Numerical Simulation Study on the Fracture Process of CFRP-Reinforced Concrete

Abstract

To investigate the crack extension mechanism in CFRP-reinforced concrete, this paper derives analytical expressions for the external load and crack opening displacement in the fracture process of CFRP concrete beams based on the crack emergence toughness criterion and the Paris displacement formula as the theoretical basis. A numerical iterative method was used to computationally simulate the fracture process of CFRP-reinforced concrete beams and to analyze the effect of different initial crack lengths on the fracture process. The research results indicate that the numerical simulation results of the crack initiation load are in good agreement with the test results, and the crack propagation curves and the test results are basically consistent before the CFRP-concrete interface peels off. The numerical results of ultimate load are lower than the test results, but it is safe for fracture prediction in actual engineering. With the increase in the initial crack length, the effect of the initial crack length on the critical effective crack propagation length is more obvious.

1. Introduction

As the basic material of modern building structure, it is difficult to avoid cracking problems during long-term service, and this has become a universal problem in the field of structural engineering [1,2,3,4]. Carbon fiber-reinforced polymers (CFRPs) are widely used for repairing and strengthening concrete structures with cracks due to their high strength, light weight, corrosion resistance, and convenient construction [5,6]. Despite the significant advantages and considerable development potential of carbon fiber reinforcement technology (CFRT) in the repair and strengthening of concrete structures, it still faces challenges and limitations, especially in crack extension mechanisms and nonlinear fracture behavior. Therefore, it is particularly important to study the relevant mechanical properties of CFRP concrete structures, especially the fracture properties, and reveal the crack propagation law of CFRP concrete structures, which is conducive to the improvement of CFRP reinforcement technology.

In recent years, the fracture properties of CFRP concrete structures have been widely investigated through both experimental testing and numerical modeling. Martin [7] proposed a new fracture analysis technique to evaluate the bonding effect of CFRP concrete structure, achieving a reinforcement balance between strength and fracture energy absorption. Kheyroddin et al. [8] analyzed the optimum tensile and compression bar model for CFRP concrete columns using finite element analysis, providing support for the improvement of CFRP reinforcement techniques. Elghandour et al. [9] investigated the fracture properties of CFRP concrete beams using numerical methods and quantified the contribution of CFRP to the shear capacity. Mensah et al. [10] investigated the effect of different bond parameters on the mechanical properties of the CFRP–concrete interface; analyzed the damage mode, ultimate load, load–slip, strain distribution, and bond–slip relationship between the CFRP and concrete interface; and revealed the fracture mechanism of CFRP concrete. Hejazi et al. [11] established a numerical model of crack extension in CFRP beams based on nonlinear fracture mechanics, revealing the crack resistance mechanism of CFRP reinforcement on the crack path of concrete structures and providing a theoretical basis for engineering applications. Maazoun et al. [12] developed a finite element model for analyzing the bond–slip behavior of CFRP concrete under static and blast loading and predicted the debonding damage modes, strain distributions, and layered loads, revealing the dynamic enhancement effect on bond–slip properties due to stress wave propagation and high strain rates. Shang et al. [13] investigated the fracture properties and damage modes of CFRP concrete beams using the digital image correlation method, proposed a method for evaluating the reinforcement effect of CFRP concrete beams, and utilized a research method to investigate the influence of CFRP bond defects on the reinforcement effect. Wang et al. [14] investigated the crack development law of CFRP concrete under different bending conditions; analyzed the cracking rate, maximum crack height, and crack morphology of the specimens at different stages; and established a calculation formula of the maximum crack width of CFRP concrete beams. Li et al. [15] proposed a new fatigue crack extension equation for CFRP RC beams, established a finite element model considering material nonlinearity and performance degradation under cyclic loading, and analyzed the fatigue crack extension behavior.

The above studies show that the simulation of the nonlinear fracture behavior of CFRP concrete is insufficient; the nonlinear simulation of the whole process of crack extension (from initiation to instability) is still imperfect; the numerical simulation of the interface peeling behavior has insufficient accuracy due to the strong nonlinearity of CFRP concrete in the fracture process [16,17,18], rendering it difficult to truly reflect the crack extension law. Therefore, the fracture criterion of CFRP concrete is constructed by considering the stress intensity factor resulting from external loading, cohesion, and CFRP crack-resistant action, and the bilinear softening principal structure is used to describe the tensile softening of the concrete and the shear-softening relationship at the CFRP–concrete interface. In addition, a staged iterative algorithm is developed to combine the Paris displacement formula and Gauss–Chebyshev integrals to realize the simulation of the full fracture process, which breaks through the limitations of the traditional finite element method in dealing with the nonlinearity of the interface. The method in this paper will break through the limitations of the traditional finite element method in the interface nonlinear problem, and it is more suitable for the reinforcement design of general building structures.

2. Computational Model

The crack model of a CFRP concrete beam is shown in Figure 1. The force acting on the crack surface consists of three components: the external load P, the cohesive force σ(x) on the crack surface in the fracture process zone, and the CFRP tensile stress σ_F. The initial crack length is a₀, the critical crack length is a_c, the height of the specimen is h, and the width of the specimen is b.

2.1. Calculation Formula of Stress Intensity Factor

According to Figure 1, the stress intensity factor $K_{\mathrm{IC}}$ at the crack tip of the CFRP concrete beam is superimposed by stress intensity factors $K_{\mathrm{IP}}$, $K_{\mathrm{I}\sigma }$, and $K_{\mathrm{IF}}$ resulting from the external load, cohesive force, and CFRP tensile stress.

$$K_{\mathrm{IC}}=K_{\mathrm{IP}}+K_{\mathrm{I}\sigma }+K_{\mathrm{IF}}$$

(1)

Referring to the literature [19,20], the stress intensity factor generated by the external load at the crack tip can be calculated using Equation (2):

$$\left\{\begin{array}{l}{K}_{\mathrm{IP}}={\displaystyle \frac{P}{b{h}^{1/2}}}k\left({\displaystyle \frac{a_{\mathrm{c}}}{h}}\right)\\ k\left({\displaystyle \frac{a_{\mathrm{c}}}{h}}\right)=3.675\left[1-0.12\left({\displaystyle \frac{a_{\mathrm{c}}}{h}}-0.45\right)\right]{\left(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}}\right)}^{-3/2}\end{array}\right.$$

(2)

The stress intensity factor generated by cohesion at the crack tip can be calculated by the infinite-length equal-width central zone crack model [21,22], as shown in Figure 2, which is given by the following:

$$\left\{\begin{array}{l}{K}_{\mathrm{I}\sigma }=-{\displaystyle {\int }_{a_0}^{a_{\mathrm{c}}}\frac{2\sigma (x)}{\sqrt{\pi a}}\frac{G(\frac{x}{a_{\mathrm{c}}},\frac{a_{\mathrm{c}}}{h})}{{\left(1-\frac{a_{\mathrm{c}}}{h}\right)}^{3/2}{(1-{(\frac{x}{a_{\mathrm{c}}})}^2)}^{1/2}}\mathrm{d}x}\\ G({\displaystyle \frac{x}{a_{\mathrm{c}}}},{\displaystyle \frac{a_{\mathrm{c}}}{h}})=g_1({\displaystyle \frac{a_{\mathrm{c}}}{h}})+g_2({\displaystyle \frac{a_{\mathrm{c}}}{h}}){\displaystyle \frac{x}{a_{\mathrm{c}}}}+g_3{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2{\displaystyle \frac{x}{a_{\mathrm{c}}}}+g_4{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^3{\displaystyle \frac{x}{a_{\mathrm{c}}}}\end{array}\right.$$

(3)

where

$$\left\{\begin{array}{l}{g}_1({\displaystyle \frac{a_{\mathrm{c}}}{h}})=0.46+3.06{\displaystyle \frac{a_{\mathrm{c}}}{h}}+0.84{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^5+0.66{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2\\ g_2({\displaystyle \frac{a_{\mathrm{c}}}{h}})=-3.52{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2\\ g_3({\displaystyle \frac{a_{\mathrm{c}}}{h}})=6.17-28.22{\displaystyle \frac{a_{\mathrm{c}}}{h}}+34.54{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2-14.39{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^3-{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^{3/2}-5.88{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^5-2.64{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2\\ g_4({\displaystyle \frac{a_{\mathrm{c}}}{h}})=-6.63+25.16{\displaystyle \frac{a_{\mathrm{c}}}{h}}-31.04{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2+14.41{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^3-2{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^{3/2}+5.04{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^5+1.98{({\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2{(1-{\displaystyle \frac{a_{\mathrm{c}}}{h}})}^2\end{array}\right.$$

The stress intensity factor resulting from the crack-resistant action of CFRP can be calculated by referring to the infinite-length equal-width central zone crack model, which equates the action of CFRP on concrete as a pair of concentrated forces [23] acting at the bottom of the beam, which is given by

$$\left\{\begin{array}{l}{K}_{\mathrm{IF}}={\displaystyle \frac{2t_{\mathrm{F}}{\sigma }_{\mathrm{F}}}{\sqrt{\pi a_0}}}F\left({\alpha }_{\mathrm{c}}\right)\\ F\left({\alpha }_{\mathrm{c}}\right)={\displaystyle \frac{3.52}{{\left(1-{\alpha }_{\mathrm{c}}\right)}^{3/2}}}-{\displaystyle \frac{4.35}{{\left(1-{\alpha }_{\mathrm{c}}\right)}^{1/2}}}+2.13\left(1-{\alpha }_{\mathrm{c}}\right), {\alpha }_{\mathrm{c}}=x/a_{\mathrm{c}}\end{array}\right.$$

(4)

2.2. Calculation Formula of Crack Opening Displacement

The crack opening displacements ${\delta }_P(x)$, ${\delta }_{\sigma }(x)$, and ${\delta }_F(x)$ caused by external loads, cohesive forces, and CFRP tensile stresses can also be calculated using linear superposition via Equation (5).

$$\delta (x)={\delta }_P(x)+{\delta }_{\sigma }(x)+{\delta }_F(x)$$

(5)

The displacement expression can be calculated using the Paris displacement formula [24], assuming that there is a crack-containing unit-thickness thin plate subjected to an external force P. A pair of virtual forces F is introduced in the direction of the line connecting D₁ and D₂, as shown in Figure 3:

$$\delta ={\displaystyle \frac{2}{E’}}{\displaystyle {\int }_x^{a_{\mathrm{c}}}{K}_{IP}}{\displaystyle \frac{\partial K_{IF}}{\partial F}}\mathrm{d}\xi$$

(6)

where for the plane stress, $E’=E$. For the plane strain, $E’=E(1-{\nu }^2)$. E is the elastic modulus of concrete, and $\nu$ is Poisson’s ratio; $K_{IP}$ and $K_{IF}$ are the stress intensity factors produced by P and F on the crack tip, respectively.

The crack opening displacement caused by the external load is

$${\delta }_P(x)={\displaystyle \frac{2}{E’}}{\displaystyle {\int }_x^{a_{\mathrm{c}}}\frac{P}{b{h}^{1/2}}k\left(\frac{a_{\mathrm{c}}}{h}\right) \times }{\displaystyle \frac{2}{\sqrt{\pi \xi }}}{\displaystyle \frac{G\left(\frac{x}{\xi },\frac{\xi }{h}\right)}{{\left(1-\frac{\xi }{h}\right)}^{3/2}{(1-{(\frac{x}{\xi })}^2)}^{1/2}}}\mathrm{d}\xi$$

(7)

where $\xi$ is the integrating variable.

The crack opening displacement caused by cohesion is

$${\delta }_{\sigma }(x)=-{\displaystyle \frac{2}{E’}}{\displaystyle {\int }_x^{a_{\mathrm{c}}}\frac{2}{\sqrt{\pi \xi }}\frac{G\left(\frac{x}{\xi },\frac{\xi }{h}\right)}{{\left(1-\frac{\xi }{h}\right)}^{3/2}{(1-{(\frac{x}{\xi })}^2)}^{1/2}}\mathrm{d}\xi }\times {\displaystyle {\int }_{a_0}^{\xi }\frac{2\sigma (\eta )}{\sqrt{\pi \xi }}\frac{G\left(\frac{\eta }{\xi },\frac{\xi }{h}\right)}{{\left(1-\frac{\xi }{h}\right)}^{3/2}{(1-{(\frac{\eta }{\xi })}^2)}^{1/2}}\mathrm{d}\eta }$$

(8)

where $\eta$ is the integrating variable.

The crack opening displacement caused by the CFRP crack-arresting effect is

$${\delta }_P(x)={\displaystyle \frac{2}{E’}}{\displaystyle {\int }_x^{a_{\mathrm{c}}}\frac{2t_{\mathrm{F}}{\sigma }_{\mathrm{F}}}{\sqrt{\pi a_0}}F\left({\alpha }_{\mathrm{c}}\right) \times }{\displaystyle \frac{2}{\sqrt{\pi \xi }}}{\displaystyle \frac{G\left(\frac{x}{\xi },\frac{\xi }{h}\right)}{{\left(1-\frac{\xi }{h}\right)}^{3/2}{(1-{(\frac{x}{\xi })}^2)}^{1/2}}}\mathrm{d}\xi$$

(9)

where t_F is the thickness of CFRP.

2.3. Softening Constitutive Relation

The tensile softening constitutive relation between cohesion and crack opening displacement can be modeled by a bilinear model [25,26], as shown in Figure 4, which is given by

$$\sigma =\left\{\begin{array}{l}{f}_{\mathrm{t}}-(f_{\mathrm{t}}-{\sigma }_{\mathrm{s}})\delta /{\delta }_{\mathrm{s}}, 0\le \delta \le {\delta }_{\mathrm{s}}\\ {\sigma }_{\mathrm{s}}({\delta }_0-\delta )/({\delta }_0-{\delta }_{\mathrm{s}}), {\delta }_{\mathrm{s}}\le \delta \le {\delta }_0\\ 0, \delta \ge {\delta }_0\end{array}\right.$$

(10)

where ${\sigma }_{\mathrm{s}}$, ${\delta }_0$ and ${\delta }_{\mathrm{s}}$ are material parameters, which can be calculated according to ${\sigma }_{\mathrm{s}}=f_{\mathrm{t}}/3$, ${\delta }_0=3.6G_F/f_{\mathrm{t}}$ and ${\delta }_{\mathrm{s}}=0.8G_F/f_{\mathrm{t}}$ in reference [27]. $f_{\mathrm{t}}$ is the tensile strength of concrete; $G_F$ is the fracture energy of the specimen.

The fracture failure of the CFRP concrete beam is caused by the peeling failure of the CFRP–concrete interface rather than the tensile failure of CFRP. The crack-arresting effect of CFRP comes from the CFRP–concrete interface’s shear stress, and the interfacial shear stress is used to replace the CFRP tensile stress in this paper. Therefore, it is necessary to consider the ontological shear-softening relationship at the CFRP–concrete interface, which is also modeled by the bilinear model [28,29], as shown in Figure 5, and given by

$$\tau =\left\{\begin{array}{l}{\tau }_u\delta /{\delta }_1, 0\le \delta \le {\delta }_1\\ {\tau }_u\left({\delta }_{\mathrm{F}}-\delta \right)/\left({\delta }_{\mathrm{F}}-{\delta }_1\right), {\delta }_1\le \delta \le {\delta }_{\mathrm{F}}\\ 0, \delta \ge {\delta }_{\mathrm{F}}\end{array}\right.$$

(11)

where ${\tau }_u$ is the peak value of the shear stress; $\delta$ is the interface’s slip; ${\delta }_1$ is the slip corresponding to ${\tau }_u$; ${\delta }_{\mathrm{F}}$ is the corresponding critical slip when the shear stress is 0. Referring to the literature [30], the calculations can be performed according to ${\tau }_u=1.5{\beta }_w{f}_{\mathrm{t}}$, ${\delta }_1=0.0195{\beta }_w{f}_{\mathrm{t}}$, and ${\delta }_f=1.1088\sqrt{f_{\mathrm{t}}}/t_u$.

3. Simulation Methodology and Solution Step

The three-point bending fracture test of the CFRP concrete beam simulated in this paper is shown in Figure 6, which has a size of 100 mm × 100 mm × 400 mm (h × b × l), and its moment equilibrium condition can be obtained as

$$M={\displaystyle \frac{PS}{2}}+{\displaystyle \frac{WS}{4}}$$

(12)

The numerical simulation of the fracture process of CFRP concrete beams can be divided into two parts: before and after crack initiation. It is necessary to first calculate the value of the crack initiation load, which can be theoretically derived by the flat cross-section assumption [31], and the equilibrium conditions for the mid-span cross-section are shown in Figure 7.

According to the force equilibrium condition and moment equilibrium condition in Figure 7, the following equations can be obtained:

$$b\left[-{\displaystyle \frac{{\sigma }_{\mathrm{c}}\left(h-h_{\mathrm{c}}-a_0\right)}{2}}+{\displaystyle \frac{f_{\mathrm{r}}{h}_{\mathrm{c}}}{2}}\right]+{\sigma }_{\mathrm{F}}b{h}_{\mathrm{F}}=0$$

(13)

$$M=b\left[{\displaystyle \frac{{\sigma }_{\mathrm{c}}{\left(h-h_{\mathrm{c}}-a_0\right)}^2}{3}}+{\displaystyle \frac{f_{\mathrm{r}}{h}_{\mathrm{c}}^2}{3}}\right]+{\sigma }_{\mathrm{F}}b{h}_{\mathrm{F}}\left(h_{\mathrm{c}}+a_0\right)$$

(14)

where $f_{\mathrm{r}}$ is the flexural tensile strength of an uncracked concrete beam; ${\sigma }_{\mathrm{c}}$ is the compressive stress of the concrete compression zone; ${\sigma }_{\mathrm{F}}$ is the tensile stress of CFRP; a₀ is the initial crack length of the concrete beam.

According to the equilibrium condition of the mid-span section in Figure 7, the relationship between ${\sigma }_{\mathrm{c}}$ and ${\sigma }_{\mathrm{F}}$ can be obtained as follows:

$${\sigma }_{\mathrm{c}}={\displaystyle \frac{h-h_{\mathrm{c}}-a_0}{h_{\mathrm{c}}}}{f}_{\mathrm{r}}$$

(15)

$${\sigma }_{\mathrm{F}}={\displaystyle \frac{E_{\mathrm{F}}\left(a_0+h_{\mathrm{c}}\right)}{E_{\mathrm{c}}{h}_{\mathrm{c}}}}{f}_{\mathrm{r}}$$

(16)

Substituting Equations (15) and (16) into Equation (13) yields

$${\Phi }_{11}\left(h_{\mathrm{c}}\right)=0$$

(17)

Substituting Equations (15) and (16) into Equation (14) yields

$${\Phi }_{12}\left(P,h_{\mathrm{c}}\right)=0$$

(18)

The crack initiation load $P_{\mathrm{theory}}^{\mathrm{ini}}$ can be calculated by associating Equations (17) and (18), and the calculation flowchart is shown in Figure 8.

After calculating the crack initiation load, the crack initiation toughness $K_{\mathrm{IC}}$ and the crack opening displacement corresponding to the crack initiation point can be calculated by substituting them into Equations (2) and (11). After crack initiation of the concrete beam, the effective crack length is used as the expansion increment [32,33], and the external loads and crack opening displacements are calculated step by step using numerical iteration with the coupling of Equations (1), (5), (10) and (11). The specific iteration steps are as follows:

(1)

We are arbitrarily given an initial value of the displacement distribution on the cohesive zone and in the bonded zone at the CFRP–concrete interface.

(2)

Calculate the stress distribution in the cohesive and interfacial bonding zone according to Equations (10) and (11).

(3)

Calculate the external load P at that time step by substituting $K=K_{\mathrm{IC}}$ and the cohesion and shear stress distributions into Equation (1).

(4)

Substitute the external load P, cohesive force, and shear stress into Equations (2)–(9), respectively, to solve for the new displacement distribution.

(5)

Determine whether the calculated new displacement distribution satisfies the convergence condition: $\max {\left|{\delta }_i(x)-{\delta }_{i-1}(x)\right|}_{a_0\le x\le a_c}<\Delta$ (the Δ value is set to 0.01 mm). If it is not satisfied, repeat steps (2) through (4) for iterative calculations.

(6)

Finally, when the effective crack length is less than the height of the concrete beam, the extension is increased for iterative calculation until the crack extends to the edge of the specimen to stop the calculation and output the results.

The computational process of the extension phase is shown in Figure 9, and the corresponding computational program is prepared. Numerical integration was performed using the Gauss–Chebyshev integral formula. The calculation results show that different initial displacements can converge quickly.

4. Validation of Numerical Simulation Results

To verify the reasonableness of the methodology, CFRP concrete beam specimens with different initial crack lengths under static loading in the literature [34] were used for verification calculations. The initial crack lengths of the concrete beams were 20 mm, 30 mm, 40 mm, 50 mm, and 60 mm, respectively. The compressive strength of concrete was 30MPa, the CFRP thickness was 0.167 mm, the CFRP tensile strength was 3400 MPa, and the elastic modulus of CFRP was 227 Gpa. The crack initiation load, ultimate load, and corresponding crack mouth opening displacement of CFRP concrete beams with different initial crack lengths were calculated (the iteration numbers of the calculation process are 22, 20, 28, 32, and 27, respectively) and compared with data means in reference [34], as shown in

Table 1. Comparison between numerical calculation results and test results.

	a0	Pini/kN	CMODini/mm	Pul/kN	CMODul/mma	ac/mm
Computed	20 mm	6.938	0.030	15.620	1.611	55.731
Simulated		6.827	0.024	12.997	0.291	59.122
Computed	30 mm	6.933	0.038	16.844	1.652	65.366
Simulated		6.901	0.037	14.297	0.291	64.361
Computed	40 mm	6.215	0.040	17.016	1.784	72.194
Simulated		5.966	0.058	15.698	0.338	69.781
Computed	50 mm	4.104	0.041	16.843	1.904	81.533
Simulated		3.988	0.039	15.148	0.358	74.132
Computed	60 mm	1.996	0.017	16.143	1.423	85.561
Simulated		1.904	0.019	14.913	0.371	79.682

. Based on the numerical simulation results, the load–crack mouth opening displacement (P-CMOD) curves were plotted, as shown in Figure 10.

As can be seen in Table 1, the theoretical values of the crack initiation load derived using the calculation process in Figure 8 are in good agreement with the test results, indicating that the crack initiation load calculated using the method in this paper is of high accuracy. However, there are some differences between the numerically simulated ultimate load and the test results: The simulated value is small, but the variation law with the initial crack length is the same. This is due to the fluctuation of the CFRP–concrete interface in the actual test, where the ultimate load can appear at a point in the fluctuation phase, while for the numerical simulation, the unstable peeling phase is difficult to represent, which can be found from the P-CMOD curves in Figure 10.

From Figure 10, it can be seen that the P-CMOD curves of numerical simulation are similar to the test results before the crack initiation of concrete beams, and the external load P increases with the increase in CMOD. When CFRP concrete beams start cracking, the concrete virtual crack expands, and CFRP plays the role of crack arresting. The external load P increases nonlinearly with CMOD, but the increase slows down. As the concrete cracks continued to expand, a small decrease in the external load P occurred. When CFRP exerts its crack-arresting effect, the external load P increases subsequently, and the numerically simulated curves show that P increases up to the ultimate load and then does not increase any more, and it gradually tends to decrease linearly with the concrete cracks’ extension; this occurs because the method used in this paper is unable to simulate the exfoliation of the CFRP–concrete interface. In fact, the peeling process at the CFRP–concrete interface is a complex nonlinear process, and the accurate prediction of the interfacial shear stress distribution and the concrete cracks extension process needs to be further investigated.

In addition, comparing the fracture parameters of CFRP concrete beams with different initial crack lengths, it can be seen that as the initial crack length increases, the critical crack length shows an increasing trend. To further analyze the effect of the initial crack length on the deformation capacity of CFRP concrete, the critical effective crack extension length $\Delta a_{\mathrm{c}}$ is calculated, and the means of the numerical and test results are shown in Figure 11.

As can be seen from Figure 11, the variation law of the numerical and test results is consistent, and the critical effective crack extension length decreases with an increase in the initial crack length, which is due to the smaller bearing area, resulting in a decrease in the effective crack extension length. However, the numerical results decrease linearly with increasing initial crack length, with a more pronounced downward trend, which may be due to the limitations of numerical simulations for boundary size effects. When the initial crack length is 30mm, the numerical results and the test results are in good agreement, and the error of the numerical method gradually increases with an increase in the initial crack length, indicating that the numerical simulation method can serve the purpose of accurate prediction for the structure with the short initial crack length, and the calculation is simple and fast, providing a technical method for engineering applications.

5. Sensitivity Analysis of Key Parameters

To verify the rationality of the model in this paper and the reliability of the assumptions, the sensitivity analysis of the critical parameters is carried out in this section using the single-parameter perturbation method. Seven critical parameters were selected for sensitivity analysis, including the initial crack length (a₀), concrete tensile strength (f_t), interface shear strength (τ_u), concrete fracture energy (G_F), CFRP modulus of elasticity (E_F), CFRP thickness (t_F), and critical slip at the interface (δ_F). A ±15% perturbation range is set for each parameter to assess the extent of its effect on the ultimate structural load and the critical effective crack extension length. The normalized sensitivity coefficient method shown in Equation (19) is used for sensitivity calculation:

$$s={\displaystyle \frac{\left(Y_{\max }-Y_{\min }\right)/Y_{\mathrm{reference}}}{\left(X_{i,\max }-X_{i,\min }\right)/X_{i,\mathrm{reference}}}}$$

(19)

where Y is the output variable, and X_i is the input parameter.

According to the typical range classification of the sensitivity coefficients, when S_i > 1, the degree of parameter influence is highly sensitive, and its small changes can lead to significant fluctuations in the output results. When 0.5 < S_i ≤ 1, the degree of parameter influence is moderately sensitive, and its influence is linearly related to the magnitude of change. When S_i ≤ 0.1, the parameter can be considered to have a negligible effect on the resultant output. The critical parameters were calculated as shown in

Table 2. Sensitivity analysis results of critical parameters.

Parameter	Si (Pul)	Conclusion	Si (Δac)	Conclusion
a0	1.82	Highly sensitive	2.15	Highly sensitive
ft	1.25	Highly sensitive	0.78	Moderately sensitive
τu	0.93	Moderately sensitive	1.42	Highly sensitive
GF	0.68	Moderately sensitive	1.10	Highly sensitive
EF	0.45	Minimally sensitive	0.32	Minimally sensitive
tF	0.30	Minimally sensitive	0.25	Minimally sensitive
δF	0.18	Minimally sensitive	0.21	Minimally sensitive

.

As can be seen in Table 2, the initial crack length has the most significant effect on the ultimate load and the critical effective crack extension length, which is consistent with the fracture mechanics theory stating that the crack size directly affects the stress intensity factor. The sensitivity of the concrete’s tensile strength to the ultimate load is high, but the effect on crack propagation is relatively small, indicating that its load-bearing capacity is not ductile. The sensitivity of the interfacial shear strength to crack extension is significantly higher than its effect on the ultimate load, indicating that the bonding properties of the CFRP–concrete interface play a key role in inhibiting crack extension. In contrast, the sensitivity of the CFRP elastic modulus and thickness is relatively low, indicating that the effect of simply increasing CFRP stiffness or thickness on the lifting bearing capacity is limited, resulting in significant deviations in the prediction of structural response after interface debonding and indirectly explaining the difference between simulation results and test results. Therefore, further research needs to incorporate finer interfacial modeling and multi-field coupling analysis to improve the accuracy of debonding prediction.

6. Conclusions

In this paper, the expressions for the external load and crack opening displacement during crack extension in CFRP concrete beams were derived based on the extension criterion of crack initiation toughness and the Paris displacement formula. The crack initiation load, ultimate load, crack opening displacement, and P-CMOD curves were simulated, and the effects of different initial crack lengths on the fracture parameters were analyzed and compared with the test results in the literature [34]. The following conclusions can be drawn:

(1)

The crack initiation load of CFRP concrete beams can be accurately predicted using the calculation method in this paper, and the numerical simulation results are in good agreement with the test results, providing a new method for the determination of the crack initiation load.

(2)

A staged iterative algorithm combining the Paris displacement formulation and Gauss–Chebyshev integrals was developed to realize the simulation of the full fracture process, which breaks through the limitations of the traditional finite element method in dealing with interface nonlinearities. There are some errors in the numerical methods used in this paper when calculating the ultimate load and its corresponding crack opening displacement. The numerical results are lower than the test results, indicating that the calculation results are conservative, but they can provide a safe guarantee for the prediction of fracture behavior in engineering, which can not only improve the fracture theory of CFRP concrete but also provide a reliable analytical method.

(3)

The numerical results of the critical effective crack extension length of CFRP concrete beams under static loading decrease linearly with an increase in the initial crack length, which is consistent with the variation law of the test results. When the initial crack length is 30 mm, the numerical results are in good agreement with the test results, indicating that the numerical simulation method in this paper can serve the purpose of accurate prediction for structures with short initial crack lengths.

(4)

The sensitivity analysis of seven critical parameters by the single-parameter perturbation method showed that the initial crack length and concrete tensile strength had the most significant effect on the ultimate load and critical crack extension length. The initial crack growth can reduce the bearing capacity of concrete, and tensile strength can delay crack propagation. The interface strength and fracture energy of concrete have a significant effect on inhibiting crack propagation, but they have little effect on the bearing capacity. The elastic modulus and thickness sensitivity of CFRP are low, especially after interface debonding. Therefore, priority should be given to controlling initial cracking, improving concrete tensile properties, and optimizing interfacial treatments rather than over-reliance on CFRP dosage in practical engineering.