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Correlation between ultrasonic shear wave velocity and Poisson’s ratio for isotropic solid materials
Acta Materialia 51 (2003) 2417–2426 www.actamat-journals.com
Correlation between ultrasonic shear wave velocity and Poisson’s ratio for isotropic solid materials Anish Kumar a,∗, T. Jayakumar a, Baldev Raj a, K.K. Ray b a
Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India b Indian Institute of Technology, Kharagpur 721302, India Received 19 August 2002; received in revised form 23 January 2003; accepted 23 January 2003
Abstract A new correlation between ultrasonic shear wave velocity and Poisson’s ratio has been established for isotropic solid materials, based on the data generated experimentally and collected from the literature. Poisson’s ratio has been found to decrease with increasing ultrasonic velocity in various solid materials such as metals and alloys, ceramics and glasses, intermetallics and polymers. The slope of the plot of the ultrasonic velocity against Poisson’s ratio is found to be almost constant for any given alloy system with different microstructures associated with various heat treatments, alloying elements, grain size, temperature effect, etc. Further, it has been demonstrated that ultrasonic shear wave velocity is a better parameter for materials characterization as compared to longitudinal wave velocity. 2003 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Ultrasonic velocity; Poisson’s ratio; Modulus; Elastic constant
1. Introduction Elastic constants are physical properties that relate stress and strain. Since solids resist both volume change and shape change, they have at least two independent elastic constants. A low symmetry crystal may have as many as 21 independent elastic constants, but polycrystalline isotropic materials (i.e. texture free materials, as considered in the present study) can be characterized by only two independent elastic constants. Many functional and
Corresponding author. Tel.: +91 (4114) 280208; fax: +91 (4114) 280356. E-mail address: [email protected] (A. Kumar). ∗
differential relationships between elastic constants and other elastic-constant pairs have been reported [1–4], which emerge from the theory of elasticity and from the physical meanings of the constants. However useful these relationships may be, they only represent the functions relating a dependent variable to two independent variables. Two independent elastic constants are required to characterize the isotropic materials with the presently available relations. Hence, identification of any new relationship that reduces the number of required independent elastic constants to one is beneficial. This hypothesis is based on the belief that there must be a hidden relationship between the two independent elastic constants that is based on the microstructure, atomic bonding, electronic
1359-6454/03/$30.00 2003 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. doi:10.1016/S1359-6454(03)00054-5
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configurations, etc. A number of researchers have given the empirical relationships to relate shear modulus (G), Young’s modulus (E) and bulk modulus (B) for various elements [1–3]. Poisson’s ratio, defined as the lateral contraction per unit breadth divided by the longitudinal extension per unit length in simple tension [5], is reported to provide more information about the character of the bonding forces than any of the other elastic coefficients [4]. Poisson’s ratio (n) is related to E and G by n⫽
E ⫺1 2G
(1)
and to ultrasonic longitudinal wave velocity (V L) and shear wave velocity (V T) by n⫽
V2L⫺2V2T . 2(V2L⫺V2T)
(2)
V L and V T are related to E and G by
冉
冊
4 3rV2T V2L⫺ V2T 3 E⫽ (V2L⫺V2T)
(3)
G ⫽ rV2T
(4)
where r is the density. The differential relationships of n with E and G and with V L and V T can be obtained by differentiating Eqs. (1) and (2), respectively. The differential forms of the above equations are:
冉
冊 冉
dE dG 1 dn ; ⫽ ⫺ n (1⫺2G / E) E G
(5)
冊
dVL dVT 2V2LV2T dn ⫺ . ⫽ 2 n (VL⫺V2T)(V2L⫺2V2T) VL VT
(6)
It can be seen from Eqs. (5) and (6), that the variation in Poisson’s ratio depends upon the variations in E and G and their relative values (Eq. (5)). Similarly, the relative variation in V L and V T also affects the variation in Poisson’s ratio (Eq. (6)). It is to be noted that in any material, E and G, as well as V L and V T, tend to vary in the same direction (increase or decrease). Hence, it is difficult to deduce from Eqs. (5) and (6) whether the variation of n would be in the same or opposite
direction to that of the elastic constants, E and G. The nature of the variation of n would therefore depend on whether G or E is affected more by metallurgical variables. If the rate of increase of E (or V L) is more than that of G (or V T), then the right hand side of Eqs. (5) and (6) will be positive, and hence n will vary in the same direction as that of E and G (or V L and V T). Similarly, if the rate of increase of G (or V T) is more than that of E (or V L), the right hand side of Eqs. (5) and (6) will be negative and hence n will vary opposite to E and G (or V L and V T). From the first law of thermodynamics, it can be seen that E, B and G are positive, and since B = E / 3(1⫺2n) and G = E / 2(1 + n), it follows that ⫺1 ⬍ n ⬍ 0.5. Further, even though the laws of thermodynamics are not violated by a hypothetical isotropic elastic solid with ⫺1 ⬍ nⱕ0, such negative values have been observed in foam structures and anisotropic materials only [6]. Thus, in isotropic solids, n is bounded practically by 0 and 1/2. For n = 1 / 2, no volume change occurs during deformation. Koister and Franz [4] have reviewed in detail the effect of various metallurgical parameters, such as alloying of Zn to Cu, alloying of Cu to Ag and alloying of Ni to Fe, etc, on Poisson’s ratio. They have also reviewed various methods for the determination of Poisson’s ratio and have concluded that the ultrasonic velocity measurement based dynamic method, which gives the adiabatic Poisson’s ratio, is the most accurate of all methods. This dynamic method involves the propagation of a low amplitude stress (ultrasonic) wave (⬍10 kPa) through the specimen and the measurement of transit time of the stress wave, which is then used for the determination of the elastic properties [1]. Other methods for the measurement of Poisson’s ratio include static methods, such as measurement of change in volume and length, determination of Young’s and shear moduli from bending and torsion tests, testing of helical spring and dynamic methods like measurement of natural frequencies of vibration, vibration of a spring under load and resonance frequencies of flexural vibrations [4]. Recently, Tervo et al. [7] have studied the effect of nitrogen alloying on elastic coefficients of austenitic stainless steels and showed that nitrogen in
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excess of 1.5 at.% increases the shear modulus and decreases the Poisson’s ratio. In the present study, the authors have studied the effect of ultrasonic shear wave velocity on Poisson’s ratio. The data have been collected from the literature for various metals, alloys, intermetallics, ceramics, glasses and polymers. Further, it is always reported in the textbooks of metallurgy and materials science that elastic moduli and Poisson’s ratio are material properties and insensitive to the microstructural features. However, in this paper, we have also shown the effect of the microstructural features on elastic moduli and Poisson’s ratio, by carrying out the ultrasonic velocity measurements in various alloy systems, such as Ti–6Al– 4V, Ti–4.5Al–3Mo–1V, Inconel 625, Cu alloys, ferritic steels and austenitic steels, with different microstructural conditions. It has been shown for the first time that the Poisson’s ratio decreases with increase in the elastic moduli and ultrasonic velocities for various solid isotropic materials. Further, it has been demonstrated that ultrasonic shear wave velocity is a better parameter for materials characterization as compared to longitudinal wave velocity. 2. Data from literature Table 1 is a compilation of the shear wave velocity and Poisson’s ratio for different isotropic materials. The shear modulus, Young’s modulus and density values for different elements are collected from the literature [5,8–11], and have been used for determining the Poisson’s ratio and ultrasonic velocities using Eqs. (1), (3) and (4), respectively. For the temperature effect on the elastic parameters, the temperature dependent E, G and n have been taken from Ledbetter’s review on elastic properties [1]. The temperature dependent density values have been taken from ASM Metals Handbook [12] for calculating the respective ultrasonic velocities. The ultrasonic velocities and density values for various ceramics, glasses and polymers have been taken from the ASTM NDT Handbook on Ultrasonic Testing [13] and used for calculating n, E and G using Eqs. (1), (3) and (4), respectively. The elastic constants for various intermetallics were reported by Koch et al. [14].
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Table 1 Ultrasonic shear wave velocity (V T) and Poisson’ s ratio (n) for different materials VT
n
7900 7900 7200 6900 5160 3400 6280 3980 5900 6360 4862 5900 6000 6100 3800
0.094 0.139 0.168 0.156 0.181 0.208 0.222 0.222 0.225 0.231 0.246 0.249 0.277 0.289 0.299
3260 3660 2795 1990
0.305 0.313 0.319 0.333
Polymers p1 Perspex p2 Lucite p3 Teflon p4 PVC hard
1430 1260 550 1060
0.311 0.358 0.399 0.378
Set A elements g1 As g2 Bi g3 Cd g4 Ce g5 Dy g6 Er g7 Gd g8 Ho g9 La g10 Nd g11 Pm g12 Pr g13 Sb g14 Se g15 Sm g16 Tb g17 Te g18 Tm g19 U g20 Y g21 Zn
696.6 1107.7 1482.0 1446.7 1709.9 1757.4 1668.8 1719.4 1509.3 1533.9 1574.2 1503.0 1728.1 876.2 1649.6 1636.1 1601.3 1823.7 2413 2411 2454
No. Name Ceramics and glasses c1 SiC c2 SiC c3 SiC c4 SiC c5 TiC c6 Porcelain c7 AlN c8 WC c9 Al2O3 c10 Al2O3 c11 Clay c12 Al2O3 c13 Si3N4 c14 Sapphire c15 ZrO2, thermally aged c16 Marble c17 ZrO2, sintered c18 UO2 c19 Ice
0.44 0.30 0.30 0.24 0.25 0.24 0.26 0.23 0.28 0.28 0.28 0.28 0.28 0.33 0.27 0.26 0.39 0.21 0.23 0.24 0.25 (continued on next page)
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Table 1 (continued) No. Name
VT
n
Rest elements r1 Ag r2 Al r3 Au r4 Be r5 Ca r6 Co r7 Cr r8 Cu r9 Fe r10 Hf r11 Ir r12 Mg r13 Mo r14 Nb r15 Ni r16 Os r17 Pb r18 Pd r19 Pt r20 Re r21 Rh r22 Ru r23 Sc r24 Sn r25 Sr r26 Ta r27 Ti r28 Tl r29 V r30 W r31 Zr
1691.1 3103.2 1182.8 8451.5 2185 2902.9 4013.3 2319.7 3227.1 1501.3 3044.9 3127.5 4410.8 2105.7 2920.9 3133.5 702.7 1913 1700.7 2910 3471.1 3739.7 3116.9 1569.2 4816 2035.7 3124.5 486.1 2773.5 2892 2251.3
0.37 0.35 0.44 0.032 0.31 0.31 0.21 0.34 0.29 0.37 0.26 0.29 0.31 0.4 0.31 0.25 0.44 0.39 0.38 0.3 0.26 0.3 0.28 0.36 0.28 0.34 0.32 0.45 0.37 0.28 0.33
Intermetallics i1 HfV2 i2 MoSi2 i3 NbCr2 i4 NbSi2 i5 NiAl i6 TaSi2 i7 Ti3Al i8 TiAl
1885.6 5819.9 3266.0 5049.8 3474.0 4051.3 3683.9 4356.0
0.39 0.15 0.34 0.18 0.30 0.19 0.27 0.23
3. Experimental details To generate the specimens with different microstructures and hence different elastic properties, various specimens of titanium alloys (Ti–6Al–4V and Ti–4.5Al–3Mo–1V), superalloy Inconel 625, various ferritic steels (modified 9Cr–1Mo steel and
2.25Cr–1Mo steel) and AISI type 316 austenitic stainless steel have been given different heat treatments. Specimens of titanium alloys have been solution annealed at various temperatures followed by water quenching and tempering at various temperatures up to 10 h to generate different amounts of α, β and α⬘ phases [15]. Similarly, ferritic steel specimens have been solution annealed from different temperatures to generate different amounts of ferrite and martensite phases [16]. Various specimens of service exposed and solution annealed Inconel 625 alloy have been thermally aged at various temperatures to generate different intermetallic precipitates and carbides [17]. Various specimens of AISI type 316 stainless steel have been solution annealed at different temperatures for different durations to generate specimens with varying grain size [18]. Ultrasonic velocities were measured using 5 MHz shear and 15 MHz longitudinal beam transducers, to keep the wavelength almost similar in both the ultrasonic wave modes. The wavelength of the ultrasonic waves and the grain size of the polycrystalline materials, for which data are generated experimentally, are listed in Table 2. The details of the experimental setup for ultrasonic velocity measurements have been given elsewhere [16]. Poisson’s ratio (n) has been calculated using Eq. (2) using the measured ultrasonic longitudinal (V L) and shear (V S) wave velocities. The maximum scatter in the measurement of ultrasonic longitudinal and shear wave velocities is found to be ±2.5 and ±1.5 m/s, respectively, and the corresponding maximum scatter in n is ±0.0005. 4. Results and discussion 4.1. Data from literature Figs. 1 and 2 show the variation in n with E and G, respectively, for 55 metallic elements. It can be seen from Figs. 1 and 2 that n shows a decreasing trend with increase in E and G except for a few elements with high moduli (E ⬎ 300 GPa and G ⬎ 130 GPa) (Figs. 1 and 2). It can be seen from Figs. 1 and 2 that the elements that show exception to this decreasing trend are mainly the heavier
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Table 2 Values of slope and correlation coefficients (corr. coeff.) for correlations between Poisson’s ratio and shear wave velocity for various materials (grain size and wavelength of ultrasonic waves used for materials studied experimentally are also included) Material
Data source
All materials Literature Set A elements Literature and polymers Rest of the Literature materials Intermetallics Literature Ti-alloys Experimental Inconel 625 Experimental 316 Stainless Experimental steel Ferritic steels Experimental Temperature effect Cu Literature Ti Literature Ni Literature Al Literature
Grain size (µm)
Wavelength (µm)
Intercept
Slope × 10⫺4
Corr. coeff.
Longitudinal
Shear
– –
– –
– –
0.5 0.5
0.61 1.39
0.73 0.89
–
–
–
0.5
0.54
0.88
– 30–400 60 30–150
– 430–450 420 410
– 580–640 620 625
0.51 0.65 0.6 0.65
0.66 1.053 0.94 1.21
0.94 0.998 0.99 0.994
20–130
425
650
0.58
0.90
0.99
– – – –
– – – –
– – – –
0.52 0.54 0.59 0.533
0.76 0.86 0.80 0.60
0.99 0.99 0.99 0.99
Fig. 1. Variation in Poisson’s ratio with Young’s modulus for elements.
Fig. 2. Variation in Poisson’s ratio with shear modulus for elements.
elements. The elements towards the left hand side are from column IIb (Zn and Cd), Va (As, Sb and Bi) and VIa (Se and Te) of the periodic table and all lanthanides and actinides (let us call all of them as set A elements). This indicates that elements in set A need to be dealt with separately and density
also plays an important role in the correlation of n with other elastic constants. Hence, an attempt has been made to correlate n with ultrasonic velocities, which are functions of r also, along with G and/or E. Figs. 3 and 4 show the variation in ultrasonic longitudinal and shear wave velocities,
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(identified as 왕 and 왖 in Figs. 1–4) show lower value of Poisson’s ratio for the same shear wave velocity as compared to others. These materials include polymers and the elements in set A. For all materials other than polymers and elements in set A, Poisson’s ratio can be correlated linearly with shear wave velocity with a correlation coefficient (R) of 0.88, as per Eq. (7) (Fig. 5 and Table 2), n ⫽ 0.5⫺0.543 ⫻ 10⫺4 VT (R ⫽ 0.88, VT in m / s), Fig. 3. Variation in ultrasonic longitudinal wave velocity with Poisson’s ratio for metals and alloys, ceramics and glasses, polymers and intermetallics.
(7)
and for polymers and elements in set A, Poisson’s ratio can be correlated linearly with shear wave velocity with correlation coefficients of 0.89 as per Eq. (8) (Fig. 5 and Table 2), n ⫽ 0.5⫺1.392 ⫻ 10⫺4 VT (R ⫽ 0.89, VT in m / s).
(8)
If we consider only the set A elements, then Poisson’s ratio can be correlated linearly with shear wave velocity for elements in set A as per Eq. (9) (Fig. 6 and Table 2), n ⫽ 0.5⫺1.363 ⫻ 10⫺4 VT (R ⫽ 0.86, VT in m / s),
(9)
and for rest of the elements, the correlation can be expressed as per Eq. (10) (Fig. 6 and Table 2),
Fig. 4. Variation in ultrasonic shear wave velocity with Poisson’s ratio for metals and alloys, ceramics and glasses, polymers and intermetallics.
respectively, with Poisson’s ratio for different solid materials. The limit of the ultrasonic shear wave velocity and the corresponding Poisson’s ratio has been shown as ∗ (star) in Fig. 4. It can be seen from these figures that both the ultrasonic velocities decrease with increase in Poisson’s ratio. Further, it can also be seen that ultrasonic shear wave velocity shows better correlation with Poisson’s ratio as compared to ultrasonic longitudinal wave velocity. Even though for all the materials, the Poisson’s ratio decreases with increase in ultrasonic shear wave velocity, some of the materials
Fig. 5. Variation in ultrasonic shear wave velocity with Poisson’s ratio for metals and alloys, ceramics and glasses, polymers and intermetallics, divided into two groups.
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Fig. 6. Variation in ultrasonic shear wave velocity with Poisson’s ratio for various elements, divided into two groups.
n ⫽ 0.5⫺0.661 ⫻ 10⫺4 VT (R ⫽ 0.92, VT in m / s).
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Fig. 7. Variation in ultrasonic shear wave velocity with Poisson’s ratio for intermetallics.
(10)
In the above correlations, it has been taken into account that the increase in n tends towards the value of the extreme condition, i.e. the value for V T (=0) and n (=0.5). This can be visualized clearly from Figs. 5 and 6. The above correlations clearly indicate that n varies in the direction opposite to that of the other elastic constants, but the error involved in using these correlations is high. The error involved can be reduced if different alloy systems are dealt with separately. In view of this, the correlations between V T and n are dealt with separately for intermetallics, titanium alloys, ferritic steels, stainless steel and nickel base alloy. Fig. 7 shows the variation in Poisson’s ratio with shear wave velocity for intermetallics. The slope of the curve (0.66) is very close to that for the variation of all the materials (Fig. 4), and the correlation coefficient has increased to 0.94. 4.2. Data generated experimentally for microstructural changes Fig. 8 shows the variation in Poisson’s ratio with shear wave velocity for titanium alloys (Ti–4.5Al– 3Mo–1V and Ti–6Al–4V), nickel base superalloy (Inconel 625), ferritic steels (modified 9Cr–1Mo and 2.25Cr–1Mo steels) and AISI type 316 stain-
Fig. 8. Variation in ultrasonic shear wave velocity with Poisson’s ratio for microstructural variation in various alloy systems.
less steel. The ultrasonic velocities and elastic moduli of Ti–6Al–4V are found to be higher as compared to that for Ti–4.5Al–3Mo–1V in all the microstructural conditions, and the corresponding Poisson’s ratios are lower for Ti–6Al–4V. The higher moduli in Ti–6Al–4V are ascribed to the higher amount of α-phase stabilizing elements (Al) [15]. Ti alloys solutionized at intermediate temperatures (~1123 K) followed by water quenching exhibited the minimum moduli because of the presence of unstable β phase [15]. The alloys solutionized at lower (⬍1123 K) and higher (⬎1123 K) temperature ranges exhibited comparatively higher moduli because of the presence of stable α
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and β phases and α⬘ martensite, respectively. Tempering of the alloys subsequent to water quenching from different temperatures increased the elastic moduli. Increase in elastic moduli (ultrasonic velocity) has been always found to be associated with decrease in Poisson’s ratio for both of these Ti alloys. It can be seen from Fig. 8 that for both the Ti alloys, the increase in ultrasonic shear wave velocity can be linearly correlated with decrease in Poisson’s ratio by the same correlation, with very high correlation coefficient, as shown in Eq. (11) and Table 2, n ⫽ 0.65⫺1.053 ⫻ 10⫺4 VT (R ⫽ 0.998,
VT in m / s).
(11)
Fig. 8 also shows the variation in ultrasonic shear wave velocity with Poisson’s ratio for Ni base superalloy Inconel 625. In this alloy, ultrasonic velocity was found to be lowest in the solution annealed condition and it increased with the precipitation of intermetallic phases [17]. Poisson’s ratio has been found to decrease linearly with increase in ultrasonic shear wave velocity in this alloy also. In AISI type 316 stainless steel, ultrasonic velocities have been found to decrease with increase in grain size [19]. It can be seen from Fig. 8 that decrease in ultrasonic velocity with grain size led to the increase in Poisson’s ratio. Similarly, ferritic steel specimens solutionized at different temperatures exhibited different ultrasonic velocities due to different amounts of ferrite and martensite present. The specimen with martensite exhibited lowest moduli (/ultrasonic velocities) and highest Poisson’s ratio as compared to that with the ferrite. 4.3. Analysis of temperature effect using literature data Fig. 9 shows the effect of temperature on Poisson’s ratio and ultrasonic velocity for copper, aluminum, titanium and nickel. Increase in the temperature decreases the ultrasonic velocity and increases the Poisson’s ratio in all these metals. Table 2 also gives the slope and intercept for the linear correlation of ultrasonic velocity and Poisson’s ratio for these metals. It can be seen from Figs. 8 and 9 and Table 2
Fig. 9. Variation in ultrasonic shear wave velocity with Poisson’s ratio for effect of temperature in various metals.
that the slopes of the correlations for any given alloy system are almost similar. Even though the intercepts of the correlations were found to be more than 0.5 (limit of the Poisson ratio), this is only due to the restricted range of the data for any given alloy system. With the help of the abovegiven examples, it can be established that Poisson’s ratio decreases always with increase in ultrasonic shear wave velocity. This can also be seen in the review of Koister and Franz [4] for the effect of different metallurgical parameters on the variation of Poisson’s ratio and other elastic coefficients. It has been reported that the Poisson’s ratio decreased with increasing Young’s and shear moduli, and it was maximum when elastic moduli were minimum and vice versa [4]. This can be seen very clearly in the variation of different elastic coefficients for copper–zinc alloys (Fig. 10) and for iron–nickel alloys (Fig. 11) [4]. In α solid solution regime of copper–zinc system, with increase in Zn (up to ~38%), shear modulus decreases and Poisson’s ratio increases slowly (Fig. 10). The β phase (~50% Zn) shows sharp increase in Poisson’s ratio and decrease in shear modulus. The γ phase (~60%) exhibits the maximum value of shear modulus and minimum of Poisson’s ratio. Beyond this, the shear modulus decreases, whereas Poisson’s ratio increases slowly. In iron–nickel system (Fig. 11), Young’s modulus decreases up to ~40% of Ni, and beyond that it increases. The variation in Poisson’s ratio exhibits the opposite trend: it increases up to ~40% of Ni and then decreases.
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perpendicular planes (propagation and vibration) in the case of shear wave propagation as compared to one in the case of longitudinal wave. It may be noted that care was taken to use nearly constant wavelengths for longitudinal and shear waves to avoid difference in interaction based on wavelength. Further, as ultrasonic shear wave velocity is less than (about half of that of) longitudinal wave velocity, the transit time of shear wave is more than (about double of) that of longitudinal wave velocity for the same thickness of the specimen. Hence, the error in the measurement of shear wave velocity is less as compared to that for longitudinal wave velocity. As the shear wave velocity is affected more and the error in the measurement is also less, it can be deduced that ultrasonic shear wave velocity is a better parameter for material/ microstructural characterization as compared to longitudinal wave velocity.
Fig. 10. Young’s (E), shear (G), and compressional (K) moduli and Poisson’s ratio (n) for copper–zinc alloys [4].
5. Conclusions The correlation of Poisson’s ratio with ultrasonic longitudinal and shear wave velocities for isotropic solid materials indicates that
Fig. 11. Young’s modulus (E) and Poisson’s ratio (n) for iron–nickel alloys [4].
With the establishment of the fact that Poisson’s ratio varies in a direction opposite to that of the ultrasonic velocities and elastic (Young’s and shear) moduli, reanalyzing Eqs. (5) and (6) leads to the inference that ultrasonic shear wave velocity (/shear modulus) is affected more than ultrasonic longitudinal wave velocity (/Young’s modulus). This may be attributed to the involvement of two
1. the Poisson ratio decreases with increase in ultrasonic velocities, 2. the shear wave velocity is affected more than the longitudinal wave velocity due to any microstructural variation or temperature effect, 3. the slope of the correlation is larger for polymers and the elements from column IIb (Zn and Cd), Va (As, Sb and Bi) and VIa (Se and Te) of the periodic table and all lanthanides and actinides as compared to the rest of the materials, 4. the slope of the correlation is almost constant for various alloy systems for changes in microstructure due to various heat treatments, alloying effects, grain size effect, temperature effect, etc., and 5. ultrasonic shear wave velocity is a better parameter for materials characterization as compared to longitudinal wave velocity.
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Acknowledgements The authors are grateful to Prof. Robert W. Cahn, Department of Materials Science and Metallurgy, University of Cambridge, UK, for critical review of the paper and useful suggestions. The authors are thankful to Dr. C.S. Sundar, Materials Science Division, and Dr. E. Mohan Das and Dr. V. Raju, Physical Metallurgy Section, Indira Gandhi Centre for Atomic Research (IGCAR), for important discussions on the paper. Thanks are also due to Dr. S.L. Mannan, Associate Director, Materials Development Group, and Mr. P. Kalyanasundaram, Head, Division for PIE and NDT, IGCAR, for their support.
References [1] Ledbetter HM. Elastic properties. In: Reed RP, Clark AF, editors. Materials at low temperatures. Metals Park (OH 44073): American Society for Metals, 1983. p. 1–45. [2] Ledbetter HM. Mater. Sci. Eng. 1977;27:133. [3] Gorecki T. Mater. Sci. Eng. 1979;43:225. [4] Koister W, Franz H. Metall. Rev. 1961;6(21):1. [5] Kaye GWC, Laby TH. Tables of physical and chemical constants, 15th ed. London (UK): Longman, 1993. [6] Lakes R. Science 1987;235:1038.
[7] Tervo J, Tarasenko A, Hanninen H. Mater. Sci. For. Trans. Tech. Publ. Uetikon-Zuerich Switz. 1999;318:41. [8] http://www.webelements.com. 2002. [9] James AM, Lord MP. Macmillan’s chemical and physical data. London (UK): Macmillan, 1992. [10] Samsonov GV, editor. Handbook of the physiochemical properties of the elements. New York (USA): IFI-Plenum, 1968. [11] Lide DR, editor. Chemical Rubber Company handbook of chemistry and physics. 79th ed. Boca Raton (FL, USA): CRC Press, 1998. [12] Properties and selection of metals. Metals handbook. 8th ed., vol. 1. Metals Park (OH): American Society for Metals. 1961. [13] Mclntire P, editor. Ultrasonic testing. 2nd ed. Nondestructive testing handbook, vol. 7. USA: ASTM; 1991. p. 83-6. [14] Koch CC, Liu CT, Stoloff NS, editors. High-temperature ordered intermetallic alloys. Materials research society symposium proceedings, vol. 39. Pittsburgh: MRS, 1985. [15] Kumar A, Jayakumar T, Raj B, Ray KK. Characterization of solutionizing behaviour in titanium alloys by ultrasonic velocity measurements. In: Proceedings of review of progress in quantitative NDE, WWU, Bellingham, USA, July 2002 [in press]. [16] Kumar A, Jayakumar T, Laha K, Rao KBS, Raj B. Metall. Mater. Trans. A 2002;33:1617. [17] Kumar A, Shankar V, Rao KBS, Jayakumar T, Raj B. Philos. Mag. A 2002;82(13):2529. [18] Kumar A, Jayakumar T, Raj B. Philos. Mag. A 2000;80(11):2469. [19] Palanichamy P, Joseph A, Jayakumar T, Raj B. NDT & E Int. 1995;28:179.
Correlation between ultrasonic shear wave velocity and Poisson’s ratio for isotropic solid materials Anish Kumar a,∗, T. Jayakumar a, Baldev Raj a, K.K. Ray b a
Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India b Indian Institute of Technology, Kharagpur 721302, India Received 19 August 2002; received in revised form 23 January 2003; accepted 23 January 2003
Abstract A new correlation between ultrasonic shear wave velocity and Poisson’s ratio has been established for isotropic solid materials, based on the data generated experimentally and collected from the literature. Poisson’s ratio has been found to decrease with increasing ultrasonic velocity in various solid materials such as metals and alloys, ceramics and glasses, intermetallics and polymers. The slope of the plot of the ultrasonic velocity against Poisson’s ratio is found to be almost constant for any given alloy system with different microstructures associated with various heat treatments, alloying elements, grain size, temperature effect, etc. Further, it has been demonstrated that ultrasonic shear wave velocity is a better parameter for materials characterization as compared to longitudinal wave velocity. 2003 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Ultrasonic velocity; Poisson’s ratio; Modulus; Elastic constant
1. Introduction Elastic constants are physical properties that relate stress and strain. Since solids resist both volume change and shape change, they have at least two independent elastic constants. A low symmetry crystal may have as many as 21 independent elastic constants, but polycrystalline isotropic materials (i.e. texture free materials, as considered in the present study) can be characterized by only two independent elastic constants. Many functional and
Corresponding author. Tel.: +91 (4114) 280208; fax: +91 (4114) 280356. E-mail address: [email protected] (A. Kumar). ∗
differential relationships between elastic constants and other elastic-constant pairs have been reported [1–4], which emerge from the theory of elasticity and from the physical meanings of the constants. However useful these relationships may be, they only represent the functions relating a dependent variable to two independent variables. Two independent elastic constants are required to characterize the isotropic materials with the presently available relations. Hence, identification of any new relationship that reduces the number of required independent elastic constants to one is beneficial. This hypothesis is based on the belief that there must be a hidden relationship between the two independent elastic constants that is based on the microstructure, atomic bonding, electronic
1359-6454/03/$30.00 2003 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. doi:10.1016/S1359-6454(03)00054-5
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configurations, etc. A number of researchers have given the empirical relationships to relate shear modulus (G), Young’s modulus (E) and bulk modulus (B) for various elements [1–3]. Poisson’s ratio, defined as the lateral contraction per unit breadth divided by the longitudinal extension per unit length in simple tension [5], is reported to provide more information about the character of the bonding forces than any of the other elastic coefficients [4]. Poisson’s ratio (n) is related to E and G by n⫽
E ⫺1 2G
(1)
and to ultrasonic longitudinal wave velocity (V L) and shear wave velocity (V T) by n⫽
V2L⫺2V2T . 2(V2L⫺V2T)
(2)
V L and V T are related to E and G by
冉
冊
4 3rV2T V2L⫺ V2T 3 E⫽ (V2L⫺V2T)
(3)
G ⫽ rV2T
(4)
where r is the density. The differential relationships of n with E and G and with V L and V T can be obtained by differentiating Eqs. (1) and (2), respectively. The differential forms of the above equations are:
冉
冊 冉
dE dG 1 dn ; ⫽ ⫺ n (1⫺2G / E) E G
(5)
冊
dVL dVT 2V2LV2T dn ⫺ . ⫽ 2 n (VL⫺V2T)(V2L⫺2V2T) VL VT
(6)
It can be seen from Eqs. (5) and (6), that the variation in Poisson’s ratio depends upon the variations in E and G and their relative values (Eq. (5)). Similarly, the relative variation in V L and V T also affects the variation in Poisson’s ratio (Eq. (6)). It is to be noted that in any material, E and G, as well as V L and V T, tend to vary in the same direction (increase or decrease). Hence, it is difficult to deduce from Eqs. (5) and (6) whether the variation of n would be in the same or opposite
direction to that of the elastic constants, E and G. The nature of the variation of n would therefore depend on whether G or E is affected more by metallurgical variables. If the rate of increase of E (or V L) is more than that of G (or V T), then the right hand side of Eqs. (5) and (6) will be positive, and hence n will vary in the same direction as that of E and G (or V L and V T). Similarly, if the rate of increase of G (or V T) is more than that of E (or V L), the right hand side of Eqs. (5) and (6) will be negative and hence n will vary opposite to E and G (or V L and V T). From the first law of thermodynamics, it can be seen that E, B and G are positive, and since B = E / 3(1⫺2n) and G = E / 2(1 + n), it follows that ⫺1 ⬍ n ⬍ 0.5. Further, even though the laws of thermodynamics are not violated by a hypothetical isotropic elastic solid with ⫺1 ⬍ nⱕ0, such negative values have been observed in foam structures and anisotropic materials only [6]. Thus, in isotropic solids, n is bounded practically by 0 and 1/2. For n = 1 / 2, no volume change occurs during deformation. Koister and Franz [4] have reviewed in detail the effect of various metallurgical parameters, such as alloying of Zn to Cu, alloying of Cu to Ag and alloying of Ni to Fe, etc, on Poisson’s ratio. They have also reviewed various methods for the determination of Poisson’s ratio and have concluded that the ultrasonic velocity measurement based dynamic method, which gives the adiabatic Poisson’s ratio, is the most accurate of all methods. This dynamic method involves the propagation of a low amplitude stress (ultrasonic) wave (⬍10 kPa) through the specimen and the measurement of transit time of the stress wave, which is then used for the determination of the elastic properties [1]. Other methods for the measurement of Poisson’s ratio include static methods, such as measurement of change in volume and length, determination of Young’s and shear moduli from bending and torsion tests, testing of helical spring and dynamic methods like measurement of natural frequencies of vibration, vibration of a spring under load and resonance frequencies of flexural vibrations [4]. Recently, Tervo et al. [7] have studied the effect of nitrogen alloying on elastic coefficients of austenitic stainless steels and showed that nitrogen in
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excess of 1.5 at.% increases the shear modulus and decreases the Poisson’s ratio. In the present study, the authors have studied the effect of ultrasonic shear wave velocity on Poisson’s ratio. The data have been collected from the literature for various metals, alloys, intermetallics, ceramics, glasses and polymers. Further, it is always reported in the textbooks of metallurgy and materials science that elastic moduli and Poisson’s ratio are material properties and insensitive to the microstructural features. However, in this paper, we have also shown the effect of the microstructural features on elastic moduli and Poisson’s ratio, by carrying out the ultrasonic velocity measurements in various alloy systems, such as Ti–6Al– 4V, Ti–4.5Al–3Mo–1V, Inconel 625, Cu alloys, ferritic steels and austenitic steels, with different microstructural conditions. It has been shown for the first time that the Poisson’s ratio decreases with increase in the elastic moduli and ultrasonic velocities for various solid isotropic materials. Further, it has been demonstrated that ultrasonic shear wave velocity is a better parameter for materials characterization as compared to longitudinal wave velocity. 2. Data from literature Table 1 is a compilation of the shear wave velocity and Poisson’s ratio for different isotropic materials. The shear modulus, Young’s modulus and density values for different elements are collected from the literature [5,8–11], and have been used for determining the Poisson’s ratio and ultrasonic velocities using Eqs. (1), (3) and (4), respectively. For the temperature effect on the elastic parameters, the temperature dependent E, G and n have been taken from Ledbetter’s review on elastic properties [1]. The temperature dependent density values have been taken from ASM Metals Handbook [12] for calculating the respective ultrasonic velocities. The ultrasonic velocities and density values for various ceramics, glasses and polymers have been taken from the ASTM NDT Handbook on Ultrasonic Testing [13] and used for calculating n, E and G using Eqs. (1), (3) and (4), respectively. The elastic constants for various intermetallics were reported by Koch et al. [14].
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Table 1 Ultrasonic shear wave velocity (V T) and Poisson’ s ratio (n) for different materials VT
n
7900 7900 7200 6900 5160 3400 6280 3980 5900 6360 4862 5900 6000 6100 3800
0.094 0.139 0.168 0.156 0.181 0.208 0.222 0.222 0.225 0.231 0.246 0.249 0.277 0.289 0.299
3260 3660 2795 1990
0.305 0.313 0.319 0.333
Polymers p1 Perspex p2 Lucite p3 Teflon p4 PVC hard
1430 1260 550 1060
0.311 0.358 0.399 0.378
Set A elements g1 As g2 Bi g3 Cd g4 Ce g5 Dy g6 Er g7 Gd g8 Ho g9 La g10 Nd g11 Pm g12 Pr g13 Sb g14 Se g15 Sm g16 Tb g17 Te g18 Tm g19 U g20 Y g21 Zn
696.6 1107.7 1482.0 1446.7 1709.9 1757.4 1668.8 1719.4 1509.3 1533.9 1574.2 1503.0 1728.1 876.2 1649.6 1636.1 1601.3 1823.7 2413 2411 2454
No. Name Ceramics and glasses c1 SiC c2 SiC c3 SiC c4 SiC c5 TiC c6 Porcelain c7 AlN c8 WC c9 Al2O3 c10 Al2O3 c11 Clay c12 Al2O3 c13 Si3N4 c14 Sapphire c15 ZrO2, thermally aged c16 Marble c17 ZrO2, sintered c18 UO2 c19 Ice
0.44 0.30 0.30 0.24 0.25 0.24 0.26 0.23 0.28 0.28 0.28 0.28 0.28 0.33 0.27 0.26 0.39 0.21 0.23 0.24 0.25 (continued on next page)
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Table 1 (continued) No. Name
VT
n
Rest elements r1 Ag r2 Al r3 Au r4 Be r5 Ca r6 Co r7 Cr r8 Cu r9 Fe r10 Hf r11 Ir r12 Mg r13 Mo r14 Nb r15 Ni r16 Os r17 Pb r18 Pd r19 Pt r20 Re r21 Rh r22 Ru r23 Sc r24 Sn r25 Sr r26 Ta r27 Ti r28 Tl r29 V r30 W r31 Zr
1691.1 3103.2 1182.8 8451.5 2185 2902.9 4013.3 2319.7 3227.1 1501.3 3044.9 3127.5 4410.8 2105.7 2920.9 3133.5 702.7 1913 1700.7 2910 3471.1 3739.7 3116.9 1569.2 4816 2035.7 3124.5 486.1 2773.5 2892 2251.3
0.37 0.35 0.44 0.032 0.31 0.31 0.21 0.34 0.29 0.37 0.26 0.29 0.31 0.4 0.31 0.25 0.44 0.39 0.38 0.3 0.26 0.3 0.28 0.36 0.28 0.34 0.32 0.45 0.37 0.28 0.33
Intermetallics i1 HfV2 i2 MoSi2 i3 NbCr2 i4 NbSi2 i5 NiAl i6 TaSi2 i7 Ti3Al i8 TiAl
1885.6 5819.9 3266.0 5049.8 3474.0 4051.3 3683.9 4356.0
0.39 0.15 0.34 0.18 0.30 0.19 0.27 0.23
3. Experimental details To generate the specimens with different microstructures and hence different elastic properties, various specimens of titanium alloys (Ti–6Al–4V and Ti–4.5Al–3Mo–1V), superalloy Inconel 625, various ferritic steels (modified 9Cr–1Mo steel and
2.25Cr–1Mo steel) and AISI type 316 austenitic stainless steel have been given different heat treatments. Specimens of titanium alloys have been solution annealed at various temperatures followed by water quenching and tempering at various temperatures up to 10 h to generate different amounts of α, β and α⬘ phases [15]. Similarly, ferritic steel specimens have been solution annealed from different temperatures to generate different amounts of ferrite and martensite phases [16]. Various specimens of service exposed and solution annealed Inconel 625 alloy have been thermally aged at various temperatures to generate different intermetallic precipitates and carbides [17]. Various specimens of AISI type 316 stainless steel have been solution annealed at different temperatures for different durations to generate specimens with varying grain size [18]. Ultrasonic velocities were measured using 5 MHz shear and 15 MHz longitudinal beam transducers, to keep the wavelength almost similar in both the ultrasonic wave modes. The wavelength of the ultrasonic waves and the grain size of the polycrystalline materials, for which data are generated experimentally, are listed in Table 2. The details of the experimental setup for ultrasonic velocity measurements have been given elsewhere [16]. Poisson’s ratio (n) has been calculated using Eq. (2) using the measured ultrasonic longitudinal (V L) and shear (V S) wave velocities. The maximum scatter in the measurement of ultrasonic longitudinal and shear wave velocities is found to be ±2.5 and ±1.5 m/s, respectively, and the corresponding maximum scatter in n is ±0.0005. 4. Results and discussion 4.1. Data from literature Figs. 1 and 2 show the variation in n with E and G, respectively, for 55 metallic elements. It can be seen from Figs. 1 and 2 that n shows a decreasing trend with increase in E and G except for a few elements with high moduli (E ⬎ 300 GPa and G ⬎ 130 GPa) (Figs. 1 and 2). It can be seen from Figs. 1 and 2 that the elements that show exception to this decreasing trend are mainly the heavier
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Table 2 Values of slope and correlation coefficients (corr. coeff.) for correlations between Poisson’s ratio and shear wave velocity for various materials (grain size and wavelength of ultrasonic waves used for materials studied experimentally are also included) Material
Data source
All materials Literature Set A elements Literature and polymers Rest of the Literature materials Intermetallics Literature Ti-alloys Experimental Inconel 625 Experimental 316 Stainless Experimental steel Ferritic steels Experimental Temperature effect Cu Literature Ti Literature Ni Literature Al Literature
Grain size (µm)
Wavelength (µm)
Intercept
Slope × 10⫺4
Corr. coeff.
Longitudinal
Shear
– –
– –
– –
0.5 0.5
0.61 1.39
0.73 0.89
–
–
–
0.5
0.54
0.88
– 30–400 60 30–150
– 430–450 420 410
– 580–640 620 625
0.51 0.65 0.6 0.65
0.66 1.053 0.94 1.21
0.94 0.998 0.99 0.994
20–130
425
650
0.58
0.90
0.99
– – – –
– – – –
– – – –
0.52 0.54 0.59 0.533
0.76 0.86 0.80 0.60
0.99 0.99 0.99 0.99
Fig. 1. Variation in Poisson’s ratio with Young’s modulus for elements.
Fig. 2. Variation in Poisson’s ratio with shear modulus for elements.
elements. The elements towards the left hand side are from column IIb (Zn and Cd), Va (As, Sb and Bi) and VIa (Se and Te) of the periodic table and all lanthanides and actinides (let us call all of them as set A elements). This indicates that elements in set A need to be dealt with separately and density
also plays an important role in the correlation of n with other elastic constants. Hence, an attempt has been made to correlate n with ultrasonic velocities, which are functions of r also, along with G and/or E. Figs. 3 and 4 show the variation in ultrasonic longitudinal and shear wave velocities,
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(identified as 왕 and 왖 in Figs. 1–4) show lower value of Poisson’s ratio for the same shear wave velocity as compared to others. These materials include polymers and the elements in set A. For all materials other than polymers and elements in set A, Poisson’s ratio can be correlated linearly with shear wave velocity with a correlation coefficient (R) of 0.88, as per Eq. (7) (Fig. 5 and Table 2), n ⫽ 0.5⫺0.543 ⫻ 10⫺4 VT (R ⫽ 0.88, VT in m / s), Fig. 3. Variation in ultrasonic longitudinal wave velocity with Poisson’s ratio for metals and alloys, ceramics and glasses, polymers and intermetallics.
(7)
and for polymers and elements in set A, Poisson’s ratio can be correlated linearly with shear wave velocity with correlation coefficients of 0.89 as per Eq. (8) (Fig. 5 and Table 2), n ⫽ 0.5⫺1.392 ⫻ 10⫺4 VT (R ⫽ 0.89, VT in m / s).
(8)
If we consider only the set A elements, then Poisson’s ratio can be correlated linearly with shear wave velocity for elements in set A as per Eq. (9) (Fig. 6 and Table 2), n ⫽ 0.5⫺1.363 ⫻ 10⫺4 VT (R ⫽ 0.86, VT in m / s),
(9)
and for rest of the elements, the correlation can be expressed as per Eq. (10) (Fig. 6 and Table 2),
Fig. 4. Variation in ultrasonic shear wave velocity with Poisson’s ratio for metals and alloys, ceramics and glasses, polymers and intermetallics.
respectively, with Poisson’s ratio for different solid materials. The limit of the ultrasonic shear wave velocity and the corresponding Poisson’s ratio has been shown as ∗ (star) in Fig. 4. It can be seen from these figures that both the ultrasonic velocities decrease with increase in Poisson’s ratio. Further, it can also be seen that ultrasonic shear wave velocity shows better correlation with Poisson’s ratio as compared to ultrasonic longitudinal wave velocity. Even though for all the materials, the Poisson’s ratio decreases with increase in ultrasonic shear wave velocity, some of the materials
Fig. 5. Variation in ultrasonic shear wave velocity with Poisson’s ratio for metals and alloys, ceramics and glasses, polymers and intermetallics, divided into two groups.
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Fig. 6. Variation in ultrasonic shear wave velocity with Poisson’s ratio for various elements, divided into two groups.
n ⫽ 0.5⫺0.661 ⫻ 10⫺4 VT (R ⫽ 0.92, VT in m / s).
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Fig. 7. Variation in ultrasonic shear wave velocity with Poisson’s ratio for intermetallics.
(10)
In the above correlations, it has been taken into account that the increase in n tends towards the value of the extreme condition, i.e. the value for V T (=0) and n (=0.5). This can be visualized clearly from Figs. 5 and 6. The above correlations clearly indicate that n varies in the direction opposite to that of the other elastic constants, but the error involved in using these correlations is high. The error involved can be reduced if different alloy systems are dealt with separately. In view of this, the correlations between V T and n are dealt with separately for intermetallics, titanium alloys, ferritic steels, stainless steel and nickel base alloy. Fig. 7 shows the variation in Poisson’s ratio with shear wave velocity for intermetallics. The slope of the curve (0.66) is very close to that for the variation of all the materials (Fig. 4), and the correlation coefficient has increased to 0.94. 4.2. Data generated experimentally for microstructural changes Fig. 8 shows the variation in Poisson’s ratio with shear wave velocity for titanium alloys (Ti–4.5Al– 3Mo–1V and Ti–6Al–4V), nickel base superalloy (Inconel 625), ferritic steels (modified 9Cr–1Mo and 2.25Cr–1Mo steels) and AISI type 316 stain-
Fig. 8. Variation in ultrasonic shear wave velocity with Poisson’s ratio for microstructural variation in various alloy systems.
less steel. The ultrasonic velocities and elastic moduli of Ti–6Al–4V are found to be higher as compared to that for Ti–4.5Al–3Mo–1V in all the microstructural conditions, and the corresponding Poisson’s ratios are lower for Ti–6Al–4V. The higher moduli in Ti–6Al–4V are ascribed to the higher amount of α-phase stabilizing elements (Al) [15]. Ti alloys solutionized at intermediate temperatures (~1123 K) followed by water quenching exhibited the minimum moduli because of the presence of unstable β phase [15]. The alloys solutionized at lower (⬍1123 K) and higher (⬎1123 K) temperature ranges exhibited comparatively higher moduli because of the presence of stable α
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and β phases and α⬘ martensite, respectively. Tempering of the alloys subsequent to water quenching from different temperatures increased the elastic moduli. Increase in elastic moduli (ultrasonic velocity) has been always found to be associated with decrease in Poisson’s ratio for both of these Ti alloys. It can be seen from Fig. 8 that for both the Ti alloys, the increase in ultrasonic shear wave velocity can be linearly correlated with decrease in Poisson’s ratio by the same correlation, with very high correlation coefficient, as shown in Eq. (11) and Table 2, n ⫽ 0.65⫺1.053 ⫻ 10⫺4 VT (R ⫽ 0.998,
VT in m / s).
(11)
Fig. 8 also shows the variation in ultrasonic shear wave velocity with Poisson’s ratio for Ni base superalloy Inconel 625. In this alloy, ultrasonic velocity was found to be lowest in the solution annealed condition and it increased with the precipitation of intermetallic phases [17]. Poisson’s ratio has been found to decrease linearly with increase in ultrasonic shear wave velocity in this alloy also. In AISI type 316 stainless steel, ultrasonic velocities have been found to decrease with increase in grain size [19]. It can be seen from Fig. 8 that decrease in ultrasonic velocity with grain size led to the increase in Poisson’s ratio. Similarly, ferritic steel specimens solutionized at different temperatures exhibited different ultrasonic velocities due to different amounts of ferrite and martensite present. The specimen with martensite exhibited lowest moduli (/ultrasonic velocities) and highest Poisson’s ratio as compared to that with the ferrite. 4.3. Analysis of temperature effect using literature data Fig. 9 shows the effect of temperature on Poisson’s ratio and ultrasonic velocity for copper, aluminum, titanium and nickel. Increase in the temperature decreases the ultrasonic velocity and increases the Poisson’s ratio in all these metals. Table 2 also gives the slope and intercept for the linear correlation of ultrasonic velocity and Poisson’s ratio for these metals. It can be seen from Figs. 8 and 9 and Table 2
Fig. 9. Variation in ultrasonic shear wave velocity with Poisson’s ratio for effect of temperature in various metals.
that the slopes of the correlations for any given alloy system are almost similar. Even though the intercepts of the correlations were found to be more than 0.5 (limit of the Poisson ratio), this is only due to the restricted range of the data for any given alloy system. With the help of the abovegiven examples, it can be established that Poisson’s ratio decreases always with increase in ultrasonic shear wave velocity. This can also be seen in the review of Koister and Franz [4] for the effect of different metallurgical parameters on the variation of Poisson’s ratio and other elastic coefficients. It has been reported that the Poisson’s ratio decreased with increasing Young’s and shear moduli, and it was maximum when elastic moduli were minimum and vice versa [4]. This can be seen very clearly in the variation of different elastic coefficients for copper–zinc alloys (Fig. 10) and for iron–nickel alloys (Fig. 11) [4]. In α solid solution regime of copper–zinc system, with increase in Zn (up to ~38%), shear modulus decreases and Poisson’s ratio increases slowly (Fig. 10). The β phase (~50% Zn) shows sharp increase in Poisson’s ratio and decrease in shear modulus. The γ phase (~60%) exhibits the maximum value of shear modulus and minimum of Poisson’s ratio. Beyond this, the shear modulus decreases, whereas Poisson’s ratio increases slowly. In iron–nickel system (Fig. 11), Young’s modulus decreases up to ~40% of Ni, and beyond that it increases. The variation in Poisson’s ratio exhibits the opposite trend: it increases up to ~40% of Ni and then decreases.
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perpendicular planes (propagation and vibration) in the case of shear wave propagation as compared to one in the case of longitudinal wave. It may be noted that care was taken to use nearly constant wavelengths for longitudinal and shear waves to avoid difference in interaction based on wavelength. Further, as ultrasonic shear wave velocity is less than (about half of that of) longitudinal wave velocity, the transit time of shear wave is more than (about double of) that of longitudinal wave velocity for the same thickness of the specimen. Hence, the error in the measurement of shear wave velocity is less as compared to that for longitudinal wave velocity. As the shear wave velocity is affected more and the error in the measurement is also less, it can be deduced that ultrasonic shear wave velocity is a better parameter for material/ microstructural characterization as compared to longitudinal wave velocity.
Fig. 10. Young’s (E), shear (G), and compressional (K) moduli and Poisson’s ratio (n) for copper–zinc alloys [4].
5. Conclusions The correlation of Poisson’s ratio with ultrasonic longitudinal and shear wave velocities for isotropic solid materials indicates that
Fig. 11. Young’s modulus (E) and Poisson’s ratio (n) for iron–nickel alloys [4].
With the establishment of the fact that Poisson’s ratio varies in a direction opposite to that of the ultrasonic velocities and elastic (Young’s and shear) moduli, reanalyzing Eqs. (5) and (6) leads to the inference that ultrasonic shear wave velocity (/shear modulus) is affected more than ultrasonic longitudinal wave velocity (/Young’s modulus). This may be attributed to the involvement of two
1. the Poisson ratio decreases with increase in ultrasonic velocities, 2. the shear wave velocity is affected more than the longitudinal wave velocity due to any microstructural variation or temperature effect, 3. the slope of the correlation is larger for polymers and the elements from column IIb (Zn and Cd), Va (As, Sb and Bi) and VIa (Se and Te) of the periodic table and all lanthanides and actinides as compared to the rest of the materials, 4. the slope of the correlation is almost constant for various alloy systems for changes in microstructure due to various heat treatments, alloying effects, grain size effect, temperature effect, etc., and 5. ultrasonic shear wave velocity is a better parameter for materials characterization as compared to longitudinal wave velocity.
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Acknowledgements The authors are grateful to Prof. Robert W. Cahn, Department of Materials Science and Metallurgy, University of Cambridge, UK, for critical review of the paper and useful suggestions. The authors are thankful to Dr. C.S. Sundar, Materials Science Division, and Dr. E. Mohan Das and Dr. V. Raju, Physical Metallurgy Section, Indira Gandhi Centre for Atomic Research (IGCAR), for important discussions on the paper. Thanks are also due to Dr. S.L. Mannan, Associate Director, Materials Development Group, and Mr. P. Kalyanasundaram, Head, Division for PIE and NDT, IGCAR, for their support.
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[7] Tervo J, Tarasenko A, Hanninen H. Mater. Sci. For. Trans. Tech. Publ. Uetikon-Zuerich Switz. 1999;318:41. [8] http://www.webelements.com. 2002. [9] James AM, Lord MP. Macmillan’s chemical and physical data. London (UK): Macmillan, 1992. [10] Samsonov GV, editor. Handbook of the physiochemical properties of the elements. New York (USA): IFI-Plenum, 1968. [11] Lide DR, editor. Chemical Rubber Company handbook of chemistry and physics. 79th ed. Boca Raton (FL, USA): CRC Press, 1998. [12] Properties and selection of metals. Metals handbook. 8th ed., vol. 1. Metals Park (OH): American Society for Metals. 1961. [13] Mclntire P, editor. Ultrasonic testing. 2nd ed. Nondestructive testing handbook, vol. 7. USA: ASTM; 1991. p. 83-6. [14] Koch CC, Liu CT, Stoloff NS, editors. High-temperature ordered intermetallic alloys. Materials research society symposium proceedings, vol. 39. Pittsburgh: MRS, 1985. [15] Kumar A, Jayakumar T, Raj B, Ray KK. Characterization of solutionizing behaviour in titanium alloys by ultrasonic velocity measurements. In: Proceedings of review of progress in quantitative NDE, WWU, Bellingham, USA, July 2002 [in press]. [16] Kumar A, Jayakumar T, Laha K, Rao KBS, Raj B. Metall. Mater. Trans. A 2002;33:1617. [17] Kumar A, Shankar V, Rao KBS, Jayakumar T, Raj B. Philos. Mag. A 2002;82(13):2529. [18] Kumar A, Jayakumar T, Raj B. Philos. Mag. A 2000;80(11):2469. [19] Palanichamy P, Joseph A, Jayakumar T, Raj B. NDT & E Int. 1995;28:179.