## Abstract

When polycrystalline materials deform at elevated temperatures under low applied stresses by the stress directed migration of vacancies, specific features need consideration in the bending of thin beams since a relatively high stress variation may arise across individual planar grain boundaries in addition to the variation that exists between boundaries of differing inclination. The features depend on grain and beam geometry and expressions are derived for their effect on the rate of deflection of cantilevered beams. Experiments were carried out on beams of high purity copper 100 and 250 μm thick. The cantilever profiles supported the theoretical approach and showed creep rates linearly dependent on stress at rates in accord with predictions based on a diffusional creep process. A further indication of this process was the associated strain localization that resulted in fracture of 100 nm thick alumina coatings applied to some of the beams. The analysis shows how relationships change as the beam thickness approaches the grain size and permits an evaluation of the rate of beam deflection under small bending moments in terms of grain and beam dimensions.

## 1. Introduction

Polycrystalline materials can deform at elevated temperatures by a variety of mechanisms. Nabarro (1948) and Herring (1950) first developed the theory, relevant to behaviour under low stresses, that involves the stress directed generation and migration of vacancies (Burton 1977). In the theory, individual grain boundaries act either as a vacancy source or sink such that a flux of vacancies exists between boundaries of different inclination. Recently, Burton (2002, 2003) has dealt with the situation where a single grain boundary experiences a variation of stress, with tensile and compressive stresses acting over different areas, as is the case when a bending moment is applied. In these circumstances, different regions of the same grain boundary can act either as sources or sinks for vacancies with the further implication that there must be relative rotation between grains on either side of the boundary. An analysis of this situation, both for diffusion in the grain boundary and through the lattice, has allowed the rate of this rotation to be evaluated (Burton 2003) when the boundary between adjacent grains extends over the entire cross-section of a thin, but relatively broad, beam with the long boundary dimension parallel to the axis of rotation.

For thicker beams or with smaller grain sizes, where several grains occupy the beam cross-section, it is more appropriate to analyse the creep deformation in terms of the tensile and compressive stresses acting, respectively, on either side of the neutral axis to create vacancy fluxes within individual grains between differently oriented grain boundaries.

There is also a third, more complicated, situation where several grains occupy the beam width but with each grain extending through the thickness (the smallest dimension) of the beam.

A preliminary experimental study is reported to provide validation of the form of analysis that is developed, with respect to various beam and grain dimensions, to evaluate the rate of bending of a beam when a diffusional creep process is operative. It is implicit in the work that the creep strain rate is linearly dependent upon stress. The technique described here allows this aspect to be confirmed by demonstrating that the geometrical form of the deformation follows a pattern that is identical with that of elastic deformation. Finally, the implications of the study are discussed whereby it is concluded that extrapolations of diffusional creep rates in bending must be analysed in terms of formulae that are dependent on grain size and beam thickness and on the relationship between them.

## 2. Theory of the experimental method

### (a) Elastic deflection

The test geometry comprises a beam of thickness *h*, width *b* and length *L* cantilevered horizontally from a vertical sample holder as shown in figure 1. The applied load is thus due to the self-weight of the unsupported length of the beam. This loading results in a tensile stress in the direction of the beam length at its upper surface, highest adjacent to the support, with a numerically equal corresponding compressive stress at the lower surface. These stresses decrease to zero at the unsupported end of the beam. The bending moment *M*_{x}, giving rise to the elastic and subsequent creep deflection, varies with the distance *x* along the beam, measured horizontally from the supported end and is given by(2.1)where *ρ* is density of the beam material and *g* the acceleration due to gravity.

Neglecting transverse shear stresses that are negligible, a small bending moment causes elastic deflection readily calculable from elasticity theory (Cottrell 1971) such that, at a distance *x* along the beam, its radius of curvature *R*_{x}=*EI*/*M*_{x}, where *E* is Young's modulus of the beam material and *I* is the second moment of area equal to *bh*^{3}/12. Since the beam is cantilevered horizontally, 1/*R*_{x}=d^{2}*y*/d*x*^{2} so that(2.2)Integrating twice with respect to *x* and noting the conditions that *y* and d*y*/d*x*=0 at *x*=0, we obtain the vertical deflection *y* of the beam at a distance *x* from the support such that(2.3)From equation (2.2), the maximum curvature 1/*R*_{x=0}=6*ρgL*^{2}/*Eh*^{2} and the maximum tensile strain (occurring at the top surface of the beam) *ϵ*=*h*/2*R*_{x=0}. Thus, the maximum tensile stress in the beam is . Appropriate dimensions, listed in table 1, were derived for beams employed in the experimental programme through the application of the above equations at room temperature with *E*=130 GPa and *ρ*=8960 kg m^{−3} such that the maximum stress was kept below 0.6 MPa and the maximum elastic deflection *y*_{m} at *x*=*L* was less than 5 μm. This would be increased but to a value less than 7 μm due to a reduced value of *E* in the tests to be described later at a temperature of 950 °C.

To use this approach for creep evaluation, we need to consider the vacancy flux patterns that develop as these are influenced by the size and geometry of the grains and of the beam. In the present study, the beam thickness *h* was always small in relation to its breadth *b*. Three cases, illustrated in figure 2, that are determined by grain and beam dimensions will be considered separately in the following sections.

### (b) The situation for creep where grains are equiaxed with *d*≪*h*

In this case, when the grain size *d* is small in relation to beam thickness *h*, as shown schematically in figure 2*a*, the variation in stress across any individual grain boundary is negligible compared with the variation that exists between grain boundaries of differing orientations.

It follows for this case, where *d*≪*h*, that the principal driving forces are tensile and compressive stresses creating fluxes between differently oriented boundaries, and so it is appropriate to use the familiar diffusional creep equation (Nabarro 1948; Herring 1950) where creep rate at temperature *T* under a stress *σ* is given by(2.4)where *D* is the bulk diffusion coefficient, *Ω* the atomic volume and *k* Boltzmann's constant. *B* is a dimensionless constant dependent upon grain geometry with a value for equiaxed grains of about 10. At relatively low temperatures compared with the melting point, when grain boundary diffusion becomes the prominent flux path (Coble 1963), *D* is replaced by *D*_{g}*w*/*d* where *D*_{g} is the grain boundary diffusion coefficient and *w* the grain boundary width. In conjunction with this, the value of *B* is increased by roughly five times. These modifications can readily be incorporated into the analysis if required but will not be considered here.

Now it was noted from equations (2.2) and (2.3) that the tensile strain *ϵ*_{x} at a point *x* on the upper surface along the beam is given by *h*/2*R*_{x} and 1/*R*_{x}=d^{2}*y*/d*x*^{2}; thus the accumulation of creep strain in time *t* is given by(2.5)where *σ*_{x} is the tensile stress at the beam surface, shown in §2*a* to be of magnitude 3*ρg*(*L*−*x*)^{2}/*h*.

Substituting for *σ*_{x}, putting *B*=10, and integrating equation (2.5) twice with respect to *x*, it follows from the boundary conditions d*y*/d*x* and *y* both zero at *x*=0, that the deflection *y* of the beam from the horizontal at a distance *x* from the support is given by(2.6)With regard to further use of equation (2.6) and its subsequent variants, it is convenient to denote the factor in the first bracket by *K* so that the equation is abbreviated to(2.7)*K* is then derived directly from experimental measurements of *L*, *x* and *y* and compared with theoretical predictions from the parameters in the first bracket of equation (2.6). In the tabulation of data in table 1, we use the suffix ‘e’ for the experimental values derived from equation (2.7) so that *K* is then denoted by *K*_{e} and the theoretical values are denoted by *K*_{n} as calculated from the parameters in equation (2.6), which is derived directly from the Nabarro equation (2.4).

### (c) Where all grain boundaries are perpendicular to the longest dimension of the beam, with breadth *b* and thickness *h*, separated by distance *d* along the beam with *d*≫*h*

In this situation, illustrated in figure 2*b*, in contrast with that considered in §2*b*, the crucial feature is the dominating influence of a stress gradient across each individual grain boundary. The maximum applied tensile stress across a grain boundary is adjacent to its intersection with the top surface of the beam with the maximum compressive stress adjacent to its intersection with the lower surface. This pattern, however, becomes modified through the internal stress redistribution consequent on diffusional creep.

Burton (2003) has recently presented solutions for rates of rotation between grains that arise through grain boundary and lattice diffusion in bicrystals with a rectangular grain interface subjected to a bending moment with axis parallel to the interface. For conditions of high homologous temperature, as here, for which the contribution of grain boundary diffusion creep (Coble 1963) is negligible, the resulting expression for the angular rotation rate between adjacent grains is(2.8)where *α* is a dimensionless numerical constant equal to 4750 and *M*_{x} is the bending moment. Now *θ*=(d^{2}*y*/d*x*^{2})*d*, so substituting for *M*_{x} from equation (2.1) and for *θ* from the integration of equation (2.8) with respect to *t*, we obtain(2.9)On integrating twice with respect to *x*,(2.10)By analogy with equation (2.6), a new value for *K* can be calculated under the conditions stated with its magnitude predicted from the parameters in the first bracket of equation (2.10). Since this calculation of *K* is based on the diffusion induced grain rotation theory of Burton (2003), this theoretical derivation will be termed *K*_{b}.

### (d) Where several grains lie within the breadth of the beam with each grain occupying the entire beam thickness such that *d*<*b* but greater than *h*

Here, we have an intermediate case, lying between the two cases already examined, illustrated schematically in figure 2*c*. Its analysis presents difficulties that cannot yet be fully overcome for the grain boundaries lie at angles to the axis of the bending moment. With each grain occupying the through thickness of the beam, it is expected that the vacancy fluxes will be primarily determined by the stress gradients along individual boundaries rather than by the stress variation between different boundaries. Thus, the form of analysis required is most closely related to that in §2*c*.

Overall in this situation, the grain boundaries through the beam thickness will be subject to smaller stress gradients than envisaged in §2*b* so that their effectiveness in enabling grain rotation will be correspondingly reduced. It will suffice here to assume that equation (2.10) will continue to be valid when a dimensionless numerical coefficient *β* is inserted as a factor in the right-hand side of the equation. This factor will depend on any preferred grain boundary orientation but should have a definitive value for a random inclination averaged over a large number of grains. Its theoretical evaluation seems feasible though complex and has not yet been attempted.

The preliminary experiments described in §3 go some way in providing a validation of the general theoretical approach, together with an experimental evaluation of *β*.

## 3. Experimental

Sheets of high purity (greater than 99.999%) copper of thickness 100 and 250 μm were cut into rectangular pieces 6 mm wide to form cantilevers with unsupported length between 9 and 22 mm. After grain size measurement, each cut piece was clamped in turn at one end to the post of a rectangular sample holder machined from oxygen-free high conductivity (OFHC) copper or graphite. The assembled cantilever specimen was placed in a silica tube containing a titanium getter, which was then evacuated, back-filled with 400 mbar of argon, and sealed off. The assembly was placed in a horizontal tube furnace at 950 °C for the test duration of 16, 32 or 64 h.

The profile of each crept beam was determined by using the graduated eyepiece scale of an optical microscope with reference to its point of attachment and the horizontal lower face of the sample holder, to which it was made parallel prior to creep testing. Thus, the vertical deflection *y* from the original position was determined at 500 μm intervals in the horizontal *x* direction. The value of *K*_{e} was then obtained from equation (2.7) from the beam profile. Figure 3 linearizes the measurements of a profile on the basis of equation (2.7) and illustrates the excellent agreement with this equation.

The geometry of elastic deformation is replicated through a corresponding linear relationship between stress and strain rate in this creep situation.

Some specimens were creep tested after applying an alumina coating 100 nm thick to one surface of the beam to obtain evidence of any strain localization indicative of diffusional creep (Jaeger & Gleiter 1978). The coating was applied by plasma sputtering from an aluminium target in an argon/oxygen atmosphere, following plasma pre-treatment to clean the surface. The sample reached a maximum temperature of 150 °C during this deposition. The required duration of deposition was determined by first carrying out a long duration test (10 min at 570 V and 7 A), which gave a uniform and adhesive coating of thickness approximately 1.6 μm measured by a nano indentation technique. Based on this, a duration of deposition of 40 s was used to achieve the desired coating thickness of approximately 100 nm. Sample surfaces were examined by optical and scanning electron microscopy after creep.

## 4. Results

The values of *K*=*K*_{e} determined experimentally are listed in table 1 along with the beam and grain dimensions. Grains were essentially equiaxed in planes parallel to the large beam surfaces with these grain dimensions exceeding the beam thickness by factors between 1.4 and 2.8, corresponding to the situation in figure 2*c*. The theoretically calculated values of *K* from the first bracket of equation (2.6) that we now label *K*_{n} and similarly from equation (2.10) now labelled *K*_{b} are also listed for comparison. The data used in these calculations were: the lattice self-diffusion coefficient of copper at 950 °C, *D*=7.7×10^{−14} m^{2} s^{−1} and the atomic volume *Ω*=1.18×10^{−29} m^{3} (Frost & Ashby 1982), the acceleration due to gravity *g*=9.81 m s^{−2} and Boltzmann's constant *k*=1.38×10^{−23} J K^{−1}. It is apparent that in all cases the experimental value *K*_{e} lies between the theoretical derivations *K*_{n} and *K*_{b}. This is to be anticipated since the fluxes implicit in the calculation of *K*_{n} are not expected to provide the dominant contribution to creep. Correspondingly, the calculated value of *K*_{b} is expected to be an overestimate since it is based on the condition that depletion and deposition occurs along individual grain boundaries, all parallel to the axis of bending. As considered earlier, the experimental condition lies most closely to that detailed in §2*d*, where grain boundaries lie at all angles but always perpendicular to the plane of the largest beam surfaces. Thus, their overall contribution to the rate of bending is reduced to some fraction *β* of the estimate in equation (2.10). This parameter is the only one that has not yet been evaluated theoretically but the present results permit its experimental assessment. The addition of equations (2.6) and (2.10) with the introduction of the factor *β* leads to the relationship *K*_{e}=*K*_{n}+*βK*_{b}, and so the value of *β* in table 1 is calculated from . Not unexpectedly, there is some scatter in these results since there will be differences in grain boundary orientations between the different specimens. When the beams, however, are from the same sheet, as for the four specimens in rows 3–6 of table 1, the scatter is considerably reduced. More experiments are required to narrow down the estimate of *β* to a value corresponding to a fully randomized grain structure but averaging the results listed in table 1 would indicate that *β* has a value of about 0.06.

The experiments further confirmed the linear dependence of strain rate upon stress since the beams were subjected to stresses ranging from zero at the free end of the beam to a maximum at the supported end. Figure 3 confirms that the beam profile corresponded to this relationship described by equation (2.7).

The 100 nm alumina coating applied to some of the beams did not influence the creep performance significantly (e.g. compare *K*_{e} of test G in table 1 with those of tests I and J, where a coating was applied) and the surface appearance after test was often complex. As illustrated in figure 4, there was, however, evidence of preferential cracking of the coating at those grain boundaries across which the highest tensile stresses operated. This would be expected from the highly localized strain in the coating resulting from atom deposition at these substrate boundaries.

## 5. Discussion

The results of the experimental study provide support for the theoretical approach proposed for the analysis of the process of diffusional creep in bending. This analysis supplements the well-established standard early treatments of the effects of tensile and compressive stresses that took into account only the fluxes between grain boundaries of different orientation. The present study highlights the need to incorporate and extend the recent theoretical study of Burton (2002, 2003), when the grain size measured on the largest beam surface approaches or exceeds the dimension of the beam thickness.

Two limiting cases, represented by figure 2*a*,*b*, have been considered here that can be analysed without the incorporation of any new parameters and lead, respectively, to equations (2.6) and (2.10). These different formulations arise, not through any fundamental change in phenomena, but through a definitive change in flux patterns that can be appreciated by reference to figure 5. It is noted in figure 5*a*, for a grain at the upper surface, that fluxes are essentially between differently orientated grain boundaries, tending mainly towards horizontal directions when the grain size is small in relation to the beam thickness. This situation forms the basis of the derivation of equation (2.6). In contrast, when single grains occupy the entire beam cross-section, with the boundaries between them lying parallel to the axis of the bending moment as in figure 2*b*, the predominantly nearly vertical flux pattern takes the form illustrated by figure 5*b*. The analysis of this situation results in equation (2.10). Consideration of the internal stress redistribution (Burton 2003), through a computer-based analysis, indicates that the fluxes involve some withdrawal and deposition of vacancies, respectively, from the upper and lower surfaces of the beam to complement the major fluxes generated by different regions of the grain boundary acting as vacancy sources and sinks. In the intermediate case, where each grain occupies the entire thickness of the beam, but with several grains across its breadth, the grain boundaries do not lie generally parallel to the axis of the bending moment, and so these fluxes will be substantially reduced. To cope with this situation it is suggested in §2*d* that a dimensionless numerical parameter *β* is introduced and this is the only parameter that still requires theoretical evaluation.

The fundamental scientific basis now seems established but when the grain size *d* in planes parallel to the beam surfaces approaches the beam thickness *h* then the two flux patterns described will interact to produce new three-dimensional fluxes. Further progress might be achieved by computational methods, to provide a comprehensive picture of these fluxes that exist both between and along the grain boundaries. The two extreme cases, however, are amenable to the mathematical procedures given here and these provide useful limiting values that would establish the validity of any future overall analysis. In the meantime, it seems not unreasonable to assume that approximate answers for the intermediate cases may be obtained by addition of equations (2.6) and (2.10) with a new factor *β* inserted into the latter equation to provide a single description of the processes involved. Rounding off the numerical constant to 2400 and incorporating this additional factor *β* into equation (2.10) for the case where each grain extends through the thickness of the beam but with many grains occupying the breadth of the beam, we arrive at a combined equation(5.1)From equation (5.1) it is apparent that the predominant fluxes occur between grain boundaries of different inclination when 40*βd*<*h* whereas the fluxes are primarily along individual vertical boundaries when 40*βd*>*h*. For the condition represented by figure 2*c*, the value of *β* will depend upon any preferred grain boundary inclination. The present experimental study suggests that *β* has an approximate value of about 0.06 for random orientation, so it is likely that the transition in predominance of vertical fluxes occurs when *h* is reduced to a size approximately 2*d*.

The deflection *y* plotted against the inverse square of the beam thickness 1/*h*^{2} from equations (2.6) and (2.10), with the factor *β* incorporated, and (5.1) is shown in figure 6. Curve (a), from equation (2.6), represents the effect of the fluxes between differently oriented grain boundaries and curve (b), from equation (2.10), the effect of fluxes centred along the vertical boundaries. Note how the beam deflection is enhanced when the beam thickness *h* is small. Curve (c) is constructed to show the combined effect, with the assumption that the respective contributions of the mainly horizontal and mainly vertical fluxes are approximately additive. This illustrates the enhanced importance of beam thickness *h* as its magnitude decreases such that the deflection eventually becomes proportional to 1/*h*^{3}. When these features are taken into account, the creep deflection of thin beams under small bending moments may be evaluated over a wide range of beam and grain dimensions.

It is implicit throughout the discussion that the analysis depends on the operation of a diffusional creep process. Where other creep mechanisms may predominate, the analysis will be inapplicable. For pure aluminium, there is no confirmed evidence for the occurrence of diffusional creep and the alternative grain size independent mechanism proposed by Harper & Dorn (1957) has been questioned (Blum & Maier 1999; McNee *et al*. 2001). Copper was specifically chosen for the current experimental programme because it is established that its grain boundaries can act as vacancy sources and sinks (Barnes *et al*. 1958), which is a pre-requisite for the occurrence of diffusional creep in polycrystals. There is also substantial data on the effect of grain size in accord with the predictions of this mechanism (Burton & Greenwood 1969). Measurements of creep cavity growth in copper (Hull & Rimmer 1959; Hanna & Greenwood 1982) provide additional evidence of vacancy diffusion. In the present experimental study, three distinct observations are notable. The first is that the beam profiles after creep retained the same geometry as that pertaining to their deformation at lower temperatures in the elastic range. This feature established the linear dependence of the rate of creep strain with stress. The second is that the creep rates are consistent with a diffusional creep process. The third is the supporting observation of the cracking of the thin brittle alumina coatings applied to some of the specimens. Cracks in the coatings occurred predominantly above grain boundaries of the substrate in the top surface nearly perpendicular to the beam length where a highly localized strain is expected to develop through atom deposition under the tensile stress.

## 6. Conclusions

The study has provided a formulation for the bending of thin beams under low stresses at elevated temperatures when a diffusional creep process operates. A limited experimental investigation has validated the occurrence of this process and given support to the theoretical treatment, though a more extensive programme would be required to support the full range of behaviour that the theory predicts. In particular, the theoretical development has identified the importance of beam and grain geometry. When the grain size is small in comparison with the beam thickness, then it is appropriate to analyse the rate of beam deflection in terms of standard diffusional creep formulae represented by equation (2.4) for tension and compression applied, respectively, to the upper and lower halves of the beam separated by the neutral axis. When the grain size approaches the dimension of the beam thickness, however, a new formulation is required since the controlling diffusional fluxes are then concentrated along each individual vertical boundary rather than between different grain boundaries. Burton (2003) has provided a basis for such analysis by considering the case where diffusion takes place along a single grain boundary lying parallel to the axis of the bending moment, noting that this causes a rotation between the adjacent grains.

The present study has highlighted a third case where the individual grains occupy the beam thickness but several grains exist across the breadth of the beam. This implies that the grain boundaries then lie perpendicular to the large beam surfaces at all angles to the axis of the bending moment and it is proposed that the contribution to the rate of deflection caused by vertically oriented fluxes is then reduced by a dimensionless numerical factor *β*. The present experimental study has permitted an approximate evaluation of its magnitude. The scientific basis established for its role may permit its eventual theoretical evaluation by computational methods.

The formulation derived should allow the deflection of thin beams, when diffusional creep is operative, to be calculated for all beam and grain geometries. One of the important consequences of this approach is the enhancement of the beam defection rate when the grain dimensions are such that each individual grain occupies the entire thickness of the beam.

## Acknowledgements

This work formed part of a Ph.D. research programme by V.S. with the support of a project studentship through a UK EPSRC Research Grant (GR/N00296) and a University of Sheffield Fee Bursary.

## Footnotes

↵† Present address: Otto von Guericke Universität, Institut für Werkstoff und Fügetechnik, Postfach 4120, 39016 Magdeburg, Germany.

- Received December 15, 2005.
- Accepted February 22, 2006.

- © 2006 The Royal Society