*4.1. Stage I—Determination of Acoustoelastic Constant*

The tests included four series of masonry units with thickness within the range of 180–240 mm and different classes of density: 400 kg/m3, 500 kg/m3, 600 kg/m3, and 700 kg/m3, each 20 masonry units were randomly selected. Six cores with a diameter of 59 mm and the height of 120 mm were taken from each type of the masonry unit using a drill. They were used to determine fundamental properties of tested autoclaved aerated concrete (AAC). All drilled cores were dried until constant weight at temperature of 105 ± 5 ◦C. Then, two vertical and horizontal electro-resistant tensometers were fixed to side surfaces of cylindrical specimens to measure deformations and determine modulus of elasticity E within the range of 0.1–0.33 σmax and Poisson's ratio υ at the level of 0.33 σmax. Tests were conducted using the testing machine, in which an increment in load was controlled manually, and the reading range of the dynamometer was 100 kN. Mean mechanical parameters obtained for all tested types of masonry units are shown in Table 1. The presented results from testing density were taken from the paper [12].

**Table 1.** Fundamental characteristics of masonry units.


Besides the cores used to determine properties of AAC, four series of six cuboid specimens each (24 specimens in total) were drilled using a diamond saw 4. The specimens had dimensions of 100 × 100 × 100 mm, and were used as basic specimens for determining the strength *f* <sup>B</sup> in accordance with Appendix B to the standard EN 771-4 [42]) harmonized with the European standard PN-EN 1996-1-1:2010 [43].

All specimens drilled from blocks to determine the correlation between vertical stresses and ultrasound velocity, were air-dried until constant weight at a temperature of 105 ± 5 ◦C (for at least 36 h). That way, the impact of moisture content on AAC was eliminated [13,44]. Generally, it tends to reduce significantly compressive strength and change velocity of the ultrasonic wave propagation [12].

The ultrasonic technique, commonly applied to test strength of concrete and masonry, was used to determine velocity of ultrasonic waves in AAC [45,46]. Ultrasonic testing was conducted on the block specimens 100 × 100 × 100 mm drilled from masonry units—Figure 3. The specimens in air-dry conditions and relative humidity w/wmax = 0% were used for testing. Each series of elements included at least six specimens, and 24 specimens in total were tested. PUNDIT LAB (Proceq SA, Schwerzenbach, Switzerland) instrument was used for tests. Exponential transducers with the waveguide length *L* = 50 mm, diameters ø1 = 4.2 mm and ø2 = 50 mm, and frequency 54 kHz were employed. The measurement accuracy of passing time of the ultrasonic wave was equal to ±0.1 μs. The used methodology of testing and equipment were typical for ultrasonic tomography for concrete and masonry [47,48]. Each specimen was placed on transducers of the testing machine (type FORM+TEST Prüffsysteme MEGA 3 with the range of 100 kN, class 1, reading accuracy ±1%) through the vibration isolation washer and steel sheet of 3 mm thickness. The steel plate and the vibration isolation washer were placed on the top surface of the specimen. Vibration isolation was necessary for eliminating possible vibrations that could affect the results from measurements of ultrasonic waves. Then, the transducers were applied to opposite walls and the passing time of wave was measured using the transmission method. The transducers were in contact with the specimens at an angle of 90◦ within distance between the transducers measured every time with accuracy up to 1 mm. The tests were conducted for various loading of the specimens and the force was scaled every 2.5 kN.

**Figure 3.** A test stand for measuring ultrasonic wave velocity in compressed specimens: (**a**) specimen geometry and elements of the stand, (**b**) geometry of exponential transducer, (**c**) a test stand; *1*—tested AAC specimen 100 × 100 × 100 mm, *2*—exponential transducers, *3*—cables connecting transducers with recording equipment, *4*—recording equipment, *5*—steel sheet, 15 mm thick, 6—vibration isolation, *7*—heads of testing machine.

The selected results from measurements and maximum values of stress σ3max are presented in Table 2. There are also empirical values of the longitudinal wave obs*c*p0 without the participation of compressive stress and ratios of normal stress σ3/σ3max, for which the measurements are presented in a tabular form. Figure 4a illustrates results from measured velocities of ultrasounds as the ratio (*c*p–obs*c*p0)/ obs*c*p0 expressing the relative increment in ultrasound velocity as a function of stress σ3. Relative increments in velocity of ultrasonic waves are presented in Figure 4b as a function of relative compressive stress σ3/σ3max.

As in previous tests [12], the specimens dried until constant weight demonstrated an increase in ultrasound velocity with increased density of AAC under stress σ<sup>3</sup> = 0. Velocity obs*c*p0 increased to 1875 m/s in concrete of a nominal class of 400 kg/m3, and to 2225 m/s in concrete with density of 700 kg/m3. Increased compressive stress in all specimens caused nearly proportional drop in ultrasound velocity. Under relatively low stress when 0 ≤ σ<sup>3</sup> ≤ 0.25σ3max, values of ultrasound velocity decreased by 2–4% when compared to obscp0. When normal stress increased to the level of 0.25σ3max <sup>≤</sup> <sup>σ</sup><sup>3</sup> <sup>≤</sup> 0.50σ3max, the velocity of ultrasounds decreased by 5–7% when compared to the reference value of 0.25σ3max. Under greater values of relative stress 0.50σ3max ≤ σ<sup>3</sup> ≤ 0.75σ3max, the greatest percentage drop in propagation of ultrasonic waves by 9–11% was found in concrete with nominal densities of 400 and 500 kg/m3. The reduction in velocity of ultrasonic waves by 7–9% was observed in the specimens made of concrete with density of 600 and 700 kg/m3. No clear reduction in wave velocity in concrete with densities of 600 and 700 kg/m3 was observed for the stress level, at which slight noise was heard in the specimens and local crushing was apparent within the stress range of 0.75σ3max ≤ σ<sup>3</sup> ≤ 0.95σ3max. The relative velocity of ultrasounds decreased by 11–12% in other specimens. In conclusion, a nearly linear drop in relative velocity of longitudinal ultrasonic wave was observed regardless of AAC density. The maximum reduction in relative velocity of ultrasounds was directly proportional to AAC density and changed within the range of 7–12%. As in tests conducted on metals [38,40], linear relationships were obtained, which defined the reduction in velocity of ultrasonic wave propagation as a function of applied normal stress. Considering the relationship (23), accurate physical relationships can be determined:

$$\begin{split} V\_{113}^2 = V\_0^2 + \frac{1}{3\mathcal{K}\_0\rho\_0} \Big[ \frac{2\lambda}{\mu} (\lambda + 20\mu + m) - 2l \Big] \sigma\_3 \to c\_p^2 = c\_{p0}^2 + \frac{1}{3\mathcal{K}\_0\rho\_0} \Big[ \frac{2\lambda}{\mu} (\lambda + 20\mu + m) - 2l \Big] \sigma\_3\\ c\_p^2 - \alpha^{obs} c\_{p0}^2 = \left( c\_p - c\_{p0} \right) \big( c\_p + c\_{p0} \Big) \approx \left( c\_p - c\_{p0} \right) 2c\_{p0}, \end{split} \tag{35}$$
 
$$\frac{\left( c\_p - c\_{p0} \right) 2c\_{p0}}{\varepsilon\_{\%}} = \frac{1}{6\lambda\eta\alpha\varepsilon\_{p0}} \Big[ \frac{2\lambda}{\mu} (\lambda + 20\mu + m) - 2l \Big] \sigma\_3 \to \frac{\left( c\_p - c\_{p0} \right)}{\varepsilon\_{\%}} = \frac{\frac{\lambda}{\mu} (\lambda + 20\mu + m) - 2l}{(\lambda + 2\mu)(3\lambda + 2\mu)} \sigma\_3.$$


**Table 2.** Test results for ultrasound velocity in AAC at various compressive stresses.

The relationship after transformation can be expressed as:

$$\frac{\left(c\_p - c\_{p0}\right)}{c\_{p0}} = \frac{\left(t\_{p0} - t\_p\right)}{t\_p} = \frac{\frac{\lambda}{\mu}(\lambda + 20\mu + m) - 2l}{(\lambda + 2\mu)(3\lambda + 2\mu)}\sigma\_3 = \beta\_{113}\sigma\_{3\nu} \tag{36}$$

where β<sup>113</sup> is the acoustoelastic effect [40] related to the longitudinal wave perpendicular to the direction of the applied load.

If cp0 in the relationship (36) is replaced with the value determined in the tests, then the relationship illustrated in Figure 4a is obtained. By dividing both sides of the Equation (36) by the value of maximum stress σ3max, the following relationship is developed:

$$\frac{\left(c\_p - c\_{p0}\right)}{c\_{p0}} = \frac{\left(t\_{p0} - t\_p\right)}{t\_p} = \gamma\_{113} \frac{\sigma\_3}{\sigma\_{3\text{max}}}.\tag{37}$$

where γ<sup>113</sup> = β<sup>113</sup> σ3max can be called the relative acoustoelastic coefficient.

**Figure 4.** Results from measuring velocity of the longitudinal ultrasonic wave: (**a**) relative change in velocity of longitudinal wave as a function of compressive stress, (**b**) relative change in velocity of longitudinal wave as a function of relative compressive stresses.

The introduction of coefficient β<sup>113</sup> considerably simplifies practical applications. By using relative values of passing time of the wave, the effect of wave scattering and other related effects described under point 2.2 could be neglected. If *c*p0 in the relationship (37) was replaced with the value determined in the tests, then the relationship illustrated in Figure 4b was obtained. It was adequate to know the coefficient γ<sup>113</sup> to determine the maximum value of compressive stresses corresponding to normalized compressive strength of the masonry unit *f* Bw in air-dry conditions. The obtained values of coefficients β<sup>113</sup> and γ<sup>113</sup> for straight lines determined from Equations (36) and (37) as a function of density are presented in Figure 5.

**Figure 5.** Values of coefficients β<sup>113</sup> and γ<sup>113</sup> as a function of AAC density.

Empirical relationships developed from obtained results were proposed to express values of coefficients β<sup>113</sup> and γ<sup>113</sup> as a function of AAC density at (w = 0)

$$\begin{array}{l} \rho\_{113} = 1.39 \times 10^{-4} \rho - 0.104, \ R^2 = 0.995, \\ \rho\_{113} = 1.72 \times 10^{-4} \rho - 0.206, \ R^2 = 0.923 \\ \text{when } 397 \, \frac{\text{kg}}{\text{m}^3} \le \rho \le 674 \, \frac{\text{kg}}{\text{m}^3} \end{array} \tag{38}$$

*Materials* **2020**, *13*, 2852

The practical applications required taking into account moisture content of AAC. The paper [12] demonstrated that the maximum moisture content in concrete depended on nominal density. At the density increase in the range from ρ = 397 kg/m<sup>3</sup> to 674 kg/m3, the maximum moisture content was varying within *w*max = 53.3–89.9%, which made it possible to determine a straight line of the least square in the following form:

$$w\_{\text{max}} = -1.23 \times \frac{\rho}{1000} + 1.34 \text{, when } 397 \, \frac{\text{kg}}{\text{m}^3} \le \rho \le 674 \, \frac{\text{kg}}{\text{m}^3} \tag{39}$$

Moreover, relative changes in velocity of longitudinal ultrasonic waves were shown by the relationships illustrated in Figure 6.

**Figure 6.** Relationship between velocity of ultrasonic wave propagation, moisture content and density: (**a**) relative changes in velocity of longitudinal wave as a function of relative moisture content w/wmax, (**b**) values of coefficients as a function of AAC density.

The tests were used to develop the following relationships including velocity *c*pw in wet AAC with reference to AAC in air-dry conditions *c*p:

$$\begin{array}{ll} \frac{\varepsilon\_{\rm F}}{c\_{\rm r}} = 0.569 \frac{\rm w}{w\_{\rm max}} - 0.818 + 1, \text{ when } 397 \frac{\rm kg}{\rm m} \leq \rho \leq 446 \frac{\rm kg}{\rm m},\\ \frac{\varepsilon\_{\rm F}}{c\_{\rm r}} = 0.483 \frac{\rm w}{w\_{\rm max}} - 0.671 + 1, \text{ when } 462 \frac{\rm kg}{\rm m} \leq \rho \leq 532 \frac{\rm kg}{\rm m^3},\\ \frac{\varepsilon\_{\rm F}}{c\_{\rm r}} = 0.366 \frac{\rm w}{w\_{\rm max}} - 0.504 + 1, \text{ when } 562 \frac{\rm kg}{\rm m^3} \leq \rho \leq 619 \frac{\rm kg}{\rm m^3},\\ \frac{\varepsilon\_{\rm F}}{c\_{\rm r}} = 0.323 \frac{\rm w}{w\_{\rm max}} - 0.434 + 1, \text{ when } 655 \frac{\rm kg}{\rm m^3} \leq \rho \leq 725 \frac{\rm kg}{\rm m^3}. \end{array} \tag{40}$$

After taking into account the obtained results, values of empirical coefficient defined the following linear relationships:

$$\begin{aligned} a &= 9.187 \times 10^{-4} \rho + 0.932, \text{ when } 397 \, \frac{\text{kg}}{\text{m}^3} \le \rho \le 674 \, \frac{\text{kg}}{\text{m}^3}. \\ b &= 1.416 \times 10^{-3} \rho - 1.373, \text{ when } 397 \, \frac{\text{kg}}{\text{m}^3} \le \rho \le 674 \, \frac{\text{kg}}{\text{m}^3}. \end{aligned} \tag{41}$$
