*2.2. Propagation of Ultrasonic Waves in Porous Material*

Biot is regarded as the initiator of works on the theory and studies on ultrasonic waves in porous materials [16,17]. According to the theory, there are two compressional waves in the wet porous material—P1-wave (the fast wave) and P2-wave (the slow wave). Further works [18,19] have confirmed Biot's hypothesis. Other research works refer to other phenomena, including reflections and refractions, which are significant for testing and diagnosing materials. Currently, different aspects concerning wave propagation in the porous medium are examined. The issue of wave propagation and scattering in the inhomogeneous material is presented in, inter alia, the papers [20,21]. The works [22,23] present the mathematical model of propagation of low-frequency surface waves–the Stoney waves, in the porous material. Another paper [24] describes experiments on absorption and propagation of ultrasonic

waves in materials with dual porosity, whereas the work [25] demonstrates test on the propagation of Rayleigh waves at liquid–solid interfaces.

Concrete, like rock media, is not ideally elastic. Therefore, the wave equation cannot be directly applied to this medium (12). The imperfect elasticity of concrete causes internal friction that transforms a part of energy into heat causing scattering and dispersion of velocity of elastic waves. The mathematical presentation of imperfection of the elastic medium is described in different ways. For example, the equation of the perfectly elastic medium can be replaced with the system of equations describing stresses and deformations. The equation of the plane longitudinal wave moving and scattered in the imperfectly elastic medium takes the following form:

$$
\rho\_0 \frac{\partial^2 \xi}{\partial t^2} = \frac{1}{\beta\_{\rm ad}} \frac{\partial^2 \xi}{\partial \mathbf{x}^2} + \eta \frac{\partial^3 \xi}{\partial \mathbf{x}^2 \partial t} \tag{14}
$$

where βad—adiabatic compressibility coefficient, η = η + <sup>4</sup> 3η —viscosity coefficient composed of coefficients (η") of bulk and shear viscosity (η ).

Generally, the solution to the wave Equation (14) is expressed as:

$$\mathcal{L}(\mathbf{x},t) = A e^{-\alpha\_{\mathbb{P}}\mathbf{x}} e^{i\omega \left(t - \frac{\mathbf{x}}{c}\right)} + B e^{\alpha\_{\mathbb{P}}\mathbf{x}} e^{i\omega \left(t + \frac{\mathbf{x}}{c}\right)} \tag{15}$$

where: *A*, *B*—integration constants, αη—integration constant depending on the value of viscosity coefficient, *c*—wave velocity, x—coordinate of wavefront, ω—wave frequency.

The velocity of longitudinal waves in viscoelastic medium can be described as:

$$c\_p = \sqrt{\frac{K}{P0}} \sqrt{\frac{2(1+\omega^2r^2)\left(\sqrt{1+\omega^2r^2}-1\right)}{\omega^2r^2}}\tag{16}$$

where *r* = ηβ*ad*, *K* = 1/β*ad*.

The velocity of waves in the inhomogenuous granular medium, despite being the material constant, is related to its physical properties—density, elasticity defined by the Lamé coefficients also depends on wave scattered by the medium, wave frequencies, the medium structure, etc. Therefore, velocity not regarded as the constant value in contrast to the propagation of waves in perfectly elastic media. Granular media, such as rocks, concrete, or mortar, are characterized by:


Scattering of the elastic wave in granular media depends on many factors—mechanical and thermal processes caused by the propagating wave. There are three main reasons for energy loss during wave scattering:


The overall wave scattering is the sum of mentioned elements:

$$
\delta = \delta\_r + \delta\_T + \delta\_{\mathbb{R}\_\prime} \tag{17}
$$

The role of each of the three factors above in ultrasonic wave scattering in the homogeneous granular medium depends on the frequency and structure of that medium characterized by:

• dimensions of the matrix grains,


Wave velocity in granular media characterized by a large coefficient of wave scattering can be calculated from the following dependence:

$$C = C\_0 \sqrt{1 - \left(\frac{\delta}{a\nu}\right)^2} \tag{18}$$

where *C*0—wave velocity in the linear-elastic medium, δ—total scattering coefficient, and ω—wave frequency.

The velocity of wave propagation in granular materials changes within a wide range and is subjected to fluctuations depending on the type of components and their distribution. It is caused by different values of elasticity constants *E G* ν demonstrated by individual components of granular materials. Therefore, we obtain a certain mean velocity that results from the percentage contribution of velocity to individual components. Determining ultrasound velocity for different specimens cannot be neglected in that case. Greater scattering and more problems related to signal recording are expected in specimens with longer wave paths. Hence, the use in NDT methods requires the conversion of wave velocities.

### **3. Stress Measurements Using an Ultrasonic Technique**

Material stress can affect velocity of the acoustic wave due to inhomogeneity and anisotropy of the material. That effect has been described for the first time by seismologist Biot [26] and experimentally verified by Hughes and Kelly [27] and Bergman and Shahbender [28]. It is demonstrated that the static stress can change velocity of the acoustic wave in the medium, and that effect is called the acoustoelastic (AE) effect [29,30].

The acoustoelastic effect is based on the relationship between the velocity of transverse wave propagation and stress in solid bodies found by Benson and Raelson in the 1970s [14]. Since then, this aspect has been widely developed [31–33]. The impact of stress on the velocity of transverse wave propagation is determined by the direction of wave propagation with reference to the stress direction and wave polarization. A change in the polarization plane depends on stress, similarly to a light wave in the elastooptic effect. Its mechanism was theoretically described on the basis of the non-linear theory of solid deformation [27]. According to that theory, constant elasticity of higher orders (than those observed in the theory of linear elasticity) was responsible for nonlinear effects. The propagation velocity in the stressed body can be expressed as the sum of velocities in the tensionless stress (σ = 0) and its change (increment) caused by stress. That change can be defined as dependent on stress including constant characteristics of elasticity of second or third order.

In accordance with the infinite deformation of elastic materials by Murnaghan [34], the stressdeformation relationship should be described by the function of free energy *W*s defined as [27,35]:

$$\mathcal{W}\_{\rm s} = \frac{1}{2}(\lambda + 2\mu)\mathbf{I}\_1^2 - 2\mu\mathbf{I}\_2 + \frac{1}{3}(l + 2m)\mathbf{I}\_1^3 - 2m\mathbf{I}\_1\mathbf{I}\_2 + n\mathbf{I}\_3 \tag{19}$$

where: λ*,* μ—Lamé constants, *lmn*—elasticity constants of second and third order by Murnaghan, *I*1, *I*2, *I*3—deformation invariants.

Taking into account the principle of energy conservation, Hooke's law can be expressed as:

$$
\rho \,\delta \mathcal{W}\_{\mathbb{S}} = \sigma\_{i\circ} \frac{\partial \delta u\_i}{\partial u\_j},
\tag{20}
$$

where δ*W* and δ*ui* mean finite increments in the function of free energy and displacement area, ρ is density after deformation. The combination of Equations (19) and (20) produces the acoustoelastic equation, which binds the static load with velocity of the elastic wave under hydrostatic pressure *P*:

$$\begin{array}{l} \rho\_0 c\_p^2 = \lambda + 2\mu - \frac{P}{3\lambda + 2\mu}(6l + 4m + 7\lambda + 10\mu) \\\rho\_0 c\_T^2 = \mu - \frac{P}{3\lambda + 2\mu}(3m + 0, 5n + 3\lambda + 6\mu) \end{array} \tag{21}$$

where: *c*<sup>p</sup> and *c*<sup>T</sup> are velocity of longitudinal and transverse waves respectively, a ρ0—body density in the tensionless state.

Thus, the hydrostatic level of stress can be defined from Equation (20) [36] by measuring velocity of the longitudinal and transverse waves—Figure 1a. In the case of uniaxial stress, wave velocity depends on the direction of the stress and the square of velocity on Figure 1b-1f is as follows:

$$V\_{111}^2 = \frac{\lambda + 2\mu}{\rho\_0} - \frac{\sigma\_1}{3K\_0\rho\_0} \left[ \frac{\lambda + \mu}{\mu} (4\lambda + 10\mu + 4m) + \lambda + 2l \right] \tag{22}$$

$$V\_{113}^2 = \frac{\lambda + 2\mu}{\rho\_0} + \frac{\sigma\_3}{3K\_0\rho\_0} \left[ \frac{2\lambda}{\mu} (\lambda + 20\mu + m) - 2l \right] \tag{23}$$

$$\mathcal{V}\_{131}^2 = \mu - \frac{\sigma\_1}{3K\_0\rho\_0} \left[ 4\lambda + 4\mu + m + \frac{\lambda n}{4\mu} \right] \tag{24}$$

$$V\_{133}^2 = \mu - \frac{\sigma\_3}{3K\_0\rho\_0} \left[\lambda + 2\mu + m + \frac{\lambda n}{4\mu}\right] \tag{25}$$

$$V\_{132}^2 = \mu + \frac{\sigma\_2}{3K\_0\rho\_0} \left[2\lambda - m + \frac{n}{2}\frac{\lambda}{2}\frac{n}{\mu}\right] \tag{26}$$

where: *K*<sup>0</sup> = *<sup>E</sup>* <sup>3</sup>(1−2ν) <sup>=</sup> <sup>2</sup>μ+3<sup>λ</sup> <sup>3</sup> .

Knowing velocity of the ultrasonic wave in the loaded material and elasticity constants of the first (λ μ), second and third order (*mnl*) normal stresses can be determined. Measurements of wave velocity do not cause any problems except for small specimens (due to high sensitivity of the recording equipment). However, determining material constants *m, n,* and *l* is difficult.

Using the equation [27], the precise method of determining material constants was presented in the papers [37,38]. Velocities of longitudinal and transverse waves under the uniaxial stress are presented in the following form:

$$
\rho\_0 V\_{11}^2 = \lambda + 2\mu + \frac{\sigma\_1}{E} [5\lambda + 10\mu + 2l + 4m - 2\nu(\lambda + 2l)] \to V\_{11}^2 = V\_0^2 \left(1 + 2a\_{11}\frac{\sigma\_1}{E}\right) \tag{27}
$$

$$\rho\_0 \rho\_{12}^2 = \mu + \frac{\sigma\_1}{E} \left[ \lambda + 4\mu + m - \nu \left( 2\lambda + 2\mu + 2m - \frac{n}{2} \right) \right] \to V\_{11}^2 = \left( \frac{\mu}{\rho\_0} \right)^2 \left( 1 + 2a\_{12} \frac{\sigma\_1}{E} \right) \tag{28}$$

$$\rho\_0 V\_{21}^2 = \mu + \frac{\sigma\_1}{E} \left[ \lambda + \mu + m - \nu \left( 2\lambda + 4\mu + 2m - \frac{n}{2} \right) \right] \to V\_{21}^2 = \left( \frac{\mu}{\rho\_0} \right)^2 \left( 1 + 2\alpha\_{21} \frac{\sigma\_1}{E} \right) \tag{29}$$

$$
\rho\_0 V\_{22}^2 = \lambda + 2\mu + \frac{\sigma\_1}{E} [\lambda + 2l - \nu(6\lambda + 10\mu + 4l + 4m)] \to V\_{22}^2 = V\_0^2 \left(1 + 2a\_{22} \frac{\sigma\_1}{E}\right) \tag{30}
$$

$$
\rho\_0 V\_{23}^2 = \mu + \frac{\sigma\_1}{E} \left[ \lambda + m - \frac{n}{2} - \nu (2\lambda + 6\mu + 2m) \right] \rightarrow V\_{21}^2 = \left( \frac{\mu}{\rho\_0} \right)^2 \left( 1 + 2\alpha\_{23} \frac{\sigma\_1}{E} \right) \tag{31}
$$

**Figure 1.** Identification of velocity of ultrasonic waves in isotropic material: (**a**) hydrostatic compression by pressure P, (**b**) longitudinal wave under stress σ1, (**c**) transverse wave under stress σ3, polarized in planes 1–3 (**d**) transverse plane under stress σ1, polarized in planes 1–3, (**e**) transverse plane under stress σ3, polarized in planes 1–3, (**f**) longitudinal wave under stress σ2, polarized in planes 1–3.

Tests on the specimens of two lengths were the base to formulate relationships for determining constants α11, α22, α12, α21, and α<sup>23</sup> from the following equations:

$$\alpha\_{11} = 1 - \frac{E}{\sigma\_1} \left[ \frac{L\_1}{L\_2 - L\_1} \left( \frac{\Delta t\_1}{t\_{01}} \right) - \frac{L\_1}{L\_2 - L\_1} \left( \frac{\Delta t\_2}{t\_{02}} \right) \right] \tag{32}$$

where: *L*1—length of specimens "1" and "2" used for calibration, Δ*t*<sup>1</sup> = *t*<sup>1</sup> − *t*01—difference in passing time of wave in specimen "1" after deformation (*t*01) and before deformation (*t*01), Δ*t*<sup>2</sup> = *t*<sup>2</sup> − *t*02—difference in passing time of wave in specimen "2" after deformation (t02) and before deformation (*t*02).

The equation for determining other material constants is as follows:

$$
\alpha\_{1\dot{j}} = -\nu - \frac{E}{\sigma\_1} \left(\frac{\Delta t}{t\_0}\right) \tag{33}
$$

where Δ*t* = *t*<sup>1</sup> − *t*<sup>01</sup> is the difference in passing time of the wave in the specimen after and before its deformation.

My own research indicated the linear nature of changes in the ratio of passing time of the wave Δ*t*/*t*<sup>0</sup> in relation to stress increase. Determining other constants consisted in solving the following system of equations:

$$\begin{aligned} l &= \frac{(2a\_{11} - 5)(\lambda + 2\mu)}{2(1 - 2\nu)} - \frac{2w - \nu\lambda}{1 - 2\nu}, \; m = \left[ \frac{a\_{11} - a\_{22}}{2(1 + \nu)} - 1 \right] (\lambda + 2\mu) - \frac{\mu}{2} \\\ n\_{12} &= \frac{2}{\nu} [-(a + 4\nu) + 2\nu(a + \mu) + 2\mu a\_{12}], \; n\_{21} = \frac{2}{\nu} [-(a + 2\nu) + 2\nu(a + 2\mu) + 2\mu a\_{21}] \\\ n\_{23} &= 2[a - 2\nu(a + 3\mu) - 2\mu a\_{23}] \end{aligned} \tag{34}$$

where: *a* = λ + *m*.

Figure 2 shows changes in increment of propagation time of longitudinal and transverse waves described in the paper by Takahashi [38], who based this on his experience of developing and patenting the measuring apparatus to determine directly constants *l, m,* and *n* [39].

**Figure 2.** Selected test results for acoustoelastic effect: (**a**) ratio of changes in velocity of waves of different length obtained from tests (adapted from [38]), (**b**) changes in velocity of longitudinal, transverse, and Rayleigh waves obtained from tests (adapted from [40]).

As expected, the greatest increments in wave velocity were observed for longitudinal waves in the direction of stress. In addition, surface waves could be used to detect changes in stress states. The smallest gradients of velocity were obtained for transverse waves. As expected, the greatest increments in wave velocity were observed for longitudinal waves perpendicular to the stress direction. An increase in compressive stress caused an increase in wave velocity. Similar relationships were observed for waves propagating perpendicularly to the stress direction. Theoretical principles of the acoustoelastic effect are relatively well documented in the literature. There is also an apparatus to determine elasticity constants *l, m,* and *n* of the third order for metals and plastic in accordance with procedures described in, among others, papers [37,38,41]. Diagnosing stress states in structures using the NDT method requires the information on load direction and defined gradient of changes in velocity of longitudinal or transverse wave Knowing Muraghan coefficients is not essential.

### **4. Test Program and Results**

The test program was divided into two stages. Stage I included the material tests on specimens made of autoclaved aerated concrete (AAC) to determine density ρ0, elasticity modulus E and Poisson's ratio υ. Each cube specimen was subjected to axial compression until the failure and velocity of the longitudinal wave were determined at different normal stresses. The obtained results were used to determine linear correlations describing a σ–Cp relationship. In stage II, nine models of masonry walls were tested in axial compression. The velocity of the longitudinal wave was measured at different values of vertical loads. Then, vertical loads were determined on the basis of a correlation curve obtained during stage I. To interpret the results, they were compared with numerical calculations for 3D models of the masonry wall.
