CAPACIDAD DE SOPORTE DEL TERRENO DE FUNDACIÓN

The application of LEFM modelling to fracture problems is subject to a stress-strain state where no plasticity exists or its effects are negligible. As a result of this, a good number of approaches have been followed in the past in order to establish the extent to which plasticity may be present in general fracture problems. For a finite size slit crack in an infinite plate subjected to a tension stress, a , in the mode I direction, the stress in the loading direction, CFy, at a distance r from the crack tip along the x axis can be written [Ewalds et Wanhill, 1984]:

This equation assumes that the stresses are infinite at the crack tip (r —» 0). Using this equation for the stress, the length, r, over which the yield stress is exceeded will yield the size of the plastic zone. Substitution of the yield stress, Cy,, for Gy, will yield a value of r (ry), which can be used to describe the length ahead of the crack tip over which the stress field proposed in equation (3.1.1) exceeds Gy,. In fact, the stresses obtained for r<ry are seen to be well above Gy, and tend to as r approaches 0. Since the stresses cannot be infinite in an elastic-perfectly plastic situation, some assumptions need to be made in order to establish more accurately the size and shape of the plastic zone.

This turns out to be an extremely difficult problem to solve, and therefore attempts have been made to simplify the matter by selecting an arbitrary shape and then calculating the plastic zone length (ie. Irwin or Dugdale [Ewalds et Wanhill, 1984]), or by retaining a first approximation of the plastic zone size and then establishing the plastic zone shape. Since stresses in the y direction are above Gy^

for all x<Fp (the plastic zone length along the x axis), and materials deform plastically at stresses above Cy^, this linear elastic solution cannot be applied as it stands. Irwin suggested that a crack of length a behaves as if it were of length a + Tp, with an additional plastic zone tp, thus allowing for stress redistribution after local yielding, and ending up with a plastic zone length twice the original r^ The typical solutions proposed by Irwin are:

1

\PysJ

3ji_\^ysj

{plane stress) (3.1.2)

{plane strain)

To establish the plastic zone shape, the elastic-plastic boundary needs to be computed using a yield criterion. Using Von Mises’ yield criterion, Broek [Broek, 1974] calculated the plane stress plastic zone shape for the same slit crack problem described above.

. , ( 6 = 0) = ^ (3.1.3)

Thus showing the same length as the Irwin plastic zone along the x axis but a slightly larger length at 90", giving rise to the well known kidney shape.

Thus far, only the plastic zone due to monotonie loading has been considered. However, since fatigue loading is cyclic, the existence of a reversed or cyclic plastic zone has been postulated [Rice, 1967] and observed experimentally [Davidson et Lankford, 1976]. The cyclic plastic zone size can be obtained using an "effective yield strength", usually the cyclic yield stress, that can then be substituted into equation (3.1.1). Rice used a superposition argument to show that under reversed loading the Cy, should be replaced by twice its value (assuming Cy, in compression = <Tys in tension). Thus equation (3.1.1) yields a cyclic plastic zone four times smaller than the monotonie plastic zone.

A number of experimental techniques have been used in the past in order to show the existence of a monotonie and cyclic plastic zone ahead of the crack tip. These include microhardness testing, transmission electron microscopy and in some cases etching [Lankford et al, 1977]. In each of these methods, the cyclic zone boundary is delineated from the monotonie plastic zone as a discontinuity in the radially measured values of strain-induced damage.

When a cracked body is subjected to tension stress, this stress causes its surfaces to distance themselves. It has been shown that as the distance from the crack tip increases, the vertical displacement, or crack opening displacement (COD), v, also increases. The actual crack opening at the crack tip (CTOD), 5^, is given by (using Irwin’s plastic zone [Ewalds et Wanhill, 1984]):

<- 4 K}

Alternatively, the Dugdale approach comes out to be:

5, = y — (3.1.5)

Gys

Direct measurements of crack tip opening displacements showed the Dugdale approximation to hold for a number of cases.

Under cyclic loading, however, unloading may introduce compressive stresses in the plastic zone, even under prevalent global tension stress. This has been very well illustrated in an FEA by Newman [Newman, 1977]. In this analysis, a cracked plate is subjected to a CA saw-tooth stress wave with Aa=0.2CTyj, and R=0. On the assumption that the material behaviour is elastic-perfectly plastic, the critical strain to fracture is 0.2 and using limited-load increments, it was seen that initial loading from a condition of zero stress to 0.2Cyg resulted in a crack growth increment of over 0.15mm (aq=28mm). Subsequent unloading and reloading resulted in further increments of less than 0.05mm. Furthermore, the crack surfaces were in contact for over a quarter of that subsequent loading part of the cycle. The analysis also shows that cycling between V-0.2Gys results in a much larger crack increment as a result of a lower crack opening stress.

Furthermore, VA loading introduces variations in CTOD that alter the K^o, thus affecting subsequent cycling [Newman, 1977]. These findings are in agreement with simple CTOD calculations: Let us take for instance a cycle with a peak SIP, Kj, followed by another cycle with a peak SIP, Kj, where K2=2Ki. The CTOD will then increase according to (using Irwin’s analysis):

To give a value 4 times the previous CTOD and, in addition, any subsequent cycling at a peak SIP value of is subject to a plastic zone four times larger than before.

It is also interesting to note that the model used to obtain the equation for the CTOD can now be used with a crack length a + 4r^. Thus the Irwin analysis would yield:

Ô, = (3.1.7)

Substituting rp f o r ^ f ^ l , and neglecting the rp^ term yields:

Which is twice the previous CTOD at Kj. This may provide us with an explanation as to why after overload microscopical crack growth acceleration is observed prior to the retardation undergone on entering the region subjected to compressive stresses in the monotonie plastic zone.

It now remains to be seen what phenomena are involved in the cracking taking place just ahead of the crack tip. A very interesting contribution to the problem was that of Hahn et al [1972]. In their study, PCGR’s in the order of lO'^mm/c were observed in a 460MPa Gy^ Pe-3Si steel. Subsequent etching of the specimens allowed the authors to identify a large monotonie plastic zone and a much smaller (about 5 times) reversed cyclic plastic zone. If we consider a point in the crack plane but far away from the crack tip, as the crack front advances towards this position the following conclusions may be drawn: the material at this point has endured 10^ to lO'^ cyclic microstrain cycles (ie 0<Aep<10‘^), as this materials enters the monotonie

plastic zone, a further 200 cycles at 10'^<Aep<10'^ act upon it and a final -1 0 cycles in the COD affected zone at strain ranges of up to AOp=l are inflicted before fracture occurs.

Hahn et al’s approach emphasises the fact that the material undergoing fracture has experienced a number of cycles under elastic stresses, a lesser amount of cycles at plastic stresses and a limited amount of cycles of large strain cycles prior to cracking. While this is an important consideration, it does not provide us with an explanation of why cracking occurs at the crack tip. Furthermore, it is impossible to say whether cracking took place as a result of the large strain cycles alone or whether the microstrain cycles also played an important role in the fatigue crack growth mechanism. At this stage it may be useful to have recourse to fractographic evidence of the well known beach marks on specimens fractured under cyclic loading conditions. The ripples observed in this beach marks have been the subject of theories attempting to shed some light into the problem. Given the great diversity of materials and loading environments of the experiments described in the existing literature, it is unlikely that a single mechanism be common to all of them. However, it is interesting to look into some of the studies that may be relevant to the present work.

Observations of the fracture surfaces of specimens tested under the loading described in Fig.5.2 [McMillan et Pelloux, 1967] showed a ridge or saw-tooth profile with alternating bright and dark bands. The relative width of the bright to the dark band is relatively uniform for CA cycling. The bright band of the striation was flat and generally featureless while the dark band had a rumpled appearance with fine

wavy slip lines parallel to the crack front indicating heavy deformation. Changes of maximum load or amplitude result in changes in striation pattern. Making use of this fact, McMillan et Pelloux used a number of programmed sequences to come up with a number of findings. In the first instance, they observed that fatigue cracking only took place during a load rise or opening (bright band) and subsequent unloading or closing of the crack results in heavy deformation of the two fracture surfaces near the crack tip giving rise to the dark, rumpled band of the striation and erases in part the bright appearance of the fracture surface created during the loading part of the cycle. The ratio of the dark to bright striation depends on the crack tip loading versus unloading amplitudes. The deformation mechanisms acting at the crack tip on unloading are still not clearly understood but this findings are compatible with crack tip sharpening observations during unloading [Laird et Smith, 1962]. A second

finding was the fact that while the bright band of the striation slopes upwards from the crack plane, the dark band slopes down towards it, thus forming the saw-tooth pattern. This is also compatible with suggestions that crack growth is by shear decohesion along the edge dislocations at +/-45“ angles [Tomkins, 1980].

Forsyth et Ryder [1961] have suggested that these ridges are created by necking of a ligament formed between the crack tip and the microvoids developed ahead of the crack tip because of the higher triaxial stress conditions encountered there, thus giving rise to peaks along the crack front surface. Later observations on a similar alloy [Laird et Smith, 1962], lead to suggestions that these ridges are not peaks, but rather troughs. Close examination of the crack tip at various stages within a load cycle show that under sinusoidal wave loading the crack is a sharp slit during the final compression part of the cycle. As tension loading is applied, the crack begins to deform by the Bauschinger effect revealing a sharp notch. Increasing tensile stress results in a more rounded crack tip until pronounced ears form around the crack tip (about 60“ principal stress [Williams, 1955]). When the rounded-off crack is compressed, crack tip sharpening results in each ear giving rise to a trough in the wake of the crack.