# Analyzing the Bond Behavior of Fiber-Reinforced Polymer (FRP) Bars Embedded in Engineered Cementitious Composites (ECCs) with the Nonlocal Continuum Rod Model

Authors: Qiuxiang Li; Mingfu Fu; Banghua Xie
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this study, a nonlocal elastic rod model is applied to analytically evaluate the bond behavior between fiber-reinforced polymer (FRP) bars and engineered cementitious composites (ECCs). The second-order differential equation, which is based on nonlocal elasticity theory, governs the bond behavior of the FRP bars along the bond length. The classical elasticity model is a special case of the nonlocal model. The solution of the second-order differential equation can be obtained by substituting three-stage linear bond stress-slip relationship of the FRP bars. The slip values (solution of the second-order differential equation) within the bond length calculated by the nonlocal continuum rod model are affected by the nonlocal parametere0a. The results from a case study show that the maximum pullout force decreases when the nonlocal size effect is considered, thereby providing a closer approximation of the experimental data than the existing local model.


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## Body

## 1. Introduction

Fiber-reinforced polymer (FRP) bars are used as an alternative reinforcement for concrete structures due to their high strength, light weight, and high corrosion resistance. Due to the crack bridging effect provided by dispersed fiber reinforcements, engineered cementitious composites (ECCs) feature good postcracking resistance and ductility and sustain multiple stable microcracks during loading; in contrast, major cracks form in normal concrete (NC) during loading [1]. Hence, a combination of the advantages of FRPs and ECCs would be beneficial for construction projects [2–5]. The bond mechanism between FRP bars and ECC is a critical design parameter that controls the performance of FRP/ECC structures at serviceability limit states. Similar to steel bars, the bond behavior of FRP bars is usually presented by a relationship between the shear stress of the bond (τ) and the relative displacement (slip) between the bar and the matrix. Malvar [6] proposed the first bond-slip constitutive model for GFRP based on many experimental data. Later, the modified Bertero–Eligehausen–Popov (mBEP) model was presented by Cosenza et al. [7] for FRP bars, which adopted the bond-slip constitutive law proposed by Eligehausen et al. [8]. Furthermore, some scholars have updated these models to suit the bond properties of FRP bars in different materials and environments [9, 10]. In the abovementioned bond-slip relationships, the bond stress is the average bond stress within the bond length zone, and the slip corresponds to the amount of slip at the loaded end or free end. Hence, these relationships represented the overall bond-slip behavior, whereas the influence of the anchoring position is not considered. Mi [11] obtained local bond-slip curves during a pullout experiment by installing a strain probe at different positions that were predrilled inside an FRP bar. Based on the obtained local bond-slip constitutive relationship, the mechanical behavior of the FRP bar was theoretically analyzed and calibrated with the finite element method. Mazaheripour et al. [12] proposed a local bond-slip model with four stages, which they used to calculate the pullout force of FRP bars in steel fiber-reinforced self-compacting concrete (SFRSC); they verified the results of their model through a comparison with experimental data. However, there were still some differences between the theoretical calculation results and the experimental results.Nonlocal elasticity theory [13] is an integrated approach that accounts for characteristic length scale effects and considers the stress at a point in a material as a function of strain within a finite surrounding volume. One of the advantages of this approach is the analysis of uneven stress distributions caused by irregular surfaces and microcracks. Based on this model, many studies have examined the size-dependent effects on the mechanical behavior of bolts [14], anchors [15], rods [16–18], beams [19–21], and plates [22–24]. As shown in [25, 26], the stress in the FRP is not evenly distributed within the bond length zone. In addition, the FRP surface has irregularities, cracks, and defects owing to differences in the production processes of different manufacturers and the lack of standardization for surface characteristics. Therefore, these factors can be considered by the nonlocal size effect rather than being ignored or replaced by additional experimental data when analyzing the mechanical behavior of the reinforcement. Hence, there are sufficient reasons to assess the influence of the nonlocal size effect on the mechanical behavior of FRP bars. To the best of the authors’ knowledge, no research has been reported on the influence of nonlocal effects on the bond behavior between FRP bars and ECC.In this study, we confined our attention to the influence of the nonlocal size effect on the bond behavior between FRP bars and ECC. A size-dependent elastic model was developed to analyze the mechanical bond behavior between FRP bars and ECC. Based on nonlocal elasticity theory, the second-order differential equation that governs the bond behavior of the bar along the bond length was derived. The proposed nonlocal model, which combines the three linear bond stress-slip relationships in [11], was used to theoretically analyze the bond behavior of FRP bars with infinite/finite bond lengths. Finally, an engineering case study was carried out to evaluate the influence of nonlocal size parameters on the slip and pullout force within the bond length of the FRP bars.


## 2. Governing Equations of the Nonlocal Elastic Model

### 2.1. Nonlocal Continuum Rod Model

Based on classical elastic continuum mechanics, the nonlocal elasticity theory proposed by Eringen and Edelen [27, 28] reveals that the stress field at a desired point can be expressed by a function of the strain field of all points existing in the configuration. Hence, the essence of Eringen’s nonlocal theory is the representation of the strains in terms of stresses and stress gradients. Hereinafter, the FRP bar is considered an elastic rod. For a one-dimensional problem in which the longitudinal coordinates are represented by x, Eringen’s nonlocal constitutive relation can be expressed as [29–31](1)1−e0a2∂2∂x2σxx=Eεxx,where e0 is the small length scale coefficient, a is an internal characteristic length (e.g., lattice length or bond length), σxx is the nonlocal axial stress, εxx is the axial strain, and E is the longitudinal modulus of elasticity. The nonlocal parameter e0a is introduced to consider the significance of the nonlocal elastic stress field.


### 2.2. Governing Equations

Before the governing equations can be derived, the following considerations must be taken into account. (1) Within the bond length, the FRP bar exhibits elastic deformation only. In contrast with steel bars, FRP bars have a relatively low Young’s modulus and do not have a yielding stage in the stress-strain response, which creates additional challenges in the flexural behavior of concrete members reinforced with FRP bars. To satisfy the requirements for serviceability limit states, FRP bars must exhibit elastic deformation only. (2) In this study, the FRP bars and ECC are considered to be always in contact. The good postcracking resistance and ductility response of the FRP/ECC system results in the formation of multiple stable microcracks during pullout failure, whereas major cracks form in NC. Therefore, only the bond behavior between the FRP bars and ECC during pullout failure is studied. (3) The ECC exhibits and sustains much less deformation than the FRP bars and can therefore be ignored. According to the force diagram of the embedded FRP bar in Figure1, the relationship between the bond shear stress τ of the FRP/ECC interface and the axial stress σxx of the bar can be expressed as(2)τδxπddx=Aσxx+dσxx−Aσxx.Figure 1
Force diagram of the FRP bar.This formula can be further simplified as(3)dσxxdx−4dτδx=0,where d and A are the diameter and the cross-sectional area of the FRP bar, respectively, and δx represents the slip between the FRP bar and surrounding ECC at a distance of x from the free end.By neglecting the deformation of the surrounding concrete, the slipδx can be represented by an integral of the strain in the FRP bonded segment. The resulting equation can be expressed as(4)δx=δF+∫εxxdx,where δF represents the slip between the FRP bar and surrounding ECC at the free end.Using equations (1)–(4), a governing equation for the bond behavior along the bond length in the nonlocal elastic model can be found in terms of slip. This equation is expressed as(5)∂2δx∂x2−J⋅1−e0a2∂2∂x2⋅τδx=0,where J=4/Ed. Equation (5) is a fundamental equation in the nonlocal rod model that represents the bond behavior along the bond length of the FRP bar. When e0a=0, equation (5) degenerates into the classical elastic model shown in [11, 12].


## 2.1. Nonlocal Continuum Rod Model

Based on classical elastic continuum mechanics, the nonlocal elasticity theory proposed by Eringen and Edelen [27, 28] reveals that the stress field at a desired point can be expressed by a function of the strain field of all points existing in the configuration. Hence, the essence of Eringen’s nonlocal theory is the representation of the strains in terms of stresses and stress gradients. Hereinafter, the FRP bar is considered an elastic rod. For a one-dimensional problem in which the longitudinal coordinates are represented by x, Eringen’s nonlocal constitutive relation can be expressed as [29–31](1)1−e0a2∂2∂x2σxx=Eεxx,where e0 is the small length scale coefficient, a is an internal characteristic length (e.g., lattice length or bond length), σxx is the nonlocal axial stress, εxx is the axial strain, and E is the longitudinal modulus of elasticity. The nonlocal parameter e0a is introduced to consider the significance of the nonlocal elastic stress field.


## 2.2. Governing Equations

Before the governing equations can be derived, the following considerations must be taken into account. (1) Within the bond length, the FRP bar exhibits elastic deformation only. In contrast with steel bars, FRP bars have a relatively low Young’s modulus and do not have a yielding stage in the stress-strain response, which creates additional challenges in the flexural behavior of concrete members reinforced with FRP bars. To satisfy the requirements for serviceability limit states, FRP bars must exhibit elastic deformation only. (2) In this study, the FRP bars and ECC are considered to be always in contact. The good postcracking resistance and ductility response of the FRP/ECC system results in the formation of multiple stable microcracks during pullout failure, whereas major cracks form in NC. Therefore, only the bond behavior between the FRP bars and ECC during pullout failure is studied. (3) The ECC exhibits and sustains much less deformation than the FRP bars and can therefore be ignored. According to the force diagram of the embedded FRP bar in Figure1, the relationship between the bond shear stress τ of the FRP/ECC interface and the axial stress σxx of the bar can be expressed as(2)τδxπddx=Aσxx+dσxx−Aσxx.Figure 1
Force diagram of the FRP bar.This formula can be further simplified as(3)dσxxdx−4dτδx=0,where d and A are the diameter and the cross-sectional area of the FRP bar, respectively, and δx represents the slip between the FRP bar and surrounding ECC at a distance of x from the free end.By neglecting the deformation of the surrounding concrete, the slipδx can be represented by an integral of the strain in the FRP bonded segment. The resulting equation can be expressed as(4)δx=δF+∫εxxdx,where δF represents the slip between the FRP bar and surrounding ECC at the free end.Using equations (1)–(4), a governing equation for the bond behavior along the bond length in the nonlocal elastic model can be found in terms of slip. This equation is expressed as(5)∂2δx∂x2−J⋅1−e0a2∂2∂x2⋅τδx=0,where J=4/Ed. Equation (5) is a fundamental equation in the nonlocal rod model that represents the bond behavior along the bond length of the FRP bar. When e0a=0, equation (5) degenerates into the classical elastic model shown in [11, 12].


## 3. Bond Stress-Slip Relationship

In [11], twenty pullout experiments were performed with round/ribbed deformation FRP (carbon FRP (CFRP) and glass FRP (GFRP)) bars, and a three-stage linear relationship was obtained from the results, as shown in Figure 2. This relationship is used as the bond shear stress-slip τ−δ relationship between the FRP bars and ECC in this study. This τ−δ relationship can be divided into three stages: an elastic stage, a softening stage, and a frictional stage. These stages are described hereinafter:(6)τδx=τ0+τm−τ0δ1δ,0≤δ≤δ1elastic,τm−τm−τfδ2−δ1δ−δ1,δ1≤δ≤δ2softening,τf,δ≥δ2frictional.Figure 2
Bond shear stress-slip relationship.To account for the chemical and micromechanical properties of the materials and interfaces involved in this study, a horizontal segment is used at the beginning of the curve0−τ0 to represent the initial shear stress before relative sliding occurs between the FRP bar and ECC. The elastic stage represents the portion of the curve in which the shear stress increases from the initial bond shear stress τ0 to the bond stress at a slip of δ1τm. As damage accumulates in the bar/concrete interface, the bond stress starts decreasing as slip increases. This softening stage, which is attributed to friction and micromechanical interlocking along the length of the bond, is represented by the branch of the curve that ends at a slip of δ2; the residual bond shear stress (τf) is found at this point. For δ>δ2, the bond stress is constant due to the friction between the FRP bar and ECC.


## 4. Theoretical Pullout Force of FRP Bars with Infinite/Finite Bond Lengths Based on the Nonlocal Model

### 4.1. Theoretical Analysis with Infinite Bond Length

If the bond length of the FRP embedded in the ECC is infinite, the pullout process can be described by introducing the proposedτ−δ relationship in equation (6) into equation (5). Note that δx denotes the slip distribution along the bar, LtrδL denotes the required bond transfer length, and Fx represents the pullout forces in each section of the bar, which are determined for specific values of slip at the loaded end δL. These concepts and the definition of the local reference systems in the elastic xe, softening xs, and frictional xf bond stages which are illustrated in Figure 3.Figure 3
Pullout process of an FRP bar with an infinite bond length (definition of local reference systems).


#### 4.1.1. Elastic Stage

When the slip at the loaded end is0≤δL≤δ1, equation (5) is solved in the local reference system of xe, and the solution becomes [12](7)δexe=C1eeλx+C2ee−λx−C3e,where 1/λ2=δ1+Je0a2τm−τ0/Jτm−τ0.Note that the particular solution of equation (5) is(8)C3e=δ1τ0τm−τ0.This solution was obtained by imposing the following boundary conditions on equation (7):(9)δe=0atxe=0,δe=δLatxe=LtreδL,where LtreδL is the bond transfer length corresponding to the elastic stage δL≤δ1. The integration constants can be obtained as(10)C1e=δL+C3e1−e−λLtreδL⋅1eλLtreδL−e−λLtreδL,(11)C2e=C3e−C1e.The equilibrium equation was imposed along the bond length, i.e.,Fx=LtrδL=πd⋅∫0LtrδLτxedxe. Hence, LtreδL can be calculated by(12)LtreδL=1λarcosh1+δLCe3.Moreover, the pullout force at a certain value of imposed slip can be obtained as(13)Fx=LtrδL=Fex=LtreδL=πd∫0LetrδLτxedxe=EAdδedxeLtreδL0.Then, the maximum bond transfer length (Ltr1) and maximum pullout force (Fe1) in the elastic stage can be determined by imposing a loaded end slip of δ1:(14)Ltr1=LtreδL=δ1,(15)F1e=Fexe=Ltr1.


#### 4.1.2. Softening Stage

When the imposed slip isδ1≤δL≤δ2, the process has entered the second stage. After introducing the appropriate function from equation (6) into equation (5), the solution in the local coordinate system of xs is expressed as [12](16)δsxs=C1s⋅sinκ⋅xs+C2s⋅cosκ⋅xs+C3s,with κ2=Jτm−τf/δ2−δ1/1+e0a2Jτm−τf/δ2−δ1.Therefore, ifβ2=Jτm−τf/δ2−δ1, the particular solution of the differential equation becomes(17)C3s=Jτmβ2+δ1.The boundary conditions used in the softening stage were(18)δs=δ1atxs=0,δs=δLatxs=LtrsδL,where LtrsδL is the bond transfer length in the softening stage δ1≤δL≤δ2. The integration constants can be obtained as(19)C1s=1sinκ⋅LtrsδL⋅δL−δ1+Jτmβ2⋅cosκ⋅Ltrs−1,(20)C2s=δ1−C3s=−Jτmβ2.The equilibrium condition was imposed along the bond length, i.e.,Fx=LtrδL=πd⋅∫0LtrsδLτxsdxs+F1e, to derive the bond transfer length as a function of δL. Note that LtrsδL can be expressed as [12](21)LtrsδL=1κ⋅arcsinBsDs2+Bs2+arcsinCsDs2+Bs2,where Bs=πd⋅τm/κ(22)Cs=πd⋅κJ⋅δL−C3s,Ds=F1e.The overall bond transfer length at the end of the softening stage is(23)LtrδL=Ltr1+LtrsδL.Moreover, the pullout force for a particular value of imposed slip in this stage can be calculated as(24)Fsx=LtrsδL=πd∫0LtrsδLτxsdxs=EAdδsdxsLtrsδL0.Hence, the total pullout force becomes(25)Fx=LtrδL=F1e+Fsxs=LtrsδL.The maximum bond transfer length and maximum pullout force in this stage can also be calculated by imposingδL equal to δ2 in equations (21) and (24), respectively:(26)Ltr2=LtrsδL=δ2,(27)F2s=Fsxs=Ltr2.The total force at the end of the softening stage becomes(28)F2=Fx=LtrδL=δ2=F1e+F2s.


#### 4.1.3. Frictional Stage

The frictional stage corresponds to the range in which the slip at the loaded endδL≥δ2. In this stage, the bond shear stress is a constant value τf because a stable amount of interfacial friction is generated between the FRP bar and the surrounding ECC. Therefore, the solution of equation (5) is a polynomial function in the local reference system of xs, which can be expressed as(29)δfxf=C1fxf2+C2fxf+C3f.Note that(30)C1f=Jτf2,and the boundary conditions are(31)δf=δ2atxf=0,δf=δLatxf=LtrfδL,where LtrfδL is the bond transfer length for δL>δ2. By imposing these boundary conditions on equation (29), the integration constants can be expressed as(32)C2f=δL−δ2−C1fLtrfδL2LtrfδL,(33)C3f=δ2.Using the equilibrium equation, i.e.,Fx=LtrδL=πd⋅∫0LtrfδLτxfdxf+F2s+F1e, LtrfδL can be expressed as(34)LtrfδL=F2s−F2s+4C1fδL−δ20.52C1f.The pullout force for any value ofδL>δ2 can be obtained from(35)Ffx=LtrfδL=πd∫0LtrfδLτfdxf=πd⋅LtrfδL⋅τf.The total pullout force is(36)Fx=LtrδL=F1e+F2s+Ffxf=LtrfδL.


### 4.2. Theoretical Analysis with Finite Bond Length

From the analysis presented in Section4.1, in the case of infinite bond length, the pullout process of an embedded FRP bar in ECC can be analyzed by substituting the bond stress-slip relationship into equation (5) and imposing boundary conditions for slip at the end of the bond transfer length Ltr. However, when the embedded FRP has a finite bond length, the analysis needs to be discussed. When the finite bond length Lf exceeds the bond transfer length, the pullout force for any value of δL can be directly calculated with equations (15), (25), and (36), similar to the approach used with an infinite bond length. However, when δF>0, i.e., Lf<Ltr, two or more bond-slip stages occur simultaneously within Lf, and calculating pullout force F is not straightforward due to the complexity of the expression. Bianco et al. [32] and Mazaheripour et al. [12] verified that the slip distribution δx for infinite bond length conditions could also be applied to finite bond length conditions. This means that the expression of δx is the same in both cases; however, different boundary conditions are used in these two cases.The next study is based on the works carried out by Mazaheripour et al. [12] on SFRSC beams reinforced with GFRP bars. The pullout force F in the case of a finite bond length Lf is determined for any value of δL by substituting a variable boundary that does not impose the same bond transfer length LtrδL corresponding to any complete stage. Therefore, for each bond stage, new integration constants in equations (7), (16), and (29) are under the conditions at the free and loaded ends (i.e., δ=δF at x=0 and δ=δL at x=Lf). Then, a new δx can be calculated over Lf. Finally, the pullout force F and the slip at the free end δF can be determined by considering different configurations of the proposed bond stages over the bond length.


#### 4.2.1. Only One Bond Stage Acting onLf

(1) Elastic Stage Only (Figure4(a)). When δL≤δ1 and 0<δF<δ1, the elastic stage is the only bond stage acting on Lf. Accordingly, equations (10) and (11) become(37)C1e=δL−δFe−λLf+C3e⋅1−e−λLf⋅1eλLf−e−λLf,(38)C2e=δF+C3e−C1eFigure 4
Configurations ofFx and δx when only one bond stage is acting on Lf: (a) elastic stage only, (b) softening stage only, and (c) frictional stage only.
(a)(b)(c)Using the equilibrium conditions,δF can be expressed by δL:(39)δF=δL+C3ecoshλLf−C3e.The pullout force can then be calculated by equations (37) and (38):(40)Fexe=Lf=Aσxxxe=Lf=EA⋅C1eδL⋅λeλLf−C2eδL⋅λe−λLf.(2) Softening Stage Only (Figure4(b)). When δF>δ1 and δL≤δ2, the softening stage is the only bond stage acting on Lf. The integration constants in equations (19) and (20) can be expressed as(41)C1s=1sinκ⋅Lf⋅δL−δF+J⋅τmβ2⋅cosκ⋅Lf−1,(42)C2s=δF−C3s.Then,δF can be expressed as a function of δL with the equilibrium condition:(43)δF=δL−J⋅τmβ2cosκ⋅Lf+C3s−δ1.Therefore,F in this stage can be obtained from equations (41) and (42):(44)Fsxs=Lf=πd∫0Lfτxsdxs=EAdδsdxs0Lf.(3) Frictional Stage Only (Figure4(c)). When δL>δ2 and δF>δ2, the friction stage is the only bond stage acting on Lf. The integration constants in equations (32) and (33) can be expressed as(45)C2f=δL−δF−C1fLf2Lf,(46)C3f=δF.Then,δF can be obtained by(47)δF=δL−C1fLf2.Moreover,F can be obtained by(48)Ffxf=Lf=Lfπdτf.


#### 4.2.2. Two or More Bond Stages Acting onLf

In the present study, when two or more bond stages are acting onLf, a numerical strategy is adopted to obtain the slip and force (or bond stress) distributions. Hence, by taking a small increment in the imposed slip at the ith step of the calculation (i.e., δLi=δLi−1+ΔδL) and initially using the value of the pullout force at the last converged step of the calculation (i−1), for each bond stage (i.e., Fei−1,Fsi−1andFfi−1), a new value of the pullout force (Fi=Fi−1+ΔF) can be calculated.(1) Both Elastic and Softening Stages Acting onLf (Figure 5(a)). For δLi, Ltrsi can be calculated by equation (21), where F1e is replaced by Fei−1. When obtaining Ltrsi, Ltrei is simply derived from Lf−Ltrsi. Therefore, δFi and Fei can be calculated by equations (39) and (40), respectively, at xe=Ltrei. Fsi can also be calculated from equation (44) at xs=Ltrsi. The total pullout force becomes (49)Fi=Fei+Fsi.Figure 5
Configurations forFx and δx when two bond stages are acting on Lf: (a) both elastic and softening stages and (b) both softening and frictional stages.
(a)(b)(2) Both Softening and Frictional Bond Stages Acting onLf (Figure 5(b)). For δLi, Ltrfi can be calculated by means of equation (34), where F1e is null and F2s is substituted by Fsi−1. When obtaining Ltrfi, Ltrsi is derived by Lf−Ltrfi. Therefore, δFi and Fsi can be calculated by equations (43) and (44), respectively, at xs=Ltrsi; Ffi can also be calculated from equation (48) at xf=Ltrfi. The total pullout force becomes(50)Fi=Fsi+Ffi.(3) Elastic, Softening, and Frictional Stages Acting onLf (Figure 6). When δL>δ2 and Lf>LtrδL, the three proposed bond stages occur simultaneously within Lf. By imposing a small increment in δL, the same strategy can also be applied in this case to determine Ltrsi and Ltrfi by initially taking F1e and F2s as Fei−1 and Fsi−1, respectively. Then, Ltrei can be obtained by(51)Ltrei=Lf−Ltrsi−Ltrfi.Figure 6
Configurations forFx and δx when the elastic, softening, and frictional stages are simultaneously acting on Lf.Hence, the total pullout force can be expressed as(52)Fi=Fei+Fsi+Ffi.A flowchart of the proposed analytical-numerical algorithm is presented in Figure7. In all of the above cases, once Ltri is obtained for all active bond stages (e.g., the elastic, plastic, and softening stages are all active for the case shown in Figure 5(a)), the values of Fei−1,Fsi−1, and Ffi−1 are replaced by the new calculated values of Fei,Fsi, and Ffi, respectively. Then, Ltri is recalculated until the error becomes less than the tolerance adopted for ΔLtr. This calculation loop is also illustrated in the flowchart.Figure 7
Flowchart of the nonlocal continuum rod model.


## 4.1. Theoretical Analysis with Infinite Bond Length

If the bond length of the FRP embedded in the ECC is infinite, the pullout process can be described by introducing the proposedτ−δ relationship in equation (6) into equation (5). Note that δx denotes the slip distribution along the bar, LtrδL denotes the required bond transfer length, and Fx represents the pullout forces in each section of the bar, which are determined for specific values of slip at the loaded end δL. These concepts and the definition of the local reference systems in the elastic xe, softening xs, and frictional xf bond stages which are illustrated in Figure 3.Figure 3
Pullout process of an FRP bar with an infinite bond length (definition of local reference systems).


### 4.1.1. Elastic Stage

When the slip at the loaded end is0≤δL≤δ1, equation (5) is solved in the local reference system of xe, and the solution becomes [12](7)δexe=C1eeλx+C2ee−λx−C3e,where 1/λ2=δ1+Je0a2τm−τ0/Jτm−τ0.Note that the particular solution of equation (5) is(8)C3e=δ1τ0τm−τ0.This solution was obtained by imposing the following boundary conditions on equation (7):(9)δe=0atxe=0,δe=δLatxe=LtreδL,where LtreδL is the bond transfer length corresponding to the elastic stage δL≤δ1. The integration constants can be obtained as(10)C1e=δL+C3e1−e−λLtreδL⋅1eλLtreδL−e−λLtreδL,(11)C2e=C3e−C1e.The equilibrium equation was imposed along the bond length, i.e.,Fx=LtrδL=πd⋅∫0LtrδLτxedxe. Hence, LtreδL can be calculated by(12)LtreδL=1λarcosh1+δLCe3.Moreover, the pullout force at a certain value of imposed slip can be obtained as(13)Fx=LtrδL=Fex=LtreδL=πd∫0LetrδLτxedxe=EAdδedxeLtreδL0.Then, the maximum bond transfer length (Ltr1) and maximum pullout force (Fe1) in the elastic stage can be determined by imposing a loaded end slip of δ1:(14)Ltr1=LtreδL=δ1,(15)F1e=Fexe=Ltr1.


### 4.1.2. Softening Stage

When the imposed slip isδ1≤δL≤δ2, the process has entered the second stage. After introducing the appropriate function from equation (6) into equation (5), the solution in the local coordinate system of xs is expressed as [12](16)δsxs=C1s⋅sinκ⋅xs+C2s⋅cosκ⋅xs+C3s,with κ2=Jτm−τf/δ2−δ1/1+e0a2Jτm−τf/δ2−δ1.Therefore, ifβ2=Jτm−τf/δ2−δ1, the particular solution of the differential equation becomes(17)C3s=Jτmβ2+δ1.The boundary conditions used in the softening stage were(18)δs=δ1atxs=0,δs=δLatxs=LtrsδL,where LtrsδL is the bond transfer length in the softening stage δ1≤δL≤δ2. The integration constants can be obtained as(19)C1s=1sinκ⋅LtrsδL⋅δL−δ1+Jτmβ2⋅cosκ⋅Ltrs−1,(20)C2s=δ1−C3s=−Jτmβ2.The equilibrium condition was imposed along the bond length, i.e.,Fx=LtrδL=πd⋅∫0LtrsδLτxsdxs+F1e, to derive the bond transfer length as a function of δL. Note that LtrsδL can be expressed as [12](21)LtrsδL=1κ⋅arcsinBsDs2+Bs2+arcsinCsDs2+Bs2,where Bs=πd⋅τm/κ(22)Cs=πd⋅κJ⋅δL−C3s,Ds=F1e.The overall bond transfer length at the end of the softening stage is(23)LtrδL=Ltr1+LtrsδL.Moreover, the pullout force for a particular value of imposed slip in this stage can be calculated as(24)Fsx=LtrsδL=πd∫0LtrsδLτxsdxs=EAdδsdxsLtrsδL0.Hence, the total pullout force becomes(25)Fx=LtrδL=F1e+Fsxs=LtrsδL.The maximum bond transfer length and maximum pullout force in this stage can also be calculated by imposingδL equal to δ2 in equations (21) and (24), respectively:(26)Ltr2=LtrsδL=δ2,(27)F2s=Fsxs=Ltr2.The total force at the end of the softening stage becomes(28)F2=Fx=LtrδL=δ2=F1e+F2s.


### 4.1.3. Frictional Stage

The frictional stage corresponds to the range in which the slip at the loaded endδL≥δ2. In this stage, the bond shear stress is a constant value τf because a stable amount of interfacial friction is generated between the FRP bar and the surrounding ECC. Therefore, the solution of equation (5) is a polynomial function in the local reference system of xs, which can be expressed as(29)δfxf=C1fxf2+C2fxf+C3f.Note that(30)C1f=Jτf2,and the boundary conditions are(31)δf=δ2atxf=0,δf=δLatxf=LtrfδL,where LtrfδL is the bond transfer length for δL>δ2. By imposing these boundary conditions on equation (29), the integration constants can be expressed as(32)C2f=δL−δ2−C1fLtrfδL2LtrfδL,(33)C3f=δ2.Using the equilibrium equation, i.e.,Fx=LtrδL=πd⋅∫0LtrfδLτxfdxf+F2s+F1e, LtrfδL can be expressed as(34)LtrfδL=F2s−F2s+4C1fδL−δ20.52C1f.The pullout force for any value ofδL>δ2 can be obtained from(35)Ffx=LtrfδL=πd∫0LtrfδLτfdxf=πd⋅LtrfδL⋅τf.The total pullout force is(36)Fx=LtrδL=F1e+F2s+Ffxf=LtrfδL.


## 4.1.1. Elastic Stage

When the slip at the loaded end is0≤δL≤δ1, equation (5) is solved in the local reference system of xe, and the solution becomes [12](7)δexe=C1eeλx+C2ee−λx−C3e,where 1/λ2=δ1+Je0a2τm−τ0/Jτm−τ0.Note that the particular solution of equation (5) is(8)C3e=δ1τ0τm−τ0.This solution was obtained by imposing the following boundary conditions on equation (7):(9)δe=0atxe=0,δe=δLatxe=LtreδL,where LtreδL is the bond transfer length corresponding to the elastic stage δL≤δ1. The integration constants can be obtained as(10)C1e=δL+C3e1−e−λLtreδL⋅1eλLtreδL−e−λLtreδL,(11)C2e=C3e−C1e.The equilibrium equation was imposed along the bond length, i.e.,Fx=LtrδL=πd⋅∫0LtrδLτxedxe. Hence, LtreδL can be calculated by(12)LtreδL=1λarcosh1+δLCe3.Moreover, the pullout force at a certain value of imposed slip can be obtained as(13)Fx=LtrδL=Fex=LtreδL=πd∫0LetrδLτxedxe=EAdδedxeLtreδL0.Then, the maximum bond transfer length (Ltr1) and maximum pullout force (Fe1) in the elastic stage can be determined by imposing a loaded end slip of δ1:(14)Ltr1=LtreδL=δ1,(15)F1e=Fexe=Ltr1.


## 4.1.2. Softening Stage

When the imposed slip isδ1≤δL≤δ2, the process has entered the second stage. After introducing the appropriate function from equation (6) into equation (5), the solution in the local coordinate system of xs is expressed as [12](16)δsxs=C1s⋅sinκ⋅xs+C2s⋅cosκ⋅xs+C3s,with κ2=Jτm−τf/δ2−δ1/1+e0a2Jτm−τf/δ2−δ1.Therefore, ifβ2=Jτm−τf/δ2−δ1, the particular solution of the differential equation becomes(17)C3s=Jτmβ2+δ1.The boundary conditions used in the softening stage were(18)δs=δ1atxs=0,δs=δLatxs=LtrsδL,where LtrsδL is the bond transfer length in the softening stage δ1≤δL≤δ2. The integration constants can be obtained as(19)C1s=1sinκ⋅LtrsδL⋅δL−δ1+Jτmβ2⋅cosκ⋅Ltrs−1,(20)C2s=δ1−C3s=−Jτmβ2.The equilibrium condition was imposed along the bond length, i.e.,Fx=LtrδL=πd⋅∫0LtrsδLτxsdxs+F1e, to derive the bond transfer length as a function of δL. Note that LtrsδL can be expressed as [12](21)LtrsδL=1κ⋅arcsinBsDs2+Bs2+arcsinCsDs2+Bs2,where Bs=πd⋅τm/κ(22)Cs=πd⋅κJ⋅δL−C3s,Ds=F1e.The overall bond transfer length at the end of the softening stage is(23)LtrδL=Ltr1+LtrsδL.Moreover, the pullout force for a particular value of imposed slip in this stage can be calculated as(24)Fsx=LtrsδL=πd∫0LtrsδLτxsdxs=EAdδsdxsLtrsδL0.Hence, the total pullout force becomes(25)Fx=LtrδL=F1e+Fsxs=LtrsδL.The maximum bond transfer length and maximum pullout force in this stage can also be calculated by imposingδL equal to δ2 in equations (21) and (24), respectively:(26)Ltr2=LtrsδL=δ2,(27)F2s=Fsxs=Ltr2.The total force at the end of the softening stage becomes(28)F2=Fx=LtrδL=δ2=F1e+F2s.


## 4.1.3. Frictional Stage

The frictional stage corresponds to the range in which the slip at the loaded endδL≥δ2. In this stage, the bond shear stress is a constant value τf because a stable amount of interfacial friction is generated between the FRP bar and the surrounding ECC. Therefore, the solution of equation (5) is a polynomial function in the local reference system of xs, which can be expressed as(29)δfxf=C1fxf2+C2fxf+C3f.Note that(30)C1f=Jτf2,and the boundary conditions are(31)δf=δ2atxf=0,δf=δLatxf=LtrfδL,where LtrfδL is the bond transfer length for δL>δ2. By imposing these boundary conditions on equation (29), the integration constants can be expressed as(32)C2f=δL−δ2−C1fLtrfδL2LtrfδL,(33)C3f=δ2.Using the equilibrium equation, i.e.,Fx=LtrδL=πd⋅∫0LtrfδLτxfdxf+F2s+F1e, LtrfδL can be expressed as(34)LtrfδL=F2s−F2s+4C1fδL−δ20.52C1f.The pullout force for any value ofδL>δ2 can be obtained from(35)Ffx=LtrfδL=πd∫0LtrfδLτfdxf=πd⋅LtrfδL⋅τf.The total pullout force is(36)Fx=LtrδL=F1e+F2s+Ffxf=LtrfδL.


## 4.2. Theoretical Analysis with Finite Bond Length

From the analysis presented in Section4.1, in the case of infinite bond length, the pullout process of an embedded FRP bar in ECC can be analyzed by substituting the bond stress-slip relationship into equation (5) and imposing boundary conditions for slip at the end of the bond transfer length Ltr. However, when the embedded FRP has a finite bond length, the analysis needs to be discussed. When the finite bond length Lf exceeds the bond transfer length, the pullout force for any value of δL can be directly calculated with equations (15), (25), and (36), similar to the approach used with an infinite bond length. However, when δF>0, i.e., Lf<Ltr, two or more bond-slip stages occur simultaneously within Lf, and calculating pullout force F is not straightforward due to the complexity of the expression. Bianco et al. [32] and Mazaheripour et al. [12] verified that the slip distribution δx for infinite bond length conditions could also be applied to finite bond length conditions. This means that the expression of δx is the same in both cases; however, different boundary conditions are used in these two cases.The next study is based on the works carried out by Mazaheripour et al. [12] on SFRSC beams reinforced with GFRP bars. The pullout force F in the case of a finite bond length Lf is determined for any value of δL by substituting a variable boundary that does not impose the same bond transfer length LtrδL corresponding to any complete stage. Therefore, for each bond stage, new integration constants in equations (7), (16), and (29) are under the conditions at the free and loaded ends (i.e., δ=δF at x=0 and δ=δL at x=Lf). Then, a new δx can be calculated over Lf. Finally, the pullout force F and the slip at the free end δF can be determined by considering different configurations of the proposed bond stages over the bond length.


### 4.2.1. Only One Bond Stage Acting onLf

(1) Elastic Stage Only (Figure4(a)). When δL≤δ1 and 0<δF<δ1, the elastic stage is the only bond stage acting on Lf. Accordingly, equations (10) and (11) become(37)C1e=δL−δFe−λLf+C3e⋅1−e−λLf⋅1eλLf−e−λLf,(38)C2e=δF+C3e−C1eFigure 4
Configurations ofFx and δx when only one bond stage is acting on Lf: (a) elastic stage only, (b) softening stage only, and (c) frictional stage only.
(a)(b)(c)Using the equilibrium conditions,δF can be expressed by δL:(39)δF=δL+C3ecoshλLf−C3e.The pullout force can then be calculated by equations (37) and (38):(40)Fexe=Lf=Aσxxxe=Lf=EA⋅C1eδL⋅λeλLf−C2eδL⋅λe−λLf.(2) Softening Stage Only (Figure4(b)). When δF>δ1 and δL≤δ2, the softening stage is the only bond stage acting on Lf. The integration constants in equations (19) and (20) can be expressed as(41)C1s=1sinκ⋅Lf⋅δL−δF+J⋅τmβ2⋅cosκ⋅Lf−1,(42)C2s=δF−C3s.Then,δF can be expressed as a function of δL with the equilibrium condition:(43)δF=δL−J⋅τmβ2cosκ⋅Lf+C3s−δ1.Therefore,F in this stage can be obtained from equations (41) and (42):(44)Fsxs=Lf=πd∫0Lfτxsdxs=EAdδsdxs0Lf.(3) Frictional Stage Only (Figure4(c)). When δL>δ2 and δF>δ2, the friction stage is the only bond stage acting on Lf. The integration constants in equations (32) and (33) can be expressed as(45)C2f=δL−δF−C1fLf2Lf,(46)C3f=δF.Then,δF can be obtained by(47)δF=δL−C1fLf2.Moreover,F can be obtained by(48)Ffxf=Lf=Lfπdτf.


### 4.2.2. Two or More Bond Stages Acting onLf

In the present study, when two or more bond stages are acting onLf, a numerical strategy is adopted to obtain the slip and force (or bond stress) distributions. Hence, by taking a small increment in the imposed slip at the ith step of the calculation (i.e., δLi=δLi−1+ΔδL) and initially using the value of the pullout force at the last converged step of the calculation (i−1), for each bond stage (i.e., Fei−1,Fsi−1andFfi−1), a new value of the pullout force (Fi=Fi−1+ΔF) can be calculated.(1) Both Elastic and Softening Stages Acting onLf (Figure 5(a)). For δLi, Ltrsi can be calculated by equation (21), where F1e is replaced by Fei−1. When obtaining Ltrsi, Ltrei is simply derived from Lf−Ltrsi. Therefore, δFi and Fei can be calculated by equations (39) and (40), respectively, at xe=Ltrei. Fsi can also be calculated from equation (44) at xs=Ltrsi. The total pullout force becomes (49)Fi=Fei+Fsi.Figure 5
Configurations forFx and δx when two bond stages are acting on Lf: (a) both elastic and softening stages and (b) both softening and frictional stages.
(a)(b)(2) Both Softening and Frictional Bond Stages Acting onLf (Figure 5(b)). For δLi, Ltrfi can be calculated by means of equation (34), where F1e is null and F2s is substituted by Fsi−1. When obtaining Ltrfi, Ltrsi is derived by Lf−Ltrfi. Therefore, δFi and Fsi can be calculated by equations (43) and (44), respectively, at xs=Ltrsi; Ffi can also be calculated from equation (48) at xf=Ltrfi. The total pullout force becomes(50)Fi=Fsi+Ffi.(3) Elastic, Softening, and Frictional Stages Acting onLf (Figure 6). When δL>δ2 and Lf>LtrδL, the three proposed bond stages occur simultaneously within Lf. By imposing a small increment in δL, the same strategy can also be applied in this case to determine Ltrsi and Ltrfi by initially taking F1e and F2s as Fei−1 and Fsi−1, respectively. Then, Ltrei can be obtained by(51)Ltrei=Lf−Ltrsi−Ltrfi.Figure 6
Configurations forFx and δx when the elastic, softening, and frictional stages are simultaneously acting on Lf.Hence, the total pullout force can be expressed as(52)Fi=Fei+Fsi+Ffi.A flowchart of the proposed analytical-numerical algorithm is presented in Figure7. In all of the above cases, once Ltri is obtained for all active bond stages (e.g., the elastic, plastic, and softening stages are all active for the case shown in Figure 5(a)), the values of Fei−1,Fsi−1, and Ffi−1 are replaced by the new calculated values of Fei,Fsi, and Ffi, respectively. Then, Ltri is recalculated until the error becomes less than the tolerance adopted for ΔLtr. This calculation loop is also illustrated in the flowchart.Figure 7
Flowchart of the nonlocal continuum rod model.


## 4.2.1. Only One Bond Stage Acting onLf

(1) Elastic Stage Only (Figure4(a)). When δL≤δ1 and 0<δF<δ1, the elastic stage is the only bond stage acting on Lf. Accordingly, equations (10) and (11) become(37)C1e=δL−δFe−λLf+C3e⋅1−e−λLf⋅1eλLf−e−λLf,(38)C2e=δF+C3e−C1eFigure 4
Configurations ofFx and δx when only one bond stage is acting on Lf: (a) elastic stage only, (b) softening stage only, and (c) frictional stage only.
(a)(b)(c)Using the equilibrium conditions,δF can be expressed by δL:(39)δF=δL+C3ecoshλLf−C3e.The pullout force can then be calculated by equations (37) and (38):(40)Fexe=Lf=Aσxxxe=Lf=EA⋅C1eδL⋅λeλLf−C2eδL⋅λe−λLf.(2) Softening Stage Only (Figure4(b)). When δF>δ1 and δL≤δ2, the softening stage is the only bond stage acting on Lf. The integration constants in equations (19) and (20) can be expressed as(41)C1s=1sinκ⋅Lf⋅δL−δF+J⋅τmβ2⋅cosκ⋅Lf−1,(42)C2s=δF−C3s.Then,δF can be expressed as a function of δL with the equilibrium condition:(43)δF=δL−J⋅τmβ2cosκ⋅Lf+C3s−δ1.Therefore,F in this stage can be obtained from equations (41) and (42):(44)Fsxs=Lf=πd∫0Lfτxsdxs=EAdδsdxs0Lf.(3) Frictional Stage Only (Figure4(c)). When δL>δ2 and δF>δ2, the friction stage is the only bond stage acting on Lf. The integration constants in equations (32) and (33) can be expressed as(45)C2f=δL−δF−C1fLf2Lf,(46)C3f=δF.Then,δF can be obtained by(47)δF=δL−C1fLf2.Moreover,F can be obtained by(48)Ffxf=Lf=Lfπdτf.


## 4.2.2. Two or More Bond Stages Acting onLf

In the present study, when two or more bond stages are acting onLf, a numerical strategy is adopted to obtain the slip and force (or bond stress) distributions. Hence, by taking a small increment in the imposed slip at the ith step of the calculation (i.e., δLi=δLi−1+ΔδL) and initially using the value of the pullout force at the last converged step of the calculation (i−1), for each bond stage (i.e., Fei−1,Fsi−1andFfi−1), a new value of the pullout force (Fi=Fi−1+ΔF) can be calculated.(1) Both Elastic and Softening Stages Acting onLf (Figure 5(a)). For δLi, Ltrsi can be calculated by equation (21), where F1e is replaced by Fei−1. When obtaining Ltrsi, Ltrei is simply derived from Lf−Ltrsi. Therefore, δFi and Fei can be calculated by equations (39) and (40), respectively, at xe=Ltrei. Fsi can also be calculated from equation (44) at xs=Ltrsi. The total pullout force becomes (49)Fi=Fei+Fsi.Figure 5
Configurations forFx and δx when two bond stages are acting on Lf: (a) both elastic and softening stages and (b) both softening and frictional stages.
(a)(b)(2) Both Softening and Frictional Bond Stages Acting onLf (Figure 5(b)). For δLi, Ltrfi can be calculated by means of equation (34), where F1e is null and F2s is substituted by Fsi−1. When obtaining Ltrfi, Ltrsi is derived by Lf−Ltrfi. Therefore, δFi and Fsi can be calculated by equations (43) and (44), respectively, at xs=Ltrsi; Ffi can also be calculated from equation (48) at xf=Ltrfi. The total pullout force becomes(50)Fi=Fsi+Ffi.(3) Elastic, Softening, and Frictional Stages Acting onLf (Figure 6). When δL>δ2 and Lf>LtrδL, the three proposed bond stages occur simultaneously within Lf. By imposing a small increment in δL, the same strategy can also be applied in this case to determine Ltrsi and Ltrfi by initially taking F1e and F2s as Fei−1 and Fsi−1, respectively. Then, Ltrei can be obtained by(51)Ltrei=Lf−Ltrsi−Ltrfi.Figure 6
Configurations forFx and δx when the elastic, softening, and frictional stages are simultaneously acting on Lf.Hence, the total pullout force can be expressed as(52)Fi=Fei+Fsi+Ffi.A flowchart of the proposed analytical-numerical algorithm is presented in Figure7. In all of the above cases, once Ltri is obtained for all active bond stages (e.g., the elastic, plastic, and softening stages are all active for the case shown in Figure 5(a)), the values of Fei−1,Fsi−1, and Ffi−1 are replaced by the new calculated values of Fei,Fsi, and Ffi, respectively. Then, Ltri is recalculated until the error becomes less than the tolerance adopted for ΔLtr. This calculation loop is also illustrated in the flowchart.Figure 7
Flowchart of the nonlocal continuum rod model.


## 5. Case Study and Discussion

To illustrate the bond behavior with the nonlocal elastic model, FRP bars embedded in ECC were subjected to pullout tests in [11]. The parameters and experimental results are shown in Table 1. The maximum pullout forces calculated using the classical elastic model for CFRP bars and GFRP bars were 3.09 kN and 2.30 kN, respectively. We confined our attention to the influence of the nonlocal parameter e0a on δx and the pullout force F with a finite bond length Lf.Table 1
Values of the parameters and experimental results.
SpecimensParametersExperimental resultsEc (GPa)Es (GPa)ds (mm)b (mm)Lf (mm)τ0 (MPa)τm (MPa)τf (MPa)δ1 (mm)δ2 (mm)Fmax (kN)Fres (kN)C3030180.1101001000.0310.720.552.432.942.28G503032101001000.140.740.460.613.842.21.46Note. C30 represents the specimens containing CFRP bars embedded in ECC with a compressive strength of 30 MPa, whereas G50 represents the specimens containing GFRP bars embedded in ECC with a compressive strength of 50 MPa.


### 5.1. Introduction of Free-End SlipδF to the Nonlocal Elastic Model

As shown in Section4.2, when the FRP bar only experiences one bond stage, the slip at the free end δF can be determined by equations (39), (43), and (47) for any value of imposed slip in the case of finite bond length. A comparison of the nonlocal elastic model and the classical elastic model shows that the free-end slip exhibits differences when the FRP bar experiences different debonding processes, as shown in Figures 8–10.Figure 8
Normalized free-end slipδFe0a for various values of e0a/Lf and representative values of δL when the elastic stage was the only bond stage acting on the FRP bar.Figure 9
Normalized free-end slipδFe0a for various values of e0a/Lf and representative values of δL when the softening stage was the only bond stage acting on the FRP bar.Figure 10
Normalized free-end slipδFe0a for various values of e0a/Lf and representative values of δL when the frictional stage was the only bond stage acting on the FRP bar.Note thatδFe0a is the free-end slip under the nonlocal elastic model, whereas δF is the free-end slip under the classical elastic model. In the elastic stage (Figure 8), due to the nonlocal size effect, the free-end slip is slightly larger in the nonlocal elastic model than in the classical elastic model, and the growth trend is positively related to the nonlocal parameter e0a and inversely related to the slip at the loaded end δL. The nonlocal effect within the elastic stage is more obvious in the early portion of the stage. For the softening stage in Figure 9, the free-end slip under the nonlocal elastic model decreases as the nonlocal parameter increases. This trend is more significant with a small value of imposed slip δL. For the frictional stage in Figure 10, the free-end slip is not affected by the nonlocal parameters because the constant stress field does not depend upon the nonlocal parameter e0a (as confirmed by Benvenuti and Simone [33]).Nonlocal effects affect the slipδx over Lf in the elastic and softening stages. Therefore, the maximum pullout force, which only appears in the first two stages of the debonding process, needs further analysis.


### 5.2. Analysis of the Pullout ForceF

The pullout force-loaded end slip curve based on the nonlocal elastic model is similar to that in the classical elastic model; however, the curve based on the nonlocal model is closer to the experimental values due to the influence of the nonlocal effects. The pullout force-slip relationship recorded from the experiments and that determined by the nonlocal bond model (e0a/Lf=0.3) are compared in Figure 11. Figures 12 and 13 show the experimental and calculated values of the pullout force for the CFRP and GFRP bars with an imposed slip of δL, respectively.Figure 11
Comparison between the theoretical and experimentalF−δL relationships for the deformed FRP bars.Figure 12
Pullout force versus loaded end slip for deformed CFRP bars.Figure 13
Pullout force versus loaded end slip for deformed GFRP bars.Similar to the classical elastic model, a three-stage (ascending, descending, and stationary stages) linear relationship exists between the pullout forceF and imposed slip δL under the nonlocal model, as shown in Figure 11. When the maximum pullout force occurs during the debonding process, the loaded end slip is δ1<δL<δ2. Consequently, the value is reduced, and the corresponding slip is small, which is attributable to the influence of the nonlocal parameter. Hence, because of the nonlocal effect on the slip over the bond length, the pullout force is inevitably affected. Figures 12 and 13 show that the maximum load decreases due to nonlocal effects, whereas the surrounding values increase slightly. The nonlocal parameters have a discrete effect and are closer to the experimental values. To determine suitable nonlocal parameters for different types of FRP bars, the corresponding experimental values must be reviewed. For the two FRP bars discussed in the case, the nonlocal parameter of the CFRP bar is 0.9 and that of the GFRP bar is 0.6. In short, comparing the experimental results of FRP pullout in [11], it is known that the results predicted by the proposed nonlocal model are accurate.


## 5.1. Introduction of Free-End SlipδF to the Nonlocal Elastic Model

As shown in Section4.2, when the FRP bar only experiences one bond stage, the slip at the free end δF can be determined by equations (39), (43), and (47) for any value of imposed slip in the case of finite bond length. A comparison of the nonlocal elastic model and the classical elastic model shows that the free-end slip exhibits differences when the FRP bar experiences different debonding processes, as shown in Figures 8–10.Figure 8
Normalized free-end slipδFe0a for various values of e0a/Lf and representative values of δL when the elastic stage was the only bond stage acting on the FRP bar.Figure 9
Normalized free-end slipδFe0a for various values of e0a/Lf and representative values of δL when the softening stage was the only bond stage acting on the FRP bar.Figure 10
Normalized free-end slipδFe0a for various values of e0a/Lf and representative values of δL when the frictional stage was the only bond stage acting on the FRP bar.Note thatδFe0a is the free-end slip under the nonlocal elastic model, whereas δF is the free-end slip under the classical elastic model. In the elastic stage (Figure 8), due to the nonlocal size effect, the free-end slip is slightly larger in the nonlocal elastic model than in the classical elastic model, and the growth trend is positively related to the nonlocal parameter e0a and inversely related to the slip at the loaded end δL. The nonlocal effect within the elastic stage is more obvious in the early portion of the stage. For the softening stage in Figure 9, the free-end slip under the nonlocal elastic model decreases as the nonlocal parameter increases. This trend is more significant with a small value of imposed slip δL. For the frictional stage in Figure 10, the free-end slip is not affected by the nonlocal parameters because the constant stress field does not depend upon the nonlocal parameter e0a (as confirmed by Benvenuti and Simone [33]).Nonlocal effects affect the slipδx over Lf in the elastic and softening stages. Therefore, the maximum pullout force, which only appears in the first two stages of the debonding process, needs further analysis.


## 5.2. Analysis of the Pullout ForceF

The pullout force-loaded end slip curve based on the nonlocal elastic model is similar to that in the classical elastic model; however, the curve based on the nonlocal model is closer to the experimental values due to the influence of the nonlocal effects. The pullout force-slip relationship recorded from the experiments and that determined by the nonlocal bond model (e0a/Lf=0.3) are compared in Figure 11. Figures 12 and 13 show the experimental and calculated values of the pullout force for the CFRP and GFRP bars with an imposed slip of δL, respectively.Figure 11
Comparison between the theoretical and experimentalF−δL relationships for the deformed FRP bars.Figure 12
Pullout force versus loaded end slip for deformed CFRP bars.Figure 13
Pullout force versus loaded end slip for deformed GFRP bars.Similar to the classical elastic model, a three-stage (ascending, descending, and stationary stages) linear relationship exists between the pullout forceF and imposed slip δL under the nonlocal model, as shown in Figure 11. When the maximum pullout force occurs during the debonding process, the loaded end slip is δ1<δL<δ2. Consequently, the value is reduced, and the corresponding slip is small, which is attributable to the influence of the nonlocal parameter. Hence, because of the nonlocal effect on the slip over the bond length, the pullout force is inevitably affected. Figures 12 and 13 show that the maximum load decreases due to nonlocal effects, whereas the surrounding values increase slightly. The nonlocal parameters have a discrete effect and are closer to the experimental values. To determine suitable nonlocal parameters for different types of FRP bars, the corresponding experimental values must be reviewed. For the two FRP bars discussed in the case, the nonlocal parameter of the CFRP bar is 0.9 and that of the GFRP bar is 0.6. In short, comparing the experimental results of FRP pullout in [11], it is known that the results predicted by the proposed nonlocal model are accurate.


## 6. Conclusion

The theoretical analysis of the debonding of FRP with a nonlocal model allows us to have a deeper understanding of the bonding mechanism of FRP bars and ECC. Inspired by Eringen’s nonlocal theory, the proposed nonlocal model considers the influence of the deformation variables on the bonding stress in a certain area of the research site, which is not available in the existing models. When the nonlocal parametere0a=0, the second-order differential governing equation over the bond length of the FRP bar, which is based on the nonlocal elastic model, degenerates into existing classical model, as demonstrated in [11, 12]. Substituting the bond stress-slip relationship into the governing equation, the solution δx is an expression containing nonlocal parameters. Affected by these nonlocal parameters, the slip of the elastic stage increases and decreases in the softening stage but does not change in the frictional stage. Because the nonlocal parameter mainly affects the first two stages of the debonding process, the theoretically calculated pullout force is reduced and is more consistent with the experimental data. Therefore, it is more convincing and more accurate to analyze the bond behavior of FRP bars with a nonlocal model. However, the nonlocal parameters are not uniform for different types of FRP bars.



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*Source: 1710364-2020-07-20.xml*