Third Order Elastic Constants (TOEC)

Third order elastic constants were introduced to describe the nonlinear stress-strain model. For an isotropic solid there are 2 Second Order Elastic Constants (SOEC) and 3 Third Order Elastic Constants (TOEC) needed to describe and define the properties of the material. The second order elastic constants are often known as the Lamé constants λ and μ and the third order constants are given the convention l, m and n. These constants can then be related to acoustic velocity for a given stress, this is the basis of the acoustoelastic measurement. Inversely it is possible to determine the TOEC experimentally by measuring the velocity of longitudinal and shear acoustic waves through a material with varying orientation and stress.

4.4.2 Acoustoelasticity

If we consider two types of acoustic waves, longitudinal and shear, and consider two directions of propagation, parallel and perpendicular to stress, then we can start to consider what effect stress on acoustic wave velocity will have.

Figure 4.2: Wave Type and Polarisation Notation with Reference to the Direction of Stress.

The equations for the acoustic velocity of each of these wave types and orientations with stress have already been derived and are shown below. Equation 4.18 gives this relationship for a longitudinal wave travelling parallel to the applied stress:

𝜌₀𝑐_𝑥𝑥² = 𝜆 + 2𝜇 + 𝜎

3𝐵[2𝑙 + 𝜆 +𝜆 + 𝜇

𝜇 (4𝑚 + 4𝜆 + 10𝜇)] 4.18

Where 𝜌0 denotes unstressed density, 𝑐𝑥𝑥 is the speed of sound, the first subscript denoting wave propagation direction, the second subscript denoting the polarisation direction. 𝜎 is the uniaxial stress and 𝐵 is the bulk modulus. The second order elastic constants are given by λ, μ and the third order constants are given by l, m and n. Equation 4.19 gives the relationship for a longitudinal wave travelling perpendicular to the stress.

𝜌₀𝑐_𝑦𝑦² or 𝜌₀𝑐_𝑧𝑧² = 𝜆 + 2𝜇 − 𝜎

3𝐵[2𝑙 −2𝜆

𝜇 (𝑚 + 𝜆 + 2𝜇)] 4.19

Equation 4.20 gives the relationship for a shear wave with polarisation perpendicular to the stress travelling parallel to the stress.

𝜌₀𝑐_𝑥𝑦² or 𝜌₀𝑐_𝑥𝑧² = 𝜇 + 𝜎

3𝐵[𝑚 +𝜆𝑛

4𝜇+ 4𝜆 + 4𝜇] 4.20

Equation 4.21 gives the relationship for a shear wave with polarisation parallel to the stress travelling perpendicular to the stress.

𝜌₀𝑐_𝑦𝑥² or 𝜌₀𝑐_𝑧𝑥² = 𝜇 + 𝜎

3𝐵[𝑚 +𝜆𝑛

4𝜇+ 𝜆 + 2𝜇] 4.21

Equation 4.22 gives the relationship for a shear wave with polarisation perpendicular to the stress travelling perpendicular to the stress.

A number of previous studies have determined the SOECs, TOECs and densities for a range of steels. Some of these values, along the bulk modulus calculated using Equation 4.23 are given in Table 4.1.

Steel, unspecified (Allen &Sayers, 1984) StE 355 (Lüthi, 1990) StE 42 (Si-Chaib, Menad, Djelouah, & Bocquet, 2001) DIN 22NiMoCr37 (Si-Chaib, Menad, Djelouah, &Bocquet, 2001) E 295 (Si-Chaib, Menad, Djelouah, & Bocquet, 2001) Low Carbon Steel (Muir, 2009) Rail Steel (Bray D. M., 1976) Nickel-Steel S/NTV(Crecraft, 1967)

λ (GPa) 112.8 109 110.4 109.1 110 111 115.8 109.0

Table 4.1: SOEC, TOEC and Density for a Number of Steels.

The values for the first steel listed in Table 4.1 where used to calculate the relative change in the speed of sound when a material is under tension for both longitudinal and shear waves propagating both parallel and perpendicular to the direction of the stress. These values show that the wave types relating to Equations 4.18 to 4.21 have a decreasing wave velocity with increasing tensile stress, while the wave polarisation relating to Equation 4.22 is the only one which has an increasing wave velocity with increasing tensile stress. The relative change in the speed of sound for each wave type in a material experiencing a tensile stress are plotted in Figure 4.3.

The figure shows that a longitudinal wave propagating in the direction of stress has the largest variation of acoustic velocity, followed by a shear wave travelling perpendicular to the direction of stress but polarised in the direction of stress. It is also interesting to note that the acoustic velocity of a shear wave travelling and polarised perpendicular to the stress direction will increase with tensile stress.

Figure 4.3: Percentage change in Speed of Sound for Various Wave Types and Polarisations in a Standard Steel subjected to Tensile Stress in the x-direction.

Figure 4.3 shows that over the range of stresses calculated the various equations can be approximated as linear. The relationship between stress and the change in a materials time of flight can be characterised by the acoustoelastic constant. This can be derived from the TOEC as shown below.

For uniaxial stress acting in the x-direction (𝜎𝑥), as shown in Figure 4.2 the triaxial principal strains are given by:

𝜀_𝑥= 𝜀 𝜀_𝑦= 𝜀_𝑧 = −𝜈𝜀

4.24 4.25 Where 𝜀𝑥, 𝜀𝑦 and 𝜀𝑧 are the components of strain in the x, y and z directions respectively and 𝜈 is Poisson’s ratio. From this and Equations 4.18 to 4.22 the speed of sound with respect to strain can be calculated (Bray D. M., 1976).

𝜌₀𝑐_𝑥𝑥² = 𝜆 + 2𝜇 + [4(𝜆 + 2𝜇) + 2(𝜇 + 2𝑚) + 𝜈𝜇(1 + 2𝑙 𝜆⁄ )]𝜀 4.26a 𝜌₀𝑐_𝑦𝑦² = 𝜌₀𝑐_𝑧𝑧² = 𝜆 + 2𝜇 + [2𝑙(1 − 2𝜈) − 4𝜈(𝑚 + 𝜆 + 2𝜇)]𝜀 4.26b 𝜌₀𝑐_𝑥𝑦² = 𝜌₀𝑐_𝑥𝑧² = 𝜇 + [4𝜇 + 𝜈(𝑛 2⁄ ) + 𝑚(1 − 2𝜈)]𝜀 4.26c 𝜌₀𝑐_𝑦𝑥² = 𝜌₀𝑐_𝑧𝑥² = 𝜇 + [(𝜆 + 2𝜇 + 𝑚)(1 − 2𝜈) + (𝑛𝜈 2⁄ )]𝜀 4.26d

𝜌₀𝑐_𝑦𝑧² = 𝜌₀𝑐_𝑧𝑦² = 𝜇 + [(𝜆 + 𝑚)(1 − 2𝜈) − 6𝜈𝜇 − (𝑛𝜈 2⁄ )]𝜀 4.26e Following from this we can calculate the change in time of flight using the method developed by Chen et. al. (Chen, Mills, & Dwyer-Joyce, 2015) and detailed below. The speed of sound in the unstrained state is given by:

ρ₀(𝑐_𝑥𝑥)₀²= 𝜆 + 2𝜇 4.27

Where (𝑐𝑥𝑥)₀ is the unstrained longitudinal speed of sound in the x-direction. Next by differentiating Equation 4.26a we get:

ρ₀(𝑐_𝑥𝑥)_𝑃𝑑(𝑐_𝑥𝑥)_𝑃= [2(𝜆 + 2𝜇) + (𝜇 + 2𝑚) + 𝜈𝜇(1 + 2𝑙 𝜆⁄ )]d(𝜀_𝑥)_𝑃 4.28 If the change in the speed of sound is small the following assumption can be made.

ρ₀(𝑐_𝑥𝑥)_𝑃(𝑐_𝑥𝑥)₀≈ ρ₀(𝑐_𝑥𝑥)₀²= 𝜆 + 2𝜇 4.29 Dividing Equation 4.28 by 4.29 gives the change in the speed of sound in relation to the strain.

d(𝑐_𝑥𝑥)_𝑃⁄(𝑐_𝑥𝑥)₀= [2 +(𝜇 + 2𝑚) + 𝜈𝜇(1 + 2𝑙 𝜆⁄ )

𝜆 + 2𝜇 ] d(𝜀_𝑥)_𝑃 4.30

This can be rearranged to leave the right hand side of this equation dependant on material properties alone. It is useful to substitute the term 𝛼𝑥𝑥 in place of these properties.

𝛼𝑥𝑥 = 2 +(𝜇 + 2𝑚) + 𝜈𝜇(1 + 2𝑙 𝜆⁄ )

𝜆 + 2𝜇 4.31

Where 𝛼_𝑥𝑥 is the acoustoelastic constant for a longitudinal wave propagating in the x-direction.

Substituting into Equation 4.30 and rearranging yields:

𝛼𝑥𝑥 =d(𝑐_𝑥𝑥)_𝑃⁄(𝑐_𝑥𝑥)₀

d(𝜀_𝑥)_𝑃 4.32

From Equation 4.32 we can get the change in speed of sound for a given change in the strain.

Δ𝑐_𝑥𝑥 = 𝛼_𝑥𝑥(𝑐_𝑥𝑥)₀Δ𝜀_𝑥 4.33

Taking the change in the speed of sound as:

Δ𝑐_𝑥𝑥 = (𝑐_𝑥𝑥)₀− (𝑐_𝑥𝑥)_𝑃 4.34

And the change in the strain as:

Δ𝜀_𝑥 = −(𝜀_𝑥)_𝑃 4.35

Combining Equations 4.33 to 4.35 gives the change of speed of sound with the change in the strain.

(𝑐_𝑥𝑥)_𝑃= (𝑐_𝑥𝑥)₀(1 − 𝛼_𝑥𝑥Δ𝜀_𝑥) 4.36

The change in time of flight caused by the change in speed of sound due to stress alone can be calculated from the difference between the time-of-flight of the stressed and unstressed states, Equation 4.37.

(Δ𝑡_𝑥𝑥)_E= 2𝑑₀

(𝑐_𝑥𝑥)₀− 2𝑑₀

(𝑐_𝑥𝑥)_𝑃 4.37

Substituting in Equation 4.36 and rearranging results in:

(Δ𝑡_𝑥𝑥)_E= 2𝑑₀

(𝑐_𝑥𝑥)₀( −𝛼_𝑥𝑥Δ𝜀_𝑥

1 − 𝛼_𝑥𝑥Δ𝜀_𝑥) 4.38

It is useful to substitute the deflection in the place of the strain. The deflection, δ, is given in terms of strain by the following

δ = Δ𝜀_𝑥𝑑₀ 4.39

This gives the change in Time-of-Flight due to the acoustoelastic effect alone, in terms of the deflection, acoustoelastic constant, initial length and unstressed speed of sound.

(Δ𝑡_𝑥𝑥)_E= −2𝑑₀𝛼_𝑥𝑥δ

(𝑐_𝑥𝑥)₀(𝑑₀− 𝛼_𝑥𝑥δ) 4.40

This gives the change in Time-of-Flight due to the acoustoelastic effect alone in terms of the deflection, acoustoelastic constant, initial length and unstressed speed of sound of a longitudinal wave. The effect is slight, resulting in a 0.05% increase in the speed of sound within a typical steel for a compressive load of 100MPa. Because the acoustoelastic constant is just an expression of material properties, the same equation can be applied to a shear wave by substituting in the shear speed of sound and acoustoelastic constant, 𝛼𝑥𝑦.