STEARY FATIGUE CRACK GROWTH TENSILE LOAD
UNDER
N. C. HUANO and Y. C. LI ~~rtment
of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA,
Ab&rnctFatigue crack speed is analysed based on the criterion of total aecumtdative plastic work. Closure of crack surface during unloading is included in the study. Fatigue crack speed per cycle of loading is found to be dependent on two parameters km = k/R; and R = &,,I& where Kr is the fatigue fracture toughness defbred by the condition that di/dN approaches infinity as km approaches unity. Our theoretical fatigue crack speeds are compared with experimental data.
EVER SINCE the i~~~uction of an empirical formula by Paris and Erdogan[l] for ~l~~atjn8 the speed of steady fatigue crack growth, theoretics prediction of fatigue crack growth has become a challenging problem in mechanics. Weertman[Z] proposed a criterion of cumulative plastic displacement for material faihrre. He derived a fourth power law for the speed of fatigue crack growth based on a model of a fracture process zone of the Dugdale type131ahead of the crack tip. The idea of a reverse plastic zone during the unloading process was first pointed out by Rice for the case of a stationary crack[4]. Lardner studied the problem of fatigue under shear load and also found a locked dislocation region near the tip of the fracture process zone during unloading[S]. More recently, Huang studied the fatigue crack growth under antiplane shear deformation based on the total plastic work criterion for material failureIdf. He suggested that the total plastic work for material failure consists of (i) the work done to a material point due to the gradual opening motion of the fracture process zone and (ii) the work due to the oscillatory motion of the boundary of the fracture process zone caused by the loading and unloading process. The problem of tensile fatigue is more complicated than the ease of shear fatigue on account of the behavior of the crack closure which is observed from Elber’s
[email protected] Budiansky and Hutchinson analysed a quasistatic plane stress problem of tensile fatigue associated with small scale ~eiding~8}~Their study was based on the Dugdale model of fracture process zone associated with a locked dislocation zone of the Lardner type near the tip of the zone. Within the fracture process zone, the material is plastically stretched up to the point of failure. After failure, the stretched material is attached to the surface of the crack and forms a thin layer. This attached layer of stretched material can cause crack closure during unloading. The final thickness of the attached layer is determined by the state of maxims compression as the stress intensity factor reaches the ~nimum value I&,. A boundary value problem can be formulated for different values of K. At K = I,&, 2 0, the final thickness of the attached material layer as well as the sizes of the contact region of the crack and the region of the reverse plastic deformation can be determined from the conditions of finite values of the normal stress along the crack line. It is found that the size of the reverse plastic zone increases as the value of K decreases from the maximum value Km=, Budiansky and Hutchinson evaluated the values of stress intensity factors for the crack surface in contact &, and for the crack surface in full opening KOpen. They suggested that the speed of fatigue crack growth might be a function of (AK),, defined by either K,,,  KOwnor K,,,  KC,,,. But no specific expression for the fatigue crack speed has been given, In this paper, we shah employ the results of Budiansky and Hutehinson’s analysis to study the fatigue crack speed based on the criterion of total a~~ulative plastic workc6f. We expect that the fatigue crack speed would approach infinity as K& approaches the fatigue fracture toughness &. We are interested in finding the relation of fatigue crack growth per cycle of loading dads to the ratios k, = ~~/~~ and R =Kd,/K"m. Our theoretical fatigue crack speeds are cornp~~ with experimental data.
Ep&I 33/%K
477
478
N. C. HUANG
and Y. C. LI
BASIC EQUATIONS Let us consider a problem of plane stress with a crack subjected to periodic tensile mode deformation. We shall assume small scale yielding and regard the crack to be semiinfinite on the negative real axis as shown in Fig. 1. The stress intensity factor K varies periodically between K,,,,, and Kmin2 0. When K = K,,,,,, the crack is opened and there is a fracture process zone of the Dugdale type at 0 < x Q w ahead of the crack tip, where
and by is the yield stress in tension. The material at any position (x, 0) in the fracture process zone is stretched to a length S,(x). Let us denote < = x/w and AM(t) = 6, (x)/&(O). We have 1 f (1  4)“2 When the value of K decreases from K,,, to J&t) the opened crack will begin to close. For further decrease in K, the closed region grows. When K = K m,n2 0, the closed region is at 0 2 x 2 b, where b < 0 and there is a reverse plastic zone within the fracture process zone at 0 < x G a. Mixed boundary value problems of this type can be analysed by the complex variable technique. Let z = x + iy and d(z) and 1+9(z)be complex Muskhelishvili potentials. Let us denote a = a/w,
p = b/w,
and
i =2/w,
5 =x/w
3 F=;&(z).
Budiansky and Hutchinson[8] find that at K = Kmin,
’ K5 Ott5  a)1’c26
+
5i
s1
dt; + n(A + Rr)
d5
) (3)
where R = Kmin/Km,xand A is a constant to be determined. For given R, the values of a, fl and A can be determined from the conditions of bounded magnitudes of a,(i) at [ = B, 0 and u on the real axis. Our numerical results agree with those given by Budiansky and Hutchinson[8]. At K = Kmin, denote the stretched length of the material point in the reverse plastic zone by 6(x). Let us introduce A(<) = S(x)/&(O). We have dA z
1 =
27ci
(F,  F_) = [(~ _
+A
B)S;a _ 5)1,/2
I [(' 
= s il
Fig. 1. Geometry
“[(’ “i’i”;‘)““* dt
f
Is Pb('
0

')1"*f&)&
_
(A
+
r5
of the problem
for K = K,,, b 0
R()
for
0
(4)
479
Steady fatigue crack growth under tensile load
where branch lines are along ( co, 8) and (0, a). Thus the stretched length of the material point within the reverse plastic zone at K = K,i” is (5) where continuity conditions on A and dA/dt at { = 0 and 6 = a are imposed. Equations (4) and (5) can be evaluated by numerical integration. TOTAL ACCUMULATIVE PLASTIC WORK CRITERION Let us consider a material point in the fracture process zone. The total plastic work done to the material point can be divided into two parts. The crack opening plastic work is due to the propagation of the crack. Its magnitude increases as the crack tip propagates toward the material point. The crack opening plastic work attains its maximum value when the stretched length is a maximum. It is Y = CJY &l(O)
(6)
as the crack tip reaches the material point. The other part of the plastic work is the oscillatory plastic work induced by the oscillatory motion of the boundary of the fracture process zone at 0 < x < a. Since S(x) is a function of x, the oscillatory plastic work depends on the relative position of the material point with respect to the fracture process zone. As crack propagates forward, the material point moves backward relative to the crack tip. The oscillatory plastic work is the accumulation of the plastic work done at the material point as it moves backward through the entire fracture process zone. For each cycle of oscillation, the crack tip propagates forward by a distance di/dN. Hence the plastic work done to the material point within 0 < x < a through one cycle of oscillation can be approximated by dW, = 2 3
[6,(x)  6(x)] dx,
(7)
dN where 6(x) is the stretch at K = I&, and the factor 2 stems from the unloading and reloading process in each cycle. Note that in the derivation of eq. (7), we have used the property that the size of the reverse plastic zone is the largest at K = &in. The total oscillatory plastic work is therefore wz=2J$
a [6,(x)  6 (x)] dx. s0
(8)
dN
Let WC be the critical value of the total accumulative plastic work for material failure. It is a physical quantity which can be measured experimentally. Hence we have the following condition for crack propagation: w,+
w,=
WC.
(9)
Let us define the plane stress fatigue fracture toughness by Kf =
(EW,)“2.
In the following, we assume Kf to be a constant. Denote T= 4acI/(aK:), (6), (8), (9) and (lo), we find
(10) k,,, = K,,,,/K,. From eqs
(11) where
F,(k)=& m
(12)
480
N. C. HUANG and Y. C. LI 0.005
0.05
0.004
0.04
0.003
0.03
F*(R)
F, (RI
0.002
0.02
0.001
0.01
0
0
0.2
0.4
R=
0.6
0 1.0
0.8
Kmin yimY
Fig. 2. F,(R) and F>(R) curves.
and WR>
=
’ I&,(5> AC01dt = a (1 5)‘12_5n 2 S[0
I+” ‘)“’ A(r) 1  (1  <)“2I
s0
I* dc
03)
After A(6) is determined by eq. (S), I;;(R) can be evaluated numerically according to eq. (13). Equation (11) suggests that the speed of steady fatigue crack growth is determined by two dimensionless parameters k, and R. As K,,,,, approaches K,, the fatigue crack speed approaches infinity. In engineering applications, it is customary to express the fatigue crack speed in terms of the range of stress intensity factor. Let Ak = (K,,,  Km;,)/&. Equation (1 I) can also be expressed in terms of bk and R as dT
(Ak)”
dN=(lR)*(A/c)*
When km<<1, the fatigue
F,(R) (I’
crack speed can be approximated
by
60
~___~.__LII 1j / i!j// / 0.01
0.1
I
i]
1
$(
mm/kc)
Fig. 3. Comparison of the theoretical fatigue crack speeds with the experimental values for 2024.T3alloy, uy = 40 kg/mm2, Kf= I30 kg/mnP2.
Steady fatigue crack growth under tensile load
where F3(R) = F,(R)/( 1  Rr. Equation (15) is the well known speed of fatigue crack growth. In Fig. 2, F,(R) and F,(R) are It can be shown that if we ignore the closure of fatigue crack, can still be represented by eq. (1 1), where F,(k,) is also given
F,(R)=$(l  R)4.
481
fourth power law for the steady plotted against R. the speed of fatigue crack growth by eq. (12), but
06)
Hence, F,(R) becomes a constant equal to 0.04167. It is seen that the fatigue crack speed is decreased by a considerable amount due to the effect of crack closure. This decrease in fatigue speed is caused by the reduction in both the size of the reverse plastic zone at K = &, and the amplitude of the oscillatory motion of the boundary of the reverse plastic zone. For example, when R = 0, without the crack closure, a = 0.25 and Am  A(0) = OS. While in the case with the crack closure, a = 0.09286 and A, (0)  A(0) = 0.1438. In Fig. 3, our theoretical speeds of fatigue crack growth are compared with the experimental values given by Broak[9] for 2024T3 aluminum alloy with R = 0.1 to R = 0.5. It is found that in the region of high values of K,,, our theoretical values check well with experimental data. However, when the value of Kma,,is low, our theory only provides a lower boundary for &/dN. A possible cause of the deviation between theory and experiment stems from the dependence of & on K,,,=. Note that the threshold value of dl/dN is not shown in our theory in view of the rigidideally plastic behavior of the Dugdale model employed in the analysis. In conclusion, this paper presents a simple theory for the prediction of the speed of fatigue crack propagation based on the criterion of accumulative plastic work. The theory can be improved by considering a model with more complicated constitutive relations for the defo~ation of the fracture process zone. The dependence of & on K,, must also be taken into consideration. REFERENCES [I] P. C. Paris and F. Erdogan A critical analysis of crack propagation laws. J. boa. Engng 85, 528534 (1963). [2] J. We&man, Rate of growth of fatigue cracks as calculated from the theory of in~t~i~l dislocation distributed
on a plane. Zn?. f. Fracture h4ech. 2, 460467 (1966). [3] D. S. Dugdale, Yielding of steel sheets containing siits. L Mech. Phys. SoUs 8, 100108 (1960). [4] J. R. Rice, Mechanics of crack tip deformation and extension by fatigue. Fatigue Crack Propagation,ASTM STP 415, 247309 (1967). [S] B. A. Lardner, A dislocation model for fatigue crack growth in metals. Phil. Mug. 17, 7182 (1968). [6] N. C. Huang, Fatigue crack growth under antiplane shear mode deformation. Theor. uppl. Fracture Me& 19231239 (1988). [7] W. Elber, Fatigue crack dosure under cyclic tension. &gng Fructwe Mech. 2, 3745 (1970). [S] B. Budiansky and J. W. Hutchinson, Analysis of closure in fatigue crack growth. J. uppl. Me&. 4!!, 267276 (1978). [9] D. Broak, Elementary Engineering Frucfure Mechanics, 3rd revised edn, p. 254. Martinus Nijhoff Publishers, The Hague. (Received 25 August 1988)