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DualHorizon Peridynamics Analysis of Debonding Failure in FRPtoConcrete Bonded Joints
International Journal of Concrete Structures and Materials volume 13, Article number: 26 (2019)
Abstract
A dualhorizon peridynamics (DHPD) model is proposed for the simulation of debonding process in fiberreinforced polymer (FRP)toconcrete bonded joints. In the novel procedure of implementing DHPD in the framework of finite element method (FEM), truss elements are employed to represent the bonds and dualbonds. The quadtree approach is utilized to generate the multiscale discretization and a volume correction scheme based on the background grid is proposed for the nonuniform grid. A benchmark numerical example is performed to test the accuracy and efficiency of the developed model in analysis of the bond behavior. The predicted results are consistent with the experimental findings, the FEM results and the analytical solutions. Additionally, these results demonstrate that the bond strength and the debonding ductility are visibly affected by concrete strength and the thickness of FRP plate, while the adhesive thickness has no significant impact on the debonding behavior.
1 Introduction
Fiberreinforced polymer (FRP) is a kind of high performance material with high strengthtoweight ratio, good durability and ease of application. The external bonding of FRP to concrete members has been accepted as an effective technology to strengthen and retrofit concrete structures (Täljsten 1996; Ronagh and Baji 2014). In FRP strengthened concrete structures, the interfacial bond between FRP and concrete is often the weakest region and can contribute to the debonding failure. Therefore, ensuring the safety of FRP strengthened concrete structures necessitates a realistic description of the bond behavior.
Over the past two decades, extensive experimental studies have been conducted to investigate the bond behavior in FRP strengthened concrete structures. Generally, the shear tests of FRPtoconcrete bonded joints are the most commonly used test methodology, in which an FRP plate is bonded to a concrete prism and subjected to tension (Chen et al. 2001; Yao et al. 2005; AliAhmad et al. 2006; Carrara et al. 2011; Benzarti et al. 2011; Wu and Jiang 2013; Kabir et al. 2017). Typical debonding failure in the shear test is observed generally in concrete at a few millimeters from the adhesive layer. Unfortunately, the data directly measured from the shear tests always have significant variations due to bending effect of the thin FRP plate and irregularity of aggregates in concrete. In order to reduce the scattering of the directly measured results, substantial analytical models were developed (Yuan et al. 2004; Dai et al. 2005; Zhou et al. 2010; Liu and Wu 2012; Pan and Wu 2014; Moein and Tasnimi 2016). However, because such analytical models are based on empirical parameter calibrations and the theory of elasticity, they are limited to describe the nonlinear debonding process.
Apart from experimental and analytical studies, the numerical simulation is a convenient and powerful alternative for the study of the interfacial bond. In current available numerical methods, finite element method (FEM) is generally used and existing FEM studies can be classified into two approaches. One approach is the interface modeling approach, in which the bond behavior is modeled using a layer of interface elements (Diab and Wu 2007; Chen et al. 2011; Sun et al. 2017a, b). The other is the direct modeling approach, in which debonding is simulated by modeling the failure of concrete adjacent to the adhesive layer (Lu et al. 2005a, 2006; Pham and AlMahaidi 2007; Tao and Chen 2015; Xu et al. 2015). The success of the former approach depends on the constitutive law of the interface elements; as a result, it is not truly predictive. In the latter approach, the principle of equivalent stiffness, namely the thickness of the FRP plate is amplified and the modulus of elasticity is modified to keep the axial stiffness of FRP constant, is often utilized to enlarge the size of elements in FEM model. Consequently the computational cost is significantly reduced, however, the equivalence can trigger numerical errors. Additionally, in FEM, singularities in the presence of discontinuities caused by cracks usually are unavoidable, which makes the construction of a corresponding numerical model cumbersome and timeconsuming.
Alternatively, a new nonlocal method of continuum called peridynamics (PD) has been developed by Silling (2000), Silling et al. (2007) to overcome the limitations of traditional FEM. PD employs spatial integral equations rather than partial differential equations, making it more suitable for solving practical problems involving discontinuities and significant nonlocal effects. It has been applied successfully to a wide range of traditional problems including crack propagation in brittle materials (Ha and Bobaru 2011), the fracture of concrete structures (Li and Guo 2018), composites delamination (Hu et al. 2012), thermal diffusion (Bobaru and Duangpanya 2012) and flow in porous media (Jabakhanji and Mohtar 2015).
In the original PD formulation developed by Silling, the discretization is uniform and the horizon sizes are constant in the whole domain. However, heterogeneous materials and structures, such as FRP strengthened concrete structures necessitate the nonuniform and multiscale discretization to model different constitutive materials separately and reduce the computational cost. Recently, a dualhorizon peridynamics (DHPD) formulation was proposed by Ren et al. (2016) and Rabczuk and Ren (2017) to incorporate nonuniform material point distribution and variable horizons into PD. DHPD can completely solve the issues of spurious wave reflections and simulate the crack propagation in composite materials. Therefore, DHPD is suitable for simulating debonding failure of FRP strengthened concrete structures.
In present paper, DHPD is extended and applied for investigating debonding failure in FRPtoconcrete bonded joints. To keep the efficiency of FEM without losing the generality of DHPD, DHPD is implemented in the framework of FEM and the quasistatic simulations are carried out by solving the equilibrium equations. Furthermore, in order to reduce the computational cost and increase the accuracy, a nonuniform discretization and a volume correction scheme are proposed. The remainder of the paper is organized as follows. Section 2 is devoted to a brief review of PD and DHPD. In Sect. 3, the numerical implementation of DHPD within the framework of FEM is proposed. In Sect. 4, a DHPD model of FRPtoconcrete bonded joints is established and a comparison between DHPD predictions with test results is carried out to validate the proposed method on debonding failure. In Sect. 5, the effects of concrete strength, the FRP thickness and the thickness of the adhesive layer on the bond behavior are evaluated. Finally, some concluding remarks are addressed in Sect. 6.
2 Theoretical Basis of Peridynamics
2.1 Basic Equations of PD and Constitutive Modeling
Peridynamics is a nonlocal continuum theory, which assumes that every material point interacts with all others within its horizon. Horizon with radius of δ refers to the size of nonlocal interaction and such interaction is called bond. The PD equation of motion at a material point x in the continuum R_{0} and time t is defined as
where \(\varvec{f} = \varvec{f}\left( {\varvec{x},\varvec{x^{\prime}},\varvec{u}\left( {\varvec{x},t} \right),\varvec{u}\left( {\varvec{x^{\prime}},t} \right),t} \right)\) is the pairwise force function of the bond \(\varvec{xx^{\prime}}\), which describes the interactions between x and any point \(\varvec{x^{\prime}}\) within its horizon, \(H_{\varvec{x}} = \left\{ {\varvec{x^{\prime}} \in R_{0} \;\varvec{x}  \varvec{x^{\prime}}\text{} \le \delta } \right\}\) is the horizon of material point x, \(\left\ \cdot \right\\) is the Euclidean norm, ρ is the density, u is the displacement, \(\varvec{\ddot{u}}\) is the acceleration, \(\text{d}V_{{\varvec{x^{\prime}}}}\) is the volume element, b is the prescribed loading force density.
For brittle microelastic materials, the pairwise force function f of the bond xx′ is expressed as
where \(\xi = \varvec{x}  \varvec{x^{\prime}}\) and \(\varvec{\eta}= \varvec{u}  \varvec{u^{\prime}}\) denote the relative position and the relative displacement of material points at x and x′in the reference configuration, respectively; \(\varvec{\eta}+ \xi\) represents the current relative position between the material points; c is the micromodulus function and denotes the stiffness of the bond. In present study, the two dimensional (2D) conical micromodulus function is adopted, which takes the form of
where E is Young’s modulus; ν is Poisson’s ratio, which is equal to 1/3 for a 2D continuum. s is the bond stretch, which is defined as
In order to incorporate damage into the constitutive model, a critical bond stretch s_{0} is introduced. Once the critical bond stretch is reached, the bond breaks and consequently the bond forces vanish. A historydependent function μ is used to implement it, which is defined as
The local damage at material point x is defined as the ratio of the number of broken bonds to the total number of bonds relevant to point x,
The value of φ(x, t) ranges from 0 to 1. Zero denotes virgin material, while one indicates complete damage.
2.2 DualHorizon Peridynamics
In DHPD formulation, the horizon H_{x} is still the neighborhood of material point x with the horizon radius of δ_{x}, where the bond xx′ will exert the direct force f_{xx′} on x. Based on Newton’s third law, x′ will suffer the reaction force − f_{xx′}. The dualhorizon \(H^{\prime}_{x} = \left\{ {\varvec{x}^{\prime} \in R_{0} \varvec{x} \in H_{{\varvec{x}}^{\prime}} } \right\}\) is a set of points whose horizons contain x. The bond from the dualhorizon is named as dualbond. Similarly, the dualbond x′x will exert the direct force f_{x′x} on x′ and x will suffer the reaction force − f_{x′x}. When uniform discretization and constant horizon are used, DHPD reduces to the traditional PD (Ren et al. 2016). The equation of motion, the bond force and the dualbond force in DHPD are given as
Note that the micromodulus c(x, δ_{x}) depends on the horizon radius of material point x and takes half of the corresponding micromodulus used in the constanthorizon PD. Furthermore, the bond and the dualbond break independently considering that they may have different critical stretches.
3 Finite Element Discretization of DualHorizon Peridynamics
Macek and Silling (2007) pointed out that the basis equations of PD are consistent with FEM code architectures. Additionally, compared with FEM, PD and DHPD are computationally expensive. Therefore, to gain the efficiency from FEM and exploit the generality of DHPD, we implement DHPD into the framework of FEM. The implementation process is comprised of two steps, namely, generating nonuniform grid and solving governing equations.
3.1 Nonuniform Grid
The value of horizon depends on the physical nature of the application being modeled (Silling and Askari 2005). In order to achieve acceptable accuracy, the horizon radius has to be determined in accordance with the lowest material point resolution locally required. However, the dense material point distribution and the sufficiently small horizon size can result in longer computational time. For sake of computational efficiency, nonuniform modeling is required, in which the regions of interest utilize dense material points and the other regions coarse material points.
The quadtree approach is one of the most popular algorithms to generate new material points in the implementation of multiscale modeling, as shown in Fig. 1. Bobaru et al. (2009), Bobaru and Ha (2011) firstly introduced the quadtree approach in PD to implement the adaptive refinement process. Unfortunately, the so called hanging nodes are generated in their works, which would trigger the quadrature errors. Dipasquale et al. (2014) inserted new material points in a recursive manner at the mid points of the lines connecting adjacent material points of the same refinement level. The generation approach is simple. However, the properties of the interface material points, such as the shape and the size, are modified. Consequently, the properties of material points need to update, which is complex to implement, especially for the case of successive refinements. In present study, the quadtree approach is adopted, whereas the hanging nodes are eliminated. The details of the process are as follows.
Firstly, geometry of the specimen is meshed by uniform quadrate solid elements in commercial FEM software ABAQUS and the edge length is ∆x. In what follows, we export the nodal data of the mesh and take the center (x_{c}, y_{c}) and the area of each element as the position and the associated area of the corresponding material point, respectively, as shown in Fig. 2a. If an element is refined, four subelements will be generated and their areas are one quarter of the original. Furthermore, the centers and vertexes of the subelements could be calculated from the center and vertexes of the original element. It can be noticed from Fig. 2b that the positions of all material points are at the centers of their associated domains, that is, the hanging nodes are not generated. In the same pattern, level 2 points can be generated from level 1 points, level 3 points from level 2 points and so on. Hence, the nonuniform discretization is obtained.
After discretization, the mesh of bonds are generated by a preprocessor in MATLAB, which connects each material point to all others within its horizon. The horizon of each material point is defined as three times the corresponding edge length.
3.2 Numerical Implementation
3.2.1 Volume Correction Scheme
For numerical purpose, the integral Eq. (7) is discretized into the following form
where \(N_{{H_{x} }}\) is the number of material points included in the horizon of point x_{i}; \(N_{{H^{\prime}_{x} }}\) is the number of material points whose horizons include point x_{i}; ΔV_{j} is the portion of the volume of point x_{j} covered by the horizon of point x_{i}. When uniform quadrate discretization is adopted, the volume of point x_{j} usually takes the value of ΔV_{j} = Δx_{j}^{2}. Considering that some points are partially covered, a correction scheme of volume fraction for point x_{j} were proposed by Bobaru and Ha (2011) as
When using nonuniform grid, however, Eq. (10) will still introduce certain numerical errors. To improve the accuracy of volume approximation, we propose a method based on the background grid. The main idea of the method is to split the associated domain of point x_{j} into many subdomains based on the background grid, calculate the volume fraction of each subdomain covered by the horizon of point x_{i} with Eq. (10), and summate the covered volumes of all subdomains. The summation is the portion of the volume of point x_{j} covered by the horizon of point x_{i}.
Take the grid with 2 levels in Fig. 3a as an example. First, all the points at level 0 are refined to level 1 points and the inheritance relationship between old and new points are recorded. Then, the grid with all level 1 points (the blue dashed lines in Fig. 3b) is the background grid. With the background grid, calculating the volume fraction ΔV_{j} of point x_{j} covered by the horizon of point x_{i} is actually to calculate and summate the covered volume of all subdomains. In the same way, the covered volume of other points could be obtained. It is a remarkable fact that the points \(\varvec{x}_{j} ,\varvec{x}_{k} ,\varvec{x}_{m} ,\;\varvec{x}_{n}\) are out of the horizon of point x_{i}, and their distances from point x_{i} are more than δ + 0.5Δx. If only Eq. (10) is utilized without the background grid, the covered volume of the four points (the black shadows in Fig. 3b) are zero, which will trigger large errors. Furthermore, the denser the background grid is, the more accurate the volume approximation is; contradictorily, the more expensive the computational cost is. Therefore, to balance the accuracy and efficiency, the background grid in level (k + 1) is used to calculate the volume covered by the horizon of point x_{i} in level k in the following simulations.
3.2.2 Solving Algorithms for QuasiStatic Problem
In its original formulation, PD has been developed to solve dynamic problems, such as dynamic crack propagation in brittle materials. In recent years, some researchers have applied PD in static problems with two methods. Kilic and Madenci (2010), Huang et al. (2015) introduced an adaptive dynamic relaxation method into the PD equations of motion for quasistatic fracture analysis. Gerstle et al. (2007), Zaccariotto et al. (2015) converted the PD equations of motion into equilibrium equations by letting the acceleration be zero, and solved the equilibrium equations in the framework of FEM. The former method is timeconsuming and the computational efficiency and accuracy is dependent on the artificial damping. In present study, truss elements are used to represent the bonds and dualbonds, and the quasistatic simulations are carried out by solving the equilibrium equations.
In the case of quasistatic motion, the acceleration is zero. Hence, Eq. (9) is rewritten into Eq. (11) or Eq. (12) in the form of FEM.
where [K] is the global stiffness matrix, \(\left\{ \varvec{u} \right\} = [u_{1} ,v_{1} , \cdots u_{i} ,v_{i} , \cdots u_{n} ,v_{n} ]^{\text{T}}\) is the vector of nodal displacements and n is the number of material points, \(\left\{ \varvec{F} \right\} = [f_{1x} ,f_{1y} , \cdots f_{ix} ,f_{iy} , \cdots f_{nx} ,f_{ny} ]^{\text{T}}\) is the vector of external forces, in which f_{ix} = b_{ix}ΔV_{i}.
\(\left[ {\bar{k}^{p} } \right]^{e}\) and \(\left[ {k^{p} } \right]^{e}\) are the element stiffness matrix of the bond x_{i}x_{j} in local and global coordinates, respectively; and take the forms of
where [T^{e}] is the coordinate transformation matrix.
The pairwise force functions of the bond x_{i}x_{j} and the dualbond x_{j}x_{i} take the following forms
where [k^{p}]^{e}, {u^{p}}^{e}, [k^{q}]^{e} and {u^{q}}^{e} denote element stiffness matrix and the vector of element nodal displacements of the bond x_{i}x_{j} and the dualbond x_{j}x_{i} in global coordinates, respectively.
The global stiffness matrix [K] can be assembled by the superimposing method in FEM. Let row (2i − 1) and (2i) about point x_{i} in [K] be [K_{i}], which can be expressed as
An inhouse 2D peridynamic code in MATLAB is used to solve the equilibrium equations. With the microelastic damage model described in Eq. (5), damage is incorporated into the model. Once its critical stretch is reached, the bond fails and the global stiffness matrix is updated.
4 Simulation of the Shear Test in FRPtoConcrete Bonded Joints
4.1 DHPD Modeling
As a new approach in analysis of the bond behavior between FRP and concrete, the validity and accuracy of the proposed DHPD approach need to be verified. Similar to the reliabilitybased analysis with a large database of joint tests in literatures (Zhang et al. 2018; Shi et al. 2015), the single shear tests reported in Yao et al. (2005) were employed to compare with the DHPD predictions. Yao et al. (2005) reported a total of 72 tests, but 16 of them failed in other modes rather than debonding in concrete and 4 of them were designed to investigate the effect of the loading offset so they were excluded in present study. The reported specimens had two kinds of FRP (CFRP and GFRP), the FRP width ranging from 25 to 100 mm, the bond length increasing from 75 to 240 mm, concrete cylinder strength varying from 19 to 27 MPa and the height of free concrete edge changing from 5 to 120 mm. As shown in Fig. 4, a concrete prism bonded with an FRP plate was adopted in the DHPD simulations, in which the geometry of the concrete prism was 350 mm (L) × 150 mm (H), and h_{c} ranged from 5 to 120 mm. In the numerical model, the specimen was restrained vertically along the base and horizontally along part of the right edge. To reduce the computational cost, the three dimensional specimen was modeled as a plane stress problem and the nonuniform grid was adopted in present study. The grid spacing of concrete increases from the top to the bottom of the concrete prism, and the maximum and minimum grid spacing were 1 mm and 0.125 mm, respectively. The grid spacing of the adhesive layer and FRP plate was 0.0625 mm and 0.03125 mm, respectively. The material properties for specimens were given in Table 1.
In order to correctly model real materials, some material properties have to be mathematically transformed to equivalent parameters used in DHPD. The micromodulus of bonds in FRP, the adhesive layer and concrete could be obtained from Eq. (3). The critical stretch of each phase is equal to the corresponding ultimate tensile strain. It is worth noting that the material model of DHPD bonds in all three phases is assumed to be linearly elastic brittle. Once the critical stretch is exceeded, the bond is broken and removed from the analysis.
4.2 Comparison of DHPD Predictions with Test Results
In present study, the shear test was modeled as a plane stress problem, while the actual behavior is threedimensional. Therefore, the width ratio factor β_{w} proposed by Chen and Teng (2001) was adopted to correct the discrepancy. The predicted load, displacement, stress and strain in the FRP plate were all adjusted with the width ratio factor β_{w}. The width ratio factor took the form of
where b_{f} and b_{c} are the widths of the FRP plate and the concrete prism, respectively.
Figure 5 shows the comparison of DHPD predictions with test results. It can be clearly seen that the DHPD predicted results are overall in very close agreement with the 52 test data. The typical specimen (specimen II5) is chosen from the aforementioned database for further comparison between the DHPD prediction, test data, FEM results and the analytical solution.
Damage contour of debonding failure for specimen II5 is shown in Fig. 6, in which only the part of concrete near the adhesivetoconcrete interface is exhibited for easier observation. It can be observed that debonding failure occurs in concrete at a small distance (1–4 mm) beneath the adhesivetoconcrete interface. Additionally, the distribution of damage is not uniform along the bond interface and the cracks are at about 45° to the horizontal. These phenomena are consistent with the experimental observations (Yao et al. 2005; AliAhmad et al. 2006; Carrara et al. 2011; Benzarti et al. 2011; Wu and Jiang 2013) and the FEM results (Lu et al. 2005a, 2006; Pham and AlMahaidi 2007; Tao and Chen 2015; Xu et al. 2015).
A comparison of the load–displacement curves for specimen II5 between the experiment, the FEM simulation, the analytical solution and present simulation is shown in Fig. 7. It is clear that the numerically predicted response agrees well with the experimental and FEM results. Point A and B marked on the curves represent the softening initiation and the debonding initiation, respectively. Initially, the load increases linearly with the displacement until Point A is reached. Subsequently, the concrete under the adhesivetoconcrete interface near the loaded end exhibits damage and the softening phase initializes. When Point B is reached, the FRP starts to debond from concrete and thereafter the debonding propagates rapidly towards the free end of the FRP. It is worth noting that, similar to the trend of the experimental and FEM results, the load increases slowly after the debonding initiation, which is probably caused by the friction in the debonded zone and the stochastic nature of the cracking process.
Figure 8 shows the normal stress distributions of FRP plate in the experiment, the FEM simulation, the analytical study, and present DHPD simulation at Point A, B and C marked in Fig. 7. There is a very close agreement between the four curves at Point A. At Point B and C, the DHPD predictions are consistent with the FEM and analytical curves and in agreement with the experimental results in terms of the overall trend, despite the large fluctuations in the experimental results. Fig 9 shows the comparison of the normal strain distributions in FRP plate between the experiment and present simulation. Similar to the normal stress, the DHPD predictions are in agreement with the experimental results in overall trend and the test strain fluctuates as a result of probable measurement errors and plate bending due to the thinness of the plate and the roughness along the crack.
Bondslip curves for specimen II5 at different distances from the free end are shown in Fig. 10, in which the interfacial slip of a particular point is defined as the relative displacement between the FRP plate and concrete at that point, and the shear stress at the corresponding location of the interface is calculated from the normal stress in the FRP plate using the following relationship (Wu and Jiang 2013; Dai et al. 2005; Lu et al. 2005a; Nakaba et al. 2001)
where σ_{f} and t_{f} are the normal stress and the thickness of the FRP plate, respectively; x is the distance from the free end; \(\text{d}x\) is the incremental distance used for calculation of the shear stress.
It can be seen from Fig. 10 that the trends of bond slip curves at different distances from the free end are consistent, while the numerical difference of the maximum shear stresses is large. The nonuniform bondslip relationship is in accordance with the test findings (Dai et al. 2005; Chajes et al. 1996; Yun et al. 2008), the FEM result (Lin and Wu 2016) and the analytical study (Zhou et al. 2010). The nonuniformity is related to the variation of the depth of failure plane. When the failure depth increases, the maximum shear stress increases (Lin and Wu 2016). To further verify the accuracy of the bondslip relationship, the mean curve of all curves in Fig. 9 is calculated and compared with existing bondslip models. It can be observed from Fig. 11 that the mean curve predicted by DHPD simulation matches well with existing bondslip models(Wu and Jiang 2013; Dai et al. 2005; Nakaba et al. 2001; Savoia et al. 2003; Lu et al. 2005b).
5 Discussion
As mentioned above, the main failure mode of FRPtoconcrete bonded joints in the shear tests is concrete failure. Therefore, the mechanical performances of concrete play a dominant role in the bond behavior. Additionally, the bond behavior can be also affected by the geometric and material properties of FRP and the adhesive. In this section, the effects of concrete strength, the thickness of the FRP plate and the adhesive layer on the bond behavior were studied with the proposed DHPD model.
5.1 Effect of Concrete Strength
In order to investigate the effect of concrete strength on the bond behavior, four groups of DHPD models for specimen II5 with concrete strengths ranging from C20 to C50 were adopted. From C20 to C50, the tensile strength is 1.1 MPa, 1.43 MPa, 1.71 MPa, 1.89 MPa, and the elastic modulus is 25.5 GPa, 30 GPa, 32.5 GPa, 34.5 GPa, respectively.
Figure 12 illustrates the load–displacement curves with different concrete strengths. It is evident that the ultimate load and the debonding ductility increase with the increasing concrete strength. It has been generally accepted that the debonding failure occurs in concrete and thus the bond behavior is usually controlled by the behavior of concrete. Higher concrete strength contributes to stronger resistance to debonding, and consequently the ultimate load and the debonding ductility are higher. Furthermore, the maximum shear stress at a particular point of the interface denotes the bond strength at that point. The average maximum shear stress of all material points in the interface versus concrete strength is shown in Fig. 13. It can be observed that the average maximum shear stress almost increases linearly with concrete strength, which demonstrates that the bond strength increases with the increasing concrete strength.
5.2 Effect of the Thickness of FRP Plate and Adhesive Layer
To study the effect of the FRP thickness on the bond behavior, three groups of DHPD models with different FRP thicknesses (0.165 mm, 0.33 mm and 0.495 mm) were adopted.
Figure 14 shows the load–displacement curves with different FRP thicknesses. It can be clearly seen that the ultimate load increases with the increasing FRP thickness, while the debonding ductility significantly decreases. These phenomena match well with the test results (Zhang and Smith 2013) and the numerical simulations (Sun et al. 2017a, b), which further demonstrates the accuracy and efficiency of the proposed DHPD model. Additionally, the decrease of the debonding ductility indicates that the efficiency of the FRP plate is reduced when a thicker FRP plate is used.
Figure 15 illustrates the predicted load–displacement curves with different adhesive thickness, in which the adhesive thicknesses are 0.5 mm, 1 mm and 2 mm, respectively. As shown in Fig. 15, the ultimate load slightly increase with the adhesive thickness. In addition, the average maximum shear stresses for the three adhesive thicknesses have small difference and are 6.82 MPa, 6.9 MPa and 6.96 MPa, respectively; that is, the adhesive thickness has no significant effect on the bond strength. Therefore, the effect of the adhesive layer can be regarded as mainly playing a role in transferring surely shear stresses from concrete to the FRP plate.
6 Conclusions
A DHPD model implemented in the framework of FEM with multiscale discretization and a volume correction scheme have been developed for the simulation of debonding process in FRPtoconcrete bonded joints in this paper. A benchmark numerical example was implemented to validate the proposed model in the analysis of the debonding process. The predicted results including concrete failure pattern, loadslip curve, FRP stress distribution, FRP strain distribution were consistent with the experimental findings, the FEM simulations and the analytical solutions. Furthermore, the effects of concrete strength, the FRP thickness and the adhesive thickness on the bond behavior were investigated. The main conclusions could be drawn as follows.

1.
The DHPD model of FRPtoconcrete bonded joints combines the efficiency of FEM with the generality of DHPD, and thus can effectively simulate the debonding process. The model lays a certain foundation for further research of the bond behavior.

2.
The nonuniform discretization generated from the quadtree approach can significantly reduce the computational cost. The volume correction scheme based on the background grid can enhance the accuracy when the nonuniform grid is used. Moreover, the utilization of the background grid need to balance the accuracy and efficiency.

3.
The concrete failure pattern and the local bond strength along the interface is nonuniform. The nonuniformity is consistent with the experimental observations, which probably caused by the friction in the debonded zone and the stochastic nature of the cracking process.

4.
The bond strength and the debonding ductility are distinctly affected by concrete strength and the FRP thickness, while the adhesive thickness has no significant effect on the bond behavior. The bond strength and the debonding ductility increase with the increasing concrete strength. For a thicker FRP plate, the bond strength will increase and the debonding ductility will decrease.
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Authors’ contributions
LG conceived and designed the study, reviewed and edited the manuscript. WL performed the simulations and wrote the manuscript. Both authors read and approved the final manuscript.
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Funding supports for this work by the National Natural Science Foundation of China (Nos. 51578142 and 51478108) and the Priority Academic Program Development of Jiangsu Higher Education Institutions are gratefully acknowledged.
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The authors declare that they have no competing interests.
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Li, W., Guo, L. DualHorizon Peridynamics Analysis of Debonding Failure in FRPtoConcrete Bonded Joints. Int J Concr Struct Mater 13, 26 (2019). https://doi.org/10.1186/s4006901803286
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DOI: https://doi.org/10.1186/s4006901803286