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Effect of interphase thickness on the composite elastic property with different interphase types: (a) elastic modulus E 1, (b) elastic modulus E 2, (c) Poisson’s ratio ν 12, (d) Poisson’s ratio ν 23 and (e) shear modulus G 12.

Based on the theoretical model in Section 2, four types of interphase models were established, including inhomogeneous transverse isotropy (ITI), inhomogeneous isotropy (II), homogeneous transverse isotropy (HTI), and homogeneous isotropy (HI). The composite elastic properties calculation methods with different interphase types were used to predict the elastic properties of T300/BSL914C, and the effect of interphase thickness on the prediction results was analyzed. The elastic properties and their determination methods of interphase are shown in Table 6.

The element convergence verification in accordance with the elastic modulus E 2. The blue bar shows the results for different numbers of elements, and the red line shows the trend of results.

In this model, the inhomogeneous interphase is divided into n layers of uniform interphase firstly, and the thickness of every layer is t/n. Then, a two-phase RVE model could be composed of the fiber and the first sub-interphase layer, and its elastic modulus could be obtained by the bridging model. And then, as an equivalent fiber, it forms a new two-phase RVE model with the next sub-interphase layer. Finally, the elastic modulus of composites with inhomogeneous interphase could be obtained by cycling calculation, as shown in Figure 4. The theoretical calculation framework is shown in Figure 5. And the equivalent flexibility tensor S k of the material in cycle k could be expressed as equation (16). Substituting equation (7) into equations (10)–(14), the elastic modulus of composites in the kth cycle can be obtained.

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Carbon fiber modulus of elasticityformula

However, scholars found that there is an interphase region with a thickness of tens to hundreds of nanometers between fiber and matrix through experiments [12,13,14], and its mechanical properties are different from those of the fiber and matrix and change in a gradient from matrix to the fiber [15,16]. To analyze the effect of interphase characteristics on the properties of composites, three-phase micromechanical models containing interphase have been established [17,18,19,20,21,22,23]. Wang et al. [2] established a three-phase representative volume element (RVE) model with interphase by pre-inserted cohesive layer between fiber and matrix and analyzed the effect of interphase modulus, thickness, and Poisson’s ratio on the mechanical properties of CFRP. Bohayra Mortazavi [24] assumed that the interphase thickness is related to the fillers radius, and there are perfect bonding contacts between fiber, matrix, and interphase. Kari et al. [25] used square and hexagon three-phase RVE models, respectively, to evaluate the effect of interphase stiffness and volume fraction on the overall properties of unidirectional CFRP and spherical inclusion composites with random distribution.

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The calculation results of theoretical models with four different interphase types show that the interphase type has an important influence on the elastic property of CFRP composites and the prediction error of elastic property caused by interphase type increases rapidly with the increase in interphase thickness. When the ratio of interphase thickness to the fiber radius is less than 0.04, the influence of interphase type on elastic property of CFRP composites can be ignored. When the ratio rise to 0.09, which the prediction error is more than 10%, it is necessary to accurately consider the properties of composite interphase in order to improve the accuracy of prediction results.

Carbon fiberyield strength

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Based on the Mori-Tanaka model, considering the consistency of the bridging matrix, Huang [36] modified the calculation method of the bridging matrix and proposed the bridging model. By adding bridging parameters α and β , the explicit expression of the bridging matrix is given. The modified bridging model greatly simplifies the expression and improves the calculation accuracy. The bridging matrix a of the bridging model could be expressed as

The characteristic of interphase has a significant influence on the macroscopic performance of carbon fiber-reinforced plastics (CFRP). To investigate the effect of interphase on composite elastic modulus, a representative volume element (RVE) of unidirectional CFRP with inhomogeneous interphase was established. Based on the bridging model, a theoretical calculation method of composite elastic modulus was given. The elastic modulus of T300/BSL914C composites was obtained by the theoretical method. Results are in good agreement with the finite element method and experimental data. Four types of interphase models were given including inhomogeneous transversely isotropic, inhomogeneous isotropic, homogeneous transversely isotropic, and homogeneous isotropic. The results demonstrate that interphase type has an influence on the prediction of CFRP composites’ elastic modulus. With the increase of thickness, the prediction error of elastic modulus caused by interphase type increases rapidly. Furthermore, the relationship between composite elastic modulus and interphase thickness and stiffness is analyzed. With the increase in thickness, the changes in shear modulus G 12 and Poisson’s ratio ν23 are more evident than in other elastic properties, and with the enhancement of interphase stiffness, the increase of G 12 is the most significant.

Carbon fiberdensity

where l is the length of RVE, R f is the fiber radius, t is the thickness of interphase, t = R i − R f , and V f is the volume fraction of fiber.

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The unidirectional CFRP composite is formed by the periodic arrangement of RVE. The stress and strain fields meet the continuity and periodicity [40,41]. The PBC were applied to the RVE model to simulate the composite macroscopic deformations of axial tensile, transverse tensile, and XZ in-plane shear [13,15,22]. The boundary conditions of tensile and shear deformation were set as equations (20)–(22), where Z (x, y, l z ) is the displacement of points on the plane (x, y, l z ) in the Z direction, and ε z is the strain of the model in the Z direction.

Through FE simulation, the mechanical properties of composites can be obtained [38,39]. In this section, the finite element method (FEM) and theoretical calculation method were used to simulate the elastic properties of unidirectional T300 carbon fiber-reinforced BSL914C polymer (T300/BSL914C) composites, and the comparative analyses of results against experimental data were presented. The elastic properties of T300 fiber and BSL914C epoxy resin are listed in Table 3.

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The results of four interphase models show that the elastic properties of CFRP composites are linearly related to the interphase thickness. With the thickness increasing, the equivalent moduli E 1, E 2, and G 12 of the composites increase. Poison’s ratios ν 12 and ν 23 decrease. And the variation range of G 12 and ν 23 are significantly higher than other elastic property parameters.

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It is assumed that there are perfect bonding conditions between fiber, interphase, and matrix. The elastic properties change continuously at the inner ( r = R f ) and outer ( r = R i ) boundaries of the interphase [1]. The elastic properties of the interphase at the boundaries are listed in Table 2.

Effect of interphase thickness on the composite elastic property parameter calculation errors: (a) the effect of interphase type on composite elastic properties with the interphase thickness of 0.1 μm, and (b) the effect of interphase type on composite elastic properties with the interphase thickness of 0.5 μm.

The relationship between three-dimensional stress tensor σ and strain tensor ε and flexibility tensor S could be expressed as

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The FE model of RVE with inhomogeneous interphase was established, as shown in Figure 6. The geometric dimensions of the model are l x × l y × l z = 8 × 8 × 10 μm. The diameter of the T300 fiber is 7 μm. The fiber volume fraction V f is 0.6. The interphase thickness is 0.1 μm, which is divided into five layers on average. The interphase elastic properties of sub-layers are listed in Table 4. Where Z of the coordinate axis represents the axial direction of the fiber, X and Y represent the transverse direction.

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Von Mises stress contour plot of RVE modal: (a) tensile deformation in the Z direction, (b) tensile deformation in the X direction, and (c) the shear deformation of the plane ZX.

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Most of the studies, which deal with the three-phase RVE model of composites, are restricted to the isotropic interphase model, and the assumption of isotropic interphase is different from the actual mechanical properties of interphase to a certain extent, which affects the prediction accuracy of macro mechanical properties of composites and limits the application scope of this kind of model. To accurately describe the interphase characteristics and investigate the influence of interphase on the elastic modulus of CFRP, an RVE model with inhomogeneous interphase was established. Using the discretization method, the interphase was divided into several transverse isotropic layers along the radial direction to consider the non-uniform interphase mechanical properties. Based on the bridging model and the concept of equivalent fiber, a calculation method of composite equivalent mechanical properties was proposed. The effects of the interphase characteristics (material type, thickness, stiffness, etc.) on the composite elastic modulus were discussed. This study provides a theoretical calculation method for the macro elastic modulus of CFRP, which considers the influence of interphase mechanical properties in more detail. Compared with the FE method, this method could rapidly predict the elastic properties of CFRP composites with different kinds of interphase. And it could be useful to guide the design of the RVE model with interphase and provide guidance for the optimization design of CFRP composites

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The interphase thickness and stiffness have an important effect on the composite elastic properties G 12 and ν 23. The results show that with the increase of interphase thickness, the variation range of G 12 and ν 23 is significantly higher than that of other properties. The elastic modulus of the composites is enhanced in varying degrees with the interphase properties increased, and the increase of G 12 is the most significant compared with other modulus.

The results show that with the increase of interphase thickness, the elastic properties of CFRP with different interphase properties gradients have different change rates (Figure 15). With the increase in interphase thickness, the modulus E 1, E 2, and G 12 of the linear function increase faster than that of the cubic function. Poisson’s ratios ν 12 and ν 23 of the cubic function decrease faster than those of the linear function.

Based on the traditional two-phase composite theory, the stress and strain tensors of homogenized composites meet equations (3) and (4) in the initial stage of loading [35,36].

The variation of composite elastic property with different interphase types when the interphase thickness increases from 0.1 to 0.5 μm.

Considering the inhomogeneity of interphase mechanical properties, the interphase was discretized into a multi-layer concentric cylinder structure along the radial direction, and every sub interphase layer is assumed to be a uniform transversely isotropic material [8], as shown in Figure 3. The mechanical properties of sub-interphase layer are obtained based on the equation (15). Where the subscript j represents the sub-interphase layer j.

The effect of interphase thickness on composite elastic properties with different interphase types was analyzed, as shown in Figures 11 and 12. When the thickness increases from 0.1 to 0.5 μm, the calculation results of different interphase types are significantly different. The increase of E 1 of isotropic interphase is less than 1% and that of transverse isotropic interphase is more than 5%. The elastic modulus E 2 has the same increase for different interphase types. Poisson’s ratio ν 12 of transversely isotropic interphase decreased by 4.3% and that of isotropic interphase decreased by 8.1 and 9.9%, respectively. Poisson’s ratio ν 23 of the interphase type HTI decreases by 26%, which is higher than that of the interphase type II by 19%. The ν 23 of interphase types ITI and HI have the same decrease, which is 24%. The increase in shear modulus G 12 of the interphase type HTI by 32% is significantly higher than that of the II by 16%. The increases of G 12 of ITI and HI interphase types are the same, which are 21%.

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In this section, the effect of interphase stiffness on the CFRP composite elastic properties was analyzed by changing the gradient of mechanical properties in the interphase region. It is assumed that the interphase is an inhomogeneous transversely isotropic material, and its properties change in cubic function, quadratic function, and linear function, respectively, as shown in Figure 14.

The PBC was imposed on the corresponding surface in the form of node coupling pairs and constraint equations. Under the action of the PBC, the parallel planes of the RVE model have the same stress and deformation fields, and the stress contour plots of the RVE are shown in Figure 7.

The calculation results of mechanical property of four interphase types RVE models show that when the interphase thickness is 0.1 μm, the calculation error caused by the interphase type is less than 3%, and it increases rapidly with the increase of interphase thickness, and the maximum error increases from 2.7 to 16.2% as shown in Figure 13. Based on the method of Linear Interpolation, the conclusions can be obtained, when the ratio of interphase thickness to the fiber radius t R f is less than 0.04, which error is less than 5%, the influence of interphase type on the calculation results of elastic property of CFRP composites can be ignored, and when the ratio is more than 0.09, which error is more than 10%, it is necessary to accurately consider the properties of composite interphase in order to improve the accuracy of prediction results.

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Substituting the bridging matrix a into equation (7), the equivalent elastic modulus of composites is derived based on equation (8) [36]. The explicit expression of composite elastic modulus could be expressed as

Comparison of errors in the prediction results of composite elastic properties by different calculation methods compared with the experiment data.

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Compared with the experimental data, the prediction errors of elastic property of RVE models with different interphase types are shown in Figure 10. For axial elastic modulus E 1 and shear modulus G 12, the model of interphase type II has the highest prediction accuracy, which error is less than 1%, and the error of interphase type HTI is the largest, which is 4%. For the transverse elastic modulus E 2, the errors of interphase ITI and HI are 10.6 and 10.7% respectively, which are less than those of interphase types II and HTI (11.3%). For Poisson’s ratio ν 12, the prediction accuracy of transversely isotropic interphase type is higher than that of isotropic. For the Poisson’s ratio ν 23, the errors of the four interphase types are about 26%.

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Carbon fiber modulus of elasticitygraph

Interphase controls the transfer of stress and strain between fiber and matrix and affects the macro mechanical properties of composites [25,43,44]. In this section, the effects of interphase type and stiffness on the calculation results of elastic modulus of T300/BSL914C composites and the trend of composite equivalent elastic properties with interphase thickness were analyzed.

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Effect of interphase thickness on composite properties with different interphase mechanical property gradients: (a) elastic modulus E 1, (b) elastic modulus E 2, (c) Poisson’s ratio ν 12, (d) Poisson’s ratio ν 23, and (e) shear modulus G 12.

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Interphase plays an important role in the stress–strain transfer between fiber and matrix. It is of great significance for predicting the properties of composites to the accurate description of interphase mechanical properties [26,27,28]. However, the interphase is a non-uniform region with a complex structure and thin thickness [29], and the mechanical properties are variable from the fiber surface to the matrix surface [30,31]. It is difficult to obtain the complete mechanical properties of interphase, including thickness, stiffness, and strength. To study the effect of interphase on composite properties, the mechanical properties of interphase need to be assumed and simplified to a certain extent [2,21]. Sun et al. [1] simplified the interphase as an isotropic material, adopted the exponential gradient model to describe the variation of interphase properties and took the average value of the model as the mechanical properties of interphase, and established a three-phase RVE model. Compared with the traditional two-phase model, the modified RVE model adding the interphase region has a higher accuracy of simulation results. Lim et al. [15] assumed that the interphase properties were constant and gradient functions, respectively, and analyzed the effect of mass fractions of fiber on the equivalent elastic modulus E c of composites by Mori Tanaka and the finite element (FE) model. The results showed that interphase parameters must be considered carefully when predicting E c using the FE model. Wang et al. [32] regarded the interphase as a non-uniform isotropic region and assigned its properties as exponential variation and established an RVE model with interphase. The interphase thickness was divided into 10 layers of pixels along the radius directions to consider the non-uniform mechanical properties of the interphase. And Wang studied the macroscopic mechanical behavior of unidirectional CFRP by using a fast Fourier transform simulation. Compared to the RVE model with a zero-thickness interface, the stress–strain results of the three-phase RVE model are in good agreement with the experiment data.

where F represents the composite elastic property; a and b are constants; r is the distance from the position to the fiber axis.

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The transverse strain ε y and shear strain γ xz were loaded on the RVE, respectively. The equivalent elastic modulus E y , Poisson’s ratio ν xy and shear modulus G xz could be obtained from equations (26)–(28).

In the three-phase RVE model, the fiber and matrix are assumed to be transverse isotropic and isotropic, respectively. It has been demonstrated that the mechanical properties of the interphase region decrease as it transitioned from the fiber to the matrix [15,16]. In this article, it is assumed that interphase region properties exhibit a gradient as shown in Figure 2, and its variation obeys the power function [22,31], as shown in equation (2). The elastic properties of component constituents are listed in Table 1, where E is the elastic modulus, G is the shear modulus, ν is Poisson’s ratio, the superscripts f, m, and i represent the fiber, matrix, and interphase, respectively. The subscript 1 represents the fiber axis direction; 2 and 3 donate the transverse direction.

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Effect of the increase of interphase properties on composite mechanical properties when the interphase thickness is 0.5 μm.

Loaded the axial strain ε z on the RVE model, the sum of reaction forces F z (x, y, 0) in the Z direction of the plane (x, y, 0) and the average displacements Y (x, 0, z) and Y (x, l y , z) were obtained by FE simulation. The equivalent elastic modulus E z and Poisson’s ratio ν zy of the RVE model could be calculated based on the equations (24) and (25).

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Figure 8 shows the result of element convergence verification and the elastic modulus E 2 becomes stable when the number of elements reached 0.74 million. The FEM and theoretical calculation results are compared with the results given in reference [22] and experimental data of T300/BSL914C composites [42]. The results are shown in Table 5. The calculation results of the bridging model for E z , E y , G xz , ν zy , and ν xy are close to the experimental data; the errors are 2.4, 11.3, 1.8, 8.2, and 26.3% respectively, as shown in Figure 9. In general, the predicted results of the elastic properties calculation method of CFRP composites with inhomogeneous interphase based on the bridging model are in good agreement with the experimental data.

Considering the internal stress tensors at the interface between fiber and matrix are equal, there is a bridging matrix a making the stress tensors of the fiber and matrix satisfy equation (5). And the constitutive equation of fiber, matrix, and composites could be expressed as the equation (6).

The distribution of fibers in the matrix is approximately periodic, and the relationship between the micro characteristics and the macro mechanical properties of composites can be obtained by selecting an appropriate RVE model [33,34]. Because there is a transition region between fiber and matrix, a non-uniform interphase region was added between fiber and matrix. Assuming that fibers are regularly arranged in CFRP composites [21], a square RVE model was established (Figure 1). The model consists of three phases, namely fiber, interphase, and matrix, respectively.

In this article, the RVE model of CFRP with inhomogeneous interphase was constructed, and the theoretical calculation method of composite elastic modulus considering the inhomogeneity of interphase was established. The calculation results of this method are agreed with the FEM results and experimental data, indicating that the method is effective for predicting the elastic modulus of CFRP.

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Carbon fiber-reinforced plastics (CFRP) are widely used in aerospace, weapons, and other fields because of their excellent mechanical property and design ability [1,2,3]. However, it is challenging to predict the mechanical properties due to the complicated micro-structures and insufficient knowledge of the interface between fiber and matrix. Several micromechanical theoretical models have been developed to predict the elastic property of composites, including the Mori-Tanaka model [4,5,6], self-consistent model [7], and bridging model [8,9,10]. Based on Eshelby’s equivalent inclusion theory and ideal interface assumption, several homogenization methods are presented to evaluate the mechanical properties of the fiber-reinforced composite [11]. The relationship between microstructure characteristics and macro properties is established by the two-phase models, which brings great convenience to the design of composites.

where σ and ε are the average stress and strain tensors of composites respectively, and V is the volume fraction. The superscript, f and m denote fiber and matrix, respectively.

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Data availability statement: The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.

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The mechanical properties of composites with different property gradient interphases are compared when the thickness of the interphase is 0.5 μm, as shown in Figure 15. It can be included that when the interphase properties increase, the macroscopic mechanical properties of the composites are enhanced in varying degrees. E 1 and E 2 increased by 6.9 and 6.4% respectively, and the increase in Poison’s ratio is less than 5%. The increase of G 12 is significantly higher than that of other elastic properties, which is 13.4% (Figure 16).

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