**1. Introduction**

The ultrasonic technique [1–4] is used in spectroscopy, defectoscopy, evaluation tests, coagulation, dispergation, sonoluminescence cavitation, and chemical reactions. Ultrasounds can be also applied to crush, form hard media, bond, solder, wash, extract, and dry substances. Another important application of these ultrasonic techniques includes stress measurements in metal constructions. Ultrasonic methods of measuring stress use the acoustoelastic effect (AE), i.e., the correlation between the stress and the velocity of acoustic wave propagation.

The ultrasonic pulse velocity (UPV) is a method applied to cement (concrete) and ceramic materials. This method is used to determine a setting time, changes in the elasticity modulus, and to test the compressive strength (only with the applied minor non-destructive

**Citation:** Jasi ´nski, R.; Stebel, K.; Kielan, P. Use of the AE Effect to Determine the Stresses State in AAC Masonry Walls under Compression. *Materials* **2021**, *14*, 3459. https:// doi.org/10.3390/ma14133459

Academic Editor: Krzysztof Schabowicz

Received: 15 May 2021 Accepted: 18 June 2021 Published: 22 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

(MDT), technique) [5,6]. Besides the AE tests conducted on isotropic materials, the current experience and theoretical analyses of construction materials are related to the anisotropy effect on the wave propagation. The tests mainly include almost isotropic, moderately and strongly anisotropic metamaterials at shear strain and standard deformation. Reference [7] describes the quantitative effect of finite deformations with reference to their magnitude and load direction. Strain-induced instabilities cause negative increments in the phase velocity just as in the case of the isotropic materials. It was also demonstrated that shear strains did not change the velocity of longitudinal waves as the material volume was stable. On the other hand, the effect of high values of deformations on the propagation of acoustic waves in repetitive network materials was explored in the paper [8]. The deformations were found to significantly affect the frequency waves and the phase velocity. In particular, the phase velocity for the hexagonal network strongly decreased under finite compressive deformations. The effective density was shown to have an important impact on the dispersion relation and band diagrams under the application of incremental deformation over the lattice unit cell. Additionally, the theoretical analyses [9,10] are made on mechanical wave propagation in the infinite two-dimensional periodic lattices using Floquet-Bloch. Conclusions derived from the tests can be applied to research on orthotropic construction materials (composites, composite panels, homogeneous masonry structures, etc.). The paper [11] showed that the acoustoelastic effect (AE) [12] also occurs in autoclaved aerated concrete (AAC). The stress state in the wall made of AAC masonry units was determined on the basis of the conducted analyses.

This paper describes the tests aimed at determining the complex state of stresses in masonry units made of autoclaved aerated concrete. As in [11], verification tests were performed on small parts of the wall subjected to axial compression. Their aim was to define empirical relationships of mean hydrostatic stresses *P*, and then normal stresses *σ*<sup>1</sup> and *σ*<sup>3</sup> in the wall, in which the AE effect was observed [12]. Taking into account the AAC vulnerability to moisture content [13] which deteriorates insulation and strength parameters, the analyses included both density and relative humidity of this material. This paper demonstrates a practical application of the AE effect in testing masonry structures which was described in previous works of the author [11,14,15]. The tests were divided into two stages. In the first stage of the tests, experiments were performed on 24 small cuboidal specimens (180 mm × 180 mm × 120 mm) of autoclaved aerated concrete with nominal densities of 400, 500, 600, and 700 kg/m3. The obtained results were used for determining the acoustoelastic constant *δ<sup>P</sup>* that showed the relationship between mean values of hydrostatic stress *P* and velocity of the longitudinal wave *cp*.

Stage II included nine wall models [11] made of AAC masonry units which had a nominal density of 600 kg/m3. They were subjected to axial compression The velocity of ultrasonic wave *cp* in the masonry units was measured. The complex stress state in the wall was examined using the relationships established in stage I. Then, the linear elastic FEM models was applied to match the *P*–*cp* relationship. Knowing stress values *σ*<sup>3</sup> determined in [11], the levels of normal stress *σ*<sup>1</sup> could be determined.

#### **2. Theoretical Bases of the AE Method**

Stress in the material can affect velocity of the acoustic wave because of inhomogeneity and anisotropy. That effect was theoretically described for the first time in the paper [16], and the experimental verification was presented in the papers [17,18]. The static stress was found to have an impact on changes in the velocity values of the acoustic wave in the medium. This pattern has been known as the acoustoelastic (AE) effect [19,20].

This effect, whose theoretical background was described in the paper [12,21], specifies the relationship between stress and velocity of transverse wave propagation. As from then, this subject has significantly evolved [22–24]. The normal stresses have an impact on a change in the velocity of longitudinal and transverse waves (as in the elastooptic effect involving light waves), which is determined by the direction of wave propagation over the direction, the stress, and the wave polarization. Following the theory of solid

deformation [17], the higher orders elasticity constants (neglected in the linear theory of elasticity) which describe the non-linear effects, should be taken account during the analysis of the AE effect. A sum of velocities in the tensionless state (*σ* = 0) and its change (an increment) as a result of the stress (strain) expresses the velocity of ultrasonic wave propagation.

In accordance with the Murnaghan theory [25], the function of free energy *Ws* defined below [17,26], is described by the stress-deformation relationship

$$\mathcal{W}\_{\mathfrak{s}} = \frac{1}{2}(\lambda + 2\mu)I\_1^2 - 2\mu I\_2 + \frac{1}{3}(l + 2m)I\_1^3 - 2mI\_1I\_2 + nI\_3. \tag{1}$$

where: *λ*, *μ*–Lamé constants, *l*, *m*, *n*–elasticity constants of second and third order by Murnaghan, *I*1, *I*2, *I*3–deformation invariants.

Following the principle of energy conservation, Hooke's law can be given by

$$
\rho \delta \mathsf{W}\_s = \sigma\_{i\dot{j}} \frac{\partial \delta u\_i}{\partial u\_j},
\tag{2}
$$

where *δW* and *δui* mean finite increments in the function of free energy and the displacement area, *ρ* is density after deformation (in the stressed body). This AE equation specifies the relationship between the static load and the elastic wave velocity under hydrostatic conditions (that is, under the hydrostatic stress *P*)–Figure 1

$$c\_p^2 = \frac{\lambda + 2\mu}{\rho\_0} - \frac{P}{\rho\_0(3\lambda + 2\mu)}(6l + 4m + 7\lambda + 10\mu),\tag{3}$$

$$c\_t^2 = \underbrace{\begin{array}{c} \mu\\ \rho\_0 \end{array}}\_{\rho\_0} - \frac{P}{\rho\_0(3\lambda + 2\mu)}(3m + 0, 5n + 3\lambda + 6\mu)\,,\tag{4}$$

where  $c\_{\mathcal{P}}$  and  $c\_{\mathcal{I}}$  are velocity of longitudinal and transverse waves respectively, and  $\rho\_0$  body density in the tensions state,  $P$ -hydroxystatic stress defined as  $\underline{P} = \frac{1}{3}(\sigma\_1 + \sigma\_2 + \sigma\_3)$ .

The Equation (2) [27] can be used to determine the stress *P*. For that purpose velocity of the longitudinal and transverse waves is measured. The squared velocities of waves at uniaxial stress states are expressed by these equations

$$V\_{111}^2 = c\_{p0}^2 - \frac{\sigma\_1}{3K\_0 \rho\_0} \left[ \frac{\lambda + \mu}{\mu} (4\lambda + 10\mu + 4m) + \lambda + 2l \right],\tag{5}$$

$$V\_{113}^2 = c\_{p0}^2 + \frac{\sigma\_3}{3K\_0\rho\_0} \left[ \frac{2\lambda}{\mu} (\lambda + 20\mu + m) - 2I \right],\tag{6}$$

$$V\_{131}^2 = c\_{t0}^2 - \frac{\sigma\_1}{3K\_0\rho\_0} \left[ 4\lambda + 4\mu + m + \frac{\lambda n}{4\mu} \right],\tag{7}$$

$$V\_{133}^2 = c\_{t0}^2 - \frac{\sigma\_3}{3K\_0\rho\_0} \left[ \lambda + 2\mu + m + \frac{\lambda n}{4\mu} \right],\tag{8}$$

$$V\_{132}^2 = c\_{t0}^2 + \frac{\sigma\_2}{3K\_0\rho\_0} \left[2\lambda - m + \frac{n}{2}\frac{\lambda}{2}\frac{n}{\mu}\right],\tag{9}$$

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

%&'( *c*2 *t*0

The velocity of the ultrasonic wave (in the deformed material), elastic constants of the first (*λ*, *μ*), second and third order (*m*, *n*, *l*), whose detection is the most difficult in the tests, are used to determine the normal stresses in the material.

**Figure 1.** Directions of the stress and the ultrasonic wave in the isotropic material at the state of hydrostatic compression.

The theoretical background of the AE effect has been adequately proved. Also, the suitable equipment is employed to determine the elastic constants of the third order *l*, *m*, *n* for metal and plastic materials following the procedures presented in e.g., the papers [28–31]. Knowing the direction of the exerted load and a gradient of changes in the longitudinal or transverse wave velocity is required to examine the stress states with the NDT technique.

The proposed procedures can be easily applied to the laboratory tests, however, their use under the in-situ conditions can be troublesome. Hence, a relative increment in the longitudinal wave velocity [32] (knowing the Murnaghan coefficients is not required) is more favorable for practical applications and it can be obtained from the following relationship (based on the Equation (3))

$$\begin{aligned} c\_p^2 - c\_{p0}^2 &= -P \frac{(6l + 4m + 7\lambda + 10\mu)}{3\nu\_0 \xi\_0} \to (c\_p - c\_{p0})(c\_p + c\_{p0}) = -P \frac{(6l + 4m + 7\lambda + 10\mu)}{3\nu\_0 \xi\_0}, \\ \text{assuming that } c\_p + c\_{p0} &\approx 2c\_{p0} \to (c\_p - c\_{p0})2c\_{p0'} \\ \text{the following was obtained} \\ \frac{(c\_p - c\_{p0})}{c\_{p0}} &= -P \frac{(6l + 4m + 7\lambda + 10\mu)}{6\nu\_0 \xi\_0 c\_{p0}}, \\ \text{taking account the following terms} \\ K\_0 &= \frac{2\mu + 3\lambda}{3}; c\_{p0}^2 = \frac{\lambda + 2\mu}{\rho\_0}, \\ \text{Finally, we obtain} \\ \frac{(c\_p - c\_{p0})}{c\_{p0}} &= -P \frac{(6l + 4m + 7\lambda + 10\mu)}{2(2\mu + 3\lambda)(\lambda + 2\mu)} = P\delta\_P. \end{aligned} \tag{10}$$
  $\delta v\_0$  is the AE coefficient expression the relationship between a relative increment in the  $\delta v\_0$  is the AE coefficient expression for the relationship between a relative increment in the  $\delta v\_0$  is the AE coefficient expression for the relationship between a relative increment in the  $\delta v\_0$ .

where *δ<sup>P</sup>* is the AE coefficient expressing the relationship between a relative increment in the longitudinal wave and the mean hydrostatic stresses.

The relative AE coefficient can be expressed as

$$\frac{(c\_p - c\_{p0})}{c\_{p0}} = \frac{P}{P\_{\text{max}}} \eta\_P. \tag{11}$$

where *η<sup>P</sup>* is the relative AE coefficient expressing the relationship between a relative increment in the longitudinal wave and the relative mean of hydrostatic stresses.

The paper [11] defined values of the AE coefficients under the uniaxial compression in the form

$$\frac{\left(\mathcal{c}\_p - \mathcal{c}\_{p0}\right)}{\mathcal{c}\_{p0}} = \beta\_{113}\sigma\_3. \tag{12}$$

$$\frac{\left(\mathcal{c}\_p - \mathcal{c}\_{p0}\right)}{\mathcal{c}\_{p0}} = \gamma\_{113} \frac{\mathcal{c}\_3}{\mathcal{c}\_{3\text{max}}}.\tag{13}$$

where

$$\begin{array}{l} \beta\_{113} = 1.39 \cdot 10^{-4} \rho - 0.104 \text{ \textdegree R}^2 = 0.995, \\ \gamma\_{113} = 1.72 \cdot 10^{-4} \rho - 0.206, \text{ \textdegree R}^2 = 0.923, \end{array} \tag{14}$$

when 397 kg <sup>m</sup><sup>3</sup> <sup>≤</sup> *<sup>ρ</sup>* <sup>≤</sup> <sup>674</sup> kg m3 .

#### **3. Program of Own Research**

Following the procedure described in the paper [11] the tests were divided into two stages. In the first stage of the tests, the biaxial compression was exerted until the failure of the specimens with dimensions of 180 mm × 180 mm × 120 mm. Velocity of the longitudinal wave was determined under different hydrostatic stress *P*. The tests were conducted in a test stand specially prepared to test the specimens and simultaneously control their deformations by the non-contact technique of Digital Image Correlation DIC. The obtained results were the base to determine the linear correlations of the *cp*–*P* relationship. The test results for nine masonry models under axial compression, described in [11], were used in the stage II to determine at first mean hydrostatic stress, and then the normal stress *σ*<sup>1</sup> which was parallel to the plane of bed joints. The test results for the complex state of stresses were compared with the results for the linear-elastic FEM models. Then, the method was validated.

#### **4. Test Results**

#### *4.1. Stage I-Determination of Acoustoelastic Constant*

#### 4.1.1. Physical and Mechanical Properties of Autoclaved Aerated Concrete AAC

The tests included four series of masonry units with a thickness within the range of 180–240 mm and different classes of density: 400, 500, 600, and 700 [33], which were the subject of tests presented in the paper [11]. Six cores with a diameter of 59 mm and a height of 120 mm were cut out from the masonry units. They were used to determine the fundamental properties of the test autoclaved aerated concrete (AAC). All the cores were dried until constant weight at a temperature of 105 ± 5 ◦C. The modulus of elasticity *E* and Poisson's ratio *ν* were determined for the core specimens. Mean mechanical parameters obtained for all the tested types of masonry units are shown in Table 1. The results from testing density and compressive strength of the specimens 100 mm × 100 mm × 100 mm were taken from [14].


**Table 1.** Fundamental characteristics of masonry units as defined in the papers [11,14].

Apart from the core specimens of AAC masonry units, also 24 rectangular specimens having dimension of 180 mm × 180 mm × 120 mm were cut out and used in the stage I. To determine the correlation between mean hydrostatic stress *P* and ultrasonic velocity, all the specimens were air-dried at a temperature of 105 ± 5 ◦C for at least 36 h until constant weight. That way the impact of moisture content on AAC was eliminated [15,34]. Generally, moisture content tends to significantly reduce compressive strength and change velocity of the ultrasonic wave propagation [14].

#### 4.1.2. Test Stand and Procedure

The velocity of ultrasonic waves was determined by the method of transmission [11,35,36]. Velocity of ultrasonic waves in 180 mm × 180 mm × 120 mm specimens taken from the masonry units, was measured at the specially prepared test stand—Figure 2. The

test stand for testing biaxial compression consisted of two vertical columns 1 made of a set of two channel profiles 120 with a length of 1000 mm and connected at the bottom with a spandrel beam 2 made of three I-beams 140 with a top spandrel beam 3 which was made of an I-beam 200 with a length of 1000 mm and reinforced with ribs. Inside dimensions between spandrel beams and the column were 820 mm in a vertical plane, and 810 mm in a horizontal plane. Openings with the spacing of 75 mm were made in vertical columns 1 and in the spandrel beam 3 to change its position. The hydraulic actuator 4 with an operating range of 500 kN was pin jointed to the top spandrel beam. A draw-wire displacement converter 5 of SWH-1-B-FK-01 type with the TRA50-SA1800WSC01 encoder (TWK-ELEKTRONIK GmbH, Düsseldorf, Germany) was attached to the side wall of the actuator. The hydraulic actuator was connected to the hydraulic power unit "A" (Zwick Roell Company Group, Ulm, Germany) with a pressurized pipeline, to which the pressure transmitter P30 was attached (WIKA SE & Co. KG, Klingenberg, Germany) 6. Its operating range was 0–1000 bar and the reading accuracy was 1 bar. The hydraulic actuator 7 with an operating range of 500 kN was sliding jointed to one vertical column. A draw-wire displacement converter 8 (SWH-1-B-FK-01 type with the TRA50-SA1800WSC01 encoder) was fixed to the actuator. The actuator was connected to the hydraulic power unit "B" (Hydac International GmbH, Sulzbach/Saar, Germany) with a pressurized pipeline, to which the pressure transmitter P30 was attached-9 The research model 10 was placed between Teflon washers 11 and steel plates 12 with ball joints.

This test stand was a complex system designed and prepared by the authors [37]. This design is copyrighted [38]. The advanced control algorithms had to be applied as many non-linearities were present in the subsystems. These algorithms ensured the proper interactions between elements of the test stand. Due to the continuous improvement of these algorithms [39,40], the test stand performance is characterized by high repeatability as proper feedback is ensured among the following components of the system:


The block scheme in Figure 3 illustrates the interactions between individual elements of the system. The IT system with the hydraulic system generated stresses *σ*<sup>1</sup> and *σ*<sup>3</sup> of the same value and were used to read the ultrasonic wave path recorded with the PUNDIT LAB+ instrument. The ARAMIS 6M system was used to control deformations and observe crack images in individual specimens. Collecting data from different subsystem in one IT system ensured an additional option for the tests due to the time correlation of many data and their cause–effect relations. When different systems were combined, the set tasks were performed in a more effective way compared to individual subsystems [42–45].

**Figure 2.** Test stand for measuring the acoustoelastic effect under the biaxial stress state (described in the text).

**Figure 3.** Block diagram of the test stand.

The tests were conducted on the specimens which were dried to constant weight and which had relative humidity *w*/*w*max = 0%. The tests included at least 6 specimens of the same density, and 24 specimens in total were tested (Figure 4a,c,d). The PUNDIT LAB+ instrument (Proceq SA, Schwerzenbach, Switzerland), which was integrated with the IT system of the test stand, was used to measure velocity of ultrasonic waves. The point measurements were taken with the exponential transducers the frequency 54 kHz (Figure 4e). The measurement accuracy of passing time of the ultrasonic wave was equal to ±0.1 μs. Each specimen was placed between the plates of the test stand using Teflon washers of 10 mm in thickness. Compressive stress *σ*<sup>3</sup> was generated in the vertical direction. In the horizontal direction, in which the normal stress *σ*<sup>1</sup> was generated, Teflon plates, and then steel sheet were placed on the lateral sides to generate loading. The measuring templates were placed to end face of each specimen (Figure 4b) in the next step. The passing time of the wave was measured with transducer which were in put (at 90◦) into the openings of the measuring templates. Each time a distance was measured between the transducers with an accuracy of 1 mm. An increment in stress values could be uniform by controlling loads exerted in both vertical and horizontal directions by hydraulic actuators 'A' and 'B'. Velocities of ultrasonic waves were read every 5 kN (for the specimens with nominal densities of 400 and 500 kg/m3) and every 10 kN (the specimens with nominal densities of 600 and 700 kg/m3). A view of the test stand in operation is shown in Figure 5.

#### 4.1.3. Test Results

There were not any models with damaged front face during the loading cycle. Prior to the failure crack was heard and noticeable cracks were observed on the surface. Noticeable cracks were also found on the specimen surfaces under loading that preceded the failure. Debonding of external surface of each test element was observed at failure. It revealed a type of the specimen damage with clearly truncated pyramids that were connected in the center of the specimen. Passing time of the ultrasonic wave using the transmission method was measured at 25 points of each specimen at the following stress values: 0, ~0.25*P*max, ~0.50*P*max, ~0.75*P*max, *P*max. Examples of the obtained maps showing passing time of the wave are illustrated in Figures 6–9.

Velocities of ultrasonic waves in all the specimens were significantly disturbed in the edge areas. Noticeably lower wave velocities were observed in these areas. The results referred to 16 points (as shown in the template–Figure 4a): A1–A5, B1, B5, C1, C5, D1, D5, E1, E5, and F1–F5. The observed disturbances were caused by the immediate vicinity of loaded edges of the specimens and the recorded wave reflection at the edge, and also by local damage to the material during the loading phase. The highest homogeneity of the results was found in central areas of each specimen at nine points B1–B3, C1–C3, D1–D3, and E1–E3. Table 2 presents the measurement results for the ultrasonic wave with reference to the mean and maximum values of hydrostatic stress *P*max. The table below presents velocities of the longitudinal wave obscp0 determined at free state and relative mean values of hydrostatic stress *<sup>P</sup>*/*P*max. The measurements expressed as (*cp* − obs*cp*0)/obs*cp*<sup>0</sup> ratio of a relative increment in ultrasounds as a function of stress *P* are shown in Figure 10a. Figure 10b illustrates the relative rate of an increase of ultrasonic wave velocity rise over the relative of compressive stresses *P*/*P*max.

**Figure 4.** Measurements of ultrasonic wave velocity in biaxially compressed specimens: (**a**) components of stress states and the position of the measuring template; (**b**) geometry of the measuring template; (**c**,**d**) the test specimen; (**e**) the exponential transducer; 1—the AAC specimen 180 mm × 180 mm × 120 mm, 2—exponential transducers, 3—cables connecting transducers with recording equipment, 4—the measuring template, 5—PUNDIT LAB+ recording equipment.

**Figure 5.** View of the test stand in operation: (**a**) overall view, (**b**) specimen view; 1—the test stand with the fixed actuators; 2—a test element; 3—the hydraulic system 'A'; 4—the hydraulic system 'B'; 5—the measurement and control interface; 6—IT system; 7—cameras of the ARAMIS 6M system; 8—PUNDIT LAB+ instrument.

**Figure 6.** Maps of passing time the ultrasonic wave in the model 400/1 at selected loading levels: (**a**) *P* = 0, (**b**) *P* = *P*max.

**Figure 7.** Maps of passing time the ultrasonic wave in the model 500/1 at selected loading levels: (**a**) *P* = 0, (**b**) *P* = *P*max.

**Figure 8.** Maps of passing time the ultrasonic wave in the model 600/1 at selected loading levels: (**a**) *P* = 0, (**b**) *P* = *P*max.

**Figure 9.** Maps of passing time the ultrasonic wave in the model 700/1 at selected loading levels: (**a**) *P* = 0, (**b**) *P* = *P*max.

**Figure 10.** 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.


**Table 2.** Test results for ultrasonic wave velocity in AAC under various mean hydrostatic stresses determined at central points (B1–B3, C1–C3, D1–D3, E1–E3) of each specimen.

> The tests showed that AAC density had an impact on velocities of ultrasonic waves, which confirmed the previous tests [14]. At stress values *P* = 0, velocities of ultrasonic waves increased in the specimens dried until constant weight. This increase was proportional to densities of AAC under stress The longitudinal wave velocity obs*cp*<sup>0</sup> in the AAC units of the minimum nominal density of 400 kg/m<sup>3</sup> was equal to 1875 m/s and increased to 2225 m/s in concrete characterized by the highest density of 700 kg/m3. Velocities of longitudinal waves noticeably dropped as means stresses *P* increased in all the units. Under relatively low stress when 0 ≤ *P* ≤ 0.25*P*max, values of ultrasounds decreased by 3–9% to the value obs*cp*0. At slightly higher values of hydrostatic stress 0.25*P*max ≤ *<sup>P</sup>* ≤ 0.50*P*max the ultrasonic wave velocities dropped by 9–16% (with reference to the base value). Higher stress values 0.50*P*max ≤ *P* ≤ 0.75*P*max in concrete having nominal densities of 400 and 500 kg/m3 caused the highest percentage drop in the velocities of ultrasonic waves by 23–24%. Ultrasonic wave velocities dropped by 13–17% in more dense masonry units made of AAC (600 and 700 kg/m3). In opposition to lower hydrostatic loads, no clear reduction in wave velocity was observed at the stress level preceding the failure when local cracking and crushing were found within the stress range of 0.75*P*max ≤ *P* ≤ ~*P*max. For concretes with lower density, the velocity was reduced by 32–39%, whereas the velocity drop by 19–22% was found in concretes having density of 600 and 700 kg/m3. As in the tests under uniaxial stress state [11], a nearly linear drop in the relative velocity of longitudinal ultrasonic wave was observed at any density of AAC. A drop in velocity was practically 1.5–2.0 times higher than in the tests [11] on the specimens under uniaxial stress state and subjected to stress *σ*3. The resulting biaxial stress state confirmed the linear correlation

which specified a reduced velocity of ultrasonic wave over mean hydrostatic stress. This effect was noted during the tests on AAC [11] and metals [29,32].

Table 3 presents coefficients of the linear correlation of the relative velocity of ultrasonic waves as a function of mean hydrostatic stress which are shown in Figure 10. Regression lines based on values of AE coefficients contained values of AE (*δ, η*) coefficients and density of AAC, which are illustrated in Figure 11. The coefficients were determined at moisture content of AAC *w* = 0. Additionally, values of coefficients *β*113, *γ*<sup>113</sup> determined in the tests on uniaxial compression which are described in the paper [11], are shown in Figure 11.


**Table 3.** Values of AE coefficients for concrete of specific densities.

**Figure 11.** Values of coefficients δ and η as a function of AAC density.

$$
\delta = 5.068 \cdot 10^{-4} \rho - 0.635, \,\mathrm{R}^2 \,\, = 0.991,\tag{15}
$$

$$
\eta = 6.91 \cdot 10^{-4} \rho - 0.64 \text{ } \text{R}^2 \\
= 0.976 \tag{16}
$$

when 397 kg <sup>m</sup><sup>3</sup> <sup>≤</sup> *<sup>ρ</sup>* <sup>≤</sup> <sup>674</sup> kg m3 .

Walls in real structures have moisture content *w* > 0 and the effect of this factor has to be taken into account. Considering the results from own research [14] and the procedure described in the paper [11], the empirical relationship was defined to determine velocities of UV waves under air-dry conditions *cp* (at *w* = 0) based on the equation

$$\frac{c\_{pw}}{c\_p} = a\left(\frac{w}{w\_{\text{max}}}\right)^2 + b\left(\frac{w}{w\_{\text{max}}}\right) + 1 \to c\_p = \frac{1}{c\_{pw}} \left[ a\left(\frac{w}{w\_{\text{max}}}\right)^2 + b\left(\frac{w}{w\_{\text{max}}}\right) + 1 \right],\tag{17}$$

where *cpw*—velocity of ultrasonic wave in wet AAC in the unloaded state *P* = 0; *cp* velocity of ultrasonic wave in dry (*w* = 0) AAC in the unloaded state *P* = 0; *w*—relative humidity of AAC; *w*max—maximum relative humidity of AAC [14] calculated from the following relationship

$$w\_{\text{max}} = -1.23 \frac{\rho}{1000} + 1.34 \text{, when } 397 \,\frac{\text{kg}}{\text{m}^3} \le \rho \le 674 \,\frac{\text{kg}}{\text{m}^3}. \tag{18}$$

*a, b*–empirical coefficients dependent on density were

$$\begin{aligned} a &= 9.187 \cdot 10^{-4} \rho + 0.932, \text{ when } 397 \, \frac{\text{kg}}{\text{m}^3} \le \rho \le 674 \, \frac{\text{kg}}{\text{m}^3}. \\ b &= 1.416 \cdot 10^{-3} \rho - 1.373, \text{ when } 397 \, \frac{\text{kg}}{\text{m}^3} \le \rho \le 674 \, \frac{\text{kg}}{\text{m}^3}. \end{aligned} \tag{19}$$

### *4.2. Stage II-Testing Models under Compression*

Stage II involved small models of the masonry already used in the tests described in the previous paper [11]. The models of 500 mm × 726 mm × 180 mm in dimensions were composed of three layers of masonry units made of AAC of nominal density of 600 kg/m3. They were connected with thin bed joints laid in the commercial mortar with a strength *f* <sup>m</sup> = 6.10 N/mm2 [46] and the nominal class M5 [47]. Models (nine specimens)—divided into three series marked as I, II, and III—were tested. The models differed in the presence or lack of head joints. The models of series I did not have the head joint, whereas the unfilled head joint in the central layer was at mid-length or 1/4 length of the masonry unit in other series II and III. An overall view of tests specimens of the series I, II, and III is shown in Figure 12.

**Figure 12.** Geometry of models made of AAC tested in stage II: (**a**) models of series I without head joint, (**b**) models of series II with head joint at mid-length of the masonry unit, (**c**) models of series III with head joint at 1/4 length of the masonry unit; 1—masonry units, 2—bed joints, 3—head joints.

All the models were subjected to monotonic compression perpendicular to the plane of bed joints by exerting the uniform increment in the shift of the testing machine piston– Figure 13. The mean normal stress *σ*<sup>3</sup> was calculated as a ratio of the exerted load *F* and the area of bed face of the masonry unit *<sup>A</sup>* (*<sup>A</sup>* = 180 mm × 500 mm= 90,000 mm2). For two models from each series [11] velocities of ultrasonic waves *cp* were measured at the following values: 0, 0.25*σ*3max, 0.50*σ*3max, and 0.75*σ*3max. In the stage I, waves were measured using the method transmission–Figure 13a. The template was used to ensure the coaxiality of the transducers. The tests are described in details in the paper [11]. Vertical strains were measured during the tests on all the models except for I-3, II-3, III-3 series, using the digital-image correlation system ARAMIS 6M (GOM GmbH, Braunschweig, Germany) [48–51]. The main tests were preceded by determination of apparent density *ρ*<sup>0</sup> (at air-dry state) and relative humidity *w* in AAC. Then the maximum moisture content

*w*max was calculated from the Equation (19). Table 4 presents the main results from material tests and the results from main tests as crack-inducing stress *σ*3cr, and maximum stress *σ*3max.

Considering density (*ρ*<sup>0</sup> = 587–597 kg/m3) and relative humidity (*w* =4.5–6.0%), the research model were regarded as nearly homogeneous. In all the models a nearly proportional increase in deformations was noticed at increasing loading. Cracks were formed at failure stress of ca. > 90%. They were detected at horizontal edges of the masonry units and in the extended head joints. The failure was gentle. An increase in the width of vertical cracks and spalling of external parts of the masonry units were noticed— Figure 13b,c. The passing time *tp* of the ultrasonic wave was measured at defined load levels (then the strength testing machine was stopped). Calculating the velocity of the wave propagation from the relationship *cp* = *L*/*tp* (*L* = 180 mm) was the next step. The synthetic test results for all measuring points and the points located at mid-height of each masonry unit are shown in Table 5, and the partial results can be found in the paper [11].

**Figure 13.** The procedure employed in Stage II to test the AAC wall models: (**a**) measurements of velocity of the ultrasonic wave at different stress value *σ*3, (**b**) selected models at failure, (**c**) vertical strains of selected wall models under stress *σ*3max; 1—masonry units, 2—ultrasonic transducers, 3—templates to arrange symmetrically ultrasonic transducers.

As presented in the paper [11], passing time of the ultrasonic wave through the models under zero loads was characterized by some variability. The longest passing time was usually recorded in central parts of the elements. Distinct disturbances described by different passing times of the wave were noticed at vertical edges of the masonry units and at bed joints. Passing times were consistent in the central areas of the units in spite of disturbed edge areas. The obtained variation coefficient was rather low within a range of 1.4–1.6% even though all the measurements were considered (even from the disturbed areas). A clear increase in the passing time of the ultrasonic wave in all the models was observed when the loads increased up to 0.25*σ*3max. The coefficient of variation was rather low within a range of 1.0–1.3% as in the case of lower stress values. An increase in loads to 0.50*σ*3max and 0.75*σ*3max resulted in a gradual increase in the mean time of propagation for almost all measuring points. The calculated coefficients of passing time of the wave were close to the values noticed at previous loading values and amounted to ca. 1.4%.


**Table 4.** Summary of mean results from the tests on the models.

**Table 5.** Results from measuring propagation of ultrasonic waves.

