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Plastic zones about fatigue cracks in metals
InrJ Fatigue11
No 2 (1989) pp 107-115
P l a s t i c z o n e s a b o u t f a t i g u e c r a c k s in metals G. Nicoletto
The plastic areas surrounding fatigue cracks produced by constant amplitude loading in various metallic materials have been studied by Moir6 interferometry. Two distinct plastic zones have been identified and their sizes in the direction perpendicular to the crack plane correlated with previous theoretical and experimental results. Differences have been addressed referring to specimen geometry and mechanical properties of materials. Moir6 interferometry has also been used to investigate the effect of overloads on plastic zone development. Once the monotonic and cyclic plastic zone evolutions were outlined, they favourably correlated with predictions according to F~hring/Seeger's fatigue crack growth model. Key words: fatigue crack growth; monotonic plastic zone; cyclic plastic zone; aluminium alloy; steels; Moir6 interferometry; overload; effective stress intensity factor range
Notation a B E
f ij K
gmax Kol
Ko~ gre~ AK AK~ Ni
r R S t T ui IV ct am 0tc ]3 ~,i t;th
Oo oc v
crack length non-dimensional specimen geometry dependent factor Young's modulus reference grating frequency dummy indices ( = 1,2) Mode I stress intensity factor maximum stress intensity factor overload stress intensity factor crack opening stress intensity factor residual stress intensity factor nominal stress intensity factor range effective stress intensity factor range Moir~ fringe order along xi-direction radial distance of plastic zone boundary from crack tip stress ratio ( = Smin/Srnax) far-field stress specimen thickness first non-singular stress term acting in the x t-direction in-plane displacement component in the xidirection specimen width non-dimensional plastic zone size coefficient for the x2-direction monotonic plastic zone size coefficient for the x2-direction cyclic plastic zone size coefficient for the x2-direction fatigue crack propagation parameter in-plane strain residual strain threshold monotonic yield stress cyclic yield stress Poisson's ratio
Fracture mechanics analysis of subcritical crack propagation is based on the concept that a parameter such as the stress intensity factor, K, derived from an elastic stress analysis of a stationary crack, describes the remote loading and geometry effects on the crack tip deformation even in the presence of small scale plasticity. 1 The semi-empirical approach based on the stress intensity factor range, AK, has proved itself highly successful in many practical applications and is now incorporated in design codes. Various plasticity-related aspects, however, have gained prominence in recent years and must be addressed to extend the range of applicability of the fracture mechanics methodology and its predictive capability in fatigue. The crack closure mechanism of long cracks, short crack vs long crack behaviour and similitude concepts used to predict actual component behaviour on the basis of laboratory tests and load interaction effects during variable amplitude loading are examples of responses which are heavily affected by the elasticplastic behaviour of materials. Theoretical and numerical modelling of the mechanics of cyclic crack growth is complex and needs experimental corroboration. The inelastic regions about a fatigue crack are also difficult to study experimentally because of their small sizes and severe strain concentrations in the presence of nonlinear material responses. In the last few years the present author has been developing and using Moir6 interferometry in the study of the mechanics of fatigue crack growth. 3-4 The technique, which represents one of the latest developments in experimental mechanics methods, visualizes contour maps of in-plane displacements with a sub-micron measuring sensitivity. Its geometrical response to deformation renders it suited also for investigation of material inelastic behaviour. Although various aspects of the mechanics of crack deformation can be investigated with this technique (k Ref. 5), the present paper examines the results of experiments performed on three materials (two steels and an Al-alloy) with the aim of assessing the nature and extent of the plastic
0142-11231891020107-09 S3.00 © 1989 Butterworth & Co (Publishers) Ltd Int J Fatigue March 1989
107
zones about fatigue cracks. The behaviour of the plastic zones as detected by Moire interferometry in an single overload experiment is also considered.
Plasticity about a fatigue crack Before presenting the experimental results relative to fatigue cracks, it is worth giving a brief synopsis of concepts and models of crack tip plasticity which have been derived from the analysis of a stationary crack in an elastic-plastic solid under monotonic loading. Because of its significance to subcritical propagation of long fatigue cracks, the definition of small-scale yielding is given. A small-scale yielding condition is defined by a cracked elastic-plastic body subjected to a monotonic load sufficiently low that the plastic zone at the crack tip is contained well within the region over which the elastic singularity fields dominate, t In this case and for Mode I loading, the stress intensity factor K is the field parameter which incorporates the effects of far-field loading, geometry and crack size and is expected to uniquely control the extent of crack tip yielding (see Fig. la). According to the small-scale yielding approximation for an elastic-perfectly-plastic material model the following nondimensional relationship can be established:
rl(Kl~oV
=
a
(ao/E, v)
(1)
The parameter 0t depends markedly on the stress state (ie plane strain usplane stress) and is assumed to have a negligible dependence on the mechanical properties of the material. Estimates of plastic zone shapes and sizes have been obtained theoretically for various material models, stress states and modes of loading. 6 Numerical modelling of both plane strain and plane stress conditions have also appeared in the past. 78 During cyclic loading, the assumption of proportional
plastic flow which is satisfactory for monotonic loading may not be valid. Nonetheless, in order to identify the parameters which are important during loading, unloading and reloading of a stationary crack as a simplified model of a fatigue crack, a plastic superposition method was used in Ref. 1. As schematically shown in Fig. lb, the development of two plastic regions characterized by cyclic straining with positive mean strain and reversed plastic deformation, respectively, and named monotonic and cyclic plastic zones (the latter is shaded in Fig. lb) was predicted. To estimate their respective extensions, results for the monotonic loading case were used. The basic conclusions of this simple model are that the plastic zone sizes, r, are related to the maximum stress intensity factor K.~. and to the stress intensity factor range, AK, as follows; r~/(Kmax/Go):
= am(oo/E,
v)
(2)
r'/(AK/oo)2 = ac(oc/E,v, R)
(3)
The superscripts m and c stand for monotonic and cyclic, respectively. Cyclic stresses, strains and displacements also depend only on AK and not on Kin,, for R >/ 0. For R = 0 and cyclic yield stress equal to 20"o, Rice 1 predicted a ratio of 1/4 between cyclic and monotonic plastic zone sizes. These theoretical studies have yielded valuable information on the nature and extent of material plasticity under monotonic and cyclic loading cases. In contrast to a stationary crack, however, a fatigue crack involves a subcritical propagation through an active plastic zone ahead of its tip which leaves behind a wake of inelastically deformed material as shown in Fig. lc. Cyclic loading conditions induce a residual stress distribution ahead of the crack tip, 9 while the wake of permanently deformed material behind the crack tip is responsible for the phenomenon of anticipated crack closure. 10 Identification of these mechanisms represents an attempt to rationalize the notion that even in fatigue under
S
Y
11/ 3(1
Xl
0
-I
S V Sm.x t/
sm,.
time, t
a
t
b
t
c
Fig. 1 Schematic representation of smell scale yielding for; (a) a monotonically loaded stationa~ crack; (b) a cyclically loaded stationary crack; and (c) a fatigue crack
108
Int J Fatigue M a r c h 1 9 8 9
constant amplitude loading the nominal AK imposed on the specimen may differ from the effective AK, AK,a, which contributes to crack extension. Definitions of AK, a depend on the mechanism assumed to be operative (ie AK, a = K~,~ - Kop where Kop defines the load level at which crack opening takes place or AK~ = Km~ - K~ where K ~ accounts for the previous cycling). When plastic zone sizes about a fatigue crack (Fig. lc) are measured by the experimental methods described here, Equations (2) and (3) derived for the situation shown in Fig. lb are not correct. The following modifications are therefore proposed to Equations (2) and (3): rm/(gmax/Oo) z = (lm(oo/E, V) r~/(AK/oo)"
= a ° (odE,
v, R) p(AK=dAK)
(4)
(5)
where AK~ is not explicitly related to a specific mechanism. Since, from Equation (1) plastic zone size is related to the square of K, ~ = (AK~a/AK)2 is a reasonable assumption. When fatigue crack growth is conducted at controlled amplitude loading and at low AK so that plane strain conditions prevail and phenomena such as crack closure may be neglected, then ~ m 1 in Equation (5) and theoretical estimates of a m and a c obtained for a cyclically loaded stationary crack (Fig. lb) may be used for correlating with experimental values.
Approach The approach of this paper is experimental and based on the use of Moir~ interferometry.3"4 Moir~ interferometry uses a high-frequency diffraction grating solidly attached to the specimen surface as the deformation-sensitive element and examined, upon load application, with two collimated beams of laser light impinging symmetrically at prescribed angles on the specimen surface according to the scheme in Fig. 2. Although in Fig. 2 a linear grating is considered for simplicity, crossed line gratings can be used and investigation for two perpendicular directions may be carried out. Moir6 interferometry is equivalent, as far as fringe interpretation
is concerned, to the Moir~ effect generated by the superposition of two sets of regularly spaced features such as lines or dots. It is characterized by a full-field geometrical response which is independent of the inelastic or anisotropic response of the material. In-plane displacement components, Hi, can be evaluated in the field of view according to the following simple relationships: u, = N~/f
i = 1,2
(6)
By using high frequency diffraction gratings, Moir6 interferometry reaches a sub-micron displacement sensitivity. Strain distributions, sii, can be indirectly determined by applying to the fringe patterns relative to two orthogonal directions the following finite-difference relationships ~j ~ (AglA,cj + A u / ~ I 2 = (! l a x i + I / A x , ) / 2 f
i,j = 1,2
(7)
where Ax~ are the distances in the i-direction between two neighbouring interference fringes. The experimental evidence relative to three metallic materials is examined; namely a commercial aluminium alloy (7075-T6), 4 Fe-3Si steel (Si:3.63, C:0.026, N:0.002, O:0.017 wt%) and a NiCrMoV steel (C:0.24, Ni:2.7, Cr:1.5, Mo:0.5, V: 0.1 wt %). The yield stress of these materials are Oo = 480 MPa, Oo = 460 MPa and Oo = 650 MPa, respectively. Young's modulus, E ~ 200 GPa for the two steels and E ~ 70 GPa for the Al-alloy. Compact-tension specimens were used (width = 25mm, thickness = 10ram) for the silicon steel whereas single-edge-notched-tension (width = 25mm, thickness = 5mm) were used for the NiCrMoV steel and the Al-alloy. The elastic-perfectly-plastic material model was representative of the uniaxial response of all the three materials. Fatigue crack propagations for R ~ 0.1 were conducted on a computer-controlled, servo-hydraulic testing machine. The Molt6 interferometric analyses required a vibration-free environment. Therefore, specimen loading was achieved with a rigid, screw-driven loading rig bolted onto a holographic table which also carried the optical system. 3 Fringe patterns were recorded on photographic film for various stages o f crack growth and load level and were subsequently examined under high magnification with a transmission profile projector.
Results and discussion Constant amplitude loading Determination of the plastic zones from fringe patterns rding
Fig. 2 Scheme for Moir6 inte~ferometry
Int J Fatigue March 1989
The possibility of using Moir~ interferometry for studying the fatigue plastic zones and determining the small-scale yielding parameters, ct, was first demonstrated in Ref. 4 on the 7075-T6 aluminium alloy. The plastic zones ahead of the crack tip and those in the wake of the crack schematically shown in Fig. lc were determined using different procedures. While the plastic zone size perpendicular to the crack plane can be rather easily defined, the plastic zone sizes ahead of the crack tip require a preliminary evaluation of the strain distributions. Inspection of the theoretical boundaries of the plastic zones in small-scale in yielding for an elastic-perfectly-plastic material obeying Von Mise's yield criterion shown in Fig. 311 reveals that the difference in size due to the stress state
100
0.4-
. . . .
ing regions and the kink in the fringes related to the boundary between cyclic and monotonic plastic zones.
Po Pe
% 0.2-
7075- 7"6 aluminium alloy \ I
I
012 ......
.i
I
0.'4
/1
X'I
Co
(K/°°) ~
Fig. 3 Size and shape of plastic zones in elestic-parfectly-plastic solid obeying Von Mises' yield criterion (P~ plane stress; P~ plane
strain) is significant ahead of the crack tip while it is apparently minor in the direction perpendicular to the crack plane. Therefore only the plastic zone size perpendicular to the crack plane (in the x2-direction ) will be considered in this paper since this measurement can be expected to be fairly representative of the through-the-thickness plastic behaviour. On the other hand, plastic zone sizes ahead of the crack tip would be drastically affected by the free-surface response. The typical experimental evidence of the method, namely interference fringe patterns depicting u2-displacements for no-load and live-load conditions for a controlled amplitude loading fatigue crack in NiCrMoV steel are shown in Figs 4a and 4b. The material plasticity left in the wake of the advancing crack is associated with the residual displacements shown in Fig. 4a. The extension of the monotonic plastic zone can be evaluated from fringe spacing in the x2-direction and upon definition of a threshold residual strain ~;,~ (Equation 7). The presence of the inner zone where reversed plasticity occurs can be more readily identified from live-load patterns such as that of Fig. 4b. It reveals that fringes approaching the crack surfaces undergo an abrupt change in their behaviour. This peculiar effect was attributed in Ref. 4 to the different strain histories of the two neighbour-
This material was the first to be investigated with Moir~ interferometry and the results have been discussed in detail. 4 Only the main conclusions relevant to the plastic zone extension in the x2-direction are reviewed here. Referring to the small-scale yielding approximation and to the non-dimensional parameters, a, defined in Fxluations (2) and (3), average values for the monotonic plastic zone were ctm = 0.91 for e,h = 0.001 and a m = 0.58 for e,h = 0.003; for the cvclic plastic zone a c = 0.2. The monotonic values correlated well with previous estimates of a m for the same material obtained using the electron channelling pattern analysis technique 12 and apparently confirmed the 4 to 1 ratio between monotonic and cyclic plastic zone sizes predicted by Rice 1 for R ~ 0. Nonetheless, they showed a marked disagreement with theoretical predictions for both plane stress and plane strain states such as those of Fig. 3.
NiCrMoV steel Similar experiments have been conducted on a NiCrMoV steel. From evidence such as that of Figs 4a and 4b, the extensions of the cyclic and of the monotonic plastic zones in the x2-direction were easily estimated at various locations behind the crack tip. The corresponding non-dimensional coefficients, 0t, where then computed for different crack lengths and are plotted in Fig. 5. These results refer to intermediate stress intensities (AK = 19-24 MPax/-~" and R = 0.1) and ~;,h = 0.003. Theoretical predictions of the plastic zone shape were originally obtained for Mode III loading. 6 The circular plastic zone ahead of the crack tip predicted for Mode III loading
Fig. 4 Typical Moir~ interferometric fringe patterns about a fatigue crack in NiCrMoV steel for: (a) no-load u= field; and (b) live-load u= field;
110
Int J Fatigue M a r c h 1 9 8 9
__(
Fe- 3Si steel
NiCrMoV Steel
- - o - - Monotonic
I ntermed late A K
- - O - - Cyclic
_(
- - o- - Monotonic
Fe-3Si Steel Low A K
~-
0.2
---0.
~
~0~
0 ~
~. ~0----
O--
---0
.j1/27r
__---
O . . . .
0 . . . .
"13 . . . . .
O ------
J:]
0
0.1
t ~
tt
=-
-
~1/8~ Crack length Fig. 5. Non-dimensional plastic zone size coefficients for two steels tested at different stress intensity ranges
and plane stress was generally confirmed by the corresponding Mode I case depicted in Fig. 3. Since measurements are in the x2-direction , the relevant non-dimensional parameter pertains to the radius of the circle. The theoretical plane stress estimate u m = 1/2x is therefore included in Fig. 5. If the model of Ref 1 is applied, u c = urn/4 = 1/87t for R ~ 0. This value is also introduced in Fig. 5. In contrast to the previous results for the Al-alloy4 the theoretical predictions are found to correlate favourably with experimental results for the NiCrMoV steel, both for monotonic and cyclic plastic zones. The ratio of the two plastic zone sizes in the x2-direction also agrees with the prediction of Rice's model. This suggests that the conditions selected for fatigue crack growth could be assessed with the simple cyclically loaded stationary crack model of Fig. lb.
The small-scale yielding approximation tends to the exact solution in the limit as the AK level tends to 0. The corresponding plastic zone size, however, may be gready reduced, thus increasing the experimental difficulties in visualizing it. Determination of the extent of plasticity remains important for the study of the mechanisms of crack growth at nearthreshold stress intensities at which interaction of plasticity and material microstructure plays a fundamental role. With the aim of evaluating the feasibility of revealing even minuscule plastic zones with Moir/~ interferometry, experiments were conducted on Fe-3Si steel tested at low AK levels. This material was selected because it was found to present well-defined plastic zones at intermediate AK. t3 In an etching study, the non-dimensional parameters of Equations (2) and (3) were found to be in good agreement with theoretical estimates. At low stress intensity levels (AK = 9-11 M P a ~ ) , the plasticity-induced residual displacements were so contained that almost no interference fringe could be observed in the field of view after unloading (see the no-load uzdisplacement field of Fig. 6a). From these fringe patterns it was difficult to determine even the extent of the monotonic plastic zone. To enhance its visibility the following optical approach was followed: a dominant fringe pattern given by a system of packed parallel fringes nearly perpendicular to the crack plane was introduced by rotation of one of the gratings. The extent of the monotonic plastic zone could be estimated by the amount of distortion of the otherwise rectilinear fringes as shown in Fig. 6b. From Fig. 6b, only the external (monotonic) zone could be detected with confidence. The appropriate non-dimensional coefficients of Equation (2) were determined at various locations behind the crack tip and are plotted in Fig. 5. The correlation with the theoretical prediction of Ref. 1 is also good.
Assessment
of constant
amplitude
Examination of previous experimental results reveals that the experimental estimates of the non-dimensional parameters
Fig. 6 Moir6 interferometric fringes about a fatigue crack in Fe-3Si =eel for: (a) no-load u= field; and (b) linear carrier ~ to no-load u= field
Int J Fatigue M a r c h 1 9 8 9
loading
auperpoaed
111
are not in close agreement among themselves or with theoretical predictions. The differences are relatively small between the two steels but are large between the steels and the AIalloy. The aim of this section is to address these differencies in the ct values on the basis of an assessment of the contributing factors. The results for the two steels will be considered first and discrepancies between observations in the steels and the M-alloy will be addressed subsequently. If consideration is made of the basic Equation (1), and since the two steels have similar mechanical properties (v and cs,,/E) and R ratios, the discrepancy of Fig. 5 suggests a specimen geometry-dependent effect. Equation (1) is derived from the assumption that for small-scale yielding conditions only the singular term in the stress field is actually contributing to crack tip yielding. After the numerical analyses of Ref. 14, and the subsequent assessment of small-scale yielding approximation of Rice is it has become evident that the first non-singular elastic stress term acting parallel to the crack plal.e plays a fundamental role in defining the shape and extension of the monotonic plastic zone. This non-singular stress term (also called T-term 15) depends on specimen geometry and crack length. Hence, Equation (1) is strictly valid for T = 0 (equibiaxially stressed sheet). A more general formulation of Equation (1) is rl(Kl(So) 2 = (z (Tlao, ~o/E, v)
(8)
Analogous modifications can also be carried over to Equations (2) and (3). While the actual effect on crack growth rate is still debated, crack tip stress biaxiality is expected to have a marked effect on plastic zone size, stress and strain within the plastic boundary for monotonic loading according to the analysis of Ref. 15. The inherent stress biaxiality at the crack tip of common uniaxially loaded specimen geometries has prompted the work of Leevers and Radon 16 who have performed numerical analyses of various fracture specimen geometries of different crack lengths. The behaviour of the non-dimensional parameter B defined as
/
c
0.5
0.25
112
a/W
°~'V z
-0.25
SENT
-0.5
a .... B = - 1.04 ~ B = 0 B = 0.52
0.2
/
/
\
f
\
/
l I I
/
/
%
ii
b
0
/I /
I
(9)
with normalized crack length for the single-edge-notchedtension and compact-tension specimen geometries are presented in Fig. 7a. The compact-tension geometry is characterized by positive T values while single-edge-notch-test specimens reveal a strong dependence on crack length and negative T values for relatively small a/W/ratios. The plastic zone shapes of the plane strain analyses of Ref. 14 are presented in Fig. 7b to evaluate the effect of the T-term on the normalized plastic zone sizes. The smoothed finite element results show a marked extension of the plastic zone, especially in the x2-direction, for negative T values and a contraction of its size for positive T values with respect to the condition of T = 0 implicitly assumed in Equations (1) to (3). The present observations can be applied only qualitatively to the Moirt~ results since the plane strain assumption is not appropriate for analysing free surface measurements. In view of the results presented in Fig. 3, however, it is believed that the trend shown in Fig. 7b may be significant even for the plane stress state, especially in the x2-direction. The results shown in Fig. 5 for the NiCrMoV steel were obtained on a single-edge-notched-tension geometry and for crack lengths which, from Fig. 7a, are characterized by high negative T values. From Fig. 7b negative T values are seen to yield larger plastic zones. Therefore, the non-
/
0.~5
0.1
B = Tx/-ffE/K
@-.--O"---@
//
!
-0.1
!
0.1
0 x1
( K/oo)2 Fig. 7 (a) Dependence of non-singular stre~ term factor B on specimen geometry and crack length for compact-tension (CT) and single-edge-notched tension (SENT) specimens; and (b) dependence of plane strain plastic zone shapes and sizes on non-singular stress term factor B
dimensional factor ¢tm for this case is expected to be larger than the theoretical value associated with T = 0, as observed in Fig. 5.When the results for the Fe-3Si steel are considered, an analogous assessment can be attempted. From Fig. 7a the compact-tension specimen geometry is characterized by positive non-singular stresses, which as shown in Fig. 7b result in a plastic zone size smaller than for the T = 0 condition. Therefore, the experimentally-based non-dimensional factor, a, is expected to be smaller than the theoretical value for T = 0 in accordance with the evidence of Fig. 5. These opposite specimen-geometry-dependent corrections of the results would result in a convergence of the Moir/~ interferometric value, a m, for the two steels, demonstrating their independence from the stress intensity level.
Int J Fatigue M a r c h 1 9 8 9
Although qualitative at this stage, upon correction of the results for specimen geometry an excellent agreement between the experimental results and the theoretical values (a m = 1/2~ and ctc -- 1/8~ in Fig. 5) is expected. As far as the results in the 7075-T6 aluminium alloy are concerned the assessment is somewhat more complex. The agreement between the independent results obtained by Lankford and Davidson tz with the electron channelling pattern analysis technique and the author 4 with Moir6 interferometry on the same aluminium alloy at similar stress intensities confirms the relevance of the observations. Since single-edge-notched-tension specimens were used in both studies, the T-term effect invoked previously can only partially explain the marked difference in o values with the theoretical estimates and the observations in steels. Going back to Equation (1), only the mechanical properties Oo/E and v of the material may be considered in the assessment of the Al-alloy results. While v = 0.3 for aluminium alloys and steels, the Oo/E ratios are rather different: 0.0067 for the aluminium alloy,0.0033 for the NiCrMoV steel and 0.0026 for the Fe-3Si steel. A plastic zone size measurement is dependent on the arbitrary selection of a residual strain threshold value, e,h, needed to define the plastic boundary. This selection is largely dependent on the technique used. In the measurements carried out so far, a value of e,h "~ 0.003 was assumed in accordance with previous studies. If the elastic-perfectly-plastic material model is considered, a characteristic strain is given by the yield strain, Oo/E. The value of a,h ,~ 0.1303 is approximately the value of Oo/E for a steel. Conversely, the same threshold value is about half of the Oo/E ratio for the aluminium alloy. To further clarify this effect, the residual displacements were re-examined and an a m vs ~,h relationship for the 7075-T6 aluminium alloy determined. In Fig. 8, the (/m value for ~;,h = oo/E can be extrapolated to a value between 0.2 and 0.3. In order to give a meaningful comparison with the theoretical predictions, this value should also be corrected to account for the specimen-geometry dependence of am. In the single-edge-notched-tension specimen, the characteristic negative T-stress tends to yield larger plastic zones. As a qualitative conclusion, once residual strain threshold and specimen geometry effects are considered, the corrected a m value tends to be in closer agreement with the theoretical value of 1/27t verified by the experiments in the two steels.
-.\
E
7075-T6
0.1
I
I
,
,
t
,
I
0.001
I
I
+th
I
I
,
II
I
O0
,
,
I
0.01
E Fig. 8 Dependence of non-dimensional plastic zone size coefficent. o, on residual strain threshold ¢= in 7076-T6 aluminium alloy
Int J Fatigue March 1989
Overload
experiment
For fatigue crack growth under variable amplitude loading conditions, so called load interaction effects, which are not accounted for by the linear damage summation hypothesis, become important. The classic example of a non-linear effect during variable amplitude loading is given by the growth retardation following the imposition of an overload in a constant amplitude loading sequence. Most prediction methods use plastic zone sizes as representative parameters of the extent of overload affected fatigue crack growth. An experimental investigation of the evolution of the plastic zones under such conditions is therefore important for an assessment of the physical relevance of these models. A study of the 7075-T6 aluminium alloy has already been reported, s Here, analogous results for the NiCrMoV steel are discussed. The loading history considered was extremely simple; a single tensile overload (K,~/K~ = 1.6) was applied during a constant amplitude loading crack propagation. The specimen was subsequendy removed from the testing rig and the no-load (residual) and live-load fringe patterns were recorded; they are presented in Fig. 6. The evolution of the monotonic plastic zone is readily determined from the no-load pattern. As demonstrated by the previous observations in this material for controlled amplitude loading conditions, the fringe distortions near the crack surfaces are associated with the boundary of the cyclic plastic zone. The evolution of the boundaries of the monotonic and cyclic plastic zones with crack length following overload application is shown on the top of Fig. 10a. Asymmetry about the crack plane of the plastic zone shapes confirms previous observations obtained with different techniques 17 and suggests an important role for mixed-mode effects during fatigue crack growth. The evolution of the plastic zone sizes with crack length is also included in Fig. 10a. The evolution of the cyclic plastic zone sizes is characterized by a gradual reduction to a minimum size immediately after overload application and by a subsequent progressive increase to approximately pre-overload size. In accordance with the observations of Ref. 5, its size is found to qualitatively change during crack propagation as the crack growth rate is expected to change after overload application. By assuming that ctc = 1/8~ in this material and that 13 = (AK,a/AK) 2, an estimate of AKa~ vs crack length can be obtained and is plotted on the bottom of Fig. 10a. An assessment of the experimental findings was attempted using an analytical fatigue crack growth model. The Ftihring-Seeger continuum mechanics model was selected from the analytical/numerical fatigue crack growth models and used for correlation purposes because stress and displacement variations during every consecutive half-cycle as well as evolution of crack tip plasticity during overload experiments have been reported in detail.m The FiihringSeeger model is based on the Dugdale-Barenblatt description of crack tip plasticity which was extended to cope with cyclic loading. It simulates crack growth by incremental material separation and stress redistribution under the assumption of an elastic-perfectly-plastic material behaviour and includes the closure phenomenon in the determination of the AK,~ contributing to crack growth. The evolution of the characteristic quantities during an overload experiment are shown in Fig. 10b. In contrast to Fig. 10a, the plastic zone sizes are relative to the xt-direction. Although the predicted evolutions of the monotonic and of the cyclic plastic zone sizes according to the model have a similar trend, it is the cyclic plastic zone size evolution that most accurately
113
Fig. 9 Moir~ intorferometric fringes about a fatigue crack in NiCrMoV steel grown under single overload conditions for: (a) no-load ua field; and (b) live-load u= field ~
~ "
/Monotonic
\
*~ i
!
- - - ~ Cyclic .
.
.
...---~.
Overload
---
'~
{/"
.
.
.
.
.
.
x1
~
.7.6-
~
.5-
N
.2
]
I
~
I
'
.s
~~
/ ~
Monotonic
.1"
aol
CL
{
I
40,
_~ 30,
a
g
ao,
7 a
(mm)
L I1~
C r a c k length ao=
Fig. 10 Qualitative comparison of: (a) Moir~ interferometric observations of plastic zone sizes during an overload expafiment; and (b) predictions according to the FOhring/Seeger model
reflects the evolution of AK~, based on the closure mechanism. The qualitative comparison of Moir~ interferometric observations and the behaviour predicted by the FiihringSeeger fatigue crack growth model which is predicted by inspection of Fig. 10 can be considered satisfactory. A difference between the two is, however, the high peak in AKaf following overload predicted by the Fiihring-Seeger model and not evident in the Moir~ interferometric results. The
114
response of the model can be explained by referring to the residual crack tip blunting phenomenon following overload application which prevents crack closure upon unloading. This accelerating phenomenon is physically countered by the inevitable resistance to crack tip resharpening. Since, in contrast to the Fiihring-Seeger fatigue crack model, no retarding mechanism is explicitly involved in the use of cyclic plastic zone size to predict AKa~ evolution, the Moir~ interferometric response after overload application is the more realistic. Verification of the assumption that the cyclic plastic zone size is directly related to AK~a is potentially important from the experimental point of view. In fact, it may be feasible to propagate a crack according to a variable amplitude loading history on a testing machine and to extract from a unique Moir~ interferometric pattern the evolution of AKCf,vs crack length. This information can be used subsequently for an assessment of predicted behaviour according to various crack growth models.
Conclusions
The plastic wakes surrounding fatigue cracks in an Al-alloy, a Fe-3Si steel and a NiCrMoV steel have been studied using high sensitivity Moir~ interferometry. Full field in-plane displacement patterns were used in determining the extent of the monotonic plastic zone. Non-dimensional plastic zone size parameters were evaluated and compared with theoretical predictions. Discrepancies between experimental and theoretical results have been addressed, introducing the effects of specimen geometry and the ~,,/E ratio. Finally, the change in the plastic zone size following a single overload application was determined and found to resemble the expected crack growth rate variation. Using certain assumptions, the experimental evolution of AK~ with
Int J Fatigue M a r c h 1 9 8 9
crack length has been determined. The Moir~ interfcrometrybased results qualitatively confirmed the physical relevance of Fiihring-Seeger's fatigue crack growth model.
10.
Elber, W. "Fatigue crack closure under cyclic tension', Engng Fract Mach 2 (1970) pp 37-45
11.
Hutchinson, J.W, "Fundamentals of the phenomenological theory of nonlinear fracture mechanics' Trens ASME J Appl MachSO (1983) pp 1042-1051
References
12.
Rice, J.R. "Machen;cs of crack tip deformation and extension by fatigue" Fatigue Crack Propagation, ASTM STP 415 (American Society for Testing and Materials, 1987) pp 2463O9
Dsvidson, D.L. end Lankford, J,'Fatigue crack tip plastic strain in high-strength aluminium alloys' Fatigue Frect Engng Mater Struct 3 4 (1980) pp 289-303
13.
2.
Kanninen, M.F. and Popelar, C.H. Advanced Fracture Mechanics (Oxford University Press, New York, 1985) p 498
Hahn, G.T., Hosglsnd, R.G. and Rosenfield. A.R. 'Local yielding attending fatigue crack growth' Metallurgical Trans 3 (1972) pp 1189-1202
14.
3.
Nicoletto, G. 'Experimental crack tip displacement analysis under smell scale yielding conditions' IntJ Fatigue8 2 (1986) pp 83-89
Laruon, S.G. and Cerllmon, A.J. 'Influence of nonsingular stress terms and specimen geometry on small-scale yielding at crack tips in elastic plastic materials' J Mech Phys So/idsZ! (1973) pp 263-277
4.
Nicoletto, G. 'Fatigue crack tip strains in 7075-T6 aluminium alloy' Fatigue Fract Engng Mater StructlO 1 (1 987) pp 37-49
15.
Rice, J.R. 'Limitations to the small-scale yielding approximation for crack tip plasticity' J Mech Phys Solids 22 (1973) pp 17-26
5.
Nicoletto, G. 'Fatigue crack tip mechanica in 7075-T6 aluminium alloy by high sensitivity field displacement measurements" Nonlinear Fracture Mechanics: Volume I - TimeDependent Fracture, ASTM STP 995 ed. A. Saxena at al (American Society for Testing and Materials, PA, 1989) pp 415--432
16.
Leevers. P.S. and Radon. J.C. 'Inherent stress biaxielity in various fracture specimen geometries' Int J Fracture 19 (1982) pp 311-322
17.
Lankford, J. and Davidson, D.L. 'Fatigue crack tip plesticity associated with overloads and subsequent cycling' Trans ASME J Engng Mater Techno198 1 (1976) pp 17-23
18.
F/ihring, H. 'Berechnung von elastisch-plestischen Beanspruchungsablfiufen in Dugdale-Risscheiben mit Rissufercontakt auf der Grundlage nichtlinearer Schwingungbruchmechanik' (Institut fiJr Statik und Stahlbau, TH Darmstadt, Heft 30, 1977)
1.
6.
Rice, J.R. 'Fracture: an advanced treatise' in H. Liebowitz (ed) Vol. I1: Mathematical Fundamentals (Academic Press, New York 1968) pp 192-314
7.
levy, N., Marcel, P.V., Otrtergren, W.J. end Rice, J.R. 'Small scale yielding near a crack in plane strain: a finite element analysis' IntJ FrectMach7 2 (1971) pp 143-156
8.
N a r N i m h e n , R. and Rosakis, A.J. 'A finite element analysis of small-scale yielding near a stationary crack under plane stress" SM Report 86-21 (California Institute of Tach, Pasadena, 1986)
9.
Klesnil, M. and Lukas, P. Fatigue of Metallic Materials (Elsevier, Amsterdam, 1980)
Int J Fatigue M a r c h 1989
Authors G. Nicoletto is with the Dipartimento di Ingegneria deUe Costruzioni Meccaniche, Universul, degli Studi di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy.
115
No 2 (1989) pp 107-115
P l a s t i c z o n e s a b o u t f a t i g u e c r a c k s in metals G. Nicoletto
The plastic areas surrounding fatigue cracks produced by constant amplitude loading in various metallic materials have been studied by Moir6 interferometry. Two distinct plastic zones have been identified and their sizes in the direction perpendicular to the crack plane correlated with previous theoretical and experimental results. Differences have been addressed referring to specimen geometry and mechanical properties of materials. Moir6 interferometry has also been used to investigate the effect of overloads on plastic zone development. Once the monotonic and cyclic plastic zone evolutions were outlined, they favourably correlated with predictions according to F~hring/Seeger's fatigue crack growth model. Key words: fatigue crack growth; monotonic plastic zone; cyclic plastic zone; aluminium alloy; steels; Moir6 interferometry; overload; effective stress intensity factor range
Notation a B E
f ij K
gmax Kol
Ko~ gre~ AK AK~ Ni
r R S t T ui IV ct am 0tc ]3 ~,i t;th
Oo oc v
crack length non-dimensional specimen geometry dependent factor Young's modulus reference grating frequency dummy indices ( = 1,2) Mode I stress intensity factor maximum stress intensity factor overload stress intensity factor crack opening stress intensity factor residual stress intensity factor nominal stress intensity factor range effective stress intensity factor range Moir~ fringe order along xi-direction radial distance of plastic zone boundary from crack tip stress ratio ( = Smin/Srnax) far-field stress specimen thickness first non-singular stress term acting in the x t-direction in-plane displacement component in the xidirection specimen width non-dimensional plastic zone size coefficient for the x2-direction monotonic plastic zone size coefficient for the x2-direction cyclic plastic zone size coefficient for the x2-direction fatigue crack propagation parameter in-plane strain residual strain threshold monotonic yield stress cyclic yield stress Poisson's ratio
Fracture mechanics analysis of subcritical crack propagation is based on the concept that a parameter such as the stress intensity factor, K, derived from an elastic stress analysis of a stationary crack, describes the remote loading and geometry effects on the crack tip deformation even in the presence of small scale plasticity. 1 The semi-empirical approach based on the stress intensity factor range, AK, has proved itself highly successful in many practical applications and is now incorporated in design codes. Various plasticity-related aspects, however, have gained prominence in recent years and must be addressed to extend the range of applicability of the fracture mechanics methodology and its predictive capability in fatigue. The crack closure mechanism of long cracks, short crack vs long crack behaviour and similitude concepts used to predict actual component behaviour on the basis of laboratory tests and load interaction effects during variable amplitude loading are examples of responses which are heavily affected by the elasticplastic behaviour of materials. Theoretical and numerical modelling of the mechanics of cyclic crack growth is complex and needs experimental corroboration. The inelastic regions about a fatigue crack are also difficult to study experimentally because of their small sizes and severe strain concentrations in the presence of nonlinear material responses. In the last few years the present author has been developing and using Moir6 interferometry in the study of the mechanics of fatigue crack growth. 3-4 The technique, which represents one of the latest developments in experimental mechanics methods, visualizes contour maps of in-plane displacements with a sub-micron measuring sensitivity. Its geometrical response to deformation renders it suited also for investigation of material inelastic behaviour. Although various aspects of the mechanics of crack deformation can be investigated with this technique (k Ref. 5), the present paper examines the results of experiments performed on three materials (two steels and an Al-alloy) with the aim of assessing the nature and extent of the plastic
0142-11231891020107-09 S3.00 © 1989 Butterworth & Co (Publishers) Ltd Int J Fatigue March 1989
107
zones about fatigue cracks. The behaviour of the plastic zones as detected by Moire interferometry in an single overload experiment is also considered.
Plasticity about a fatigue crack Before presenting the experimental results relative to fatigue cracks, it is worth giving a brief synopsis of concepts and models of crack tip plasticity which have been derived from the analysis of a stationary crack in an elastic-plastic solid under monotonic loading. Because of its significance to subcritical propagation of long fatigue cracks, the definition of small-scale yielding is given. A small-scale yielding condition is defined by a cracked elastic-plastic body subjected to a monotonic load sufficiently low that the plastic zone at the crack tip is contained well within the region over which the elastic singularity fields dominate, t In this case and for Mode I loading, the stress intensity factor K is the field parameter which incorporates the effects of far-field loading, geometry and crack size and is expected to uniquely control the extent of crack tip yielding (see Fig. la). According to the small-scale yielding approximation for an elastic-perfectly-plastic material model the following nondimensional relationship can be established:
rl(Kl~oV
=
a
(ao/E, v)
(1)
The parameter 0t depends markedly on the stress state (ie plane strain usplane stress) and is assumed to have a negligible dependence on the mechanical properties of the material. Estimates of plastic zone shapes and sizes have been obtained theoretically for various material models, stress states and modes of loading. 6 Numerical modelling of both plane strain and plane stress conditions have also appeared in the past. 78 During cyclic loading, the assumption of proportional
plastic flow which is satisfactory for monotonic loading may not be valid. Nonetheless, in order to identify the parameters which are important during loading, unloading and reloading of a stationary crack as a simplified model of a fatigue crack, a plastic superposition method was used in Ref. 1. As schematically shown in Fig. lb, the development of two plastic regions characterized by cyclic straining with positive mean strain and reversed plastic deformation, respectively, and named monotonic and cyclic plastic zones (the latter is shaded in Fig. lb) was predicted. To estimate their respective extensions, results for the monotonic loading case were used. The basic conclusions of this simple model are that the plastic zone sizes, r, are related to the maximum stress intensity factor K.~. and to the stress intensity factor range, AK, as follows; r~/(Kmax/Go):
= am(oo/E,
v)
(2)
r'/(AK/oo)2 = ac(oc/E,v, R)
(3)
The superscripts m and c stand for monotonic and cyclic, respectively. Cyclic stresses, strains and displacements also depend only on AK and not on Kin,, for R >/ 0. For R = 0 and cyclic yield stress equal to 20"o, Rice 1 predicted a ratio of 1/4 between cyclic and monotonic plastic zone sizes. These theoretical studies have yielded valuable information on the nature and extent of material plasticity under monotonic and cyclic loading cases. In contrast to a stationary crack, however, a fatigue crack involves a subcritical propagation through an active plastic zone ahead of its tip which leaves behind a wake of inelastically deformed material as shown in Fig. lc. Cyclic loading conditions induce a residual stress distribution ahead of the crack tip, 9 while the wake of permanently deformed material behind the crack tip is responsible for the phenomenon of anticipated crack closure. 10 Identification of these mechanisms represents an attempt to rationalize the notion that even in fatigue under
S
Y
11/ 3(1
Xl
0
-I
S V Sm.x t/
sm,.
time, t
a
t
b
t
c
Fig. 1 Schematic representation of smell scale yielding for; (a) a monotonically loaded stationa~ crack; (b) a cyclically loaded stationary crack; and (c) a fatigue crack
108
Int J Fatigue M a r c h 1 9 8 9
constant amplitude loading the nominal AK imposed on the specimen may differ from the effective AK, AK,a, which contributes to crack extension. Definitions of AK, a depend on the mechanism assumed to be operative (ie AK, a = K~,~ - Kop where Kop defines the load level at which crack opening takes place or AK~ = Km~ - K~ where K ~ accounts for the previous cycling). When plastic zone sizes about a fatigue crack (Fig. lc) are measured by the experimental methods described here, Equations (2) and (3) derived for the situation shown in Fig. lb are not correct. The following modifications are therefore proposed to Equations (2) and (3): rm/(gmax/Oo) z = (lm(oo/E, V) r~/(AK/oo)"
= a ° (odE,
v, R) p(AK=dAK)
(4)
(5)
where AK~ is not explicitly related to a specific mechanism. Since, from Equation (1) plastic zone size is related to the square of K, ~ = (AK~a/AK)2 is a reasonable assumption. When fatigue crack growth is conducted at controlled amplitude loading and at low AK so that plane strain conditions prevail and phenomena such as crack closure may be neglected, then ~ m 1 in Equation (5) and theoretical estimates of a m and a c obtained for a cyclically loaded stationary crack (Fig. lb) may be used for correlating with experimental values.
Approach The approach of this paper is experimental and based on the use of Moir~ interferometry.3"4 Moir~ interferometry uses a high-frequency diffraction grating solidly attached to the specimen surface as the deformation-sensitive element and examined, upon load application, with two collimated beams of laser light impinging symmetrically at prescribed angles on the specimen surface according to the scheme in Fig. 2. Although in Fig. 2 a linear grating is considered for simplicity, crossed line gratings can be used and investigation for two perpendicular directions may be carried out. Moir6 interferometry is equivalent, as far as fringe interpretation
is concerned, to the Moir~ effect generated by the superposition of two sets of regularly spaced features such as lines or dots. It is characterized by a full-field geometrical response which is independent of the inelastic or anisotropic response of the material. In-plane displacement components, Hi, can be evaluated in the field of view according to the following simple relationships: u, = N~/f
i = 1,2
(6)
By using high frequency diffraction gratings, Moir6 interferometry reaches a sub-micron displacement sensitivity. Strain distributions, sii, can be indirectly determined by applying to the fringe patterns relative to two orthogonal directions the following finite-difference relationships ~j ~ (AglA,cj + A u / ~ I 2 = (! l a x i + I / A x , ) / 2 f
i,j = 1,2
(7)
where Ax~ are the distances in the i-direction between two neighbouring interference fringes. The experimental evidence relative to three metallic materials is examined; namely a commercial aluminium alloy (7075-T6), 4 Fe-3Si steel (Si:3.63, C:0.026, N:0.002, O:0.017 wt%) and a NiCrMoV steel (C:0.24, Ni:2.7, Cr:1.5, Mo:0.5, V: 0.1 wt %). The yield stress of these materials are Oo = 480 MPa, Oo = 460 MPa and Oo = 650 MPa, respectively. Young's modulus, E ~ 200 GPa for the two steels and E ~ 70 GPa for the Al-alloy. Compact-tension specimens were used (width = 25mm, thickness = 10ram) for the silicon steel whereas single-edge-notched-tension (width = 25mm, thickness = 5mm) were used for the NiCrMoV steel and the Al-alloy. The elastic-perfectly-plastic material model was representative of the uniaxial response of all the three materials. Fatigue crack propagations for R ~ 0.1 were conducted on a computer-controlled, servo-hydraulic testing machine. The Molt6 interferometric analyses required a vibration-free environment. Therefore, specimen loading was achieved with a rigid, screw-driven loading rig bolted onto a holographic table which also carried the optical system. 3 Fringe patterns were recorded on photographic film for various stages o f crack growth and load level and were subsequently examined under high magnification with a transmission profile projector.
Results and discussion Constant amplitude loading Determination of the plastic zones from fringe patterns rding
Fig. 2 Scheme for Moir6 inte~ferometry
Int J Fatigue March 1989
The possibility of using Moir~ interferometry for studying the fatigue plastic zones and determining the small-scale yielding parameters, ct, was first demonstrated in Ref. 4 on the 7075-T6 aluminium alloy. The plastic zones ahead of the crack tip and those in the wake of the crack schematically shown in Fig. lc were determined using different procedures. While the plastic zone size perpendicular to the crack plane can be rather easily defined, the plastic zone sizes ahead of the crack tip require a preliminary evaluation of the strain distributions. Inspection of the theoretical boundaries of the plastic zones in small-scale in yielding for an elastic-perfectly-plastic material obeying Von Mise's yield criterion shown in Fig. 311 reveals that the difference in size due to the stress state
100
0.4-
. . . .
ing regions and the kink in the fringes related to the boundary between cyclic and monotonic plastic zones.
Po Pe
% 0.2-
7075- 7"6 aluminium alloy \ I
I
012 ......
.i
I
0.'4
/1
X'I
Co
(K/°°) ~
Fig. 3 Size and shape of plastic zones in elestic-parfectly-plastic solid obeying Von Mises' yield criterion (P~ plane stress; P~ plane
strain) is significant ahead of the crack tip while it is apparently minor in the direction perpendicular to the crack plane. Therefore only the plastic zone size perpendicular to the crack plane (in the x2-direction ) will be considered in this paper since this measurement can be expected to be fairly representative of the through-the-thickness plastic behaviour. On the other hand, plastic zone sizes ahead of the crack tip would be drastically affected by the free-surface response. The typical experimental evidence of the method, namely interference fringe patterns depicting u2-displacements for no-load and live-load conditions for a controlled amplitude loading fatigue crack in NiCrMoV steel are shown in Figs 4a and 4b. The material plasticity left in the wake of the advancing crack is associated with the residual displacements shown in Fig. 4a. The extension of the monotonic plastic zone can be evaluated from fringe spacing in the x2-direction and upon definition of a threshold residual strain ~;,~ (Equation 7). The presence of the inner zone where reversed plasticity occurs can be more readily identified from live-load patterns such as that of Fig. 4b. It reveals that fringes approaching the crack surfaces undergo an abrupt change in their behaviour. This peculiar effect was attributed in Ref. 4 to the different strain histories of the two neighbour-
This material was the first to be investigated with Moir~ interferometry and the results have been discussed in detail. 4 Only the main conclusions relevant to the plastic zone extension in the x2-direction are reviewed here. Referring to the small-scale yielding approximation and to the non-dimensional parameters, a, defined in Fxluations (2) and (3), average values for the monotonic plastic zone were ctm = 0.91 for e,h = 0.001 and a m = 0.58 for e,h = 0.003; for the cvclic plastic zone a c = 0.2. The monotonic values correlated well with previous estimates of a m for the same material obtained using the electron channelling pattern analysis technique 12 and apparently confirmed the 4 to 1 ratio between monotonic and cyclic plastic zone sizes predicted by Rice 1 for R ~ 0. Nonetheless, they showed a marked disagreement with theoretical predictions for both plane stress and plane strain states such as those of Fig. 3.
NiCrMoV steel Similar experiments have been conducted on a NiCrMoV steel. From evidence such as that of Figs 4a and 4b, the extensions of the cyclic and of the monotonic plastic zones in the x2-direction were easily estimated at various locations behind the crack tip. The corresponding non-dimensional coefficients, 0t, where then computed for different crack lengths and are plotted in Fig. 5. These results refer to intermediate stress intensities (AK = 19-24 MPax/-~" and R = 0.1) and ~;,h = 0.003. Theoretical predictions of the plastic zone shape were originally obtained for Mode III loading. 6 The circular plastic zone ahead of the crack tip predicted for Mode III loading
Fig. 4 Typical Moir~ interferometric fringe patterns about a fatigue crack in NiCrMoV steel for: (a) no-load u= field; and (b) live-load u= field;
110
Int J Fatigue M a r c h 1 9 8 9
__(
Fe- 3Si steel
NiCrMoV Steel
- - o - - Monotonic
I ntermed late A K
- - O - - Cyclic
_(
- - o- - Monotonic
Fe-3Si Steel Low A K
~-
0.2
---0.
~
~0~
0 ~
~. ~0----
O--
---0
.j1/27r
__---
O . . . .
0 . . . .
"13 . . . . .
O ------
J:]
0
0.1
t ~
tt
=-
-
~1/8~ Crack length Fig. 5. Non-dimensional plastic zone size coefficients for two steels tested at different stress intensity ranges
and plane stress was generally confirmed by the corresponding Mode I case depicted in Fig. 3. Since measurements are in the x2-direction , the relevant non-dimensional parameter pertains to the radius of the circle. The theoretical plane stress estimate u m = 1/2x is therefore included in Fig. 5. If the model of Ref 1 is applied, u c = urn/4 = 1/87t for R ~ 0. This value is also introduced in Fig. 5. In contrast to the previous results for the Al-alloy4 the theoretical predictions are found to correlate favourably with experimental results for the NiCrMoV steel, both for monotonic and cyclic plastic zones. The ratio of the two plastic zone sizes in the x2-direction also agrees with the prediction of Rice's model. This suggests that the conditions selected for fatigue crack growth could be assessed with the simple cyclically loaded stationary crack model of Fig. lb.
The small-scale yielding approximation tends to the exact solution in the limit as the AK level tends to 0. The corresponding plastic zone size, however, may be gready reduced, thus increasing the experimental difficulties in visualizing it. Determination of the extent of plasticity remains important for the study of the mechanisms of crack growth at nearthreshold stress intensities at which interaction of plasticity and material microstructure plays a fundamental role. With the aim of evaluating the feasibility of revealing even minuscule plastic zones with Moir/~ interferometry, experiments were conducted on Fe-3Si steel tested at low AK levels. This material was selected because it was found to present well-defined plastic zones at intermediate AK. t3 In an etching study, the non-dimensional parameters of Equations (2) and (3) were found to be in good agreement with theoretical estimates. At low stress intensity levels (AK = 9-11 M P a ~ ) , the plasticity-induced residual displacements were so contained that almost no interference fringe could be observed in the field of view after unloading (see the no-load uzdisplacement field of Fig. 6a). From these fringe patterns it was difficult to determine even the extent of the monotonic plastic zone. To enhance its visibility the following optical approach was followed: a dominant fringe pattern given by a system of packed parallel fringes nearly perpendicular to the crack plane was introduced by rotation of one of the gratings. The extent of the monotonic plastic zone could be estimated by the amount of distortion of the otherwise rectilinear fringes as shown in Fig. 6b. From Fig. 6b, only the external (monotonic) zone could be detected with confidence. The appropriate non-dimensional coefficients of Equation (2) were determined at various locations behind the crack tip and are plotted in Fig. 5. The correlation with the theoretical prediction of Ref. 1 is also good.
Assessment
of constant
amplitude
Examination of previous experimental results reveals that the experimental estimates of the non-dimensional parameters
Fig. 6 Moir6 interferometric fringes about a fatigue crack in Fe-3Si =eel for: (a) no-load u= field; and (b) linear carrier ~ to no-load u= field
Int J Fatigue M a r c h 1 9 8 9
loading
auperpoaed
111
are not in close agreement among themselves or with theoretical predictions. The differences are relatively small between the two steels but are large between the steels and the AIalloy. The aim of this section is to address these differencies in the ct values on the basis of an assessment of the contributing factors. The results for the two steels will be considered first and discrepancies between observations in the steels and the M-alloy will be addressed subsequently. If consideration is made of the basic Equation (1), and since the two steels have similar mechanical properties (v and cs,,/E) and R ratios, the discrepancy of Fig. 5 suggests a specimen geometry-dependent effect. Equation (1) is derived from the assumption that for small-scale yielding conditions only the singular term in the stress field is actually contributing to crack tip yielding. After the numerical analyses of Ref. 14, and the subsequent assessment of small-scale yielding approximation of Rice is it has become evident that the first non-singular elastic stress term acting parallel to the crack plal.e plays a fundamental role in defining the shape and extension of the monotonic plastic zone. This non-singular stress term (also called T-term 15) depends on specimen geometry and crack length. Hence, Equation (1) is strictly valid for T = 0 (equibiaxially stressed sheet). A more general formulation of Equation (1) is rl(Kl(So) 2 = (z (Tlao, ~o/E, v)
(8)
Analogous modifications can also be carried over to Equations (2) and (3). While the actual effect on crack growth rate is still debated, crack tip stress biaxiality is expected to have a marked effect on plastic zone size, stress and strain within the plastic boundary for monotonic loading according to the analysis of Ref. 15. The inherent stress biaxiality at the crack tip of common uniaxially loaded specimen geometries has prompted the work of Leevers and Radon 16 who have performed numerical analyses of various fracture specimen geometries of different crack lengths. The behaviour of the non-dimensional parameter B defined as
/
c
0.5
0.25
112
a/W
°~'V z
-0.25
SENT
-0.5
a .... B = - 1.04 ~ B = 0 B = 0.52
0.2
/
/
\
f
\
/
l I I
/
/
%
ii
b
0
/I /
I
(9)
with normalized crack length for the single-edge-notchedtension and compact-tension specimen geometries are presented in Fig. 7a. The compact-tension geometry is characterized by positive T values while single-edge-notch-test specimens reveal a strong dependence on crack length and negative T values for relatively small a/W/ratios. The plastic zone shapes of the plane strain analyses of Ref. 14 are presented in Fig. 7b to evaluate the effect of the T-term on the normalized plastic zone sizes. The smoothed finite element results show a marked extension of the plastic zone, especially in the x2-direction, for negative T values and a contraction of its size for positive T values with respect to the condition of T = 0 implicitly assumed in Equations (1) to (3). The present observations can be applied only qualitatively to the Moirt~ results since the plane strain assumption is not appropriate for analysing free surface measurements. In view of the results presented in Fig. 3, however, it is believed that the trend shown in Fig. 7b may be significant even for the plane stress state, especially in the x2-direction. The results shown in Fig. 5 for the NiCrMoV steel were obtained on a single-edge-notched-tension geometry and for crack lengths which, from Fig. 7a, are characterized by high negative T values. From Fig. 7b negative T values are seen to yield larger plastic zones. Therefore, the non-
/
0.~5
0.1
B = Tx/-ffE/K
@-.--O"---@
//
!
-0.1
!
0.1
0 x1
( K/oo)2 Fig. 7 (a) Dependence of non-singular stre~ term factor B on specimen geometry and crack length for compact-tension (CT) and single-edge-notched tension (SENT) specimens; and (b) dependence of plane strain plastic zone shapes and sizes on non-singular stress term factor B
dimensional factor ¢tm for this case is expected to be larger than the theoretical value associated with T = 0, as observed in Fig. 5.When the results for the Fe-3Si steel are considered, an analogous assessment can be attempted. From Fig. 7a the compact-tension specimen geometry is characterized by positive non-singular stresses, which as shown in Fig. 7b result in a plastic zone size smaller than for the T = 0 condition. Therefore, the experimentally-based non-dimensional factor, a, is expected to be smaller than the theoretical value for T = 0 in accordance with the evidence of Fig. 5. These opposite specimen-geometry-dependent corrections of the results would result in a convergence of the Moir/~ interferometric value, a m, for the two steels, demonstrating their independence from the stress intensity level.
Int J Fatigue M a r c h 1 9 8 9
Although qualitative at this stage, upon correction of the results for specimen geometry an excellent agreement between the experimental results and the theoretical values (a m = 1/2~ and ctc -- 1/8~ in Fig. 5) is expected. As far as the results in the 7075-T6 aluminium alloy are concerned the assessment is somewhat more complex. The agreement between the independent results obtained by Lankford and Davidson tz with the electron channelling pattern analysis technique and the author 4 with Moir6 interferometry on the same aluminium alloy at similar stress intensities confirms the relevance of the observations. Since single-edge-notched-tension specimens were used in both studies, the T-term effect invoked previously can only partially explain the marked difference in o values with the theoretical estimates and the observations in steels. Going back to Equation (1), only the mechanical properties Oo/E and v of the material may be considered in the assessment of the Al-alloy results. While v = 0.3 for aluminium alloys and steels, the Oo/E ratios are rather different: 0.0067 for the aluminium alloy,0.0033 for the NiCrMoV steel and 0.0026 for the Fe-3Si steel. A plastic zone size measurement is dependent on the arbitrary selection of a residual strain threshold value, e,h, needed to define the plastic boundary. This selection is largely dependent on the technique used. In the measurements carried out so far, a value of e,h "~ 0.003 was assumed in accordance with previous studies. If the elastic-perfectly-plastic material model is considered, a characteristic strain is given by the yield strain, Oo/E. The value of a,h ,~ 0.1303 is approximately the value of Oo/E for a steel. Conversely, the same threshold value is about half of the Oo/E ratio for the aluminium alloy. To further clarify this effect, the residual displacements were re-examined and an a m vs ~,h relationship for the 7075-T6 aluminium alloy determined. In Fig. 8, the (/m value for ~;,h = oo/E can be extrapolated to a value between 0.2 and 0.3. In order to give a meaningful comparison with the theoretical predictions, this value should also be corrected to account for the specimen-geometry dependence of am. In the single-edge-notched-tension specimen, the characteristic negative T-stress tends to yield larger plastic zones. As a qualitative conclusion, once residual strain threshold and specimen geometry effects are considered, the corrected a m value tends to be in closer agreement with the theoretical value of 1/27t verified by the experiments in the two steels.
-.\
E
7075-T6
0.1
I
I
,
,
t
,
I
0.001
I
I
+th
I
I
,
II
I
O0
,
,
I
0.01
E Fig. 8 Dependence of non-dimensional plastic zone size coefficent. o, on residual strain threshold ¢= in 7076-T6 aluminium alloy
Int J Fatigue March 1989
Overload
experiment
For fatigue crack growth under variable amplitude loading conditions, so called load interaction effects, which are not accounted for by the linear damage summation hypothesis, become important. The classic example of a non-linear effect during variable amplitude loading is given by the growth retardation following the imposition of an overload in a constant amplitude loading sequence. Most prediction methods use plastic zone sizes as representative parameters of the extent of overload affected fatigue crack growth. An experimental investigation of the evolution of the plastic zones under such conditions is therefore important for an assessment of the physical relevance of these models. A study of the 7075-T6 aluminium alloy has already been reported, s Here, analogous results for the NiCrMoV steel are discussed. The loading history considered was extremely simple; a single tensile overload (K,~/K~ = 1.6) was applied during a constant amplitude loading crack propagation. The specimen was subsequendy removed from the testing rig and the no-load (residual) and live-load fringe patterns were recorded; they are presented in Fig. 6. The evolution of the monotonic plastic zone is readily determined from the no-load pattern. As demonstrated by the previous observations in this material for controlled amplitude loading conditions, the fringe distortions near the crack surfaces are associated with the boundary of the cyclic plastic zone. The evolution of the boundaries of the monotonic and cyclic plastic zones with crack length following overload application is shown on the top of Fig. 10a. Asymmetry about the crack plane of the plastic zone shapes confirms previous observations obtained with different techniques 17 and suggests an important role for mixed-mode effects during fatigue crack growth. The evolution of the plastic zone sizes with crack length is also included in Fig. 10a. The evolution of the cyclic plastic zone sizes is characterized by a gradual reduction to a minimum size immediately after overload application and by a subsequent progressive increase to approximately pre-overload size. In accordance with the observations of Ref. 5, its size is found to qualitatively change during crack propagation as the crack growth rate is expected to change after overload application. By assuming that ctc = 1/8~ in this material and that 13 = (AK,a/AK) 2, an estimate of AKa~ vs crack length can be obtained and is plotted on the bottom of Fig. 10a. An assessment of the experimental findings was attempted using an analytical fatigue crack growth model. The Ftihring-Seeger continuum mechanics model was selected from the analytical/numerical fatigue crack growth models and used for correlation purposes because stress and displacement variations during every consecutive half-cycle as well as evolution of crack tip plasticity during overload experiments have been reported in detail.m The FiihringSeeger model is based on the Dugdale-Barenblatt description of crack tip plasticity which was extended to cope with cyclic loading. It simulates crack growth by incremental material separation and stress redistribution under the assumption of an elastic-perfectly-plastic material behaviour and includes the closure phenomenon in the determination of the AK,~ contributing to crack growth. The evolution of the characteristic quantities during an overload experiment are shown in Fig. 10b. In contrast to Fig. 10a, the plastic zone sizes are relative to the xt-direction. Although the predicted evolutions of the monotonic and of the cyclic plastic zone sizes according to the model have a similar trend, it is the cyclic plastic zone size evolution that most accurately
113
Fig. 9 Moir~ intorferometric fringes about a fatigue crack in NiCrMoV steel grown under single overload conditions for: (a) no-load ua field; and (b) live-load u= field ~
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Fig. 10 Qualitative comparison of: (a) Moir~ interferometric observations of plastic zone sizes during an overload expafiment; and (b) predictions according to the FOhring/Seeger model
reflects the evolution of AK~, based on the closure mechanism. The qualitative comparison of Moir~ interferometric observations and the behaviour predicted by the FiihringSeeger fatigue crack growth model which is predicted by inspection of Fig. 10 can be considered satisfactory. A difference between the two is, however, the high peak in AKaf following overload predicted by the Fiihring-Seeger model and not evident in the Moir~ interferometric results. The
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response of the model can be explained by referring to the residual crack tip blunting phenomenon following overload application which prevents crack closure upon unloading. This accelerating phenomenon is physically countered by the inevitable resistance to crack tip resharpening. Since, in contrast to the Fiihring-Seeger fatigue crack model, no retarding mechanism is explicitly involved in the use of cyclic plastic zone size to predict AKa~ evolution, the Moir~ interferometric response after overload application is the more realistic. Verification of the assumption that the cyclic plastic zone size is directly related to AK~a is potentially important from the experimental point of view. In fact, it may be feasible to propagate a crack according to a variable amplitude loading history on a testing machine and to extract from a unique Moir~ interferometric pattern the evolution of AKCf,vs crack length. This information can be used subsequently for an assessment of predicted behaviour according to various crack growth models.
Conclusions
The plastic wakes surrounding fatigue cracks in an Al-alloy, a Fe-3Si steel and a NiCrMoV steel have been studied using high sensitivity Moir~ interferometry. Full field in-plane displacement patterns were used in determining the extent of the monotonic plastic zone. Non-dimensional plastic zone size parameters were evaluated and compared with theoretical predictions. Discrepancies between experimental and theoretical results have been addressed, introducing the effects of specimen geometry and the ~,,/E ratio. Finally, the change in the plastic zone size following a single overload application was determined and found to resemble the expected crack growth rate variation. Using certain assumptions, the experimental evolution of AK~ with
Int J Fatigue M a r c h 1 9 8 9
crack length has been determined. The Moir~ interfcrometrybased results qualitatively confirmed the physical relevance of Fiihring-Seeger's fatigue crack growth model.
10.
Elber, W. "Fatigue crack closure under cyclic tension', Engng Fract Mach 2 (1970) pp 37-45
11.
Hutchinson, J.W, "Fundamentals of the phenomenological theory of nonlinear fracture mechanics' Trens ASME J Appl MachSO (1983) pp 1042-1051
References
12.
Rice, J.R. "Machen;cs of crack tip deformation and extension by fatigue" Fatigue Crack Propagation, ASTM STP 415 (American Society for Testing and Materials, 1987) pp 2463O9
Dsvidson, D.L. end Lankford, J,'Fatigue crack tip plastic strain in high-strength aluminium alloys' Fatigue Frect Engng Mater Struct 3 4 (1980) pp 289-303
13.
2.
Kanninen, M.F. and Popelar, C.H. Advanced Fracture Mechanics (Oxford University Press, New York, 1985) p 498
Hahn, G.T., Hosglsnd, R.G. and Rosenfield. A.R. 'Local yielding attending fatigue crack growth' Metallurgical Trans 3 (1972) pp 1189-1202
14.
3.
Nicoletto, G. 'Experimental crack tip displacement analysis under smell scale yielding conditions' IntJ Fatigue8 2 (1986) pp 83-89
Laruon, S.G. and Cerllmon, A.J. 'Influence of nonsingular stress terms and specimen geometry on small-scale yielding at crack tips in elastic plastic materials' J Mech Phys So/idsZ! (1973) pp 263-277
4.
Nicoletto, G. 'Fatigue crack tip strains in 7075-T6 aluminium alloy' Fatigue Fract Engng Mater StructlO 1 (1 987) pp 37-49
15.
Rice, J.R. 'Limitations to the small-scale yielding approximation for crack tip plasticity' J Mech Phys Solids 22 (1973) pp 17-26
5.
Nicoletto, G. 'Fatigue crack tip mechanica in 7075-T6 aluminium alloy by high sensitivity field displacement measurements" Nonlinear Fracture Mechanics: Volume I - TimeDependent Fracture, ASTM STP 995 ed. A. Saxena at al (American Society for Testing and Materials, PA, 1989) pp 415--432
16.
Leevers. P.S. and Radon. J.C. 'Inherent stress biaxielity in various fracture specimen geometries' Int J Fracture 19 (1982) pp 311-322
17.
Lankford, J. and Davidson, D.L. 'Fatigue crack tip plesticity associated with overloads and subsequent cycling' Trans ASME J Engng Mater Techno198 1 (1976) pp 17-23
18.
F/ihring, H. 'Berechnung von elastisch-plestischen Beanspruchungsablfiufen in Dugdale-Risscheiben mit Rissufercontakt auf der Grundlage nichtlinearer Schwingungbruchmechanik' (Institut fiJr Statik und Stahlbau, TH Darmstadt, Heft 30, 1977)
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Rice, J.R. 'Fracture: an advanced treatise' in H. Liebowitz (ed) Vol. I1: Mathematical Fundamentals (Academic Press, New York 1968) pp 192-314
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levy, N., Marcel, P.V., Otrtergren, W.J. end Rice, J.R. 'Small scale yielding near a crack in plane strain: a finite element analysis' IntJ FrectMach7 2 (1971) pp 143-156
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N a r N i m h e n , R. and Rosakis, A.J. 'A finite element analysis of small-scale yielding near a stationary crack under plane stress" SM Report 86-21 (California Institute of Tach, Pasadena, 1986)
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Int J Fatigue M a r c h 1989
Authors G. Nicoletto is with the Dipartimento di Ingegneria deUe Costruzioni Meccaniche, Universul, degli Studi di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy.
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