Micro-XCT-based finite element method for predicting the elastic modulus of needle carbon-fiber-reinforced ceramic matrix composites

2.1 Construction of the FEM model

The pixel size can reach as low as 1 μm in current XCT technology. This size is sufficiently small to provide detailed geometrical information of a CMC’s microstructure because the diameter of a carbon fiber is approximately 6 μm. However, the size of an X-ray sensor is finite. The sensor size of a normal XCT device is approximately 1024×1024 in pixels, which means the maximum size of a reconstructed virtual 3D volume is approximately 1×1×1 mm³ if the pixel size is 1 μm. In general, the needle CMCs contains about 36 needle fiber bundles in a 10 mm×10 mm region. That means the distance between centers of two fiber bundles is at least 1.67 mm. Thus, a 1×1×1 mm³ virtual 3D volume cannot include even one minimum “periodic” structure. In addition, the random distribution of voids may disturb the periodicity of microstructure. Thus, to enable a larger model that contains more structures, the pixel size must be increased. On the other hand, if a lower resolution is used, such as 10 μm or larger, we will obtain a larger virtual 3D volume; however, it will be difficult to detect defects smaller than 10 μm. Thus, we choose 5 μm as the optimum pixel size with which the virtual 3D volume will big enough to describe the average mechanical behavior of material and the resolution of images is as high as possible. The fiber, matrix, and voids can be distinguished clearly under this condition, and the model size is theoretically 5(x)×5(y)×3.5(z) mm³, which includes sufficient information to reflect the properties of CMCs. The “y” and “z” denote the fiber directions in 0° and 90° plies, respectively. The “x” denotes the direction of needle fiber bundle.

Figure 1 shows the 2D slice sequence for a needle C/SiC composite and the corresponding image series with color marks. The color marks denote the fiber direction of the regions. In this study, the fiber direction in regions marked by red denotes fibers along the x-direction, while the fiber directions in regions marked by green and blue identify fibers along the y- and z-directions, respectively. On the basis of these 2D slice sequences, we can construct two virtual 3D volumes, as shown in Figure 2. Next, we constructed a 3D cuboid mesh that contained l 3D eight-node elements in width, h elements in height, and w elements in depth. Each of the elements of the 3D cuboid mesh corresponded to the pixel of the virtual 3D volume created by XCT. The primary direction of the element is determined on the basis of the color of the corresponding pixel in the virtual 3D volume with color marks, while the elastic moduli of each element are determined by the gray level of the corresponding pixel in the original 3D volume.

Figure 1:

2D slice sequence for the needle C/SiC composite and the corresponding image series with color marks.

Figure 2:

Relationship between the FEM model and the virtual 3D volumes.

2.2 Material properties determined from the images

The material under study is assumed to consist of fiber bundles and the matrix. The composition of the matrix is not pure SiC; instead, it is composed of SiC and voids. The gray level of the elements within the matrix regions reflects the volume fraction of the voids. The composition of a fiber bundle is also not a pure carbon fiber bundle; instead, it is composed of carbon fiber, SiC matrix, and voids. The gray level of the elements in this region reflects the volume fraction of fiber, SiC, and voids. As a result, the mixture principle is used to determine the effective elastic moduli of the elements.

In the fiber bundle regions, if the gray level of a point approaches zero, it means this point is void; if the gray level approaches g_b, which is the average gray level of fiber bundle, it means this point is filled by fiber bundle; if the gray level approaches g_m, which is the average gray level of matrix, it means this region is filled by the SiC matrix. In addition, we assume that the relation between gray level and the elastic properties is linear. On the basis of this assumption, we use Eqs. (1) and (2) to calculate the elastic properties of an element. It was assumed that the pixel gray value of each element is g, while that of the matrix is g_m, that of the fiber bundle is g_b, and that of the void is g_v. If g≤g_b, then the effective elastic parameters of an element can be calculated by

where E denotes the elastic parameters of the element and E^b and E^v denote the elastic parameters of the fiber bundles containing the matrix and the voids, respectively.

If g>g_b, then the effective elastic parameters of the element can be calculated by

where E^m denotes the elastic parameters of SiC. It is worth noting that the gray level of the fiber bundle (g_b) is defined to be the average gray level of the fiber bundle. The gray level of the matrix (g_m) is the average gray level of the region in which the nanoindentation measurement is performed. The detail of nanoindentation measurement is given in Section 3.3.

In the regions of the matrix, the gray level of pure SiC matrix should be constant. However, the matrix usually contains voids (or defects) and other impurities that are induced by the chemical vapor infiltration (CVI) process. That is why the gray level of the matrix varies greatly. As the volume fraction of impurities is very low, so the gray level of the matrix reflects the volume fraction of voids. If the gray level of matrix approaches zero, it means this region is void. If the gray level of the matrix approaches g_m, it means this region is filled of the SiC matrix. On the basis of this assumption, the effective elastic parameters of the element can be calculated using Eq. (3).

In Eqs. (1) to (3), E can be replaced by E₁, E₂, E₃, μ₁₂, μ₂₃, μ₁₃, G₁₂, G₂₃, and G₁₃. Particularly, E1v=E2v=E3v=Ev, G12v=G23v=G13v=Gv, μ12v=μ23v=μ13v=μv, E1m=E2m  =  E3m  =  Em,    G12m  =  G23m  =  G13m  =  Gm,    μ12m  =   μ23m   =  μ13m  =  μm. The directions denoted by 1, 2, and 3 are equal to directions x, y and z, which are presented in Figure 1.

On the basis of Ref. [3], the longitudinal elastic moduli of the fiber bundle can be calculated by Eq. (4a), as

The transverse elastic modulus can be determined by Eq. (4b), as

The formulations of shear modulus and Poisson’s ratio are similar with Eq. (4a) as

In Eqs. (4a) to (4d), a1 denotes the longitudinal direction of fiber, while a2 and a3 denote the transverse direction of fiber. “pyc” stands for pyrolytic carbon interphase. G and μ denote the shear moduli and Poisson’s ratio, respectively. v_fb, v_mb, and v_pyc are the volume fractions of the carbon fiber, matrix, and pyrolytic carbon interphase within the fiber bundle, respectively.

2.3 Homogenization process

On the basis of the homogenization theory, the macro-stress (σ̅) and the macro-strain (ε̅) are defined as the average of the stress and the strain of the RVE as

where ε, σ, and V denote strain, stress, and volume, respectively.

The effective elastic constitutive equations of the composite are

where 1, 2, and 3 denote directions x, y, and z, respectively.

If the six groups of boundary conditions presented in Eq. (7) are applied to the FEM model of the RVE in sequence, by performing FEM analysis, we obtain six groups of strain and stress fields of the RVE:

Next, the six groups of effective strain and stress are calculated by using Eq. (5). Because the flexibility matrix will not change with different boundary conditions, the six groups of effective strains and stresses all satisfy Eq. (6) as

Thus, the effective flexibility matrix of the composite can be calculated by the following equation as

All of the engineering elastic parameters can be calculated by

3.1 Material system

The material is a CVI Cf/SiC composite, provided by CAS Institute of Metals. The carbon fiber used as the material to produce the preform is T700-6K manufactured by Toray Company, with an average diameter of 6 μm and a fiber volume fraction of 30%. Figure 3 shows the cross section of the material and the fiber bundle. We can see that many voids are distributed randomly between fiber bundles and that carbon fibers are surrounded by a pyrolytic carbon interphase. Some tiny pores also can be seen in the cross section of the fiber bundle.

Figure 3:

Micrographs of the Cf/SiC composite: (A) distribution of voids and fiber bundles and (B) distribution of fibers and the matrix, and the porosity within the fiber bundle.

3.2 Tensile and shear test

Dog-bone-shaped specimens were used in the tensile experiments, as shown in Figure 4B. The size and geometry of a specimen is shown in Figure 4A. The tensile test was performed by using a WDW-100 electromechanical testing system (HRJ LTD, Jinan, China) to obtain the stress-strain response at room temperature. The maximum load of the machine is 50 kN, and the accuracy of the force is ±0.5%.

Figure 4:

Specimens and system for the tensile test: (A) specimen, (B) specimen with strain extensometer, (C) WDW-100 electromechanical testing system, and (D) the size of specimen (unit: mm).

Shear tests were performed according to ASTM C1292-10 at room temperature. The shear modulus was determined by loading a test specimen in the form of a rectangular flap with symmetric centrally located V-notches using a mechanical testing machine and a four-point asymmetric fixture (shown in Figure 5). The shear strain was obtained through strain rosettes. The loading speed was 0.02 mm/min. The shear modulus was calculated according to Eq. (11):

Figure 5:

Fixture and a specimen for the shear test.

where G denotes the engineering shear modulus; F_s denotes the applied load; A denotes the area of the cross section of the double V-notches; γ denotes the engineering shear strain; and ε_0°, ε_90°, and ε_45° denote the strain in directions 0°, 90°, and 45°, respectively.

Tensile tests were conducted on three specimens, and the results are shown in Figure 6A. Within 0–500 μm, the strain-stress curves present a good linear form. According to the definition of Young’s modulus, the slope of the strain-stress curves is the Young’s modulus. Thus, the tensile modulus values are 73.98, 75.98, and 73.32 GPa for the three specimens, and the average of modulus is 74.43 GPa. Additionally, shear tests were conducted on the three specimens, with the results shown in Figure 6B; the shear modulus values for the three specimens are 30, 36, and 40 GPa, and the average of the results is 35.3 GPa.

Figure 6:

Experimental results of (A) the tensile test and (B) the shear test.

3.3 Nanoindentation test

On the basis of Eqs. (1) to (4), E_f, E_m, E_pyc, v_fb, v_mb, and v_pyc are required to determine the elastic moduli of the fiber bundle. The elastic behavior of carbon fiber is stable, so we use the value of E_f from Ref. [15]. Because E_pyc is much lower than E_f and E_m, the variance of E_pyc will not affect the elastic moduli of the fiber bundle significantly. Thus, we also use the value of E_pyc reported in Ref. [15]. E_m is an important part of the elastic moduli of the fiber bundle. However, the elastic moduli of CVI SiC are very different from that of pure SiC and are affected by the process conditions. The value of E_m from other researchers may be different from that of our material. As a result, in this study, we used nanoindentation to determine the elastic moduli of the CVI SiC matrix directly. The nanoindentation method has been used by Diss et al. [16]. The nanoindentation measurements were performed at Huazhong University of Science and Technology using a Hysitron nanoindentor. The results are listed in Table 1, where 1 denotes the longitudinal direction of the fiber, 2 denotes the transverse direction of the fiber, and 3 denotes the direction perpendicular to 1 and 2.

3.4 Volume fraction of the components

Even though the highest resolution of XCT is about 0.5 μm, this resolution is too low to obtain an image that gives details of the fiber bundle. Thus, we use a scanning electron microscopy graph, as shown in Figure 7, to perform volume fraction analysis of components. We assume that the region shown in Figure 7A is big enough to represent each section of the fiber bundle statistically. By performing the gray-scale analysis of Figure 7A, we know that the volume ratio of the fiber with pyc coating is 61.72%, the ratio of voids is 4.13%, and the ratio of the matrix is 34.15%. On the basis of Figure 7B, we know that the thickness of the pyc coating is 0.5 μm, and the diameter of the fiber with the coating is 7 μm. Thus, v_fb and v_pyc are equal to 45.4% and 16.36%, while v_mb is equal to 34.15%. The volume fractions measured here are close to the values reported in Ref. [3].

Figure 7:

Cross section of the fiber bundle: (A) phase diagram cross-section of the fiber bundle; (B) cross section of the fiber with coating.

3.5 XCT process

To rebuild the FEM model of a RVE, we first used a “YXLON-Y.Cheetah” X-ray microtomography system to obtain the slice images of the material. A sample measuring 3.5(x)×5(y)×3.5(z) mm³ in volume was scanned at different X-ray energy levels, and the best resolution was obtained at 90 keV. The scanning details are presented in Table 2.

Even though the sensor of the XCT device is 1024×1024 pixels, we can only obtain a model with 1004(x)×1004(y)×720(z) pixels, because the data near the boundary of sensor is invalid. Then, we cut a part of the original model with 642(x)×530(y)×322(z) pixels to perform FEM analysis. Figure 8 illustrates some of the slice images. The voids and fiber bundles are clearly distinguishable in the 2D images. The fiber bundles were observed to be nearly periodically distributed in the composite. Figure 9 shows the slices with color marks. Red denotes the needle fiber (x direction), green denotes the fiber bundle in the 0° layers (y direction), and blue denotes the fiber bundle in the 90° layers (z direction).

Figure 8:

Slices obtained from CT imaging of the needle carbon-fiber-reinforced CMC.

Figure 9:

Slices with color marks.

4.1 Elastic moduli of the needle Cf/SiC composite

On the basis of the elastic moduli listed in Table 1 and the volume fractions presented in Section 3.4, we can calculate the elastic parameters of a fiber bundles by using Eq. (4), which are listed in Table 3.

Model reconstruction and the homogenization process presented in Section 2 are executed by a C code program named “CMCsXCT”. An FEM model of the needle composite was constructed by CMCsXCT on the basis of the slice images and the corresponding images with color marks obtained by the XCT process presented in Section 3.5.

The original XCT model for FEM analysis is 642(x)× 530(y)×322(z) in pixels. As we translate each pixel to an element, the total number of elements in this model is about 10⁸. It is nearly impossible to perform FEM analysis of a model with this huge number of elements because of the computational cost. Thus, this method cannot be used to handle high-resolution XCT images directly. We need an approximation technique to reduce the scale of the model. In this study, to reduce the number of elements, we compress the XCT model to be 161(x)×133(y)×80(z) in pixel by the image compress technique of the famous XCT postprocess software “VG studio”. Thus, the total number of elements is 1,713,040. The FEM model is illustrated in Figure 10. The primary direction of the material of each element is determined on the basis of the images with the color marks. The elastic moduli of each element are determined by using Eqs. (1) to (4). The gray levels of the fiber bundle, matrix, and void (g_b, g_m, and g_v) are equal to 10, 18, and 0, respectively.

Figure 10:

FEM model of the RVE: (A) model without the matrix (only fiber bundles); (B) model with the matrix.

By performing the homogenization process presented in Section 2.3, we obtained the elastic moduli of the material, which are listed in Table 4. Note that the predicted results fit the experimental results well. In Table 4, subscript 1 denotes the direction of x, and subscripts 2 and 3 denote the directions of y and z, respectively.

4.2 Optimal RVE size

The size of the RVE is one of the most important parameters in the micromechanics method. The RVE should be large enough to contain sufficient information about the microstructure. However, a larger RVE corresponds to a longer computing time. As a result, selection of a proper RVE size is required. In this study, we used a moving window method to determine the proper size of the RVE. As shown in Figure 11, a window is moving within the slices randomly. The region within the window is selected to be one sample of the RVE. Twenty samples are selected from each window size. By performing the micro-XCT-based finite element method developed in this study, we calculate the elastic moduli of these 20 samples, and then, the standard deviation (STD) of the elastic parameters of these 20 samples is obtained. Table 5 lists the STD of E₂₂ and G₁₂ for six window sizes. The STDs decrease significantly with the increasing RVE size. A size of 111×83 is appropriate because the STD at this size is small enough; in addition, the RVE of this size includes sufficient information about the global elastic behavior.

Figure 11:

Window moving randomly in slices.

However, the moving window process consists of hundreds of FEM analyses to determine a proper window size, which takes too long to process on a PC. Because the volume fraction is the main factor that affects the elastic modulus, the volume average method can be used to determine a proper window size. We defined the average elastic parameter of an RVE as

where E̅ denotes the average elastic parameter, N is the total number of pixels within the RVE, and Eⁱ is the elastic parameter at the ith pixel. Eⁱ is determined by using Eqs. (1) to (4).

Figure 12 shows the trend of the STD for different RVE sizes. We can see that the results of the volume average method are consistent with the FEM results. The STD calculated by the volume average method and by the FEM both converge with increasing RVE size. The calculation time of the volume average method to produce Figure 12 is only a few seconds, while the calculation time of the FEM to produce a similar result is approximately 15 h.

Figure 12:

STD for different RVE sizes, obtained by the volume average method and the finite element method.

On the basis of Figure 12, we chose 111×83 as the optimal RVE size, which is the knee point of the curve of the STD versus RVE size. The results of the micro-XCT-based finite element method of this RVE size are listed in Table 6. We can see that the results of 111×83 agree well with the results of 161×133.