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The resonance method used to determine the elastic properties is a non-destructive, dynamic technique characterised by its high precision. It is applicable to all materials which can be stimulated to mechanical oscillation, see Figure 1. This state-of-the-art process is suitable for determining elastic constants of materials with isotropic, cubic or transverse-isotropic mechanical behaviour in a temperature range from -30°C to 1650°C (12–14). In order to derive these properties with a high degree of precision from the characteristic frequencies (of oscillation) on specimens using the resonance method, it is necessary to know the mathematical relationships between these quantities as exactly as possible. The frequency equations derived from the basic theory of oscillating beams, which are commonly used for such evaluations, do not give the required accuracy. The necessary relationships can therefore only be derived on the basis of the known three dimensional Equation of motion from the linear theory of elasticity. Under the condition that the body is ideally elastic, homogeneous and isotropic, we derive for Young’s modulus (E) and Poisson’s ratio (V):

(i)

where = displacement vector, ρ = density

Fig. 1

The solutions of this system of differential equations must also fulfil the boundary conditions, that is: zero stress over the complete surface in the practical experimental arrangement.

If the partial spectra of only the torsional and longitudinal oscillations are evaluated, we obtain the frequency Equations:

(ii)

with F_Tn = 1 for circular cylindrical beams, G = modulus of rigidity, 1 = beam length, n = order

(iii)

where the factor F_Ln for circular cylindrical beams is derived from the Equation:

(iv)

(J ₀, J ₁ are Bessel functions of the first kind, a = radius)

If F_Ln² from Equation (iv) is developed into a power series in ε_n, we obtain:

(v)

with k_t = −½ V² and

Equation (v) thus obtained shows clearly the dependence of the factor F_Ln on V and na/1 which is caused by the coupling of the longitudinal and transverse oscillations (dispersion). However, it also shows that the accuracy of the basic theory (F_Ln =1) is insufficient and that the more precise modelling permits the determination of Young’s modulus and Poisson’s ratio (V_D) from a measured partial spectrum of the longitudinal characteristic frequencies alone. The modulus of rigidity can be determined from the measured partial spectrum of the torsional oscillations according to Equation (ii).

The temperature dependence of the elastic constants was determined in a high temperature furnace. The cylindrical sample beam is supported on alumina knife-edges, on the right of each diagram in Figure 1. The oscillations were generated using a network analyser, transformed into mechanical oscillations via piezo sensors (on the left of each diagram) and transmitted to the beam via fine alumina fibre couplers. The oscillations of the sample are detected via a further alumina coupler attached to a second piezo sensor (not shown) and transmitted back to the network analyser for processing. The alumina fibre coupler is placed at the centre of the circular end surface of the sample if longitudinal oscillations are to be analysed (left-hand diagram) or at the circumference of the end surface for torsional oscillations (right-hand diagram). The resonant frequencies and the half-peak width of the amplification function (determining damping) can be recorded. The sample beam requires time to achieve a stable temperature between measurements to avoid errors.

The elastic constants, Young’s modulus E, the modulus of rigidity G and Poisson’s ratio V were measured on platinum, iridium and rhodium and on alloys of platinum with 10,20 and 30 weight per cent of iridium and rhodium at both room temperature and elevated temperatures, by the resonance method. Poisson’s ratio, V, was determined as V_D from the dispersion of the characteristic longitudinal frequencies and also as V_E/G from the relationship V_E/G = E/(2G) – 1. If the two values are the same the sample is isotropic or quasi-isotropic.

All the materials could be measured at temperatures where the loss factor of internal friction (damping), d, was not greater than 10^-2. At higher values of loss factor it was not possible to determine the resonance point reliably from the amplification function*.

Measurements with reproducible results were possible up to 800°C and in the case of repetition up to 900°C. Both Young’s modulus and the modulus of rigidity of platinum show a steady decrease with increasing temperature, see Figure 2. This is partly in contrast to earlier determinations (2, 20) which showed a steady decrease in Young’s modulus from 174 GPa at room temperature to 168 GPa at 400°C during a first measurement, followed by a decrease to 146 GPa at 500°C and then a steady decrease to 135 GPa at 700°C. This effect was found to be irreversible. Repeat measurements showed a Young’s modulus of 155 GPa at room temperature which decreased continuously to 127 GPa at 800°C. The current measured values given in Table I were determined on as-cast platinum rods, and show relatively good agreement with the repeat determinations and with values measured at temperatures ≥ 500°C (2). The irreversible decrease in Young’s modulus found in the earlier work was apparently due to a deformation structure in the material which was removed by recrystallisation during the measurement.

Fig. 2

It is interesting that the values of Young’s modulus determined at room temperature on the specimen with the apparently deformed structure correspond reasonably well with the values in the literature (4, 9, 16), whereas the values determined on platinum in the recrystallised state (155 GPa) and the as-cast state (165 GPa) are lower. Furthermore, Young’s modulus was found to be dependent on the purity of the platinum. On undeformed specimens, the following values were determined: 169 GPa with 99.99% Pt, 172 GPa with 99.95% Pt and 177 GPa with 99.9% Pt.

The value for Poisson’s ratio determined from the dispersion of the longitudinal characteristic frequencies V_D is approximately constant over the whole temperature range, whereas the value of Poisson’s ratio determined from the elastic moduli V_E/G decreases slightly with increasing test temperature. The difference between V_D and V_E/G indicates some influence from anisotropy which may be related to the primary solidification structure.

At room temperature, Young’s modulus for rhodium (373 GPa to 384 GPa (2)) is considerably higher than that for platinum. With increasing temperature Young’s modulus decreases in an approximately linear manner to 280 GPa (at 1000°C (2) and 248 GPa (at 1200°C. The modulus of rigidity also shows a linear decrease with increasing temperature.

A comparison of the current measurements, also carried out on forged and subsequently machined rhodium rods (Table II and Figure 3), and earlier investigations (2) shows that for Young’s modulus, the earlier measurements are reproducible at about 10 GPa higher than current values. The earlier values for Poisson’s ratio V_D and V_E/G differ by only about 5 per cent (2), while in the current measurements the difference is 12 to 15 per cent. This means that the anisotropy is significantly less for those samples with the higher Young’s modulus. This difference is presumably related to the fact that the earlier samples (2) were more severely deformed by forging because a larger ingot size had been used. The values for Young’s modulus given in the literature (4, 7) also indicate that the microstructure is relatively severely deformed.

Fig. 3

Iridium has the highest Young’s modulus of all face-centred cubic metals and the highest modulus of rigidity of all metals. The elastic properties E, G, V_D and V_E/G measured on iridium in the as-cast state are summarised in Table III. Young’s modulus and the modulus of rigidity decrease linearly from room temperature with increasing temperature, see Figure 4. At 1000°C the modulus of rigidity was still 170 GPa and Young’s modulus 417 GPa. Young’s modulus could be measured up to 1300°C (382 GPa).

Fig. 4

The values for Poisson’s ratio V_D and V_E/G increase with increasing test temperature. The difference between the two values was about 18 per cent. This indicates marked anisotropy associated with the primary as-cast microstructure. A comparison of these results with previous investigations (2) shows that deformation by hot rolling leads to somewhat higher values for Young’s modulus (E_RT = 532 GPa, E_1000°C = 424 GPa) and the modulus of rigidity (G_RT = 223 GPa, G_1000°C = 173 GPa).

These prior values correspond relatively well with data from the literature (4, 9, 16). However, although the increase in Poisson’s ratio with increasing temperature measured by both sets of investigations corresponds qualitatively fairly closely, more substantial discrepancies are determined between V_D and V_E/G (∼ 35 per cent), thus indicating a high degree of anisotropy caused by the deformation microstructure from the hot rolling.

The elastic properties E, G, V_D and V_E/G determined for alloys Pt-10%Rh, Pt-20%Rh and Pt-30%Rh as a function of temperature for specimens in the as-cast condition, are presented in Table IV. Young’s modulus and the modulus of rigidity decrease linearly with increasing temperature, see Figures 5a and 5b. The values for Poisson’s ratio V_D and V_E/G show only slight differences which become negligible at high rhodium concentrations, Figure 5c. In contrast to the large discrepancies found for the pure metals, these small differences may be due to the influence of solid solution formation during the development of the primary cast microstructure. The damping showed maxima in specific regions for the various alloys. This indicates a miscibility gap in the binary Pt-Rh system similar to that shown in Figure 6 (17).

Fig. 5

Fig. 6

The higher values in the literature for Young’s modulus at room temperature (4, 18) have a high probability of being attributable to prior deformation of the specimens. Figure 7 shows the effect of rhodium content on Young’s modulus of specimens in the as-cast condition at various test temperatures. The greatest effect on Young’s modulus due to rhodium additions is observed for concentrations of up to ∼ 10 weight per cent. The rate of increase is less marked at higher rhodium contents. A similar effect has been found for the stress-rupture strength of Pt-Rh alloys (19).

Fig. 7

The elastic properties E, G, V_D and V_E/G determined on specimens of as-cast alloys Pt-10%Ir, Pt-20%Ir and Pt-30%Ir are shown in Table V as functions of temperature. Young’s modulus and the modulus of rigidity decrease linearly with increasing temperature, see Figure 8. The differences between the values for Poisson’s ratio V_D and V_E/G are somewhat greater for the Pt-Ir alloys than for the Pt-Rh alloys. At this stage, it is not clear why the difference for Pt-20%Ir is so large. The behaviour of the Pt-Ir alloys also indicates a maximum in damping corresponding to the miscibility gap (Figure 6b (17, 20)). This maximum was more clearly distinguished than that found in the Pt-Rh system.

Fig. 8

Fig. 8(c)

In Figure 9 the influence of the iridium content on Young’s modulus at various test temperatures is shown for as-cast specimens. The modulus increases nearly linearly with iridium content up to 30 weight per cent. Comparing values for Young’s modulus shows generally good agreement with results of prior investigations (2) and data from the literature (4, 18). The relatively small discrepancies are attributable to different processing conditions.

Fig. 9

The results of investigations carried out using the resonance method show that Young’s modulus and the modulus of rigidity of platinum, rhodium and iridium and various platinum alloys in the as-cast condition decrease linearly with increasing test temperature. The gradients of the lines are dependent on the compositions of the alloys.

The microstructural state of the material resulting from prior deformation influences in particular the magnitude of Young’s modulus and the anisotropic behaviour of Poisson’s ratio. Poisson’s ratio is also influenced by the state of the primary as-cast microstructure.

A marked increase in damping was observed in the regions of the miscibility gaps. This suggests that the resonance method could be a sensitive technique for determining miscibility gaps in materials which can be subjected to mechanical oscillations and whose basic damping, d, is less than 10^-3 (21). Further microstructural and crystallographic investigations are required to confirm these correlations.