# Measurement of third-order elastic constants and stress dependent coefficients for steels

## Abstract

### Background

There has been little discussion of the third-order elastic constants of steels in the literature until now. In this study, the precise second- and third-order elastic constants of polycrystalline steels were measured under adiabatic and isothermal conditions.

### Method

To measure the minute change in the propagation time of the elastic wave corresponding to the tensile stress, the uniform and isotropic specimens were processed with high precision, the measuring instruments were strictly calibrated, and the temperature of the measurement chamber was kept constant. The author proposes an experimental formula to obtain the third-order elastic constants of steels. The stress dependent coefficients *α*_{ ij } in this formula are absolutely necessary to obtain the third-order elastic constants.

### Results

The obtained stress dependent coefficients clearly indicated that there is a special relationship between the directions of stress and that of the oscillation of the elastic wave. When the frequency direction of the elastic wave matched the direction of the applied stress, *α*_{ ij } became a larger negative value. Lamè constants and Murnaghan’s third-order elastic constants *ℓ*,*m,n* were obtained for four types of steels.

### Conclusions

The second- and third-order elastic constants under adiabatic conditions were smaller than those under isothermal conditions. Oscillation of crystal lattice is nonlinear and this is observed as the third-order elastic constants. Therefore, it is possible to obtain the knowledge on the internal stress and the thermal properties of the materials. This is also the basis of theoretical discussion of the thermal expansion coefficients.

## Keywords

Second- and third-order elastic modulus Elastic wave Stress dependent coefficient Polycrystalline material## Abbreviations

- AISI
American iron and steel institute

## Introduction

The first study of the theory and measurement of the third-order elastic constants of practical materials was published by Hughes and Kelly (1953). D.Lazarus reported the third-order elastic constants of the single crystals like *KCl*, *Cu* etc by measureing propagation velocity of elastic wave under hydrostatic pressure (Lazarus 1949) and it may useful in comparison of the theories of finite strain proposed by Murnagham (1951). R.N.Thurston published a paper on the theoretical analysis of the propagation of elastic wave (Thurston and Brugger 1964). D.M. Egle et al. carried out the measurement of the third-order elastic constants for rail steel using Hughes’s result (Egle and Bray 1976). S.Takahashi got the U.S.patent on the stress measurement and its equipment according to the method of the present paper (Takahashi 2007). T.Batemen et al. reported that the third-order elastic constant of semiconductor was related to thermal expansion coefficient and Gűneisen constants (Bateman et al. 1961). As described above, the knowledge on the third-order elastic constants contributes to the study of physical properties of various materials.

This paper describes the measurement of the third-order elastic constants of four common steels. The stress applied to the specimen was increased stepwise, and the velocity of the elastic wave was measured at every step using a high accuracy measuring technique. The shape and dimensions of the specimen were controlled as precisely as possible and the measuring equipment was also accurately calibrated. The change in room temperature was kept to 1°C or less during the measurement. Second- and third-order elastic constants contribute to the change in the propagation velocity of the elastic wave caused by stress (Hughes and Kelly 1953; Takahashi and Motegi 2015).

The author proposed a simple equation of the propagation velocity under stress by introducing the coefficient *α*_{ ij } including the second- and third-order elastic constants. The value of *α*_{ ij } can be obtained from the measured stress and the change ratio in propagation velocity. The coefficients *α*_{ ij } is absolutely necessary to obtain the third-order elastic constants. When the frequency direction of the elastic wave matched the direction of the applied stress, the value of *α*_{ ij } became more negative. This means that the coincidence of the frequency direction with the applied stress greatly contributed to the propagation velocity of the wave. The value of *α*_{ ij } is based on the stress-strain relation. Therefore it is necessary to obtain the vales of *α*_{ ij } on this relationship.

## Experimental method

### Test specimen

*V*

_{11}, while that of the transvers wave to direction 1 and vibration in direction 2 was expressed as

*V*

_{12}. For the propagation velocity V, the same subscripts express the longitudinal wave, while different subscripts express the transverse wave.

*T*

_{11}represents the tensile stress in the direction of the long axis of the specimen. Table 1 lists the chemical compositions of the S20C(AISI 1020), S30C(AISI 1030), S40C(AISI 1039) and S50C(AISI 1049) test specimens.

Chemical composition of S20C to S50C specimens(wt%)

Sample | C | Si | Mn | Ni+Cr |
---|---|---|---|---|

S20C | 0.22 | 0.29 | 0.52 | |

S30C | 0.32 | 0.31 | 0.81 | |

S40C | 0.40 | 0.28 | 0.78 | 0.12 |

S50C | 0.49 | 0.31 | 0.83 | 0.12 |

### Stress dependent coefficients of elastic wave *α* _{ ij }

*V*

_{ ij }of the elastic wave in the specimen under an applied stress of

*T*

_{11}is expressed as

*V*

_{0}is the propagation velocity of the elastic wave under non-loaded state,

*α*

_{ ij }is the stress dependent coefficient of the elastic wave, and E is Young’s modulus. The expression of

*V*

_{ ij }by Hughes and Kelly (1953) and the authors Takahashi and Motegi (2015) is

*λ*,

*μ*are Lamè constants,

*ℓ*,

*m,n*are the Murnaghan’s third order elastic constants,

*ν*is Poisson’s ratio and

*ρ*

_{0}is the density in the non-deformed state. The formula (1) is based on an equation previously introduced by the authors Takahashi and Motegi (2015) and it can be rewritten as

*α*

_{11}is given as follows,

*α*

_{22},

*α*

_{21},

*α*

_{12}and

*α*

_{23}can be obtained.

*α*

_{ ij }is used to obtain Murnaghan’s third-order elastic constants

*ℓ*,

*m,n*as follows.

here *a*= *λ*+*m*

### Measurement of the propagation velocity of elastic wave in the applied stress direction

*L*

_{ a },

*L*

_{ b }are the total length of the specimens under stress free conditions, and

*t*

_{ a },

*t*

_{ b }are the propagatioon time under applied stress.

The differences in propagation time are written as *Δ**t*_{ a }=*t*_{ a }−*t*_{0a} and *Δ**t*_{ b }=*t*_{ b }−*t*_{0b}.

*L*

_{ m }is the length of the gauge part in the non-loaded state,

*L*

_{ g }is the grip length under the applied stress,

*V*

_{ g }is the average velocity of the wave passing through grip part. In the case of a longitudinal wave propagating in the applied stress direction of the long specimen,

*Δ*

*t*

_{ a }/

*t*

_{0a}can be written using an approximate calculation as

*Δ**t*_{ a }/*t*_{0a} and *Δ**t*_{ b }/*t*_{0b} can be obtained by measuring the time of the propagating elastic wave.

*α*

_{11}is given as

The transvers wave of *α*_{12} is also obtained from applying the measured *Δ**t*_{ a }/*t*_{0a} and *Δ**t*_{ b }/*t*_{0b} data to the formula (9).

### Measurement of propagation velocity of elastic wave in direction orthogonal to the tensile stress axis

*ν*is Poisson’s ratio. From the approximate calculation of formula(10), the coefficient related to the elastic wave in the direction of 2

*j*is given as

The coefficient *α*_{22} for longitudinal waves and *α*_{21}, *α*_{23} for transverse waves can be obtained from formula (11) using the respective value of \(\left (\frac {\Delta t}{t_{0}}\right)\) from the measurements of S20C to S50C specimens.

### Measurement equipment

*mm*

^{2}in size, were used as transducers for the longitudinal and tansvers waves.

## Results

*α*

_{ ij }was obtained by measurering the gradient of the stress vs change ratio of the propagation time curve shown in Fig. 4 to formula (9) and (11) for long and short specimens.

*α*

_{ ij }obtained by the stress-strain curve of tencile testing machine were set as values under isothermal conditions. On the other hand, these obtained from the measured values of the propagating time of the elastic wave were taken as values of adibatic conditions. The measured values of

*α*

_{11},

*α*

_{12},

*α*

_{21},

*α*

_{22}and

*α*

_{23}in the isothermal and adiabatic states are shown in Table 2. These were different owing to the different elastic constants measured by the tensile test or from the adiabatic elastic wave. The second-order elastic constants of the isothermal and adiabatic measurements are shown in Table 2 alongside the third-order elastic constants.

Coefficient *α*_{ ij } and second- and third-order elastic constants in the isothermal and adiabatic states

(A) Stress dependent coefficients of | ||||||||

| S20iso | S20adi | S30iso | S30adi | S40iso | S40adi | S50iso | S50adi |

| -1.484 | -1.450 | -1.620 | -1.507 | -1.823 | -1.689 | -1.887 | -1.764 |

| 0.232 | 0.227 | 0.161 | 0.143 | 0.164 | 0.159 | 0.194 | 0.173 |

| 0.282 | 0.291 | 0.157 | 0.193 | 0.128 | 0.159 | 0.059 | 0.098 |

| -1.209 | -1.197 | -1.443 | -1.393 | -1.448 | -1.393 | -1.388 | -1.342 |

| 0.408 | 0.399 | 0.116 | 0.100 | 0.035 | 0.024 | 0.092 | 0.076 |

(B) Lamè constants and Young’s modulus(× 10 | ||||||||

| 115 | 114 | 98 | 94 | 113 | 110 | 132 | 126 |

| 83 | 82 | 86 | 82 | 85 | 82 | 85 | 82 |

E | 214 | 211 | 217 | 207 | 218 | 210 | 223 | 213 |

(C) Murnaghan’s third order elastic constants(× 10 | ||||||||

| -940 | -932 | -797 | -795 | -916 | -914 | -1064 | -1058 |

| -346 | -335 | -355 | -318 | -402 | -362 | -434 | -390 |

| -1161 | -1025 | -1120 | -1105 | -1012 | -1003 | -1058 | -1038 |

| -1192 | -1187 | -1420 | -1373 | -1331 | -1297 | -1233 | -1189 |

| -619 | -601 | -553 | -505 | -550 | -503 | -580 | -530 |

| 7.833 | 7.829 | 7.840 | 7.835 | ||||

| 0.291 | 0.290 | 0.267 | 0.267 | 0.286 | 0.287 | 0.304 | 0.304 |

*α*

_{11},

*α*

_{12},

*α*

_{21},

*α*

_{22}and

*α*

_{23}of each specimen. It was clear that

*α*having coordinates 11 or 21 showed a larger negative value. It is considered that the agreement between the stress and oscillation direction of the elastic wave greatly displaced

*α*

_{ ij }towards a more negative value. The measured values of the third-order elastic constants of each specimen in adiabatic state are shown in Fig. 6.

## Discussion

The main goal of this study was to accurately measure the change in the propagation time of elastic waves in the material with stress and obtain a mathematical formula connecting theory and experiments to derive the third-order elastic constants. An empirical formula consisting of the stress dependent coefficient *α*_{ ij } related with the third-order elastic constants was obtained based on the mathematical formula derived by Hughes and Kelly (1953) and the present author Takahashi and Motegi (2015). Care was taken in preparing the specimens to precise dimensions, using well-calibrated measument equipment, and maintaining good temperature control during the measurement to obtain precise *α*_{ ij } data.

The basis of this experiment is to measure the value of *α*_{ ij }. The value of *α*_{ ij } can be obtained from the gradient of changing ratio of elastic wave propagation time to the applied stress. Therefore from this viewpoint, *α*_{ ij } should be determined. When the direction of the stress matches the direction of the vibration wave, *α*_{ ij } has a larger negative value compared with in the other cases. It means the decrease of propagation velocity of the elastic wave. Thus, this coefficient *α*_{ ij } also has a relationship to the propagation velocity of the elastic waves. Murnaghan’s third order elastic constants could be calculated using the obtained coefficients *α*_{ ij }. As described above, of the obtained third-order elastic constants, *ℓ*, *m* and *n* were negative for all specimens.

Table 2(B)(C) show the differences between isothermal and adiabatic elastic constants. The difference in the third-order elastic constants are larger than the difference in the second-order ones. O.M.Krasinikov reported no significant difference between isothermal and adiabatic elastic constants (Krasilnikov 1977). However, there is a relatively large difference between *m* and *n* in isothermal state and adiabatic state.

## Conclusions

- 1.
An experimental methods for measuring third-order elastic constants accurately and relatively easy was described.

- 2.
Measurements of the change ratio of propagation time of the elastic wave with respect to the change in the applied stress, stress-strain curves, were performed for four types of practical steel specimens.

- 3.
A formula relating the experimentally measured values to theory was proposed. This formula consisted of stress, Young’s modulus, the propagation velocity of the elastic wave, stress dependent coefficient

*α*_{ ij }, the values of all of which were measurable. - 4.
The two types of speimens with identical grip sizes but different gauge lengths were prepared to eliminate the influence of the grip parts and to apply uniform stress.

- 5.
A formula for

*α*_{ ij }was proposed using data obtained from long and short specimens. - 6.
*α*_{ ij }was obtained from the gradient of the relationship between the change ratio of the propagation time and stress. - 7.
*α*_{ ij }is not only a coefficient necessary to obtain the third-order elastic constants but also provide other information on the behavior of elastic wave propagation. The values of*α*_{11}and*α*_{21}where the frequency direction of the elastic wave matched the stress axis, were negative and larger in magnitude than those of the other coefficients. This meant a deceleration of the propagation of the elastic wave. - 8.
The Murnaghan’s third-order elastic constants

*ℓ*,*m,n*were obtained for the four types of practical steels under isothermal and adiabatic conditions and those were negative values. - 9.
The differences between the values of Lamè constants

*λ*,*μ*and the Young’s modulus E measured under isothermal and adiabatic conditions were not very large, but a relatively large difference was found in the case of*m*and*n*in third-order modulus. - 10.
The present study of the third-order elastic constants of materials will greatly contribute to understanding internal stress, thermal properties, Grűneisen constants and so on of steel, semiconductor and non ferrous metals. The author got U.S.Patent on the stress measurement and its equipment (Takahashi 2007).

## Notes

### Acknowledgements

Not applicable.

### Funding

Not applicable.

### Availability of data and materials

Data and materials are available.

### Authors’ contributions

The author read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Consent for publication

Author agrees to publication.

### Competing interests

Not applicable.

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