Keywords

1 Introduction

Composite material is a kind of multiphase material composed of two or more materials. Compared with raw material, composite material has the advantages of high specific stiffness, high specific strength and designability [1]. The fibers are mixed with the matrix to form fiber bundles, which is woven and mixed with the matrix to form a woven composite material, as shown in Fig. 1.

Fig. 1
A schematic representation of the fiber bundles. It is composed of fiber and matrix in a cylindrical manner. The specific portion of the fiber bundle with fiber and matrix is zoomed in.

Composition of fiber bundles

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In order to make better use of composite materials, it is necessary to understand the stiffness and strength properties of composite materials, and the relationship between these properties and the components of composite materials. In order to meet the requirements of engineering, the mechanical properties of fiber bundles can be predicted to predict the properties of composites, which can provide reference for the design and optimization of composites.

For predicting the mechanical properties of fiber bundles, there are mainly theoretical formulas and finite element methods. In the classical RVE, fiber distribution is considered to be uniform and uniform. Jha [2] established a cell model with uniform fiber distribution, studied the effects of fiber geometry and fiber volume fraction on the mechanical properties of composites. It was found that the fiber geometry had an effect on the transverse Young’s modulus, the influence on the axial Young’s modulus is negligible. Wang [3] used the finite element method to establish the unit cell model of micro-fiber bundle with uniform and random distribution of fibers in carbon fiber bundles, and carried on the analysis to it, the equivalent elastic modulus and strength of the fiber bundle are predicted, and the difference of the damage caused by the arrangement is compared. Vignoli [4] established the prediction model of tensile strength and shear strength of unidirectional fiber composites, and compared with other prediction models, it was found that the prediction accuracy of this method was higher. Han [5] established a microscopic RVE model of random distribution of fibers and studied the damage initiation and evolution of unidirectional fiber composites under various loads, the main failure modes of axial compression are fiber buckling and matrix collapse. Therefore, It is important to predict the modulus and strength of fiber bundles and analyze the damage evolution of fiber bundles by using the fiber random distribution cells.

In this paper, the mechanical properties of fiber bundles are obtained by establishing a representative volume element finite element model and applying periodic boundary conditions, compared with the classical Chamis model, the method is proved to be effective. The damage evolution of the micro-scale cell is analyzed, and the generation and development of the damage of the fiber bundle are explained.

2 Periodic Boundary Conditions

For the periodic arrangement of micro fiber bundle RVE models, two conditions should be met simultaneously at the cell boundary: (1) deformation coordination, and (2) stress continuity, as shown in Fig. 2.

Fig. 2
A schematic diagram exhibits the R V E boundary conditions. It is represented by the square box, which is divided into 4 quadrants, and the curved lines are drawn along the boundary of the 4 quadrants. Quadrant 3 is labeled u asterisk.

Periodic boundary conditions of RVE

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For a cell model with parallel and paired boundaries, Xia [6] proposed a periodic boundary conditions suitable for finite element analysis, which requires that the meshes on opposite surfaces are the same. A periodic displacement field is applied to the surface of the RVE representative volume element, which can obtain the relevant effective elastic coefficiency based on the microscopic mechanical response. The expression form of periodic displacement field is:

$$u_{i} = \varepsilon_{ik} x_{k} + u_{i}^{*}$$
(1)

where \(\varepsilon_{ik}\) is the average strain of the single cell structure, \(x_{k}\) is the position function at any position within the single cell, and \(u_{i}^{*}\) is the strain correction. The boundary conditions on one side of the rectangular cuboid cell structure can be expressed as:

$$u_{i}^{j + } = \varepsilon_{ik} x_{k}^{j + } + u_{i}^{*}$$
(2)

The boundary conditions of its opposite surface are:

$$u_{i}^{j - } = \varepsilon_{ik} x_{k}^{j - } + u_{i}^{*}$$
(3)

Subtracting the boundary conditions on one side from the boundary conditions on the other side yields:

$$u_{i}^{j + } - u_{i}^{j - } = \varepsilon_{ik} \left( {x_{k}^{j + } - x_{k}^{j - } } \right)$$
(4)

3 The Solution of Equivalent Elastic Constants of RVE Model

3.1 Establishment of RVE Model

The research object of this article is carbon fiber reinforced resin based fiber bundles, usually assuming that carbon fibers are unidirectionally distributed in the matrix and have a circular cross-section. Fiber bundles can be regarded as many single cell structures arranged periodically in three spatial directions. Therefore, a representative volume element (RVE) can be selected to establish a single cell model with random fiber distribution to calculate the performance of fiber bundles.

Using the Random Sequence Adsorption (RSA) method [7], a cubic RVE model was established for the random distribution of fibers in the matrix, as shown in Fig. 3, with a unit cell size of 35 μm × 35 μm × 35 μm. This article studies the case where the volume content of fibers in fiber bundles is 65.12%. It is specified that the fiber arrangement direction is in the x direction and the vertical fiber direction is in the y and z directions. The model is meshed by voxel grid, and the element type is C3D8R (eight node linear Hexahedron element, reduced integral, hourglass control), a total of 27,000 elements. The RVE model modeling parameters are shown in Table 1.

Fig. 3
A 3 D model exhibits the fabrication of microscale R V E. The fiber phase is incorporated into the matrix phase to form R V E model.

Micro-scale RVE

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Table 1 Modeling parameters of the RVE
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The RVE includes the matrix phase and the fiber phase, as shown in Fig. 3, the matrix is an isotropic material, and the carbon fiber is a transverse anisotropic material. All material parameters are taken from reference [8, 9], as shown in Tables 2 and 3.

Table 2 Parameters of epoxy matrix materials
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Table 3 Parameters of TC33 carbon fiber materials
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3.2 Solution of Equivalent Elastic Constant

Apply the periodic boundary conditions mentioned in Sect. 2 to the RVE model to obtain the stress nephogram in all directions as shown in Fig. 4.

Fig. 4
6 cloud diagrams of the R V E model at different stretching and shearing directions. The fiber phase has the highest value, and the matrix phase has the lowest value. a and b. The tensile modulus lies in the Y and Z directions. c to f. The shear modulus lies in the X Y, X Z, and Y Z directions.

RVE stress cloud diagram: a X-direction stretching, b Y-direction stretching, c Z-direction stretching, d XY direction shearing, e XZ direction shearing, f YZ direction shearing

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From Fig. 4a, it can be seen that when subjected to tension along the fiber direction, the RVE has the maximum modulus, and the fiber bears most of the load. Compared with the other stress nephogram in Fig. 4, the tensile modulus in Y and Z directions and the shear modulus in XY, XZ and YZ directions are relatively small, and the fiber is no longer subject to most forces. From the Stress–strain curve of the results, the prediction data of equivalent elastic constants of carbon fiber bundles can be obtained as shown in Table 4.

Table 4 Prediction of elastic constants of fiber bundles in all directions
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The Chamis theoretical model [10] is a classic model for the equivalent mechanical constants of composite materials, in which the prediction formula for the equivalent elastic constants of unidirectional composite materials is as follows:

$$\left\{ {\begin{array}{*{20}l} {E_{1} = V_{f} E_{f1} + \left( {1 - V_{f} } \right)E_{m} } \\ {E_{2} = E_{3} = \frac{{E_{m} }}{{1 - \sqrt {V_{f} } \left( {1 - {{E_{m} } \mathord{\left/ {\vphantom {{E_{m} } {E_{f2} }}} \right. \kern-0pt} {E_{f2} }}} \right)}}} \\ {G_{12} = G_{13} = \frac{{G_{m} }}{{1 - \sqrt {V_{f} } \left( {1 - {{G_{m} } \mathord{\left/ {\vphantom {{G_{m} } {G_{f12} }}} \right. \kern-0pt} {G_{f12} }}} \right)}}} \\ {G_{23} = \frac{{G_{m} }}{{1 - \sqrt {V_{f} } \left( {1 - {{G_{m} } \mathord{\left/ {\vphantom {{G_{m} } {G_{f23} }}} \right. \kern-0pt} {G_{f23} }}} \right)}}} \\ {\mu_{12} = \mu_{13} = V_{f} \mu_{f12} + \left( {1 - V_{f} } \right)\mu_{m} } \\ {\mu_{23} = \frac{{E_{2} }}{{2G_{23} }} - 1} \\ \end{array} } \right.$$
(5)

where \(E_{{{\text{f}}1}}\), \(E_{{{\text{f}}2}}\) are axial and transverse elastic modulus of fiber, respectively. \(G_{{{\text{f}}12}}\), \(G_{{{\text{f}}23}}\) are axial and transverse shear modulus of fiber, respectively. \(E_{{\text{m}}}\) is matrix elastic modulus. \(G_{{\text{m}}}\) is matrix shear modulus. \(\mu_{{\text{m}}}\) is matrix Poisson ratio. \(V_{{\text{f}}}\) is fiber volume fraction.

The predicted elastic constants in all directions of the fiber bundle obtained from the theoretical model are shown in Table 4. Because the established finite element model is a random distribution of fibers, the predicted fiber bundles are anisotropic materials. However, in Chamis theory, it is assumed that the fibers are uniformly arranged, and the predicted fiber bundles are transversely isotropic materials. So for finite element models, there are differences in modulus between the transverse and axial directions, while in theoretical predictions, the transverse modulus is equal to the axial modulus. Compare the average values of the axial and transverse modulus results of the finite element method with the Chamis theory results, and the numerical values and relative errors are shown in Table 4.

From the relative errors in Table 4, it can be seen that the results of finite element simulation have a certain degree of accuracy. The error along the fiber direction is the smallest, while the error in the vertical fiber direction is slightly larger. The reason is that in the unit cell model built in this article, the fibers are randomly distributed, while in Chamis theory, the fibers are assumed to be uniformly and neatly arranged, and the random arrangement of fibers is closer to the actual situation. Moreover, the Shear modulus does not consider the influence of the interface between the fiber and the matrix, which will lead to a certain gap in the results.

4 RVE Model Strength Analysis

4.1 Material Performance Degradation Plan and Analysis Process

Fiber bundles are composed of fibers and matrices, and VUMAT subroutines are written to first determine the type of unit and run different strength standards for different units. Among them, epoxy resin matrix is considered an isotropic linear elastic material, and various strength standards applicable to isotropic materials can determine matrix damage. If it is a matrix unit, the Von Mises standards [11] is used to determine the failure of the unit. The formula is:

$$\sqrt {\frac{1}{2}\left[ {\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{3} - \sigma_{1} } \right)^{2} } \right]} \le \left[ \sigma \right]$$
(6)

If it is judged as a fiber unit, use the maximum stress standards to determine the damage of the fiber. When any of the three directions in space reaches the strength value of the fiber in that direction, the fiber begins to fail. The specific formula is as follows:

$$\max \left\{ {\frac{{\sigma_{1} }}{X}} \right.,\frac{{\sigma_{2} }}{Y},\left. {\frac{{\sigma_{3} }}{Z}} \right\} = 1$$
(7)

where \(X_{ift}\), \(X_{ifc}\), \(S_{ijf}\) are the tensile strength, compressive strength and shear strength of the fiber in three directions.

Based on the above damage evolution theory, the fiber and matrix constitutive equation, failure criteria and stiffness reduction programs were written in Fortran language, and the damage evolution process of cells was simulated using the VUMAT subroutine in ABAQUS. The damage evolution of materials is a process of gradual accumulation. After loading the model, calculate the strain, stiffness matrix and stress, and judge whether the material fails according to the failure criteria. If there is no failure, update the stress and strain, and if there is failure, update the stiffness matrix.

4.2 Analysis of Numerical Simulation Results

Based on the above research, the microscopic unit cell model is established, periodic boundary conditions are applied, and the connection between the loading point (reference point RP) and the corresponding node is established, as shown in the figure. Apply X, Y, and Z directions to a single cell, with a displacement of 20% of the cell’s side length, which is 7 μm setting unit deletability and setting unit deletion variables, using explicit analysis, can effectively simulate the degradation and failure of such materials. The obtained stress–strain results are shown in the Figs. 5, 6, 7, and 8.

Fig. 5
A graph plots stress versus strain. The y axis ranges from 0 to 2500, and the x axis ranges from 0 to 2. A triangular wave is plotted through (0, 0), (1.5, 2250), (1.5, 0), and (1.62). Values are estimated.

Axial tensile stress–strain curve

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Fig. 6
A graph plots stress versus strain. The y axis ranges from 0 to 1200, and the x axis ranges from 0 to 2.5. A triangular wave is plotted through (0, 0), (1.65, 1250), (1.65, 0), and (1.82, 0). Values are estimated.

Axial compression stress–strain curve

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Fig. 7
A line graph plots stress versus strain. The y axis ranges from 0 to 40, and the x axis ranges from 0 to 3. A curve begins at (0, 0), increases gradually to the right, and ends at (2.6, 32). Values are estimated.

Transverse tensile stress–strain curve

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Fig. 8
A line graph plots stress versus strain. The y axis ranges from 0 to 160, and the x axis ranges from 0 to 7. A curve begins at (0, 0), increases gradually to the right, and ends at (6.1, 105). Values are estimated.

Transverse compressive stress–strain curve

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The stress–strain curve in the axial direction is almost linear, because the fibers are the main force under axial force, and the strength is also determined by the fibers. When subjected to lateral forces, the matrix is mainly subjected to force, so the stress–strain curve exhibits a non-linear characteristic. The highest point of the Stress–strain curve is taken as the strength value, and the predicted results are shown in Table 5.

Table 5 Fiber bundle strength in all directions
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From Fig. 9, it can be seen that due to the much greater tensile strength of TC33 carbon fiber in the axial direction than the resin matrix, during the progressive damage process, the resin matrix will first undergo damage, followed by subsequent fiber damage and failure. Strain in Fig. 9a ε = 0.54%, the fibers and matrix have not yet been damaged, and the fibers bear most of the tensile load. The performance of the matrix gradually deteriorates, and the elastic modulus slowly decreases. When strain ε = 1.31%, as shown in Fig. 9b, the matrix has been damaged, but the material still has load-bearing capacity as the fibers have not been damaged. When loaded into ε = 1.49%, the carbon fiber loses its load-bearing capacity, and the fiber bundle immediately fails after the fiber breaks, resulting in a decrease in stress to 0. As shown in Fig. 9c, when ε = 1.51%, the single cell model has completely failed, and both the fiber and matrix have lost their load-bearing capacity.

Fig. 9
Three 3 D models of the R V E at different strains epsilon = 0.54, 1.31, and 1.51%. a. The fiber phase has the highest value, and the matrix phase has the lowest value. The matrix phase is completely damaged in b, while the fiber phase is damaged in c.

Damage evolution of the RVE under axial tension: a ε = 0.54%, b ε = 1.31%, c ε = 1.51%

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The Chamis theoretical model also proposes a strength prediction formula for composite materials:

$$\left\{ {\begin{array}{*{20}l} {X_{{\text{T}}} = V_{{\text{f}}} X_{{{\text{fT}}}} } \\ {X_{{\text{C}}} = V_{{\text{f}}} X_{{{\text{fC}}}} } \\ {Y_{{\text{T}}} = S_{{{\text{mT}}}} \left( {1 - \left( {\sqrt {V_{{\text{f}}} } - V_{{\text{f}}} } \right)\left( {1 - E_{m} /E_{{{\text{f}}2}} } \right)} \right)} \\ {Y_{{\text{C}}} = S_{{{\text{mC}}}} \left( {1 - \left( {\sqrt {V_{{\text{f}}} } - V_{{\text{f}}} } \right)\left( {1 - E_{m} /E_{{{\text{f}}2}} } \right)} \right)} \\ {S_{12} = S_{13} = S_{{{\text{mS}}}} \left( {1 - \left( {\sqrt {V_{{\text{f}}} } - V_{{\text{f}}} } \right)\left( {1 - G_{m} /G_{{{\text{f}}12}} } \right)} \right)} \\ {S_{23} = S_{{{\text{mS}}}} \left( {1 - \sqrt {V_{{\text{f}}} } /\left( {1 - G_{m} /G_{{{\text{f}}23}} } \right)} \right)/\left( {1 - V_{{\text{f}}} /\left( {1 - G_{{\text{m}}} /G_{{{\text{f}}23}} } \right)} \right)} \\ \end{array} } \right.$$
(8)

where \(X_{{{\text{f}}T}}\), \(X_{{{\text{fC}}}}\) are fiber tensile strength and compressive strength, \(S_{{{\text{mT}}}}\), \(S_{{{\text{mC}}}}\), \(S_{{{\text{mS}}}}\) are tensile strength, compressive strength and Shear strength of matrix.

According to the Chamis strength prediction model, the strength of the fiber bundle in all directions was calculated, and the results are shown in Table 5. X, Y, Z are directions, subscript T is the tensile strength, C is compressive strength, S is the shear strength, 1, 2, 3 are shear in x, y, z directions.

For the prediction of axial strength, the finite element simulation results are in good agreement with the Chamis theory results, as the tensile and compressive loads along the fiber direction are mainly borne by the fibers and are not significantly related to the cross-sectional shape and distribution of the fibers. For the prediction of lateral strength, the Chamis model assumes that the fibers are uniformly and neatly arranged, while the fiber random arrangement model in this article is closer to the actual situation. However, the influence of the interface between fibers and matrix on strength was not considered in the model in this article, which will result in certain errors. Overall, the relative error between the strength prediction results in each direction and the theoretical model is within an acceptable range, verifying the feasibility of the prediction method in this paper.

And from Table 5, it can be seen that the axial modulus and axial tensile strength of fiber bundle materials are the highest, with 149.86 GPa and 2239.74 MPa, respectively, while the transverse modulus and tensile strength are only 9.97 GPa and 41.14 MPa. It indicates that fiber bundles have good mechanical properties in the fiber direction, while other directions have poor performance. In engineering applications, it is necessary to consider the differences in mechanical properties in different directions and make more effective use of materials.

5 Conclusions

This article uses the representative volume unit method to obtain the equivalent modulus of fiber bundle cells, and combines the constant degradation method to predict the progressive damage process. The damage evolution of the microscopic model and the equivalent strength of the fiber bundle are obtained, and the following conclusions are drawn:

  1. (1)

    The prediction results of the mechanical properties of micro fiber bundles using finite element simulation method are in good agreement with the classical Chamis theoretical model, verifying the effectiveness of the finite element method.

  2. (2)

    The axial modulus and strength of fiber bundle materials are much greater than the transverse modulus and strength. This indicates that fiber bundle materials have good mechanical properties in the fiber direction, while other directions have poor performance. In engineering applications, it is necessary to consider the differences in mechanical properties in different directions and make more effective use of materials.

  3. (3)

    By using a progressive damage model, the microscopic damage evolution process of fiber bundles is obtained, where fibers play the main load-bearing role, while the matrix plays the role of transmitting stress. The failure of fiber bundles is determined by the failure of fibers.