Introduction. Numerous studies have been conducted on the identification of systems and the detection of damage in brittle materials by using a variety of methods developed for this purpose. Basically, the damage detection methods are divided into two general categories: the destructive and the non-destructive identification methods. The destructive methods using the structural sampling examine the extent of damage or even failure in the materials’ specimens. Today, the non-destructive methods have become very popular due to their cost-effectiveness. The non-destructive methods are also divided into local and general. The local injury detection methods have some limitations, including the penetration depth of the waves and the location of the injury to conduct the test [1]. The destructive methods based on the linear elastic fracture mechanics have been used by many researchers to study the fracturing mechanism of the concrete and rock structures [2,3,4]. Benedetti et al. [5] proposed an approximate solution for the problem of a vibrating damaged RC beam with opening–closing (breathing) cracks. Methods based on examining vibration (dynamic) properties of a structure to detect and estimate the possible damage can be classified as methods based on the use of intensification frequency, shape of modes, frequency response function (FRF), and curvature of the modes, modal strain energy, dynamic softness, damping, Ritz vectors, time-frequency analyses, updating the finite element model, and wave propagation theory [6].

For example, several experimental and numerical studies have been devoted to explaining the fracture and damage mechanisms of rock and concrete structures [7,8,9,10,11,12,13,14,15,16,17]. In this study, the cracking mechanism of cubic specimens with single crack has been studied both experimentally and numerically. The damage location and convergence diagram of the differential evolutionary algorithm under mode I, II, III and IV of loading are investigated in the pre-cracked concrete cubic specimens (containing single crack in the central part of the specimen) with different inclination angles. The MATLAB and the ABAQUS software are used in this study to numerically simulate the breakage mechanism based on the concrete damaged plasticity (CDP) model. The crack propagation patterns obtained numerically are compared with the corresponding experimental results showing the validity of the accomplished numerical results.

1. Methodology. The crack propagation paths of the single-crack cubic specimens have been experimentally investigated. The same experimental specimens are numerically modeled by a versatile finite element code known as ABAQUS. It has been shown that the corresponding numerical results are in good agreement with their experimentally observed counterparts. In this research, it is attempted to study the breakage process of the pre-cracked cubic specimens of concrete. Therefore, the damage location and the convergence diagram of differential evolutionary algorithm under four modes I (opening mode), mode II (shearing mode), mode III (tearing mode), mode IV (in-plane axial mode or in-plane buckling mode) in the pre-cracked cubic specimens of concrete. In the experimental work, the fracture paths were studied for single-cracked cubic specimens with different inclination angles. In these samples, different damage types have been defined and all the results are presented in separate tables. Also, the parametric analysis and the optimization algorithm are presented by considering 3% noise during the analysis.

A proper mixture of Portland Pozzolana cement (PPC), fine sands and water are to produce the pre-cracked concrete specimens in the form of cubes with sides of 150 mm. Table 1 gives the mechanical properties of the prepared concrete specimens tested in the laboratory before inserting the cracks

Table 1. Ingredient Ratios and Mechanical Properties of the Concrete Specimens
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2. Application of the Structural Response in the Time Domain to Identify the Damage. Time-domain responses of structures are dynamic responses that can be measured directly and at a lower cost compared to the other dynamic responses. To obtain the response over time, the structures must first be solved according to the principles of structural dynamics. If the total applied force to the structure is F (t ), the second-order differential equation with several degrees of freedom governing the dynamic behavior of structures can be written as [13]:

$$ M\ddot{X}(t)+C\dot{X}(t)+ KX(t)=F(t), $$
(1)

where M, C, and K are the matrices of mass, damping, and stiffness of the structure and \( \ddot{X}(t) \), \( \dot{X} \) (t), and X (t ) are the acceleration vectors, velocity, and displacement vectors of the structure at moment t, respectively. Numerical integration methods can be divided into two broad categories of explicit and implicit methods in which the most famous approaches are: (i) the central difference method; (ii) the Hubert method; (iii) the Wilson-theta method; (iv) the Newmark method.

The first method is basically an explicit method while the other three methods belong to the implicit methods. In general, the explicit conditional methods are stable, and if a sufficiently small step is not selected, the answers will diverge, while the implicit methods are unconditionally stable with proper selection of the relevant parameters. On the other hand, two types of errors are introduced in numerical integration, where, the first type of error is related to the amplitude decay and the second one is related to the period elongation of the system response.

In this respect, the Newmark method is more appropriate than the Wilson-theta and Hubert methods. In this method, the error related to the amplitude decay is insignificant and only the elongation period is considered. Therefore, the Newmark method is used to solve the desired equations [15].

Acceleration is a time-dependent function of a structure containing very comprehensive and useful information that can be used to identify the possible damage in the structure. It means that any damage occurrence will lead to changes in dynamic responses including acceleration of structures.

As it can be seen in Fig. 1, the accelerated response of the damaged structure compared to the undamaged structure has been changed.

Fig. 1.
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Acceleration responses in the undamaged and damaged structures throughout the range of usage.

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To show the changes more clearly, the two intervals (0.2, 0.1) and (0.5, 0.3) were examined (Fig. 2).

Fig. 2.
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Acceleration responses in the undamaged and damaged structures in the range of (a) 0.1 to 0.2 and (b) 0.3 to 0.5.

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2.1. Failure Identification of the Optimization Method. The purpose of this study is to identify the local damages of structures by the optimization method. The general form of the optimization problem related to fault identification can be expressed as follows:

$$ {\displaystyle \begin{array}{c}\mathrm{Find}:{X}^T=\left\{{x}_1,{x}_2,\dots, {x}_n\ \right\},\\ {}\operatorname{Minimize}:w(X)\ \left({X}^1\mathbf{\le}{x}_i\mathbf{\le}{X}^u\right)\end{array}} $$
(2)

where X is the damage variable vector containing the location and severity of the unknown damages, Xl and Xu are, respectively, the lower and upper limits of the failure vector, and w is the objective function that must be minimized.

2.2. The Objective Function. The objective function can be considered as one of the most important parts of an optimization problem. The objective function is a measure by which the convergence value of the algorithm and the stop time of the algorithm execution are determined. It is up to the user to select the appropriate target function according to the specific circumstances of the damage identification issue. In this study, according to the multiple damage reliability criterion [4], the objective function was defined as follows:

$$ w(X)=\frac{\left|{a}_d^Ta(X)\right|}{\left({a}_d^T{a}_d\right)\left(a{(X)}^Ta(X)\right)}, $$
(3)

where ad is the acceleration vector of the damaged structure and a(X ) is the acceleration vector of the analytical model.

If acceleration was measured at more than one point in the structure, the accelerations obtained at different points were arranged as a series of columnar vectors.

2.3. Implementation Steps of the Proposed Damage Identification Method. The steps for detecting damage in structures using time range responses and metaphysical optimization method are performed as follows:

Step 1: Apply a single, impact load to the structure as shown in Fig. 3. Of course, in programming the computer codes, the program can receive any kind of the desired load.

Fig. 3.
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Load P (kN) entered into the structure versus the time t (s).

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Step 2: Analyze the damaged structure using the Newmark method and extract acceleration of the structure at the two desired points according to Fig. 4. For a cantilever beam, according to Eq. (1), the accelerations at points 1 and 2 are obtained.

Fig. 4.
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The force applied and the acceleration obtained in a cantilever beam.

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Step 3: form the objective function consisting of the structural acceleration response in the damage state and the acceleration of the analytical model according to Eq. (3).

Step 4: Minimize the objective function formed in the previous step using the metaphysical optimization algorithm and determine the location and extent of damage.

3. Finite Element Analyses of the Structural Damage. The standard finite element code known as ABAQUS is used in this study to numerically simulate the inelastic behavior of concrete based on the concrete damaged plasticity (CDP) model. The tensile and compressive loading conditions in concrete may be modeled by considering the isotropic damaged elasticity concept [18]. Table 2 lists the CDP parameters used in the finite element method (ABAQUS).

Table 2. The Concrete Material Definition in Finite Element Analyses
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The geometries of the centered cracks with different inclination angles in the cubic concrete specimens are illustrated in Fig. 5.

Fig. 5.
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The geometries of cubic specimens with a single centered inclined crack. Here and in Figs. 6 and 7: (a) φ= 45°; (b) φ= 60°; (c) φ= 35°; (d) φ= 75°.

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In the experimental work, the uniaxial compression loading was conducted on cubic specimens of concrete materials (containing a single crack in the central part of the specimen) with different inclination angles. Such specimens prepared from PCC, fine sands, and water were tested in a compressive testing apparatus in a concrete mechanics laboratory.

The fracture patterns of these four cases including the tensile cracks initiated from the crack tips and propagated in the applied load direction are shown in Fig. 6.

Fig. 6.
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The failure and fracture patterns of cubic concrete specimens with a single centered crack at different inclination angles.

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Then, some of the experimental works were simulated numerically via the ABAQUS software. Crack propagation in the cubic concrete specimens with a single centered crack at different inclination angles were numerically modeled, and these results were compared with the corresponding experimental ones (Fig. 6).

Comparing the numerical results in Fig. 7a–d with the corresponding experimental results given in Fig. 6a–d, respectively, their close correlation was observed.

Fig. 7.
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The numerically simulated cubic concrete specimens with a center crack oriented at different inclination

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3.1. Cubic Specimens with a Single Centered Crack Inclined at 45_. Damage in elements of the pre-cracked cubic specimens was modeled via the elastic modulus reduction. Parameters required for the optimization are presented in Table 3.

Table 3. Parameters Required for Optimization
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To examine the accuracy of the proposed method, three different modes (modes I, II, and III) were investigated and presented in Table 4.

Table 4. Different Modes of Loading Conditions for Considering Damage in Cubic Specimens with a Single Centered Cracks Inclined at 45°
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Considering a damping coefficient of 5%, the optimization continued until all iterations were reached kmax=1000 or the target function value did not change significantly after 100 iterations. The experimental and numerical failure patterns of cubic specimens with a single centered crack inclined at 45° are presented in Fig. 8.

Fig. 8.
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Predicting damage location under mode I (a), II (b), and III (c) loading conditions in a cubic specimen with a single central crack inclinated at 45°.

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Moreover, the convergence diagrams of the optimization algorithm for these three modes can also be predicted. Figure 9 depicts the diagrams of damage identification in a single-cracked specimen with a crack inclination angle of 45° under mode I, II, and III loading conditions.

Fig. 9.
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Convergence diagram of differential evolutionary algorithm under mode I (a), mode II (b), and mode III (c) loading conditions in a single-cracked specimen with crack inclination angle of 45°.

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As can be seen in Fig. 9, the proper damage locations and their intensities are detected by the proposed method with a high accuracy, regardless of the noises. These results confirm the high effectiveness of the proposed method for predicting the structural damage to the concrete specimens.

According to the convergence diagrams, it can be observed that with 300 iterations, the algorithm reaches convergence at the last 100 iterations and no significant changes can be seen at higher iterations.

3.2. Cubic Specimens with a Single Centered Crack Inclined at 60°. To examine the accuracy of the proposed method, four different modes of damage were investigated as presented in Table 5.

Table 5. Different Damage Modes Observed in Cubic Specimens with a Single Centered Crack Inclinated at 60°
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Considering a damping coefficient of 5%, the optimization was continued until all iterations were reached kmax=1000 or the target function value did not change significantly after 100 iterations.

The damage detection diagrams for the cubic concrete specimens with a centered single crack inclined at 60° and considering I, II, III, and IV modes are shown in Fig. 10.

Fig. 10.
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Predicting damage location under loading modes of I (a), II (b), III (c), and IV (d) in cubic specimens with a single centered crack inclinated at 60°.

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Moreover, Fig. 11a–d show the convergence diagrams of the optimization algorithm for the concrete specimens under these four loading modes.

Fig. 11.
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Convergence diagrams for differential evolutionary algorithm in concrete specimens with centered single cracks inclined at 60° under loading modes I (a), II (b), III (c), and IV (d).

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According to Figs. 11, the proposed method detects the damage location and intensity with a high accuracy, regardless of the noise effect. These results confirm high effectiveness of the proposed method. Again, it can be observed that at 300 iterations, the algorithm reached convergence at the last 100 iterations with no significant change in the results.

3.3. Cubic Specimens with a Single Centered Crack Inclined at 35°. To examine the accuracy of the proposed method presented in this study, various modes of damage process are investigated and presented in Table 6. Results of each mode are presented with the simulation on MATLAB and ABAQUS softwares. In this mode, damping coefficient has been considered as 5% and optimization continues until total number of iterations reaches kmax= 3000 or the value of target function does not change significantly after 500 iterations.

Table 6. Different Damage Intensities under Three Modes of Loading for Cubic Specimens with a Single Cenered Crack Inclined at 35°
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Damage identification diagrams in cubic specimens with crack inclination angle of 35° under the modes I, II, and III loading conditions are shown in Fig. 12. The optimization algorithm convergence diagrams of the specimens for these modes of loading are graphically presented in Fig. 13.

Fig. 12.
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Predicting damage location for concrete specimens under modes I (a), II (b), and III (c) loading conditions considering the centered single-cracked cubic specimens with crack inclination angle of 35°.

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Fig. 13.
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Convergence diagrams of differential evolutionary algorithm for modes I (a), II (b), and III (c) loading conditions in the centered single-cracked cubic specimens with crack inclination angle of 35°.

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As shown in Fig. 13, the proposed method can accurately identify the damage intensity regardless of noise effects. It can also be observed that with 800 iterations, the function reached convergence at the last 500 iterations.

3.4. Cubic Specimens with a Single Centered Crack Inclined at 75°. To examine the accuracy of the results, three damage modes were considered as presented in Table 7. In these damage modes, the damping coefficient was preset at 5%, and the optimization process continued until the total number of iterations reached kmax=1000 or the target function value did not change significantly after 200 iterations.

Table 7. The Three Modes of Loadings Considered for Damages in the Centered Single-Cracked Cubic Specimens with the Crack Inclination Angle of 75°
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Figure 14 illustrates the damage identification diagrams for cubic specimens with a center crack inclination angle of 75° under the loading modes of I, II, and III. Figure 15 depicts their corresponding optimization algorithms for the convergence diagrams. According to the results shown in these figures, the proposed method properly identified the damages’ location and intensity with a high accuracy, regardless of the noise effects.

Fig. 14.
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Predicting damage locations for modes I (a), II (b), and III (c) of loading conditions in the centered single-cracked cubic specimens with a crack inclination angle of 75°.

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Fig. 15.
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The convergence diagrams of differential evolutionary algorithm for modes I (a), II (b), and III (c) loading conditions in the centered-single cracked cubic specimens with crack inclination angle of 75°.

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Conclusions. In this study, the damage location and convergence diagram of the differential evolutionary algorithm under modes I, II, III or IV of loading conditions in the cubic pre-cracked concrete specimens containing a single centered crack with different inclination angles were investigated. The MATLAB and ABAQUS software were used to numerically simulate the breakage mechanism based on the concrete damaged plasticity (CDP) model. The crack propagation mechanisms obtained numerically were compared with the corresponding experimental results showing the validity of the numerical analyses. The damage acceleration in a concrete specimen was used as a criterion for detecting the structural damages. To calculate the acceleration, the Newmark method was used. At first, the damage detection problems were reduced to optimization problems, which were then solved using the differential evolutionary algorithm. In order to examine the effectiveness of the proposed method and the damage detection process, the modeling was carried out with the reduction of the elastic modulus of the elements. The results showed a high effectiveness of the proposed method in identifying the location and intensity of the damages in concrete specimens.

Acknowledgments. This work was financially supported by National Natural Science Foundation of China (Grant No. 51608117), Key Specialized Research and Development Breakthrough Program in Henan province (Grant No. 192102210051).