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The dependence of the fracture toughness of high strength steel on crack velocity
Scripta
METALLURGICA
Vol. 12, pp. i i 0 1 - i i 0 6 , 1978 P r i n t e d in the U n i t e d States
Pergamon
Press, inc
THE DEPENDENCE OF THE FRACTURE TOUGHNESS OF HIGH STRENGTH STEEL ON CRACK VELOCITY Z. BILEK Brown University, Division of Engineering, Providence Rhode Island, 02912, USA
(Received September 6, 1978) (Revised October 13, 1978) INTRODUCTION The fracture failure of pressurized pipelines, bridge girders or ship hulls can be prevented by stopping propagating cracks before the structural integrity of the unit is lost. To stop a crack in a structure requires a dynamic stress intensity factor in the structure and the material's dynamic fracture toughness KID . Three extensive programs in which substantial experimental KID data have been obtained are those of Hahn etal. (1-6) at the Battelle blemorial Institute, Burns (7-9) at the University of Rochester and Bilek etal. (I0,II) at the Institute of Physical Metallurgy. In each of these programs fast fracture is produced by wedging apart the arms of a precracked double cantilever beam dcb specimen. The Battelle experiments use a slowly advancing wedge with fast fracture initiating from a blunt starter-notch, so that crack propagation occurs essentially under the condition of fixed displacements of the loading pins against which the opening wedge bears. In Burns' experiments, a massive falling wedge separates the arms of the specimen at a rapid constant rate, and fracture starts from a short, sharp crack. The latter experiments generally result in rather lower crack velocities and the propagation of longer cracks than do the Battelle experiments, in which crack arrest usually takes place. Both techniques described above are currently used at Institute of Physical Hetallurgy to study the temperature and crack velocity dependence of KID for various structural materials over a wide range of crack velocities. All of the above investigations use one dimensional beam theory to calculate KID from the experimental measurements. The present paper reviews the dcb specimen model being used for experimental data analysis at the Institute of Physical Hetallurgy. The equations governing crack motion and the fracture criterion in a dcb specimen are derived through use of Hamilton's principle for non-conservative systems, from the Timoshenko theory of elastic beams (including the effect of transverse and rotary inertia). Details relating to steady motion are emphasized. The developed model is applied to investigate the dynamic fracture toughness of SAE 4340 steel under both slow and rapid wedging loading. The results show a dependence of KID on a in good agreement with other data in the literature (4,5,17). A ~DEL
OF THE DCB SPECIHEN USING A TIMOSHENKO BE/DI ON A RIGID FOUNDATION
The g e o m e t r y o f t h e dcb specimen l o a d e d by a t i m e d e p e n d e n t f o r c e Q(t) and a b e n d i n g moment H(t) i s s c h e m a t i c a l l y i l l u s t r a t e d i n F i g . 1. The l o a d i n g w i l l c a u s e a h o r i z o n t a l c r a c k to p r o p a g a t e from l e f t t o r i g h t . In e l a b o r a t i n g t h e m a t h e m a t i c a l model, b e c a u s e o f symmetry we s h a l l c o n s i d e r o n l y t h e u p p e r h a l f o f t h e specimen i n t h e r e c t a n g u l a r c o o r d i n a t e s y s t e m x , y . The o r i g i n f o r t h i s s y s t e m i s a t t h e l o a d e d end o f t h e s p e c i m e n , as shown i n F i g . 1. The m o t i o n o f t h e c r a c k a t any time must be i n agreement w i t h H a m i l t o n ' s p r i n c i p l e for non-conservative systems (T-V) ÷
Z k=l
.
at = 0
(1)
where T r e p r e s e n t s t h e k i n e t i c e n e r g y and V t h e p o t e n t i a l e n e r g y o f t h e s y s t e m , F k t h e k t h g e n e r a I i z e d d i s s i p a t i v e f o r c e , and qk t h e k t h g e n e r a l i z e d c o o r d i n a t e . A c c o r d i n g t o t h e Timoshenko t h e o r y o f e l a s t i c beams, t h e k i n e t i c e n e r g y i s ra(t)
2
2
0
Permanent a d d r e s s : I n s t i t u t e o f P h y s i c a l M e t a l l u r g y , C z e c h o s l o v a k Academy o f S c i e n c e s , 616 62 Brno, Zizkova 22, C z e c h o s l o v a k i a . oo36-97~s/78/1211ol-o65o2.oo/o Copyright ( e ) 1978 P e r g a m o n P r e s s
1102
FRACTURE
TOUGHNESS
OF H I G H
STRENGTH
STEEL
Vol.
12, No. 12
where A:wh is the cross-sectional area and P is the mass density; u(x,t), ~(x,t) denote average deflection of the cross section and mean angle of rotation of the cross section about the neutral axis, respectively. The potential energy arises from strain energy of the bent part of the beam, the reversible surface energy which is calculated by integrating the reversible specific surface energy Yo over the fracture area and from work done by the applied shear force Q(t) and the applied bending moment M(t)
fa (t)
V = ~
I
+
( ax
-*(x,t)
dx + Yo
wa(t) -
qCt)u(O,t) - M ( t ) ¢ ( 0 , t )
0
(3)
where E i s Young's modulus, v i s P o i s s o n r a t i o , moment o f i n e r t i a f o r a r e c t a n g u l a r beam o f h e i g h t e n e r g y yp g i v e s a g e n e r a l i z e d d i s s i p a t i v e f o r c e
G : E / 2 ( l + v ) and I = wh3/12 i s t h e h . The i r r e v e r s i b l e p a r t o f t h e s u r f a c e
F1 : Wyp
(4)
By i n s e r t i n g (2) to (4) i n t o I1amilton's p r i n c i p l e (1) we obtain, f o l l o w i n g some v a r i a t i o n a l manipulations (12) : a2u(x,t)
~,(x,t)
@x
@x 2
:L~2u(x,t)
C~ Bt2
, c ~ = TG ' c ~ : TE (s)
aZO(x,t)
+ TA__C~c~ [I'Bu(x't)~)x ~(x,t).l : --C~I@2~(x,t)Bt 2
~x2 GA[3U(X,t) and f o r
- ¢(x,t) 1 : QCt) , El @¢(x,t) @x : M(t) for )
x : o
(6)
X : a(t)
(~u(a,t)
12 C_~
c~
A~ T
[;~(t)]2l "l%,]'L
I~ (a,t)]2[i f~(t)]2"] 2Vw J _ -L
LJJ--
T
(7) '
where y is the specific fracture surface energy and is given by the sum of the reversible and irreversible specific fracture surface energies:
Y :
Yo + Yp "
(8)
In addition, the symmetry of the problem and the assumed rigid support of the beams at the crack tip specify the deflection and the rotation of the deflected beam at the crack tip (i.e., x = a(t)) as zero uCa(t),t) = 0
, tg(aCt),t) : 0
(9)
The Timoshenko beam equations (5) represent a special case of more the general equations, derived by Kanninen (13), for a beam on an elastic foundation. Equation (7) gives the fracture criterion at the crack tip. For long slender dcb specimens, such as used in (7-9), the equations (5) may be reduced to one equation of Bernoulli-Euler beam motion and the fracture criterion (7) then yields the condition on the bending moment at the crack tip El ~2u(a't) ax 2
: (2ywEI) ½
recently derived by Bilek and Burns (12). maximum crack velocity in dc~ specimen to allows virtually unlimited a(t) .
(I0) It is interesting to point out that (7) restricts C L although the less general expression (I0)
Vol.
12,
No.
12
FRACTURE
TOUGHNESS
OF H I G H
STRENGTH
STEEL
1103
Equations (5) r e p r e s e n t a h y p e r b o l i c system which i s r e a d i l y s o l v a b l e n u m e r i c a l l y with the boundary and i n i t i a l c o n d i t i o n s corresponding t o t h e c u r r e n t experimental procedures by using a l e a s t squares or a f i n i t e d i f f e r e n c e t e c h n i q u e . In p a r t i c u l a r , f o r f i x e d displacement loading (u(O,t) = c o n s t . ) or c o n s t a n t opening r a t e loading (u(O,t) ~ t ) , t h e s p e c i f i c f r a c t u r e energy y (and hence KID ) may be c a l c u l a t e d from t h e recorded crack l e n g t h dependenc~ on t i m e. Also the crack v e l o c i t y p r o f i l e may be found i f the f u n c t i o n y(~) (or KID(a)) is specified. Details of the numerical formulation and of the computer program will be discussed in a forthcoming publication. DYNAMIC FRACTURE TOUGI~ESS OF SAE 4340 STEEL Studies of unstable and stable crack propagation over a wide range of crack velocities were carried out in order to establish KID - a dependence of SAE 4340 high strength steel in the Qg& T conditions: oil quenched at 870°C and tempered at 385UC for the lh(~y s = 1367~m-') . The unstable crack propagation study employed dcb specimens which were slowly wedge loaded as described in (I-5,11). Essentially, the identical specimen geometry used recently to investigate SAE 4340 steel properties (3,4) was applied here, however the sDecimen thickness was increased to 25mm to meet plane strain requirements (ASTM E399) at high crack velocities. The crack velocity was varied by changing the root radius of the starting notch from 0.2 to imm; crack velocities over 800m/s were achieved with imm notch radius and suddenly released externally applied transverse compression. The compression, applied ever circular zones of 1 0 ~ (Fig. i) in diameter on each face of the specimen, stabilizes the crack (14] and results in a higher potential energy accumulation which promotes a higher level of a . Crack velocities were measured using crack propagation gages of type ~ CPC 03. During a typical test,.a steady-state velocity was maintained from the start until shortly before arrest. Only for a > I000 m/s did the crack velocity despose somewhat between fracture initiation and crack arrest. This behavior is consistent with earlier results on P ~ (II) and it is obviously related to strong crack velocity dependence of y (15). The dynamic fracture toughness values KID evaluated by the model described in the previous section are shown in Fig. 2 for fifteen specimens. Rapidly,wedged dcb specimens were used to produce a stable crack propagation with crack . o velocities a < 150 m/s . Rigid loading of the test piece was achieved by uslng a 25 wedge firmly mounted on the bottom of the moving carriage of a drop weight fracture machine used for NDT testing. Specimens of outside dimensions 55mm high by 25mm wide by 380mm long, with deep side grooves similar to the specimens designed in (7,8) were used. The thickness of 12.Smm across the grooved section was sufficient to provide valid plane strain KID data. The R.F. current technique developed by Carlsson (16) was applied to measure crack length versus time. The crack velocity in dcb specimens opened at a constant rate varies continuously throughout the test since the crack travels with a flexure wave. The crack velocity thus decreases as the crack length increases so KID may be recorded as a function of a in a single test. The KID data calculated numerically from a Timoshenko beam model were identical, within numerical errors, with data obtained directly from the analytic Bernoulli-Euler beam model (12). The KID results are plotted in Fig. 2 for six specimens. DISCUSSION The KID measurements for both independent experimental techniques are reported in Fig. 2 as a function of crack velocity a . In the same figure the Battelle data (4) are plotted, on the same steel, as well as the data derived by Angelino (17) for a similar steel VCN-100, from three point bend tests (TPB). The good agreement shown supports the dcb specimen model developed in this paper and indicates that the assumption of a rigid beam foundation, which somewhat simplifies the KID calculations, might be acceptable. The rapid wedging KID data show a negative slope at low velocities and a non-pronounced minimum at a ~ 60 m/s, as suggested in (17). In addition, the KID data determined from stable (rapid wedging) and unstable (slow wedging) measurements agree well in the valley of the KID - a curve, demonstrating a significant similarity in these two experimental techniques discussed already by \lalluck and King (18). Crack arrest was observed in slow wedging experiments at which the crack velocity was below I000 m/s. Fracture toughness at crack arrest KIA was calculated from a simple steady state solution of equations (5) for dcb specimens opene~ by a wedge of constant velocity V -
1104
FRACTURE
TOUGHNESS
OF H I G H
STRENGTH
STEEL
Vol.
12,
No.
12
Fig. 3. The solution of equations [5) may be assumed in the form u(x,t) = uCx-Vt), ~(x,t) = ~[x-Vt) and u = H for x - Vt, and ~(x,t)~ = 0 for x = Vt + a . Using and substitution ~ = x-Vt , following some simple manipulations, we find
U(~) = s i n S a
(1-a2/C~)Ba cosSa[82(l-a2/C~)
~ cosSa
* H
(11)
HBCI-&2/C~) ~(a) =
(cos6a - cosSa)
gzn8~ - ( 1 - a 2 / C )
(12)
ga c o s g a
~ = ~v~dT (1-12/c~)-1/2(1-~2/C~)-1/2 Equations
( 1 1 , 1 2 ) and t h e f r a c t u r e
Esin a
-
criterion
C7) y i e l d t h e r e l a t i o n s h i p
8a cosBo1 Cla /C )
b e t w e e n a, V and y :
I"a. 2 2I E =
C13)
6
By expandin~ the trigonometric functions in (13) into series and neglecting small terms for V ~ 0 we find the relationship ~ r the crack length a A corresponding to the constant specimen opening displacement H :
~--~--- 0.0917 - 8Cs2
.
(14)
~w
Slow wedging measurements are essentially performed under fixed displacement conditions. Since this displacement is recorded during the test, the specific surface energy y at crack arrest and thus KIA may be calculated from eauation (14) from the measured crack length a A . The average value of KIA = 67.SHNm -3/2 was found to be nearlz independent of crack velocity for SAE 4340 steel and slightly higher than KIC = 6 2 ~ m "3~2 . This result confirms recent findings at Battelle (3-5). |bwever, the KIA value calculated from (14) does not reflect the entire history of the crack propa~ation event and may not be interpreted as a material constant (15). CONCLUSION A very simple model of the dcb dynamic crack g r o ~ h propagation specimen is studied. The main assumption on which the model is based is that the arms of the dcb specimen deform as Timoshenko beams on rigid ~undations. An analysis of the dynamics of crack propagation was used to calculate the dynamic plane strain fracture toughness ~ r a propagating crack in the dcb specimens as a ~nction of crack velocity at room temperature. It has been shown that KID for SAE 4340 steel in the q & T conditions decreases for increasing crack velocity to a minimum of KID = 37t~m -3/2 at a = 6 ~ / s and increases sharply above 100HNm "3/2 at a • i000 m/s . ACKNOWLEDGEMENTS The financial support of the National Academy of Sciences is gratefully acknowledged. The author is also indebted to Professors L.B. Freund and J.R. Rice for their helpful discussions. REFERENCES
i. 2. 3. 4. S. 6. 7. 8.
R.G. l l o a g l a n d , A.R. R o s e n f i e l d and G.T. Hahn, Her. T r a n s . 3,123 ( 1 9 7 2 ) . G.T. llahn, R.fi. l[oagland and A.R. R o s e n f i e l d , P r e c . I n t . Conf. on P r o s p e c t s o f F r a c t u r e Hechanics, Delft, The Netherlands, 267 (1974). G.T. Hahn, R.G. IIoagland, A.R. Rosenfield and R. Sejnoha, Her. Trans. 5, 475 (1974). C.T. Hahn et al., BHI Report 1937, Battelle-Columbus Labs., Columbus, Ohio, 13 (1975). G.T. Hahn, R.G. Hoagland and A.R. Rosenfield, Hat. Trans., 7, 43 (1976). C.T. Hahn and H.Fo Kanninen, Prec. 4th Int. Conf. on Fracture, Waterlooj Canada I, 193 (1977) S.J. Burns and Z. Bilek, Het. Trans. 4, 975 (1973). S.J. Burns and C.L. Chow, AST?| STP 627, 228 (1976).
Vol.
9. 10. ii. 12. 13. 14. 15. 16. 17. 18.
12, No.
FRACTURE T O U G H N E S S
12
OF H I G H
STRENGTH
STEEL
ii05
Z. Bilek and S.J. Burns, Proc. Int. Conf. on Dynamic Crack Propagation, Bethlehem, U.S.A., 371 (1972). Z. Bilek and P. Kyselak, Proc. Int. Conf. on Haterials Behavior under Impulsive Loading, Brno, Czechoslovakia, 34 (1978). Z. Bilek and P. Kyselak, Proc. National Conf. on Fracture, Boboty, 18 (1978). Z. Bilek and S.J. Burns, J. Mech. Phys. Solids, 22, 85 (1974). H.F. Kanninen, Int. J. Fracture, 10, 415 (1974). D.O. Harris, Eng. Fract. Hech. 4, 277 (1972). L.B. Freund, J. ~lech. Phys. Solids 25, 69 (1977). J. Carlsson, Trans. Royal Inst. Tech., 189, I, (1962). G.C. Angelino: AST~| STP G27, 392 (1976). J.F. rlalluck and W.W. King, Int. J. Fracture 13, 655 (1977).
y,
MItI Q(t)
•
do
2h
o
f o(t) W
Q(t) FIG. 1 Schematic illustration of a moment M ( t ) , i n c l u d i n g t h e t h e c o o r d i n a t e s y s t e m . The i s a p p l i e d t o promote h i g h e r
(dcb) specimen opened by a f o r c e Q(t) and a b e n d i n g n o t a t i o n u s e d t o d e s c r i b e t h e specimen g e o m e t r y and shaded c i r c l e shows t h e a r e a where a c o m p r e s s i v e f o r c e c r a c k v e l o c i t i e s d u r i n g u n s t a b l e jumps.
1106
FRACTURE
TOUGHNESS
OF HIGH STRENGTH
STEEL
Vol.
12, No. 12
150
x - Present Study, rapid wedging o - Present Study, slow wedging z~- Hohn et ol ( 4 ) , slow wedging • A n g e l i n o ( 1 7 ) , T P B test 0
I00
'E z
|
KIc= 62 MNrn'3/2
ci bC
50
Xxxx-__x
•"
8
"V~"
xxox I
O
I
t
i I
i
I
10
I I
100
l
I
I I
I
I
I I
£000
o [mls] FIG. 2 Velocity dependence of values. of the wide range of a KID.
A logarithmic abscissa is used because
y
2h
V
w
FIG. 3 A dcb specimen opened by a wedge of constant height moving with constant speed.
METALLURGICA
Vol. 12, pp. i i 0 1 - i i 0 6 , 1978 P r i n t e d in the U n i t e d States
Pergamon
Press, inc
THE DEPENDENCE OF THE FRACTURE TOUGHNESS OF HIGH STRENGTH STEEL ON CRACK VELOCITY Z. BILEK Brown University, Division of Engineering, Providence Rhode Island, 02912, USA
(Received September 6, 1978) (Revised October 13, 1978) INTRODUCTION The fracture failure of pressurized pipelines, bridge girders or ship hulls can be prevented by stopping propagating cracks before the structural integrity of the unit is lost. To stop a crack in a structure requires a dynamic stress intensity factor in the structure and the material's dynamic fracture toughness KID . Three extensive programs in which substantial experimental KID data have been obtained are those of Hahn etal. (1-6) at the Battelle blemorial Institute, Burns (7-9) at the University of Rochester and Bilek etal. (I0,II) at the Institute of Physical Metallurgy. In each of these programs fast fracture is produced by wedging apart the arms of a precracked double cantilever beam dcb specimen. The Battelle experiments use a slowly advancing wedge with fast fracture initiating from a blunt starter-notch, so that crack propagation occurs essentially under the condition of fixed displacements of the loading pins against which the opening wedge bears. In Burns' experiments, a massive falling wedge separates the arms of the specimen at a rapid constant rate, and fracture starts from a short, sharp crack. The latter experiments generally result in rather lower crack velocities and the propagation of longer cracks than do the Battelle experiments, in which crack arrest usually takes place. Both techniques described above are currently used at Institute of Physical Hetallurgy to study the temperature and crack velocity dependence of KID for various structural materials over a wide range of crack velocities. All of the above investigations use one dimensional beam theory to calculate KID from the experimental measurements. The present paper reviews the dcb specimen model being used for experimental data analysis at the Institute of Physical Hetallurgy. The equations governing crack motion and the fracture criterion in a dcb specimen are derived through use of Hamilton's principle for non-conservative systems, from the Timoshenko theory of elastic beams (including the effect of transverse and rotary inertia). Details relating to steady motion are emphasized. The developed model is applied to investigate the dynamic fracture toughness of SAE 4340 steel under both slow and rapid wedging loading. The results show a dependence of KID on a in good agreement with other data in the literature (4,5,17). A ~DEL
OF THE DCB SPECIHEN USING A TIMOSHENKO BE/DI ON A RIGID FOUNDATION
The g e o m e t r y o f t h e dcb specimen l o a d e d by a t i m e d e p e n d e n t f o r c e Q(t) and a b e n d i n g moment H(t) i s s c h e m a t i c a l l y i l l u s t r a t e d i n F i g . 1. The l o a d i n g w i l l c a u s e a h o r i z o n t a l c r a c k to p r o p a g a t e from l e f t t o r i g h t . In e l a b o r a t i n g t h e m a t h e m a t i c a l model, b e c a u s e o f symmetry we s h a l l c o n s i d e r o n l y t h e u p p e r h a l f o f t h e specimen i n t h e r e c t a n g u l a r c o o r d i n a t e s y s t e m x , y . The o r i g i n f o r t h i s s y s t e m i s a t t h e l o a d e d end o f t h e s p e c i m e n , as shown i n F i g . 1. The m o t i o n o f t h e c r a c k a t any time must be i n agreement w i t h H a m i l t o n ' s p r i n c i p l e for non-conservative systems (T-V) ÷
Z k=l
.
at = 0
(1)
where T r e p r e s e n t s t h e k i n e t i c e n e r g y and V t h e p o t e n t i a l e n e r g y o f t h e s y s t e m , F k t h e k t h g e n e r a I i z e d d i s s i p a t i v e f o r c e , and qk t h e k t h g e n e r a l i z e d c o o r d i n a t e . A c c o r d i n g t o t h e Timoshenko t h e o r y o f e l a s t i c beams, t h e k i n e t i c e n e r g y i s ra(t)
2
2
0
Permanent a d d r e s s : I n s t i t u t e o f P h y s i c a l M e t a l l u r g y , C z e c h o s l o v a k Academy o f S c i e n c e s , 616 62 Brno, Zizkova 22, C z e c h o s l o v a k i a . oo36-97~s/78/1211ol-o65o2.oo/o Copyright ( e ) 1978 P e r g a m o n P r e s s
1102
FRACTURE
TOUGHNESS
OF H I G H
STRENGTH
STEEL
Vol.
12, No. 12
where A:wh is the cross-sectional area and P is the mass density; u(x,t), ~(x,t) denote average deflection of the cross section and mean angle of rotation of the cross section about the neutral axis, respectively. The potential energy arises from strain energy of the bent part of the beam, the reversible surface energy which is calculated by integrating the reversible specific surface energy Yo over the fracture area and from work done by the applied shear force Q(t) and the applied bending moment M(t)
fa (t)
V = ~
I
+
( ax
-*(x,t)
dx + Yo
wa(t) -
qCt)u(O,t) - M ( t ) ¢ ( 0 , t )
0
(3)
where E i s Young's modulus, v i s P o i s s o n r a t i o , moment o f i n e r t i a f o r a r e c t a n g u l a r beam o f h e i g h t e n e r g y yp g i v e s a g e n e r a l i z e d d i s s i p a t i v e f o r c e
G : E / 2 ( l + v ) and I = wh3/12 i s t h e h . The i r r e v e r s i b l e p a r t o f t h e s u r f a c e
F1 : Wyp
(4)
By i n s e r t i n g (2) to (4) i n t o I1amilton's p r i n c i p l e (1) we obtain, f o l l o w i n g some v a r i a t i o n a l manipulations (12) : a2u(x,t)
~,(x,t)
@x
@x 2
:L~2u(x,t)
C~ Bt2
, c ~ = TG ' c ~ : TE (s)
aZO(x,t)
+ TA__C~c~ [I'Bu(x't)~)x ~(x,t).l : --C~I@2~(x,t)Bt 2
~x2 GA[3U(X,t) and f o r
- ¢(x,t) 1 : QCt) , El @¢(x,t) @x : M(t) for )
x : o
(6)
X : a(t)
(~u(a,t)
12 C_~
c~
A~ T
[;~(t)]2l "l%,]'L
I~ (a,t)]2[i f~(t)]2"] 2Vw J _ -L
LJJ--
T
(7) '
where y is the specific fracture surface energy and is given by the sum of the reversible and irreversible specific fracture surface energies:
Y :
Yo + Yp "
(8)
In addition, the symmetry of the problem and the assumed rigid support of the beams at the crack tip specify the deflection and the rotation of the deflected beam at the crack tip (i.e., x = a(t)) as zero uCa(t),t) = 0
, tg(aCt),t) : 0
(9)
The Timoshenko beam equations (5) represent a special case of more the general equations, derived by Kanninen (13), for a beam on an elastic foundation. Equation (7) gives the fracture criterion at the crack tip. For long slender dcb specimens, such as used in (7-9), the equations (5) may be reduced to one equation of Bernoulli-Euler beam motion and the fracture criterion (7) then yields the condition on the bending moment at the crack tip El ~2u(a't) ax 2
: (2ywEI) ½
recently derived by Bilek and Burns (12). maximum crack velocity in dc~ specimen to allows virtually unlimited a(t) .
(I0) It is interesting to point out that (7) restricts C L although the less general expression (I0)
Vol.
12,
No.
12
FRACTURE
TOUGHNESS
OF H I G H
STRENGTH
STEEL
1103
Equations (5) r e p r e s e n t a h y p e r b o l i c system which i s r e a d i l y s o l v a b l e n u m e r i c a l l y with the boundary and i n i t i a l c o n d i t i o n s corresponding t o t h e c u r r e n t experimental procedures by using a l e a s t squares or a f i n i t e d i f f e r e n c e t e c h n i q u e . In p a r t i c u l a r , f o r f i x e d displacement loading (u(O,t) = c o n s t . ) or c o n s t a n t opening r a t e loading (u(O,t) ~ t ) , t h e s p e c i f i c f r a c t u r e energy y (and hence KID ) may be c a l c u l a t e d from t h e recorded crack l e n g t h dependenc~ on t i m e. Also the crack v e l o c i t y p r o f i l e may be found i f the f u n c t i o n y(~) (or KID(a)) is specified. Details of the numerical formulation and of the computer program will be discussed in a forthcoming publication. DYNAMIC FRACTURE TOUGI~ESS OF SAE 4340 STEEL Studies of unstable and stable crack propagation over a wide range of crack velocities were carried out in order to establish KID - a dependence of SAE 4340 high strength steel in the Qg& T conditions: oil quenched at 870°C and tempered at 385UC for the lh(~y s = 1367~m-') . The unstable crack propagation study employed dcb specimens which were slowly wedge loaded as described in (I-5,11). Essentially, the identical specimen geometry used recently to investigate SAE 4340 steel properties (3,4) was applied here, however the sDecimen thickness was increased to 25mm to meet plane strain requirements (ASTM E399) at high crack velocities. The crack velocity was varied by changing the root radius of the starting notch from 0.2 to imm; crack velocities over 800m/s were achieved with imm notch radius and suddenly released externally applied transverse compression. The compression, applied ever circular zones of 1 0 ~ (Fig. i) in diameter on each face of the specimen, stabilizes the crack (14] and results in a higher potential energy accumulation which promotes a higher level of a . Crack velocities were measured using crack propagation gages of type ~ CPC 03. During a typical test,.a steady-state velocity was maintained from the start until shortly before arrest. Only for a > I000 m/s did the crack velocity despose somewhat between fracture initiation and crack arrest. This behavior is consistent with earlier results on P ~ (II) and it is obviously related to strong crack velocity dependence of y (15). The dynamic fracture toughness values KID evaluated by the model described in the previous section are shown in Fig. 2 for fifteen specimens. Rapidly,wedged dcb specimens were used to produce a stable crack propagation with crack . o velocities a < 150 m/s . Rigid loading of the test piece was achieved by uslng a 25 wedge firmly mounted on the bottom of the moving carriage of a drop weight fracture machine used for NDT testing. Specimens of outside dimensions 55mm high by 25mm wide by 380mm long, with deep side grooves similar to the specimens designed in (7,8) were used. The thickness of 12.Smm across the grooved section was sufficient to provide valid plane strain KID data. The R.F. current technique developed by Carlsson (16) was applied to measure crack length versus time. The crack velocity in dcb specimens opened at a constant rate varies continuously throughout the test since the crack travels with a flexure wave. The crack velocity thus decreases as the crack length increases so KID may be recorded as a function of a in a single test. The KID data calculated numerically from a Timoshenko beam model were identical, within numerical errors, with data obtained directly from the analytic Bernoulli-Euler beam model (12). The KID results are plotted in Fig. 2 for six specimens. DISCUSSION The KID measurements for both independent experimental techniques are reported in Fig. 2 as a function of crack velocity a . In the same figure the Battelle data (4) are plotted, on the same steel, as well as the data derived by Angelino (17) for a similar steel VCN-100, from three point bend tests (TPB). The good agreement shown supports the dcb specimen model developed in this paper and indicates that the assumption of a rigid beam foundation, which somewhat simplifies the KID calculations, might be acceptable. The rapid wedging KID data show a negative slope at low velocities and a non-pronounced minimum at a ~ 60 m/s, as suggested in (17). In addition, the KID data determined from stable (rapid wedging) and unstable (slow wedging) measurements agree well in the valley of the KID - a curve, demonstrating a significant similarity in these two experimental techniques discussed already by \lalluck and King (18). Crack arrest was observed in slow wedging experiments at which the crack velocity was below I000 m/s. Fracture toughness at crack arrest KIA was calculated from a simple steady state solution of equations (5) for dcb specimens opene~ by a wedge of constant velocity V -
1104
FRACTURE
TOUGHNESS
OF H I G H
STRENGTH
STEEL
Vol.
12,
No.
12
Fig. 3. The solution of equations [5) may be assumed in the form u(x,t) = uCx-Vt), ~(x,t) = ~[x-Vt) and u = H for x - Vt, and ~(x,t)~ = 0 for x = Vt + a . Using and substitution ~ = x-Vt , following some simple manipulations, we find
U(~) = s i n S a
(1-a2/C~)Ba cosSa[82(l-a2/C~)
~ cosSa
* H
(11)
HBCI-&2/C~) ~(a) =
(cos6a - cosSa)
gzn8~ - ( 1 - a 2 / C )
(12)
ga c o s g a
~ = ~v~dT (1-12/c~)-1/2(1-~2/C~)-1/2 Equations
( 1 1 , 1 2 ) and t h e f r a c t u r e
Esin a
-
criterion
C7) y i e l d t h e r e l a t i o n s h i p
8a cosBo1 Cla /C )
b e t w e e n a, V and y :
I"a. 2 2I E =
C13)
6
By expandin~ the trigonometric functions in (13) into series and neglecting small terms for V ~ 0 we find the relationship ~ r the crack length a A corresponding to the constant specimen opening displacement H :
~--~--- 0.0917 - 8Cs2
.
(14)
~w
Slow wedging measurements are essentially performed under fixed displacement conditions. Since this displacement is recorded during the test, the specific surface energy y at crack arrest and thus KIA may be calculated from eauation (14) from the measured crack length a A . The average value of KIA = 67.SHNm -3/2 was found to be nearlz independent of crack velocity for SAE 4340 steel and slightly higher than KIC = 6 2 ~ m "3~2 . This result confirms recent findings at Battelle (3-5). |bwever, the KIA value calculated from (14) does not reflect the entire history of the crack propa~ation event and may not be interpreted as a material constant (15). CONCLUSION A very simple model of the dcb dynamic crack g r o ~ h propagation specimen is studied. The main assumption on which the model is based is that the arms of the dcb specimen deform as Timoshenko beams on rigid ~undations. An analysis of the dynamics of crack propagation was used to calculate the dynamic plane strain fracture toughness ~ r a propagating crack in the dcb specimens as a ~nction of crack velocity at room temperature. It has been shown that KID for SAE 4340 steel in the q & T conditions decreases for increasing crack velocity to a minimum of KID = 37t~m -3/2 at a = 6 ~ / s and increases sharply above 100HNm "3/2 at a • i000 m/s . ACKNOWLEDGEMENTS The financial support of the National Academy of Sciences is gratefully acknowledged. The author is also indebted to Professors L.B. Freund and J.R. Rice for their helpful discussions. REFERENCES
i. 2. 3. 4. S. 6. 7. 8.
R.G. l l o a g l a n d , A.R. R o s e n f i e l d and G.T. Hahn, Her. T r a n s . 3,123 ( 1 9 7 2 ) . G.T. llahn, R.fi. l[oagland and A.R. R o s e n f i e l d , P r e c . I n t . Conf. on P r o s p e c t s o f F r a c t u r e Hechanics, Delft, The Netherlands, 267 (1974). G.T. Hahn, R.G. IIoagland, A.R. Rosenfield and R. Sejnoha, Her. Trans. 5, 475 (1974). C.T. Hahn et al., BHI Report 1937, Battelle-Columbus Labs., Columbus, Ohio, 13 (1975). G.T. Hahn, R.G. Hoagland and A.R. Rosenfield, Hat. Trans., 7, 43 (1976). C.T. Hahn and H.Fo Kanninen, Prec. 4th Int. Conf. on Fracture, Waterlooj Canada I, 193 (1977) S.J. Burns and Z. Bilek, Het. Trans. 4, 975 (1973). S.J. Burns and C.L. Chow, AST?| STP 627, 228 (1976).
Vol.
9. 10. ii. 12. 13. 14. 15. 16. 17. 18.
12, No.
FRACTURE T O U G H N E S S
12
OF H I G H
STRENGTH
STEEL
ii05
Z. Bilek and S.J. Burns, Proc. Int. Conf. on Dynamic Crack Propagation, Bethlehem, U.S.A., 371 (1972). Z. Bilek and P. Kyselak, Proc. Int. Conf. on Haterials Behavior under Impulsive Loading, Brno, Czechoslovakia, 34 (1978). Z. Bilek and P. Kyselak, Proc. National Conf. on Fracture, Boboty, 18 (1978). Z. Bilek and S.J. Burns, J. Mech. Phys. Solids, 22, 85 (1974). H.F. Kanninen, Int. J. Fracture, 10, 415 (1974). D.O. Harris, Eng. Fract. Hech. 4, 277 (1972). L.B. Freund, J. ~lech. Phys. Solids 25, 69 (1977). J. Carlsson, Trans. Royal Inst. Tech., 189, I, (1962). G.C. Angelino: AST~| STP G27, 392 (1976). J.F. rlalluck and W.W. King, Int. J. Fracture 13, 655 (1977).
y,
MItI Q(t)
•
do
2h
o
f o(t) W
Q(t) FIG. 1 Schematic illustration of a moment M ( t ) , i n c l u d i n g t h e t h e c o o r d i n a t e s y s t e m . The i s a p p l i e d t o promote h i g h e r
(dcb) specimen opened by a f o r c e Q(t) and a b e n d i n g n o t a t i o n u s e d t o d e s c r i b e t h e specimen g e o m e t r y and shaded c i r c l e shows t h e a r e a where a c o m p r e s s i v e f o r c e c r a c k v e l o c i t i e s d u r i n g u n s t a b l e jumps.
1106
FRACTURE
TOUGHNESS
OF HIGH STRENGTH
STEEL
Vol.
12, No. 12
150
x - Present Study, rapid wedging o - Present Study, slow wedging z~- Hohn et ol ( 4 ) , slow wedging • A n g e l i n o ( 1 7 ) , T P B test 0
I00
'E z
|
KIc= 62 MNrn'3/2
ci bC
50
Xxxx-__x
•"
8
"V~"
xxox I
O
I
t
i I
i
I
10
I I
100
l
I
I I
I
I
I I
£000
o [mls] FIG. 2 Velocity dependence of values. of the wide range of a KID.
A logarithmic abscissa is used because
y
2h
V
w
FIG. 3 A dcb specimen opened by a wedge of constant height moving with constant speed.