*3.2. Statistical Damage Model for FRC in Uniaxial Tension* 3.2.1. Influence Mechanism of Fiber

A large number of experimental studies [1–9,12] have shown that, the influencing factors mainly include: the strength of the matrix concrete, the type of fiber, the length diameter ratio and the volume ratio of fiber, the bonding strength between fiber and matrix, and the distribution and orientation of the fiber in the matrix. The existing theories mainly focus on fiber and matrix, and study the interaction between single fiber and matrix.

## (1) Fiber spacing theory

Romualdi and Batson [13] analyzed the limitation and restraint mechanism of steel fiber to concrete crack, and put forward the theory of fiber spacing to explain the mechanism of fracture enhancement of the SFRC. This theory holds that the existence of fiber could significantly reduce the size and quantity of micro-cracks, reduce the stress concentration degree of the crack tip, and restrain the occurrence and expansion of micro-cracks. The key to fiber reinforcement is the average spacing of the fibers. With the increase of average spacing, the fiber's ability to restrain the crack initiation and expansion will be greater, and the strength of the SFRC will be higher. The size of the average spacing of the fibers depends on the number of active fibers in the volume of the unit matrix.

### (2) Reinforcement rules for composite materials

Swamy [14] proposed the reinforcement rules for composite materials based on the mechanics principle of composite materials. The SFRC is simplified as fiber and concrete two-phase composite materials, and the properties of the composite material are cumulative for each phase. Due to the non-homogeneity of ordinary concrete structure, irregular stress concentration would occur within the matrix when the structure is pulled. When the ultimate tensile strength of ordinary concrete is less than the tensile stress of the stress concentration point, the stress concentration point will create cracks. Since the tensile strength of the steel fibers is much higher than the tensile strength of the concrete matrix, the incorporation of steel fiber can effectively inhibit and delay the initiation and expansion of micro-cracks in ordinary concrete.

In other words, these randomly distributed fibers can effectively prevent the expansion of micro-cracks in concrete and delay the formation of macro-cracks. Hence, it can significantly improve the macroscopic mechanical properties of concrete, such as tensile, bending, impact, fatigue resistance, and so on. The main factors that influence fiber behavior include: the species, geometric features, and content of the fibers, the bonding properties of the fibers to the concrete matrix, distribution and orientation of the fibers in matrix. Most of the existing constitutive models for FRC are adopted from the macroscopic phenomenological mathematical expressions (as shown in Table 1), of which parameters lack a clear physical meaning. Therefore, it is difficult for them to reflect the influence of fiber content on the mesoscopic damage evolution of the concrete.


**Table 1.** Theoretical stress–strain models of SFRC under tension in literature.

### 3.2.2. Practical Expressions of the IPBS

According to the statistical damage theory mentioned above, the macroscopic mechanical behavior (nominal/effective stress–strain curve) of concrete under uniaxial tension, is determined by the cumulative evolution process of the fracture and yield damage modes in a meso-scale, as shown in Figure 5. The whole process includes the homogeneous damage accumulation stage and local failure stage. Two characteristic states are distinguished, namely peak nominal stress state and critical state. *ε*cr and *ε*<sup>u</sup> denote the strains of critical state and ultimate state in the nominal stress–strain curve.

**Figure 5.** Relationship between constitutive behavior on macro-scale and damage evolution process on meso-scale.

As the probability density functions corresponding to fracture and yield damage in the meso-scale respectively, *q*(*ε*) and *p*(*ε*), may be subject to complex statistical distribution laws in the true case, such as Weibull, Normal, and other distributions. Considering the complexity of the problem, *q*(*ε*) and *p*(*ε*) are both assumed to obey the independent triangle distribution form in a specific calculation, as shown in Figure 5. Analyses [35–38] show that the true stress–strain test curves could be well fitted and the evolution mechanism of non-homogeneous damage on the meso-scale could be well revealed, when *q*(*ε*) and *p*(*ε*) are adopted by the simplified triangular distributions. *ε*<sup>a</sup> is the initial damage strain. *ε*<sup>h</sup> is the strain corresponding to the peak value of *p*(*ε*). *ε*<sup>b</sup> is the strain corresponding to the maximum yield damage state, and also to the peak value of *q*(*ε*). *ε*<sup>c</sup> is the strain corresponding to the maximum fracture damage state. It satisfies *ε*<sup>b</sup> = *ε*cr and *ε*<sup>c</sup> = *ε*<sup>u</sup> in uniaxial tension.

Due to the softening segment of the nominal stress–strain curve corresponding to the local failure stage having obvious size effect, the critical state is suggested as the final failure point of the constitutive model in this paper. Hence, the following content in the paper only discusses the constitutive behavior of concrete in the homogeneous damage stage.

At the homogeneous damage stage, *σ*<sup>N</sup> and *σ*<sup>E</sup> can be obtained by Equations (1)–(4), where *q*(*ε*) and *p*(*ε*) can be expressed as the following:

$$q(\boldsymbol{\varepsilon}) = \begin{cases} 0 & (\boldsymbol{\varepsilon} \le \boldsymbol{\varepsilon}\_{\mathsf{a}})\\ \frac{2H(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}\_{\mathsf{a}})}{\left(\boldsymbol{\varepsilon}\_{\mathsf{b}} - \boldsymbol{\varepsilon}\_{\mathsf{a}}\right)^{2}} & (\boldsymbol{\varepsilon}\_{\mathsf{a}} < \boldsymbol{\varepsilon} \le \boldsymbol{\varepsilon}\_{\mathsf{b}}) \end{cases} \tag{12}$$

$$p(\varepsilon) = \begin{cases} 0 & (\varepsilon \le \varepsilon\_{\mathsf{a}}) \\ \frac{2(\varepsilon - \varepsilon\_{\mathsf{a}})}{(\varepsilon\_{\mathsf{h}} - \varepsilon\_{\mathsf{a}})(\varepsilon\_{\mathsf{b}} - \varepsilon\_{\mathsf{a}})} & (\varepsilon\_{\mathsf{a}} < \varepsilon \le \varepsilon\_{\mathsf{h}}) \\ \frac{2(\varepsilon\_{\mathsf{b}} - \varepsilon)}{(\varepsilon\_{\mathsf{b}} - \varepsilon\_{\mathsf{h}})(\varepsilon\_{\mathsf{b}} - \varepsilon\_{\mathsf{a}})} & (\varepsilon\_{\mathsf{h}} < \varepsilon \le \varepsilon\_{\mathsf{b}}) \end{cases} \tag{13}$$

where *H* = *D*R(*ε*b) is the fracture damage value relevant to *ε*b.

Define *S* as the energy absorbing capability, which represents the energy absorbed by concrete in the process of stress and deformation. The expression is as follows:

$$S = \int\_0^\varepsilon \sigma\_\mathcal{N} d\varepsilon \tag{14}$$

$$S\_{\mathbf{p}} = \int\_0^{\varepsilon\_{\mathbf{p}}} \sigma\_{\mathbf{N}} d\varepsilon \tag{15}$$

$$S\_{\rm cr} = \int\_0^{\varepsilon\_{\rm cr}} \sigma\_{\rm N} d\varepsilon \tag{16}$$

where *S*<sup>p</sup> and *S*cr are the energy absorption capacity corresponding to peak nominal stress state and critical state, respectively.

### 3.2.3. Influence of the Fibers on Mesoscopic Damage Mechanism

Statistical damage theory suggests that the change on the macroscopic nonlinear stress–strain behavior of concrete under a complex environment is essentially caused by the microscopic damage mechanism. The fiber reinforcement effect changes the composition and characteristics of the concrete microstructures, the nucleation and growth of microcracks, and the cumulative evolution process of damage, which finally lead to the change on the macro-nonlinear stress–strain behavior of concrete. The above effects can be summarized into two aspects: (1) changes on the composition and mechanical properties of microstructure, measured by *E*0; (2) changes on the pattern and law of the initiation and propagation of microcracks; in other words, changes on the cumulative evolution process of the two meso-damage modes (yield and fracture), measured by *ε*a, *ε*h, *ε*b, and *H*, which determine the shape of the triangle probability distribution of *q*(*ε*) and *p*(*ε*).

As shown in Figure 6, it is assumed that the microstructure characteristics and mesodamage evolution process of concrete under different fiber contents obey a certain regularity, and the above changes at the meso-level lead to the strengthening of the macroscopic constitutive behavior of concrete. Make *E*0, *ε*a, *ε*h, *ε*b, and *H* as a function of fiber volume fraction *ρ* (%) (here, does not consider the influence of other factors, such as fiber type and geometric features), the expression is as follows:

$$\begin{cases} E\_0 = f\_1\left(\rho\right) \\ \varepsilon\_\mathbf{a} = f\_2\left(\rho\right) \\ \varepsilon\_\mathbf{h} = f\_3\left(\rho\right) \\ \varepsilon\_\mathbf{b} = f\_4\left(\rho\right) \\ H = f\_5\left(\rho\right) \end{cases} \tag{17}$$

**Figure 6.** The influence of the fiber volume fraction on the damage mechanism of concrete.

As mentioned above, according to these five parameters, we can determine the uniaxial tensile stress–strain behaviors of concrete with different fiber content, and also the laws of the evolution of the mesoscopic damage mechanism.

### 3.2.4. Determination of Model Parameters

Five parameters need to be determined for each nominal stress–strain curve under uniaxial tension, they are *E*0, *ε*a, *ε*h, *ε*b, and *H*. *E*<sup>0</sup> can be obtained directly from the test curve, taken as the secant modulus from 0.2–0.4 times of peak stress point in the ascending part of the curve to the original point. *ε*a, *ε*h, *ε*b, and *H* can be obtained by the multivariate regression analysis of the genetic algorithm module in the Matlab toolbox. The solution procedure is summarized as follows:


## **4. Results and Discussion**

The presented model is validated by using two sets of uniaxial tensile tests for steel fiber-reinforced concrete with different fiber contents reported by Han et al. [22] and Gao [20]. The relevant experimental information are summarized in Table 2. The rationality and applicability of the presented model are verified, and the damage evolution mechanism of steel fiber concrete in uniaxial tension is discussed.


**Table 2.** Summary of uniaxial tensile test information.

### *4.1. Comparison with the Test by Han et al., 2006*

Han et al. [22] conducted a uniaxial tensile test for a steel fiber-reinforced concrete specimen. The specimen is of dumbbell shape with a total length of 450 mm. The length of the middle tensile region is 170 mm, with a cross section of 100 mm × 100 mm. A large-end steel fiber with a length-to-diameter ratio of 44.34 is adopted. The volume fractions of the steel fiber *ρ* are 0%, 0.5%, 1.0%, 1.5%, 2.0%, and 2.5%. The theoretical nominal stress–strain curves of steel fiber concrete with different fiber contents calculated by the presented model, corresponding to the homogeneous damage phase, are shown in Figure 7a (the curve with a fiber volume fraction of 1.5% is removed due to the dispersion of the experimental data.). The curves include the ascending and partial descending segments of the stress–strain behavior of the fiber-reinforced concrete. They show a good fitting effect compared to the test data. The predicted effective stress–strain curves are shown in Figure 7b. The relevant calculation parameters are listed in Table 3, where *R* 2 is the correlation coefficient. For this proposed model, the entire process of deformation and failure of concrete under uniaxial tension is understood from the viewpoint of effective stress. In the uniform damage stage, the nominal stress *σ*<sup>N</sup> first increases and then decreases, involving with a peak nominal stress state (so-called strength state). The effective stress *σ*<sup>E</sup> increases monotonously and reaches its maximum at the critical state. After the critical state, the specimen enters the local failure stage characterized by macroscopic crack propagation. The 3D envelopes of *σ*N-*ε* and *σ*E-*ε* curves predicted by the presented model are shown in Figure 7c,d. They clearly show the variation trend of the curves of concrete with different fiber contents, and the shape of the curves show a good similarity rule, especially the connection part between the ascending and descending segments. With the increase of steel fiber volume ratio *ρ* from 0% to 2.5%, the values of *σ*<sup>N</sup> and *ε* corresponding to the peak nominal stress state and the values of *σ*<sup>E</sup> and *ε* corresponding to the critical state increase significantly.

The relationship curves of *ρ*-*ε*a, *ε*h, *ε*<sup>b</sup> are shown in Figure 8a, which represent the evolution law of the yield damage mode on a mesoscale. The curve of *H*-*ρ* is shown in Figure 8b, which depicts the evolution law of the fracture damage mode on a mesoscale. The relationship curve of *E*0-*ρ* is shown in Figure 8c. The aforementioned five parameters show an obvious regularity with the increase in steel fiber content, in which *ε*h, *ε*b, and *H* increase linearly, whereas *E*<sup>0</sup> decreases linearly. The fitting curves and fitting expressions are also shown in the figures.

Figure 9a shows the curves of the evolution process of fracture damage variable *D*<sup>R</sup> with different steel fiber contents, respectively. *D*<sup>R</sup> is closely related to microcrack density, and its evolution process under uniaxial tension is significantly delayed with the increase in fiber content from 0% to 2.5%, which is consistent with the physical background in the microstructure.

*Crystals* **2021**, *11*, x FOR PEER REVIEW 16 of 25

**Figure 7.** Stress–strain curves under uniaxial tension: (**a**) nominal stress–strain curve (Plan view); (**b**) effective stress–strain curve (Plan view); (**c**) envelope of the nominal stress–strain curve (3D view); (**d**) envelope of the effective stress–strain curve (3D view). **Figure 7.** Stress–strain curves under uniaxial tension: (**a**) nominal stress–strain curve (Plan view); (**b**) effective stress–strain curve (Plan view); (**c**) envelope of the nominal stress–strain curve (3D view); (**d**) envelope of the effective stress–strain curve (3D view).


The relationship curves of


a h b

 、 、

in Figure 8b, which depicts the evolution law of the fracture damage mode on a mesoscale.

are shown in Figure 8a, which represent

*H* -  is shown

 


*E*0 - 

ters show an obvious regularity with the increase in steel fiber content, in which

*E*0

is shown in Figure 8c. The aforementioned five parame-

decreases linearly. The fitting curves and fitting ex-

h , b ,

The relationship curve of

and *H* increase linearly, whereas

pressions are also shown in the figures.

**Figure 8.** Influence of fiber volume fraction on the characteristic parameters: (**a**) *ρ-εa*, *εh*, *εb* curves; (**b**) *H*-*ρ* curve; (**c**) *E*0-*ρ* curve. **Figure 8.** Influence of fiber volume fraction on the characteristic parameters: (**a**) *ρ-ε*a, *ε*h, *ε*<sup>b</sup> curves; (**b**) *H*-*ρ* curve; (**c**) *E*<sup>0</sup> -*ρ* curve.

**Figure 9.** Evolution curves of damage variable and evolution factor: (**a**) *D*R*-ε* curves; (**b**) *E*v-*ε* curves. **Figure 9.** Evolution curves of damage variable and evolution factor: (**a**) *D*R*-ε* curves; (**b**) *E*v-*ε* curves.

failure stage characterized by macroscopic crack growth.

cal behavior of concrete for adding steel fibers with different contents.

*E*v

*E*v

yield and cannot endure much effective stress, then the concrete specimen enters the local

The physical meanings of the above-mentioned characteristic parameters are clear. The prediction results can be used to explore the internal relations among the physical background, the mesodamage evolution mechanism, and the macro-nonlinear mechani-

Local stress concentration will occur in the concrete matrix under tensile load due to the heterogeneity in the microstructure, which will lead to the initiation and expansion of microcracks. After adding steel fiber to the concrete matrix, the composition and mechanical properties of the microstructure will change, and the initiation and penetration of microcracks in the concrete matrix will be significantly inhibited and delayed. The evolution and accumulation process of the mesodamage modes will consequently change. For the

from 0% to 2.5%,

to decrease from 3.481 × 10 GPa to 3.046 × 10 GPa. The nominal

increases linearly from 0.993 × 10−4 to 2.211 × 10−4, and

linearly from 1.311 × 10−4 to 2.441 × 10−4. For the fracture damage mode, *H* increases linearly from 0.260 × 10−4 at 0% to 0.450 × 10−4 at 2.5%. The above-mentioned four parameters can be used to determine the shape of triangular probability distribution corresponding to the mesoscopic yield and fracture damage evolution process, which can demonstrate intuitionistic physical pictures to people to understand the entire process on a mesoscale.

The macroscopic nonlinear stress–strain behavior of concrete is determined by the mechanical properties of the microstructure and the evolution of mesodamage. After add-

and mechanical properties of the microstructure will change, which will lead the initial

stress–strain curves generally show "strengthened" features due to the change in the

mesodamage evolution process characterized by the four parameters (

represents the exerting degree of the underly-

= 1, is reached, all the microbars in IPBS will

a 

varying from 0% to 2.5%, the composition

a , h , b 

increases from 0.771 ×

b 

increases

, and

*H*

In Figure 9b, evolutionary factor

in fiber content. When the critical state,

(1) Damage mechanism on a mesoscale

yield damage mode, with the increase in

(2) Mechanical behavior in macroscale

ing steel fiber to the concrete matrix, with

*E*0

, h 

10−4 to 1.181 × 10−4

elastic modulus

In Figure 9b, evolutionary factor *E*<sup>v</sup> represents the exerting degree of the underlying mechanical capacity of materials, and its evolution process is delayed with the increase in fiber content. When the critical state, *E*<sup>v</sup> = 1, is reached, all the microbars in IPBS will yield and cannot endure much effective stress, then the concrete specimen enters the local failure stage characterized by macroscopic crack growth.

The physical meanings of the above-mentioned characteristic parameters are clear. The prediction results can be used to explore the internal relations among the physical background, the mesodamage evolution mechanism, and the macro-nonlinear mechanical behavior of concrete for adding steel fibers with different contents.

### (1) Damage mechanism on a mesoscale

Local stress concentration will occur in the concrete matrix under tensile load due to the heterogeneity in the microstructure, which will lead to the initiation and expansion of microcracks. After adding steel fiber to the concrete matrix, the composition and mechanical properties of the microstructure will change, and the initiation and penetration of microcracks in the concrete matrix will be significantly inhibited and delayed. The evolution and accumulation process of the mesodamage modes will consequently change. For the yield damage mode, with the increase in *ρ* from 0% to 2.5%, *ε*<sup>a</sup> increases from 0.771 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 1.181 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , *<sup>ε</sup>*<sup>h</sup> increases linearly from 0.993 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 2.211 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , and *<sup>ε</sup>*<sup>b</sup> increases linearly from 1.311 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 2.441 <sup>×</sup> <sup>10</sup>−<sup>4</sup> . For the fracture damage mode, *H* increases linearly from 0.260 <sup>×</sup> <sup>10</sup>−<sup>4</sup> at 0% to 0.450 <sup>×</sup> <sup>10</sup>−<sup>4</sup> at 2.5%. The above-mentioned four parameters can be used to determine the shape of triangular probability distribution corresponding to the mesoscopic yield and fracture damage evolution process, which can demonstrate intuitionistic physical pictures to people to understand the entire process on a mesoscale.

(2) Mechanical behavior in macroscale

The macroscopic nonlinear stress–strain behavior of concrete is determined by the mechanical properties of the microstructure and the evolution of mesodamage. After adding steel fiber to the concrete matrix, with *ρ* varying from 0% to 2.5%, the composition and mechanical properties of the microstructure will change, which will lead the initial elastic modulus *E*<sup>0</sup> to decrease from 3.481 × 10 GPa to 3.046 × 10 GPa. The nominal stress– strain curves generally show "strengthened" features due to the change in the mesodamage evolution process characterized by the four parameters (*ε*a, *ε*h, *ε*b, and *H*) with the increase in fiber content. The nominal stresses corresponding to the peak nominal stress state and the critical state, *σ*N,p and *σ*N,cr, increase from 3.134 and 2.612 MPa to 4.713 and 3.251 MPa, respectively. The tensile strains corresponding to the peak nominal stress state and the critical state, *<sup>ε</sup>*N,p and *<sup>ε</sup>*N,cr, increase from 1.051 <sup>×</sup> <sup>10</sup>−<sup>4</sup> and 1.303 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 1.801 <sup>×</sup> <sup>10</sup>−<sup>4</sup> and 2.442 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , respectively. No significant change is observed in the slope of the descending branch of the nominal stress–strain curve, and the ductility performance is significantly improved.

Figure 10 shows the change curves of energy absorption capacity S with fiber volume fraction, where *S*<sup>p</sup> and *S*cr represent the energy absorption capacities corresponding to the peak nominal stress state and the critical state, respectively. With the increase in fiber content *ρ* from 0% to 2.5%, *S*<sup>p</sup> increases from 0.172 to 0.479 kPa, *S*cr increases from 0.266 to 0.749 kPa, and the difference between *S*cr and *S*<sup>p</sup> increases from 0.094 to 0.270 kPa. In previous studies, *S*<sup>p</sup> was commonly used to characterize the energy absorption capacity of concrete before failure. In the present study, *S*cr is suggested to characterize the energy absorption capacity of concrete before failure. This approach allows full consideration of the potential mechanical properties of the material.

the potential mechanical properties of the material.

**Figure 10.** Influence of fiber volume fraction on the energy absorption capacity. **Figure 10.** Influence of fiber volume fraction on the energy absorption capacity.

### *4.2. Comparison with the Test by Gao, 1991 4.2. Comparison with the Test by Gao, 1991*

inal stress state and the critical state,

nal stress state and the critical state,

formance is significantly improved.

tent 

Gao [20] also conducted a uniaxial tensile test for steel fiber-reinforced concrete specimen. The rectangular block with size of 100 mm × 100 mm × 500 mm is used in the experiment. Deformed steel bars with a diameter of 20 mm are inserted into both ends of the specimen. Fiber type is the melt-drawn steel fiber with length-to-diameter ratio of 50. The volume fractions of the steel fiber are 0.5%, 1.0%, 1.5%, and 2.0%. Gao [20] also conducted a uniaxial tensile test for steel fiber-reinforced concrete specimen. The rectangular block with size of 100 mm × 100 mm × 500 mm is used in the experiment. Deformed steel bars with a diameter of 20 mm are inserted into both ends of the specimen. Fiber type is the melt-drawn steel fiber with length-to-diameter ratio of 50. The volume fractions of the steel fiber *ρ* are 0.5%, 1.0%, 1.5%, and 2.0%.

) with the increase in fiber content. The nominal stresses corresponding to the peak nom-

and N,cr

to 4.713 and 3.251 MPa, respectively. The tensile strains corresponding to the peak nomi-

and

× 10−4 to 1.801 × 10−4 and 2.442 × 10−4, respectively. No significant change is observed in the slope of the descending branch of the nominal stress–strain curve, and the ductility per-

0.749 kPa, and the difference between *S*cr and *S*<sup>p</sup> increases from 0.094 to 0.270 kPa. In previous studies, *S*<sup>p</sup> was commonly used to characterize the energy absorption capacity of concrete before failure. In the present study, *S*cr is suggested to characterize the energy absorption capacity of concrete before failure. This approach allows full consideration of

Figure 10 shows the change curves of energy absorption capacity S with fiber volume fraction, where *S*p and *S*cr represent the energy absorption capacities corresponding to the peak nominal stress state and the critical state, respectively. With the increase in fiber con-

from 0% to 2.5%, *S*<sup>p</sup> increases from 0.172 to 0.479 kPa, *S*cr increases from 0.266 to

N,cr 

, increase from 3.134 and 2.612 MPa

, increase from 1.051 × 10−4 and 1.303

N,p

N,p 

As shown in Figure 11a, the predicted nominal stress–strain curves relevant to the homogeneous damage phase agree well with the test curves. The predicted effective stress–strain curves are shown in Figure 11b. The calculation parameters are listed in Table 4. The 3-D envelopes of N - and E - curves predicted are shown in Figure 11c,d, in order to better show the variation trend of the curves. It clearly shows that the shape of the curves has obvious similarity law with the increase in the volume ratio of steel fiber. With the increase of from 0.5% to 2.0%, the stresses and strains relevant to the peak nominal stress state and the critical state increase significantly. The change of the slope of the descending section of the nominal stress–strain curves is not obvious, mean-As shown in Figure 11a, the predicted nominal stress–strain curves relevant to the homogeneous damage phase agree well with the test curves. The predicted effective stress–strain curves are shown in Figure 11b. The calculation parameters are listed in Table 4. The 3-D envelopes of *σ*N-*ε* and *σ*E-*ε* curves predicted are shown in Figure 11c,d, in order to better show the variation trend of the curves. It clearly shows that the shape of the curves has obvious similarity law with the increase in the volume ratio of steel fiber. With the increase of *ρ* from 0.5% to 2.0%, the stresses and strains relevant to the peak nominal stress state and the critical state increase significantly. The change of the slope of the descending section of the nominal stress–strain curves is not obvious, meanwhile the ductility is further improved.

while the ductility is further improved. Figure 12a shows the relationship curves of *ρ*-*ε*a, *ε*h, *ε*b. With *ρ* ranging from 0.5% to 2.0%, *<sup>ε</sup>*<sup>a</sup> decreases linearly from 0.394 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 0.194 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , *ε*<sup>h</sup> decreases linearly from 0.164 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 0.030 <sup>×</sup> <sup>10</sup>−<sup>4</sup> , meanwhile *<sup>ε</sup>*<sup>b</sup> increases linearly from 1.601 <sup>×</sup> <sup>10</sup>−<sup>4</sup> to 2.411 <sup>×</sup> <sup>10</sup>−<sup>4</sup> . Figure 12b shows the relationship curves of *H*-*ρ*, *H* decreases linearly from 0.202 at 0.5% to 0.162 at 2.0%. Figure 12c shows the change curve of *E*<sup>0</sup> with *ρ*, which has no obvious changes. The fitting curves and fitting functions of the above five parameters are also shown in the figures.

Figure 13a shows the evolution curves of *D*R, which characterize the cumulative evolution process of the fracture damage mode of concrete on a mesoscale, respectively. With the increase of *ρ* from 0.5% to 2.0%, the evolution process of *D*<sup>R</sup> has been significantly delayed.

Figure 13b shows the evolution curves of *E*v, which characterizes the release process of potential mechanical capacity of concrete. Its evolution process has accelerated in early and then obviously delayed in later, with the increase of steel fiber content. When it reaches the critical state, *E*<sup>v</sup> = 1. It indicates that the potential mechanical capacity of the material to be fully released, and then the concrete specimen enters into the local failure stage. The whole process embodies the conversion from quantitative change to qualitative change.

**Figure 11.** Stress–strain curves under uniaxial tension: (**a**) nominal stress–strain curve (Plan view); (**b**) effective stress– strain curve (Plan view); (**c**) envelope of the nominal stress–strain curve (3D view); (**d**) envelope of the effective stress– strain curve (3D view). **Figure 11.** Stress–strain curves under uniaxial tension: (**a**) nominal stress–strain curve (Plan view); (**b**) effective stress–strain curve (Plan view); (**c**) envelope of the nominal stress–strain curve (3D view); (**d**) envelope of the effective stress–strain curve (3D view).

**Table 4.** Results for calculation parameter. **Table 4.** Results for calculation parameter.

a 

0.5% to 2.0%,

from 0.164 × 10−<sup>4</sup>


decreases linearly from 0.394 × 10−4 to 0.194 × 10−4

b 


*H* - 

a h b

 、 、

. With

increases linearly from 1.601 × 10−4 to 2.411

, h 

, *H* decreases linearly from 0.202

ranging from

decreases linearly

 

to 0.030 × 10−4, meanwhile

Figure 12a shows the relationship curves of

× 10−4. Figure 12b shows the relationship curves of

**Figure 12.** Influence of fiber volume fraction on the characteristic parameters: (**a**) *ρ-ε*a, *ε*h, *ε*<sup>b</sup> curves; (**b**) *H*-*ρ* curve; (**c**) *E*<sup>0</sup> -*ρ* curve.

**Figure 13.** Evolution curves of damage variable and evolution factor: (**a**) *D*R*-ε* curves; (**b**) *E*v-*ε* curves. **Figure 13.** Evolution curves of damage variable and evolution factor: (**a**) *D*R*-ε* curves; (**b**) *E*v-*ε* curves.

*E*v

from 0.5% to 2.5%, *S*<sup>p</sup> increases from 0.161 to 0.345 kPa, *S*cr increases from 0.266

cess of potential mechanical capacity of concrete. Its evolution process has accelerated in

material to be fully released, and then the concrete specimen enters into the local failure stage. The whole process embodies the conversion from quantitative change to qualitative

to 0.548 kPa; and the difference between *S*cr and *S*<sup>p</sup> increases from 0.105 to 0.203 kPa. The results show that the potential mechanical properties of materials could be fully considered if the energy absorption capacity corresponding to the critical state is adopted.

, which characterizes the release pro-

. With the increase in the fiber

Figure 13b shows the evolution curves of

 *S*cr *S*<sup>p</sup>

reaches the critical state,

change.

content

0.0

0.1

0.2

0.3

0.4

*S* (kPa)

0.5

0.6

0.7

Figure 14 shows the change curves of *S*p and *S*cr with

0.5 1.0 1.5 2.0

**Figure 14.** Influence of fiber volume fraction on the energy absorption capacity.

The nonlinear stress–strain behavior of fiber concrete is determined by many factors, which include water/cement contents, fiber type and content, aggregate source, additive type, specimen type, loading method and so on. Therefore, although the aforementioned two sets of tests are both for steel fiber concrete specimens, there are great differences both

*ρ* (%)

0.0

0.1

*D*R

0.2

0.3

Figure 14 shows the change curves of *S*<sup>p</sup> and *S*cr with *ρ*. With the increase in the fiber content *ρ* from 0.5% to 2.5%, *S*<sup>p</sup> increases from 0.161 to 0.345 kPa, *S*cr increases from 0.266 to 0.548 kPa; and the difference between *S*cr and *S*<sup>p</sup> increases from 0.105 to 0.203 kPa. The results show that the potential mechanical properties of materials could be fully considered if the energy absorption capacity corresponding to the critical state is adopted. Figure 14 shows the change curves of *S*p and *S*cr with . With the increase in the fiber content from 0.5% to 2.5%, *S*<sup>p</sup> increases from 0.161 to 0.345 kPa, *S*cr increases from 0.266 to 0.548 kPa; and the difference between *S*cr and *S*<sup>p</sup> increases from 0.105 to 0.203 kPa. The results show that the potential mechanical properties of materials could be fully considered if the energy absorption capacity corresponding to the critical state is adopted.

cess of potential mechanical capacity of concrete. Its evolution process has accelerated in early and then obviously delayed in later, with the increase of steel fiber content. When it

material to be fully released, and then the concrete specimen enters into the local failure stage. The whole process embodies the conversion from quantitative change to qualitative

*E*v

0.0

0.2

0.4

0.6

*E*<sup>v</sup>

(**a**) (**b**) **Figure 13.** Evolution curves of damage variable and evolution factor: (**a**) *D*R*-ε* curves; (**b**) *E*v-*ε* curves.

Figure 13b shows the evolution curves of

*E*v

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5

= 1. It indicates that the potential mechanical capacity of the

*ε* (×10-4)

, which characterizes the release pro-

*ρ =* 0.5% *ρ =* 1.0% *ρ =* 1.5% *ρ =* 2.0%

*Crystals* **2021**, *11*, x FOR PEER REVIEW 22 of 25

**Figure 14.** Influence of fiber volume fraction on the energy absorption capacity. **Figure 14.** Influence of fiber volume fraction on the energy absorption capacity.

The nonlinear stress–strain behavior of fiber concrete is determined by many factors, which include water/cement contents, fiber type and content, aggregate source, additive type, specimen type, loading method and so on. Therefore, although the aforementioned two sets of tests are both for steel fiber concrete specimens, there are great differences both The nonlinear stress–strain behavior of fiber concrete is determined by many factors, which include water/cement contents, fiber type and content, aggregate source, additive type, specimen type, loading method and so on. Therefore, although the aforementioned two sets of tests are both for steel fiber concrete specimens, there are great differences both in macroscopic stress–strain behavior and in mesoscopic damage evolution. The experimental results and theoretical analysis show that, in the same group of tests, when only the content of steel fiber is changed, the macroscopic constitutive behavior and mesoscopic damage evolution process of fiber concrete show good regularity with the change of fiber content.

### **5. Conclusions**

0.0 0.5 1.0 1.5 2.0 2.5

change.

*ρ =* 0.5% *ρ =* 1.0% *ρ =* 1.5% *ρ =* 2.0%

*ε* (×10-4)

reaches the critical state,


modes on a meso-scale. This model contains two kinds of feature parameters (*E* and *ε*a, *ε*h, *ε*b, *H*) with clear physical meanings, and has the ability to effectively reflect the above changes on meso-scale. Calculations were conducted to simulate the two sets of steel fiber concrete tensile tests in the literature. The experimental and theoretical analysis results show that, when only the fiber content is changed, the shape of the macroscopic nominal stress–strain curve will show a good law of similarity. With the increase of the fiber content, the values of stress and strain corresponding to the peak nominal stress state and the critical state linearly increase, and the curvature of the connecting part of the ascending and descending branch of the nominal stess–strain curve has the changing trend of gradual and orderly. Meanwhile, the characteristic parameters *ε*a, *ε*h, *ε*b, *H*, representing the two types of damage evolution of yield and fracture on a meso-scale, have obvious linear variation law with the change of fiber content. Through this model, the link among the physical mechanism, the mesoscopic damage mechanism and the macroscopic nonlinear constitutive behavior are effectively established.

3. The macroscopic constitutive behavior of FRC is a complex process of multiple factors. The influence factors include water/cement contents, source of aggregate, fiber type and content, type of additive, specimen size, loading mode, etc. Due to the limitation of the length of articles and test data, only two groups of steel fiber concrete test data are adopted in the validation analysis. Whether this constitutive model could be applicable to the analysis of the influence of other factors on the macroscopic mechanical behavior of fiber concrete, remains to be further researched later.

**Author Contributions:** Methodology, W.B.; data curation, validation and writing—original draft preparation, J.G. and S.H.; writing—review and editing, C.Y. and X.L.; funding acquisition, C.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant number "51679092, 51779095"; National Key R&D Program of China, grant number "2018YFC0406803" and the Science Technology Innovation Talents in Universities of Henan Province, China, grant number "20HASTIT013".

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are contained in this article.

**Acknowledgments:** The author thanks Yao Xianhua and Li Lilie of the School of Civil Engineering and Communication of North China University of Water Resources and Electric Power for their technical support.

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

### **References**


**Xuefeng Zhang <sup>1</sup> , Huiming Li <sup>1</sup> , Shixue Liang 2,\* and Hao Zhang <sup>1</sup>**


**Abstract:** This paper studies the behavior of lattice girder composite slabs with monolithic joint under bending. A full-scale experiment is performed to investigate the overall bending resistance, deflection and the final crack distribution of latticed girder composite slab under uniformly distributed load. A finite element model is given for the analysis of the latticed girder composite slabs. The effectiveness and correctness of the numerical simulations are verified against experimental results. The experimental and numerical studies conclude that the lattice girder composite slabs conform to the requirement of existing design codes. A parametric study is provided to investigate the effects of lattice girder with following conclusions: (a) the lattice girder significantly increases the stiffness of the slab when comparing with the precast slab without reinforcement crossing the interface; (b) the additional reinforcement near the joint slightly increases the stiffness and resistance, while it prevents damage near the joint.

**Keywords:** precast concrete structure; lattice girder semi-precast slabs; bending resistance; FE modelling; concrete damage
