Advanced Search
RSS LinkedIn Twitter

Journal Archive

Platinum Metals Rev., 2001, 45, (2), 74

High Temperature Mechanical Properties of the Platinum Group Metals

Elastic Properties of Platinum, Rhodium and Iridium and Their Alloys at High Temperatures

  • By Jürgen Merker a*
  • David Lupton a
  • Michael Töpfer b
  • Harald Knake b
  • a
    W. C. Heraeus GmbH & Co. KG, Hanau, Germany
  • b
    Friedrich Schiller University, Jena, Germany
  • * now with KM Europa Metal AG, Osnabrück, Germany

SHARE THIS PAGE:

Article Synopsis

The platinum group metals are well suited for use at extremely high temperatures under mechanical loads and simultaneous corrosive attack. They have high melting points, excellent chemical stability and are highly resistant to oxidation. When using these materials in the design of components it is necessary to have data available on their elastic properties as a function of temperature. In this paper, investigations are presented into the temperature dependence of Young’s modulus, the modulus of rigidity and Poisson’s ratio for platinum, platinum alloys, rhodium and iridium. Measurements were carried out at the Friedrich Schiller University, Jena, using a resonance technique. Influences from both the microstructure and the alloying elements on the elastic properties and their temperature dependence were found.

Platinum group metals (pgms) and in particular platinum alloys are indispensable in many fields of industrial application because of their outstanding physical and chemical properties. Components made from these materials are frequently subjected to extremely complex mechanical loading at high temperatures, often being simultaneously exposed to corrosive attack. A major aspect in the design of components to be used, for example, in the glass industry, in aerospace technology and in single crystal growing is to ensure optimum service life while using the least possible quantity of noble metal. In addition to data on the stress-rupture strength and creep properties (1), the design engineer requires values for the elastic properties of these materials up to very high temperatures.

However, very little data on the temperature dependence of the elastic constants of the platinum metals and their alloys is found in the literature. Apart from the published investigations (23), a current monograph gives the elastic properties of platinum alloys at room temperature (4). The elastic moduli of pure pgms as a function of temperature are given in the same publication (4) with reference to work carried out by Reinacher in the 1960s (57), and published more recently (8). Comprehensive work on the temperature dependence of the elastic moduli of metals and alloys was published by Köster in the 1940s (911). However, in view of the state of technical development at that time, these results can only be regarded as a guide.

Experimental Procedure

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 (1214). 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 elastic properties of metal and alloy samples determined at various temperatures in a high temperature furnace. The beam is supported on alumina knife-edges. Oscillations are generated with the aid of a network analyser, transformed into mechanical oscillations by piezo sensors and transmitted to the beam via alumina fibre couplers

The elastic properties of metal and alloy samples determined at various temperatures in a high temperature furnace. The beam is supported on alumina knife-edges. Oscillations are generated with the aid of a network analyser, transformed into mechanical oscillations by piezo sensors and transmitted to the beam via alumina fibre couplers

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:

Frequency of torsional oscillations

(ii)

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

Frequency of longitudinal oscillations

(iii)

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

(iv)

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

If FLn2 from Equation (iv) is developed into a power series in εn, we obtain:

(v)

with kt = −½ V2 and

Equation (v) thus obtained shows clearly the dependence of the factor FLn 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 (FLn =1) is insufficient and that the more precise modelling permits the determination of Young’s modulus and Poisson’s ratio (VD) 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 VD from the dispersion of the characteristic longitudinal frequencies and also as VE/G from the relationship VE/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*.

Elastic Properties of Platinum

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.

Table I

Temperature Dependence of the Elastic Properties E, G and v for Platinum

T, °C E, GPa VD G, GPa VE/G
25 164.6 0.396 54.2 0.518
200 159.3 0.389 52.9 0.506
400 153.3 0.401 51.1 0.500
500 149.1 0.403 50.0 0.491
600 145.6 0.406 48.9 0.489
700 141.9 0.409 47.7 0.487
800 137.8 0.396 46.6 0.479
900 132.7 0.399

Fig. 2

Temperature dependences of: (a) Young’s modulus, E, and the modulus of rigidity, G, for platinum; (b) Poisson ‘s ratio, v, for platinum. The value VD was determined from the dispersion of the characteristic longitudinal frequencies; while vD was determined from the relationship VE/G = E/(2G) – l

Temperature dependences of: (a) Young’s modulus, E, and the modulus of rigidity, G, for platinum; (b) Poisson ‘s ratio, v, for platinum. The value VD was determined from the dispersion of the characteristic longitudinal frequencies; while vD was determined from the relationship VE/G = E/(2G) – l

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

Elastic Properties of Rhodium

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 VD and VE/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.

Table II

Temperature Dependence of the Elastic Properties E, G and v for Forged Rhodium

T, °C E, GPa VD G, GPa VE/G
25 372.4 0.266 151.7 0.227
200 355.8 0.268 144.3 0.233
400 332.1 0.267 134.2 0.237
500 321.4 0.274 129.5 0.241
600 310.4 0.278 124.7 0.245
700 299.4 0.282 120.3 0.246
800 291.0 0.287 116.2 0.252
900 281.6 0.293 111.9 0.258
1000 271.5 0.296 107.3 0.265
1100 260.6 0.294
1200 246.9 0.296

Fig. 3

Temperature dependence of:

(a) the elastic properties E and G for forged rhodium

(b) Poisson’s ratio for forged rhodium

Temperature dependence of:  (a) the elastic properties E and G for forged rhodium  (b) Poisson’s ratio for forged rhodium

Elastic Properties of Iridium

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, VD and VE/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).

Table III

Temperature Dependence of the Elastic Properties E, G and v of As-cast Iridium

T, °C E, GPa VD G, GPa VE/G
25 525.5 0.254 218.2 0.204
200 507.4 0.260 209.9 0.209
400 483.6 0.261 199.4 0.213
500 472.7 0.265 194.3 0.216
600 461.2 0.268 189.5 0.217
700 450.5 0.271 184.5 0.221
800 439.9 0.275 179.7 0.224
900 429.5 0.279 174.9 0.228
1000 417.5 0.281 170.3 0.226
1100 406.1 0.279
1200 394.4 0.286
1300 384.2 0.309

Fig. 4

Temperature dependence of:

(a) the elastic properties E and G for as-cast iridium

(b) Poisson’s ratio for as-cast iridium

Temperature dependence of:  (a) the elastic properties E and G for as-cast iridium  (b) Poisson’s ratio for as-cast iridium

The values for Poisson’s ratio VD and VE/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 (ERT = 532 GPa, E1000°C = 424 GPa) and the modulus of rigidity (GRT = 223 GPa, G1000°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 VD and VE/G (∼ 35 per cent), thus indicating a high degree of anisotropy caused by the deformation microstructure from the hot rolling.

Elastic Properties of Platinum-Rhodium Alloys

The elastic properties E, G, VD and VE/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 VD and VE/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).

Table IV

Elastic Properties E, G, vD and vE/G for As-cast Platinum-Rhodium Alloys at Selected Temperatures

T, °C Pt-10%Rh Pt-20%Rh Pt-30%Rh
E, GPa VD G, GPa VE/G E, GPa VD G, GPa VE/G E, GPa VD G, GPa VE/G
25 212.6 0.365 78.0 0.363 245.9 0.342 91.6 0.342 277.7 0.324 104.8 0.325
200 206.3 0.368 75.4 0.368 236.6 0.346 87.8 0.347 265.7 0.330 99.9 0.330
400 197.9 0.372 72.1 0.372 224.7 0.351 83.3 0.349 251.0 0.334 94.0 0.335
500 193.3 0.376 70.5 0.371 218.8 0.353 80.9 0.352 243.9 0.338 91.1 0.339
600 188.7 0.376 68.7 0.373 213.0 0.355 78.6 0.355 236.6 0.340 88.2 0.341
700 183.9 0.378 66.9 0.374 207.2 0.358 76.3 0.358 229.5 0.343 85.5 0.342
800 179.2 0.379 65.2 0.374 201.0 0.359 74.1 0.356 222.1 0.345 82.7 0.343
900 175.0 0.383 63.4 0.380 195.5 0.360 72.0 0.358 215.7 0.346 80.0 0.348
1000 169.7 0.381 189.8 0.362 69.8 0.360 209.3 0.350 77.5 0.350
1100 164.9 0.385 184.6 0.367 67.7 0.363 202.8 0.352 74.7 0.357
1200 179.2 0.380 195.4 0.358

Fig. 5

Dependence of:

(a) Young’s modulus on temperature for as-cast Pt-Rh alloys

(b) the modulus of rigidity on temperature for as-cast Pt-Rh alloys

(c) Poisson’s ratio on temperature for as-cast Pt-Rh alloys

Dependence of:  (a) Young’s modulus on temperature for as-cast Pt-Rh alloys  (b) the modulus of rigidity on temperature for as-cast Pt-Rh alloys  (c) Poisson’s ratio on temperature for as-cast Pt-Rh alloys

Fig. 6

Phase diagram of the binary systems:

(a) Pt-Rh system (17); (b) Pt-Ir system (17, 20)

Phase diagram of the binary systems:  (a) Pt-Rh system (17); (b) Pt-Ir system (17, 20)

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

Dependence of Young’s modulus on rhodium content for as-cast Pt-Rh alloys at various temperatures

Dependence of Young’s modulus on rhodium content for as-cast Pt-Rh alloys at various temperatures

Elastic Properties of Platinum-Iridium Alloys

The elastic properties E, G, VD and VE/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 VD and VE/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.

Table V

Elastic Properties E, G, vD and vE/G for As-cast Platinum-Iridium Alloys at Selected Temperatures

T, °C Pt-10%lr Pt-20%lr Pt-30%lr
E, GPa VD G, GPa VE/G E, GPa VD G, GPa VE/G E, GPa VD G, GPa VE/G
25 202.3 0.378 73.4 0.378 233.3 0.368 85.5 0.364 263.3 0.346 97.5 0.350
200 196.6 0.382 71.1 0.382 224.8 0.368 82.2 0.367 253.6 0.351 93.6 0.352
400 188.3 0.382 68.1 0.382 214.3 0.371 78.2 0.370 240.8 0.354 88.6 0.359
500 183.9 0.384 66.4 0.385 209.0 0.373 76.2 0.371 234.7 0.356 86.2 0.361
600 178.8 0.381 64.8 0.381 201.6 0.379 73.9 0.364 228.5 0.358 83.9 0.362
700 173.6 0.382 62.8 0.381 196.2 0.378 71.9 0.364 222.5 0.361 81.5 0.365
800 170.7 0.389 58.1 192.3 0.384 70.1 0.372 216.1 0.359 79.3 0.363
900 166.4 0.391 186.9 0.378 68.2 0.370 210.2 0.363 76.9 0.367
1000 162.2 0.396 182.5 0.386 66.2 0.378 204.5 0.368 74.7 0.369
1100 157.1 0.400 176.9 0.387 64.1 0.380 198.5 0.368 72.5 0.369
1200 150.8 0.393 171.1 0.386 192.2 0.372
1300 165.0 0.393 185.3 0.374
1400 176.8 0.375

Fig. 8

Dependence of: (a) Young’s modulus on temperature for as-cast Pt-Ir alloys; (b) the modulus of rigidity on temperature for as-cast Pt-lr alloys

Dependence of: (a) Young’s modulus on temperature for as-cast Pt-Ir alloys; (b) the modulus of rigidity on temperature for as-cast Pt-lr alloys

Fig. 8(c)

Dependence of Poisson’s ratio on temperature for as-cast Pt-Ir alloys

Dependence of Poisson’s ratio on temperature for as-cast Pt-Ir alloys

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

Dependence of Young’s modulus on iridium content for as-cast Pt-lr alloys at various temperatures

Dependence of Young’s modulus on iridium content for as-cast Pt-lr alloys at various temperatures

Conclusions

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.

BACK TO TOP

References

  1. 1
    B. Fischer, A. Behrends, D. Freund, D. F. Lupton and J. Merker, Platinum Metals Rev ., 1999, 43, ( 1 ), 18
  2. 2
    D. F. Lupton, J. Merker and B. Fischer, 3rd European Precious Metals Conf., Florence, Italy, 17–19 September, 1997, Eurometaux, Brussels
  3. 3
    J. Merker, D. F. Lupton, W. Kock and B. Fischer, 18th Int. Congress on Glass, San Francisco, U.S.A., July 5–10, 1998
  4. 4
    AG Degussa (ed.), “ Edelmetall-Taschenbuch ”, Höthig-Verlag, Heidelberg, Germany, 1995
  5. 5
    G. Reinacher, “ Iridium ”, eds. E. Rabald and D. Behrens, Dechema-Werkstoff-Tabelle, “ Physikalische Eigenschaften ”, Frankfurt, 1966
  6. 6
    G. Reinacher, “ Platin—Platinlegierungen ”, op. cit ., (Ref. 5)
  7. 7
    G. Reinacher, “ Rhodium ”, op. cit ., (Ref. 5)
  8. 8
    M. Winter, The Periodic Table on the website http://www.webelements.com/, University of Sheffield, England
  9. 9
    W. Köster, Z. Metallkd, 1948, 39, 1
  10. 10
    Köster and W. Rauscher, Z. Metallkd, 1948, 39, 111
  11. 11
    W. Köster, Z. Metallkd ., 1948, 39, 145
  12. 12
    H. Knake, W. Meuche and H. Reichardt, Wiss. Z. d Friedrich-Schiller-Universität Jena, Math.-Naturwiss.-Reihe, 1981, 30, ( 6 ), pp. 955 – 960
  13. 13
    H. Knake, H. Reichardt and M. Töpfer, op. cit ., (Ref. 12), pp. 949 – 953
  14. 14
    A. Jordanov and H. Knake, Deutsche Gesellschaft für zerstörungsfreie Prüning, 1996, 55, pp. 9 — 16
  15. 15
    H. Knake, S. Schüssler and M. Töpfer, Versuchsanleitung: Ermitdung elastischer Kennwerte (Experimental Guide: The Determination of Elastic Properties), Dept of Applied Mechanics, Technical Institute, Friedrich Schiller University of Jena
  16. 16
    TAPP Database, E. S. Microware Inc., Hamilton, OH, 1991
  17. 17
    T. B. Massalski (ed.) et al., “ Binary Alloy Phase Diagrams ”, 2nd Edition Plus Updates, ASM International, Ohio/National Institute of Standards and Technology, 1996
  18. 18
    F. Aldinger and A. Bischoff, in “ Festigkeit und Verformung bei hoher Temperatur ”, B. Ilschner (ed.), DGM-Informationsgesellschaft, Oberursel, Germany, 1983, pp. 161 – 195
  19. 19
    D. F. Lupton, J. Merker, B. Fischer and R. Völkl, 24th Annual Conf. of International Precious Metals Institute, Williamsburg, U.S.A., 11–14 June, 2000
  20. 20
    S. N. Tripathi and M. S. Chandrasekharaiah, J. Less-Common Met ., 1983, 91, 251
  21. 21
    H. Knake and M. Töpfer, Thüringer Werkstofftag of the Technical University of Ilmenau, 15 March 1999

The Authors

Jürgen Merker was a Development Project Manager with W. C. Heraeus GmbH & Co. KG until May 2000. His main activities were in the processing and characterisation of platinum materials and the pgms for high temperature applications.

David Lupton is a Development Manager of the Engineered Materials Division with W. C. Heraeus GmbH & Co. KG. He is particularly involved in the metallurgy of the pgms, refractory metals and other special materials.

Michael Töpfer is a Technical Research Assistant in the Technical Institute of the Friedrich Schiller University of Jena. His major field of interest is the determination of material properties by dynamic oscillation techniques.

Harald Knake is Professor of Applied Mechanics at the Friedrich Schiller University of Jena and specialises in the elastic properties of materials.

Read more from this issue »

BACK TO TOP

SHARE THIS PAGE: