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On the theory of fatigue crack growth
Enpmmring
6rac1ure
MechanirJ.
1972. Vol. 4. pp. 219-230.
ON THE THEORY
Pergamon
Press.
Primed
OF FATIGUE
in Great
Brirain
CRACK
GROWTH
GENADY P. CHEREPANOV Moscow Mining Institute. Moscow. USSR
HABIL HALMANOV Institute of Mechanical Problems. Academy of Sciences. Moscow. USSR Abstract-The theoretical study is advanced to describe quantitatively the crack propagation under cyclic loads. The main assumptions accepted are: (a) a crack grows by loading during every cycle, and (b) the specific dissipation energy is a material constant. The latter contains. the known concept by Irwin and Orowan as a limiting case of steady crack growth by monotonous loading. The theory offered does not depend on the details of micromechanisms of fatigue crack tip fracturing. Apart from fracture toughness K,. only one new material constant enters into the firmI expression for the fatigue crack growth velocity. An analysis of test dataconfirms this relationship within the error limits.
NOTATION
d A’
stress intensity factor m~imum and minimum stress intensity factors during a cycle, respectively fatigue crack extension rate fracture toughness new material constant yield stress Young’s modulus Poisson’s ratio crack length plastic zone size number of cycles intensity of tangential stresses mean intensity of pIastic deformations specific total plastic work plastic work for 6K = 0 piastic work for 61= 0 dimensioniess functions constants AK = Km,, -Km,.
R = K,,,IKm,.
1. I~ODUC~ON one hundred years ago Weler discovered the phenomenon of fatigue in metals, and subsequent researches revealed the cause leading to premature fracture of structures under cyclic loads, namely, slow crack extension. The last fifteen years have seen the rapid development of studies on the subcritical extension of fatigue cracks. After the papers [l-3] appeared. it became clear that the stress intensity factor following from Irwin’s linear fracture mechanics was the parameter which controls the subcritical crack growth rate. The works [4.51 and others cleared up the possibility of using the elastic-plastic continuum in such studies. In 1965, Paris [6J offered the 4th power relationship giving a good fit of test data for great variety of materials under not-too-high loads. It is of interest how Johnson and Paris[7]t commented ABOUT
tThis reviewing paper contains the most complete bibliography on fatigue crack growth prior to f968. 219
220
G. P. CHEREPANOV
and
H. HALcMANOV
this law three years later: “The fact that such a simple law can fit data from materials with such widely different microstructures is in itself a curiosity! Perhaps this means that the controlling mechanism of crack growth is similar for all of these materials which would certainly imply control above the microstructurai level.” The authors hope that this paper will represent one more justification of such a viewpoint. on
2. THEORETICAL CONSIDERATION A crack in an elastic body begins to extend only after the stress intensity factor K attains the value K,[8]. The propagation of cracks under the action of cyclic loads cannot be. evidently. explained on the basis of an elastic body model. Let us now consider the case of quasi-static propagation of cracks in an ideal elastic-plastic body under the effect of cyclic load which is given by some timeperiodic function[9. 101. In such cases, a plastic domain with a characteristic linear size d will be formed at the crack tip. We shall confine ourselves only to the case of ‘hne’ structure of the crack tip, i.e. when the plastic zone is small as compared to the characteristic linear dimension of the body (e.g. crack length. 0. Under that assumption the stress and strain distribution at distances from the crack tip, large as compared to d but small as compared to the characteristic linear dimension of the body. will depend on K . But the crack growth rate dlldN under cyclic loads may depend only on the values of K,,, and Kmin in one cycle, on the number of cycles M, on the total plastic work y* per unit surface and on the material constants. viz. Young’s modulus E, yield stress cr,, and Poisson’s ratio v. Moreover. N can be regarded as a continuous variable as its value is very large. Dimensional analysis gives dl
dN=z*
K’
(I)
where Y is a dimensionless function. In order to find out this dependence, we study the behavior of crack tip on increasing K from Km, to K,,, under the assumption that some deformation prehistory was preceding the initial state at K = K,i,, resulting in the residual (initial) stresses and strains. Energy equation
Let us consider the fine structure of a crack tip. The value of d will depend on K. v and on the prehistory. Let o0 denote the parameter of initial stresses which are responsible for the prehistory of loading and crack development co,, 4 v,). Dimensional analysis yields E. us,
(2)
where a, (us/E, u,/E, v) is a dimensionless The total plastic work 2y,,61 represents
function. the sum of two terms
Zy,Sl = 6W + aA,,.
The quantity yzi;represents
in essence the total energy dissipation rate.
(3)
211
Fatigue crack growth
The first term 6W’. equal to the liberated elastic energy, corresponds to K (and, consequently. to d. according to equation (2)) being constant when 1 increases by 81. In other words, 6W is the irreversible deformation work done by the plastic zone as it displaces in the direction of the crack extension as a rigid one (Fig. 1). Moreover. SW depends only on K. E. crsr 61. V. oO. The simple estimate is: (4) where CV.~ ( CT,/E. CT,,/E. v) is a dimensionless function. Equation (4) can be also obtained from dimensional analysis considerations. The second term &I, is the irreversible deformation work due to the increase in plastic region during loading and is not related to crack growth. Thereby. the length I being constant when K increases by 6K (Fig. 1). Moreover. &I, = TT,&S will depend
Fig. 1. The scheme of the plastic zone-crack development.
only on oS. crO.E, K. v, SK, where T is the intensity of tangential stresses. 6S, increment of the volume of the plastic region per unit thickness. TP. mean intensity of plastic deformations in 6s. As T depends only on oS and co. &S on 6d and TP on us. crO.E. V.the simple estimate leads to (5) Equation (5) can be also obtained from dimensional analysis considerations. Therefore, we have the following energy equation: (6) Equation (6) can be written also in a form resembling the formulation of an elastic-plastic body
of yielding law
According to equation (7). the quantity K will grow monotonically and tend asymptotically to some constant value K, when I increases only if K, is a finite value. That
222
G. P. CHEREPANOV
and H. HALMANOV
asymptotic r&gime corresponds to steady crack growth and obeys, as is seen, to the known energy concept by Irwin and Orowan. Fatigue crack extension rate Now we shall accept that the quantity yt and, consequently, K,:2 are material constants. For unsteady crack growth, this is a new concept modifying the energy concept for a steady case. Quantities cr,, E, K,. v will in general depend on the preliminary deformation. however, for the sake of simplicity we shall disregard such a dependence. One can neglect the dependence of (Yeand (w4on the second argument. This is firstly due to the fact that co =z gS and us/E is very small for all structural materials. Further the continuous functions aI (o,/E, co/E. VI and (Y~(u.JE. uo/E. Y) tend to some finite limits as r. + 0 corresponding to the monotonous loading at zero initial stresses. Thus, the function 13~in equation (7) can be considered constant. Let us introduce a new material constant p
Assuming that I does not vary in unloading from K,,, to K,i,, one integrates the equation (7) from K,,, to K,,,. The expression for increment Al in one loading cycle results in
Now by passing on to continuous
variables, we obtain (10)
This final law contains two constants /3 and K, which must be determined experimentally. If /3 and K, are known, one can easily find dl/dN by means of the diagram in Fig. 2 illustrating the formula ( 10). We shall accept that Kminis equal to zero in equation (10) if K,,,< 0 because the crack is closed up if compressed (except in the vicinity of the crack tip) and the stress concentration at the tip vanishes. The quantity p in equation (10) has the dimension of length, and it characterizes the crack length increment Al under cyclic loadings (it is equal roughly to Al when K increases from zero to K,.). In most cases, the magnitude K, is very close to the fracture toughness K,,,except for the cases when the fracture takes place during small number of loading cycles or for thin plates (in these cases K, can depend on the plate thickness and width). It should be noted that during monotonous ioading, the critical stress intensity factor corresponding to the beginning of unstable fracture is less than the asymptotic one which corresponds to the steady-state crack extension [ 10, 111. They coincide only for the case of very brittle fracture or. what is the same in essehce, for very large dimensions of a body.
223
Fatigue crack growth
Fig. 2. The diagram illustrating the theoretical law for fatigue crack extension rate.
J
b3 Crack
lO-2 growth
rote
IO-' dl/dN,
I
IO
mm/cycle
Fig. 3. Theoretical curves for various p at K,,, = 0
3. ANALYSIS
OF-TEST
DATA
In Fig. 3. the curves
are plotted
with the help of the formula
(10)for several values of /I and for K,i, = 0.
224
G. P. CHEREPANOV
and H. HALMANOV
This system of curves will be used further in determining p by the superposition method. Below, we shall consider only thost papers which give the results in the invariant variables df/dN. Kmin, K,,,. Fatigue cracks under multi-cyclic loads
Donaldson and Anderson[31 studied the relation between dl/dN and K basing on large amount of experimental data. The clearest correlation was obtained for aluminium alloys 2024-T3 and 7075T6. In most cases the experimental values of olnln/o,,,, varied approximately in the range 0. 2-O-O while the respective values of K,,,,/K,,,, would probably vary within the same range and this variation can be neglected by virtue of equation ( 10). in other words. we can take K,i, to be equal to zero. The averaged values of experimental data[31 are presented in Fig. 4. The dotted
i 10-j
IO-* Crack
Fig. 4. Averaged experimental
16’ growth
IO-rate
dl/dN,
IO-
,
‘9
73m/cycle
data[3] for aluminium alloy 2024-T3 and 7075-T6 and their theoretical curves.
curve relates to alloy 7075T6 and the dotted segments show the approximate scattering of points at K = const for every series of experiments and the corresponding confidence probability of 0.98. The dashed curve represents the case of alloy 2024-T3. and the scattering of points is shown by dashed segments at the same value of confidence probability. The solid curves on the same figure were drawn with the help of equation ( 10) at the following values of parameters: for alloy 7075-T6 p = O-15 mm. K,. = 90 kg/mm’;’ and for alloy 2024-T3 p = 0. IO mm. K,. = 90 kg/mmzz;‘. The papers by Paris [6] and Pearson[ 121 contain test data for a large number of alloys of aluminium. molybdenum. titanium and other metals. Unlike previous authors. they presented their results in the form of some analytical relations. Paris suggested the relationship -$-=
C,t&LK)‘(&LK = K,,,,-IV,,,)
\l I)
125
Fatigue crack growth
which approximates very well the experimental points of aluminium alloys 2024-T3 and 7075-T6 in the velocity interval ranging from lo+ to lo+ mm/cycle. For the same materials Pearson gives the following approximation: dl= dhT
C (4k’)“‘G L’
which he investigated in the velocity interval from 1O-’ to low3 mm/cycle. Probably, this interval is too small to derive any analytical expression. The Paris’ formula can be obtained directly from the equation (10) for Kmin = 0 at K,,,IJ?,. 2 O-5. i.e. when N is refatively great. what was precisely the case in the experiments considered. On substituting for the right hand side of equation (10) Taylor series, one gets
or, by neglecting higher order terms dl z=P
K&-K& K.,4
.
(14)
If Kmin = 0, then (15) The law f IO) exhibits good agreement in the range of interest also with the empirical three-constant Forman’s formula [ 131 (16) The results of such an illustrative comparison for C, = 2.13 X 1013, n = 3.21 in equation (16) and for p = 0.02 mm in equation f lo), are shown in Fig. 5. The values of K, and R are equal to 140 kg/mm3’* and 0.5, respectively. The values of constants in equation ( 16) were taken from [ 141. The maximum relative deviation of these curves does not exceed 20-30 per cent. It should be noted that in dependencies of such a kind one must evidently compare the less stationary variable. i.e. dlidN in the case considered. Fatigue cracks under low-cyclic loads
For the case of fatigue cracks under low-cyclic loads (i.e. when K is close to K,) the experimental points deviate essentially from the empirical curve (11) toward higher powers of K due to the increased role of the higher order terms in equation (IO). Carman an d Katlin I1 51 carried out their experiments with martensitic ageing steel samples 250 and 300 (in Fig. 6 the data of steel 250 are shown by dotted points, while that of steel 300 by triangles). The curves drawn with the help of (IO) at fi = O-2 mm, K, = 710 kg/mm3’? for steel 250 and at p = 0.09 mm and K, = 675 kglmm3’2 for steel JLf.M.
Vol. 4. No. 2-C
276
G. P. CHEREPANOV
I6
Id6
and H. HALMANOV
6’
Id’
Crack
growth
rate
lO-2
dl/dN,
10-l
I
mm/cycle
Fig. 5. Forman’s empirical and theoretical curves.
/ -5
IO
/I/I
/ -4
IO
/IL
,,,!
-3
IO
Crack
Fig. 6. Experimental
data[l5]
/
I/,,
-2
I,,!
-I
IO
IO
growth rate
dl/dN,
I
IO
mm/cycle
for steel 50 and 300. and their comparison curves.
with theoretical
300 are shown in Fig. 6 by solid lines. The cause of discrepancy between the theory and test data for steel 300 is unclear. The data obtained by Yangf16J on low-cyclic fatigue with some alloys of ~u~~nium and steel are shown in Fig. 7. He approximated his data using a power function:
-!&-= c,p
(I-?)
227
Fatigue crack growth
Theory
&I
Al 2024-T6
8 0 34mm KS=@2 %m ‘5
IO=,
f I,Omm
Kc=463
G
=40mm
Kc=700
Gm3$
IO”
lo’3
Crock growth
Fig. 7. Comparison
of theoretical
3
Id’ rate
dL/dN,
Id’
curves with experimental
data1161
2024-T6and steels 3 10 and 30 1.
AL 7079 A15456-H32l
id”
IO-'
IO+
Crock growth
Fig. 8. Averaged experimental
I
IO
mmfcycie
IO+ rate dl/dN,
for aluminiumalloy
T6 K,=125G+z Kc*163s
“$
IO--'
I
IO
mm/cycle
data[l7] for aluminium alloys 7075T6 and 5456-H321 and the respective theoretical curves.
with n = 5 for the aluminium alloy and n = 7 for the steels 3 10 and 30 1. On comparing these test data with the theoretical law, it was found that the equation (10) describes the data for the following values of parameters: p = 0.34 mm. K, = 192 kg/mm3’* for aluminium alloy. p= l+Omm.K,= 463 kg/mm3’* for steel 3 10. and p = 4.0 mm, K, = 700 kg/mm3’* for steel 30 1. According to Clark f 171. the crack rate dl/dN for alumin~um alloy 7079-T6 and Ni-MO-V steel ahoy is proportional to K3 if low5 mm/cycle SEd//dj\’ * lo-” mmlcycie
228
G. P. CHEREPANOV
1 -I---
0.8
and H. HAUIANOV
1 //
Theory Qee, HP9_4Pj
.-.
Steel
Ni-MO-Vst8el
1
id'
/
/
//!
/
/
01:
IO-*
,
lO-5 Crack
Fig.
9. Averaged experimental
///
,
IOgraWh
K,=460,;;(;;;KQ 5jz
HPQ-4-25
Kc = 520 srn f
/
,/I
/
)/,
10-I rate dl/dN,
I
I
i IO
mm/cycle
data[l7] for Ni-Mo-V steel alloy and alloy HP 9-2-25 and the corresponding theoretical curves.
(Figs. 8 and 9). in this interval his results are in agreement with the curves of (10) for: j3 = O-01 mm, K, = 125 kgjmm3’2 for ahrminium ahoy 7079fT6, p = 0.03 mm, K, = 520 kg/mm3/2 for Ni-MO-V steel alloy. Clark used a broken line (Figs. 8 and 9) with exponential index n = 2 at K/K, =Z0% and n = 5.2 at K/K, > 0.6 to approximate the data obtained for aluminium alloy 5456-H321, and the equation ( 10) describes these data at the following values of parameters: @= 0.03 mm, K, = 163 kglmm3J2 (continuous line in Fig. 8). The following exponential values for the steel HP-g-425 were also suggested[ 171: n = 26at K/K,e 043 andn = 9at K/K,> 0.8. The corresponding theoretical curve calculated from equation ( 10) at p = O-02 mm and Kc= 460kg/mm is shown in Fig. 9 by the solid line, On the contrary, Markochev [ 181 suggested an entirely different formula to describe this dependence basing on the data obtained for alloys D 16T. D 16T- 1, B-95 : &
-
Af exp (BK,,,).
(18)
His appro~mation curves are shown in Fig. 10. These data are in agreement with the theoretical curves obtained from equation ( 10) at the following values of constants: p = O-17 mm, K, = 200 kg/mm3/* for Al-ahoy D16T; p = 0.32 mm, K, = 200 kg/mm3’2 for Al-alloy D 16T- 1; p = 0.6 mm, K, = 200 kg/mm3/* for steel B-95.
The above investigation? makes it possible to draw the following conchzsions: (I) The fatigue crack growth is governed by the same mechanism for all materials t
Possibleenvironmental.
time and
other effects are omitted in this paper.
229
Fatigue crack growth
-
Theory
--
Al
--X-X--
AL D16T-1 L,eI
D16T
-----
Steel B-95
10--
IO -
Crock growth role dl/dN,
Fig. 10. Comparison
I
IO
mm/cycle
of theoretical curves with averaged experimental D16T. D16T-1 and B-95.
data[ 181 for alloys
manifesting plastic properties. This mechanism does not depend on the details of micromechanisms of fracture at crack tip (analogously with &-concept). (2) The fatigue crack extension is caused by the sub-critical stable growth of a crack during loading interval of a cycle. The sub-critical stable growth is inherent for all elastic-plastic bodies. (3) The analytical relationship obtained gives a satisfactory fit of test data for various rCgimes and materials. REFERENCES [I] Fi9Cli ps. [2]
M. P. Gomez and W. E. Anderson. A rational analytic theory of fatigue. Trend Engng 13.
A. J. McEvily. Jr. and W. lllg. The rate of crack propagation
in two aluminium alloys. NASA T-ech.
Nore No. 4394 (1958).
[3] D. R. Donaldson and W. E. Anderson, Crack propagation behavior of some airframe materials. Proc. Crack Propagation Symp.. Cranjeld. England (196 1). [4] F. A. McClintock, On the plasticity of the growth of fatigue cracks. In: Fracture of Solids. (E&ted
by D. C. Druckerand J. J. Gilman). Interscience. New York (1963). [5] J. R. Rice, Mathematical analysis in the mechanics of fracture. In: Fracture. Vol. II. Academic Press. New York (1968). [6] P. C. Paris. The fracture mechanics approach to fatigue. Proc. 10th Sagamore Conf.. p. 107. Syracuse University Press (1965). [71 H. H. Johnson and P. C. Paris, Sub-critical flow growth. Engng Fracture Me& 1.3 (1968). [8] G. R. Irwin. Fracture. In: Handbuch der Physik. Springer-Verlag. Bd. 6, p. 55 1. Berlin (1958). 191 G. P. Cherepanov, Cracksin solids. 1nt.J. Soiids Sfrucf. 4.811 (1968). [ 101 G. P. Cherepanov. Crack growth under cyclic loads. Priikladnayo Mechanika i 7echnicheskaya Phy-
sika 6.64 (1968). [I 11 G. P. Cherepanov.
On crack propagation in solids. Inr. J. Solids Srruct. 5.863 (1969). [12J S. Pearson. Fatigue crack propagation in metals. Nature, 211.1077 (1966). [I31 R. G. Fonnan, V. E. Keamey and R. M. Engle. Numerical analysis of crack propagation in cyclic loaded structures. ASME Paper No. 66-Met-4 (1966). I141 C. M. Hudson and J. T. Scardina. Effect of stress ratio on fatigue-crack growth in 7075-T6 aluminium alloy sheet. Engng Fracrure Mech. 1.429 (1968). [ 151 C. M. Carman, J. M. Katlin, Low cycle fatigue crack propagation characteristics of high strength steels. Trans.ASMED;.J.bas.Engng88.792(1966).
‘30
G. P. CHEREPANOV
and H. HALMANOV
[ 161 C. T. Yang. A study of the low-cyclic crack propagation. Truns. ASME D; J. bus. Engng 89.487 ( 19671. [ 171 W. G. Clark, Jr., Subcritical crack growth and its effect upon the fatigue characteristics of structureai alloys. Engng Fracture Me&. 1,385 (1968). 1181 V. M. Markochev, Methods and studies on kinetics of macrofractures in sheet materials under single or repeated loadings. Dissertation. All-Union Research Institute for Aviation Materials (1966) (in Russian). (Received 7 Ocrober 1970)
NOTE ADDED IN PROOF The above treated theory was fithilled by one of the authors in 1966- 1967. Subsequently, it was modified to account for effect of various environments and time. Some recent test data on high-strength steels disagree with the formula (IO). However. this fact is probably explained by the presence of moisture in air.
6rac1ure
MechanirJ.
1972. Vol. 4. pp. 219-230.
ON THE THEORY
Pergamon
Press.
Primed
OF FATIGUE
in Great
Brirain
CRACK
GROWTH
GENADY P. CHEREPANOV Moscow Mining Institute. Moscow. USSR
HABIL HALMANOV Institute of Mechanical Problems. Academy of Sciences. Moscow. USSR Abstract-The theoretical study is advanced to describe quantitatively the crack propagation under cyclic loads. The main assumptions accepted are: (a) a crack grows by loading during every cycle, and (b) the specific dissipation energy is a material constant. The latter contains. the known concept by Irwin and Orowan as a limiting case of steady crack growth by monotonous loading. The theory offered does not depend on the details of micromechanisms of fatigue crack tip fracturing. Apart from fracture toughness K,. only one new material constant enters into the firmI expression for the fatigue crack growth velocity. An analysis of test dataconfirms this relationship within the error limits.
NOTATION
d A’
stress intensity factor m~imum and minimum stress intensity factors during a cycle, respectively fatigue crack extension rate fracture toughness new material constant yield stress Young’s modulus Poisson’s ratio crack length plastic zone size number of cycles intensity of tangential stresses mean intensity of pIastic deformations specific total plastic work plastic work for 6K = 0 piastic work for 61= 0 dimensioniess functions constants AK = Km,, -Km,.
R = K,,,IKm,.
1. I~ODUC~ON one hundred years ago Weler discovered the phenomenon of fatigue in metals, and subsequent researches revealed the cause leading to premature fracture of structures under cyclic loads, namely, slow crack extension. The last fifteen years have seen the rapid development of studies on the subcritical extension of fatigue cracks. After the papers [l-3] appeared. it became clear that the stress intensity factor following from Irwin’s linear fracture mechanics was the parameter which controls the subcritical crack growth rate. The works [4.51 and others cleared up the possibility of using the elastic-plastic continuum in such studies. In 1965, Paris [6J offered the 4th power relationship giving a good fit of test data for great variety of materials under not-too-high loads. It is of interest how Johnson and Paris[7]t commented ABOUT
tThis reviewing paper contains the most complete bibliography on fatigue crack growth prior to f968. 219
220
G. P. CHEREPANOV
and
H. HALcMANOV
this law three years later: “The fact that such a simple law can fit data from materials with such widely different microstructures is in itself a curiosity! Perhaps this means that the controlling mechanism of crack growth is similar for all of these materials which would certainly imply control above the microstructurai level.” The authors hope that this paper will represent one more justification of such a viewpoint. on
2. THEORETICAL CONSIDERATION A crack in an elastic body begins to extend only after the stress intensity factor K attains the value K,[8]. The propagation of cracks under the action of cyclic loads cannot be. evidently. explained on the basis of an elastic body model. Let us now consider the case of quasi-static propagation of cracks in an ideal elastic-plastic body under the effect of cyclic load which is given by some timeperiodic function[9. 101. In such cases, a plastic domain with a characteristic linear size d will be formed at the crack tip. We shall confine ourselves only to the case of ‘hne’ structure of the crack tip, i.e. when the plastic zone is small as compared to the characteristic linear dimension of the body (e.g. crack length. 0. Under that assumption the stress and strain distribution at distances from the crack tip, large as compared to d but small as compared to the characteristic linear dimension of the body. will depend on K . But the crack growth rate dlldN under cyclic loads may depend only on the values of K,,, and Kmin in one cycle, on the number of cycles M, on the total plastic work y* per unit surface and on the material constants. viz. Young’s modulus E, yield stress cr,, and Poisson’s ratio v. Moreover. N can be regarded as a continuous variable as its value is very large. Dimensional analysis gives dl
dN=z*
K’
(I)
where Y is a dimensionless function. In order to find out this dependence, we study the behavior of crack tip on increasing K from Km, to K,,, under the assumption that some deformation prehistory was preceding the initial state at K = K,i,, resulting in the residual (initial) stresses and strains. Energy equation
Let us consider the fine structure of a crack tip. The value of d will depend on K. v and on the prehistory. Let o0 denote the parameter of initial stresses which are responsible for the prehistory of loading and crack development co,, 4 v,). Dimensional analysis yields E. us,
(2)
where a, (us/E, u,/E, v) is a dimensionless The total plastic work 2y,,61 represents
function. the sum of two terms
Zy,Sl = 6W + aA,,.
The quantity yzi;represents
in essence the total energy dissipation rate.
(3)
211
Fatigue crack growth
The first term 6W’. equal to the liberated elastic energy, corresponds to K (and, consequently. to d. according to equation (2)) being constant when 1 increases by 81. In other words, 6W is the irreversible deformation work done by the plastic zone as it displaces in the direction of the crack extension as a rigid one (Fig. 1). Moreover. SW depends only on K. E. crsr 61. V. oO. The simple estimate is: (4) where CV.~ ( CT,/E. CT,,/E. v) is a dimensionless function. Equation (4) can be also obtained from dimensional analysis considerations. The second term &I, is the irreversible deformation work due to the increase in plastic region during loading and is not related to crack growth. Thereby. the length I being constant when K increases by 6K (Fig. 1). Moreover. &I, = TT,&S will depend
Fig. 1. The scheme of the plastic zone-crack development.
only on oS. crO.E, K. v, SK, where T is the intensity of tangential stresses. 6S, increment of the volume of the plastic region per unit thickness. TP. mean intensity of plastic deformations in 6s. As T depends only on oS and co. &S on 6d and TP on us. crO.E. V.the simple estimate leads to (5) Equation (5) can be also obtained from dimensional analysis considerations. Therefore, we have the following energy equation: (6) Equation (6) can be written also in a form resembling the formulation of an elastic-plastic body
of yielding law
According to equation (7). the quantity K will grow monotonically and tend asymptotically to some constant value K, when I increases only if K, is a finite value. That
222
G. P. CHEREPANOV
and H. HALMANOV
asymptotic r&gime corresponds to steady crack growth and obeys, as is seen, to the known energy concept by Irwin and Orowan. Fatigue crack extension rate Now we shall accept that the quantity yt and, consequently, K,:2 are material constants. For unsteady crack growth, this is a new concept modifying the energy concept for a steady case. Quantities cr,, E, K,. v will in general depend on the preliminary deformation. however, for the sake of simplicity we shall disregard such a dependence. One can neglect the dependence of (Yeand (w4on the second argument. This is firstly due to the fact that co =z gS and us/E is very small for all structural materials. Further the continuous functions aI (o,/E, co/E. VI and (Y~(u.JE. uo/E. Y) tend to some finite limits as r. + 0 corresponding to the monotonous loading at zero initial stresses. Thus, the function 13~in equation (7) can be considered constant. Let us introduce a new material constant p
Assuming that I does not vary in unloading from K,,, to K,i,, one integrates the equation (7) from K,,, to K,,,. The expression for increment Al in one loading cycle results in
Now by passing on to continuous
variables, we obtain (10)
This final law contains two constants /3 and K, which must be determined experimentally. If /3 and K, are known, one can easily find dl/dN by means of the diagram in Fig. 2 illustrating the formula ( 10). We shall accept that Kminis equal to zero in equation (10) if K,,,< 0 because the crack is closed up if compressed (except in the vicinity of the crack tip) and the stress concentration at the tip vanishes. The quantity p in equation (10) has the dimension of length, and it characterizes the crack length increment Al under cyclic loadings (it is equal roughly to Al when K increases from zero to K,.). In most cases, the magnitude K, is very close to the fracture toughness K,,,except for the cases when the fracture takes place during small number of loading cycles or for thin plates (in these cases K, can depend on the plate thickness and width). It should be noted that during monotonous ioading, the critical stress intensity factor corresponding to the beginning of unstable fracture is less than the asymptotic one which corresponds to the steady-state crack extension [ 10, 111. They coincide only for the case of very brittle fracture or. what is the same in essehce, for very large dimensions of a body.
223
Fatigue crack growth
Fig. 2. The diagram illustrating the theoretical law for fatigue crack extension rate.
J
b3 Crack
lO-2 growth
rote
IO-' dl/dN,
I
IO
mm/cycle
Fig. 3. Theoretical curves for various p at K,,, = 0
3. ANALYSIS
OF-TEST
DATA
In Fig. 3. the curves
are plotted
with the help of the formula
(10)for several values of /I and for K,i, = 0.
224
G. P. CHEREPANOV
and H. HALMANOV
This system of curves will be used further in determining p by the superposition method. Below, we shall consider only thost papers which give the results in the invariant variables df/dN. Kmin, K,,,. Fatigue cracks under multi-cyclic loads
Donaldson and Anderson[31 studied the relation between dl/dN and K basing on large amount of experimental data. The clearest correlation was obtained for aluminium alloys 2024-T3 and 7075T6. In most cases the experimental values of olnln/o,,,, varied approximately in the range 0. 2-O-O while the respective values of K,,,,/K,,,, would probably vary within the same range and this variation can be neglected by virtue of equation ( 10). in other words. we can take K,i, to be equal to zero. The averaged values of experimental data[31 are presented in Fig. 4. The dotted
i 10-j
IO-* Crack
Fig. 4. Averaged experimental
16’ growth
IO-rate
dl/dN,
IO-
,
‘9
73m/cycle
data[3] for aluminium alloy 2024-T3 and 7075-T6 and their theoretical curves.
curve relates to alloy 7075T6 and the dotted segments show the approximate scattering of points at K = const for every series of experiments and the corresponding confidence probability of 0.98. The dashed curve represents the case of alloy 2024-T3. and the scattering of points is shown by dashed segments at the same value of confidence probability. The solid curves on the same figure were drawn with the help of equation ( 10) at the following values of parameters: for alloy 7075-T6 p = O-15 mm. K,. = 90 kg/mm’;’ and for alloy 2024-T3 p = 0. IO mm. K,. = 90 kg/mmzz;‘. The papers by Paris [6] and Pearson[ 121 contain test data for a large number of alloys of aluminium. molybdenum. titanium and other metals. Unlike previous authors. they presented their results in the form of some analytical relations. Paris suggested the relationship -$-=
C,t&LK)‘(&LK = K,,,,-IV,,,)
\l I)
125
Fatigue crack growth
which approximates very well the experimental points of aluminium alloys 2024-T3 and 7075-T6 in the velocity interval ranging from lo+ to lo+ mm/cycle. For the same materials Pearson gives the following approximation: dl= dhT
C (4k’)“‘G L’
which he investigated in the velocity interval from 1O-’ to low3 mm/cycle. Probably, this interval is too small to derive any analytical expression. The Paris’ formula can be obtained directly from the equation (10) for Kmin = 0 at K,,,IJ?,. 2 O-5. i.e. when N is refatively great. what was precisely the case in the experiments considered. On substituting for the right hand side of equation (10) Taylor series, one gets
or, by neglecting higher order terms dl z=P
K&-K& K.,4
.
(14)
If Kmin = 0, then (15) The law f IO) exhibits good agreement in the range of interest also with the empirical three-constant Forman’s formula [ 131 (16) The results of such an illustrative comparison for C, = 2.13 X 1013, n = 3.21 in equation (16) and for p = 0.02 mm in equation f lo), are shown in Fig. 5. The values of K, and R are equal to 140 kg/mm3’* and 0.5, respectively. The values of constants in equation ( 16) were taken from [ 141. The maximum relative deviation of these curves does not exceed 20-30 per cent. It should be noted that in dependencies of such a kind one must evidently compare the less stationary variable. i.e. dlidN in the case considered. Fatigue cracks under low-cyclic loads
For the case of fatigue cracks under low-cyclic loads (i.e. when K is close to K,) the experimental points deviate essentially from the empirical curve (11) toward higher powers of K due to the increased role of the higher order terms in equation (IO). Carman an d Katlin I1 51 carried out their experiments with martensitic ageing steel samples 250 and 300 (in Fig. 6 the data of steel 250 are shown by dotted points, while that of steel 300 by triangles). The curves drawn with the help of (IO) at fi = O-2 mm, K, = 710 kg/mm3’? for steel 250 and at p = 0.09 mm and K, = 675 kglmm3’2 for steel JLf.M.
Vol. 4. No. 2-C
276
G. P. CHEREPANOV
I6
Id6
and H. HALMANOV
6’
Id’
Crack
growth
rate
lO-2
dl/dN,
10-l
I
mm/cycle
Fig. 5. Forman’s empirical and theoretical curves.
/ -5
IO
/I/I
/ -4
IO
/IL
,,,!
-3
IO
Crack
Fig. 6. Experimental
data[l5]
/
I/,,
-2
I,,!
-I
IO
IO
growth rate
dl/dN,
I
IO
mm/cycle
for steel 50 and 300. and their comparison curves.
with theoretical
300 are shown in Fig. 6 by solid lines. The cause of discrepancy between the theory and test data for steel 300 is unclear. The data obtained by Yangf16J on low-cyclic fatigue with some alloys of ~u~~nium and steel are shown in Fig. 7. He approximated his data using a power function:
-!&-= c,p
(I-?)
227
Fatigue crack growth
Theory
&I
Al 2024-T6
8 0 34mm KS=@2 %m ‘5
IO=,
f I,Omm
Kc=463
G
=40mm
Kc=700
Gm3$
IO”
lo’3
Crock growth
Fig. 7. Comparison
of theoretical
3
Id’ rate
dL/dN,
Id’
curves with experimental
data1161
2024-T6and steels 3 10 and 30 1.
AL 7079 A15456-H32l
id”
IO-'
IO+
Crock growth
Fig. 8. Averaged experimental
I
IO
mmfcycie
IO+ rate dl/dN,
for aluminiumalloy
T6 K,=125G+z Kc*163s
“$
IO--'
I
IO
mm/cycle
data[l7] for aluminium alloys 7075T6 and 5456-H321 and the respective theoretical curves.
with n = 5 for the aluminium alloy and n = 7 for the steels 3 10 and 30 1. On comparing these test data with the theoretical law, it was found that the equation (10) describes the data for the following values of parameters: p = 0.34 mm. K, = 192 kg/mm3’* for aluminium alloy. p= l+Omm.K,= 463 kg/mm3’* for steel 3 10. and p = 4.0 mm, K, = 700 kg/mm3’* for steel 30 1. According to Clark f 171. the crack rate dl/dN for alumin~um alloy 7079-T6 and Ni-MO-V steel ahoy is proportional to K3 if low5 mm/cycle SEd//dj\’ * lo-” mmlcycie
228
G. P. CHEREPANOV
1 -I---
0.8
and H. HAUIANOV
1 //
Theory Qee, HP9_4Pj
.-.
Steel
Ni-MO-Vst8el
1
id'
/
/
//!
/
/
01:
IO-*
,
lO-5 Crack
Fig.
9. Averaged experimental
///
,
IOgraWh
K,=460,;;(;;;KQ 5jz
HPQ-4-25
Kc = 520 srn f
/
,/I
/
)/,
10-I rate dl/dN,
I
I
i IO
mm/cycle
data[l7] for Ni-Mo-V steel alloy and alloy HP 9-2-25 and the corresponding theoretical curves.
(Figs. 8 and 9). in this interval his results are in agreement with the curves of (10) for: j3 = O-01 mm, K, = 125 kgjmm3’2 for ahrminium ahoy 7079fT6, p = 0.03 mm, K, = 520 kg/mm3/2 for Ni-MO-V steel alloy. Clark used a broken line (Figs. 8 and 9) with exponential index n = 2 at K/K, =Z0% and n = 5.2 at K/K, > 0.6 to approximate the data obtained for aluminium alloy 5456-H321, and the equation ( 10) describes these data at the following values of parameters: @= 0.03 mm, K, = 163 kglmm3J2 (continuous line in Fig. 8). The following exponential values for the steel HP-g-425 were also suggested[ 171: n = 26at K/K,e 043 andn = 9at K/K,> 0.8. The corresponding theoretical curve calculated from equation ( 10) at p = O-02 mm and Kc= 460kg/mm is shown in Fig. 9 by the solid line, On the contrary, Markochev [ 181 suggested an entirely different formula to describe this dependence basing on the data obtained for alloys D 16T. D 16T- 1, B-95 : &
-
Af exp (BK,,,).
(18)
His appro~mation curves are shown in Fig. 10. These data are in agreement with the theoretical curves obtained from equation ( 10) at the following values of constants: p = O-17 mm, K, = 200 kg/mm3/* for Al-ahoy D16T; p = 0.32 mm, K, = 200 kg/mm3’2 for Al-alloy D 16T- 1; p = 0.6 mm, K, = 200 kg/mm3/* for steel B-95.
The above investigation? makes it possible to draw the following conchzsions: (I) The fatigue crack growth is governed by the same mechanism for all materials t
Possibleenvironmental.
time and
other effects are omitted in this paper.
229
Fatigue crack growth
-
Theory
--
Al
--X-X--
AL D16T-1 L,eI
D16T
-----
Steel B-95
10--
IO -
Crock growth role dl/dN,
Fig. 10. Comparison
I
IO
mm/cycle
of theoretical curves with averaged experimental D16T. D16T-1 and B-95.
data[ 181 for alloys
manifesting plastic properties. This mechanism does not depend on the details of micromechanisms of fracture at crack tip (analogously with &-concept). (2) The fatigue crack extension is caused by the sub-critical stable growth of a crack during loading interval of a cycle. The sub-critical stable growth is inherent for all elastic-plastic bodies. (3) The analytical relationship obtained gives a satisfactory fit of test data for various rCgimes and materials. REFERENCES [I] Fi9Cli ps. [2]
M. P. Gomez and W. E. Anderson. A rational analytic theory of fatigue. Trend Engng 13.
A. J. McEvily. Jr. and W. lllg. The rate of crack propagation
in two aluminium alloys. NASA T-ech.
Nore No. 4394 (1958).
[3] D. R. Donaldson and W. E. Anderson, Crack propagation behavior of some airframe materials. Proc. Crack Propagation Symp.. Cranjeld. England (196 1). [4] F. A. McClintock, On the plasticity of the growth of fatigue cracks. In: Fracture of Solids. (E&ted
by D. C. Druckerand J. J. Gilman). Interscience. New York (1963). [5] J. R. Rice, Mathematical analysis in the mechanics of fracture. In: Fracture. Vol. II. Academic Press. New York (1968). [6] P. C. Paris. The fracture mechanics approach to fatigue. Proc. 10th Sagamore Conf.. p. 107. Syracuse University Press (1965). [71 H. H. Johnson and P. C. Paris, Sub-critical flow growth. Engng Fracture Me& 1.3 (1968). [8] G. R. Irwin. Fracture. In: Handbuch der Physik. Springer-Verlag. Bd. 6, p. 55 1. Berlin (1958). 191 G. P. Cherepanov, Cracksin solids. 1nt.J. Soiids Sfrucf. 4.811 (1968). [ 101 G. P. Cherepanov. Crack growth under cyclic loads. Priikladnayo Mechanika i 7echnicheskaya Phy-
sika 6.64 (1968). [I 11 G. P. Cherepanov.
On crack propagation in solids. Inr. J. Solids Srruct. 5.863 (1969). [12J S. Pearson. Fatigue crack propagation in metals. Nature, 211.1077 (1966). [I31 R. G. Fonnan, V. E. Keamey and R. M. Engle. Numerical analysis of crack propagation in cyclic loaded structures. ASME Paper No. 66-Met-4 (1966). I141 C. M. Hudson and J. T. Scardina. Effect of stress ratio on fatigue-crack growth in 7075-T6 aluminium alloy sheet. Engng Fracrure Mech. 1.429 (1968). [ 151 C. M. Carman, J. M. Katlin, Low cycle fatigue crack propagation characteristics of high strength steels. Trans.ASMED;.J.bas.Engng88.792(1966).
‘30
G. P. CHEREPANOV
and H. HALMANOV
[ 161 C. T. Yang. A study of the low-cyclic crack propagation. Truns. ASME D; J. bus. Engng 89.487 ( 19671. [ 171 W. G. Clark, Jr., Subcritical crack growth and its effect upon the fatigue characteristics of structureai alloys. Engng Fracture Me&. 1,385 (1968). 1181 V. M. Markochev, Methods and studies on kinetics of macrofractures in sheet materials under single or repeated loadings. Dissertation. All-Union Research Institute for Aviation Materials (1966) (in Russian). (Received 7 Ocrober 1970)
NOTE ADDED IN PROOF The above treated theory was fithilled by one of the authors in 1966- 1967. Subsequently, it was modified to account for effect of various environments and time. Some recent test data on high-strength steels disagree with the formula (IO). However. this fact is probably explained by the presence of moisture in air.