Keywords

1 Introduction

At present, the application of sandwich composite materials is becoming increasingly common in the field of ship building. Sandwich panel panels mainly use unidirectional fiber composite materials and woven composite materials. Compared with unidirectional fiber composite materials, the performance of woven composite materials is affected by factors such as fiber bundle bending and staggered weaving. Its structure is more complex than unidirectional fiber composite materials, and the mechanical properties such as stiffness and strength research methods are also different from traditional unidirectional fibers [1]. However, in the production and application of woven composite materials, it is necessary to conduct in-depth research on the stiffness and strength of woven composite materials starting from the material constitutive model.

Woven composite materials and unidirectional fiber reinforced composite materials have similarities in structure, with the main difference being that the fiber bundles inside the woven composite materials are crisscrossed and rub against each other [2]. This structure makes the mechanical properties of the woven composite materials more consistent in the layer plane. At present, researchers mainly obtain the stiffness and strength of woven composite materials through experimental methods and finite element methods. The experimental method measures their tensile and shear properties through tensile experiments, compression tests, in-plane shear tests, and out of plane shear tests, and calculates their engineering constants through experimental data. The multi-scale finite element simulation method, on the other hand, predicts the material parameters of unidirectional fiber reinforced composite materials by numerically simulating RVE. Both methods lack accurate understanding of the constitutive properties of woven composite materials [3, 4].

This study focuses on this current situation and adopts multi-scale finite element analysis method. Firstly, the accuracy of two theoretical calculation formulas for the engineering constants of unidirectional fiber composite materials was evaluated at the micro scale (referring to the scale that can reflect the structural characteristics of the fiber matrix, where the fiber diameter is 10–100 µm, and the fiber matrix RVE size is 50–200 µm). Subsequently, based on the engineering constant results of unidirectional fiber composite materials, plain, twill, and Three kinds of RVE (Representative Volume Element) models of 2D braided composites commonly used in satin Naval architecture are used to analyze woven composites from the mesoscopic scale (referring to the scale that can reflect the specific braided structure of braided composites, and the RVE model size of braided structures is 1–5 mm), and then obtain the stiffness and strength of woven composites [5, 6].

2 Method for Analyzing the Mechanical Properties of Composite Material Cells

2.1 Representative Volume Element and Homogenization Theory

The characterizing volume unit RVE is a typical unit selected from composite materials, which is a unit cell volume unit that can represent the microstructure characteristics of composite materials. Through the assumption of homogenization theory, its stress and strain can be considered uniform to reflect the average properties of the material. Therefore, studying the equivalent mechanical properties of macroscopic homogenization models for composite materials is equivalent to studying the mechanical properties of homogenized equivalent media corresponding to representative volume units in the material [7].

2.2 Periodic Boundary Conditions

When the RVE is loaded, the stress and strain on its symmetrical interface should be periodic and continuous rather than a simple free boundary [8]. This constraint condition can be achieved by applying Periodic boundary conditions. The Periodic boundary conditions requires that the symmetric nodes of the cell meet the following formula:

$$u_{i} - u_{i}^{\prime } = \overline{\varepsilon }_{ik} \cdot \Delta x_{k}$$
(1)

Among them, \(u_{i}\), \(u_{i}^{\prime }\) is the displacement of two symmetrical nodes, \(\overline{\varepsilon }_{ik}\) is the average strain component of a single cell, and \(\Delta x_{k}\) is a fixed value, which is equal to the relative position vector of the corresponding nodes of the single cell. This boundary condition not only satisfies the continuity of displacement on the boundary, but also ensures the continuity of stress at the boundary of the single cell [9, 10].

2.3 Finite Element Analysis Method for Woven Composite Materials

Woven fiber composite materials are composed of woven fiber bundles and matrices, and properties need to be defined separately during finite element analysis. The fiber bundles in the woven structure are infiltrated by fiber bundles and matrices, which can be regarded as transversely isotropic materials (\(E_{2} = E_{3} ,G_{12} = G_{13} ,\nu_{12} = \nu_{13}\)), the matrix is defined as an isotropic material. For woven composite materials, their woven structure makes the elastic modulus of the warp and weft lines in the fiber cloth yarn equal in both directions. In finite element calculations, defining their material properties often simplifies them as \(E_{1} = E_{2}\), \(G_{13} = G_{23}\), \(\nu_{13} = \nu_{23}\). First: Treat the fibers and matrix as isotropic materials and measure material parameters through experiments; Second: through the analysis of the micro RVE element (fiber bundle RVE), solve the Stiffness matrix and calculate the material properties; Third: solve the Stiffness matrix of the woven structure RVE in the second step, and calculate the material parameters. The fiber bundle material properties are calculated in the second step, and the matrix material properties remain unchanged [7].

2.4 Finite Element Analysis of Mesoscale Stiffness

According to the composite Strength of materials, the composite Stiffness matrix is solved, and the composite stress–strain relationship is as follows:

$$\sigma = C\varepsilon ,\varepsilon = S\sigma$$
(2)
$$\varepsilon = S\sigma \Rightarrow \left\{ {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\sigma_{z} } \\ {\tau_{xy} } \\ {\tau_{xz} } \\ {\tau_{yz} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\frac{1}{{E_{1} }}} & { - \frac{{\nu_{12} }}{{E_{2} }}} & { - \frac{{\nu_{13} }}{{E_{3} }}} & 0 & 0 & 0 \\ { - \frac{{\nu_{21} }}{{E_{1} }}} & {\frac{1}{{E_{2} }}} & { - \frac{{\nu_{23} }}{{E_{3} }}} & 0 & 0 & 0 \\ { - \frac{{\nu_{31} }}{{E_{1} }}} & { - \frac{{\nu_{32} }}{{E_{2} }}} & {\frac{1}{{E_{3} }}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{{G_{23} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{1}{{G_{31} }}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{1}{{G_{12} }}} \\ \end{array} } \right] \times \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\varepsilon_{z} } \\ {\gamma_{xy} } \\ {\gamma_{xz} } \\ {\gamma_{yz} } \\ \end{array} } \right\}$$
(3)

Abaqus is used to establish the RVE model of the meso fiber matrix, apply Periodic boundary conditions, and apply tensile loads in x, y, z directions and shear loads in xy, yz, xz directions to the cell through the displacement of the coupling point of the control surface. In the post-processing, the reaction force and displacement at the coupling point of the control surface are extracted, and the six groups of stress and strain states of the cell under six working conditions are calculated to solve the Stiffness matrix of the fiber matrix RVE. The material properties in other literature are selected for this study, and the specific material properties are shown in Table 1.

Table 1 Material parameters of fiber and matrix components [11]
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The generated unit cell is a cube with a length of 16.3 µm, a width of 6.145 µm, and a height of 3 µm, in which the fiber volume fraction is 70%. The Hexahedron unit is selected in Abaqus for grid division. No interface layer attribute is defined between the fiber and the node, and the fiber bundle and the matrix are connected in a common node manner (Fig. 1).

Fig. 1
A 3 D R V E model of fiber matrix at the mesoscale depicts the structure with fibers and matrix. It also marks the z, y, and x axes.

RVE model of fiber matrix at mesoscale

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Calculate the unit cell flexibility matrix when the fiber volume fraction is 70%. Calculate the strain in each direction of RVE based on the displacement of the coupling point and the unit cell size. Calculate the stress in the corresponding direction under each working condition based on the support reaction force and RVE size. Calculate the nine material parameters of RVE, and obtain the RVE material parameters with a fiber volume fraction of 70%, as shown in Table 2.

Table 2 Parameters of cellular materials with 70% fiber volume fraction calculated by finite element method
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3 Finite Element Analysis of Mesoscopic Scale Stiffness

3.1 Establishment and Calculation of RVE Finite Element Model for Woven Structures

Analyzing the RVE units of 2D woven composite materials at a mesoscale, three weaving methods are shown in Fig. 2, with the specified warp direction of the yarn being in the 1 direction, the weft direction being in the 2 direction, and the plane normal direction of the woven fiber cloth being in the 3 direction.

Fig. 2
Three 3 D models of woven structures R V E labeled plain weave, twill weave, and satin weave. Each model depicts different patterns and is marked with z, y, and x axes.

Schematic diagram of three 2D woven structures RVE

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In the finite element calculation, material properties are assigned to the fiber bundles and matrix of the RVE of the woven structure. According to the scanning results of the electron microscope, the internal yarns of the plain weave woven composite material are long ellipses, so an ellipse is selected as the cross-section of the finite element model yarn. The fiber bundles are transversely isotropic materials, and the matrix is isotropic materials [12]. The woven structure of the yarn causes the yarn itself to have a bending arc, Therefore, it is necessary to define the direction for yarns at different arc positions to maintain E1 always along the fiber axis in the yarn.

3.2 Calculation Results of RVE Finite Element Model for Woven Structures

Generate a unit cell model with a fiber volume fraction of 45% in Digimat software, solve the stiffness and flexibility matrix of the RVE, and calculate the nine engineering constants of the RVE. The calculation results of the woven RVE with a fiber volume fraction of 45% are as follows (Fig. 3).

Fig. 3
Three grouped column charts. Two charts plot Poisson’s ratio versus Poisson’s ratio direction, while the third chart plots Young’s module versus Young’s module direction. All charts contain columns with data for plain weave, twill weave, satin weave, and unidirectional fiber.

Finite element results of engineering constants for woven RVE

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4 Finite Element Analysis of Tensile Strength of 2D Woven Composite Materials

4.1 Establishment of Finite Element Model for 2D Woven Composite Materials

Due to the complex periodic boundary application of woven RVE, three types of voxel mesh models were established for woven RVE, with voxel mesh sizes of (0.129 mm × 0.0069 mm × 0.0069 mm), with a grid quantity of 14,400, 57,600 and 90,000, respectively. After the grid components are generated in Digit, they are imported into ABAQUS for post-processing. Since the catastrophic failure of woven RVE mainly occurs when the fiber bundle breaks, the damage of the matrix near the fiber bundle is not considered, and the Periodic boundary conditions conflicts with the ABAQUS display dynamics algorithm, Statics is comprehensively considered for calculation, and the UMAT subroutine based on 3D Hashin failure criterion is compiled for failure simulation, where the 3D Hashin failure criterion has four failure modes, They are: fiber axial tensile failure, fiber axial compression failure, fiber normal tensile failure, and fiber normal compression failure.

This study selected the material parameters and their sizes from the literature for the simulation of the woven RVE model. The specific dimensions and material parameters are shown in Tables 3 and 4 (with a fiber volume fraction of 0.722 in the fiber bundle and 0.453 in the woven composite material). Given the volume fraction and component material parameters, the material parameters of the fiber bundle can be obtained. The stiffness of the fiber bundle is directly calculated based on Chamis’s equations [13], and the strength is directly calculated based on the statistical strength theory of Zhou et al. [14]. The theory suggests that the strength of the fiber bundle is distributed according to the Weibull rule, taking into account existing formulas, and no further micro mechanical performance analysis of the fiber bundle will be conducted separately.

Table 3 Material properties of the constituents [15]
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Table 4 Strength parameters of the constituents [14]
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The load displacement curves of the finite element calculation results and experimental results are shown in Fig. 4, indicating a good agreement between the experimental curve and the simulation calculation curve.

Fig. 4
A line graph of stress in M P a versus strain. Values are estimated. Twill weave-experience (0, 0), (0.010, 380), (0.018, 670), and (0.019, 500). Twill weave F E M (0, 0), (0.010, 405), (0.0176, 700), and (0.0176, 435).

Comparison between predicted and tested stress–strain curves

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For different weaving methods, failure simulation calculations were conducted on three types of woven composite materials under the same volume fraction. The load displacement curves of plain weave, twill weave, and satin weave were obtained, as shown in Fig. 5. Catastrophic failures occurred on the fiber bundle in the tensile direction, and the main failure mode was tensile failure along the fiber axis direction. Among them, the stress distribution and damage situation of three types of woven fiber composite materials during failure are shown in Fig. 6 (with hidden resin matrix element). From the figure, it can be seen that tensile failure in the axial direction mainly occurs at the intersection of fiber bundles, where there is obvious stress concentration, and failure also occurs first in these areas.

Fig. 5
A line graph of stress in M P a versus strain. Values are estimated. Twill weave F E M (0, 0), (0.010, 400), (0.018, 690), and (0.018, 10). Satin weave F E M (0, 0), (0.010, 390), and (0.018, 650). Plain weave F E M (0, 0), (0.010, 405), (0.018, 710).

Prediction of load displacement curves under tensile action of three weaving methods

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Fig. 6
A table with diagrams compares the yarn stress cloud diagram, stress distribution of tension warp yarn, and location of fiber axial tensile damage in three different weaving methods including plain, twill, and satin.

Stress distribution and damage situation of three types of woven composite materials during failure

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5 Conclusions

This study simulates the stiffness and strength of three 2D woven composite materials by establishing a composite material RVE finite element model. The stiffness and strength of three types of woven composite materials were compared under the same volume fraction, and the following conclusions were drawn:

  • When the volume fraction of composite material fibers is the same, and the volume fraction of warp yarn and weft yarn is the same. The stiffness of 2D woven composite materials is similar.

  • When using voxel grids for finite element simulation of woven RVE, the speed is fast and the accuracy is good. The fiber volume fraction is the same. When conducting simulation analysis, as the catastrophic failure of 2D woven composite materials is mainly controlled by the tensile failure of the fiber bundle in the fiber axis direction, the damage of the matrix around the fiber bundle can be ignored to improve calculation efficiency.

  • At the same volume fraction, there are differences in the strength of the three 2D woven composite materials, with plain weave having the lowest strength and satin weave having the highest strength.