The evolution of crack-tip stresses during a fatigue overload event
Introduction
While most fatigue data is collected in a constant nominal crack-tip stress intensity range (ΔK = Kmax − Kmin), in practice it is commonplace for engineering components to experience a spectrum of loads. With this in mind, considerable research has been undertaken to characterise the transient crack growth behaviour following individual over- or underload spikes in loading. The current state of knowledge can be summarised thus [1].
- (i)
Overloads retard, while underloads accelerate, the rate of crack growth rate relative to the underlying constant amplitude rate.
- (ii)
A number of cycles (Nd) must be applied before the original steady-state crack growth rate is re-established following an over- or underload event.
- (iii)
The extent of the retardation effect depends on the overload ratio (OLR, KOmax/Kmax), the baseline value of ΔK and the load ratio (R).
- (iv)
Overloads can produce a very short initial acceleration phase prior to prolonged retardation.
These effects have previously been explained in terms of plasticity-induced closure, crack-tip blunting, residual compressive stresses and crack-tip branching or deflection. Plasticity-induced closure explanations have existed almost as long as crack closure has been proposed as a crack growth retardation mechanism [2]. The argument is based on the fact that the overload creates a larger stretch in the wake of the fatigue crack, which causes crack closure as the crack-tip progresses through the overload plastic zone. Since the crack must first progress into the overload plastic zone, this would explain why retardation is delayed. Because of the larger plastic zone and the capacity for inward movement of material at the crack-tip under plane stress, one might expect plasticity-induced closure to be more significant in plane stress than generalised plane strain [3]. Under plane stress a potential mechanism of material transfer is obvious. Since out-of-plane deformation is not constrained, material can be transferred from the thickness direction to the loading direction. However, the mechanism of material transfer postulated for plane stress is not admissible for plane strain. By definition, no net out-of-plane contraction can occur and, therefore, it has been suggested that there can be no net axial stretch of material in the plastic wake behind the crack-tip, which implies no plasticity-induced crack closure [4]. The existence of plasticity-induced crack closure under plane strain conditions has thus been a topic of intense debate [5]. Fleck and Newman have shown that closure does not occur for a bend specimen under plane strain conditions, while it does occur for the middle crack tension geometry under plane strain [6]. This may be due to the fact that the latter has a compressive T stress, while the former has a tensile T stress. Contrary to earlier work [7], Sadananda et al. [1] dismissed plasticity-induced crack closure, arguing that plasticity always acts to open rather than close the crack.
The main problem with the crack-tip blunting explanation is that while it can reduce the stress intensity at the crack-tip, it cannot easily explain delayed retardation. It has been argued that reverse yielding ahead of the crack-tip increases the size and magnitude of the compressive residual stress zone, thereby retarding crack growth. However, the largest residual stresses will occur in the immediate vicinity of the crack-tip, so it is difficult to explain delayed retardation [8]. Further, the extent of the retardation phase seems to be larger than the likely extent of the compressive zone [9]. Especially for alloys with a propensity for planar slip, tensile overloads can promote crack deflection away from mode I (pure crack-opening). As a result, there is a short burst of accelerated growth, followed by retardation due to the lower ΔKeff. However, such mechanisms are not generally observed at the crack-tip, while overload retardation is widely observed.
Regarding residual stress arguments, simple models such as Dugdale’s strip model [10] have been developed to explain the evolution of crack-opening stress. These assume that crack-tip plasticity occurs in thin strips, with crack-tip plastic displacements and crack advance calculated step-wise by considering the region broken down into strips to simulate formation of the plastic wake. The tri-axial constraints at the crack-tip are introduced into such models using a constraint factor (for which ideal values are 1 or 3 for plane stress and plane strain, respectively). The approach of Beretta and Carboni [11] is shown schematically in Fig. 1 where αc, αt and αw are the constraint factors for compressive and tensile yielding and for compressive yielding of the wake, respectively.
To date, experimental measurements of crack closure have relied mostly on either: (i) measuring some secondary property of the cracked body, such as compliance or electrical resistance, or (ii) measurement of crack opening displacements on the surface of the cracked body. There are significant difficulties in interpreting secondary data in terms of crack-tip stresses, as the surface of a cracked body experiences conditions of plane stress, in contrast to the bulk which experiences conditions more akin to plane strain.
In summary, the mechanisms of overload fatigue crack retardation are still the subject of debate. Some of the mechanisms relate to events in the crack wake, some ahead of the crack-tip. Crack-tip residual stresses under plane stress have been measured at the surface using X-rays [12]. While the measurement of stress fields in the bulk of cracked samples by neutron diffraction has been possible for some years [13], [14], [15], the spatial resolution (∼1 mm) has been insufficient to resolve the immediate crack-tip field. Recently, very narrow, but intense, beams of hard X-rays have become available at third generation synchrotron X-ray sources. This has opened up the way for much higher spatial resolutions, enabling crack-tip strains to be evaluated [16], [17], [18], [19]. Recently, elegant synchrotron X-ray diffraction (SXRD) experiments have been directed towards overload phenomena on 4 mm thick samples [20], showing evidence of the overload on crack-tip stresses for cracks showing subsequent retardation. SXRD also enables the study of cracks in much thicker samples [17], [21], presenting a unique opportunity to non-destructively determine the crack-tip elastic strain field under plane strain, i.e. in both the plastic and K-dominant zones. As high energy X-ray diffraction utilizes small diffraction angles the very high (lateral) spatial resolution can usually only be achieved in two orthogonal directions within bulk specimens. However, if sufficiently thick cracked specimens are used then, in principle, the full crack-tip stress field can be determined, since the conditions at the centre of such specimens approximate well to generalised plane strain.
This paper reports what we believe to be the first measurements of the two-dimensional crack-tip stress field in the plane strain region of thick samples at a resolution of tens of microns. In order to examine the extent to which residual stresses arise during conventional fatigue and subsequent to an overload results are reported at various points within the cycle for baseline fatigue, as well as during, after and well beyond a 100% overload event.
Section snippets
The material studied
The material used in this study is a powder metallurgy Al–Li alloy based on alloy AA5091 obtained from Aerospace Metal Composites Ltd., UK. The alloy composition was Al, 4.0% Mg, 1.2% Li, 1.0% C, 0.5% O and was produced from powders using a mechanical alloying process followed by hot isostatic pressing [22], [23]. The billet was supplied as an upset-forged 42 mm thick plate in the as forged (T1) condition. The material exhibited a yield stress of 450 MPa and a tensile strength of 505 MPa, values
Experimental procedure and results
Strain maps over the mid plane were collected at various stages of loading. For the fatigued test specimen (S1) at KIMin = 0.6 MPa√m, KIMean = 3.6 MPa√m, KIMax = 6.6 MPa√m, for the 100% fatigued overloaded (FO) test specimen (S2) at KIMin = 0.6 MPa√m, Kmean = 3.6 MPa√m, KIMax = 6.6 MPa√m and KIOMax = 2KIMax = 13.2 MPa√m and for the fatigued overloaded fatigued (FOF) specimen (S3) at KIMin = 0.6 MPa√m, KIMean = 3.6 MPa√m, KIMax = 6.6 MPa√m.
Conclusions
This paper presents two-dimensional maps of the elastic crack-opening and crack growth direction strains in the vicinity of a fatigue crack under plane strain conditions at much higher spatial resolution than has been achieved before [17]. From these measurements it has been possible to map the crack opening, transverse and normal stresses using an assumption of plane strain. Furthermore, because a large part of the diffraction profile had been recorded at each point, it was possible to use a
Acknowledgements
These experiments were undertaken in 2005 at the ESRF through the long-term project HS2252. We are grateful for the support of staff on the beamline ID15A for experimental support. M.E.F. is supported by a Grant through The Open University from the Lloyd’s Register Educational Trust. P.J.W. is grateful for helpful discussions with Prof. David Nowell.
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