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Dynamic vs. static Young's moduli: a case study
Materials Science and Engineering, A165 (1993) L 9 - L 10
L9
Letter
Dynamic vs. static Young's moduli: a case study Hassel Ledbetter Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Boulder, CO 80303 (USA) (Received December 7, 1992; in revised form December 30, 1992)
Abstract For one material, a 316LN austenitic steel, a comparison is made between the Young's modulus measured statically with usual commercial equipment and that measured dynamically at megahertz frequencies. The first gives a _+6% uncertainty; the second a + 0.5% uncertainty. The two Young's moduli, 207 GPa (static) and 202 GPa (dynamic), agree within the statistical bounds of the static measurement, 195 GPa and 219 GPa, the extreme static values being 178 GPa and 229 GPa. Simple thermodynamics that shows the difference between dynamic and static elastic constants is also presented.
Perennially among materials engineers, misunderstandings arise about dynamically measured elastic constants. A common viewpoint is that static and dynamic values differ considerably, that dynamic values always exceed static, and that dynamic values are useless (even dangerous) for engineering design. This view is unfortunate for two principal reasons: dynamic values are more accurate than static, and an increasing number of dynamic elastic constants are appearing, even for more practical materials. As shown here, from both measurement and theory sides, the difference between dynamic and static values is small. For a typical engineering alloy, 316LN austenitic steel, a comparison is given between the dynamic and static values of the principal engineering elastic constant: the Young's (extensional) modulus. Besides being a representative engineering material, 316LN has an extensive set of thorough static Young's modulus measurements. As described by Nagai [1], these emerged in round-robin studies on a single batch of material. The measurements were made at 18 institutes in six countries and resulted in 29 recommended values (see Table 1 ). On specimens from the Same batch of 316LN alloy, the Young's modulus was measured at megahertz 0921-5093/93/$6.00
frequencies using dynamic methods described previously [2]. Table 1 shows our results. Against expectations, the dynamic result is lower than the static result by about 2%, within the 6% standard deviation of the static results. The dynamic result shows a standard deviation of only 0.5%. Much experience shows that from laboratory to laboratory, dynamic elastic constant measurements vary between 0.5%-1.0%. For example, a review of the elastic constants of copper showed that the longitudinal stiffness C1~ varied 0.2% among 20 laboratories [3]. What does one expect from simple thermodynamics? The difference between dynamic and static values arises not from frequency effects (these materials are practically dispersionless), but from the fact that dynamic measurements are adiabatic while static measurements are isothermal. Landau and Lifshitz [4] gave a relationship between the adiabatic (constant entropy S) and isothermal (constant temperature T) Young's moduli E: Es = ET + E-~fl 2 T/9Cp
(1)
Here, Cp denotes specific heat at constant pressure and fl denotes volume thermal expansivity:
fl = ( 1/V)(O VIa T )p
(2)
Landau and Lifshitz also pointed out an important result for the shear modulus G: Gs = G~
(3)
The adiabatic and isothermal shear moduli are identical at all temperatures because, from the first law of thermodynamics, deformation at constant volume is adiabatic if it is isothermal. For iron, Fig. 1 shows the
TABLE 1. Dynamics and static Young's moduli of 316LN austenitic steel
No. of laboratories No. of reports Max. (GPa) Min. (GPa) Ave. (GPa) Stand. Dev. (GPa) 100"S.D./Ave.
Static
Dynamic (present)
18 29 229 178 207 12.4 6.0
1 3 203 201 202 1.0 0.5
© 1993 - Elsevier Sequoia. All rights reserved
L10 Z
Letter 20
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,~e.~
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--
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TEMPERATURE, K
Fig. 1. The difference between adiabatic and isothermal elastic constants as a function of temperature for iron. For dispersionless materials, this represents the expected difference between dynamic and static elastic constants. In this figure, B=bulk modulus, v = Poisson's ratio, E =Young's modulus, G= shear modulus.
adiabatic-isothermal elastic constant difference between 0 K and 300 K; Cp and fl values came from standard sources. Besides E and G, Fig. 1 shows two other elastic constants: bulk modulus B and Poisson's ratio v. The four interrelate: (1/E)=(1/9B)+(1/3G)
(4)
and v=(E/2G)- 1
(5)
Figure 1 shows that for iron one expects E s to exceed E x by only about 0.25%, well within the measurement uncertainty of most dynamic methods. For Poisson's ratio, the difference equals about 1.1%. The worst case, the bulk modulus, shows a predicted difference of about 1.8%, approximately the uncertainty of the dynamic method. Since Stokes in 1845, we know that B and G represent the two extreme types of elastic deformation, constant shape with volume change, and constant volume with shape change. Thus, for iron, for any imaginable elastic constant, 1.8% represents the largest possible adiabatic-isothermal difference. In summary, this study examined the difference between dynamic and static Young's moduli for a common austenitic steel. Within the uncertainty of the static measurements, no difference exists. Thermodynamics predicts that the actual difference lies below detection by usual measurement methods. Thus, when materials engineers seek useful physical property information, they should consider both dynamic and static values.
References
1 K. Nagai, in Report on Low-Temperature Structural Materials and Standards Workshop, Naka, Japan, May, 1988, Chapter 20. 2 H. Ledbetter, N. Frederick and M. Austin, J. Appl. Phys., 51 (1980) 305. 3 H. Ledbetter and E. Naimon, J. Phys. Chem. Ref. Data, 3 (1974) 987. 4 L. Landau and E. Lifshitz, Theory of Elasticity, Pergamon, London, 1959, p. 16.
L9
Letter
Dynamic vs. static Young's moduli: a case study Hassel Ledbetter Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Boulder, CO 80303 (USA) (Received December 7, 1992; in revised form December 30, 1992)
Abstract For one material, a 316LN austenitic steel, a comparison is made between the Young's modulus measured statically with usual commercial equipment and that measured dynamically at megahertz frequencies. The first gives a _+6% uncertainty; the second a + 0.5% uncertainty. The two Young's moduli, 207 GPa (static) and 202 GPa (dynamic), agree within the statistical bounds of the static measurement, 195 GPa and 219 GPa, the extreme static values being 178 GPa and 229 GPa. Simple thermodynamics that shows the difference between dynamic and static elastic constants is also presented.
Perennially among materials engineers, misunderstandings arise about dynamically measured elastic constants. A common viewpoint is that static and dynamic values differ considerably, that dynamic values always exceed static, and that dynamic values are useless (even dangerous) for engineering design. This view is unfortunate for two principal reasons: dynamic values are more accurate than static, and an increasing number of dynamic elastic constants are appearing, even for more practical materials. As shown here, from both measurement and theory sides, the difference between dynamic and static values is small. For a typical engineering alloy, 316LN austenitic steel, a comparison is given between the dynamic and static values of the principal engineering elastic constant: the Young's (extensional) modulus. Besides being a representative engineering material, 316LN has an extensive set of thorough static Young's modulus measurements. As described by Nagai [1], these emerged in round-robin studies on a single batch of material. The measurements were made at 18 institutes in six countries and resulted in 29 recommended values (see Table 1 ). On specimens from the Same batch of 316LN alloy, the Young's modulus was measured at megahertz 0921-5093/93/$6.00
frequencies using dynamic methods described previously [2]. Table 1 shows our results. Against expectations, the dynamic result is lower than the static result by about 2%, within the 6% standard deviation of the static results. The dynamic result shows a standard deviation of only 0.5%. Much experience shows that from laboratory to laboratory, dynamic elastic constant measurements vary between 0.5%-1.0%. For example, a review of the elastic constants of copper showed that the longitudinal stiffness C1~ varied 0.2% among 20 laboratories [3]. What does one expect from simple thermodynamics? The difference between dynamic and static values arises not from frequency effects (these materials are practically dispersionless), but from the fact that dynamic measurements are adiabatic while static measurements are isothermal. Landau and Lifshitz [4] gave a relationship between the adiabatic (constant entropy S) and isothermal (constant temperature T) Young's moduli E: Es = ET + E-~fl 2 T/9Cp
(1)
Here, Cp denotes specific heat at constant pressure and fl denotes volume thermal expansivity:
fl = ( 1/V)(O VIa T )p
(2)
Landau and Lifshitz also pointed out an important result for the shear modulus G: Gs = G~
(3)
The adiabatic and isothermal shear moduli are identical at all temperatures because, from the first law of thermodynamics, deformation at constant volume is adiabatic if it is isothermal. For iron, Fig. 1 shows the
TABLE 1. Dynamics and static Young's moduli of 316LN austenitic steel
No. of laboratories No. of reports Max. (GPa) Min. (GPa) Ave. (GPa) Stand. Dev. (GPa) 100"S.D./Ave.
Static
Dynamic (present)
18 29 229 178 207 12.4 6.0
1 3 203 201 202 1.0 0.5
© 1993 - Elsevier Sequoia. All rights reserved
L10 Z
Letter 20
i
I
i
t
i
k-
IRON
e~
¢~_
1.5
,~e.~
1.0
B
--
M.I ,-i,.. P-- I.~.
T
0.5
t.J I-oi3 <¢
7-, <¢
o 100
200
300
TEMPERATURE, K
Fig. 1. The difference between adiabatic and isothermal elastic constants as a function of temperature for iron. For dispersionless materials, this represents the expected difference between dynamic and static elastic constants. In this figure, B=bulk modulus, v = Poisson's ratio, E =Young's modulus, G= shear modulus.
adiabatic-isothermal elastic constant difference between 0 K and 300 K; Cp and fl values came from standard sources. Besides E and G, Fig. 1 shows two other elastic constants: bulk modulus B and Poisson's ratio v. The four interrelate: (1/E)=(1/9B)+(1/3G)
(4)
and v=(E/2G)- 1
(5)
Figure 1 shows that for iron one expects E s to exceed E x by only about 0.25%, well within the measurement uncertainty of most dynamic methods. For Poisson's ratio, the difference equals about 1.1%. The worst case, the bulk modulus, shows a predicted difference of about 1.8%, approximately the uncertainty of the dynamic method. Since Stokes in 1845, we know that B and G represent the two extreme types of elastic deformation, constant shape with volume change, and constant volume with shape change. Thus, for iron, for any imaginable elastic constant, 1.8% represents the largest possible adiabatic-isothermal difference. In summary, this study examined the difference between dynamic and static Young's moduli for a common austenitic steel. Within the uncertainty of the static measurements, no difference exists. Thermodynamics predicts that the actual difference lies below detection by usual measurement methods. Thus, when materials engineers seek useful physical property information, they should consider both dynamic and static values.
References
1 K. Nagai, in Report on Low-Temperature Structural Materials and Standards Workshop, Naka, Japan, May, 1988, Chapter 20. 2 H. Ledbetter, N. Frederick and M. Austin, J. Appl. Phys., 51 (1980) 305. 3 H. Ledbetter and E. Naimon, J. Phys. Chem. Ref. Data, 3 (1974) 987. 4 L. Landau and E. Lifshitz, Theory of Elasticity, Pergamon, London, 1959, p. 16.