How Ultrasonic Pulse-Echo Techniques and Numerical Simulations Can Work Together in the Evaluation of the Elastic Properties of Glasses

Abstract

This paper presents the numerical simulation of the ultrasonic wave transmittance utilizing the elastodynamic finite integration technique (EFIT). With this methodology, it is possible to simulate the propagation of the ultrasound in a medium with a relatively low computational cost. The capability of this technique for determining the elastic properties of fluorophosphate and the aluminosilicate glasses is described in detail. The elastic constants of the glasses were calculated from the theoretically predicted longitudinal and transversal sound velocities and compared with the corresponding experimental data. Furthermore, the calculated and experimental elastic properties of the fluorophosphate and aluminosilicate glasses were correlated with the structural peculiarities of these glasses. This simulation technique is also suitable for unveiling the existence of possible defects in the glasses by comparing the experimental and simulation data. The EFIT technique is shown to be a very useful tool in order to provide fast and easy-to-acquire data regarding also the structural characteristics of various glassy systems. This can be used in conjunction with other spectroscopic techniques which can prove to be extremely useful for the non-destructive testing of vitreous materials. The latter can prove very important when vitreous materials used in optical or optoelectronic applications need continuous monitoring in order to ensure their optimum operation and functionality with limited intervention. The main contribution of this paper is the treatment of numerical time-domain modeling of 2D acoustic wave propagation in a viscoelastic medium by implementing the elastodynamic finite integration technique (EFIT).

1. Introduction

Among acoustic techniques, ultrasonic echography stands out as a valuable tool in various fields for the non-destructive inspection of materials including neat liquids, plastics, metals, composites, concrete, and cements [1,2] and references therein. Furthermore, ultrasonic waves can be used to sono-chemically synthesize materials and for welding purposes. In terms of ultrasound intensity, the applications are separated into two main groups, namely the low- and the high-intensity ultrasonic applications. In the low-intensity or equivalent high-frequency ultrasound group, the low-energy ultrasonic waves propagate through the medium providing information dynamic analysis, rheology, and process monitoring [3]. Ultrasonic relaxation spectroscopy also utilizes low-power ultrasound waves to deliver information for the structure of the studied material in the medium- to long-range order. These studies include proton-transfer reaction [4,5,6] and association–dissociation mechanisms [7], conformations between isomers [8], complexation processes [9,10,11], backbone segmental motion and normal mode relaxation mechanisms in polymeric solutions [12,13], and other processes related to reaction engineering [14]. In the high-intensity or low-frequency ultrasound group, the sound waves propagate through the medium inducing locally high pressures and temperatures. This methodology is utilized in several laboratory and industrial applications involving processes such as emulsification, welding, cleaning, fatigue testing, chemical reactions, heat generation, and others [15]. In general, ultrasonic measurements are rapid, accurate, non-destructive in the high-frequency region, and can be fully automated for industrial uses but require experienced operators that master the specific methodology and be able to precisely interpret the experimental results. The situation in most of the cases is perplexing since the results are subject to severe disturbances including beam deviation and/or splitting, mode conversion due to the presence of anisotropic and heterogeneous domains in the material, etc. These phenomena are reflected in acoustic spectra as spurious echoes and noise.

On the other hand, the numerical simulation of ultrasonic transmission experiments is a powerful tool to evaluate the non-destructive inspections of materials performed by means of acoustic techniques or to progress the optimal design of ultrasonic inspection systems. The simulation involves the transmission of the acoustic waves into materials and the subsequent detection and analysis of the reflected or transmitted signal. The relevance of the simulation data with the experimental results is directly related to the ability of the numerical approach to handle reflection, dispersions, scattering and mode conversion phenomena, and the accuracy of the input parameters of the code [16,17].

Fluorophosphate (FP) glasses exhibit improved properties compared with the corresponding fluoride and phosphate counterparts. The addition of phosphates into the fluoride systems enhances its glass-forming ability and the final products are promising materials for laser and optics applications [18,19,20]. The short-range structure of FP glasses and the medium-range order were studied extensively by means of Raman spectroscopy [21,22]. The structure is dominated by PO₄ tetrahedra with 2, 3, and 4 non-bridging oxygen atoms in each tetrahedron corresponding to Q², Q¹, and Q⁰ species, respectively. A recent review on the structure of FP glasses can be found in [23].

Aluminosilicate glasses doped with lanthanum (LAS) exhibit high glass-transition temperature and refractive indices, high hardness, and are chemically durable [24,25]. Trivalent rare earth cations, such as La³⁺, serve as charge compensators for three neighboring aluminum oxide tetrahedra [AlO₄]⁵⁻ or for one aluminum oxide octahedron [AlO₆]⁹⁻ [26,27]. The structure is dominated by the presence of tetrahedral species, nevertheless aluminum cations can be found four-, five-, and six-fold coordinated, while the addition of lanthanum oxide in the aluminosilicate matrix induces the formation of terminal non-bridging oxygen atoms exhibiting a traditional network modifier behavior [28,29].

The present paper aims to determine the elastic properties of such glasses with the use of a numerical simulation methodology of ultrasonic transmission of solid materials performing elastodynamic finite integration techniques (EFIT). The simulation results were evaluated by means of experimental data available in the literature for two distinct families of glasses, namely the fluorophosphate and aluminosilicate glasses. The relevant elastic constants were determined from the longitudinal and transversal sound velocities. The calculated and the experimental elastic properties were compared and correlated with the structure of these glasses. To our knowledge, no study has yet compared the simulation results obtained from the numerical time-domain modeling of 2D acoustic wave propagation in a viscoelastic medium by implementing the elastodynamic finite integration technique (EFIT) with the corresponding experimental ultrasonic echography results. The numerical ultrasonic NDT model presented here can be used effectively for the inspection of 3D-embedded defects in the material.

2. Materials and Methods

2.1. Glass Formation

The preparation of the FP glasses was described in detail elsewhere [21,22]. In brief, the fluoride strontium phosphate-based glasses correspond to xSr(PO₃)₂–(1–x)(0.62MgF₂–0.38AlF₃) with x = 0, 0.04, 0.06, 0.1, 0.15, 0.2, 0.3, 0.4, 0.8, 0.9, and 1. The prepared glasses have a constant ratio of AlF₃/MgF₂. As starting materials we used high purity Sr(PO₃)₂, MgF₂, and AlF₃ that were used as received. The mole fractions of the fluorophosphate glasses are presented in

Table 1. Mole fraction of the fluorophosphate glasses used in this study.

xSr(PO3)2	(1–x)(0.62MgF2–0.38AlF3)
0.04	0.96
0.06	0.94
0.1	0.90
0.15	0.85
0.2	0.80
0.3	0.70
0.4	0.60
0.8	0.20
0.9	0.10
1	0

.

The glass formation was achieved by means of the standard melt-quenching technique. The materials were melted in crucibles at temperatures above their corresponding melting points depending on the composition and kept at these temperatures until achieving full homogenization and then fast quenched on preheated plates near the glass transition temperature to avoid internal stresses in the resulting glassy materials. Subsequently, the samples were cooled down to room temperature with a cooling rate of ∼5 K/min. Annealing was also performed for 1 h near their respective glass transition temperature. Stable glasses were not obtained in the mole fraction range 0.45 < x < 0.65. The obtained glasses were fully amorphous and with a low OH content. Before ultrasonic pulse-echo measurements, the samples were cut into circular disks with ∼10 mm diameter and 3–5 mm thickness and highly polished to facilitate the ultrasonic wave propagation in the glassy samples. The non-parallelism between the opposite faces of the studied sample is a key factor that may affect the accuracy of the sound velocity measurements. Thus, special attention was paid to the sample during cutting. The parallelism of the opposite faces was verified by means of a dial gauge.

In this study, we did not prepare the lanthanum aluminosilicate glasses. We only used the experimental longitudinal and transversal values of sound velocity reported in [30] to estimate the experimental elastic properties of these glasses and compare these values with our simulation results. In that work, the authors prepared lanthanum aluminosilicate glasses corresponding to 15La₂O₃–xAl₂O₃–(85–x)SiO₂ and 25La₂O₃–yAl₂O₃–(75–y)SiO₂, with x and y = 15, 20, 25, 30, 35 mol%. All the experimental details for the glass formation can be found in [30]. In the present work, our aim was not to prepare and study the structure of the FP and LAS glasses, but to evaluate our proposed simulation model by comparing the experimental and simulated elastic properties for two distinct glass families. Therefore, the main contribution of this paper is the treatment of the numerical time–domain modeling of 2D acoustic wave propagation in a viscoelastic medium by implementing the elastodynamic finite integration technique (EFIT). The mole fractions of the lanthanum aluminosilicate glasses are presented in

Table 2. Mole fraction of the lanthanum aluminosilicate glasses used in this study.

Sample Notation	La2O3	Al2O3	SiO2
LAS1	0.15	0.15	0.70
LAS2	0.15	0.20	0.65
LAS3	0.15	0.25	0.60
LAS4	0.15	0.30	0.55
LAS5	0.15	0.35	0.50
LAS6	0.25	0.15	0.60
LAS7	0.25	0.20	0.55
LAS8	0.25	0.25	0.50
LAS9 LAS10	0.25 0.25	0.30 0.35	0.45 0.40
SiO2	0	0	1

.

2.2. Pulse–Echo Measurements

A set of X- and Y-cut wide-band transducers (Olympus Corporation, Tokyo, Japan) was used to send and receive longitudinal and transversal ultrasonic waves in the glass samples under isothermal and isobaric conditions in a pulse-echo configuration. The same piezoelectric element was working both as signal transmitter and receiver of the echoes. A burst generator (TTi Inc., Munich, Germany) was used to excite the transducers at a frequency of 10 MHz, which is near the fundamental frequency of the piezoelectric elements. A second channel of the function generator was utilized to trigger and synchronize a Tektronix digital oscilloscope for accurate measurements of the output signal. The reference and back wall echo trail monitored in the oscilloscope were averaged over time and then sent to a computer for further analysis of the signals. A small amount of couplant, appropriate for common medical applications, was placed in the transducer–glass interface to ensure the maximum ultrasonic radio frequency (RF) signal transmittance in the glass sample.

Considering the thickness of the glass sample as d, the longitudinal u_l and transversal u_s sound velocity can be estimated as:

$$u_l=\frac{2d}{t_l} \mathrm{and} u_s=\frac{2d}{t_s}$$

(1)

where t_l and t_s are the time between two consecutive echoes of the longitudinal and transversal waves, respectively, that are monitored in the oscilloscope. For a given sample thickness, the t_l and t_s time intervals are the main sources of error in the determination of the ultrasound speed. A mathematical methodology that permits precise estimation of the time intervals is the cross-correlation technique, which provides an accuracy in sound velocity better than 0.2% [31].

Considering as r₁(t) and r₂(t) are the two amplitude-normalized signals and τ_s the time-lag between these signals, then the following relation holds:

$$r_2\left(t\right)=r_1\left(t-{\tau }_s\right)$$

(2)

Subsequently, the cross-correlation function can be estimated as:

$$R_{12}\left(\tau \right)=\frac{1}{2T}\underset{0}{\overset{T}{{\displaystyle \int }}}{r}_1\left(t\right)r_2\left(t+\tau \right)dt$$

(3)

This function reveals a global maximum at τ_s, that is the time interval between consecutive echoes. The two amplitude-normalized signals r₁(t) and r₂(t), as well as the cross-correlation function, are presented in Figure 1.

3. Numerical Simulation of Wave Propagation in Glasses

We implemented the elastodynamic finite integration technique (EFIT) in an effort to numerically simulate the ultrasound propagation in glasses taking into account dispersion, scattering, reflection, and mode conversion phenomena that may overwhelm the desired information concerning the structural properties of the material. The numerical simulation is performed in the time-domain and handles effectively the above “defect” acoustic phenomena [32,33,34] and the papers cited therein. The first step is to design the two-dimensional geometry of the glass sample or “cell”, which is subjected to the acoustic field; then, we integrate the corresponding equations over this cell. We utilized an equivalent formulation of the Kelvin–Voigt constitutive equations by considering the glass sample as a visco-elastic isotropic medium [35,36].

In the context of the Kelvin–Voigt model for an isotropic medium, the 2D linear elasto-dynamic equations of momentum conservation in terms of velocity and stress are:

$$\rho \frac{\partial u_x}{\partial t}=\frac{\partial {\tau }_{xx}}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}+f_x$$

(4a)

$$\rho \frac{\partial u_y}{\partial t}=\frac{\partial {\tau }_{xy}}{\partial x}+\frac{\partial {\tau }_{yy}}{\partial y}+f_y$$

(4b)

and

$$\frac{\partial {\tau }_{xx}}{\partial t}=\left(\lambda +2\mu \right)\frac{\partial u_x}{\partial x}+\lambda \frac{\partial u_y}{\partial y}+g_{xx}$$

(5a)

$$\frac{\partial {\tau }_{yy}}{\partial t}=\left(\lambda +2\mu \right)\frac{\partial u_y}{\partial y}+\lambda \frac{\partial u_x}{\partial x}+g_{yy}$$

(5b)

$$\frac{\partial {\tau }_{xy}}{\partial t}=\mu \left(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right)+g_{xy}$$

(5c)

where λ, μ denote the Lamé constants; ρ is the mass density; and f_ij, g_ij with i,j = [x,y] are the components of force and stress sources.

The first step in the elastodynamic finite integration technique is to perform space discretization and integration of the above equations over the integration cell (glass sample). In Figure 2, a schematic representation of the velocity and stress components in a 2D grid is presented, where the central point of the integration cell is denoted by i,j. Considering a tetragonal grid or equivalently Δx = Δy with i = [1,n] and j = [1,n], the finite integration provides the following discretized forms for Equations (4) and (5):

$$u_{x,i,j}^n=u_{x,i,j}^{n-1}+\frac{\Delta t}{\Delta x}{B}_x\left({\tau }_{xx,i+1,j}^{n-1/2}-{\tau }_{xx,i,j}^{n-1/2}+{\tau }_{xy,i,j}^{n-1/2}-{\tau }_{xy,i,j-1}^{n-1/2}\right)+\Delta t{B}_x{f}_{x,i,j}$$

(6a)

$$u_{y,i,j}^n=u_{y,i,j}^{n-1}+\frac{\Delta t}{\Delta x}{B}_y\left({\tau }_{xx,i+1,j}^{n-1/2}-{\tau }_{xx,i,j}^{n-1/2}+{\tau }_{xy,i,j}^{n-1/2}-{\tau }_{xy,i,j-1}^{n-1/2}\right)+\Delta t{B}_y{f}_{x,i,j}$$

(6b)

$$\begin{array}{cc}{\tau }_{xx,i,j}^{n+1/2}={\tau }_{xx,i,j}^{n-1/2}& +\frac{\Delta t}{\Delta x}\{\left(\lambda +2\mu \right)\left(u_{x,i,j}^n-u_{x,i-1,j}^n\right)+\lambda \left(u_{y,i,j}^n-u_{y,i,j-1}^n\right)\hfill \\ & +\left({\lambda }^{\prime }+2{\mu }^{\prime }\right)\left({\left(\frac{\partial u_x}{\partial t}\right)}_{i,j}-{\left(\frac{\partial u_x}{\partial t}\right)}_{i-1,j}\right)\hfill \\ & +{\lambda }^{\prime }\left({\left(\frac{\partial u_y}{\partial t}\right)}_{i,j}-{\left(\frac{\partial u_y}{\partial t}\right)}_{i,j-1}\right)\}+\Delta t{g}_{xx,i,j}\hfill \end{array}$$

(7a)

$$\begin{array}{cc}{\tau }_{yy,i,j}^{n+1/2}={\tau }_{yy,i,j}^{n-1/2}& +\frac{\Delta t}{\Delta x}\{\lambda \left(u_{x,i,j}^n-u_{x,i-1,j}^n\right)+\left(\lambda +2\mu \right)\left(u_{y,i,j}^n-u_{y,i,j-1}^n\right)\hfill \\ & +\left({\lambda }^{\prime }+2{\mu }^{\prime }\right)\left({\left(\frac{\partial u_y}{\partial t}\right)}_{i,j}-{\left(\frac{\partial u_y}{\partial t}\right)}_{i,j-1}\right)\hfill \\ & +{\lambda }^{\prime }\left({\left(\frac{\partial u_x}{\partial t}\right)}_{i,j}-{\left(\frac{\partial u_x}{\partial t}\right)}_{i-1,j}\right)\}+\Delta t{g}_{yy,i,j}\hfill \end{array}$$

(7b)

$$\begin{array}{cc}{\tau }_{xy,i,j}^{n+1/2}={\tau }_{xy,i,j}^{n-1/2}& +{\overline{\mu }}_{xy}\frac{\Delta t}{\Delta x}\left(u_{x,i,j+1}^n-u_{x,i,j}^n+u_{y,i+1,j}^n-u_{y,i,j}^n\right)\hfill \\ & +{\overline{\mu }}_{xy}^{\prime }\frac{\Delta t}{\Delta x}({\left(\frac{\partial u_x}{\partial t}\right)}_{i,j+1}-{\left(\frac{\partial u_x}{\partial t}\right)}_{i,j}-{\left(\frac{\partial u_y}{\partial t}\right)}_{i+1,j}\hfill \\ & -{\left(\frac{\partial u_y}{\partial t}\right)}_{i,j})+\Delta t{g}_{xy,i,j}\hfill \end{array}$$

(7c)

where λ’, μ’ are the anelastic parameters; g_i,j represent the stress sources; B_i is the effective buoyancies; and ${\overline{\mu }}_{ij}$, ${\overline{\mu }}_{ij}^{\prime }$ are the visco-elastic effective rigidity constants, respectively, given by:

$$B_x=\frac{2}{{\rho }_{i+1,j}+{\rho }_{i,j}}$$

(8)

$$B_y=\frac{2}{{\rho }_{i,j+1}+{\rho }_{i,j}}$$

(9)

$${\overline{\mu }}_{ij}=\frac{4}{\frac{1}{{\mu }_{i,j}}+\frac{1}{{\mu }_{i+1,j}}+\frac{1}{{\mu }_{i,j+1}}+\frac{1}{{\mu }_{i+1,j+1}}}$$

(10)

$${\overline{\mu }}_{ij}^{\prime }=\frac{4}{\frac{1}{{\mu }_{i,j}^{\prime }}+\frac{1}{{\mu }_{i+1,j}^{\prime }}+\frac{1}{{\mu }_{i,j+1}^{\prime }}+\frac{1}{{\mu }_{i+1,j+1}^{\prime }}}$$

(11)

The applied force is:

$$f\left(x,y,t\right)=\left[1-cos\left(\pi f_{0}t\right)\right] cos\left(2\pi f_{0}t\right)$$

(12)

where f₀ is the central frequency of the source. Equation (12) has the form of a raised cosine and is valid for $t\le 2/f_0$, while for higher values of t, the applied field is zero. In Figure 2b, the experimental and simulated ultrasonic signals are presented for direct comparison. Both signals are attenuated as a function of time in the lossy visco-elastic material following an exponential profile. Furthermore, the direct comparison between experimental and simulated ultrasonic signals also provides a direct methodology for the accurate detection of possible defects in the glasses allowing for the precise determination of their size and location.

Despite the fact that several “defect” acoustic phenomena including dispersion, scattering, reflection, and mode conversion phenomena, have been taken into consideration in the numerical simulation, there are minor differences between the experimental and the theoretically predicted spectrum. These differences are due to the non-perfect parallelism between the parallel phases of the glass samples. From the experimental point of view, the non-parallelism induces a dispersion of the acoustic beam that cannot be incorporated into the simulation procedure. An additional cause of the relative intensity differences is the shape of the initial pulse. In the numerical simulation, the shape of the pulse was raised cosine, while the experimental waveform is quite different since the probe is not ideal.

In all pulse-echo simulations, we used 5 mm absorbing layers on the left and right side of the sample. The piezoelectric elements used to transmit and receive the longitudinal and shear ultrasonic waves had a constant diameter of 9 mm. The frequency of the ultrasonic wave was fixed at 10 MHz for all simulations. The shape of the pulse was raised cosine with initial amplitude set to 1 in arbitrary units and the total simulation time was 25 μs for all simulations performed. This simulation duration was more than enough to receive a complete back wall echo train. A schematic representation of the sample modeling is presented in Figure 3. In the same scheme, the piezoelectric elements and the absorbing layers are also shown. The 2D model used in this work remains much less computationally expensive than a full 3D model.

The code for the numerical simulation of wave propagation in glasses by EFIT was developed in Octave programming language. The Octave syntax is largely compatible with MATLAB. The Octave interpreter can be run in GUI mode, as a console, or invoked as part of a shell script. All calculations were carried out in a Dell PowerEdge R420 rack server with two Intel Xeon E5-2420 processors (6-Cores) @1.90 GHz and Linux Ubuntu operating system. Each processor supports six memory slots. There are 12 memory slots in total, with each processor supporting six memory modules with maximum 384 GB of DDR3 memory @1.60 GHz supporting transfer speeds of up to 1600 MT/s. The maximum internal storage was up to 16 TB.

4. Evaluation of the Elastic Properties

The elastic properties of a glassy material of mass density $\rho$ can be estimated from the following simple equations through the longitudinal and transversal sound velocities [37].

To describe an isotropic homogeneous material, the longitudinal or constrained modulus, L, can be used, which is related with longitudinal sound velocity as:

$$L=\rho u_L^2$$

(13)

The resistance of a material to shear is evaluated through the shear modulus or rigidity, G, which is related with longitudinal sound velocity as:

$$G=\rho u_S^2$$

(14)

The incompressibility of a material under hydrostatic pressure is measured through the bulk modulus, K that is calculated based on the longitudinal and shear modulus, as:

$$K=L-\frac{4}{3}G$$

(15)

The ratio of the radial to axial strain is evaluated through the Poisson’s Ratio, σ is also given as a function of the longitudinal and shear modulus:

$$\sigma =\frac{\left(L-2G\right)}{2\left(L-G\right)}$$

(16)

The proportionality constant connecting uniaxial stress and strain is the Young’s Modulus, Y, calculated by means of the shear modulus and Poisson’s Ratio as:

$$Y=\left(1+\sigma \right)2G$$

(17)

5. Results and Discussion

5.1. Fluorophosphate Glasses

The experimental and the calculated longitudinal and transversal ultrasonic velocities as well as the elastic properties of the fluorophosphate glasses investigated in the present study are summarized in

Table 3. Experimental (upper value of each row) and calculated (bottom value of each row) values of the elastic properties of xSr(PO₃)₂–(1–x)(0.62MgF₂–0.38AlF₃) glasses.

Sr(PO3)2 mol Fraction	uL (m/s)	uS (m/s)	L (GPa)	G (GPa)	K (GPa)	σ	Y (GPa)
0.04	5262.6 ± 10.5	1375.0 ± 2.8	95.8 ± 0.4	6.54 ± 0.03	87.1 ± 0.4	0.4634 ± 0.0002	19.15 ± 0.08
	5262.7 ± 10.5	1315.9 ± 2.6	95.8 ± 0.4	5.99 ± 0.02	87.8 ± 0.4	0.4667 ± 0.0002	17.57 ± 0.07
0.06	5406.9 ± 10.8	1333.4 ± 2.7	101.6 ± 0.4	6.18 ± 0.02	93.4 ± 0.4	0.4676 ± 0.0002	18.14 ± 0.07
	5406.7 ± 10.8	1351.5 ± 2.7	101.6 ± 0.4	6.35 ± 0.03	93.2 ± 0.4	0.4667 ± 0.0002	18.63 ± 0.07
0.1	5420.5 ± 10.8	1394.2 ± 2.8	103.0 ± 0.4	6.81 ± 0.03	93.9 ± 0.4	0.4646 ± 0.0002	19.95 ± 0.08
	5420.6 ± 10.8	1355.4 ± 2.7	103.0 ± 0.4	6.44 ± 0.03	94.4 ± 0.4	0.4667 ± 0.0002	18.88 ± 0.08
0.15	5378.1 ± 10.8	1410.2 ± 2.8	102.0 ± 0.4	7.01 ± 0.03	92.7 ± 0.4	0.4631 ± 0.0002	20.53 ± 0.08
	5381.8 ± 10.8	1345.7 ± 2.7	102.2 ± 0.4	6.39 ± 0.03	93.7 ± 0.4	0.4667 ± 0.0002	18.74 ± 0.08
0.2	5377.3 ± 10.8	1412.7 ± 2.8	102.3 ± 0.4	7.06 ± 0.03	92.9 ± 0.4	0.4629 ± 0.0002	20.67 ± 0.08
	5381.4 ± 10.8	1345.1 ± 2.7	102.5 ± 0.4	6.40 ± 0.03	94.0 ± 0.4	0.4667 ± 0.0002	18.79 ± 0.08
0.3	5427.6 ± 10.9	1359.9 ± 2.7	104.1 ± 0.4	6.53 ± 0.03	95.4 ± 0.4	0.4665 ± 0.0002	19.17 ± 0.08
	5423.9 ± 10.8	1356.2 ± 2.7	103.9 ± 0.4	6.50 ± 0.03	95.3 ± 0.4	0.4667 ± 0.0002	19.06 ± 0.08
0.4	5390.2 ± 10.8	1333.4 ± 2.7	101.7 ± 0.4	6.22 ± 0.02	93.4 ± 0.4	0.4674 ± 0.0002	18.26 ± 0.07
	5390.8 ± 10.8	1347.3 ± 2.7	101.7 ± 0.4	6.35 ± 0.03	93.2 ± 0.4	0.4667 ± 0.0002	18.64 ± 0.07
0.8	4777.4 ± 9.6	1081.1 ± 2.2	73.6 ± 0.3	3.77 ± 0.02	68.6 ± 0.3	0.4730 ± 0.0002	11.10 ± 0.04
	4779.2 ± 9.6	1195.1 ± 2.4	73.7 ± 0.3	4.61 ± 0.02	67.5 ± 0.3	0.4667 ± 0.0002	13.51 ± 0.05
0.9	4770.6 ± 9.5	1052.6 ± 2.1	72.3 ± 0.3	3.52 ± 0.01	67.6 ± 0.3	0.4744 ± 0.0002	10.37 ± 0.04
	4767.2 ± 9.5	1192.3 ± 2.4	72.2 ± 0.3	4.51 ± 0.02	66.1 ± 0.3	0.4666 ± 0.0002	13.24 ± 0.05
1.0	4565.9 ± 9.1	1075.3 ± 2.2	65.7 ± 0.3	3.64 ± 0.01	60.8 ± 0.3	0.4706 ± 0.0002	10.71 ± 0.04
	4565.9 ± 9.1	1141.8 ± 2.3	65.7 ± 0.3	4.11 ± 0.02	60.2 ± 0.3	0.4667 ± 0.0002	12.05 ± 0.05

. The first row for each mole fraction corresponds with the experimental results, while the second row represents the corresponding calculated value through the proposed numerical simulation model of the wave propagation in these glasses.

The structure of the pseudo-binary fluorophosphate glasses is analogous to that reported for multicomponent mixed fluoride-phosphate glassy systems. The study of the structure of these glasses has been performed and presented in detail in [21,22]. In the present work, we only used the experimental longitudinal and transversal values of sound velocity and the experimental elastic properties of these glasses reported in the literature to compare these values with our simulation results. In an effort to facilitate the validation of the simulation procedure, we present in Figure 4a,b the experimental and the calculated values of the longitudinal and transversal sound velocities, respectively, that correspond to the fluorophosphate glasses at room temperature.

It is widely accepted that the structure of a glass material is completely different compared with the corresponding crystalline material. Glass only has a short-range order of atoms, while crystals have a long-range order. When studying the structure of the specific fluorophosphate glasses [21,22], our aim was to study in detail the structure of the amorphous phase and this is the reason why we restricted our analysis only in the glass formation range. Crystalline samples can have altered structural properties and they may lead to the abnormal propagation of waves compared to the corresponding glasses of the same composition; thus, causing additional scattering or absorption of the ultrasonic waves. This comparison of the acoustic wave propagation in crystalline and amorphous materials could not provide any new information. In terms of elastic properties, it is proper to examine separately the crystalline and the glassy systems, despite their identical chemical composition.

The incorporation of metaphosphate units into the fluoride structure results in the formation of Al(O,F)₆ octahedra that link the two sub-networks by bridging the metaphosphate and fluoride structural units [21,22] and references therein. In the high metaphosphate region, the corresponding phosphate sub-network is dominated by [PO₂(O_T)₂]⁻ units, the so-called Q² groups. In the low metaphosphate region, the corresponding network is less polymerized through the formation of pyrophosphate [PO(O_T)₃]^2– (Q¹) and isolated orthophosphate [P(O_T)₄]^3– (Q⁰) structural units [21,22] and references therein. This transition from the fluoride, which is more ionic in character, to the phosphate network, which is covalent, causes a constant increase in the glass transition temperature with increasing the phosphate content in the glasses. A maximum T_g is detected at ~80 mol% in Sr(PO₃)₂ due to the distinct crosslinking capability of the Al³⁺ and Sr²⁺ ions. The Al³⁺ cations crosslink the short metaphosphate structural units more effectively than the Sr²⁺ cations. Thus, the crosslinking through Al³⁺ ions leads to a more rigid glass structure.

The bond breaking reduces the network rigidity with the formation of discontinuities in the structure and attenuates the acoustic wave propagation in these glasses, as is presented in Figure 4.

In Figure 5, it appears that the experimental longitudinal, shear, and bulk moduli are strongly affected by the transition from the fluoride (ionic) to phosphate (covalent) limit with increasing the PO³⁻/F⁻ ratio in the glasses. The variation in the PO³⁻/F⁻ ratio leads to atomic volume rise and bulk modulus decrease. The behavior of the moduli is attributed to the variation in the coordination number with increasing the phosphate content in the glasses.

In Figure 6a, the variation in the Poisson’s ratio with the Sr(PO₃)₂ mole fraction in the fluorophosphate glasses is presented. The low values of the Poisson’s ratio imply a diminished coupling of sound waves from one mode of propagation to another. The Poisson’s ratio was found experimentally in the range from 0.46 to 0.47. These values indicate a glass network with low cross-linking density. Both bulk and Young’s moduli exhibit similar behavior as is observed in Figure 6b.

The experimental data available in the literature were compared to the simulated data. From Figure 4, Figure 5 and Figure 6, it seems that the direct comparison reveals an adequate agreement between the experimental and the numerically simulated results demonstrating the ability of the modelling approach proposed in this work to predict the elastic properties of these glasses and providing a fine description of the microstructure.

5.2. Aluminosilicate Glasses

The experimental and the calculated longitudinal and transversal ultrasonic velocities and the elastic properties of the aluminosilicate glasses investigated in [30] are summarized in

Table 4. Experimental (upper value of each row) and calculated (bottom value of each row) values of elastic properties of lanthanum aluminosilicate (LAS) and SiO₂ glasses.

Glass Name	uL (m/s)	uS (m/s)	L (GPa)	G (GPa)	K (GPa)	σ	Y (GPa)
LAS1	5883 ± 12	3180 ± 6	121.8 ± 0.5	35.60 ± 0.14	74.4 ± 0.5	0.294 ± 0.002	92.1 ± 0.4
	5888.2 ± 11.8	3062.4 ± 6.1	122.0 ± 0.5	33.01 ± 0.13	78.0 ± 0.5	0.315 ± 0.001	86.8 ± 0.4
LAS2	6128 ± 12	3329 ± 7	134.1 ± 0.5	39.56 ± 0.16	81.3 ± 0.6	0.291 ± 0.002	102.1 ± 0.4
	6156.8 ± 12.3	3342.6 ± 6.7	135.3 ± 0.5	39.89 ± 0.16	82.1 ± 0.6	0.291 ± 0.002	103.0 ± 0.4
LAS3	6144 ± 12	3345 ± 7	132.9 ± 0.5	39.39 ± 0.16	80.4 ± 0.6	0.289 ± 0.002	101.6 ± 0.4
	6131.3 ± 12.3	3201.7 ± 6.4	132.3 ± 0.5	36.08 ± 0.14	84.2 ± 0.6	0.313 ± 0.001	94.7 ± 0.4
LAS4	6381 ± 13	3413 ± 7	145.8 ± 0.6	41.70 ± 0.17	90.2 ± 0.6	0.300 ± 0.002	108.4 ± 0.5
	6368.7 ± 12.7	3294.8 ± 6.6	145.2 ± 0.6	38.86 ± 0.16	93.4 ± 0.6	0.317 ± 0.001	102.4 ± 0.4
LAS5	6413 ± 13	3449 ± 7	148.1 ± 0.6	42.82 ± 0.17	91.0 ± 0.6	0.297 ± 0.002	111.0 ± 0.5
	6421.5 ± 12.8	3447.9 ± 6.9	148.4 ± 0.6	42.80 ± 0.17	91.4 ± 0.6	0.297 ± 0.002	111.1 ± 0.5
LAS6	6015 ± 12	3130 ± 6	154.1 ± 0.6	41.73 ± 0.17	98.5 ± 0.7	0.314 ± 0.001	109.7 ± 0.5
	6026.1 ± 12.1	3009.7 ± 6.0	154.7 ± 0.6	38.59 ± 0.15	103.2 ± 0.7	0.334 ± 0.001	102.9 ± 0.4
LAS7	5871 ± 12	3110 ± 6	144.4 ± 0.6	40.53 ± 0.16	90.4 ± 0.6	0.305 ± 0.002	105.8 ± 0.4
	5895.7 ± 11.8	3116.4 ± 6.2	145.6 ± 0.6	40.69 ± 0.16	91.4 ± 0.6	0.306 ± 0.002	106.3 ± 0.4
LAS8	6240 ± 12	3339 ± 7	162.8 ± 0.7	46.60 ± 0.19	100.6 ± 0.7	0.300 ± 0.002	121.1 ± 0.5
	6223.6 ± 12.4	3221.8 ± 6.4	162.0 ± 0.6	43.39 ± 0.17	104.1 ± 0.7	0.317 ± 0.001	114.3 ± 0.5
LAS9	5923 ± 12	3208 ± 6	145.9 ± 0.6	42.81 ± 0.17	88.9 ± 0.6	0.292 ± 0.002	110.7 ± 0.5
	5941.1 ± 11.9	3086.4 ± 6.2	146.8 ± 0.6	39.63 ± 0.16	94.0 ± 0.6	0.315 ± 0.001	104.2 ± 0.4
LAS10	6117 ± 12	3261 ± 7	154.2 ± 0.6	43.81 ± 0.18	95.7 ± 0.7	0.301 ± 0.002	114.0 ± 0.5
	6126.6 ± 12.3	3157.5 ± 6.3	154.6 ± 0.6	41.08 ± 0.16	99.9 ± 0.7	0.319 ± 0.001	108.4 ± 0.4
SiO2	5944 ± 12	3748 ± 7	77.7 ± 0.3	30.90 ± 0.12	36.5 ± 0.4	0.170 ± 0.003	72.3 ± 0.3
	5943.9 ± 11.9	2967.4 ± 5.9	77.7 ± 0.3	19.37 ± 0.07	51.9 ± 0.3	0.334 ± 0.001	51.7 ± 0.2

. Also, in this case, the first row for each glass composition corresponds to the experimental results, while the second row represents the corresponding calculated value after the simulation procedure.

From Table 4, it is obvious that the longitudinal and transversal elastic constants, the Young’s modulus, and the bulk modulus are higher in aluminosilicate glasses relative to that of pure SiO₂ glass. This behavior can be explained considering that the structure of the aluminosilicate glasses is characterized by a high network rigidity. The 15La₂O₃–xAl₂O₃–(85–x)SiO₂ glasses reveal an increase in the elastic moduli with Al₂O₃ content. On the contrary, the elastic moduli of the 25La₂O₃–yAl₂O₃–(75–y)SiO₂ glasses exhibit a completely different behavior with varying the Al₂O₃ content. The addition of lanthanum ions in the structure affects the coordination number of Al³⁺, which is evidenced experimentally [30] and references therein. Indeed, Al³⁺ ions are five-fold and six-fold coordinated in the structure of these glasses. These structural characteristics are reflected in the experimental longitudinal and transversal sound velocities of the aluminosilicate glasses under ambient conditions presented in Figure 7. In Figure 8, the experimental and calculated longitudinal, shear, and bulk moduli of the LAS and SiO₂ glasses are shown. All moduli are found to be strongly affected by the Al₂O₃ content in the glasses.

Figure 9a,b present the variation in the experimental and calculated Poisson’s ratio and Young’s modulus with mole fraction for the aluminosilicate and for pure SiO₂ glasses. The Poisson’s ratio was found experimentally in the range 0.29 to 0.31. The corresponding value for pure SiO₂ is 0.16. This difference is explained considering that most of the bonds in aluminosilicate glasses are ionic compared to the mainly covalent bonds in the case of SiO₂. The number of bridging bonds per cation, which is expressed as the cross-link density of the structure, is strongly correlated with the Poisson’s ratio. When the cross-link density is equal to 2, 1, and 0, the values of the Poisson’s ratio are 0.15, 0.30, and 0.40. Since the Poisson’s ratio in the case of lanthanum aluminosilicate glasses is between 0.29 to 0.31, then the cross-link density is equal to 0.15 and the structure of the glasses is a 2D layer structure. For the 25La₂O₃–yAl₂O₃–(75–y)SiO₂ glasses, the overall structure is dominated by four-fold AlO₄ tetrahedra and six-fold AlO₆ octahedral species [30] and references therein.

As in the case of fluorophosphate glasses, the experimental data available in the literature for the lanthanum aluminosilicate glasses were compared to simulated data. Figure 7, Figure 8 and Figure 9 allow a direct comparison between the experimental and the numerically simulated results. The agreement is also adequate in this case and further validates the modelling ability to predict the elastic properties of these glasses providing a fine description of the microstructure elastic properties. The investigation validates the proposed numerical simulation procedure as a powerful tool to predict theoretically the elastic properties of glasses with completely different structural characteristics.

Several solidification phenomena occurring during the fabrication process may lead to samples with highly anisotropic and heterogeneous structures, both in terms of microstructure and elastic properties. Because of the internal structure, an ultrasonic beam traveling through an anisotropic and heterogeneous structure is subject to severe disturbances, such as beam deviation, beam splitting, and mode conversion, resulting in complex ultrasonic signals with spurious echoes and structural noise.

Nevertheless, numerical simulations have become an essential tool to demonstrate the performance of non-destructive testing (NDT). The relevance of the numerical results is directly related with the accuracy of the input data for the simulation. In this context, a complex anisotropic material is commonly represented by a large number of homogenous and anisotropic domains, and this method has been shown in the present work to be effective as it reproduces the deviation and splitting of the acoustic beam and the relevant mode conversions.

At the ultrasonic frequencies commonly used to inspect a highly anisotropic structure that are between 0.5 and 10 MHz, the corresponding wavelength ranging from 11 to 0.5 mm is similar to the grain size. The scattering of the wave by the grain boundaries will thus be a key factor. Each grain boundary is considered as a discontinuity with specific acoustic impedance, which finally will scatter the acoustic wave. Under these circumstances, the structure scatters the acoustic wave, and the ultrasonic energy is diffused away from the axis of beam propagation. From the probe point of view, the scattering by the microstructure will induce a decrease in the reflected echo amplitude which depends on the dimension (depth) of the reflector. The phenomenon is assessed by the use of an attenuation coefficient, which accounts for the anisotropic dispersion of the ultrasonic power of the beam along its path. Nevertheless, the variation in the attenuation due to the microstructure scattering, characteristic of anisotropic structures, is very hard to obtain experimentally. A common methodology is to utilize samples with various orientations of the elongated grains and to evaluate their attenuation by transmission or reflection techniques of waves under normal incidence.

Concerning future applications, by applying the procedure described in this work in orthotropic, transversely isotropic, or generally anisotropic materials, one can determine the anisotropic variations of scattering attenuation by describing the microstructure at the grain-scale based on numerical simulation. The simulation method should combine a 2D finite element code and a detailed description of the microstructure that will account for the geometric, elastic, and crystallographic properties of the anisotropic material. This way, the grain orientation, the grains’ size, and the anisotropy level are the main input parameters that are implemented for the efficient determination of the acoustic attenuation.

6. Conclusions

In the present work, the transmittance of ultrasonic wave was numerically simulated by means of elastodynamic finite integration techniques (EFIT). Two distinct families of glasses, the fluorophosphate and the aluminosilicate glasses, were used as model systems to evaluate the applicability and effectiveness of the pulse-echo simulation. The relevant elastic constants were determined from the longitudinal and transversal sound velocities. The calculated and the experimental elastic properties were compared and correlated with the structure of these glasses. The comparison between experimental and simulation results showed low deviations supporting the concept that the numerical simulation of the pulse-echo ultrasonic inspection signals is effective and applicable for the study of the elastic properties of amorphous glasses as well as crystalline bulk materials. Elastic properties are important parameters related to the mechanical behavior of solids and for this reason their prediction is an important achievement. Their evaluation through computational models could lead to the reduction in the cost of manufacturing materials while simultaneously improving their properties. Furthermore, it has been found that using this technique, one can determine the precise location and size of a possible defect in the studied samples by comparing the experimental and simulation results.

The latter can prove to be extremely important when one needs to monitor the performance of vitreous materials that are used either in optical or optoelectronic applications. The above can also stretch to defense and space applications where vitreous materials are commonly used due to their enhanced properties regarding their crystalline counter parts. Having a fast, non-destructive, and reliable technique can provide significant reductions in maintenance costs, eliminate failures due to early detection of structural flaws at the atomic level and ensure the optimum operation of critical optoelectronic and other components and infrastructures.