To verify the effectiveness of the proposed model in predicting the post-fatigue stress–strain response of concrete, the literature [59] is applied to calibrate the parameters, as follows, *E*0 − = 36.0 GPa, *f* cu− = 41.4 MPa, *A*1 − = 0, *A*2 − = 1, ε0 − = 2153.8 με, *p*− = 3.186, *A*3 − = 0.713, *A*4 − = 1.160, *A*5 − = 5.439. The predicted post-fatigue stress-strain responses are obtained in Figure 11a. Figure 11a illustrates that the post-fatigue stress–strain responses predicted by the model varies in a typical three stages way depending on the increasing of cycle ratio. In detail, the development of the initial strains (i.e., the residual strains) of the responses grows in a three stages way, and the variation of the initial (post-fatigue) stiffness and the residual strength experience in a similar way. Therefore, the proposed model is able to reproduce the post-fatigue constitutive behaviors.

### *4.6. Comparison among Proposed Model and Other Models*

The proposed model is compared with the typical damage evolution models [60,61]. The parameters are calibrated by using the experimental results in the literature [61], such that, *E*0 − = 54.5 GPa, *A*3 − = 0.6, *A*4 − = 1.01, *A*5 − = 9, *B*1 = 2.29, *B*2 = −1.23, and *B*3 = 0. Thus, the predictions are obtained in Figure 11b–d. Figures 10 and 11b illustrate that the predictions of the proposed model are more accurate than those of other damage evolution models [60,61]. In addition, the proposed model is able to reproduce the other behavior variations under fatigue loading (e.g., the development of cyclic creep (Figures 10 and 11c), stiffness degradation (Figure 11d) and post-fatigue constitutive behavior (Figure 11a), which are not well considered in the other models [60,61].

**Figure 11.** Post-fatigue stress–strain response predicted by the proposed model (*n*/*N* = 0.02~0.98, *S* = 0.66), and comparison of *D*i , cyclic creep and stiffness degradation among predicted and experimental results [61] (The graphs are redrawn by authors).

Furthermore, the proposed models are capable of predicting the variations of cyclic creep, stiffness degradation, residual strength, and the stress–strain relationship under both fatigue loading and post-fatigue loading. However, few damage models [6,41–49] took all the characteristics above into account for relevant characterizing.

Additionally, the proposed model obtains a clear physical consideration based on the micro mechanical description of damage behaviors in concrete under multi-axial loading, and proposed a behavior characterizing method based on both the above-mentioned description and a thermodynamics-based CDM method. However, in the classical damage models [44,45], the yield concept cannot coexist with the loading/unloading irreversible strain concept [62] introduced in their framework. In detail, in the yield concept [44,45], there is only one yield surface for determining the plastic strains in the material. However, in the loading/unloading irreversible strain concept [62], each loading/unloading process (i.e., a loading cycle) obtains a corresponding irreversible strain surface for the development of irreversible strains.

### **5. Conclusions**

In this work, a damage model with the concept of mode-II microcracks using thermodynamic interpretation of damage behaviors for concrete under fatigue loading was developed.

In detail, by applying the micromechanics method, a micro-cell-based damage model was introduced to understand the damage behavior. The mode-II microcracks were further introduced as a contributing part of irreversible/residual strains.

Additionally, by introducing the physical interpretation of the damage variable based on the thermodynamic method, a novel fatigue damage variable (irreversible deformation fatigue damage variable) was proposed to describe the irreversible strains. With this methodology, a continuum damage mechanics model considered both the elastic and irreversible deformation fatigue damages was developed.

It is found that the predictions of this model highly agreed with experimental results. This model is able to characterize the variations of cyclic creep, stiffness degradation, residual strength, and the post-fatigue stress-strain relationship of concrete. The model can also be used to analyze the behaviors of concrete under complex fatigue loads such as a multi-axial case.

**Author Contributions:** Conceptualization, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; methodology, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; software, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; validation, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; formal analysis, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; investigation, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; resources, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; data curation, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; writing—original draft preparation, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; writing—review and editing, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; visualization, Z.S., Z.Y., X.L. (Xiao Li), X.L. (Xiaoyong Lv) and Z.L.; supervision, Z.S. and Z.Y.; project administration, Z.S. and Z.Y.; funding acquisition, Z.S. and Z.Y.

**Funding:** This research was supported by the National Key R&D Program of China, grant number 2018YFD1100401; the National Natural Science Foundation of China, grant numbers 51808558,<PHONE_NUMBER>4, U1434204, 51378506, 51478478; the Natural Science Foundation of Hunan Province, China, grant number 2019JJ50800; the China Energy Investment Corporation, grant number SHGF-18-50.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A. Damage Variable Definition Based on Thermodynamics**

#### (1) Perfect elastic materials

The Helmholtz free energy of a perfect elastic material per unit mass is obtained as follows (see Figure A1a):

$$
\Psi \Psi = \psi + \mathcal{y} = \psi = \psi\_0^c \tag{A1}
$$

And the energy dissipation χ is introduced in this work (see Figure A1), that is,

$$
\chi = \frac{1}{2\nu} \sigma \varepsilon\_0 = \frac{1}{2\nu} \sigma \varepsilon = \psi\_0^{\mathbf{e}} \tag{A2}
$$

where χ denotes that energy dissipation exists as a typical property of the elastic behaviors, it accompanies the initial strain energy ψ0 e, and both are equal to each other like the twins, see Figure A1.

There is no damage appearing in the material, thus, it is not necessary to define a damage variable. And the stress–strain relation is described as follows:

$$
\sigma = E\_0 \varepsilon \tag{A3}
$$

### (2) Elastic deformation damage materials

The Helmholtz free energy of an elastic deformation damage material is obtained as follows (see Figure A1b):

$$
\Psi' = \psi + \mathcal{\boldsymbol{\gamma}} = \left(\psi\_0^{\mathrm{e}} + \psi^{\mathrm{de}}\right) + \mathcal{\boldsymbol{\gamma}}^{\mathrm{de}} \tag{A4}
$$

where

$$
\psi^{\rm de} = \gamma^{\rm de} = \frac{1}{2\nu} \sigma \varepsilon^{\rm de} \tag{A5}
$$

And the energy dissipation χ is derived in this work (see Figure A1), that is,

$$\chi = \frac{1}{2\nu} \sigma \varepsilon\_0 = \psi\_0^{\mathfrak{e}} \tag{A6}$$

The elastic deformation damage variable in the elastic strain space is defined in this work by two methods considering two different energy dissipation aspects, respectively, as follows:

$$D^{\rm E} = \frac{\psi^{\rm de}}{\psi} = \frac{\psi^{\rm de}}{\psi\_0^{\rm e} + \psi^{\rm de}} = \frac{\varepsilon^{\rm de}}{\varepsilon\_0 + \varepsilon^{\rm de}} = \frac{\varepsilon^{\rm de}}{\varepsilon^{\rm E}} \tag{A7}$$

$$D^{\rm E} = \frac{\Psi^{\rm de}}{\Psi + \chi} = \frac{\psi^{\rm de} + \chi^{\rm de}}{\left(\psi\_0^{\rm c} + \psi^{\rm de} + \chi^{\rm de}\right) + \chi} = \frac{\varepsilon^{\rm de}}{\varepsilon^{\rm E}}\tag{A8}$$

where εE denotes the elastic strains, εE = ε0 + εde.

The nonlinear stress–strain relation is described as follows:

$$
\sigma = \left(1 - D^{\rm E}\right) E\_{0\%} \tag{A9}
$$

The damage variable *D*E is able to be used to describe the stiffness degradation, that is (Figure A1b),

$$E = \left(1 - \frac{\varepsilon^{\rm de}}{\varepsilon}\right) E\_0 = \left(1 - D^E\right) E\_0 \tag{A10}$$

**Figure A1.** Sketch of the energy dissipation for various materials under uniaxial loading.

### (3) Irreversible deformation damage materials

The Helmholtz free energy of an irreversible deformation damage material is obtained as follows (see Figure A1c):

$$
\Psi = \psi + \gamma = \psi\_0^\mathbf{e} + \gamma^{\mathrm{di}} \tag{A11}
$$

And the energy dissipation χ is derived in this work (see Figure A1), that is,

$$
\chi = \frac{1}{2\nu} \sigma \varepsilon\_0 = \psi\_0^{\mathfrak{e}} \tag{A12}
$$

The irreversible deformation damage variable is defined in this work by the method considering energy dissipation, as follows:

$$D^{\rm i} = \frac{\Psi^{\rm di}}{\Psi + \chi} = \frac{\gamma^{\rm di}}{\left(\psi\_0^{\rm c} + \gamma^{\rm di}\right) + \chi} = \frac{\varepsilon^{\rm di}}{\varepsilon} \tag{A13}$$

where ε = ε0 + εdi.

The nonlinear stress–strain relation is described as follows:

$$
\sigma = E\_0 \left( \varepsilon - \varepsilon^{\rm di} \right) = \left( 1 - D^i \right) E\_0 \varepsilon \tag{A14}
$$

The damage variable *D*i is able to be used to describe the irreversible strain development, that is (see Figure A1c),

$$
\varepsilon^{\text{di}} = D^{\text{i}} \varepsilon \tag{A15}
$$

Additionally, when the initial stiffness approaches an infinite value ∞, the material exhibits a prefect brittle-plastic behavior (see Figure A1d). Thus, the Helmholtz free energy is derived as follows,

$$
\Psi = \psi + \gamma = \gamma = \gamma^{\text{dir}} \tag{A16}
$$

due to the energy dissipation χ = ψ0 e = 0. The irreversible deformation damage variable is obtained, as follows:

$$D^{\rm i} = \frac{\mathcal{V}^{\rm di}}{\mathcal{V} + \chi} = \frac{\mathcal{V}^{\rm di}}{\mathcal{V}^{\rm di}} = 1 \tag{A17}$$

And the irreversible strain is derived, as follows:

$$
\varepsilon^{\text{di}} = D^{\text{i}} \varepsilon = \varepsilon \tag{A18}
$$

Furthermore, the nonlinear stress-strain relation is described as follows:

$$\begin{cases} \sigma = \sigma \\ \varepsilon = \varepsilon^{\text{di}} \end{cases} \tag{A19}$$

### (4) Quasi-brittle materials (Elastic-irreversible deformation damage materials)

The Helmholtz free energy of a quasi-brittle material is obtained as follows (see Figure A1e):

$$
\Psi = \psi + \gamma = \left(\psi\_0^{\mathbf{c}} + \psi^{\mathbf{d}\mathbf{e}}\right) + \left(\gamma^{\mathbf{d}\mathbf{e}} + \gamma^{\mathbf{d}\mathbf{i}}\right) \tag{A20}
$$

where

$$
\psi^{\rm de} = \gamma^{\rm de} = \frac{1}{2\nu} \sigma \varepsilon^{\rm de} \tag{A21}
$$

And the energy dissipation χ is derived in this work (see Figure A1), that is,

$$
\chi = \frac{1}{2\nu} \sigma \varepsilon\_0 = \psi\_0^{\mathfrak{e}} \tag{A22}
$$

The elastic deformation damage variable in the elastic strain space is defined in this work by two methods considering two different energy dissipation aspects, respectively, as follows:

$$D^{\rm E} = \frac{\psi^{\rm dc}}{\psi} = \frac{\psi^{\rm dc}}{\psi\_0^{\rm c} + \psi^{\rm dc}} = \frac{\varepsilon^{\rm dc}}{\varepsilon\_0 + \varepsilon^{\rm dc}} = \frac{\varepsilon^{\rm dc}}{\varepsilon^{\rm E}} \tag{A23}$$

$$D^{\rm E} = \frac{\Psi^{\rm de}}{\Psi^{\rm e} + \chi} = \frac{\psi^{\rm de} + \gamma^{\rm de}}{\left(\psi\_0^{\rm e} + \psi^{\rm de} + \gamma^{\rm de}\right) + \chi} = \frac{\varepsilon^{\rm de}}{\varepsilon^{\rm E}}\tag{A24}$$

The elastic deformation damage variable in the total strain space is also defined in this work by the method considering energy dissipation, as follows:

$$D^{\mathbf{e}} = \frac{\Psi^{\mathrm{de}}}{\Psi + \chi} = \frac{\psi^{\mathrm{de}} + \chi^{\mathrm{de}}}{\left(\psi\_0^{\mathrm{e}} + \psi^{\mathrm{de}} + \chi^{\mathrm{de}} + \chi^{\mathrm{di}}\right) + \chi} = \frac{\varepsilon^{\mathrm{de}}}{\varepsilon} \tag{A25}$$

where the total strain ε = ε0 + εde + εdi, and the elastic strain εE = ε0 + εde.

The irreversible deformation damage variable is defined in this work by the method considering energy dissipation, as follows:

$$D^{\rm i} = \frac{\mu^{\rm qdi}}{\Psi + \chi} = \frac{\gamma^{\rm di}}{\left(\psi\_0^{\rm e} + \psi^{\rm de} + \gamma^{\rm de} + \chi^{\rm di}\right) + \chi} = \frac{\varepsilon^{\rm di}}{\varepsilon} \tag{A26}$$

The nonlinear stress–strain relation is described as follows:

$$\begin{array}{lcl}\sigma &= \left(1 - D^{\mathrm{E}}\right) E\_0 \left(\varepsilon - \varepsilon^{\mathrm{di}}\right) = \left(1 - D^{\mathrm{E}}\right) \left(1 - D^{\mathrm{i}}\right) E\_0 \varepsilon \\ &= \left[1 - \left(D^{\mathrm{e}} + D^{\mathrm{i}}\right)\right] E\_0 \varepsilon = (1 - D) E\_0 \varepsilon \end{array} \tag{A27}$$

The damage variable *D*E (or *D*e) and *D*i is able to be employed to characterize the stiffness degradation and irreversible strain development, respectively, as follows (see Figure A1e),

$$E = \left(1 - D^{\rm E}\right) E\_0 = \left(1 - \frac{D^{\rm e}}{1 - D^{\rm i}}\right) E\_0 = \frac{1 - D}{1 - D^{\rm i}} E\_0 \tag{A28}$$

$$
\varepsilon^{\text{di}} = D^{\text{i}} \varepsilon \tag{A29}
$$

Note that the energy dissipation χ presents a typical property of materials' elastic behaviors. Figure A1 and Equations (A2), (A6), (A12) and (A22) show that the energy dissipation χ is equal to the initial strain energy ψ0 e, i.e., χ = ψ0 e, and in a limit case without elastic behaviors in Figure A1d, χ = ψ0 e = 0. Additionally, the damage variable definition method considering the energy dissipation χ in Equations (A8) and (A24) is as effective as that in Equations (A7) and (A23), respectively. Therefore, it is reasonable to apply the damage variable definition method considering the energy dissipation χ in this work.

### **References**

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

### *Article*

### **Plain and Fiber-Reinforced Concrete Subjected to Cyclic Compressive Loading: Study of the Mechanical Response and Correlations with Microstructure Using CT Scanning**

### **Jesús Mínguez 1,\*, Laura Gutiérrez 2, Dorys C. González 1 and Miguel A. Vicente 1**

Received: 30 May 2019; Accepted: 25 July 2019; Published: 27 July 2019

**Abstract:** The response ranges of three principal mechanical parameters were measured following cyclic compressive loading of three types of concrete specimen to a pre-defined number of cycles. Thus, compressive strength, compressive modulus of elasticity, and maximum compressive strain were studied in (i) plain, (ii) steel-fiber-reinforced, and (iii) polypropylene-fiber-reinforced high-performance concrete specimens. A specific procedure is presented for evaluating the residual values of the three mechanical parameters. The results revealed no significant variation in the mechanical properties of the concrete mixtures within the test range, and slight improvements in the mechanical responses were, in some cases, detected. In contrast, the scatter of the mechanical parameters significantly increased with the number of cycles. In addition, all the specimens were scanned by means of high resolution computed tomography, in order to visualize the microstructure and the internal damage (i.e., internal micro cracks). Consistent with the test results, the images revealed no observable internal damage caused by the cyclic loading.

**Keywords:** fatigue; high performance concrete; fibre-reinforced high performance concrete; compressive stress; compressive modulus of elasticity; maximum compressive strain

### **1. Introduction**

Progressive improvement of the compressive properties of concrete have led to the development of more and more slender concrete structures worldwide, resulting in a progressive reduction of component weight. In consequence, cyclic loads due to variable loading (transient loads and wind, among others) have to be closely studied as tolerance margins are reduced. In some cases, fatigue loads are crucial and the design of the structural element, as in the case of concrete wind turbine towers, depends on fatigue load levels.

Fatigue in concrete can be defined as a process of mechanical degradation leading to failure. Due to cyclic loads, crack initiation and propagation within the specimen occur, until final failure. However, during the application of the variable loads, a modification of the mechanical parameters of the concrete occurs, progressively varying the structural responses of an element.

The classic way to address the problem of fatigue in concrete is to determine the number of cycles (N) that a concrete element can support under given loading conditions. "N" is also called "fatigue life." A concrete element under cyclic loads collapses after a sufficient number of cycles, even if the maximum applied stress is less than its compressive strength.

The value of N depends on both the maximum and the minimum stress applied. The most common way to represent this relationship is through S–N curves. Fatigue strength (S) is defined as the fraction of the static strength that can be supported repeatedly over a number of cycles. In general, there is a huge scatter in the results of apparently identical specimens.

Traditionally, fatigue analysis has focused on the study of fatigue life, i.e., the number of cycles that the specimen can withstand before collapse [1–10], and almost no attention has been paid to how the mechanical parameters vary with the number of cycles [11–14].

Nevertheless, those variations represent an interesting approach to fatigue analysis, as most structures are subjected to a combination of cyclic loads, resulting in mechanical degradation, and extreme static loading. A proper design should guarantee the safety of the structure, taking into account the possible reduction of the mechanical capacity of the structure due to cyclic loads.

The addition of fibers inside the concrete mass provides a better behavior of concrete under fatigue loads. Fibers bridge the cracks, resulting in a significant increase of the fatigue life. Much research has been carried out in this field [15–21], and in all cases, the results have shown a relevant improvement of the fatigue response of the concrete elements.

It is commonly accepted that a fatigue test consists of applying a cyclic load until failure, with the condition that the maximum load is always below the strength of the concrete element. However, cyclic loads applied to concrete elements will result in a progressive deterioration of the microstructure, with the initiation and propagation of microcracks. This fuzzy damage, distributed over the whole specimen, is expected to result in a progressive variation of the macroscopic response, i.e., the mechanical parameters of the concrete. At the point in time when the specimen collapses during the fatigue test, it may be understood that the residual strength of the concrete specimen is exactly the maximum load that is applied. In consequence, a progressive reduction of the strength of the concrete occurs during the cyclic test.

It must be taken into account that real concrete structures are not subjected only to cyclic loading until failure. In fact, they are usually subjected to cyclic loading, and occasionally to extreme loading episodes. Cyclic loading does not usually causes a concrete structure to collapse, but it causes a variation in its mechanical parameters, altering its behavior under extreme loading episodes.

Concrete towers for wind turbines are a good example of this fact, since they are subjected to a "permanent" cyclic loading and eventual extreme loading events. Considering this fact, Urban et al. [22] studied the variation in the compressive modulus of elasticity of concrete towers for wind turbines caused by cyclic loading, and its consequence in terms of variation in the structural response to static and dynamic loading.

This phenomenon could also affect post-tensioned concrete beams. Variations in compressive strength, compressive modulus of elasticity, and maximum compressive strain due to cyclic loading alter the structural response, especially under static loading (instantaneous elastic deflection, deferred deflection, etc.). The dynamic response of the structural element is also modified (natural frequency, maximum vertical acceleration, impact factor, etc.).

Damage caused by cyclic loads could leave a trace in the interior of the concrete specimen, such as a crack pattern, and could therefore be studied using computed tomography (CT) technology.

The CT scan is a non-destructive technique to visualize the microstructure of materials based on X-ray properties. This technology is able to define the density of each specimen voxel (volumetric pixel) by assigning a shade of gray according to voxel density. Light shades of grey correspond to high densities, whereas dark shades of grey correspond to low densities. In recent years, many authors have conducted research on concrete microstructures with this technique, and many have, in particular, focused on the fiber orientation and fiber distribution inside concrete matrices [15,21,22]. Some other research works have focused on the spatial distributions of concrete voids and their effects on macroscopic properties [23–28]. A complete, state-of-the-art work on the use of computed tomography to explore the microstructure of materials in civil and mechanical engineering is presented in Vicente et al. [29,30].

This paper focused on study of the variation of the mechanical parameters of three concrete mixtures with a number of cycles: plain, steel-fiber-reinforced, and polypropylene-fiber-reinforced high-performance concrete (HPC, SFHPC, and PFHPC, respectively). In this case, the mechanical parameters under study were compressive strength (*fc*), compressive modulus of elasticity (*Ec*), and maximum compressive strain (ε*c*,*max*). The specimens were first subjected to cyclic compression loads over five complete cycles: 0; 2000; 20,000; 200,000; and 2,000,000. The specimens were then subjected to a static compression test up to failure, and the abovementioned mechanical parameters were tested in each specimen.

This paper shows an experimental procedure by which to evaluate the variation of the main mechanical parameters due to cyclic loads, and also the results obtained in the three different mixtures with the number of cycles.

Finally, having completed the compressive test, the specimens were scanned and the internal failure mechanisms were studied, using the information provided by the CT scan to gain insight into the resistance mechanisms of the fibers and their influence on the macroscopic response of the concrete specimen.

The structure of this paper is as follows: The experimental program is presented in Section 2, the experimental results are described and discussed in Section 3, the resistance mechanisms are described in Section 4, and, finally, the conclusions are presented in Section 5.

### **2. Experimental Program**

### *2.1. Materials*

A total of three high-strength concrete mixtures were cast. They all shared the same concrete matrix, and the difference was that one of them had no fibers, while the other two had either steel or polypropylene fibers. The mixtures were identified as plain high-performance concrete (HPC), steel-fiber-reinforced high-performance concrete (SFHPC), and polypropylene-fiber-reinforced high-performance concrete (PPFHPC).

A total of 40 test specimens of each mixture were cast in the form of cylinders with a diameter of 100 mm and a height of 200 mm. Table 1 shows the mixtures that were used.

**Table 1.** Concrete mixture.

HPC is high-performance concrete; SFHPC is steel-fiber-reinforced high-performance concrete and PPFHPC is polypropylene-fiber-reinforced high-performance concrete.

The SFHPC contained a volume of 78.5 kg/m3 of Dramix 3D 45/50BL hooked-ended steel fibers (BEKAERT, Kortrijk, Belgium), each of 50 mm in length, with a diameter of 1.05 mm, giving an aspect ratio of 45, a fiber tensile strength of 1115 MPa, and a Young's modulus of 200 GPa. In the case of PPFHPC, an amount of 9.1 kg/m3 of monofilament polypropylene fibers Masterfiber 249 (BASF, Ludwigshafen am Rhein, Germany) was used. These fibers were 48 mm in length with a diameter of 0.85 mm, resulting in an aspect ratio of 56.5. The tensile strength was 400 MPa and the Young's modulus was 4.7 GPa. Both fibers were quite similar in terms of their geometry and the number of fibers inside the concrete mixture. The main difference was the structural behavior and overall stiffness.

MasterRoc MS 685 (BASF, Ludwigshafen am Rhein, Germany) nanosilica and a Glenium 52 (BASF, Ludwigshafen am Rhein, Germany) superplasticizer were considered. Siliceous aggregate was used

for both fine aggregate and coarse aggregate, with a nominal maximum aggregate size of 4 mm for fine aggregate and 12 mm for coarse aggregate.

Mixing was done in a rotary mixer and the fibers were gradually sprinkled into the drum by hand. The specimens were cured for 180 days in a curing room at a constant relative humidity of 100% and an ambient temperature of 20 ◦C. The specimens were then removed from the curing room and held under laboratory conditions until testing. All the specimens were at least 300 days old when the test campaign began. Thus, the possible strength increase during the fatigue test was minimized.

### *2.2. Testing Campaign*

As explained before, a total of 40 cylinders of each mixture were cast and divided into two series of 20 cylinders each. The two series were labeled as L-Series and H-Series, as explained later.

Additionally, three cylinders, 150 mm in diameter and 300 mm in height, were cast to define the concrete quality. These cylinders were kept in the curing room with the rest of the specimens until day 28, when they were tested under static compression. The tests were performed according to the European Standards [31,32].

The average compressive strength, *fcm*, was 75.95 MPa and the characteristic compressive strength, *fck*, was 73.0 MPa. According to Eurocode 2 [33], the strength class was therefore C70/85.

The abovementioned 40 cylinders, 100 mm in diameter and 200 mm in height, were tested at an age of 180 days, as follows. Four specimens of each series, i.e., eight cylinders, were tested under static compression until failure just before starting the cyclic testing campaign, in order to define the real loads to be applied during the cyclic tests to the rest of the specimens. Tables 2 and 3 show the average values of the main mechanical parameters of the concrete for each series. The value in brackets represents the standard deviation.

**Table 2.** Mechanical parameters of the concrete before starting the cyclic tests. L-Series.

**Table 3.** Mechanical parameters of the concrete before starting the cyclic test. H-Series.

The compressive modulus of elasticity was obtained following the method described in standard EN 12390-13 [31]. The static compression test was then performed, as per standard EN 12390-3 [32], and the compressive strength and maximum compressive strain values were extracted.

The rest of the cylinders, that is, 16 specimens of each series, were tested under cyclic load up to a pre-defined number of cycles. The main parameters of the cyclic tests were:

A third set of four specimens from each series was tested up to 200,000 cycles, and the last set of four specimens, up to 2,000,000 cycles.

• Test frequency. In all cases, the test frequency was 6 Hz.

According to Model Code 2010 [34], the expected fatigue life for the H-Series was 1.5 × 103 cycles, while the expected fatigue life for the L-Series was 1.9 × 108 cycles. These values were obtained considering the characteristic compressive strength at 28 days, i.e., 70 MPa, which was the reference value for concrete design. On the contrary, considering the characteristic compressive strength at the age when the cyclic testing was carried out, the expected fatigue life for the H-Series was 1.8 × 108 cycles, while the expected fatigue life for the L-Series was 1.2 × 1011 cycles. From 28 days to the date when the cyclic testing began, concrete compression had increased around 30%, but the theoretical fatigue life had increased around 100,000 times for the case of the H-Series and around 1000 times for the case of the L-Series.

None of the specimens broke during the cyclic testing. Once the cylinders reached the pre-defined number of cycles, they were tested under static loading, first to obtain the compressive modulus of elasticity, and then the compressive strength and the maximum compressive strain. Figure 1 shows a scheme of the test procedure. The longitudinal axis represents the time.

**Figure 1.** Testing procedure.

All the specimens were wrapped in polyvinyl chloride (PVC) cling film before the static post-cyclic tests, to prevent defragmentation once failure occurred. Moreover, an end-of-test condition was programmed into the test machine: the test ended when the vertical load dropped by more than 20% of the maximum load. In this case, the end-of-test conditions, in combination with the PVC cling film, meant that all of the specimen fragments were easily locatable.

### *2.3. CT Scanning*

Having completed the mechanical tests, all the specimens were scanned in order to visualize the microstructure of the concrete. CT scan technology generated an image for close up examination of the fiber locations and orientations in each specimen and the internal microcracks, so that the resistance mechanisms, i.e., the mechanisms used by the specimen to withstand the compressive load, could be observed.

In this research, the cubic specimens were CT scanned using a GE Phoenix v|tome|x device (General Electric, Boston, MA, USA) equipped with a 300 kV/500 W tube. The intensity-controlled X-ray source emits a cone ray that is received by an array of detectors, which measures the loss of X-ray intensity, depending on the density of the matter along its path. Using a post-processing software package, a total of 1334 2D slices with pixel sizes of 2048 × 2048 (Figure 2) were obtained throughout the

height of the specimen from the CT scan data. In this case, the horizontal resolution was approximately 75 × 75 μm2 and the vertical distance between the slices was around 75 μm. A total of 2667 images were obtained from each cylinder. After that, a 3D image of the whole specimen was generated using all the above mentioned 2D images. The post-processing software assigns a grey value to each voxel, from 0 (which belongs to the least dense voxel, i.e., voids and cracks) to 255 (which belongs to the densest voxel, i.e., steel). The scanning process results in a file that includes the X, Y, and Z Cartesian coordinates of the voxel center of gravity and an integer number, from 0 to 255, with regards to its density. The total number of voxels in a specimen was around 4.8 × 109.

**Figure 2.** Slices belonging to different mixtures. From left to right, HPC, SFHPC, and PPFHPC.

### **3. Experimental Results**

The three main mechanical parameters obtained from the static post-cyclic tests were the compressive strength (*fc*), the compressive modulus of elasticity (*Ec*), and the maximum compressive strain (ε*c*,*max*) (Figure 3).

**Figure 3.** Definition of the three main mechanical parameters.

It was assumed that the data distribution pattern of the same set of four cylinders (sharing the same stress level, the same concrete mixture, and the same number of cycles) would follow a normal distribution.

Using those hypotheses, the following three main values were obtained: the average value, showing a probability of not being exceeded by 50% (*p* = 0.50); the minimum value, showing a probability of not being exceeded by 5% (*p* = 0.05); and the maximum value, showing a probability of not being exceeded of 95% (*p* = 0.95).

The relative value was defined as the quotient between the value and the average value at 0 cycles, for better visualization of their evolution with the number of cycles. Moreover, three tendency lines were fitted (using the least square method), for the average, maximum, and minimum relative values, respectively.

These data measurements, compressive strength, compressive modulus of elasticity, and maximum compressive strain are discussed below.

### *3.1. Compressive Strength*

Figures 4–6 show the range of compressive strengths by number of cycles for the three mixtures and for both series.

**Figure 4.** Variation of compressive strength by number of cycles. HPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

**Figure 5.** *Cont*.

**Figure 5.** Variation of compressive strength by number of cycles. SFHPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

**Figure 6.** Variation of compressive strength by number of cycles. PPFHPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

Figures 4–6 show some interesting results. First, it should be highlighted that although the expected fatigue life was 1.9 × 108 cycles for the L-Series and 1.5 × 103 cycles for the H-Series, no collapse occurred during the cyclic tests. It was especially noticeable for the H-Series, where the number of cycles was, in all cases, greater than the expected fatigue life. When the expected fatigue life corresponding to the real concrete quality at the beginning of the cyclic tests is considered, the results are more consistent.

The results also showed no significant variation in the average compressive strength of the concrete by the number of cycles. In all cases, the variation was less than 7%. This behavior was observed for all the mixtures and for both series. It means that, in this case, any damage caused by cyclic loading was very small, and no appreciable mechanical consequences were observed.

On the contrary, a relevant increase of the scatter of the results was observed. Specimens not previously subjected to cyclic loads showed very low scatter, while the rest of the specimens, in general, showed a higher scatter. In general, the scatter increased more than two-fold, and this variation is quite similar in all cases.

This phenomenon can be explained as follows. Concrete specimens have internal stress caused by the curing process, shrinkage, etc. Cyclic loads release the internal stresses, resulting in microcracking. This phenomenon has been observed in previous research [35–37]. Depending on the real internal stress field, the releasing process is more or less intense and more or less microcracking occurs, resulting in different macroscopic responses.

In some cases, this internal stress relaxation is not harmful and a greater compressive strength is observed.

This scatter was higher in the case of the mixtures including fiber, which reveals that the fibers introduced additional restrictions to the concrete mixture during the curing process, resulting in more internal stress. Cyclic loads released this stress, and the macroscopic result was a higher scatter.

The mixture including steel fibers showed, in general, higher compressive strength, which denotes that steel fibers help to increase the mechanical capacity of concrete. In contrast, the mixtures without fibers and with polypropylene fibers showed similar results.

#### *3.2. Compressive Modulus of Elasticity*

Figures 7–9 show the results of the variation of the compressive modulus of elasticity with the number of cycles, for the three mixtures and for both series.

**Figure 7.** Variation of the compressive modulus of elasticity by number of cycles. HPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

**Figure 8.** Variation of the compressive modulus of elasticity with the number of cycles. SFHPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

**Figure 9.** Variation of the compressive modulus of elasticity with the number of cycles. PPFHPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

In contrast to the results depicted in Figures 4–6, Figures 7–9 show that there was no significant variation in the average compressive modulus of elasticity of concrete with the number of cycles. The L-Series showed a slight increase of the compressive modulus of elasticity, up to 15–20%. On the other hand, the H-Series showed a flat behavior, with a variation below 5%.

As with compressive strength, the compressive modulus of elasticity was not adversely affected by cyclic loading, and no relevant mechanical consequences were observed.

In a deeper analysis, it may be observed that the variation of the compressive modulus of elasticity did not follow a monotonic tendency. First, an initial slight increase of the compressive modulus of elasticity up to a maximum value was observed, which was located between 20,000 and 200,000 cycles, depending on the mixture and on the series. It then started to decrease.

In this case, an increase of the scatter of the results was observed, although it is not very relevant.

As happened with compressive strength, the internal stress released by the cyclic loads during the first cycles resulted in microcracking. In this case, this stress relaxation resulted in an increase of the compressive modulus of elasticity. Under low and moderate stress levels, which are the ones used to measure the compressive modulus of elasticity, the results show that the presence of fibers had no influence, since no different tendency was observed between the mixtures with and without fibers.

The scatter was also similar in all cases, which denotes that fibers had no relevant influence on the compressive modulus of elasticity, i.e., the evolution of the microcracking and its impact on the compressive modulus of elasticity depends on concrete matrix only. The scatter depends on concrete matrix only, resulting in a similar variation in all cases, regardless of the presence or absence of fibers, and the type of fibers.

Moreover, all the mixtures and both series showed very similar values for the compressive modulus of elasticity (in the surrounding of 40,000 MPa), implying that this parameter was not sensitive to the presence of fibers.

### *3.3. Maximum Compressive Strain*

Figures 10–12 show the results of the variation of the maximum compressive strain with the number of cycles, for the three mixtures and for the two series.

Figures 10–12 show some interesting results, complementary to those extracted from Figures 4–9. First, the results show that, on one side, there was a slight decrease of the average maximum compressive strain of concrete by number of cycles in the case of the L-Series, with a maximum variation of around 18%. On the other side, there was a slight increase of this parameter in the case of the H-Series, with a maximum variation of around 20%. However, this variation was not very relevant. As with compressive strength and the compressive modulus of elasticity, any damage caused by the cyclic loading was very slight, with no observable mechanical distress.

In a deeper analysis, it may be observed that there was an initial slight decrease of the maximum compressive strain up to a minimum value, located between 20,000 and 200,000 cycles depending on the mixture and on the series, after which it started to increase. This expected tendency was quite the opposite to the compressive modulus of elasticity.

The mixtures, in general, showed a very linear behavior up to their compressive strength. After this point, the plain concrete sample shattered and the fiber-reinforced concrete underwent dramatic softening. In this situation, and assuming that the compressive strength remained almost constant, the higher the compressive modulus of elasticity, the weaker the maximum compressive strain (see Figure 3).

In this case, increased, although not very relevant, scatter was observed.

As occurred with the previous parameters, the relaxation of the internal stress was caused by microcracking due to the cyclic loading. In this case, this stress relaxation resulted in a decrease of the maximum compressive strain.

The scatter was similar in all mixtures, meaning that the fibers had no relevant influence on the maximum compressive strain values. On the contrary, a few added differences between the maximum compressive strains were observed; the average values of that parameter varied between 0.0025 and 0.0035, showing no clear observable tendency.

**Figure 10.** Range of maximum compressive strain by number of cycles. HPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

**Figure 11.** Range of maximum compressive strain by number of cycles. SFHPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

**Figure 12.** Range of the maximum compressive strain by number of cycles. PPFHPC (**a**) L-Series. Raw values. (**b**) H-Series. Raw values. (**c**) L-Series. Maximum, minimum, and average relative values, and tendency lines. (**d**) H-Series. Maximum, minimum, and average relative values, and tendency lines.

### **4. Resistance Mechanisms**

The results described above revealed that the fibers had an influence on the fatigue behavior of the specimens, which seemed to be related to the microstructure of the mixture. In this section, the resistance mechanisms of the specimens, i.e., the mechanisms used by the specimen to withstand the compressive load, were studied.

Using CT scan technology, it was possible to observe the microstructure and the failure mechanisms within the concrete, in order to extract relevant information for a better understanding of the macroscopic response of the specimens. As explained before, the grey scale of the slides obtained through the scanning process ranged from 0 (which belongs to the least dense voxel, i.e., voids and cracks) to 255 (which belongs to the densest voxel, i.e., steel).

The static test was carried out under displacement control. During testing, the failure criterion was fixed at a 5% reduction of the load, in order to prevent shattering of the cylinders. In addition, the specimens were wrapped in plastic film, to prevent defragmentation. Consequently, only small cracks were observed in the broken cylinders.

The cracks in the HPF were mainly close to the lateral border of the cylinder. They tended to be long but few in number, and they passed through the aggregates. Only a few cracks were found in the internal cores of the specimen. This phenomenon was observed in all the specimens, regardless of the number of cycles previously applied (i.e., the damage level) and the stress level during the cyclic tests (Figures 13 and 14).

**Figure 13.** Examples of cracking in HPC specimens. L-Series under the five different sets of cycles: (**a**) 0 cycles; (**b**) 2000 cycles; (**c**) 20,000 cycles; (**d**) 200,000 cycles; (**e**) 2,000,000 cycles.

**Figure 14.** Examples of cracking in HPC specimens. H-Series the five different sets of cycles: (**a**) 0 cycles: (**b**) 2000 cycles; (**c**) 20,000 cycles; (**d**) 200,000 cycles; (**e**) 2,000,000 cycles.

(**d**) (**e**)

Figures 13 and 14 reveal that, in general, the cyclic tests resulted in low damage levels. The crack surfaces tended to be parallel to the compressive load, showing no significant influence on the behavior of the concrete under static loads.

This finding agrees with the macroscopic behavior of the specimens. The three main mechanical parameters analyzed in this paper showed a slight or null variation with the number of cycles, and no relevant internal damage was observed in the specimens.

In the case of the SFHPF, the cracks showed the same pattern as in the previous case. Similarly to the previous cases, the cracks, which tended to be long but few, were mainly close to the lateral borders of the cylinder. In this case, no fibers were detected in the internal core of the specimen. Again, no differences in the crack patterns of the specimens related to the number of cycles or the stress level during the cyclic tests were observed (Figures 15 and 16).

In this case, as steel fibers are very stiff, the damage due to cyclic loading was mainly concentrated in the lateral border region, where the fibers could not bridge the cracks, rather than within the interior. Again, the crack surfaces tended to be parallel to the compressive load and had no influence on the behavior of the concrete under static loads.

Again, an agreement between the analysis of the CT scan images and the macroscopic behavior of the specimens can be observed. On one hand, the three main mechanical parameters analyzed showed a slight or null variation with the number of cycles, and, on the other hand, no relevant internal damage was observed in the specimens.

In the case of PPFHPC, the cracks showed a similar pattern to the two previous cases, although the cracks were not only concentrated close to the lateral border, but some of them penetrated towards the internal core of the specimens. Again, the cracks tended to be lengthy, and more numerous cracks could be observed. There were no significant differences between the specimens, and neither the number of cycles nor the stress level appeared to have influenced the crack pattern (Figures 17 and 18).

**Figure 15.** Examples of cracking in specimens of SFHPC. L-Series under the five different sets of cycles: (**a**) 0 cycles; (**b**) 2000 cycles; (**c**) 20,000 cycles; (**d**) 200,000 cycles; (**e**) 2,000,000 cycles.

**Figure 16.** Examples of cracking in specimens of SFHPC. H-Series under the five different sets of cycles: (**a**) 0 cycles; (**b**) 2000 cycles; (**c**) 20,000 cycles; (**d**) 200,000 cycles; (**e**) 2,000,000 cycles.

**Figure 17.** Examples of cracking in specimens of PPFHPC. L-Series under the five different sets of cycles: (**a**) 0 cycles; (**b**) 2000 cycles; (**c**) 20,000 cycles; (**d**) 200,000 cycles; (**e**) 2,000,000 cycles.

**Figure 18.** Examples of cracking in specimens of PPFHPC. H-Series under the five different sets of cycles: (**a**) 0 cycles; (**b**) 2000 cycles; (**c**) 20,000 cycles; (**d**) 200,000 cycles; (**e**) 2,000,000 cycles.

In this case, the polypropylene fibers had a very low stiffness, and the cyclic loads resulted in internal damage. Consequently, some cracks were observed within the specimens. As in the previous cases, the crack surfaces were mainly vertical, i.e., parallel to the compressive loads, and they had no influence on macroscopic behavior under compressive loads.

It can be concluded that in all mixtures, cracks were mostly placed at the surface. This fact seems to have been due to the concrete matrix. The scale effect on concrete specimens is well known, and many examples of this phenomenon can be observed in concrete elements [38–40]. In this case, the scale effect conducted to a lower amount of coarse aggregates and fibers at the lateral border [41]. This would be the reason why the cracks mainly appeared at the surface.

As in the previous cases, the three main mechanical parameters did not show relevant variation with the number of cycles, which agrees with the internal damage observed.

For the three mechanical parameters studied in this paper, the results revealed that the scatter is a parameter clearly dependent on the number of cycles. This fact can be observed in all cases, i.e., in all the mixtures and in both series.

### **5. Conclusions**

In this paper, the range of values of three mechanical parameters of concrete, following the application of five series of cyclic compressive loading, was studied. Three concrete mixtures—plain, steel-fiber-reinforced, and polypropylene-fiber-reinforced high-performance concrete (HPC, SFHPC, and PPFHPC)—were studied. The specimens were subjected to either high stress or low stress levels, and to five different series of cycles: 0, 2000; 20,000; 200,000, and 2,000,000 cycles. All the specimens survived the cyclic test. They were then subjected to a static compression test up to failure, and the compression strength, the compressive modulus of elasticity, and the maximum compressive strain were all measured.

According to Model Code 2010 [34], the specimens were subjected to relevant fatigue damage, especially in the case of high stress level series.

In addition, before starting the cyclic tests and after the static tests, all the specimens were scanned, in order to visualize the internal damage caused by the cyclic loads. A high-resolution CT scan device was used to do so.

The mechanical results showed no relevant damage caused by the cyclic loading. Moreover, the mechanical parameters of the concrete mixtures showed no significant reduction, and even a slight improvement in some cases.

Regarding the compressive strength, the average value of this mechanical parameter did not significantly vary with the number of cycles, and, in some cases, a slight increase was observed. In addition, the scatter showed a relevant increase with the damage levels, equitable with the number of cycles.

The compressive modulus of elasticity showed a similar behavior. The L-Series showed a slight increase of the compressive modulus of elasticity, while the H-Series showed a flat behavior. Again, the scatter of the compressive modulus of elasticity significantly increased with the number of cycles.

Regarding the maximum compressive strain, the average value showed no clear tendency. In case of the L-Series, an overall decrease with the number of cycles was observed, while in the case of the H-Series, an overall increase with the number of cycles occurred. Again, the scatter showed a progressive increase with the number of cycles.

It can be noticed that for the three mechanical parameters studied in this paper, the results revealed that the scatter is a parameter clearly dependent on the number of cycles. This fact was observed in all cases, i.e., in all the mixtures and in both series.

Moreover, the variation of the mechanical parameters with the damage hardly appeared to be related to the stress level and to the concrete mixture.

The CT scan images revealed no observable internal damage (i.e., internal microcracking) due to cyclic loading, only an observably small number of long cracks. They mainly propagated close to the lateral area of the cylinders and appeared to have been caused by the static testing and not caused by the previous cyclic test, because no differences were observed between the specimens with and without cyclic damage (i.e., the ones subjected to 0 cycles).

Internal cracks were only observed for the PPFHPC specimens, which can be explained by the lower stiffness of the polypropylene fibers.

It was concluded that, contrary to the predictions of the Model Code [34], the cyclic loads tested in this research caused little or no internal damage (i.e., internal micro-cracking), and no degradation of the microstructure was observed. Both the L-Series and H-Series were less aggressive to the concrete specimens, and even at 2,000,000 cycles no relevant internal damage was observed and the variation of the mechanical properties of the concrete with the number of cycles was small. It can be highlighted that, in the case of the H-Series, the Model Code [34] predicted an expected fatigue life of 1.5 × 103 cycles, which means that all the numbers of cycles tested, i.e., 2000, 20,000, 200,000, and 2,000,000 cycles, were beyond their fatigue life. The truth is that none of the specimens collapsed during the cyclic test and their residual mechanical properties did not vary significantly with respect to the ones showed by the specimens not previously subjected to cyclic loads.

**Author Contributions:** J.M. and L.G. developed and carried out the laboratory testing, J.M. also worked on the analysis of the CT scan images, D.C.G. and M.A.V. analyzed the data and wrote the paper.

**Funding:** This research was funded by the Ministerio de Ciencia e Innovación (Spain), grant number BIA2004-08033.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

1. Hsu, T.T.C. Fatigue of plain concrete. *ACI J.* **1981**, *78*, 292–305.

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Fatigue in Concrete under Low-Cycle Tensile Loading Using a Pressure-Tension Apparatus**

### **Sayed M. Soleimani 1,\*, Andrew J. Boyd 2, Andrew J.K. Komar 3 and Sajjad S. Roudsari 4**

Received: 15 July 2019; Accepted: 6 August 2019; Published: 7 August 2019

**Abstract:** Fatigue due to low-cycle tensile loading in plain concrete was examined under different conditions using the pressure-tension apparatus. A total of 22 wet or dry standard concrete cylinders (100 mm × 200 mm) were tested. By definition, low-cycle loading refers to the concept of multiple load cycles applied at high stress levels (i.e., a concrete structure subjected to seismic loading). Results suggest that concrete samples subjected to low-cycle tensile loading will fail after a relatively low number of cycles of loading and at a lower magnitude of stress compared to the maximum value applied during cyclic loading. Furthermore, non-destructive testing was employed in order to ascertain the extent of progressive damage inflicted by tensile loading in concrete specimens. It was found that ultrasonic pulse velocity is a viable technique for evaluating the damage consequential of loads applied to concrete, including that resultant from low levels of tensile stress (i.e., as low as 10% of its maximum tensile capacity). Additionally, finite element analysis was performed on a modeled version of the pressure-tension apparatus with a sample of concrete, which has yielded similar results to the experimental work.

**Keywords:** concrete; fatigue; tension; pressure-tension apparatus; nondestructive testing; ultrasonic pulse velocity; ABAQUS FEA

### **1. Introduction**

The pressure tension testing method was originally developed by the Building Research Council as a new method of examining anisotropic loading conditions on materials [1]. The pressure tension apparatus uses standard 100 mm × 200 mm concrete cylinders as samples. Concrete core samples can also be used. The size of the sample allows this test method to be adapted to common concrete testing procedures used in industry. The test uses a pressurized gas equally applied along the curved surface of the cylinder, with rubber O-rings at either end of the specimen to hold the gas between the testing chamber and the concrete cylindrical sample surface, as shown in Figure 1. The two flat ends of the cylinder are left open to the atmosphere, which causes a net induced tension field to arise within the concrete sample parallel to the longitudinal axis of the cylinder. Komar [2] devised and implemented a novel instrumentation system that allowed for fully automated control of the loading rate and variety of loading conditions, including ramped loading at different rates, cyclic loading, constant loading rate (such as creep), and controlled unloading. This study uses the method of cyclic loading.

**Figure 1.** (**a**) Photograph of pressure-tension testing machine; (**b**) Exploded view of the pressure-tension apparatus with concrete cylinder in machine; (**c**) Cross section of pressure-tension apparatus and concrete cylinder [2].

In this test method, a tension field will be produced from a compressive load. It can be understood as a net effective stress which is consistent with principles first developed for soil mechanics [3]. Concrete is a two-phase material with a solid phase consisting of the hydrated cementitious matrix and the aggregates, and a liquid phase consisting of the pore water. The different reactions of the two phases to a biaxial stress gives rise to the pressure tension effect: the liquid phase reacts in a hydrostatic manner, in contrast to the solid phase which reacts in the directions of the applied load. All the stresses in the plane of gas loading cancel each other out, with a net tension field arising along the axis of the cylinder [4], which is shown in Figure 2. The pressure tension apparatus uniformly increases the gas pressure applied to the concrete cylinder until the point where the solid phase of the specimen can no longer remain together, at which point the concrete cylinder breaks at the weakest plane along its length. The gas pressure at the moment of failure is taken as the ultimate tensile strength of the concrete.

**Figure 2.** Pressure tension effect [2].

Rashidi et al. [5] conducted an experimental study to evaluate the strength of concrete subjected to Indirect Diametrical Tensile (IDT), Unconfined Compressive Strength (UCS) and submaximal modulus testing methods. Their experimental design involved the application of a specific strain rate of 1 mm/min on concrete specimens. IDT employs a controlled loading condition to split the cylinder into two half-cylinders. In contrast, the present experimental design uses gas pressure to achieve the same mode of failure.

Ultrasonic Pulse Velocity (UPV) provides information about the evolution of the pore space of concrete, for both hydration and changes due to durability issues. Komar [2] used UPV to track the hydration process during curing; they found UPV could distinguish between different pore structures at early ages and could track the evolution of the pore structures over the curing period. UPV can be divided into three categories: Direct (sensors opposite to each other), indirect (adjacent sensors), and semi-direct (sensors at right angles) transmission. Ultrasonic waves have been used to predict and evaluate concrete strength and its properties in several studies [6]. However, this method can also be used to detect the internal deficiencies of concrete such as cracks. In this experimental work, the UPV method is used to determine the presence of cracks in concrete after a low magnitude of tensile stress is applied. The UPV test is a measurement of the continuity of the solid part of the concrete through which the stress waves propagate, and it is well-suited for evaluating any changes to this solid structure due to ongoing deterioration. The test apparatus produces and applies a pulse into the concrete by means of a pulse transducer and receiver and encompasses the ability to accurately measure the amount of time the pulse takes to pass through the concrete.

### **2. Materials and Methods**

A total of 22 standard concrete cylinders (100 mm × 200 mm) were cast and tested. The concrete used in the cylinders was made using the mix design shown in Table 1. The specimens were molded and compacted based on ASTM C31 [7]. After 24 h of curing in their molds, the cylinders were removed and stored in ambient laboratory conditions for over 2 years.

**Table 1.** Mix design of the concrete specimens for this study (water-to-cement ratio = 0.45).

In a pressure-tension test, a destructive test method that specifically evaluates the tensile capacity of concrete, a cylindrical steel jacket is used to hold a concrete cylinder serving as the specimen. A rubber O-ring is used at each end to keep an airtight seal and prevent possible gas leakage. With the ends then left free, gas pressure is incrementally applied to the curved surface of the concrete cylinder until the specimen ultimately fractures across a single plane perpendicular to the axis of the concrete specimen. The value of gas pressure at fracture thus corresponds to the maximum tensile capacity of the concrete. Compressed air is used as the loading medium for all specimens in this experimental work. The air was pressurized using a portable air compressor with a maximum available pressure of about 17 MPa, which was stored in a pressurized gas vessel attached to the compressor and the testing machine. A variable check valve was used to keep the pressure head constant in order to minimize inconsistency in the test caused by pressure variance in the storage tank. A needle valve was used as the controller for the flow rate of the compressed gas into the testing chamber. The valve was operated by a mechanical actuator which was controlled using custom software and monitored using a digital acquisition device (DAQ). Measurement of pressure within the test chamber was obtained using a digital pressure transducer directly adjacent to the chamber, which was used as a control input in the software. The pressure transducer had a measured error of ±0.05 MPa. The DAQ had a sensing

resolution of approximately 0.1 s [2,6]. All pressure-time plots were recorded and stored, and the data point immediately preceding failure was used as the failure load.

The implemented control algorithm is a modified proportional integral derivative function which automatically varies the valve position based on the detected error between the measured pressure and the set point. The set point is an arbitrary time-dependent function, and all tests carried out in this study used a uniformly increasing pressure-time signal with a rate of about 0.02 to 0.03 MPa/second, comparable to loading rates in both ASTM C 496 and ASTM C 33 [6]. A higher rate of about 0.10 to 0.15 MPa/second was used during the cyclic loading of the specimens to expedite the process.

Rubber O-rings with a diameter of 10 mm were used to seal the interface between the test chamber and the concrete sample (Figure 1). One or two layers of polyvinylchloride tape was applied on both ends of the cylinder to provide a better seal between the O-rings and the cylinder, which occasionally had voids or deterioration defects that could cause leakage around the O-rings. The O-rings were torqued with a series of six lug nuts bearing on a steel plate in a manner that applied equal pressure on them and ensured a tight seal for the loading gas.

Pressure tension testing is affected by the moisture content of the specimens; samples with a higher water content fail at a higher tensile resistance as compared to dry specimens [2]. For dry samples, the failure mode in pressure tension testing changes from a single crack to a volumetric failure at a lower applied tensile stress [2,8].

Fatigue loading can be categorized into low-cycle and high-cycle loading. Low-cycle loading encompasses the introduction of a few high-stress load cycles such as the application of the earthquake loads on a structure. Conversely high-cyclic loading is categorized by the introduction of a large number of low-stress load cycles, such as the live load applications on the bridges and airport pavements.

The flexural test method is the most commonly used method for fatigue testing. Comparatively, compressive fatigue tests have been studied to a lesser extent. After the emergence of nonlinear fracture mechanics toward evaluating the performance of concrete, some researchers have shown interest in the fatigue characteristics of concrete in tension [9–11]. Nevertheless, there are very limited studies available on the behavior of plain concrete under direct tension fatigue loading [12–15]. The challenges associated with direct tensile testing on concrete is also a primary reason for limited and often conflicting data availability.

Furthermore, non-destructive testing (NDT) was employed to evaluate the progressive damage inflicted by tensile loading on concrete cylinders using UPV apparatus. A Proceq TICO model of automatic UPV was used for this experimental study. Three measurements of transit times along the axis of the cylinder were recorded, and the average of these three values was noted. Length measurements from the cylinders (i.e., height of the concrete cylinder specimens) were used to convert the travel times into the UPV values reported. A commercially available water-based lubricant was used to ensure a good connection between the transmitter/receiver and the greatly variable surface of the concrete samples.

In this study, two piezoelectric sensors are placed at opposite ends of the test specimen (i.e., direct method). Electronic pulses are generated by one sensor, while the other sensor measures the time taken by the pulse to transmit through the concrete. Knowing the distance traveled (i.e., the height of the concrete cylinder specimen in this case), transmission velocity will be calculated based on the resultant value and the condition of the concrete may be determined.

### **3. Experimental Results and Discussion**

### *3.1. Tensile Strengths (No Cyclic Loads)*

The first trial of samples consisted of six standard concrete cylinders loaded up to failure (Figure 3). The first two were tested in dry conditions, while the other four were tested in wet conditions (they have been submerged in water for few days before the test). The results of each individual test are presented in Table 2. The statistical analysis exhibits that the average tensile strength of the dry

specimens was lower than that of the wet specimens. In contrast, the coefficient of variation for the wet specimens is much bigger than that of the dry specimens. The recommended coefficient of variation in the ASTM C 496 standard is 14%. Given the limited number of tests conducted in this study, it seems that the pressure tension test on dry specimens will give a conservative and reliable result as compared to the wet specimens. More test results are required to attest this statement.

**Figure 3.** Broken specimen under tension using pressure-tension apparatus: (**a**) Elevation view; (**b**) Cross-sectional view.

**Table 2.** Tensile strength of concrete exposed to ambient conditions for two years.

### *3.2. Tensile Strengths (Cyclic Loads)*

Twelve wet and four dry concrete cylinders (a total of sixteen specimens) have been tested under a cyclic loading condition. Since the tensile strength of the specimen can be varied substantially for the wet specimens as shown in Table 2, finding the fatigue threshold associated with a low-cycle loading protocol can be quite challenging. The results are shown in tabular format for these specimens.

Table 3 shows the cyclic test results for two wet specimens (No. 7 and No. 8). Each one failed during the cyclic loading and under a tensile stress that was below the maximum applied tensile stress in previous cycle(s). Both specimens have not been loaded prior to the cyclic loads. It is worth mentioning that the stress at failure is also less than the average tensile strength of 4.45 MPa (as indicated in Table 2).

**Table 3.** Cyclic load results for specimens 7 and 8.

Specimens No. 9 to No. 14 have been subjected to 100 cycles of tensile loads with a maximum tensile stress of 2.23 MPa to 3.01 MPa. Then, they have been loaded up to the failure point and the tensile strengths have been recorded to see the effect of the cyclic loading on the final tensile strength. The results are tabulated in Table 4. It seems that the tensile strength of concrete will not be affected if the maximum tensile stress in the cyclic loading is below a threshold (i.e., low-cycle loading). This threshold can be more than 62% of the tensile strength (based on the limited number of test results presented in Table 4) but more test results are required for verification. The difference between these specimens and those reported in Table 3 is that they have been subjected to a cyclic loading for 100 cycles and then tested up to the failure point. The specimens in Table 3 failed during the cyclic loading.

**Table 4.** Cyclic load results for specimens 9 to 14.

Specimens No. 15 to No. 18 have been subjected to a cyclic loading with a maximum of 20 cycles. The cyclic loading started at a low level (about 50% of the average tensile strength of the wet specimens as indicated in Table 2) and the maximum load in the next 20 cycles has been increased gradually (by about 5% of the average tensile strength of the wet specimens as indicated in Table 2). The purpose of this test method was to prove that there is a threshold for the cyclic loading at which the fatigue failure will happen at a low-cycle loading. Test results for these specimens are shown in Table 5.

Table 6 shows the test results for specimens No. 19 to No. 22. They have been tested in a dry condition. The same procedure explained for testing specimens No. 15 to No. 18 has been followed for these tests. It is clear that the condition of the concrete cylinders (i.e., wet versus dry) plays an important role in the tensile capacity of the concrete in cyclic loading (Tables 3–6) and non-cyclic loading (Table 2). As an example, the cyclic loading on specimen No. 19 is shown in Figure 4, in which the specimen failed during the sixth cycle.

**Figure 4.** Cyclic loading on specimen No. 19 (failed during the 6th cycle).

**Table 6.** Cyclic load results for specimens 19 to 22.

*3.3. UPV Test Results and Discussions*

The average ultrasonic pulse velocity measurements (the mean value of three measurements) versus the applied tensile stress can be found in Figures 5–7. The UPV measurements took place after the tensile force was removed. For all cases, a general downward trend is apparent from the data, indicating decreased pulse velocities associated with an increasing of the applied tensile stresses. Higher UPV measurements are indicative of an internal structure which is relatively denser (thus allowing for faster transmission of the pulse). A downward trend as seen in Figures 4–6 shows that the application of a tensile stress on a concrete cylinder is affecting the internal structure so as to cause an increase in travel times and a lower UPV measurement.

**Figure 5.** Ultrasonic Pulse Velocity (UPV) (m/s) vs. applied tensile stress (MPa) for Specimens No. 9 to No. 14.

**Figure 6.** UPV (m/s) vs. applied tensile stress (MPa) for Specimens No. 15 to No. 18.

**Figure 7.** UPV (m/s) vs. applied tensile stress (MPa) for Specimens No. 19 to No. 22.

### **4. Finite Element Modeling**

ABAQUS FEA software, a commercially available finite-element analysis program, was used to model the concrete cylinder. The Concrete Damage Plasticity Model (CDPM) was used to define the parameters of the concrete—these parameters are computed using MATLAB toolbox [16].

For concrete modeling, 3D 8-node Linear Isoparametric Elements with reduced integration (C3D8R) have been utilized. In order to model the gas pressure, two parts are defined; first a solid element with C3D8R features; and second the Eulerian type with EC3D8R (an 8-node linear Eulerian brick, reduced integration, hourglass control). It should be noted that the solid part covers both Eulerian and concrete cylinder. Figure 8 displays the geometry of Finite Element Modeling (FEM).

**Figure 8.** Geometrical feature of the Finite Element Modeling (FEM): (**a**) Solid volume representation; (**b**) Outline representation.

The material properties for gas pressure (Eulerian Type) is considered to be Equation of State (EOS) with the type of Ideal Gas. The gas constant and ambient pressure are 278 × 108 mJ/(tonne K) and 0.101325 MPa, respectively. Also, Non-Linear Dynamic Explicit analysis is used for simulation. The contact behavior has both Tangential (frictionless) and Normal (hard contact) behaviors. The Discrete Field was converted to Volume Fraction Tools to make Eulerian part as pressure. The initial gas pressure is applied as Predefined Stress and gravity load. Also, the cyclic pressure is assigned based on the amplitude. The amplitude is displayed in Figure 9 and the stress distribution in concrete cylinder is shown in Figure 10.

**Figure 10.** Stress distribution in the FEM: (**a**) Longitudinal section; (**b**) Volumetric view.

When cyclic loading was applied at low pressure, the concrete sample did not fail in tension. Then, the maximum pressure in the cyclic loading was increased step-by-step in the FEM sample until the observation of fatigue failure after few cycles to simulate the low-cycle fatigue failure. Interestingly, the FEM sample failed after 13 cycles that is very similar to the experimental result of sample 17 (Table 5). Also, the sample in FEM failed under a pressure that was lower than the maximum applied pressure in previous cycles. FEM and experimental results of specimen No. 17 are compared in Figure 11.

**Figure 11.** Comparison between FEM and experimental results for Specimen No. 17.

### **5. Conclusions**

A total of 22 standard concrete cylinders were tested under tension using a pressure-tension apparatus. Based on the limited number of test results reported in this paper, one can conclude that the tensile strength of the dry specimens was lower than that of the wet specimens with a lower coefficient of variation in the test results. Therefore, it will be safer and more reliable to test the specimens in dry conditions.

It was shown that failure in tension due to fatigue can happen as early as the second cycle of loading if the concrete specimen has been loaded up to a high percentage of its tensile capacity (most probably more than 62% of its tensile capacity).

ABAQUS FEA software was used to model the pressure-tension test. It was shown that the failure under low-cycle tensile force can be predicted using appropriate parameters in the software. It was also found that concrete specimens subjected to a low-cycle loading protocol always fail at a lower stress level compared to the maximum value applied in the previous cycles. This was the case for both the experimental results as well as the FEM. With respect to the quantification of the effects of fatigue on the performance of concrete, comparison of the experimental and FEM results illuminates the maximum stress before failure to be 7.7% higher in FEM. Additionally, the stress at failure was 23.0% higher in FEM compared to the experimental result.

NDT was also employed to evaluate the progressive damage inflicted by tensile load in concrete using the UPV apparatus. It was shown that the application of a tensile stress on a concrete affects its internal structure and causes an increase in travel times and a lower UPV measurement. The experimental results presented in this paper prove that the UPV method can be used effectively for the detection of cracks in hardened concrete subject to tensile stresses—as low as about 0.4 MPa.

**Author Contributions:** S.M.S. and A.J.K.K. carried out the tests. S.M.S. and A.J.B. supervised the project. S.M.S. prepared the manuscript and analyzed the data. S.S.R. conducted the finite element analysis. All authors read and approved the final manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Comprehensive Evaluation of Fatigue Performance of Modified Asphalt Mixtures in Di**ff**erent Fatigue Tests**

### **Kun Li 1, Ming Huang 2,\*, Haobai Zhong 3 and Benliang Li 3**

Received: 3 April 2019; Accepted: 28 April 2019; Published: 6 May 2019

**Featured Application: Three commonly used fatigue test methods for asphalt mixtures, namely the four-point bending beam fatigue test, the two-point bending trapezoidal beam fatigue test, and an overlay tester, were selected for a fatigue test of six kinds of mixtures, including newly developed modified asphalt. The research is supposed to bring about a comprehensive evaluation of the fatigue performance of a modified asphalt mixture to provide data support for the design of fatigue in an asphalt mixture in the near future.**

**Abstract:** The four-point bending beam fatigue test (4PB), two-point bending trapezoidal beam fatigue test (Trapezoidal Beam), and Overlay Tester (OT) are used to evaluate the fatigue performance of six kinds of asphalt mixtures that are widely used in engineering, and newly developed ones. The result shows that, in all three kinds of fatigue tests, the fatigue performances of the 6% SBS (styrene-butadiene-styrene block copolymer) modified asphalt mixture is the best, and those of the 10% WPE (waxed polyethylene) + 3% SBS, 4% SBS + 0.4% PA610, and 4% SBS modified asphalt mixture are good. The fatigue performances of the warm modified mixing agent and the base asphalt mixture are the worst. An increase in SBS content can effectively improve the fatigue performance of the asphalt mixture. The fatigue performance of the SBS-modified asphalt mixture can be improved by the addition of WPE and PA610. In different tests, the ranking of fatigue performance of the asphalt mixture is similar, and the specific ranking is slightly different. The three different fatigue tests can be used simultaneously to obtain a more comprehensive and objective evaluation in the R&D process for a new modified asphalt. The three fatigue tests process shows that more precise forming and cutting technology is needed, as the strain range used in the 4PB test is very wide, and the number of samples used in each group is small. The preparation of the Trapezoidal Beam test samples is complex; the amount of test data is huge and has high precision, which is suitable for scientific research instead of a field laboratory, and the strain range of the test is moderate in the three methods. The strain range of the OT test is the narrowest; the test specimen is relatively simple to prepare, and the fatigue performance of a specific modified asphalt mixture can be obtained quickly in a simple laboratory.

**Keywords:** fatigue life; modified asphalt mixture; four-point bending beam fatigue test; two-point trapezoidal beam fatigue test; overlay tester

### **1. Introduction**

Fatigue is a well-known behavior due to the iterative loading and unloading at stress levels that are below the ultimate strength of the asphalt pavement. Thus, the fatigue performance of an asphalt mixture is an important criterion for evaluating the asphalt mixture. Some studies [1–4] have shown that the fatigue performance of a modified asphalt mixture is obviously better than that of a base asphalt mixture due to the addition of an SBS modifier. Some studies [2–7] show that the addition of PE (polyethylene), PA (polyamide), and warm mixing agents will change the fatigue performance of the asphalt mixture. Many scholars [4–12] have chosen different fatigue setups to carry out fatigue tests. However, fatigue test methods have different accuracy and pertinence in different test environments and conditions. It is necessary to compare the fatigue performance of different modified asphalt mixtures after analyzing the principles of different fatigue tests to make up for defects of understanding in design and blindness in the choice of asphalt.

In this study, the following three representative fatigue test methods are selected to test the fatigue life of six kinds of modified asphalt mixtures and base asphalt mixtures: the four-point bending beam fatigue test (4PB), which is an ASTM standard test and widely accepted in China [13]; the two-point trapezoidal beam fatigue test (Trapezoidal Beam), which is widely used in Europe [14]; and Overlay Tester (OT) tension, which is mainly used in Texas, USA [15]. Different test methods can provide a more comprehensive reference for fatigue performance evaluation of different kinds of asphalt mixtures from multiple dimensions, avoiding the evaluation deviation caused by defects in a single fatigue test. At the same time, it is easier to find a suitable fatigue method to evaluate the effects of various modifiers, especially for the newly developed ones, on the fatigue performance of asphalt mixtures, and the advantages and disadvantages of the same asphalt mixtures in different test methods can be analyzed.

### **2. Materials and Experiment Design**

### *2.1. Asphalt and Its Preparation Technology*

Six types of asphalts were selected for the experiment, including base asphalt and modified ones, which cover commonly used types in engineering and newly developed ones. The base asphalt is SK70. The modified asphalt used in the test contains two types of SBS-modified asphalt (SBS 4 and SBS 6), warm mixing agent modified asphalt (WA), waxed PE (waxed polyethylene) and SBS composite modified asphalt (WPES), and SBS + nylon fiber 610 (PA610) modified asphalt (4S4P). The modification scheme of modified asphalt is shown in Table 1, and the asphalt used in the test and its index are shown in Table 2.

**Table 2.** Asphalts of the test and their properties.

### *2.2. Aggregate and Gradation*

Aggregate, gradation, and mixture designs are as follows: coarse aggregate (particle size >2.36 mm) is made of basalt gravel, fine aggregate (particle size <2.36 mm) is made of manufactured sand, and filler is made of mineral powder. The vast majority of mineral fillers are made of limestone mainly because the limestone powder combines with asphalt strongly [16]. The aggregate indexes meet the requirements of JTG F40-2004 "Technical Specification for Asphalt Pavement Construction of Highway" [17]. The asphalt content of the asphalt mixture is 4.8%, and the gradation is AC-13. The design gradation is shown in Table 3, which contains a group of gravity data of the aggregates in different particle sizes.

**Table 3.** Gradation of the asphalt mixture and gravity of aggregates of different particle sizes.

### *2.3. Mix Design*

The optimum asphalt content for the different modified asphalt mixtures has been determined by the Marshall mix design. For consistency with the applications and studies in China [1,4,7,9,18], we chose the Marshall compaction method, T0702-2011, of Chinese specification [13]. The design indexes are shown in Table 4.

**Table 4.** Asphalt design indexes.

### *2.4. Three Fatigue Test Methods*

The 4PB fatigue test derives from the ASTM D7460 [19] standard of the United States. The mixture specimens were prepared in a trabecular (380 mm (long) × 63.5 mm (wide) × 50 mm (height)). The loading waveform is a half sinusoidal wave, the standard loading frequency is 10 Hz, the equivalent fatigue temperature of pavement in China is 15 ◦C, and the 50th loading was used as the initial stiffness modulus. When the normalized stiffness ratio *NM* reaches the maximum value, the loading times *M*, which cause fatigue failure of the specimens, is considered as the fatigue life of the asphalt mixture, in which *NM* is calculated according to the following equation:

$$N\_M = \frac{S\_\mathrm{i} \times N\_\mathrm{i}}{S\_\mathrm{0} \times N\_\mathrm{0}},\tag{1}$$

where *Si* is the modulus of stiffness under the x-th loading, *Ni* are loading times, *S*0 is the initial loading stiffness modulus (the 50th stiffness modulus is defined as the initial loading modulus), and *N*0 are the initial loading times (taken to be 50).

This method was first proposed by Rowe et al. [11] to form the standard ASTM D7460. Compared with the standard method in which the stiffness modulus changes to 50% of the initial stiffness modulus as the end of the test, the fatigue life variation coefficient is smaller, which can reflect the fatigue life of a modified asphalt mixture better [20]. The 10 beam shape samples were tested in each of the asphalt mixes, and the variance of 10% are the valid data.

The fatigue test of Trapezoidal Beam was adopted from the European EN 12697-24 [21] standard. The mixture specimens were prepared to be 25 mm × 25 mm at the top, 25 mm × 56 mm at the bottom, and 250 mm high. The loading waveform is a sinusoidal wave, the standard loading frequency is 25 Hz, and the equivalent fatigue temperature of pavement in China is 15 ◦C. The initial stiffness modulus is the 100th loading, and when the modulus of stiffness decreases to 50% of the initial value, the number of loading times is taken as the fatigue life. The three beam samples were tested in each of the asphalt mixes, and the variance of 20% are the valid data.

The Overlay Tester fatigue test adopts the TEX-248-F [15] standard of Texas, USA. The mixture specimen is a cylinder, which is 150 mm in diameter, 38 mm in height, and 76 mm in width. The loading waveform is a triangular wave, the standard loading period is 0.1 Hz, and the test temperature is 25 ◦C. The judgment for the end of the experiment is that the initial force decreases to 93% of the initial one or the loading times reach 1000. So, the number of loading times taken to reach 93% of the initial force or loading times of 1000 will be taken as the fatigue life of the specimen. The three samples were tested in each of the asphalt mixes, and the final results were averaged and are included in the chart.

Figure 1 shows the three fatigue testing apparatuses and Figure 2 shows the principles of the three fatigue test methods.

**Figure 1.** *Cont.*

**Figure 1.** Three fatigue testing apparatuses: (**a**) four-point beam (4PB) fatigue test, (**b**) two-point trapezoidal beam test (Trapezoidal Beam), and (**c**) Overlay Tester.

**Figure 2.** The loading mode and model diagram of three loading modes of the fatigue test.

### **3. Results and Discussion**

### *3.1. Fatigue Test Results*

The fatigue tests of 4PB and Trapezoidal Beam are under a different strain level, while the OT tests keep the same strain at 300 × 10−6. All tests should be repeated at least four times, and the average values of the three results with the smallest coefficient of variation should be taken as the test results. The results of all the three fatigue tests are shown in Figures 3–5.

**Figure 3.** The fatigue life of different asphalt mixtures of 4PB at varied strain levels.

**Figure 4.** The fatigue life of different asphalt mixtures of Trapezoidal Beam at varied strain levels.

**Figure 5.** The fatigue life of different asphalt mixtures of Overlay Tester (OT) at a strain of 300 × 10−6.

### *3.2. Separate Analysis of the Results*

4PB: The strain range used in the 4PB test is very wide, and number of samples used in each group is small, which requires high precision. Therefore, more precise forming and cutting technology is needed. The fatigue life of the six asphalt mixtures decreases gradually with the increase of strain. The fatigue properties of SK70 and WA are obviously poor, SBS4 is in the middle, and 4S4P, WPES, and SBS6 are the best ones. The specific performance order is slightly different with the change of strain, and WPES performs best at high strain. With the increase of SBS content, the fatigue performance of the asphalt mixture increases continuously. The addition of WPE and PA610 improves the fatigue performance of the modified asphalt mixture in varying degrees.

Trapezoidal Beam: The preparation of test samples is complex and the amount of test data is huge. It has high precision and is suitable for scientific research instead of a field laboratory. The strain range of the test is moderate in the three methods. The fatigue life of the six asphalt mixtures decreases gradually with the increase of strain, and the fatigue performance of SK70 and WA are the worst, and 4S4P, SBS4, and SBS6 are the best. The specific performance ranking varies slightly with strain, which is similar to the conclusion for 4PB. The values obtained from the Trapezoidal Beam tests are more concentrated in regions with a smaller strain than 4PB.

Overlay Tester: The strain range of the OT test is the narrowest. The test specimen is relatively simple to prepare and the test process takes less time. It is easier to popularize than the above two test methods. Similarly, the OT test again verified that the fatigue performance of SK70 and WA are poor. The fatigue performance of SBS4, 4S4P, and WPES are in the middle, and SBS6 showed the best fatigue performance. With the increase of SBS content, the fatigue life of the asphalt mixture increased. The fatigue life of 4S4P was better than that of SBS4. This result confirmed the results of the Trapezoidal Beam tests but differed from that of 4PB.

### *3.3. Comparative Analysis of Three Kinds of Fatigue Tests*

In order to analyze the comprehensive evaluation results of different asphalt mixtures in the three fatigue tests, the fatigue tests at or nearby 300 × 10−6 strain were unified and chosen to compare the fatigue life of different asphalt mixtures under the three different tests. According to the order of fatigue life of different mixtures, the mixture with the best fatigue performance scores 6 points and the worst scores 1 point. The total scores of the three test methods for various asphalt mixtures are given by analogy. The results are shown in Table 5.

**Table 5.** The fatigue life and score of different asphalt mixtures in different fatigue tests.

1. In the 4PB fatigue test, WPES has the best fatigue life, which is 19.03 times of the fatigue life of the base asphalt mixture. In the Trapezoidal Beam fatigue test, SBS6 has the best fatigue life, which is 127.48 times of the fatigue life of the base asphalt mixture. In the OT fatigue test, SBS6 has the best fatigue life, which is 3.1 times of the fatigue life of base asphalt mixture.

2. According to the total score, the fatigue performance of the asphalt mixture is ranked: SBS6 > 4S4P > WPES ≈ SBS4 > WA > SK70.

3. In the three fatigue tests, the fatigue lives of SK70 and WA are significantly lower than that of the other four asphalt mixtures, which is also identical to the experience in practical engineering [22–24], that is, the fatigue life of the base asphalt mixtures is not significantly improved by a warm mixing agent [25,26]. The fatigue performance of the asphalt mixture is improved by adding SBS, and the fatigue performance of the asphalt mixture is improved more obviously with the increase of SBS content. The addition of WPE and PA610 also improves the fatigue performance of the asphalt mixture in varying degrees. Considering fatigue performance and cost saving, these two new modification methods can replace part of the SBS-modified asphalt.

4. In the three fatigue tests, the fatigue performance of the mixture can be roughly distinguished, but the specific rankings are different. The test samples, waveforms, and frequencies are different, so the evaluation results of fatigue life values cannot be directly transformed. However, in the research and development of new materials, the three different fatigue tests can be used simultaneously to obtain a more comprehensive and objective evaluation.

### **4. Conclusions**

In this research program, test schemes are designed for the fatigue performance of several modified asphalt mixtures in three fatigue tests. According to the experimental test results and the statistical analysis findings, the following conclusions can be drawn:

1. In the 4PB fatigue test, the fatigue performance ranking is WPES > SBS6 > 4S4P > SBS4 > SK70 > WA; in the Trapezoidal Beam fatigue test, the fatigue performance ranking is SBS6 > SBS4 > 4S4P > WPE + SBS > WA > SK70; in the OT fatigue test, the fatigue performance ranking is SBS6 > SBS4 > WPE + SBS > 4S4P > SK70.

2. According to the total score, the fatigue performance of the asphalt mixture is ranked: SBS6 > 4S4P > WPES ≈ SBS4 > WA > SK70.

3. In all three fatigue tests, the fatigue lives of SK70 and WA are significantly lower than that of the other four asphalt mixtures, which means that the fatigue life of the base asphalt mixtures is not significantly improved by the warm mixing agent. The fatigue performance of the asphalt mixture is improved by adding SBS, and the fatigue performance of asphalt mixture is improved more obviously with the increase of SBS content. The addition of WPE and PA610 also improves the fatigue performance of the asphalt mixture in varying degrees. Considering fatigue performance and cost saving, these two new modification methods can replace part of SBS-modified asphalt.

4. In all three fatigue tests, the fatigue performance of the mixture can be distinguished, but it differs in specific rankings. Although the evaluation results of the three fatigue life values cannot be directly transformed, the three different fatigue tests can be used simultaneously to obtain a more comprehensive and objective evaluation in the R&D process of a new modified asphalt.

5. More precise forming and cutting technology is needed because the strain range used in the 4PB test is very wide, and number of samples used in each group is small. The preparation of the Trapezoidal Beam test samples is complex; the amount of test data is huge and has high precision, which is suitable for scientific research instead of the field laboratory, and the strain range of the test is moderate in the three methods. The strain range of the OT test is the narrowest; the test specimen is relatively simple to prepare, and fatigue performance of a specific modified asphalt mixture can be obtained quickly in a simple laboratory. It is easier to popularize than the above two test methods.

**Author Contributions:** Conceptualization, K.L.; methodology, M.H., H.Z. and B.L.; validation, K.L. and B.L.; formal analysis, M.H. and H.Z.; investigation, H.Z. and B.L.; resources, H.Z. and B.L.; data curation, M.H. and B.L.; writing—original draft preparation, H.Z. and B.L.; writing—review and editing, K.L. and M.H.; visualization, B.L.; supervision, M.H..; project administration, K.L. and M.H.; funding acquisition, K.L.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank Huang Xin from Beihang University for helpful and constructive prophase studies.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Indices to Determine the Reliability of Rocks under Fatigue Load Based on Strain Energy Method**

**Huanran Fu 1,2,3, Sijing Wang 1,2,\*, Xiangjun Pei 4 and Weichang Chen 1,2,3**

Received: 14 November 2018; Accepted: 15 January 2019; Published: 22 January 2019

**Abstract:** Rock is a complicated material which includes randomly distributed grains and cracks. The reliability of rocks under fatigue load is very important during the construction and operation of rock engineering. In this paper, we studied the deformation and failure process of red sandstone under fatigue load in a laboratory based on a new division method of strain energy types. The traditional elastic strain energy density is divided into two categories: grain strain energy density and crack strain energy density. We find that the proportion of the grain strain energy density to total strain energy density can be used as an indicator of rock yield and the proportion of the crack strain energy density to total strain energy density can be used as an indicator of rock failure. Subsequently, through extensive literature research, we found that such a phenomenon is widespread. We also find the proportion of grain strain energy density to total strain energy density when yielding is affected by rock types and elastic modulus. The proportion of crack strain energy density to total strain energy density in the pre-peak stage is stable and not affected by rock types and elastic modulus, which is about 0.04~0.13. These findings should be very helpful for rock stable state judging in rock engineering.

**Keywords:** reliability of rocks; fatigue load; strain energy; red sandstone; distribution of strain energy; indices

### **1. Introduction**

Rock is a kind of complicated material composed of mineral grains and cracks [1–4]. The deformation and failure process of rock is mainly controlled and affected by cracks [5–11]. During earthquakes, mining, and engineering construction, rock suffers deformation, producing fractures or failure as a result of fatigue load, which can impacts human lives and the environment [12–14]. Studies on the mechanical properties of rocks under fatigue loads are usually conducted using stress-based, strain-based, or strain energy-based approaches [15–18]. Due to the essence of rock material, deformation and failure is an energy-driven process [19] and energy evolution runs through the whole process of rock deformation and failure. The analyses based on the strain energy based-approach proved to be an effective approach [20–23]. In rocks, yield point and peak stress point are very important in evaluating the mechanical properties of rocks [9,24–30]. During the yield stage, rock deformation is not easily controlled by stress, and the rock structure becomes unstable. When the deformation of rock exceeds the peak stress point, the whole rock is destroyed. Therefore, it is more significant to determine whether the rock enters the yield state than whether it breaks in engineering practice. From the point of view of elastic–plastic mechanics, the mechanical energy input from outside

is divided into two parts in rock without considering heat exchange. One part is elastic strain energy, which accumulates in the rock. The other part is plastic strain energy, which leads to the cracks growing, structural changes, and unrecoverable deformations. When some kind of strain energy reaches a certain state, the rock also enters a corresponding deformation or failure stage [31]. Therefore, the study of the effects of elastic and plastic strain energy on the rock deformation process can deepen our understanding of the mechanical behavior of rocks.

Presently there are no studies in literature that described or proposed a method to determine whether rock has entered the yield stage or failure stage from the point of view of strain energy. Some studies have focused on the evolution of elastic and plastic strain energy in rock and the associated damage to rock structures [32–35]. Some studies also analyzed the external factors that affect the evolution of elastic energy and plastic strain energy [36,37]. In addition, some studies analyzed the deformation and failure behavior of rocks under different fatigue loads [13,38–40]. These studies provide a good baseline for understanding the relationship between rock and strain energy and the mechanical behavior of rocks under fatigue loads. However, most of these studies are based on the traditional elastic and plastic strain energy division method, which cannot reflect the strain energy effect of rock grains and cracks in the process of rock deformation; hence, it is difficult to make a break through. At the same time, almost all of these studies only explain the phenomena, and do not give indices of strain energy when the rock enters the yielding stage or failure. Therefore, with the further research, the new strain energy division method and the index to determine the rock from stable to unstable stage should to be discussed.

Focusing on the abovementioned questions, this study tries to give indices to determine whether rocks have entered the yielding stage or failure. Through the new classification of strain energy density types, we found that the ratio of some kinds of strain energy densities to corresponding total strain energy densities obtained by the new method could be used as the index to determine whether the rock has entered the yield stage or failed. We subsequently collected experimental data from extensive literature to confirm this finding. We believe that this discovery is of great significance in rock engineering and earthquake prediction for judging or predicting rock stability.

### **2. Classification and Calculation Methods of Rock Strain Energy by Tests Results**

### *2.1. Classifications and Methods of CalculatingStrain Energy*

It is of great importance to divide suitable strain energy types to explain the elastic–plastic transformation and the mechanical properties of rocks. Although many methods in rock researches are derived from metal, the unique properties of properties of rocks clearly makes the two materials to be treated differently in some instances. In this study, we use the strain energy density *ui* to represent the strain energy in rocks.

In conventional rock cyclic loading and unloading studies, the researchers only divided the strain energy density into elastic strain energy density *ue i* and plastic strain energy density *up i* , which as shown in Figure 1a. The deformation of rock consists of two parts: deformation of rock grains and deformation of cracks. Cai et al. [3] pointed out that during the unloading process, due to the previously compacted cracks and the opening of newly formed crack as a result of decrease in load, the slope of the unloading curve would gradually slow down. The deformation rebound of rock, at the beginning of unloading, was mainly due to the elastic deformation of grains because the crack in the rock cannot immediately change from compression to opening. Therefore, in the pre-peak stage, in order to obtain the elastic strain energy density of rock grains when unloading. We draw a line section with the slope of unloading modulus obtained at the initial stage of unloading from the unloading point *A*, and then cross the strain axis at point *C*, as the line *AC* shown in Figure 1b. So in this paper, the elastic strain energy density *ue i* is divided into two parts. One part is the elastic strain energy density of rock grains *ug i* , the second part is the crack strain energy density *uc i* . The *uc i* represents the strain energy density generated by the recoverable deformation of cracks. The method

of dividing strain energy in post-peak stage is the same as that in pre-peak stage which are shown in Figure 1c,d.

**Figure 1.** Comparison of two methods for dividing strain energy in pre-peak stage and post-peak stage: (**a**) The pattern of strain energy division in pre-peak stage which has been used in the former studies; (**b**) the new pattern of strain energy division in pre-peak stage which has been used in this study; (**c**) the pattern of strain energy division in post-peak stage which has been used in the former studies; (**d**) the new pattern of strain energy division in post-peak stage which has been used in this study.

Thus, the relationship between these kinds of strain energy density can be written as

$$
\mu\_i = \mu\_i^p + \mu\_i^c = \mu\_i^p + \left(\mu\_i^q + \mu\_i^c\right),
\tag{1}
$$

According to Figure 1b,d, the calculation method of strain energy density in a loading and unloading process can be written as follows:

$$
\mu\_i = \int\_0^{\varepsilon\_{i1}} \sigma\_{i1} d\varepsilon\_{i1} + \int\_0^{\varepsilon\_{i2}} \sigma\_{i2} d\varepsilon\_{i2} + \int\_0^{\varepsilon\_{i3}} \sigma\_{i3} d\varepsilon\_{i3} \tag{2}
$$

$$u\_i^{\varepsilon} = \int\_a^b \sigma\_{i1}^{\prime} d\varepsilon\_{i1}^{\varepsilon} + \int\_a^b \sigma\_{i2}^{\prime} d\varepsilon\_{i2}^{\varepsilon} + \int\_a^b \sigma\_{i3}^{\prime} d\varepsilon\_{i3\prime}^{\varepsilon} \tag{3}$$

$$u\_i^p = u\_i - u\_i^\varepsilon \tag{4}$$

$$
\mu\_i^{\mathcal{S}} = \mu\_{i1}^{\mathcal{S}} + \mu\_{i2}^{\mathcal{S}} + \mu\_{i3}^{\mathcal{S}} = \frac{1}{2}\sigma\_{i1}\varepsilon\_{i1}^{\mathcal{S}} + \frac{1}{2}\sigma\_{i2}\varepsilon\_{i2}^{\mathcal{S}} + \frac{1}{2}\sigma\_{i3}\varepsilon\_{i3}^{\mathcal{S}}.\tag{5}
$$

$$
\mu\_i^\varepsilon = \mu\_i^\varepsilon - \mu\_{i\cdot\prime}^\mathcal{g} \tag{6}
$$

In these equations, *u* represents strain energy density, *σ* and *σ* represent the loading stress and unloading stress respectively; *ε* represents the strain; subscript *i* represents a rock unit; numbers 1, 2, and 3 represent three principal stress directions; and superscript *p*, *e*, *g*, and *c* represent plastic strain, elastic strain, elastic strain of rock grains, and elastic strain of cracks, respectively. The letters *a* and *b* represent the strain values corresponding to the stress unloading to the lowest point and the unloading start point, respectively.

### *2.2. Testing Schemes and Experimental Data*

The test schemes adopted cyclic loading and unloading tests under different confining pressures and conducted with the MTS-815 servo-controlled testing machine in the State Key Laboratory of Geo-hazard Prevention and Geo-environment Protection (SKLGP), Chengdu, Sichuan, China. The fresh red sandstone blocks, which were taken from a quarry in Leshan, Sichuan, China were selected for this test. The blocks were cut into several cylindrical samples with 50mm in diameter and 100mm in height. The average density of dried red sandstone specimens is about 2.06 g/cm3 and the main mineral constituents are calcite (78.2%) and quartz (0.5%). The X-ray diffraction pattern is shown in Figure 2.

**Figure 2.** X-ray diffraction pattern of sandstone samples in this study.

In this study, cyclic loading and unloading tests under confining pressures of 0MPA, 5 MPa, 10 MPa, and 20 MPa were designed. By comparing the uniaxial compression deformation stages of rocks classified by some researchers [2,41,42] and the deformation stages of cyclic loading and unloading test, stress–strain curves can be divided into 5 stages: (I) crack closure, (II) elastic deformation, (III) crack initiation and stable crack growth,(IV) crack damage and unstable crack growth, and(V) failure and post-peak behavior. In general, the point between stage III and IV is called the crack damage point and be treated as the yield point and the point between stage IV and V is called the peak point.

During the cyclic loading and unloading test, the rock samples were subjected to axial force with the loading rate of 1kN/s. When the axial force reaches 10 kN, we stopped the loading and start the unloading process. The unloading force was also set at 1kN/s. In order to avoid the impact of the testing machine on the sample during the reloading process, the sample is loaded when the axial force is 1 kN in the successive unloading process. For subsequent unloading cycles, we increased the unloading force by 10 kN, i.e., during the first cycle the unloading force was set at 10 kN and was 20 kN during the second cycle. In order to obtain more cycles when the rock deformation enters the yield stage, we decreased the unloading force of subsequent cycle to 5 kN. We adopted this loading method until the rock deformation reaches the peak point and failure occurred. During post peak stage, we controlled the unloading points manually. This makes the unloading force different for each sample. Despite this approach, the difference of strain energy density evolution in the post-peak and pre-peak stage is still obvious. A representative cyclic loading and unloading stress–strain curve is shown in Figure 3.

**Figure 3.** A representative cyclic loading and unloading stress-strain curve under 5 MPa confining pressure.

The mechanical properties of red sandstone samples under different confining pressures are shown in Table 1. In Table 1 *σ*3 represents confining stress, *E* represents the average elastic modulus, the *σ*cd and *ε*cd represents the average crack damage stress and strain respectively, the *σ*p and *ε*p represents the average peak strength and strain respectively.

**Table 1.** The stress and strain at each stage of red sandstone sample's deformation under different confining pressures.

### **3. Test Results**

### *3.1. The Evolution of Strain Energy*

Based on the strain energy types mentioned above and the laboratory test results, we obtained the strain energy evolution of red sandstone samples under different confining pressures. It should be noted that the data of these kinds of strain energies in the post-peak stages cannot be obtained in uniaxial compression tests, and two sets of post-peak strain energy data are obtained in the confining pressure tests of 5 MPa. A set of post-peak strain energy data is obtained in the confining pressure tests of 10 MPa and 20 MPa, respectively. According to the results of evolution of strain energy shown in Figures 4 and 5, it can be found that the evolution of strain energies has some characteristics as follows:

**Figure 4.** The evolution of *up i* and *ue i* of red sandstone samples under different confining pressures: (**a**) Confining pressure is 0 MPa; (**b**) Confining pressure is 5 MPa; (**c**) Confining pressure is 10 MPa; (**d**) Confining pressure is 20 MPa.

(1) The characteristics of *up i* .

The *up i* grows in quadratic form during the whole deformation process. With the increase in strain, the numerical growth rate of *up i* is accelerated especially as it enters the yield stage. The value of *up i* is affected by the confining pressure, the higher the confining pressure the higher the value of *up i* .

(2) The characteristics of *ue i* .

In the crack closure stage, the growth rate of *ue i* is accelerated, and the growth rate in elastic stage is almost constant. After entering the yield stage, the growth rate of *ue i* decreases, but the value of *ue i* still increases. In the post-peak stage, the growth rate of *ue i* becomes negative. With the increase of confining pressure, the values of *ue i* increase, and the attenuation effect in the post-peak stage is also inhibited.

(3) The characteristics of *ug i* .

The value of *ug i* increases in S-shape during the deformation process and decreases after peak stress, which is similar to the stress–strain curve of rocks. The growth rate of *ug i* decreases once the sample enters the yield stage. For some samples, the value of *ug i* will still increase slightly due to confining pressure in the early post-peak stage. The value of *ug i* will also increase with the increase of confining pressure.

#### (4) The characteristics of *uc i* .

The value of *uc i* increases slightly with the increase of strain before the peak point. After entering the post-peak stage, the *uc i* of each sample is highly discrete. For most samples, the *uc i* decrease in the post-peak stage and even becomes negative, only a small number of them increase. This may be as a result of the effects of confining pressures.

**Figure 5.** The evolution of *ug i* and *uc i* of red sandstone samples under different confining pressures: (**a**) Confining pressure is 0 MPa; (**b**) Confining pressure is 5 MPa; (**c**) Confining pressure is 10 MPa; (**d**) Confining pressure is 20 MPa.

The results of Figures 4 and 5 show that under the same test conditions, the same kind of strain energy of different samples shows a highly concentrated behavior, and the rock yielding or failure is because the strain energy reaching a certain value. However, due to different confining pressures, different strain energy is required for rock entering different deformation stages, which is relatively limited in application.

### *3.2. The Distribution of Strain Energy*

Inspired by the above results, we check whether there is a relationship between the total strain energy absorbed by rock and other forms of strain energy in different deformation stages. Therefore, we divided *up i* , *ue i* , *ug i* , and *uc i* at different deformation moments by *ui* at the corresponding moment. We then obtained the distribution law of the proportion of different strain energy to the total strain energy in the entire deformation process, which is shown in Figure 6. It is worth pointing out that the first two points of each curve are in the crack closure stage, which cannot reflect the

properties of materials from the point of view of elastic-plastic mechanics, so they can be neglected in elastic–plastic analysis.

**Figure 6.** The distribution of strain energies under different confining pressures obtained after normalization: (**a**) Confining pressure is 0 MPa; (**b**) Confining pressure is 5 MPa; (**c**) Confining pressure is 10 MPa; (**d**) Confining pressure is 20 MPa.

Figure 6 and Table 1, reveal the following characteristics about the distribution of strain energies:

### *3.3. Discussions*

Based on the above results, it can be found that the evolution of strain energy is related to the deformation and failure process of red sandstone. For a sample under a confining pressure, the plastic strain energy or elastic strain energy required to reach the yield stage is almost certain. The yield conditions can be constructed by testing the values of strain energy satisfied by different rocks under different test conditions. More importantly, when the proportion of *up i* , *ue i* , *ug i* , and *uc i* in *ui* is plotted, it is found that the distribution of strain energy will appear as obvious points before or after the yield stage and pre-peak or post-peak stage.

As mentioned at the beginning in this paper, rock is a natural material composed of rock grains and cracks. Under the action of external force, the action of rock grains is to resist external force, and then weaken the damage as a result of external force to the structure. The crack in rock is a kind of existence that weakens the ability of rock to resist external force. We can assume that even though there are two opposing functions in rock, as long as their relationship is balanced, the strength of rock will increase with the increase of external forces within the energy storage range of rock grains. If this assumption is extended to the strain energy, that is, the ratio of strain energy density of different forms to total strain energy density are relatively stable before the big changes in rock structure. If the proportion of one type of strain energy density to total strain energy density changes significantly, the rock will enter the corresponding deformation stages, which is also consistent with the test results.

According to the definition of *ug i* and *uc i* , it can be seen that *ug i* represents the elastic strain energy stored in rock grains in a rock unit, which can be released with unloading or rock structure damage. The *uc i* represents the strain energy released by the recoverable deformation part of cracks. Therefore, we presume that the growth rate of the proportion of *ug i* to *ui* becomes negative, which can be used as an indicator to judge the rock entering the yield stage. We also noticed that even under different confining pressures, the ratio of *uc i* is almost the same in the pre-peak stage, which was a constant, and its ratio is about 0.06(±0.02). Therefore, we consider that whether the large-scale abrupt change of this proportion can be used as the basis for judging rock failure.

### **4. Verification and Extension of Strain Energy Index**

Due to the limitation of the type and quantity of the test samples, we cannot draw general conclusions based on the experimental results only. Therefore, we first discuss the relationship between strain energy and rock deformation mechanism, and then obtained relevant data from published literature to verify the conclusions and assumptions in previous section.

### *4.1. The Relationship between Strain Energy and the Deformationmechanism of Rocks*

Rock deformation process is related to the process of crack initiation and expansion. When the damage is within the range of rock resistance, the rock is stable, otherwise it becomes unstable. From the point of strain energy, the plastic strain energy reflects the energy consumed by the irreversible deformation of rock, including the irreversible deformation of most cracks and the displacement of a small part of rock grains. The elastic strain energy is mainly reflected by the elastic deformation of rock grains and a small number of recoverable cracks.

During the deformation of rock, the material properties of rock are activated with the closure of cracks. In the elastic stage, the elastic strain energy accumulated by rock particles is linear with the total strain energy, and the proportion of the total strain energy is relatively constant. Only after the yield stage, more and more energy is consumed by plastic deformation due to the unsteady propagation of cracks, so the proportion of elastic energy of rock grains decreases.  When the rock has failed, the rock no longer has a complete shape, and the proportion of crack strain energy changes rapidly.

### *4.2. Collection of Experimental Results from PreviousStudies*

We selected 94 rock cyclic loading and unloading tests of five types of rocks from published data for analysis. The basic information of the samples and test conditions are presented in Table 2. Although some of the data did not perform cyclic loading and unloading at the post-peak stage, it was impossible to analyze the post-peak condition, but the analysis in the pre-peak stage was still effective.

**Table 2.** Basic information of the samples and test conditions from literature.

After analyzing the obtained data, it is found that the strain energy evolution of various samples is consistent with the strain energy evolution of red sandstone obtained in Section 3. For almost all samples, the proportion of grain strain energy will decrease after entering the yield stage. In this study, we choose the elastic modulus as abscissa axis because elastic modulus is affected by confining pressure, porosity, compactness of rock grains arrangement, and other comprehensive factors, which is the general embodiment of rock elastic properties. By analyzing the collected data, we observed that the ratio of grain strain energy to total strain energy increases with the increase in elastic modulus, different rocks show the distinct classification. The proportion of *ug i* to *ui* of the same rock typeis linearly increasing with the elastic modulus. The relations between elastic modulus and proportion of *ug i* to *ui* of different rocks while entering the yield stage are shown in Figure 7.

Let y denote the ratio of *ug i* to *ui* when yielding and x denote the elastic modulus. Linear fitting of the data of different rocks in Figure 7 are given in the following equations:

The coal is

$$\mathbf{y} = 0.08278\mathbf{x} + 0.38472, \; \mathbf{R}^2 = 0.99638 \tag{7}$$

From the results of linear fitting, it can be seen that the slope of linear equation from Equation (7) to Equation (11) decreases successively, while the uniaxial compressive strength of rock increases successively. It should be noted that the uniaxial compressive strength referred to here is not the uniaxial compressive strength specific to a particular rock, but the uniaxial compressive strength commonly used to classify rocks. This is because the uniaxial compressive strength of the sandstone, in some cases, is higher than that of the marble. The classification of rock uniaxial strength is stated in China's national standards [55,56].

**Figure 7.** The relation between the proportion of grain strain energy density to total strain energy density and elastic modulus of different rock while entering the yield stage: (**a**) Coal; (**b**) Sandstone; (**c**) Basalt; (**d**) Marble; (**e**) Granite.

Through the analysis of the ratio of the *uc i* to *ui* in these rock samples, it is found that this ratio is nearly stable in a range during the pre-peak period, and will not change until failure (Figure 8). In Figure 8, the data are concentrated in the range of 0.04~0.13 and are not affected by rock type and elastic modulus. The maximum value is 0.1534 and the minimum value is 0.0366.

**Figure 8.** The proportion of crack strain energy density to total strain energy density of different rock samples in the pre-peak stage.

However, due to the limitation of sample data, this finding needs more sample data and more rock types to confirm. We believe that by analyzing enough experimental data of different rock types, we can get more accurate and comprehensive evolution and proportion of grain strain energy density and crack strain energy density to total strain energy density, and can use these two indices to predict or analyze the stability or failure of rocks.

### **5. Conclusions**

According to a series of analysis of test results and literature data, this work studied the evolution of strain energy in the process of rock deformation and its relationship with deformation mechanism classified by new strain energy types. It is observed that the proportion of grain strain energy density and crack strain energy density to total strain energy density can be used as the indices for determine rock stability or failure. The main findings of this study are as follows:

the stability of rock. The proportion of crack strain energy density to total strain energy density can be used as the index to determine rock failure.

It should be noted that these findings are based on limited data, but we believe these data have already opened the door to a new research area. Later, we will enrich these research findings by obtaining more data and popularizing it for engineering applications.

**Author Contributions:** Conceptualization, H.F. and S.W.; Methodology, H.F.; Validation, H.F., S.W. and X.P.; Formal analysis, H.F. and W.C.; Investigation, S.W. and H.F.; Resources, X.P.; Writing—original draft preparation, H.F.; Writing—review and editing, H.F. and S.W.; Visualization, H.F. and W.C.; Supervision, S.W. and X.P.; Project administration, H.F.; Funding acquisition, S.W.and X.P.

**Funding:** This research was funded by the Opening Fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology) No. SKLGP2016K001; National Key R&D Program of China No. 2017YFC1501002; the National Natural Science Foundation of China No. 41572302.

**Acknowledgments:** We would like to thank Cheng Cheng's suggestions and Afolagboye lekan olatayo's help in improving the English of manuscript. We also would like to thank the anonymous reviewers for their helpful comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Comparative Fatigue Life Assessment of Wind Turbine Blades Operating with Different Regulation Schemes**

### **Brian Loza 1, Josué Pacheco-Chérrez 1, Diego Cárdenas 1, Luis I. Minchala 2 and Oliver Probst 1,\***

Received: 18 July 2019; Accepted: 11 October 2019; Published: 31 October 2019

**Abstract:** A comparative evaluation of the fatigue damage occurring in the blades of small wind turbines, with different power regulation schemes, has been conducted for the first time. Three representative test cases were built, one based on stall regulation and two using pitch regulation. The power curves were tuned to be identical in all cases, in order to allow for a direct comparison of fatigue damage. A methodology combining a dynamic simulation of a wind turbine forced by stochastic wind speed time series, with the application of the IEC 61400-2 standard, was designed and applied for two levels of turbulence intensity. The effect of the wind regime was studied by considering Weibull-distributed wind speeds with a variety of parameter sets. Not unexpectedly, in typical wind regimes, stall regulation led to a generally higher fatigue damage than pitch regulation, for similar structural blade design, but the practical implications were smaller than thought previously. Given the need for cost-effective designs for small wind turbines, stall regulation may be a viable alternative for off-grid applications.

**Keywords:** fatigue life; small wind turbine; stall regulation; pitch regulation; aeroelastic simulation

### **1. Introduction**