Some comparisons in the behaviour of materials at elevated temperatures under uniaxial and under multiaxial stress

Some comparisons in the behaviour of materials at elevated temperatures under uniaxial and under multiaxial stress

G W G R E E N W O O D

School of Materials, University of Sheffield, Sheffield SI 3JD, UK

Abstract. Most data on the mechanical behaviour of materials at elevated temperature concerns the influence of a uniaxial stress. For many purposes such information may suffice but there is increasing awareness of its inadequacy when used in the design or service life estimation of engineering components operating under more complex stress systems. Such stress systems do not only arise from the external stresses that are applied but may also result from thermal effects, particularly at interfaces and joints, or from the special geometrical features of a component.

Some experimental techniques to provide information on creep behaviour under multiaxial stresses are described together with a discussion and evaluation of the results obtained. It is noted that data from uniaxial stress tests can be used to predict such behaviour when the material is isotropic and is not subject to volume changes, microstructural instability or creep damage. Frequently, materials do not fulfill these conditions and information is presented on the influence of some of these complicating features both on creep rate and on fracture.

Keywords. Creep under multiaxial stress; creep fracture; microstructural instability.

1. Introduction

Failure in creep can occur by excessive strain or p r e m a t u r e fracture. T h e first depends on the a t t a i n m e n t of adequate resistance to flow at high temperatures a n d the second involves the a v o i d a n c e of substantial cracking or cavity d e v e l o p m e n t d u r i n g creep deformation. These two requirements are partly interdependent b u t there are distinct differences between t h e m which are particularly manifest when a material is subjected to multiaxial stresses. Increasingly, engineers are searching for reliable design criteria that necessarily imply a provision of creep d a t a of all kinds. Further, a recognition o f the desirability of estimating the lifetime of a c o m p o n e n t in service is b e c o m i n g m o r e widespread. Section 2 of the paper is m a i n l y c o n c e r n e d with factors affecting creep resistance where a limiting strain is often defined u n d e r specific e n v i r o n m e n t a l conditions. The identification of specific creep regimes is valuable in this respect. F o r isotropic materials m a i n t a i n i n g nearly c o n s t a n t v o l u m e during creep d e f o r m a t i o n then there is m u c h evidence ( J o h n s o n et al 1962) to indicate that b e h a v i o u r u n d e r multiaxial stress c a n be predicted f r o m uniaxial data. W h e r e the material is n o t isotropic in its microstructure or where m i c r o s t r u c t u r a l changes result in a lack o f v o l u m e conservation, b e h a v i o u r u n d e r triaxial stress m a y be substantially different from the uniaxial situation a n d examples are given o f experimental m e t h o d s that d e m o n s t r a t e a n d evaluate these differences in behaviour.

Sections 3 a n d 4 are c o n c e r n e d with p r e m a t u r e fracture d u r i n g creep a n d here it is s h o w n that in some instances there is a clear relationship between d e f o r m a t i o n a n d fracture processes. I n other instances, however, the processes leading to fracture m a y 173

174 G W Greenwood

be quite different from those giving rise to matrix deformation and the extent of information in this area, together with experimental techniques of investigation, are explored.

2. Creep strength

Creep strength is defined as the resistance to flow at elevated temperatures. Such a definition, however, is difficult to quantify. The creep strength of the material must be related to the circumstances of its proposed use in terms of stress, temperature and environment. In different countries, various design codes have evolved but typically it is required that the design stress must not exceed some fraction, often about 2/3, of the stress to cause failure by reaching a maximum allowable strain in a given time period at a specific temperature. It is usual also to state that fracture must not occur before this specified strain is exceeded.

An abundance of formulae have been proposed to describe the laws of creep that relate strain rate to applied stress and temperature (Gittus 1975). This field remains one of considerable speculation and argument, but there is now a generally accepted view that much of the confusion in the past has arisen from making invalid comparisons. A much more rational approach can be adopted by comparing materials at stresses which are related to their shear modulae and at temperatures that relate to their melting temperature (Ashby 1972). Such approaches generally take the steady state, or minimum, creep rate as the strain rate in question in the formulation of these relationships. More recently (Evans and Wilshire 1985), increasing recognition has been given to the importance of the transient creep stages, to the primary range where an initially high creep rate progressively diminishes and, conversely, to the tertiary creep range of progressive creep acceleration. In some recent analyses the steady-state creep range is considered to be merely a transition between the primary and tertiary stages (Evans et al 1982). The importance of these non-steady state situations (Novotny et al 1983) must not be overlooked and it is appropriate that more recognition is currently being given to anelastic effects in creep which are proving to be of greater magnitude than was presupposed.

Nevertheless, there are two main reasons why the minimum creep rate at a specific stress and temperature must still be regarded as a parameter of significance. The first is that, apart from the total creep life and creep strain, the minimum creep rate is the parameter that can be most readily defined from the creep curve. Secondly, since it represents the slowest stage in creep, then the majority of time per unit creep displacement must be spent near this region. However, because it represents a minimum creep rate, for design purposes it must never be regarded as a uniquely determining parameter of the time to reach a particular strain.

The value of recognition of the importance of the minimum creep rate can be seen in its role in clarifying the existence of the distinct creep regimes. Of the many alloys studied, perhaps the greatest accumulation of data (Harris 1985) relates to the alloy M A G N O X AI80 consisting of magnesium-0.8wt.~o aluminium and 0.005wt.~

beryllium. This alloy was required to operate at temperatures up to 450°C compared with the melting point of the alloy of 650°C: thus the operating temperature was up to 0"8 of the melting temperature. The corresponding stresses at such elevated temperatures were naturally quite low.

Fortunately, all this data has been analysed in great detail and Harris and Jones (1963) found that the creep data could best be grouped in three regimes in which each of the parameters could be evaluated. These three regimes were primarily characterised by the stress level. In the highest stress range, greater than 15"4 MN/m 2, the relationship between the minimum creep rate ~ with grain size d and stress a was given by ~ocd-°'str 7. In the intermediate stress regime between 0.55 MN/m 2 but below 15.4 MN/m 2 the results were best described by the formula ~ oc d - ° s t r 3"5. At the lowest stress levels below 0.5 MN/m 2, the formula was of the form ~ oc d-2a.

The first two of the relations cited above were interpreted on the basis of dislocation rearrangements during creep and clearly revealed that the creep deformation occurred by dislocation glide and climb processes. By contrast, at the lowest stresses, the third relationship was identified with the type of creep first proposed by Nabarro and Herring and previously demonstrated experimentally in single crystals or in bamboo- type grain structure materials (Burton 1977).

Quite independently of the analysis of these results, another investigation on a somewhat different material provided support both for the diffusion creep process at the lowest stress range and for the alteration in microstructure which is a unique feature of deformation of this kind (Squires et al 1963). The evidence was obtained on the alloy ZR55 which consists of magnesium with 0.55 wt.% zirconium. This alloy had particles of zirconium hydride, because of its subjection to a partial pressure of hydrogen, which formed a second phase in the form of particles in the magnesium matrix. The diffusion creep mechanism implies that diffusion should take place by vacancy emission and absorption at grain boundaries and thus there should be no relative movement of precipitates with respect to each other within the grains.

However, as deformation proceeds it also follows that zones free from precipitates should begin to be formed along those grain boundaries that are nearly perpendicular to the applied stress whereas on the longitudinal boundaries, which tend to move towards each other, precipitate collection is anticipated. These features were clearly observed in the experiments of Squires et al (1963). There has since been much further work to show that in many instances precipitates can actually impede diffusion creep as well as obstruct the motion of dislocations (Harris et al 1969). Nevertheless, the importance of diffusion creep for many materials at operational temperatures remains.

It is noted from the relationships quoted that whilst the stress sensitivity of the creep rate tends to diminish at the lower stress levels, there is a corresponding increase in the importance of grain size. Nabarro-Herring creep has the inverse square relationship with grain size and for diffusion creep controlled by grain boundary diffusion, as proposed by Coble (1963), the grain size sensitivity is further enhanced to an inverse cube relationship.

A question which has only recently been addressed concerns the situation for non-equiaxed grains (Nix 1981). It is quite common for materials in a wrought condition or cooled under temperature gradients, to have grains that are no longer equiaxed and the question now arises of the effect of the direction of the stress in relationship to the grain dimensions in specific directions in the material. It can readily be seen here that uniaxial stress data must also take into account the orientation of different average grain dimensions and the response to multiaxial stresses can give rise to further features.

An analysis of the influence of grains of orthorhombic shape oriented with respect to three principal stresses has been made (Greenwood 1985) for Nabarro-Herring

176

G W Greenwood

1-0

t~/to f

0'5 I

F

f , ,

0 5.0

dtld2 ^10.0

I

Figure I. The calculated variation of diffusion creep rate i t with the largest average grain dimension d t when both are measured in the direction of the principal tensile stress. The graph is drawn such that i t has a value go when the average longitudinal and transverse grain dimensions d t and d2, respectively, are equal. A similar shaped curve is derived both for lattice and from grain boundary controlled diffusion creep.

creep. An entirely satisfactory solution can be derived in this situation which has a particularly neat form. Further, it can be shown that such a formula can totally satisfy all the requirements of deformation without volume change and indicating the lack of influence of superimposed hydrostatic pressure. The formula can be written as,

2 2 2 2

~1 = 12Of~[al(d 2 + d~)- 0.2d~ - 0.3d~]/k T(d2 d 2 + d2d 3 + d3dl).

Corresponding formulae can be written for ~2 and ~3 to meet all criteria necessary to define the creep deformation, where ~ , ~2 and ~3 are creep rates in the respective directions of the principal stresses a~, 0 2 and 0"3 and when the grain dimension respectively in these directions are

d~, d2

is atomic volume, k is Boltzmann's constant and T is absolute temperature.

It is noted that, for a given stress, the creep rate is a minimum when the grains have their longest dimension in the direction in which the stress is applied. If this is taken as a creep rate g~ under stress 0.1 when d~ >> d2 = d3 then the above formula carl be written as ~ ~

12Df~tr~/kTd 2.

It is noted that only the longest grain dimension and not the transverse grain dimension is incorporated in the formula. This provides further justification for grain shape control, as in directionally solidified alloys (Betteridge and Shaw 1987).

As yet, specific experimental information is not available for the rigorous test of the equation but circumstantial evidence exists. Perhaps the most notable concerns the behaviour of tungsten lamp filaments operating at extreme temperatures in quartz-halogen lamps. Here the orientation of the grains with respect to the superimposed shear stresses is such that diffusion paths a r e l o n g and diffusion creep rates are correspondingly slow. A number of ancilliary features are required for stability such as potassium gas bubbles to stabilise the grain boundary positions. This particular grain shape also prevents significant grain boundary sliding under the operating shear stresses which would exist if there were only a few grains across the wire cross-section.

The situation relating to anisotropy is not the only case where multiaxial

creep data is required for a complete analysis. Other situations may arise where microstructural changes such as precipitation may result in a volume change in the material. If this is so, a volumetric component must be added to the observed creep strain. This is manifest in different ways depending on the stress system to which the material is subjected. Again the importance of a multiaxial evaluation of creep strength becomes important and a description of some experimental methods is given in the next section.

3. Experimental determinations of creep rate under multiaxial stress

Pioneering experimental studies of the response to multiaxial stresses at elevated temperatures by Johnson et al (1962) have been substantially extended and a number of techniques are now available. These have recently been admirably summarised and critically assessed by Henderson and Dyson (1981). In such testing and for the purposes of the present paper, a distinction can be made between those arrangements specifically designed to measure creep rates and those in which the main aim is to study the Creep fracture process. A distinct classification is not always possible but, for example, testing under shear stresses of specimens in the form of helical coils is unsuited to fracture testing and conversely, the study of notch effects in cylindrical bars is more appropriate for fracture studies than for creep rate evaluation.

In terms of creep rate measurement, perhaps the simplest procedure to visualise, though not to undertake in practice, is that of biaxial loading using a square sheet of material with each of the edges of the square attached to thicker pieces so that two perpendicular tensile stresses may be applied (Hayhurst 1973). Equipment for such a purpose can essentially take the form of a double lever creep machine but the manufacture of specimens is necessarily complex, as is the design of any surrounding chamber for heating or environmental control.

A more flexible test (Taira and Ohtani 1970) ~ can make use of specimens in the form of thin-walled tubes with or without end loading. Such tests may be realistic insofar as a significant amount of plant operating at elevated temperatures is in the form of tubes.

A different way of utilising material in tube form is in machines which can apply combined torsion and simultaneous tension along the axis of the tube (Trampezynski etal 1980). Such an arrangement can also be readily coupled with highly sensitive and accurate monitoring of creep strain. Yet a further method involves placement of an effectively uniaxial creep testing machine within the confines of a pressure vessel such that the specimen can be simultaneously subjected to a tensile stress and a hydrostatic pressure (Clay et al 1975). Such a facility permits a variety of stress systems to be superimposed since the uniaxial stress and superimposed hydrostatic pressure are independently controlled. In practice this arrangement has proved particularly suitable for investigation of some important features of creep fracture and mention will be made later of this technique.

Much of this work on multiaxial creep testing has given confidence in the application of a Von Mises type of approach in evaluating the effective stress o-e- Such a stress is defined by

O" e = ( 1 / N / / ~ ) I'(O" 1 - - 0"2) 2 "l- (0" 2 - - 0"3) 2 "t" (0-3 - - 0-1) 2"]1/2

178 G W Greenwood

~e O.IO

frO!

0 h I ~ I I I I

-0.05~ ,

I J __ I I L L

0 400 BOO 1200

t

Figure 2. The variation of effective creep strain e, in per cent with time t in hours for a commercial Nimonic 90 alloy at 850°C. The lower curve is for tensile loading and the upper curve, showing an initial period of negative creep, is for an applied shear stress producing a similar effective stress to that induced by the tensile load. The difference in behaviour is considered to be related to volume changes in the material during the early stages of creep (Timmins et al 1986).

where, in uniaxial loading, cre = 61 and the creep rate ~1, is given by il = A ~ where A and n are constants. F o r isotropic materials, this can be generalised to determine a creep rate tensor ei~ = ( 3 / 2 ) A ~ - 1 [a o - 61jah] where ah is the hydrostatic component of stress and 6ii is a Kroneker delta.

Some important exceptions have been noted however and these give warning of the caution that must be adopted for some materials used in components in which creep criteria are of critical importance. This "is illustrated by the following example.

When conventional creep tests in uniaxial tension were made on specimens machined from bars of the commercial Nimonic 90 material which was previously given the standard commercial heat treatment of 8 h at 1080°C followed by air cooling and subsequent heat treatment for 16 h at 700°C for the precipitation of gamma prime particles, a region of negative creep was detected on testing at 850°C under a stress of 50 M Pa (Timmins et al 1986). This negative creep had a duration of several hundred hours before the creep rate became positive as illustrated in figure 2. When the same alloy was tested however using a specimen of identical material and heat treatment but in the form of a thin tube, subjected to shear stress about its axis, no such period of negative creep was observed and the creep rate was found to be of the more typical positive form.

Such results clearly indicate the different behaviour that can be manifest with a different application of stress system. The most probable interpretation of such results is that microstructural changes in the material are connected with changes in volume fraction of the gamma prime particles present. It is clear such negative volume changes would immediately become apparent on unidirectional loading whereas under pure shear stresses any volumetric change would have no effect. This is an important concept to be noted by designers of components when use is made even of well-established high temperature materials whose structure is inevitably influenced by thermal changes. The results also imply that the extrapolation of creep curves from information obtained early in creep tests must pay particular regard to the stress system under consideration.

A different situation arises with regard to anisotropic materials. Their behaviour

is not well established and here one may envisage much scope for future work, not least because of the interest in high temperature composite materials. Mathematical techniques are currently being developed to analyse such situations but practical results are few. As mentioned in §2, it has recently proved possible to undertake a complete and satisfying analysis of the effect of anisotropic grain shapes in materials undergoing Nabarro-Herring creep. It seems likely that this form of creep is more sensitive to anisotropic grain str0ctures than other creep modes but the theoretical proposals previously mentioned have not yet been subjected to rigorous experimental investigation. Since stresses are usually low in materials undergoing Nabarro-Herring creep, it follows that sensitive methods are required to evaluate the resulting strain rates. Because of this, specimens in the form of wires wound into helical coils have proved ideally suited to such studies. This geometry is also suitable to explore the behaviour at lower temperatures where the Coble creep mechanism may be expected to predominate. The effects of anisotropy of grain shape on this mechanisms have only recently been rigorously analysed (Burton and Greenwood 1985) and for a creep rate ~1 in the direction of a uniaxiat stress al with grain length d! in the same direction and a transverse grain size d2 = d 3 , it can be shown that

~a = (64 Dgwtr 1 f~/kT)/[dl d2(4d 1 + 3 d2) ].

It is noted that for stress application along the largest grain dimension and for d t >> d2, the above formula shows that

el -* 16Dgwalf~/kTd2 d2,

where Dg is the grain boundary self-diffusion coefficient and w is the grain boundary width. Here it is noted that, although the transverse grain dimension d2 appears in the formula, the creep rate is again most strongly influenced by the inverse square of the largest grain dimension as in the case of Nabarro-Herring creep. When both Coble and Nabarro-Herring creep rates are normalised to the same value when d t = d 2 , the variation of £ with d l / d 2 is almost identical and the curve in figure 1 is essentially the same for both these forms of creep.

4. Fracture in creep

For design purposes, in addition to the limitation of creep strain already mentioned, fracture or a substantial degree of interface separation should not occur in less than some specified time at the required temperature.

Some general rules have long been established for the case of a uniaxial stress.

Perhaps the best known relationship is the one proposed by Monkman and Grant which shows that ~t I is approximately constant and this is frequently found to hold for many materials independently of stress and temperature over a wide range. Such a relationship implicitly suggests that there is a unique relationship between minimum creep deformation rate ~ and the lifetime before fracture ty. If that were always the case, it would immediately imply that knowledge of the flow mechanisms during creep would also be sufficient for an understanding of the fracture process. On further examination, however, such a simplistic approach has proved to be inadequate and this inadequacy is particularly manifest when multiaxial stresses are applied.

180 G W G r e e n w o o d

Moreover, even in the uniaxial case, it was shown by Dobes and Milicka (1976) that the Monkman and Grant formula could be improved by introducing a new parameter, namely the strain to fracture, such that the new formula could be written ~ty/e I ~ c where e I is creep ductility and c is a new constant. Although this looks to be a minor modification, it does suggest that, in every case, there is need to be aware that fracture may not be uniquely determined by the creep deformation process.

Fracture during creep is now recognised (Greenwood 1978) to be generally different from the more familiar brittle fracture that occurs at low temperatures. In the latter case the propensity to fracture is largely determined by the level of stress intensity and by the rapidity of stress application. In contrast, in creep, a decrease in stress level can actually reduce the strain to failure. This reflects the important role of time-dependent processes in causing a situation in which interfacial separation can

O c c u r .

A major step in the recognition of some outstanding features of fracture at elevated temperatures was provided by observations of the cavities and cracks that may be formed as creep progresses (Gittus 198t). These almost invariably occur on grain boundaries and especially on those that are nearly perpendicular to the applied principal tensile stress. It now appears that cavity nucleation is a strain rather than a time-dependent process (Evans 1984). As in many other areas of materials science, it is the nucleation stage that is least understood. Certainly, there are instances where interfacial separation by weak adhesion between particles and matrix provides a prime site for nucleation (Harris and Roberts 1981), though it is by no means certain that cavities cannot be formed in the absence of particles. Intersection of grain boundary sliding with slip within the grains would still seem to present a possible source of nucleation (Dyson et al 1976).

Recently more attention has been given to a separation of the nucleation and the growth stages (Mintz and Mukherjee 1987). This has been achieved by a prior creation of nuclei such that, in subsequent experimental testing only the growth stages were analysed. One technique was the introduction of steam bubbles above the critical size required for cavity nucleation (Hanna and Greenwood 1982; Kutumbarao and Greenwood 1986). Here it was shown that at the lowest stresses, the amount of creep stain was nearly equal to the volume change in the material. This is clearly an extreme case, for it shows that the creep strain itself is simply produced by growth of the cavities and there is no separate requirement for grain deformation.

A more recent technique of introducing nuclei which avoids.any effects of gas solubility, induced radiation damage or prior deformation is to make use of the radioactive decay reaction from tritium to helium. By this means, very careful control of cavity size and spacing has proved possible (Mintz and Mukherjee 1987). Such experiments have confirmed a range over which cavity growth takes place primarily by vacancy diffusion along grain boundaries and they have also shown the stages at which grain deformation can contribute to cavity growth. This has led to the further development of models of cavity growth through a coupling of the effects of grain boundary diffusion and of grain plasticity (Chokshi 1987). One consequence of this, as first pointed out by Dyson (1976), is that any non-uniform distribution of cavities can only grow to the extent allowed by the compatible matrix deformation. Another aspect has been the realisation, first considered by Beere and Speight (1978), of the enhanced cavity growth that can occur when grain matrix deformation can additionally contribute as well as the vacancy condensation process.

Thus the picture emerges of creep fracture arising partly as a consequence of the general process of creep deformation during which internal damage is generated but with the further recognition that the extent of this damage is influenced by diffusion processes, particularly along grain boundaries, which are not directly related to the deformation process itself. It follows that response to multiaxial stresses may be expected to be determined by these two features and the effect of such stresses on fracture may be quite different from that on the creep deformation.

5. Experimental studies of creep fracture under multiaxial stress

Experiments where a hydrostatic pressure was imposed on an otherwise uniaxial creep test have showed the first clear indications of the different responses to multiaxial stress behaviour of deformation and of fracture processes (Evans 1984). The important studies of Hull and Rimmer (1959) showed that, in copper, a hydrostatic pressure equal to the applied tensile stress was entirely sufficient to prevent cavitation. Further work on magnesium and on a number of other materials at low stresses has also illustrated that the creep fracture lifetime and the strain to fracture can all be dramatically increased by the superimposition of a hydrostatic j~ressure (Clay et al 1975), but there is only a relatively small, if any, influence on the creep rate itself.

This clearly points to the extent to which diffusion processes rather than deformation may control cavitation. Equipment by which such studies may be carried out is illustrated in figure 3. From experiments over a relatively wide range of both tensile stress and hydrostatic pressure, however, it has become clearly apparent that the effect of the latter was progressively diminished towards the high stress ranges (Greenwood 1978). This provided clear indication of the greater importance of grain deformation processes in failure at such high stresses.

It has been noted that, where creep damage takes the form of cavities growing by diffusion, an important feature is that the volume of the specimen progressively increases. In the extreme case of pure diffusion growth, there is no lateral contraction of the specimens and so it follows that the entire cavity growth process is controlled by the principal tensile stress (Greenwood 1981). This is in marked contrast to the creep deformation mechanisms that remain governed by the Von Mises type effective stress.

Over many years, numerous experiments have been made on notched specimens and both weakening and strengthening mechanisms at a notch have been proposed (Evans 1984). The presence of a notch provides circumstances that are the opposite of experiments with superimposed hydrostatic pressure because the notch provides a state of triaxial tension in its vicinity. Such triaxial tension reduces the Von Mises effective stress but only affects the net section stress induced by uniaxial loading insofar as the cross section is reduced. Such experiments, particularly with a Bridgeman type notch, have proved very successful in elucidating fracture behaviour (Hayhurst et al 1977). A uniform triaxial state of tension is difficult to produce in laboratory experiments. A more controlled way of producing such triaxial tension has also been tried by complex specimen geometry where simultaneous loading in tension is possible through three mutually perpendicular axes. Such experiments are not easy to undertake and, in practice, proved to have some limitations but some useful observations have been made (Hayhurst and Felce 1986).

182 G W Greenwood

I E X T R A C T I O N

i

2 HOLLOW SCREW. 7 SPECIMEN JAWS. 12 PULLEY PLATE.

3 S L I P - R I N G , 8 HEATING ELEMENT. 13 PULLEY ASSEMBLY.

4 TOP P L U G . 9 FURNACE MUFFLE. I4WEIGHT ATTACHMENT PEG, 5 C Y L I N D E R . I O A N C H O R PLATE. IS B O T T O M P L U G .

Figure 3. Equipment for applying a tensile stress to a conventional creep specimen at elevated temperature within the confines of a pressure vessel to permit independent superimposition of hydrostatic pressure. The load applied to the specimen is magnified by the lever operation of a cam ~Clay et al 1975).

In notch bar experiments it is noted that initially, on loading, an elastic/plastic stress distribution is developed across the notch. With time, stress redistribution takes place due to creep until a stationary state situation is achieved. Finite element calculations have been particularly useful in this respect but it is clear that the time factor is also important in these experiments (Hayhurst et al 1977).

When stress redistribution is complete, then, for any specific geometry within the notch, both the principal normal stress and the Von Mises effective stress can be

defined. The normal stress perpendicular to the circumference of the notch is simply taken as the axial load divided by the area at the notch root. If the time before fracture at the notch is greater than that for an unnotched bar subjected to the same principal axial stress, then the material is known as notch-strengthened. Conversely, if the lifetime is shortened then the material is described as notch-weakened.

It immediately becomes apparent that, since the effective Von Mises stress is decreased within the notch, for those materials for which fracture is largely determined by deformation processes, notch-strengthening will occur. Conversely for those materials where diffusion processes control cavity growth under the influence of a principal normal stress, then a notch will have an effect related to the reduction of the section of the bar in the notch region. Because fracture modes are now well-known to be dependent on stress and temperature as well as on the type of material, it is not possible to make material classifications without taking these external effects into account. Even so, it is now apparent that pure aluminium and some of its alloys are notch-strengthened and this also applies for materials such as titanium and lead which are not subject to cavitation failure. On the other hand, for pure copper and some of its alloys in which cavities are known to form and grow easily, fracture occurs at a notch because the principal stress is highest in this region.

For many commercial alloys, it is found that their stress response falls between these two extremes of behaviour and so some coupling between creep deformation and fracture mechanisms is apparent to an extent that differs between different materials and for particular circumstances. An approach to this has been developed by proposing a general concept of creep damage and through this some useful interpretations may be made. The extent to which such creep damage occurs in different materials is then related both to the maximum principal stress at and to the effective Von Mises stress tre. The simplest formulation suggests that creep lifetime t I may be related to al and to tre in the form tloc 1 / a ~ (Dyson and McLean 1972;

Cane 1979; Lonsdale and Flewett 1981) where the values of the exponents x and y reflect the relative importance of the two types of stress. There is clearly much scope for further work in this area.

6. Conclusions

Conditions are now established under which the deformation of materials at elevated temperatures under multiaxial stresses can be predicted from data obtained through uniaxiai stress tests. These conditions require that the material shall be isotropic in its microstructure and not be subject to volume changes arising either through microstructurat instability or through the accumulation of creep damage.

It is pointed out that deformation by the diffusion creep mechanism is particularly sensitive to anisotropy of grain dimensions. Complete calculations can be made of the effect of this anisotropy for grains of simple geometry and these show that creep strength is greatest in the direction of the largest grain dimension. It is shown that when the grain aspect ratio is sufficiently large, the creep rate becomes inversely proportional to the square of the largest grain dimension for both lattice and grain boundary controlled diffusion creep.

The extent to which creep fracture is sensitive to the stress system differs substantially in different materials. When the fracture mechanism is closely related to the mode of

184 G W Greenwood

deformation by dislocation glide and climb and by grain boundary sliding, then an effective stress defined by the Von Mises equation can be of predominating importance.

Conversely, when cavity or crack growth is determined by vacancy diffusion and condensation, then the maximum principal tensile stress largely controls the fracture process. Many commercially important creep resisting materials behave in a way which is intermediate between these two extremes and this is an area requiring much further investigation.

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