CAVITATION and FRACTURE

Introduction

When a superplastic material fails during tensile deformation it is either the result of unstable plastic flow or a consequence of the growth and interlinkage of internally nucleated voids. In the former process, inhomogeneities in the cross sectional area of a test piece lead to a localised increase in the strain rate and the difference in the cross sectional area increases. The rate at which the discontinuity in the cross sectional area increases depends on both the rate of strain hardening and the strain rate sensitivity. In superplastic materials, where true strain hardening is minimal, any neck which is present will always grow, although the rate of growth decreases with increasing m. Unstable plastic flow normally results in the material pulling out to a fine point prior to failure. Where failure occurs as the result of the nucleation, growth and interlinkage of internal voids, the fracture surface is much more abrupt. The value of the strain rate sensitivity is important in determining the rate at which the voids grow and thus to some extent controls the elongation to failure in systems which undergo cavitation.

Fracture by Unstable Plastic Flow

The value of the strain rate sensitivity index, m, has been shown to have a strong effect on the ductility of superplastic materials (Fig 5.2). In general, the higher the m-value, the greater the elongation to failure [D.Woodford, 1969], although this rule is by no means universal. Rai and Grant [G.Rai & N.T.Grant, 1975], for example, observed that the strain rate at which the maximum elongation occurred in a superplastic Al-Cu eutectic alloy did not coincide with that at which the maximum strain rate sensitivity was measured (optimum strain rate). It has been suggested [A.K.Ghosh & A.Ayres,1976] that the elongation at failure depends on the “m” value at high rather than low strains, and that microstructural evolution during deformation can have a marked effect on the strain rate sensitivity.

Several relationships defining the strain at failure during superplastic flow have been derived [161-167]. The total tensile elongation to failure of a superplastic material will be comprised of both the uniform strain and that accumulated within any necks which develop during deformation.

Consider now a tensile specimen in which there is a small neck for which the initial cross-sectional area varies from ao to aao, where a<1. If the material obeys the superplastic flow law then (5.1)



The load on the tensile test piece is then (5.2)



and the strain rate (5.3)


Hence (5.4)



It can be seen from equation (5.4) that as m increases, the rate of change of the cross sectional area becomes less dependent on the magnitude of that area and any neck would grow more slowly. If, after a small increment of strain, the cross sectional area of the test piece is reduced to a, and that of the neck to ba, where b<1, then re-arranging (5.4) and integrating gives (5.5)



and (5.6)



Eliminating the left hand side of equations (5.5) & (5.6) we obtain (5.7)



The uniform strain in the test piece is simply (5.8)



and the maximum elongation to failure can be obtained by setting b equal to zero, i.e. (5.9)



If the initial discontinuity is small then there is a very strong dependence of the elongation to failure on the strain rate sensitivity, m. The variation of the elongation to failure vs. strain rate sensitivity is plotted in figure 5.2 for two values of a:- 0.99 and 0.999. It can be seen that the elongations attained for several commercial superplastic alloys lie towards the lower bound predicted by (5.9), but these are still in excess of the elongations that would be required of the material for superplastic forming (shaded area).



The uniformity of plastic flow has been measured in Zn-22Al by dividing the gauge length of a tensile specimen into a number of segments and monitoring the elongation of each segment [F.A.Mohamed & T.G.Langdon, 1981]. It can be seen from figure 5.3 that, for deformation in both Regions I and III, once a neck forms it propagates rapidly. However, for deformation in Region II, where true superplastic flow occurs, plastic flow is much more uniform, even to very high strains, and any necks which do form tend to be rather diffuse.

In the analysis of plastic flow developed above, it has been assumed that the material is structurally stable and the value of m remains constant.

Supral 220 M cs.STRAIN

Fig 5.4 measured strain rate sensitivity after SP deformation to various strains
and the variation with strain at 1.17x10-3/s

Experimental work has shown that the strain rate sensitivity can vary significantly during superplastic flow [B.Geary, 1985]. The actual variation of m with strain will depend on the (complex) interrelationship between strain rate, temperature, grain growth and strain hardening/softening. For example, in Supral 220 grain growth during deformation causes not only a reduction in the magnitude of the maximum value of the strain rate sensitivity with increasing strain but also leads to lower value of the strain rate at which that maximum. If the deformation rate is constant and higher than the strain rate at which m is initially a maximum, then the strain rate sensitivity will continually decrease with increasing strain (Fig 5.4). However, if the deformation rate is initially less than the strain rate for maximum “m” then the strain rate sensitivity would increase during the initial stages of superplastic flow, reach a maximum and then decrease. The actual magnitude of the increase in the strain rate sensitivity depends on the interrelationship between microstructural instability at the deformation temperature and strain rate.

The maximum tensile elongation attainable in superplastic materials is of little relevance when compared to the actual strains which the material would be required to achieve during a commercial superplastic forming operation, but nevertheless provides a useful guide to the superplastic deformation potential of the material. Considering only the uniform strain predicted for a given value of m, most superplastic materials would appear to have ample reserves of stretchability. Unfortunately, not all superplastic alloys pull down to a fine point at failure. Two different modes of failure in superplastic tensile test pieces are shown in figure 5.1. Both samples attained the same strain at failure, but the flat fracture surface of the lower specimen, termed pseudo-brittle, arises from the ductile tearing of tiny ligaments between regions of internal cavitation. It is the development of such cavitation damage that often leads to the premature failure of superplastic materials [M.J.Stowell,1980] which will be considered in the remainder of this secition.

General Characteristics of Cavitation

Despite the large plastic strains which can be obtained in superplastic materials it is now well established that cavitation may occur during superplastic flow. The alloys in which cavitation has been observed include those based on aluminium [39,170-174], copper [105,175-181], iron [63,66,70,71,182], lead [183], silver [184], titanium [185-187] and zinc [188-191]. The subject of cavitation in superplasticity has been reviewed extensively both from an experimental and theoretical viewpoint [192-196]. In general, cavities nucleate at the grain boundaries and their subsequent growth and coalescence invariably leads to premature failure. However, and more importantly from a practical viewpoint, the presence of cavities in superplastically formed components could adversely affect their mechanical properties. There is also evidence that cavities may develop from defects which pre-exist, and which are usually produced during the thermomechanical processing required to develop a superplastic microstructure [195].

An important requirement for cavitation during superplastic flow is the presence of a local tensile stress. Under the conditions of homogeneous compression cavitation is not observed, and cavities which are produced during superplastic tensile flow are removed during subsequent compressive flow [197]. Superplastic closed die isothermal forging of Ni-base superalloys such as IN100, starting from hot pressed powders which are heavily worked by extrusion, is used in the manufacture of turbine discs to give a sound cavity free product of uniform microstructure. It has also been demonstrated that the superimposition of hydrostatic pressure during both uniaxial and biaxial superplastic tensile flow can reduce or eliminate cavitation [198]. The cavitation damage which develops during superplastic forming can also be removed by a post-forming hot isostatic pressing (HIPping) treatment which will sinter up the voids [199]. In order to control cavitation it is therefore necessary to understand both the microstructural and deformation parameters which influence its occurrence and magnitude.

In the majority of studies on cavitation only the variation of the total volume fraction of cavities with strain at different strain rates and temperatures has been studied. It should be noted that the volume fraction of cavities, Cv, can be written as(5.10)



where Nvi is the number of cavities per unit volume having a volume Vi. The number of voids of any particular size is related to the number of pre-existing voids and to the nucleation rate, while the volume of a particular void is controlled essentially by the growth rate. However, during the latter stages of deformation when the cavitation level is high (approaching 10%) the growth process is affected by the spatial distribution of the voids through void coalescence. Hence growth becomes, in part, dependent on the nucleation process. It is therefore evident that the variation of the volume fraction of voids with strain, strain rate and temperature, will not be a simple one. The effect of changing the strain rate and temperature at which deformation occurs may be further complicated by other factors such as grain growth and/or a change in phase proportions [105,175].

Coronze
Coronze CD-638

7475
AA-7475

Supral
Supral 200

Fig 5.5 Cavity Morphology

The morphology of cavities formed during superplastic flow varies from one material to the next and even in the same materials deformed at different strain rates. In general, three types of cavities have been observed in superplastic materials (Fig 5.5). These are:
  1. Spherical voids with radii up to ~100µm.
  2. Elliptical voids elongated parallel to the tensile axis with lengths up to ~50µm and aspect ratios from 2:1 to 10:1.
  3. Groups of angular or crack like cavities each up to 10µm in length interlinked around clusters of grains.
The cavity morphologies have been cited as evidence for the operation of different void growth mechanisms. The circular section voids have often been taken to infer diffusional growth, while the elongated elliptical section voids are thought to be indicative of strain controlled growth. However, there have been few systematic metallographic studies (or quantitative analyses) of the evolution of void size and shape during superplastic flow.

Unfortunately, observations of cavities at the nucleation stage are usually difficult to make. Cavities do not form uniformly throughout the microstructure and information concerning nucleation has to be deduced from specimens in which the voids have grown to a size where they are resolvable, either by direct observation or by the measurement of a physical property of the base material. Since cavities do not always nucleate during superplastic flow and, with only very specific exceptions, will always grow during tensile superplastic deformation, the mechanisms of cavity growth will be examined prior to a consideration of the nucleation process.

Cavity Growth


Diffusion controlled growth

A cavity located on a grain boundary, whether nucleated during superplastic flow or pre-existing may grow by stress directed vacancy diffusion, by plastic deformation of the surrounding matrix or by a combination of both these mechanisms. The former process has been analysed by several workers [200-209] for the case of spherical voids under uniaxial tension and leads to a relationship of the form (5.11)



where s1 is the maximum principal stress local to the grain boundary, q is the atomic volume, g, the surface energy, l , the cavity spacing and r, the cavity radius. The driving force for cavity growth is provided by the difference in chemical potential between an atom on the stressed grain boundary and that on the free surface of the void. When the maximum principal stress is tensile, atoms move from the void surface to the grain boundary and the void will grow (Fig 5.6). If the stress is compressive then atoms move in the opposite direction and the voids will sinter.

The relationships derived in the literature differ only in the form of the last term containing solely r and l, the size and spacing of the voids. In deriving relationships of the form of equation (5.11) it has been assumed that the voids are widely spaced on fixed boundaries orientated perpendicular to the applied stress axis and are themselves small relative to the grain size of the material. It is evident from equation (5.11) that the rate of void growth decreases as the inverse square of the cavity radius and thus slows considerably as the voids grow. In the majority of superplastic materials, the voids have dimensions similar to that of the grain size. Furthermore, the cavities are usually located on sliding boundaries which are orientated randomly with respect to the applied stress. It was proposed that when the void diameter was the same as the grain size, then the growth rate would be enhanced by additional mass transfer along boundaries intersecting the surface of the void [D.A.Miller and T.G.Langdon,1979]. Such enhancement was later shown to be sensitive to the ratio of the void size to the grain size [J.Pilling et. al. 1984]. The rate of void growth is then given by [A.H.Chokshi & T.G.Langdon,1987] (5.12)



where d is the grain size. Unlike the previous equation (5.11), the rate of change of void radius with strain is independent of the void radius, r.

Equations (5.11) and (5.12), which were initially derived for the case of uniaxial tension can be written in more general terms allowing the effects of multi-axial states of stress and alternative geometries of deformation to be considered. (Commercial superplastic forming usually involves balanced biaxial or plane strain rather than uniaxial deformation). The maximum principal stress, s1, can be redefined in terms of the Von Mises equivalent stress, se, where (5.13)



and the superimposed pressure, P, such that (5.14)



where kD is a geometric factor that depends on the mode of deformation (uniaxial, balanced biaxial or plane strain) and the extent of grain boundary sliding. The determination of appropriate values for kD will be discussed later...

Plasticity or strain controlled growth


For cavity growth by deformation of the surrounding matrix in uniaxial tension, the relationship (5.15)



has been proposed, where h, the cavity growth rate parameter, is dependent on both the applied stress state and the geometry of deformation [N.Ridley & J.Pilling, 1985, M.Suery, 1985]. Unlike diffusion controlled cavity growth, the rate of void growth increases linearly with void size and is independent of the strain rate. Plasticity dominated void growth is controlled by the mean stress, sm, where (5.16)



rather than the maximum principal stress.

The parameter h in equation (5.15) has been determined theoretically by several workers for the case of uniaxial tension with values ranging from 1.22 [A.Needleman & J.R.Rice, 1980] through 1.25 to 3 [J.W.Hancock,1976] to (5.17)



[J.R.Rice and D.M.Tracey,1969] where sm is the mean stress, se the Von Mises equivalent stress, and are the principal strain rates. Rice and Tracey showed that h would be equal to 0.9 for uniaxial tension, 1.65 for plane strain and 1.92 for balanced biaxial tension in a perfectly plastic solid (m=0). Alternatively, [Budianski et al., 1982] predicted that h would take different values depending the magnitude of the strain rate sensitivity . For example, in balanced biaxial tension, which would arise in the case of blowing a hemisphere from a circular disc of material,it was calculated that h would equal 1.75 when m=0.5; 1.94 when m=0.33 and 2.64 when m=0. Stowell et al. It has also been shown experimentally [M.J.Stowell et. al., 1984] that the value of h is dependent on the strain rate sensitivity and hence will vary with the strain rate. The relationship given by Stowell et al. for uniaxial tension can be readily extended to multi-axial states of stress (A.Cocks & M.F.Ashby, 1982; J.Pilling & N.Ridley, 1984] (5.18)



or [M.Suery, 1985] (5.19)



The term containing the ratio of the mean stress, sm, to the von Mises equivalent stress, se, in equations (5.18) and (5.19) defines the triaxiality of stress local to the grain boundary. The form of this term is dependent on the geometry of deformation and may be re-written in terms of a geometric constant, kS, and the superimposed hydrostatic pressure P (5.20).



To establish the values of kS and kD appropriate to superplastic flow, two modes of deformation need to be considered. Firstly, the case of rigid grains where there is no grain boundary sliding and secondly, the case of freely sliding grains, where the state of stress on the grain boundaries is dependent on the extent of sliding. As the voids are located on the boundary they experience the local rather than the remote state of stress and thus their rate of growth is, to a large extent, dependent on the fraction of the total strain carried by the grain boundaries.

Determination of stress state local to the grain boundary

Consider first the 2-dimensional arrangement of hexagonal grains shown below (fig.5.7) (see also fig 4.7 [149]) [202]. The imposed stresses and are applied at some arbitrary angle q with respect to the orthogonal axes defined by s11 and s22.



Resolving the forces due to the remotely imposed stresses and those acting on the continuum body , then (5.21),(5.22)



If the boundaries are freely sliding then the shear stresses will be fully relaxed i.e. (5.23)



and the stress, sc, acting normal to the grain boundary will be [149,202] (5.24)



The stress on the grain boundary will vary in a periodic pattern with a repeat every 60°. The average stress on the boundary is obtained by integrating equation (5.24) over the range 0 top/6 (30°) since there is an equal probability of the hexagonal grains being present in either of the flat or upright orientations.(5.25)(5.26)



However, in a real material grain motion will occur in three dimensions not two. We now assume that there is an equal probability that the sliding boundary will lie in the plane containing s11 and s33 as in the plane containing s11 and s22, and hence s1is now taken as the average of the two orientations, i.e.(5.27)



Normalising with respect to the von Mises equivalent stress we obtain (5.28)



In order to obtain the local mean stress it is now assumed that the local deformation rate is the same as the remote deformation rate, i.e. the deviatoric component (S) of the local and remote stress fields are the same. As (5.29)



then (5.30)



Substituting equation (5.27) for s1 and solving for sm (5.31)



Now divide through by se to obtain the local degree of triaxiality (5.32)



Having defined both the remote and local extent of triaxiality in terms of the three remotely applied principal stresses it is now possible to obtain the constants kD and kS for any deformation geometry.

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