Dynamic tensile behavior of fiber reinforced materials based on fiber layering modeling

Abstract

Among the failure forms of fiber reinforced materials, interface failure caused by fiber debonding is the most common but complex failure mode. This research proposed a fiber layering modeling method based on the different mechanical properties of the outer and inner parts of fiber yarn, then investigated the mechanical behavior of fiber reinforced materials under dynamic loads with different loading speeds. Two pullout models were established, considering the layering feature of fiber caused by the matrix penetration. The simulated results were compared with the existing experimental data to determine reliable parameters values of the fiber-matrix interface. Based on the values, another pullout model and tensile model were built. Distribution and development of stress in the loading process were discussed. Results show that fiber modeling with inner and outer parts, as well as the cohesive property with reliable parameters, could effectively represent the interface characteristics of fiber reinforced materials under dynamic load.

Keywords

tensile behavior dynamic loading cohesive property numerical simulation fiber layering

Introduction

Coastal structures often suffer from dynamic loads, such as wave scouring or ships impacting, which causes severe damage in several parts of the structures. As widely accepted, cement-based material is brittle with low ductility and energy-absorbing ability (Li et al., 2023; Mousavi et al., 2019; Wang et al., 2023). Considering of such feature, fiber is often used synchronically to enhance the cracking-resistance/ductility of cementitious materials and improve the load-carrying capacity after crack initiation, which is known as fiber reinforced polymer (FRP) (Nwankwo et al., 2023; Zhou et al., 2023). The merit of FRP is the good resistance against corrosion, which can be functionally used with the marine concrete that usually faces to large amount of chloride ion (Less et al., 2023; Zhu et al., 2023). In addition to the externally bonding of FRP plate to cementitious structures for strengthening or retrofitting (Wang et al., 2022), another application is to embed the FRP short/long fibers, bars or textiles inside concrete to form the composite called fiber reinforced cementitious matrix (FRCM) (Michels et al., 2016; Tang et al., 2022). Such FRCP could be adequate composite to act as both structural strengthening (SS) and impressed charge cathodic protection (ICCP) materials for the application of sea-water seas-and concrete (SSC) with steel reinforced bars (Feng et al., 2022; Su et al., 2021).

Fiber plays a complex role in the fracture process of fiber reinforced materials, functioning as stress-transferring and energy-consumption (Amran et al., 2020; Trainor et al., 2015). In addition to enhance the cement-based materials in terms of static load, fiber also shows improvement for the mechanical properties of the materials under dynamic load (Peled, 2007; Zhu et al., 2009). Due to the different strengths of fiber, matrix and interface phases, the failure modes of fiber reinforced materials include fiber fracture, interface failure and matrix cracking (Su et al., 2023). Among them, the interface failure caused by fiber debonding is the most common failure mode, since the fiber-matrix interface is generally weaker than the fiber and matrix (Pi et al., 2021; Wang et al., 2021). Hence, several methods have been put forward to improve the strength and stiffness of the interface (Bentur, 2000; Mobasher and Cheng, 1996; Yoo and You, 2021). Especially when fiber yarn with multi-fibers is adopted for FRCM, it is reported by Zhu et al. (2020b) that the fiber yarn can be divided in to the inner and outer parts depending on the penetration of cementitious paste through the vacant spaces between fibers, which leads to an inner-outer fiber interface and a fiber-matrix interface in the fiber yarn.

With respected to interface slippage as the predominant failure mode of fiber reinforced materials (Zhu et al., 2020c), the typical pullout test is able to clarify the mechanism. Hence, experimental works have been carried out to discuss the influence by various parameters, such as the fiber’s material (Zhu et al., 2020c), geometry (Yoo et al., 2021), orientation (Chen et al., 2021), surface condition (Kim et al., 2020) and embedded length (Kim et al., 2017). In addition to the static loading condition, since the fracture morphology and load capacity of materials are greatly influenced by the loading rate (Lee et al., 2010), dynamic pullout test has also been conducted to study the influence of loading rate on the pullout behavior (Cheng et al., 2005; Xia and Wang, 1999). In these works, the mechanical properties are found to be sensitive to the loading rate and temperature, but the fracture pattern is not obviously affected. Experimental data from the fiber pullout tests can bring understanding to the mechanical property and interfacial bond failure of fiber reinforced materials. However, the test requires loading device with high precision and less fluctuation, the stability and reliability of test results are often difficult to control in reality. Furthermore, it’s hard to observe and investigate the stress distribution of fiber and matrix during the test. These drawbacks can be overcome by the numerical simulation with reliable constitutive models and rational input parameters (Zhu, 2020).

Many efforts have been paid aiming at quantitative description of the bond slip relationship of the interface based on the experimental data (Dai et al., 2005; Nakaba et al., 2001; Pang et al., 2020). However, these empirical relationships vary from each other due to the different experimental conditions and material properties, such as the properties of fiber and matrix, the details of test, and the methods of determining the bond slip relationship. Among the various schemes, the method used to determine the bond slip relationship is the primary factor. Since it’s very difficult to obtain the accurate and reliable bond slip relationship from strain measurement data, as these data normally show large discreteness (Zhu et al., 2020a). Thus, it’s necessary to carry out studies based on the bond slip relationships from formula derivation, although these relationships are usually simplified and made with assumptions. In addition, it was observed that the fiber yarn can be divided in to the inner and outer parts due to the penetration of cementitious paste (Zhu et al., 2020b). The mechanical properties of the inner and outer layers are different, so it is not appropriate to assign the same property to the two layers in the simulation analysis. Besides, slippage of the interface between the fiber’s inner and outer parts can also lead to the final failure of the FRCM material (Zhu et al., 2020b). This phenomenon can be reflected in the simulation only if the fiber layering is considered in modeling.

This research aims at the exploration of the feasibility and necessity of fiber layering modeling in the simulation of fiber reinforced materials, then investigating the mechanical behaviors of fiber reinforced materials under dynamic load with different loading speeds. Firstly, two pullout models of FRCM material were established with consideration of fiber layering induced by the matrix penetration. The simulated load-displacement curves were verified by the existing experimental results to determine the values of necessary parameters. Then, based on these inputs, another pullout model and a tensile model were built and investigated, where the distribution and development of stress in the loading process were discussed.

Theory background

Fiber-matrix interface properties

In the process of pullout, fiber is subjected to both tensile stress from the pullout load, and shear stress at the fiber-matrix interface. At the static state or uniform motion, the two forces have equal values with opposite directions, as described in equation (1):

\int A d σ = \int c τ d x

(1)

where σ is the tensile stress of fiber (MPa), τ is the bond stress of the fiber-matrix interface (MPa), A and c are the cross-sectional area (m²) and perimeter (m) of the fiber, and x is the axial distance (m).

Considering the tension behavior of fiber as linear, the force equilibrium can also be expressed as

\frac{d^{2} s}{d x^{2}} - \frac{c τ}{E A} = 0

(2)

where s and E are the slipping distance (m) and Young’s modulus (MPa) of fiber.

The displacement of loading end of fiber is the pullout displacement, which can be measured by the instrument. Meanwhile, the strain of free end is zero. Thus, the boundary conditions of fiber with pullout load can be expressed as:

s (x = 0) = u_{p}

(3)

\frac{d s}{d x} |_{x = l} = 0

(4)

where u_p is the pullout displacement (m), l is the length of fiber (m).

In the early research, the bond stress τ is assumed to distribute evenly along the pullout process, which is

τ = \frac{P_{f}}{π d l_{f}}

(5)

where P_f and l_f are the pullout load (N) and the residual embedded length (m) of fiber, d is the diameter of fiber (m). This theory is suitable for the plastic materials such as metal and polymer, since the pullout failure usually comes from the damage of matrix. Thus, the value of bond strength is determined by the properties of matrix.

τ = K (u - v)

(6)

However, for the FRCM material, the fiber-matrix interface is weaker and adhesive damage is usually the dominant failure mode. In this case, the even stress theory cannot accurately reflect the shear stress distribution of the fiber-matrix interface. Hence, when it comes to the cement-based material as matrix, the bond stress at the interface is assumed as related to the relative displacement between the fiber and matrix:where K is the elastic constant of the fiber-matrix interface, u is the axial displacement of fiber (m), and v is the displacement of matrix at the same place (m). The failure of interface is interpreted as a gradual process, where the debonding occurs near the loading end at first and then develops along the fiber.

From the viewpoint of entire pullout process, the fiber stands still although the shear stress keeps increasing at the initial stage, this elastic phase is linear in stress-strain curve, as described in equation (7):

τ = k \cdot s

(7)

where k is the initial stiffness (N/m³).

Then the debonding of fiber begins when the shear stress reaches the maximum bond stress, which is usually expressed as linear or exponential in the stress-strain curve:

τ = τ_{\max} - k_{d} \cdot (s - s_{\max})

(8)

τ = τ_{\max} - \exp (s - s_{\max})

(9)

where τ_max is the maximum bond stress (MPa), s_max is the slipping distance of fiber at the maximum bond stress point (m), and k_d is the debonding stiffness (N/m³).

And the last stage is the slipping phase, in which the fiber slips with friction. The three phases of pullout can be described with the trilinear slip relationship in Figure 1. Assuming the friction is neglectable, the three phases can be simplified and determined by three parameters: the initial stiffness k (N/m³), the maximum bond stress τ_max (MPa), and the failure displacement s₀ (m). Some complex and more accurate modeling methods (Dai et al., 2009; Garcia et al., 2014; Yamanaka et al., 2015) to represent the property of fiber-matrix interface have also been put forward.

Figure 1.

The trilinear slip relationship for the fiber-matrix interface.

Layering feature of yarn

As observed by the scanning electron microscope (SEM), the fabric embedded in the cementitious matrix is partially submerged by the paste, which divides the embedded fiber yarn into the inner and outer parts, as shown in Figure 2. According to the experimental result from Zhu et al. (2020b), the impregnation degree, which means the ratio of filaments number in the outer part to that in the total fiber yarn, is approximately 30%.

Figure 2.

The cross-section of fiber yarn embedded in the cementitious matrix.

Consequently, filaments in the inner and outer parts show different bond properties under pullout load. The outer filaments show a strong bond adhesion with matrix, while the inner filaments have no contact with matrix. Thus, when the fiber yarn is stretched, the tensile stress in the outer part is higher than the inner part. Besides, the mechanical property of the outer part also changes due to the matrix penetration. For example, the Young’s modulus of the outer part can be determined from the formula to calculate the Young’s modulus of the fiber reinforced composite:

E_{c o m} = (E_{v} \cdot V_{f} + E_{m} \cdot V_{m}) / (V_{f} + V_{m})

(10)

where E_com, E_f, E_m are the Young’s modulus of the composite, fiber and matrix respectively (GPa), V_f, and V_m are the volume ratio of the fiber and matrix in the composite. According to the SEM image of the cross-section of fiber yarn in literature (Zhu et al., 2020b), the ratio between V_f and V_m of the impregnated fiber yarn is 1:1.

In addition to the fiber-matrix interface, interface between the fiber inner and outer parts exist due to the layering feature of fiber yarn, which is noted as the “inner interface” in Figure 2. Different from the slipping characteristic of the fiber-matrix interface (“outer interface” in Figure 2) as discussed in Section 2.1, the bond of the inner interface comes from the presence of friction in between. Hence, constant bond stress is assumed for the slipping relationship of the inner interface.

Furthermore, due to the Poisson’s effect and the contraction of free space between filaments during the pullout, the inner part becomes more compact, and the exiting pressure normal to the inner interface is reduced. Thus, the bond slip relationship featured by a descending behavior can be assumed for the inner interface:

τ_{i n} = τ_{i n, \max} - k_{i n} \cdot s_{i n}

(11)

where τ_in is the bond stress of the inner interface (MPa), τ_in,_max is the maximum bond stress (MPa), s_in is the slipping distance between the inner and outer parts (m), and k_in is the stress attenuation coefficient (N/m³).

As mentioned above, due to the existence of multi-interfaces, the FRCM material may have various failure mode under the pullout load. As reported by Zhu et al. (2020b), the pullout specimen with the fiber embedded length of 30 mm failed with the slippage of the whole fiber yarn through matrix. While for the specimen with fiber embedded length of 50 mm and 70 mm, the failure mode changed to the combination of the slippage at the inner interface, and tensile rupture of the fiber outer part. Without considering the layering modeling of fiber yarn, this failure pattern cannot be reflected in the numerical simulation.

Since the fiber yarn is made up by many filaments, the bond behavior of each filament and the interaction between them must be taken into consideration to accurately model the pullout behavior of entire fiber yarn. However, such modeling method will cause a lot of computational work and is difficult to use in practice. Thus, the fiber yarn is simply modeled as the outer and inner parts, as well as the two interfaces. This simplification correlates the experimental observations, which reduces the computational consumption and synchronically capture the bonding behavior considering such matrix impregnation.

Dynamic equilibrium

The dynamic equilibrium of an object can be described by equation (12):

[M] {\ddot{u}} + {I} = {F}

(12)

where [M] is the mass matrix of the analysis structure, {

\ddot{u}

} is the acceleration vector, {I} is a vector associated with the displacement and velocity, and {F} is the load vector.

The vector {I} is used to represent the effects of damping and energy consumption, which is

{I} = [K] {u} + [C] {\dot{u}}

(13)

where [K] and [C] are the stiffness matrix and damping matrix respectively, {u} and {

\dot{u}

} are the displacement vector and velocity vector respectively.

Hence, the largest difference between the static and dynamic analysis is whether to consider the influence of inertia force on the model. The calculation equations of them in the numerical simulation are shown in Equation (14) and Equation (15) respectively:

[K] {u} = {F}

(14)

[K] {u} + [C] {\dot{u}} + [M] {\ddot{u}} = {F}

(15)

Based on the different objectives, dynamics analysis can be divided into the following categories: the transient dynamic analysis, the steady-stats dynamic analysis, the frequency analysis, the response spectrum analysis, and the random response analysis. Among them, the transient dynamic analysis is often used to analyze the mechanical response of structure under acyclic loading. It is implemented as four solvers in numerical software, that is the dynamic implicit solver, the dynamic explicit solver, the dynamic subspace solver and the modal dynamic solver. These four dynamic solvers can be adopted to respectively investigate the strong nonlinear transient response, the large-scale dynamic response with short time, the time domain analysis of linear system, and the analysis of mild nonlinear system. Unlike the static analysis, the density of material is an essential input parameter when using the dynamic solvers, since the inertia force is in proportional to the mass of the numerical model.

Validation of simulation

Model description

Since the parameters values of the fiber-matrix interface have great influence on the simulation results of the FRCM material, two verification models with the same set-up in Liu et al.’s experiment (2020) were made to determine the rational values. The models have one fiber yarn or one fiber grid embedded in the matrix.

The details of model BS and corresponding specimen in verification test are presented in Figure 3. Sizes of the fiber yarn and the mortar matrix were 65 mm × 1 mm × 0.1 mm and 30 mm × 60 mm × 15 mm respectively, where the fiber yarn was inserted into the matrix of 30 mm depth. Considering the layering feature of fiber, the cross section of the fiber yarn was modeled at 30% matrix impregnation ratio. It should be noted that the matrix impregnation ratio of 30% may not be applicable for all types of FRCM with different materials. However, this research focuses on the proposal and application of the fiber layering modeling method. Change of the value of the matrix impregnation ratio does not influence the method itself. The difference lies only in the dimensional division of the inner and outer layers of the fiber yarn’s cross-section when modeling.

Figure 3.

Size of the fiber yarn cross-section and the model BS.

No slippage was assumed between the inner and outer parts in the pullout with the fiber embedded length of 30 mm according to the research (Zhu et al., 2020b) discussed in Section 2.2. Thus, the two parts of fiber yarn were connected as “tie contact” with infinite friction, since the fiber embedded length of the models in the study was or was close to 30 mm. It is worth noting that the depths of mortar penetration in the two directions are the same in reality. In this study, the reason why the depths in the two directions were set to be different is because we want the length and width of the inner fiber cross-section to be reduced in equal proportions relative to the whole cross-section of fiber yarn. Then to discuss the difference of stress of the inner and outer part of fiber is Section 3.2.4. However, the depths of mortar penetration in the two directions are recommended to be modeled as the same in the numerical simulation without any special research purposes.

For the model BM, the grid was fabricated by 10 transverse and two longitudinal fiber yarns, as shown in Figure 4. Size of the longitudinal fiber yarn was 65 mm × 1 mm × 0.1 mm and the transverse one was 8 mm × 1 mm × 0.1 mm according to the experiment. Each fiber yarn also consisted of both inner and outer parts, which were connected as “tie contact” as well. The 12 fiber yarns were merged as a whole part. Size of the matrix was 30 mm × 60 mm × 15 mm. The fiber mesh was also inserted into the matrix with 30 mm depth.

Figure 4.

Size of the fiber mesh and the model BM.

Since no fracture of the fiber or matrix was captured in the pullout test, elastic constitutive model was used for the fiber material, and ideal elastoplastic model was chosen for the matrix with consideration of the time-dependent constitutive relationship. Material inputs of the fiber inner part and the matrix were chosen from Liu et al.’s experiment (2020). The mechanic property of the penetrated layer of fiber yarn is challenging to measure in experiments, which makes its value difficult to be determined. In this study, the Young's modulus of the fiber penetrated part was calculated by equation (10), while the density and Poisson’s ratio were simply assumed by the linear interpolation between inner parts and matrix. This is a compromise method to determine the property if there is no reliable data. Further studies on the property of fiber penetrated layer need to be carried out, such as simulation methods using sensitivity analysis, or experimental methods using nano-indentation testing. Then the relationship between strain rate and dynamic influence coefficient was input for the materials. The fiber-matrix interface was set with the cohesive property, and values of the parameters were determined by the comparation between the simulated load-displacement curves with Liu et al.’s experimental results. The parameters values used in the models are listed in Table 1.

Table 1.

The Parameter Values in the Models.

Parameters	Values
Density of the fiber inner part	1.85 × 10⁻³ g/mm³
Young’s modulus of the fiber inner part	58.8 GPa
Poisson’s ratio of the fiber inner part	0.3
Density of the fiber outer part	1.93 × 10⁻³ g/mm³
Young’s modulus of the fiber outer part	44.4 GPa
Poisson’s ratio of the fiber outer part	0.25
Density of the matrix	2 × 10⁻³ g/mm³
Young’s modulus of the matrix	30 GPa
Poisson’s ratio of the matrix	0.2
Yield stress of the matrix	3 MPa
Initial stiffness of the fiber-matrix interface¹	3200 N/mm³
Maximum bond stress of the fiber-matrix interface¹	4.5 MPa
Failure displacement of the fiber-matrix interface¹	9 mm

Note: 1. Determined by the sensitivity analysis.

Mesh size was 0.01 mm for the fiber and 0.8 mm for the matrix. Prescribed displacement loading was adopted, 10 mm displacement was applied on the loading end to represent the pullout load. In addition, the same loading rates as the ones in Liu et al.’s experiment were used, which were 25, 2.5 and 0.25 mm/min. The degrees of freedom in X, Y, Z directions of the bottom surface of the matrix (X-Y plane, Z = 0) were fixed. The models were calculated by the dynamic solver in Abaqus finite element analysis software.

Result of model BS

Determination of input values

It is generally accepted that the loading rate greatly influences the result of pullout test. Hence, simulations of the fiber yarn model with different loading rates were conducted, and the load-displacement curves of the fiber load applied end are shown in Figure 5. With the increasing of loading rate, the slopes of the ascending branch as well as the peak loads increased, while the displacements at peak load decreased. At the same time, the higher loading rate also increased the slopes of the curves descending part making it a more brittle post-peak behavior.

Figure 5.

Comparation of the load-displacement curves between simulation of the model BS and Liu’s experiment.

To determine the inputs values of various parameters for the constitutive relationships of fiber-matrix interface, the simulated pullout load-displacement curve of the fiber yarn model was compared with the result of Liu et al.’s experiment (2020) in Figure 5. The parameters including initial stiffness, maximum bond stress and the failure displacement were 3200 N/mm³, 4.5 MPa and 9 mm respectively. From comparison, the simulated load-displacement curve was found to correlate with the experimental results in the most parts, but slight difference could be seen in the debonding part. During debonding process, the fiber-matrix interface damages and results in large deformation and nonlinear characteristics, which are difficult to effectively simulate.

Stress distribution

Figure 6 shows the stress distribution of the fiber yarn in the pullout process with loading speed of 25 mm/min. Width of the fiber yarn in Figure 6 is increased in order to show the stress distribution clearly. At the first stage of pullout, large stress concentrated at the loading end. With the increasing of the pullout displacement, stress at the loading end increased and spread to the free end. When it comes to the third phase, largest stress of the pullout process existed in the part of fiber near the loading end and decreased gradually along the axial distance. After this phase, stress in the fiber yarn began to gradually decrease, because in the debonding phase, slippage occurred between the fiber and matrix, and the composite was no longer able to bear large stress.

Figure 6.

Stress distribution of the fiber yarn of the model BS.

The stress distribution of the matrix of the models BS is shown in Figure 7. In the beginning, stress appeared in the top part of the matrix, as only the upper part of the fiber was subjected to stress and transmitted to the surround matrix (He et al., 2022). The spindly large stress area was in the center of the matrix. Then, stress developed to the lower part of the matrix with the same speed of that in the fiber, and area of the maximum stress region with red color also increased. As the pullout displacement increased, stress in the matrix gradually reduced, since some parts of the fiber yarn began to de-bond and lose the ability of carrying stress.

Figure 7.

Stress distribution of the matrix in the X-Z profile of the model BS.

Parametric study

To discuss the influence of the parameters of the fiber-matrix interface on the load-displacement curve, parametric study with model BS and different parameters values was carried out. Loading rate of the model discuss in this section was kept 25 mm/min.

Effect of the initial stiffness

The initial stiffness of the interface will influence the elastic phase of the pullout process, as shown in Figure 1. Figure 8 indicates the pullout load-displacement curves with the initial stiffness of 2600, 2800, 3000 and 3200 N/mm³. It can be seen that with the increasing of the initial stiffness, the slope of the curves elastic part and the displacement at the peak load increased. However, the initial stiffness didn’t affect the peak load or the descending part of the curve, since it only controls the resistance of the fiber-matrix interface in elastic phase, but has no effect on the debonding phase.

Figure 8.

The pullout load-displacement curves with different values of the initial stiffness.

Effect of the maximum bond stress

The maximum bond stress is the shear stress at the end of the elastic phase of the trilinear slip relationship as shown in Figure 1. Figure 9 shows the pullout loads which maximum bond stresses were 3, 3.5, 4 and 4.5 MPa respectively. It is obvious that the maximum bond stress didn’t affect the slope of the curve elastic part. But the larger maximum bond stress will lead to larger displacement and load values at the end of the elastic part. What’s more, the maximum bond stress was unacted on the slope of the descending part.

Figure 9.

The pullout load-displacement curves with different values of the maximum bond stress.

Effect of the failure displacement

The failure displacement is the difference of displacement between the fiber and matrix at the interface when the fiber is totally debonded. The pullout load-displacement curves with the failure displacement of 6, 7, 8 and 9 mm are presented in Figure 10. The four curves had the same elastic part. While slopes of the descending parts were different, because the failure displacement determined the completely failure point of the interface during the debonding phase.

Figure 10.

The pullout load-displacement curves with different values of the failure displacement.

Impact of fiber layering model

The mechanical property of the outer part has changed on account of the matrix penetrated, which was reflected in the models described in section 2.2. Figure 11 compares the pullout load-displacement curves of the model BS with or without considering fiber layering. According to the figure, for each loading rate, the slope of the curve ascending part of the model considering layering was larger than that of the model not considering layering, as well as for the peak load. It shows the consideration of fiber layering in the pullout simulation will result in a higher evaluation of the bond performance, maybe due to the soft outer layer of fiber which has larger deformation contributes to transferring stress from fiber to matrix and their interface. Due to the small embedded length (30 mm) of fiber yarn in this study, the slippage between the inner and outer layers was not considered. However, when the embedded length is large enough (50 mm), the inner and outer layers of fiber yarn will slip during the pullout process, stress can transmit more sufficiently during the slippage process, which will lead to a greater difference in the load-displacement curve compared with the model without considering fiber layering.

Figure 11.

The pullout load-displacement curves with or without considering fiber layering.

The stress distribution of the fiber cross section at the peak load of the model BS is presented in Figure 12. Since the length and width of the inner fiber part were set to be reduced in equal proportion relative to the overall fiber cross-section, the different levels of stress under pullout load due to the different Young’s modulus and densities of the inner and outer parts can be easily noticed. This characteristic of stress distribution of the fiber cannot be reflected if fiber layering was not considered in the numerical model. In this model, the matrix impregnation ratio was set to be 30%, which means the proportion of the outer layer is 30% of the entire fiber yarn cross-section. If the value of impregnation ratio increases, the mechanical property of the outer fiber due to matrix impregnation will have a greater impact on the stress distribution of the fiber cross-section. When the stress gap between the inner and outer layers of fiber yarn is large enough, slippage may occur between the two parts, just as reported of the specimen with fiber embedded length of 50 mm and 70 mm (Zhu et al., 2020).

Figure 12.

Stress distribution of the fiber cross section.

Result of model BM

Load-displacement curve

Figure 13 illustrates the pullout loads with different loading rates obtained from the simulation of the model BM and Liu et al.’s experiment (2020). From the figure, with the increasing of the loading rate, larger peak load and smaller displacement at the peak load could been seen, which is consistent with the rule found in the results of the model BS in Figure 11. Besides, the descending parts of the simulated and experimental curves also show considerable difference, as the curves of the model BS. The peak load of the fiber textile model was nearly twice of that of the fiber yarn model. Considering the fiber mesh contained two longitudinal fiber yarns, the transverse fiber yarns did not provide much resistance during the pullout process.

Figure 13.

Comparation of the load-displacement curves between simulation of the model BM and Liu’s experiment.

Stress distribution

Stress distribution of the fiber mesh in the pullout process with loading speed of 25 mm/min is indicated in Figure 14. From the figure, stress propagation starting from the loading end towards the embedded end could be observed clearly. Stress increased gradually in the load-displacement curve ascending part, and the maximum stress appeared when the peak load was reached. What’s more, in the whole pullout process, transverse fiber subjected to neglectable stress in the fiber mesh. That means they provided very little resistance in the pullout process, which was consistent with the conclusion drew from the load-displacement curve. Moreover, because of the stress sustained partially by the transverse fibers, the stress of the longitudinal fibers at the intersection was less than that in the other parts.

Figure 14.

Stress distribution of the fiber mesh of the model BM.

Figure 15 shows the stress development of the matrix in the X-Z profile. At first, stress arose in the upper part of the matrix, since only the upper part of the fiber mesh sustained stress, and further transmit to the surrounding matrix. As the pullout went on, stress gradually developed to the bottom of the matrix. The maximum stress area with red color only existed at the top part in the whole process, while it went through the z direction of the matrix in the fiber yarn model in Figure 7. This was because the mesh structure constituted by multiple fiber yarns could better distribute stress evenly throughout the matrix (Sun et al., 2023). In the descending part of the load-displacement curve, stress of the whole matrix reduced.

Figure 15.

Stress distribution of the matrix in the X-Z profile of the model BM.

Analysis of unsymmetric pullout model

Model description

Some yarns in the fiber mesh may break due to the uneven loading or improper operations during the construction process of the FRCM material. In that case, only part of the longitudinal fibers in the mesh are subjected directly to the tensile load. Although the model BM discussed before has already proved the applicability of fiber layering modeling method on “mesh” aspect, in order to investigate the mechanical behavior of FRCM under unsymmetric pullout load, a mode called PM was built, as shown in Figure 16.

Figure 16.

Size of the fiber mesh and the model PM.

The fiber mesh was made up by two transverse and two longitudinal fiber yarns as shown in Figure 16. Each fiber yarn was 30 mm × 5 mm × 0.25 mm, which consisted of the outer and inner parts. The four fiber yarns were merged as a whole part. Size of the matrix was 20 mm × 50 mm × 20 mm, and the fiber mesh was inserted into the matrix with 17.5 mm distance. Elastic and ideal elastoplastic model were used for the material property of the fiber and matrix respectively, and the relationship between strain rate and dynamic influence coefficient was input for the materials. The cohesive property was chosen to describe the fiber-matrix interface. The parameters values in the model were the same as those in Table. 1.

Mesh size was 0.02 mm for the fiber mesh and 0.8 mm for the matrix. It should be noted that the load was only applied on the left longitudinal fiber yarn, on which 10 mm prescribed displacement was adopted as the pullout force. And the loading rate of 25 mm/min was used. The degrees of freedom in X, Y, Z directions were fixed of the bottom surface of the matrix (X-Y plane, Z = 0). The model was also calculated by the dynamic solver in Abaqus finite element analysis software.

Load-displacement curve

The pullout load with different displacement of the load applied end is presented in Figure 17. According to the figure, the curve kept linear in the elastic phase of the pullout process until the load reached the peak of 2720 N, and followed by the curve falling. The curve showed a second peak load of 758 N at the displacement of 4.9 mm. This may because the unsymmetric pullout load applied to the fiber mesh caused bending deformation of the fiber mesh and matrix. At this time, some parts of the fiber mesh and matrix interlocked like sawtooth at the interface, pullout of the fiber mesh needed to overcome the tangential stress at the interface as well as the normal stress, so larger pullout load was required. The bending behavior can increase the effectiveness of fiber in resisting the failure of the fiber reinforced materials.

Figure 17.

The pullout load-displacement curve.

Stress of fiber mesh

Figure 18 shows the stress distribution of the fiber mesh. In the beginning of pullout, stress in the left longitudinal fiber yarn propagated from the upper to the lower part, while almost no stress existed in the right fiber yarn, since the pullout load was applied to the left one. Next, stress in the left longitudinal fiber yarn gradually rose with the trend of spreading stress to the right one through the transverse fiber between them. The maximum stress occurred at the intersection of the left longitudinal fiber yarn and the transverse one. What’s more, large stress also appeared in the middle part of the right longitudinal fiber yarn. With the continue increasing of the pullout displacement, stress in the whole fiber mesh began to reduce, because most part of the fiber mesh had already debonded. While stress at the intersection of the left longitudinal fiber yarn and the transverse one was still large. Thus, this intersection was the weak point of the fiber mesh, at which fracture of fiber may happened if the strength of material was not enough. At the last phase of pullout, stress in most parts of the fiber mesh was rather small, while bigger stress remained in the right edge of the left longitudinal fiber yarn. In addition, since the pullout load applied to the fiber mesh was unsymmetric, the mesh had large deformation in the out-of-plane during the pullout process.

Figure 18.

Stress distribution of the fiber mesh.

As the intersection of the left longitudinal fiber yarn and the transverse one was the weak point of the fiber mesh, stress of this intersection in the pullout process was picked up in Figure 19. It is clear that the curve was composed of four phases. At the first stage, stress increased linearly as the pullout load rose. Then, stress of the intersection reduced. Because the stress spread to the lower part of the fiber mesh, other parts of the transverse fiber shared the stress of the intersection. In the third phase, stress rose again although the whole pullout load had already started to decreased, this may because the deformation of the fiber mesh outside the X-Z plane caused the intersection to be subjected to extra stresses. The end of the third phase bore the peak stress, which was the most dangerous moment for the fiber mesh in the whole pullout process. At last, as more parts of the fiber mesh debonded, stress of the intersection declined gradually.

Figure 19.

Stress of the intersection point in the pullout process.

Stress of matrix

The stress distribution of the matrix in Figure 20 is discussed below. Firstly, the stress mainly distributed in the left-center part of the matrix, since the pullout load was applied to the left fiber yarn. Then, regions of large stress with yellow color gradually spread with the increasing of the pullout displacement. Next, large stress existed in most part of the matrix except for the left upper corner. And in the fifth phase, stress in the left and right sides of the matrix started to decrease, while there was still large stress in the middle part. This corresponded to the small stress in both sides of the fiber mesh in the fifth phase, as both sides of the fiber mesh had already debonded. Furthermore, stress in the left part of the matrix faded more quickly than the right part, because the left part of the fiber mesh had more debonding interfaces that could not sustain stress to the matrix nearby. Deformation of the matrix outside the X-Z plane could also be captured in the pullout process.

Figure 20.

Stress distribution of the matrix in the Y-Z profile.

Analysis of FRCM model

Model description

Due to the existence of fiber, tensile strength of the FRCM material has been greatly increased no matter under static or dynamic load. A dynamic tensile model of FRCM slab was established, which was named as TM. The fiber textile was made up by six transverse and four longitudinal fiber yarns as shown in Figure 21. Dimension of the fiber yarns was the same as the one of the benchmark models, which consisted of the outer and inner parts. Lengths of the transverse and longitudinal fiber yarns were 18 mm and 28 mm respectively. All the fiber yarns were merged as a whole part. Size of the matrix was 28 mm × 19 mm × 2 mm. The whole mesh was inserted into the matrix.

Figure 21.

Size of the fiber mesh and the model TM.

Elastic behavior was assumed for the material property of fiber, and concrete damage plastic model was chosen for the matrix. The dilation angle, the flow potential eccentricity, fb0/fc0, K and the viscosity parameter were 30, 0.1, 1.16, 0.667 and 0.001 respectively. Then the relationship between strain rate and dynamic influence coefficient was input for the materials. To describe the characteristic of the fiber-matrix interface, the cohesive property was chosen. Other parameters values in the model were the same as the ones in Table. 1. Mesh size was 0.04 mm for fiber and 0.4 mm for matrix. Displacement of 10 mm along z direction was adopted on the left surface (X-Y plane, Z = 28 mm) of the matrix to represent the tensile force, the loading rate was 25 mm/min. The degrees of freedom in X, Y, Z directions were fixed of the right surface (X-Y plane, Z = 0). The model was also calculated by the dynamic solver in Abaqus finite element analysis software.

Load-displacement curve

As shown in Figure 22, the tensile load-displacement curve was made up of three phases. In the beginning, the fiber mesh deformed synchronously with the matrix. As a result, the curve was linear. With the increasing of the tensile load, the mortar matrix reached its elastic limitation and tensile damage appeared. At that time, tensile stress was mainly undertaken by the fiber, and some parts of the fiber mesh began to slip. Since the stress transferring function by the fiber-matrix interface still worked, and the fiber mesh could bear larger stress, the tensile load still increased. At the last phase, the fiber-mesh interface and the matrix itself damaged severely, most parts of the fiber mesh slipped along the tensile direction. As a result, tensile stress could not transfer to the fiber mesh from the matrix. Hence, softening phase could been seen in the load-displacement curve.

Figure 22.

The tensile load-displacement curve.

Stress of fiber mesh

The stress distribution of the matrix of the model TM is presented in Figure 23. At the first stage, stress occurred in the left part of the longitudinal fibers. Then stress increased and spread to the right part. With the increasing of the tensile displacement, the largest stress in the tensile process concentrated in the middle of the longitudinal fibers. If the fiber is not strong enough to resist the stress, it will break at this phase. At last, stress in the fiber gradually decreased. Since the fiber-matrix interface damaged severely at that time, which no longer had the function of transferring stress to the fiber mesh from the matrix. Additionally, in the whole tensile process, stress of the transverse fibers was always small. It shows that the longitudinal fibers which was parallel to the tensile load played the main role of load-carrying. Moreover, due to the stress sharing of the transverse fibers, stress of the longitudinal fibers at the intersection was slightly less than that in the other part.

Figure 23.

Stress distribution of the fiber mesh.

Stress of matrix

Figure 24 shows the stress distribution of the matrix in the tensile process. Firstly, stress arose in the left part of the matrix, because the left surface of the specimen was subjected to the tensile load. Then, stress in the matrix increased and two localized areas with large stress existed in the left part. Tensile failure of the matrix also occurs at this stage. As the tensile process went on, stress of the matrix began to decrease, and became quite small when the displacement reached about 3.5 mm. At that moment, the matrix was almost completely damaged by the tensile load, and no longer able to bear any stress (Gong et al., 2023). In addition, stress in the middle part of the matrix faded more quickly than the other part. In the whole process, stress of the matrix was always very small at the right end, which was the fixed end of the tensile specimen.

Figure 24.

Stress distribution of the matrix.

Conclusion

This research put forward a fiber layering modeling method based on the different mechanical properties of the outer and inner parts of fiber yarn, then investigated the mechanical behaviors of fiber reinforced materials under dynamic load with different loading speeds. Two pullout models were established for verification, with consideration of the fiber layering model caused by the matrix penetration. The simulated results were compared with the existing experimental data to determine the reliable parameters values of the fiber-matrix interface. Based on the values, an unsymmetric pullout model and a uniaxial tension model of FRCM slab were investigated. Parameter analysis was carried out to study the effects of different factors in the constitutive relationship on the pullout results. Stress distribution and development in the loading process were discussed. Some conclusions can be drawn as follows:

1. Fiber modeling with inner and outer parts, as well as the cohesive property with reliable parameters, could effectively characterize the dynamic mechanical characteristics of the fiber-matrix interface. The models investigated in the study demonstrate the applicability of fiber layering method on “fiber-mesh-FRCM” aspects.

2. In the pullout process, for the single fiber yarn, tensile stress transmitted from the loading end of the fiber to the free end gradually. The maximum stress remained at the loading end. As for the fiber textile, tensile stress generated in the force applying fiber first, and then propagated through the transverse fiber to another longitudinal one. Intersection of the longitudinal and transverse fibers was subjected to large tensile stress.

3. The inner and outer parts of the fiber bore different stress under dynamic load, due to the different strengths and Young’s modulus. Besides, the load-displacement curves of the models with or without layering showed large difference at the peak load, which also reveals the features of considering fiber layering in the pullout simulation.

4. Parameters of the fiber-matrix interface have great influence on the simulated load-displacement curve. To be specific, the initial stiffness affects the slope of the curve ascending part and the displacement at the peak load. The maximum bond stress influences the displacement at the peak load and the peak load itself. And the failure displacement affects the slope of the curve descending part.

5. The transverse and longitudinal fiber yarns were simply merged as a whole part in the modeling process, due to the unclearness of the slippage performance of the intersection of transverse and longitudinal fiber yarns. This simplification has no influence on the mechanical performance of the model BS, BM and TM which were subjected to symmetric tensile load, while it overestimates the peak load and stress during pullout process of the model PM subjected to unsymmetric pullout load.

6. The trilinear slip relationship of fiber-matrix interface was selected in the study, which may not be very accurate to represent the property of real fiber-matrix interface. In the following research, the feasibility of combining complex interface relationships with fiber layering modeling will be investigated, and modeling methods with greater accuracy will be proposed.

7. The main purpose of this study is to explore the applicability and necessity of the fiber layering modeling in the simulation of FRCM. And by the comparation between the simulated results and the existing experimental data, the modeling method of the fiber-matrix interface was determined. It laid a foundation for the simulation of the dynamic tensile properties of the fiber reinforced materials.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Key-area Research and Development Program of Guangdong Province (2019B111107002) and National Natural Science Foundation of China (51820105012).

ORCID iDs

Fuyuan Gong

Zhao Wang

Jihua Zhu

Dawei Zhang

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