Moduli Determination at Different Temperatures by an Ultrasonic Waveguide Method

Abstract

A novel ultrasonic waveguide based technique for measuring the moduli of elastic isotropic material as a function of temperature using ultrasonic guided wave modes is presented here. This technique can be utilized for measuring Young’s modulus (E) and Shear Modulus (G) of multiple material using the guided ultrasonic L(0, 1) and T(0, 1) wave modes respectively over a wide range of temperatures (demonstrated here from room temperature to 1200 °C). The specimens used in the experiments here have special embodiments (for instance, a bend) at one end of the waveguide and an ultrasonic guided wave generator (transducer) at the other end for obtaining reflected signals in a pulse-echo mode. The far end of the waveguides with the embodiment is kept inside a heating device such as a temperature-controlled furnace. The time of flight difference (δTOF) were used to measure the moduli at different temperatures. Several materials were tested and the comparison between literature values and measured values were found to be in agreement, for both elastic moduli (E and G) measurements, as a function of temperature. This technique provides significant reduction in time, effort and cost over conventional means of measurement of temperature dependence of elastic moduli.

Appendix A: Derivation of the Empirical Relationship between measured TOF vs E and G

Two methods of deriving the relationship has been described below. The gage length of the waveguide, which is the bent region as shown in Fig. 12 was considered in 2 different temperatures i.e. ambient room temperature and instantaneous temperature during measurement.

Fig. 12

The waveguide configuration with the bend region

The derivation approach assumes following:

• 1. The diameter of the waveguide is significantly small when compared to the length of the waveguide and in the wave mode is in the long wavelength regime (low frequency). Hence, the bar velocity formulation can be applied for the fundamental L(0,1) and T(0, 1) modes.

• 2. The bend region length is relatively small when compared to the remaining straight portion of the waveguide i.e. The time of flight (TOF)_v>> (TOF)_h condition is considered at vertical (v) and horizontal (h) portion of the waveguide.

• 3. The wave mode traveling is relatively non dispersive and any dispersion effect may be neglected.

• 4. The waveguide material is isotropic and homogeneous.

Method 1: Using the Velocity Ratio of the Waveguide at Different Temperature

During the transition from ambient temperature (T ₀) to the instantaneous temperature (T _i), the following relationships are applicable due to the change in temperature ΔT:

$$ Change\ in\ length\ of\ rod\ \left(\delta l\right) = {l}_0\times \alpha \times \varDelta T $$

(A1)

$$ Thermal\ Strain\kern0.5em \left({\varepsilon}_t\right) = \alpha \times \varDelta T=\frac{\updelta \mathrm{l}}{{\mathrm{l}}_0} $$

(A2)

hence, Coefficient of linear thermal expansion

$$ \left(\upalpha \right)\kern0.75em =\kern0.5em \frac{\updelta \mathrm{l}}{{\mathrm{l}}_0\times \varDelta T} $$

(A3)

Coefficient of volumetric thermal expansion of the rod

$$ \left(\beta \right)=3\alpha $$

(A4)

The density (ρ_i) of solids is always function of temperature (i) by a change in their linear dimensions and volume [56]

$$ {\uprho}_{\mathrm{i}} = \frac{\uprho_0}{1+\left(\upbeta \times \varDelta T\right)}\kern0.5em =\kern0.5em \frac{\uprho_0}{1+\left(3\upalpha \times \varDelta T\right)} = \kern0.5em \frac{\uprho_0}{1+\left(3\frac{\updelta \mathrm{l}}{{\mathrm{l}}_0}\right)} $$

(A5)

Assuming low frequency regime for the wave propagation of the L(0, 1) mode, the following relationships were used:

Ambient Longitudinal Velocity at ambient temperature

$$ \left({\mathrm{V}}_0\right) = \sqrt{\frac{{\mathrm{E}}_0}{\uprho_0}} $$

(A6)

Instantaneous Longitudinal Velocity at T₁

$$ \left({\mathrm{V}}_{\mathrm{i}}\right) = \sqrt{\frac{{\mathrm{E}}_{\mathrm{i}}}{\uprho_{\mathrm{i}}}} $$

(A7)

This leads to the expression below for the velocity ratio between the ambient velocity and the instantaneous velocity:

$$ \sqrt{\frac{\left(\frac{{\mathrm{E}}_0}{\uprho_0}\right)}{\frac{E_i}{\left(\frac{\uprho_0}{1+\left(3\ \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)}\right)}}}=\frac{{\mathrm{V}}_{0\ }}{{\mathrm{V}}_{\mathrm{i}}} from\ Eqns.\ \left(A5,\ A6\ and\ A7\right) $$

(A8)

However, the following relationship for the velocity ratio can be written as:

$$ \frac{{\mathrm{V}}_0}{{\mathrm{V}}_{\mathrm{i}}} = \frac{\left(\frac{{\mathrm{l}}_0}{{\mathrm{TOF}}_0}\right)}{\left(\frac{{\mathrm{l}}_{\mathrm{i}}}{{\mathrm{TOF}}_{\mathrm{i}}}\right)}=\frac{\left(\frac{{\mathrm{l}}_0}{{\mathrm{TOF}}_0}\right)}{\left(\frac{{\mathrm{l}}_0+{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{TOF}}_{\mathrm{i}}}\right)} $$

(A9)

Where

Instantaneous change in length (l_i) due to temperature change = l₀ + δl_i

Corresponding Instantaneous change in time (TOF_i) = TOF₀ + δTOF_i

From Eqn. (A8, A9), after simplification and using Taylor’s series expansion (up to 2^nd order only)

$$ \left(\frac{{\mathrm{E}}_0}{{\mathrm{E}}_{\mathrm{i}}}\times \frac{1}{1+\left(3\ \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)}\right)=\kern0.5em {\left(\frac{\frac{{\mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}}{\frac{{\mathrm{l}}_0+\updelta \mathrm{l}}{{\mathrm{l}}_0}}\right)}^2={\left(\frac{1+\kern0.5em \frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}}{1+\kern0.75em \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}}\right)}^2 $$

(A10)

$$ \frac{{\mathrm{E}}_0}{{\mathrm{E}}_{\mathrm{i}}}=\frac{\left(1 + \left(3\frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)\right)}{{\left(1 + \kern0.5em \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)}^2}\times {\left(1+\kern0.5em \frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}\right)}^2=\mathrm{K}\times \left(1 + 2\ \frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}+\kern0.5em {\left(\frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}\right)}^2\right) $$

(A11)

METHOD 2: Change in Time of Flight Ratio

$$ \begin{array}{l}\frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}=\left(\frac{\left(\frac{{\mathrm{l}}_0+{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{V}}_{\mathrm{i}}}\right)-\frac{{\mathrm{l}}_0}{{\mathrm{V}}_0}}{\frac{{\mathrm{l}}_0}{{\mathrm{V}}_0}}\right)\\ {}\frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}=\left(1 + \kern0.5em \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)\frac{{\mathrm{V}}_0}{{\mathrm{V}}_{\mathrm{i}}}-1\end{array} $$

(A12)

$\frac{{\mathrm{V}}_0}{{\mathrm{V}}_{\mathrm{i}}}$ from earlier case of derivation

$$ \left(1 + \frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}\right)=\left(1 + \kern0.5em \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)\times \sqrt{\frac{{\mathrm{E}}_0}{{\mathrm{E}}_{\mathrm{i}}}\times \frac{1}{\left(1+\left(3\frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)\right)}} $$

(A13)

$$ \frac{{\mathrm{E}}_0}{{\mathrm{E}}_{\mathrm{i}}}=\frac{\left(1 + \left(3\frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)\right)}{{\left(1 + \kern0.5em \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)}^2} \times {\left(1 + \frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}\right)}^2=\left(\mathbf{K}\right)\times \left(1 + 2\ \frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}+\kern0.5em {\left(\frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}\right)}^2\right) $$

(A14)

This is identical to the Eqn. A11.

In order to simplify the relationship and derive an empirical expression the following approximations are implemented K ≈ 1 and ${\left(\frac{{\updelta \mathrm{TOF}}_{\mathrm{i}}}{{\mathrm{TOF}}_0}\right)}^2\approx 0$ using the assumption that the gage length δl is small when compared to length of the waveguide l₀

The approximation K ≈ 1 is further verified for four materials Inconel, Nickel, Kanthal and Copper at different temperatures as shown in Table 5. It is believed that this approximation will be applicable to most metals.

Table 5 Calculation of the constant K for different materials

Similarly, if l>>δl , the TOF ratio is relatively small and hence the 2^nd order term can be neglected. Using these approximations, the expression in Eq. 13 can be simplified as below:

$$ {\mathbf{E}}_{\mathbf{i}}=\left(\frac{{\mathbf{E}}_0}{1+\kern0.5em 2\ \left(\frac{\boldsymbol{\updelta} \mathbf{T}\mathbf{O}{\mathbf{F}}_{\mathbf{i}}}{\mathbf{TO}{\mathbf{F}}_0}\right)}\right) $$

(A15)

From experimental results, it was also confirmed that the expression may be more generalized a as below, where the constant A for L (0,1) mode used here was found to be 2.5.

$$ {\mathbf{E}}_{\mathbf{i}}=\left(\frac{{\mathbf{E}}_0}{1+\kern0.5em \mathbf{A}\ {\left(\frac{\boldsymbol{\updelta} \mathbf{T}\mathbf{O}{\mathbf{F}}_{\mathbf{i}}}{\mathbf{TO}{\mathbf{F}}_0}\right)}_{\mathbf{L}}}\right) $$

(A16)

Similar approach is used for shear modulus (G_i) measurement of waveguide at different temperature

$$ {\mathbf{G}}_{\mathbf{i}}=\left(\frac{{\mathbf{G}}_0}{1+\kern0.5em \mathbf{B}{\left(\frac{\boldsymbol{\updelta} \mathbf{T}\mathbf{O}{\mathbf{F}}_{\mathbf{i}}}{\mathbf{T}\mathbf{O}{\mathbf{F}}_0}\right)}_{\mathbf{T}}}\right) $$

(A17)

The empirical constants B was found to be 2.25 from experiments.

$$ \mathrm{K} = \frac{\left(1 + \left(3\frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)\right)}{{\left(1 + \kern0.5em \frac{{\updelta \mathrm{l}}_{\mathrm{i}}}{{\mathrm{l}}_0}\right)}^2}\kern0.36em \mathrm{from}\ \mathrm{Eqns}.\ \left(\mathrm{A}11,\ \mathrm{A}14\right) $$

(A18)

The K ≈ 1 values are verified from different materials (α from ref [42, 48, 49, 57]) at different temperatures using Eqn. (A1).