mechanics in Engineering Application (Proceedings of International Conference

on Fracture Mechanics in Engineering Application, Bangalore, India) G.C. Sih and S.R. Valluri (eds.), Sijthoff and Noordhoff, The Netherlands (1979) 43.

17. G.R. Irwin, in Structural Mechanics (Proceedings of First Symposium on Naval Structural Mechanics, Stanford) J.N. Goodier and N.J. Hoff (eds.), Pergamon Press, London (1959) 557-594.

18. R.M. McMeeking and J.R. Rice, International Journal of Solids and Structures 11 (1974) 601-616. 19. ABAQUS, 1984, User's Manual, Version 4.5, Hibbitt, Karlsson and Sorensen, Inc., Providence, RI 02906. 20. O.C. Zienkiewicz, The Finite Element Method, 3rd ed., McGraw-Hill, London (1977). 21. J.C. Nagtegaal, D.M. Parks and J.R. Rice, Computer Methods in Applied Mechanics and Engineering 4 (1974)

153-177. 22. J.R. Rice, Transactions ASME 80, Series E, Journal of Applied Mechanics 35 (1969) 379-386. 23. J.D. Achenbach and Dunayevsky, Journal of the Mechanics and Physics of Solids 32 (1984) 89-100. 24. D. Broek, Elementary Engineering Fracture Mechanics, 4th ed., Martinus Nijhoff Publishers, Dordrecht

(1986).

R6sum6. On a utilis6 une analyse par 616ments finis ~ trois dimensions congus pour de grandes d6formations pour l'6tude de l'arrondissement de l'extr6mit6 d'une fissure de mode I dans un feuillard mince 61astoplastique. Les champs de contraintes et de d6formations au voisinage de cette extr6mit~ sont analys6s, et les r6sultats sont compar6s aux solutions obtenues en 6tat plan de tension ~t deux dimensions. On identifie en avant de l'extr+mit6 de la fissure une double bande de cisaillement qui g6n6re une zone en trois dimensions qui s'6tend devant la fissure sur plusieurs fois l'6paisseur du feuillard. Ce m6canisme conduit/t une forme irr6guli6re d'arrondissement de l'extr6mit6 de la fissure dans le plan m6dian de la t61e. On trouve des diff6rences nettes de forme d'extr6mit6 de fissure et de champs de d6formation entre la solution parfaitement plastique et la solution tenant compte d'un 6crouissage. On compare ~galement l'analyse effectu6e avec les r6snltats exp6rimentaux obtenus par d'autres chercheurs.

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Large crack tip opening in thin elastic-plastic sheets