An Investigation of Ductile Fracture Investigation covers the initiation and propagation of cracks from the viewpoint of experimental measurements and possible agreement of plastic-limit analysis with these results

by S. K. Garg and J. E. Griffith SUMMARY--Behavior of aluminum sheets during crack ~ion and propagation is determined using a highspeed camera. The observed failure mode is considered in the light of limit design and predominant energy terms.

Nomenclature A b C Co r r D Dr D~ e E f g h L PP + P~e~

= = = = = = = = = = = = = = = = = =

= = -= = = = v* =

S1 $2 t V V1

§

one-half plate width dimensionless crack length (equals C / A ) one-half crack length one-half initial crack length crack-propagation velocity m ax i m u m crack-propagation velocity flexural rigidity {equals E h 3 / [ 1 2 ( 1 - v ~) ]} external work rate internal-energy dissipation rate plastic-flow region Young's modulus cutout factor acceleration due to gravity plate thickness one-half plate length lower bound on collapse load upper bound on collapse load actual collapse load

cr ack s. T h e s e v a r i a b l e s i n c l u d e t h e m a t e r i a l structure and composition, temperature, specimen s h a p e a n d size, p r e v i o u s s t r a i n h i s t o r y , r e s i d u a l stress, s t a t e of a p p l i e d stress a n d s p e c i m e n i m p e r f ect i o n s. T h e s e v a r i a b l e s ar e also si g n i f i can t d u r i n g c r a c k p r o p a g a t i o n w h i c h m a y follow c r a c k i n i t i a t i o n . U n f o r t u n a t e l y , t h e i n d e p e n d e n t effect o f a single v a r i a b l e o n b r i t t l e f r a c t u r e is difficult t o isolate. O n e o f t h e m o s t a n n o y i n g f e a t u r e s is t h e p r e s e n c e of partially distorted or plastic zones which are l o c a t e d b o t h in t h e c r a c k - i n i t i a t i o n r e g i o n a n d a l o n g t h e surface o f s e p a r a t i o n o f a p r o p a g a t e d crack. C h a r a c t e r i s t i c m a r k i n g s i n d i c a t e t h a t c r i t i c a l s he a r stresses are o p e r a t i v e in t h e s e r e g i o n s . I n a n a t t e m p t t o b e t t e r u n d e r s t a n d t h e role o f t h e s e p l a s t i c regions, a s e p a r a t e e x p e r i m e n t a l s t u d y o f d u c t i l e f r a c t u r e w a s u n d e r t a k e n a n d is r e p o r t e d in this paper. T h e first s y s t e m a t i c s t u d y o f t h e b e h a v i o r o f

crack-instability load one-half crack extension at t vertical distance moved by loads at t arbitrary time velocity of loads velocity of sound acceleration of loads volume

= strain rates

~xP~ ~yP,

~sp = plastic-strain stress p -- density ~

b

stress

a~ = critical stress ~o = yield stress

Introduction R e c e n t e x p e r i m e n t a l e v i d e n c e f r o m s t u d i e s on " b r i t t l e " f r a c t u r e in m e t a l s , p r i n c i p a l l y steel, ind i c a t e s t h a t a l a r g e n u m b e r o f v a r i a b l e s in s o m e c o m b i n e d m a n n e r a r e r e s p o n s i b l e for i n i t i a t i o n o f S. K . Garg was Research Fellow, University of Florida, Gainesville, Fla., and J . E. Griffith was Associate Professor, North Carolina State College, Raleigh, N. C , at the time that this paper was preps,red. Paper was presented at 1963 S E S A Annual Meeting held in Boston, Mass., on .Nov. 6-8.

326 [ October 1 9 6 5

Fig. 1--Specimen shape

9

I' m

9

Fig. 2--High-speed photographs of crack propagation

fracture was conducted by Griffith 1 who proved from energy considerations that, for perfectly brittle materials (no plastic flow), fracture must occur when the tensile stress at the root of a flaw or notch approaches the value vrEw-/C0. Extensions of this criterion were made b y Orowan 2and Irwin 3for small plastic flow in otherwise an essentially brittle fracture replacing surface energy w b y plastic-flow work p. However, these extensions do not predict failure in ductile materials like soft aluminum and mild steel. An energy criterion, although necessary, is not sufficient when applied to fracture of ductile materials. I n these materials, stress concentrations are of moderate values in critical regions, and as pointed out b y Drucker, 4 plasticity rather t h a n elasticity theory governs. The present work contains an experimental in-

vestigation into the nature of ductile fracture. Of general interest is the initiation and propagation of cracks from the viewpoint of experimental measurem e n t s and possible agreement of plastic-limit analysis with these results.

Experimental Program The experimental program consisted of deadweight tension tests of thin sheets with prescribed external cuts, as shown in Fig. 1. This arrangem e n t afforded simplicity in control of initiation and p r o p a g a t i o n of failure, as well as in establishing the collapse load and crack propagation b y plastic analysis. (This analysis appears in the Appendix.) T h e testing machine was designed for central loading, and bending was eliminated b y the

Experimental Mechanics ] 327

use of vertical guides. T h e specimen was clamped between two plates at each end, the upper pair of plates being fixed to the loading frame and hence stationary. The clamping plates were made out of 1/4 in.-thick steel. F o u r 1/4 in.-diam bolts were used to clamp the test plate on each side. The lower pair of plates was clamped to a loading device constrained to move vertically only, thus straining the specimen with a uniform total deformation. A measurement of this vertical displacement was provided b y electrical strain gages m o u n t e d on a thin cantilever beam whose u n s u p p o r t e d end was displaced b y the loading device. The conventional bridge-amplifier-oscilloscope-camera arrangement provided the means to record this displacement as a function of time. Specimens were made out of 1145-0, 0.004 • 0.001in.-thick a l u m i n u m foil, with a yield strength of 3500 psi and ultimate strength of 11,500 psi. The specimens were made into rectangular sheets with dimensions of 32.5 • 8 in., and with initial external slits of length 1.5 in. on each side. Slits were made b y a sharp razor blade to ensure geometrical uniformity. The problem of following the crack growth and oscilloscope trace was solved with the help of high-speed photography, with the p r i m a r y lens of the camera directed toward the crack and the secondary lens toward the oscillospcope. A speed of 1300 frames/sec was used. I n Fig. 2 are reproduced a few of the frames illustrating the crack propagation. Note t h a t the oscilloscope trace (recorded on the right side of the frame) is t o o weak to be reproduced here.

yield stress while the remaining part of the reduced section remains essentially elastic. An increase in load is necessary to raise this stress level in the central portion until sufficient plastic flow takes place. I t is then t h a t the instantaneous crack propagation occurs. Up to this time the crack has been extended b u t arrested for each additional load increment. The mechanical behavior during crack propagation is illustrated in Fig. 3 where vertical displacement S~, crack extension $1, and vertical acceleration l~ are plotted as a function of time. Here these variables are referred to zero reference values at the onset of instantaneous crack propagation. The results of eq (A-9) of the Appendix for instantaneous propagation are applicable only to this second stage of crack propagation. D u r i n g the experimental investigation, it was observed t h a t the test specimens buckled laterally during the process of crack propagation through them. I n Fig. 4, the pictures of a specimen in its prebuckled and postbuckled configurations are reproduced. While a complete theoretical o r experimental t r e a t m e n t of such a complex phen o m e n o n as this is out of the scope of this work, an

80 84 0

Experimental Observations I n the course of this investigation, it was found more practical to concentrate on a single t y p e of specimen and, therefore, all tests were conducted under similar conditions. Although numerous tests were performed, only a typical test will be discussed T w o distinct stages of crack formation were noted, viz. (1) initiation of the crack at a load slightly higher t h a n the limit load and its growth with the addition of further increments of load, and (2) instantaneous propagation at a higher load. Initial yielding occurred at a load of 80 ib extending the crack 0.25 in. while an additional 0.25-in. extension resulted at a load of 103 lb. I n s t a n t a n e o u s crack propagation occurred at a 106-1b load level. Variation of the applied loads for initial and instantaneous crack growth from test to test was within ~=7.0 percent, which is to be expected as the material thickness m a y v a r y as m u c h as • percent. As shown in the Appendix, the calculated collapse load is 70 lb and this consistent 10-1b discrepa n c y with the experimentally measured initiation load can be explained in terms of the strain-hardening capability of the test material. The severity of the n o t c h introduces a stress concentration so t h a t the stress level in this region lies above the

328 I October 1965

-[

~

~

1

I s~ T

PLOT .~

OF

in

1/1300sec.

S~ vs. T

4o

0

--

Y

._r

I

r T

PLOT

-

15o in

111300

T in

1/1300

sec.

OF S= vs. T

400

~

~

.%

.=_

I

I

I

I

0

PLOT

FiE.3--Plots

OF

V

of $], S~ and ~/vs. t

vs. T

I 50 ser

rG

II111

.

.

.

il Fig. 4--Pre-buckled and post-buckled configurations of the specimen

a t t e m p t will be m a d e here to reach a p h y s i c a l u n d e r s t a n d i n g of the buckling a n d also to calculate the a p p r o x i m a t e energy d i s s i p a t i o n due to it b y considering the corresponding elastic case. I t is evid e n t from t h e s y m m e t r i c a l n a t u r e of b e n d i n g t h a t it could not have been possibly caused by misalignm e n t of c l a m p i n g p l a t e s or b y l a t e r a l guides. N o t e t h a t , as the plastic zone e e x p a n d s into e § $2, t h e m a t e r i a l moves l a t e r a l l y i n w a r d due to the Poisson effect. The m a t e r i a l n e a r b y it, outside the plastic region, offers resistance to t h i s m o v e m e n t , a n d hence compressive stresses or l a t e r a l c o n s t r a i n t s are introduced. This is i l l u s t r a t e d in Fig. 5. These stresses, t h o u g h e x t r e m e l y small, are perfectly c a p a b l e of p r o d u c i n g l a t e r a l b u c k l i n g in t h i n specimens. An idea of the m a g n i t u d e of t h e critical stress necessary to produce b u c k l i n g m a y be obt a i n e d b y considering a p l a t e u n d e r u n i f o r m compression w i t h b o t h ends c l a m p e d , as in t h e case e f t h e t e s t specimens. T i m o s h e n k o a n d Gere 5 give t h e following solution for ~cr in the elastic case: ~cr

K~.2D 4L~h

(1)

where K is a function of end conditions a n d p l a t e dimensions, etc. M o r e specifically, in t h e case of t e s t specimens used in t h e p r e s e n t e x p e r i m e n t a l series, K ~ 10, a n d t a k i n g , = 0.33, t h e value of critical stress is o b t a i n e d as 12 psi a n d t h a t of t h e critical load as 0.20 lb. I t is e v i d e n t t h a t this a m o u n t of l a t e r a l cons t r a i n t can easily come into p l a y due to P o i s s o n ' s effect described in the preceding p a r a g r a p h s . T h e p h e n o m e n o n is n o t noticeable in t h i c k specimens, as

.

.

.

T

.

f%

lll

Fig. 5--Introduction of taterM stresses

t h e critical stress is quite high a n d is n o t r e a c h e d by lateral constraints. T h e a p p r o x i m a t e e n e r g y d i s s i p a t e d in b u c k l i n g m a y be o b t a i n e d b y a s s u m i n g t h a t t h e l o a d (0.20 lb) moves an a v e r a g e d i s t a n c e of 1 in. t h r o u g h o u t the l e n g t h of the plate. T h e e n e r g y d i s s i p a t e d is t h e n equal to 0.20 in.-lb, which is v e r y small c o m p a r e d to p l a s t i c - s t r a i n energy or the e x t e r n a l work, a n d therefore m a y be neglected for all p r a c tical purposes. A n o t h e r c o n s i d e r a t i o n in this c o n n e c t i o n was t h e existence of a n a r r o w plastic region a t the c e n t e r of the specimen i l l u s t r a t e d in Fig. 5. A l t h o u g h no direct m e a s u r e m e n t s for distinguishing t h i s zone were made, it is of i n t e r e s t to c o m p a r e a plasticl i m i t analysis, assuming t h i s zone exists, with experim e n t a l results. A l t h o u g h the a c t u a l region m a y be quite different from t h e idealized t h i n region, a g r e e m e n t would i n d i c a t e the a p p l i c a b i l i t y of t h e m e t h o d for d e t e r m i n i n g d u c t i l e - f r a c t u r e b e h a v i o r . T h e p r e s e n t s t u d y is l i m i t e d in scope due to t h e range of t h i c k n e s s of specimens e m p l o y e d in t h e e x p e r i m e n t . However, since the t h e o r e t i c a l r e s u l t s were d e r i v e d in a more general m a n n e r , one m a y argue their v a l i d i t y to o t h e r specimens. W h e t h e r or n o t such is t h e case, in r e a l i t y , can only be d e t e r m i n e d b y f u r t h e r e x p e r i m e n t a l work. I t m u s t also be recognized t h a t t h e t e a r i n g a n d flow associated with " b r i t t l e " - f r a c t u r e crack p r o p a gation occur in d i s c o n t i n u o u s fashion, j u s t as cleavage failures do in p o l y c r y s t a l l i n e m a t e r i a l s . T h i s i n t e r m i t t e n t inelastic b e h a v i o r is d i r e c t l y coupled w i t h t h e elastic b r i t t l e f r a c t u r e in a complex m a n n e r , p r o b a b l y b e y o n d the r e a l m of cont i n u u m mechanics. Therefore, t h e r e s u l t s of t h i s

Experimental Mechanics

I 329

s t u d y are subject to the same limitations as other work based on continuum mechanics.

Conclusions T h e theoretical load necessary to start the crack propagation is found to be slightly less t h a n the experimental collapse load. Therefore, it seems reasonable to use the limit load for design purposes in ductile material like mild steel and soft aluminum. Also, the substitution in (A-9) of the observed values for S1, $2 and V gave a constant value for the plastic-flow region e, thus justifying the basic assumption a b o u t its existence. The crack propagation behavior as predicted b y eq (A-9) is sensitive to the magnitude of the region e. However, results for crack extension, S~, and vertical displacement, S~, and their time rates agree with a value of e of the order of the thickness h. As would be expected, the velocity of crack propagation is considerably less t h a n the velocity of sound, attaining a value of 34 ft/sec at the instance of complete separation of the specimen into two parts. References 1. Griffith, A. A., "'The Phenomena of Rupture and Flow in Solids," Trans. Royal Society (London), A, 221, 163-198 (1921). 2. Orowan, E., "'Fundamentals of Brittle Behavior in Metals," Contribution to a Conference on Fatigue and Fracture of Metals at M I T (1950). 3. Irwin, G., "'Fracture Dynamics," Fracturing of Metals, A . S . M . , 147166 (1948). 4. Dracker, D. C., "'An Evaluation of Current Knowledge of the Mechanics of Brittle Fracture," Ship Structure Committee Serial no. SSC-69 (1953). 5. Timoshenko, S. P., and Gere, J . M., "Theory of Elastic Stability," 2nd ed., McGraw-Hill, New York (1961). 6. Hodge, P. G., "'Plastic Limit Analysis of Structures," McGraw-Hill, New York (1959). 7. Garg, S. K., " A n Investigation of Ductile Fracture," M.S. Thesis, Univ. of Florida (1962). 8. Anderson, O. L., "'The Griffith Criterion for Glass Fracture," Fracture (Proc. Conference on Fracture at M I T ) , Editors: Averbach, Felbeck, Hahn and Thomas, John Wiley and Sons, 331-353 (1959).

APPENDIX Collapse Load Consider a plate of length 2L and width 2A having external cracks of length C (equal to b A ) on each side subjected to uniform tension (Fig. 1). The plate thickness is h. A c u t o u t factor f6 m a y be defined as the multiplier of a0 for which instability will result. To find this factor, it is necessary to determine the upper and lower bounds which closely bracket the theoretical collapse loading. Drucker 4 obtained the following value for the upper bound: f+ =

(1

-

[ b)L1

~2h_ b)~ + 8A(1

term

x/2h 8A(1 -- b) m a y be neglected in comparison to unity. Hence, f -- 1 -- b, a n d the theoretical collapse load is P = 2(A - C)h~o

Crack Propagation in Plate I n the strength tests of metals in simple tension and the drawing of hot glass tubing, it is observed t h a t the central region suffers the greatest elongation. Therefore, in the present case, it m a y be assumed t h a t plastic flow is confined to a small region e surrounding the crack. I n addition, it is assumed t h a t the material is ideally plastic so t h a t the region e has infinite ductility. Once again, consider the plate (Fig. 1) with initial cracks of length C on each side subjected to a constant load P (hanging under the action of gravity), where P is the load required initially to start the crack propagation. At any instant t after the crack propagation starts, crack instability condition is expressed b y ~o = 2(A - C0 - S1)~0h

P -- P = 2S1~oh

Exp

e ~- $2

e ~- Se $2

evP = ~ = - 2(e -~S~2)

(1-b)

For the range of thicknesses of interest here the

330 ] October 1965

~yP ~

(A-lb)

1 +8A(1 ~b) -

(A-5)

eV -

(e +

And the lower bound, as determined b y Garg 7 is

l-b
(A-4)

Expression (A-4) gives the load at a n y arbitrary time which is in excess of the load required for the crack-instability condition. I t causes the weights to move downward u n d e r the action of gravity with a velocity V. This happens simultaneously with the propagation of the crack across the plate with velocity C, and the expanding of the region e with velocity V. Then, at a n y arbitrary instant t after the start of of the crack propagation, the crack will have extended t h r o u g h a distance 2S~, and the loads moved down a distance $2. Also, the region at this instant has extended from e to e -~- $2. Making the assumption t h a t straining is limited to the plastic zone e, the strains and strain rates m a y be written ast e + $2 - e $2

(A-la)

Combining ( A - l a ) and ( A - l b )

(A-3)

where/~ is the load necessary propagate the crack at instant t. Then, from eqs (A-2) and (A-3)

~2

f - = (1 - b)

(A-2)

~zp

2

eV 2 (e + $2) 2

The internal dissipation rate, Dr, is expressed as

1),

= I" (~x~x + ~ J~ *

+ ~)dv*

The material is assumed to be incompressible.

and for the present strain rate field becomes D~

(eeVaoh -k $2) 2 (2e + S~)(A - Co - $1)

(A-6)

The external work rate,/)~, includes (a) the total potential energy-loss rate, and (b) the rate of conversion of potential energy into kinetic energy. Evidently,/)~, is given b y the difference of (a) and (b), or

D~ = 2(A - Co)haoVg - ~ g

(A-7)

I n establishing the energy criterion for crack propagation, three more energy terms are considered. I t is shown t h a t these terms are quite insignificant compared to internal energy dissipation rate and external work rate and m a y be neglected. The origin of these additional energy terms m a y be traced to (1) Kinetic energy due to displacements in front of the propagating crack, (2) Elastic strain energy stored in the specimen and its availability for crack extension, and (3) E n e r g y absorption due to lateral buckling of the specimen during the process of crack propagation. The energy contribution due to (1) above was calculated by Anderson s for a n ideally elastic and brittle material. The expression for kinetic-energy rate resulting from displacements around the crack was found to be

However, as Orowan 2 pointed out, ductile fracture is a process whose velocity is determined b y applied strain rate, and for the present dead-weight loading apparatus, the velocity V is small. Accordingly is also small, and it is possible to show t h a t eq (A-8) is of the order 10-a of eq (A 6) from dimensional analysis. In connection with energy source (2) Griffith ~ showed t h a t the elastic energy available to drive

cracks in brittle materials is approximately equal to 0.3 ~ (a~C~h/E). F r o m this expression the release rate of elastic energy becomes 0.6 ~ra~COh/E which, again, can be neglected b y comparison. The energy rate associated with lateral buckling has been previously discussed and is negligible when compared to internal-dissipation and external-work rates. Neglecting the energy terms (1), (2) and (3), the energy criteria for crack propagation m a y be expressed as follows: Internal-energy dissipation rate (/)i) = external-work rate (/)~). Substituting the values for /)i and /)~ from eq (A-6) and (A-7), respectively, in the energy criteria, the following expression for $1 m a y be obtained:

&

(e ~- $2)2(A - Co)Fe(2e + S~) e(2e + S~

2 ( g ~ V) ~, .j (A-9)

7

[_ ~-~S~)~

E q u a t i o n (A-9) gives S~ as a function of $2, l? and e. The crack-propagation velocity is now obtained b y simply differentiating eq (A-9) with respect to time. I n the preceding analysis, the existence of the plastic-flow region has been tacitly assumed. N o a t t e m p t was made or shall be made in what follows to find an empirical relationship for the plastic-flow region in terms of known or easily determined material variables. However, a few factors t h a t might influence its extent m a y be noted here in passing. These factors include test temperature, specimen dimensions (particularly thickness), initial crack length, yield point and ultimate strength of the material. The question whether an empirical relationship for the determination of e involves only the aforementioned material variables and test conditions can only be answered by a carefully planned experimental program. U p o n a little reflection, it would seem reasonable t h a t tests on specimens made out of different metals, dimensions, initial crack lengths and test t e m p e r a t u r e s should yield the m a x i m u m a m o u n t of information.

ERRA TA:

Anticlastic Behavior of Flat Plates by D. G. Bellow, G. Ford and J. S. Kennedy

We regret that an error appeared in Fig. 1 on

bQx

5Qx

page 228 of the July issue of EXPERIMENTAL

sion M~ ~- ~ x dx should be Q~ + 5xx dx, and ex-

MECHANICS.

5Mx pression Qx + 5Mx 5x dx is correctly Mx + ~xx dx.

Two of the expressions located at the lower righthand corner of the figure are incorrect. Expres-

The Editors

Experimental Mechanics I 331