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Representative volume element for non-uniform micro-structure
Computational Materials Science 24 (2002) 361–379 www.elsevier.com/locate/commatsci
Representative volume element for non-uniform micro-structure Zhaohui Shan, Arun M. Gokhale
*
School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, USA Received 9 July 2001; received in revised form 28 September 2001; accepted 28 September 2001
Abstract A representative volume element (RVE) of micro-structure is an important input for computational-mechanicsbased simulations of micro-mechanical response of heterogeneous materials such as composites. In this contribution, a methodology has been developed to arrive at a sufficiently small micro-structural window that can be regarded as a RVE of a non-uniform micro-structure of a ceramic matrix composite (CMC) containing a range of fiber sizes, and fiber-rich and -poor regions at the length scale of about 100 lm. The RVE contains about 250 fibers of 14 lm diameter average size. The absolute size of the RVE is 0.1 mm2 . The methodology involves a unique combination of quantitative characterization of geometry and spatial arrangement of micro-structural features using stereological and image analysis techniques, development of computer simulated micro-structure model that is statistically similar to the real micro-structure, finite element (FE)-based simulations of micro-mechanical response on computer-simulated microstructural widows of different sizes containing 60–2000 fibers, and FE-based simulations on large-area high-resolution digital image of the composite micro-structure containing about 2000 fibers. The RVE has the micro-structure that is statistically similar to that of the CMC having fiber-rich and -poor regions, its Young’s modulus is very close to that of the composite, and has local stress distribution that is comparable to that in the real composite under similar loading conditions. Therefore, such an RVE is useful for realistic simulations of damage initiation. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Composite; Micro-structure; FEM; RVE; Stereology; Simulations
1. Introduction The concept of representative volume element (RVE) is important in the study of mechanical
* Corresponding author. Tel.: +1-404-894-2887; fax: +1-404894-9140. E-mail address: [email protected] (A.M. Gokhale).
response of heterogeneous materials. Mathematically, RVE is the infinite length scale limit relative to the micro-scale (or the length scale of a single heterogeneity), where the material appears uniform, and therefore, the continuum concepts are applicable. RVE is useful for modeling the effect of nano- and meso-scale heterogeneities on the overall mechanical response of the macro-size components/specimens. Different research workers have defined RVE in different ways. The first formal
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 2 5 7 - 9
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definition of RVE was given by Hill [1], according to which, RVE (i) must be structurally entirely typical of whole micro-structure on average, and (ii) must contain a sufficiently large number of micro-structural heterogeneities for apparent overall moduli to be effectively independent of the surface values of traction and displacement, as long as these values are ‘macroscopically uniform’. Drugan and Willis [2] have defined RVE as the smallest volume element of a material for which the usual spatially constant ‘overall modulus’, i.e., macroscopic constitutive representation, is sufficiently accurate to represent overall constitutive response. According to Drugan and Willis [2], a micro-structural volume element that yields modulus value within 5% of the global bulk modulus can be regarded as an RVE for practical applications. Therefore, such a volume element is useful to simulate global mechanical response of material using finite element (FE)-based simulations. However, majority of FE-based simulations are performed on a micro-structural window of arbitrary small size, with an assumption that such micro-structural window is an RVE, which may be reasonable only if the micro-structure consists of periodic arrays of micro-structural features of mono-size. Unfortunately, micro-structures of many materials (including composites) do not satisfy this requirement. For example, Fig. 1 shows micro-structure of a ceramic matrix composite (CMC) that has non-uniform spatial distribution of fibers (observe the fiber-rich and -poor regions). Further, a distribution of fiber sizes exists in this micro-structure. In the earlier studies, on the basis
Fig. 1. Low magnification micrograph of the glass CMC shows the alternating fiber-rich and -poor regions in the composite micro-structure.
of computer-simulated micro-structures, it has been shown that the spatial arrangement of fibers or particles has a strong effect on the mechanical responses of the composites [3,4]. These simulations have been performed on small micro-structural windows that may not be descriptive of the corresponding RVEs. Consequently, the simulated mechanical response obtained from such small windows of micro-structures may not be representative of the overall mechanical response of the material. The problem persists even in composites having uniformly distributed mono-size fibers as well, because the simulated properties of small micro-structural windows are of stochastic nature, and depend on the window size (scale effects), especially at low length scales [5–8]. On the other hand, from computational point of view, it is not practical to perform FE-based simulation on very large micro-structural windows containing very large number of heterogeneities (for example, say millions of fibers). Therefore, it is of interest to determine a sufficiently small micro-structural window that can serve as an RVE for simulation of micro-mechanical response of a heterogeneous micro-structure that may or may not be spatially homogenous or uniform random. In this context, it is essential to point out that most of the previous studies on RVE concern the simulation of global bulk properties such as modulus; very little attention has been paid to simulation of micro-stress (and strain) distributions that are statistically representative of the corresponding distributions in macro-size specimens. For simulation of damage initiation and evolution in heterogeneous materials such as composites it is of interest to know the micro-stress (and strain) distributions at the length scale of fiber size. Therefore, it is of interest to develop a methodology to arrive at sufficiently small micro-structural window size (which can be termed as an RVE in the present context) that has the bulk properties (such as modulus), microstructure, and the micro-stress (and strain) distributions that are statistically representative of those in the macro-size specimens. Such an RVE should be useful (i) to simulate mechanical response of large components using mesh size approximately equal to or larger than RVE size, and (ii) for representative parametric studies on the effect of
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variations in the micro-structure on the micromechanical response, using micro-structural window of RVE size. It is the purpose of this contribution to develop a methodology to arrive at a sufficiently small window size that can serve as an RVE. The technique involves a combination of digital image processing, quantitative micro-structure characterization, computer-simulated microstructure models, and FE-based simulations on large-area high-resolution digital images of microstructure, and computer-simulated micro-structures. In an earlier contribution [9], a methodology has been developed to arrive at detailed microstructure model for composites having nonuniform spatial distribution of fibers consisting of fiber-rich and -poor regions, and a range of fiber sizes. The methodology yields a computer-simulated micro-structure that is statistically similar to the micro-structure of the composite. In this contribution, this computer-simulated micro-structure model is used to arrive at a sufficiently small micro-structural window size that has the microstructure, modulus, and the micro-stress (and strain) distributions that are statistically representative of a macro-scale specimen having the same micro-structure. The next section of the paper gives a brief description of processing of the composite and development of the geometric model for its micro-structure. The subsequent sections describe the simulated window and the methodology to arrive at sufficiently small microstructure window size that has the geometrical attributes, the bulk properties (such as modulus) and the micro-stress (and strain) distributions that are statistically representative of those in the macro-size specimens.
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[10]. Fig. 1 is a low magnification micrograph of the composite. Observe that the spatial arrangement of the fibers is not uniform random. The micro-structure contains fiber-rich and -poor regions due to which the number density of fibers is spatially non-uniform. Fig. 2 is a digital image montage of the micro-structure created through pixel-by-pixel matching and pasting of 120 contiguous micro-structural fields grabbed at the magnification of 500. The image montage has been digitally compressed for the presentation in Fig. 2. The montage contains about 2000 fibers; it has a resolution of 0.5 lm, and covers an area of 1 mm2 on the transverse metallographic plane of the composite specimen. Therefore, it is a large-area high-resolution micro-structural image of the size that approaches a macro-size specimen. It is extremely difficult to experimentally measure the local stress (and strain) distributions around the fibers at the micron length scale, therefore, for the development of the methodology, the micromechanical response and local stress (and strain) distributions resulting from FE-based simulations on this image montage are regarded as representative of a macro-size specimen of the composite. It is of interest to determine if there exists a
2. Background The material used in this study is a glass CMC (Nicalon/MAS-5), supplied by Corning Incorporated. It consists of MAS cordierite (2MgO– 2Al2 O3 –5SiO2 ) matrix and continuous aligned Nicalon (SiC) fibers of 14 lm average diameter having volume fraction of 0.35. The composite is manufactured by slurry-infiltration-type process
Fig. 2. Actual micro-structure of a glass CMC with unidirectional aligned SiC fibers in a glass ceramic matrix (MAS).
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sufficiently small micro-structural window having local stress (and strain) distributions, modulus and micro-structure, that are statistically similar to those of the large-area high-resolution montage. For this purpose, a computer-simulated microstructure model has been developed that has the fiber size distribution, volume fraction, number density, nearest neighbor distributions, and radial distribution function of fibers statistically similar those in the composite [9]. The model has been developed by using experimentally measured data on nearest neighbor distribution, radial distribution function, size distribution, number density, and volume fraction of fiber in the composite obtained via extensive digital image analysis [9]. Fig. 3 is the computer-simulated micro-structure of the composite obtained from such a model. The following sections describe FE-based simulations on the large-area high-resolution image montage, and the various windows from the computer simulated micro-structure model. These simulations have been subsequently utilized to arrive at a sufficiently small window of the geometric microstructure model that has local stress (and strain) distributions, modulus and micro-structure, that
are statistically similar to those of the large-area high-resolution montage.
3. Representative volume element study of the composite with non-uniformly distributed fibers In the present context, a micro-structural window is regarded as an RVE provided (i) microstructure, global mechanical properties such as modulus, and micro-mechanical response of the window are statistically similar to those of a macrosize specimen of the same material, (ii) the microstructure, simulated global mechanical properties, and micro-stress (and strain) distributions of the window do not vary with the location of the window, or with the nature of boundary conditions (for example, displacement control versus traction control), (iii) different realizations of the simulated micro-structure window have statistically similar micro-structure and micro-mechanical response, and (iv) the micro-mechanical response of the window is unique for a given loading direction, although different loading directions may yield different micro-mechanical response due to nonuniform nature of the micro-structure. It is extremely difficult to experimentally measure local stress (and strain) distributions in a real macro-size specimen, at the length scale of fiber size (14 lm). Therefore, as mentioned earlier, in this study, these local stress distributions have been obtained by performing FE-simulations on a large-area highresolution digital image of a seamless micro-structural montage of about 1 mm2 area having about 2000 fibers, but having an image resolution of about 0.5 lm [9,11–13]. 3.1. Comparison of the micro-structural geometry of a large window from simulated micro-structure and the large-area micro-structural digital image
Fig. 3. Simulated micro-structural geometric model of the CMC.
The micro-structural attributes of the large-area high-resolution image in Fig. 2 image are representative of the CMC, and therefore, they are compared with the corresponding attributes of simulated micro-structural windows of different sizes. Fig. 3 is a simulated micro-structure containing approximately the same number of fibers
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as digital image in Fig. 2. Fig. 4 compares fiber size distribution, and the first and second nearest
Fig. 4. Fiber size distribution, the first and second nearest neighbor function of fibers in the digital image and the simulated window at the same size.
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neighbor distance distributions of fibers in the simulated structure in Fig. 3 and the digital image in Fig. 2. Observe that all the three distributions (size as well as spatial) of the two micro-structures are in an excellent agreement. Therefore, simulated window (Fig. 3) has a micro-structure that is representative of the CMC. It is now of interest to determine, if smaller windows from the simulated micro-structures also have micro-structures that are statistically similar to that of the digital image of CMC (Fig. 2). In the present composite, there are fiber-rich and -poor regions. Therefore, a useful window of this micro-structure must have this typical fiber distribution pattern. The windows A– G in Fig. 5 show the windows of different sizes from the simulated micro-structure. Window-G is
Fig. 5. Different size simulated windows in the study of microstructural RVE.
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of the biggest size of Fig. 3, which contains about 2000 fibers. Window-A is of the smallest size, and it consists of one layer of fiber-rich and -poor region. This window contains about 60 fibers. Table 1 lists the fiber volume fraction, Young’s modulus. The volume fraction of the fibers in the microstructure model is measured by using image analysis. The Young’s modulus is calculated from the volume fraction of the fibers in the specific microstructural window and the Young’s modulii of the fibers and matrix using a linear interpolation method. Fig. 6 compares the fiber size distribution in windows A, B, C, G and the digital image of CMC (Fig. 2). Notice that fiber volume fraction, Young’s modulus of all the windows in Fig. 5 are very close to that of the composite, and fiber size distributions of these windows are in reasonable agreement with digital image of the micro-structure of the composite. Figs. 7 and 8 show the first and second nearest neighbor distance distributions of fiber centers in windows A and B. Observe that although the fiber volume fraction and fiber size distributions are in reasonable agreement, the first and second nearest neighbor distributions for windows A and B exhibit lot of noise, which is due to the edge effects and window size effects (these windows do not have sufficient number of fibers for stable statistical distributions). Therefore, stable statistical distributions can only be obtained by averaging over many realizations of such small windows. In other words, the micro-structure of such small windows (A and B) is expected to randomly vary from one location to another, and from one realization to another. It follows that windows A and B cannot be treated as RVE of the micro-structure. Fig. 9 compares and first and second nearest neighbor distance distributions of fiber centers in window-C and the large-area highresolution digital image of CMC (Fig. 2). The data in Table 1, Fig. 6, and Fig. 9 show that the fiber volume fraction, fiber size distribution and the first
Fig. 6. Size distribution of fibers in different simulated windows and digital image of CMC.
and the second nearest neighbor distributions for the window-C are in good agreement with the corresponding distributions for the digital image of the CMC (Fig. 2). Therefore, simulated window-C is a sufficiently small window that can serve as a micro-structural RVE for the CMC microstructure, provided the micro-structural attributes of this window do not vary significantly from one realization to another and one location to another. This window contains about 250 fibers (a factor of 8 less than the number of fibers in the digital image in Fig. 2. Due to the stochastic nature of micro-structure of the composite, there are no two micro-structural windows that can have exactly the same spatial arrangement of fibers. The fiber distribution pattern varies with the locations of the selected window. An RVE is required to have statistically equivalent micro-structural attributes regardless of its location. In the present composite, there are three characteristic fiber distribution patterns associated with windows of type C, which are shown
Table 1 Fiber volume fraction and Young’s modulus of the simulated windows and the composite Simulated window
Window-A
Window-B
Window-C
Window-G
Digital image
Volume fraction of fibers (%) Young’s modulus (GPa)
34.5 117.95
34.8 118.28
34.7 118.17
34.9 118.39
35.0 118.5
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Fig. 8. The first and second nearest neighbor function of fibers in window-B. Fig. 7. The first and second nearest neighbor function of fibers in window-A.
in Fig. 10: (i) a window that starts and ends at the boundary of fiber-rich and -poor region; (ii) a window that starts and ends at the middle fiberpoor region; and (iii) a window that starts and ends at the middle fiber-rich region. If it can be shown that type-C windows in these locations have statistically similar micro-structure, then it can be concluded that the micro-structural geometry of window-C is independent of its location (which is one of the requirements for it to be an RVE). The simulated window of type C at location (i) has been proven having the statistically equivalent micro-structural attributes of the composite in the early of this part (Table 1, Figs. 6 and 9), Table 2 and Fig. 11 make a similar comparison for type-C windows at locations (i), (ii) and (iii). Table 2,
Figs. 9 and 11 clearly reveal that the fiber volume fraction, fiber size distributions as well as the first and the second nearest neighbor distributions for type-C windows at the three locations are in good agreement of the corresponding attributes with each other, also with the large-area digital image (Fig. 2). Therefore, it can be concluded that, in the statistical sense, the micro-structure of type-C window does not vary with its location, and it is representative of the non-uniform micro-structure of the CMC. Table 2 and Fig. 12 show a similar comparison for two type-C windows obtained from two different realizations (simulations) ((a) and (d) in Fig. 10) of the computer simulated micro-structure model. From inspection of Table 2 and Fig. 12, it can be concluded that different realizations lead to statistically similar micro-structures in window-C.
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C are statistically similar to those in the large-area digital image of the composite (Fig. 2). 3.2. Finite element-based simulations of micromechanical response of the composite The following assumptions, loading conditions, boundary conditions, and constitutive properties are used for all the FE-based simulations of the present work.
Fig. 9. The first and second nearest neighbor function of fibers in window-C and digital image.
The volume fraction, the size distribution, and the first and second nearest neighbor distribution functions of fibers in the simulated type-C windows at different locations or obtained from different realizations are statistically similar and agree well with corresponding attributes of the large-area digital image of the micro-structure (Fig. 2). These results demonstrate that the simulated window at the size scale of window-C (containing two layers of fiber-rich and -poor region) is sufficiently large to have the statistically equivalent micro-structural attributes as those in the composite, and it can be regarded as a micro-structural RVE of the composite. It remains to be shown if the local stress and strain distributions in window-
1. It is assumed that both the fibers and the matrix are perfectly elastic, and there is a perfect bond between them. 2. The constitutive properties of the fibers and the matrix are given in Table 3. 3. Micro-structure model is embedded in a homogeneous medium to minimize the effects of the finite boundaries [14,15]. The elastic modulus of the homogeneous medium is equal to the over all elastic modulus of the corresponding composite micro-structure window embedded in that homogeneous medium. The volume fraction of the fibers in the micro-structure model is measured by using image analysis. The elastic modulus of the homogeneous medium is calculated from the volume fraction of the fibers in the specific micro-structural window embedded in that homogenous medium and the elastic modulii of the fibers and matrix using a linear interpolation method. In-plane isotropy is assumed for the homogenous medium. 4. In each simulation, a traction boundary condition, which contains a constant uniaxial tensile stress of 118 MPa in a transverse direction along X-axis (Fig. 13) is applied on the boundary of the homogeneous medium. 118 MPa is 80% of the fracture stress of the matrix [16]. The boundary conditions and the FE mesh of one simulation are also shown in Fig. 13. Note that the simulated micro-structural window is embedded in the homogeneous medium, the area of which is about three times that of the simulated micro-structural window. 5. The generalized plane strain elements are more suitable for handling the longitudinal constraints; therefore CGPE5 elements are utilized in the present simulations rather than CPE3
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Fig. 10. Type C windows at various locations (a–c), and from different realization (d).
Table 2 Volume fraction of fibers in the various type-C windows Simulated window
Window-C (i)
Window-C (ii)
Window-C (iii)
Window-C0 (i)
Volume fraction of fibers (%)
34.7
34.5
34.8
34.6
elements. For simulations on windows containing one to six layers of fiber-rich and -poor regions, about three to ten thousands CGPE5 elements of the same mesh density are required. Our early study about element mesh density [17] showed that such mesh density is sufficiently fine to avoid any artifacts. All the
simulations are performed with ABAQUS software package. 3.3. Results of the FE-based simulations Due to brittle nature of the matrix and assumed perfect interface, the damage initiation can
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Fig. 11. Fiber size distribution, the first and second nearest neighbor function of fibers in the type-C window at three typical locations.
be assumed to occur in the matrix, and controlled by distribution of the local maximum principal
Fig. 12. Fiber size distribution, the first and second nearest neighbor function of fibers of type-C windows from different realizations of the same model.
stress in the matrix. Therefore, in the present FE simulations, the distribution of local maximum principal stress is of interest. In this section, the FE
Z. Shan, A.M. Gokhale / Computational Materials Science 24 (2002) 361–379 Table 3 Mechanical properties of Nicalon (SiC) fiber, MAS-5 matrix and the composite
E (GPa) m
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Table 4 max Summary of rmax 1 , r1 , r1 =r1 in matrix of window-C and digital image
Fiber (Nicalon (SiC))
Matrix (MAS-5)
Simulated window
(MPa) rmax 1
r1 (MPa)
rmax 1 =r1
190 0.29
80 0.3
Window-C Digital image
224.3 228.46
109.64 109.24
2.05 2.09
Fig. 13. FE mesh overlaid on the simulated window and boundary conditions. The model is embedded in an effective media. (a) A window (b) FE mesh and boundary conditions.
simulations are performed on micro-structural windows of type C (or larger) as such windows can be regarded as micro-structural RVE of the material. Table 4 gives the values of the maximum
principal stress (r1 ), the average maximum principal stress (r1 ), and the ratio of the r1 =r1 in the matrix of the simulated window-C and the large-area digital image of the micro-structure. Table 4 reveals that the average maximum principal stresses are almost identical and the highest maximum principal stress are slightly different in these two windows. However, only one tiny element with higher stress may not cause the final failure of the composite, the extent of matrix having high local stress and the connectivity of the high stress regions are other important factors to cause the failure of composite. Therefore, the comparison of fraction of matrix experiencing high maximum principal stress (r1 ) in each window is more meaningful. The fracture strength of the matrix of this composite depends on the processing conditions, and has an average value about 140 MPa [16]. Therefore, the fraction of the matrix with maximum principal stresses higher than 140 MPa is a critical attribute of significant interest for analyzing the damage initiation in the composite. Fig. 14a compares the distribution of the maximum principal stresses in the matrix of window-C and large-area digital image, where Y-axis is the area fraction of the matrix with maximum principal stress higher than a given value. Inspection of Fig. 14a reveals excellent agreement between the patterns of local maximum principal stresses in the two simulations. Fig. 14b shows the fraction of matrix experiencing a stress higher than a given value for stresses higher than 140 MPa. These plots for the two simulations essentially represent the tail ends of the corresponding plots in Fig. 14a, and again it is observed that for any given stress level the area fractions of matrix with the stress higher than that value are within 5% of each other. Therefore, it can be concluded that the distribution of local maximum principal stress in windowC is representative of the large-area digital image
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Fig. 14. Comparison of the fraction of matrix with high maximum principal stresses of simulated window-C and digital image of CMC.
(Fig. 2). Fig. 15 shows the contour plot of the maximum principal stress in the matrix of window-C and large-area digital image. Note that higher maximum principal stresses are present in the regions where fibers are clustered and very closely spaced. These sites are the weak areas of the composite, where the cracks are likely to initiate. Observe that the elements experiencing higher stresses are more or less linked together, which is deleterious to the fracture resistance. These critical elements may form micro-cracks first and cause the final failure of the composite. It is of interest to determine if increasing the window size leads to significantly better agreement between the local stress distributions of the simulated micro-structural window and largearea digital image. Fig. 16 shows the simulated windows C–G, from computer simulated micro-
structure of CMC, which contains two to six layers of fiber-rich and -poor regions, respectively. Note that windows D–G are larger than window-C. Table 5 gives the values of the maximum principal stress (r1 ), the average maximum principal stress (r1 ), and the ratio of the r1 =r1 in the matrix of the five simulated windows and large-area digital image of CMC, showing that both the average maximum principal stress and the highest maximum principal stress does not vary significantly with scale of the simulated window, and are close to those in the matrix of the large-area digital image of CMC. The maximum principal stress distributions in the matrix of the five simulated windows and large-area digital image of CMC are compared in Fig. 17 with the focus on the higher stress part (Fig. 17b). The fraction of matrix experiencing a stress higher than a given value is similar for all windows. Therefore, window-C can be regarded as an RVE for FE simulations of damage initiation, provided the stress distributions are comparable for displacement control and traction control simulations, and they do not vary with the location of the window, or from one realization of the model to another realization of the same model, are unique for a given loading direction. Fig. 18 compares the distribution of the maximum principal stresses ðr1 ) with displacement and traction controlled boundary conditions. These data show that the differences in the distribution of the maximum principal stresses ðr1 Þ for these two conditions are very small at any stress level. The total strain energy of the matrix with the displacement and traction controlled boundary condition are also investigated, and they are 5:179 109 and 5:152 109 J/m3 , respectively. Fig. 19 shows the contour plots of the maximum principal stress (r1 ) in the matrix of type-C windows at three different typical locations in Fig. 10. Fig. 20 compares the distribution of the maximum principal stresses in the matrix for the three type-C windows. The results show that distributions of the maximum principal stresses in the matrix for the type-C windows at three locations are almost identical in the entire stress range. The contour plots of the maximum principal stress (r1 ) in the matrix of another type-C window obtained
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Fig. 15. Contour plot of the maximum principal stresses r1 (MPa) in a simulated window-C and larger area digital image.
from a different computer simulation (realization) of the same model is shown in Fig. 19(d). Fig. 21 compares the distribution of the maximum principal stresses in the matrix of the two windows from different realizations at similar location (Fig. 10(a) and (d)). Observe that the difference between the stress distributions for these two simulations is very small in the entire stress range. Table 6 also lists the values of the maximum principal stress (r1 ), the average maximum principal stress (r1 ), and the ratio of the r1 =r1 in the matrix of the type-
C windows at the three typical locations and from different realizations of the same model, showing that the average and highest maximum principal stress does not vary significantly among these typeC windows. It is now shown type-C is a window of sufficiently small size (250 fibers) for which (i) microstructure is statistically similar to that of large-area digital image, and it does not vary (in statistical sense) with the location or different realizations of the model, (ii) the distribution of local maximum
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Fig. 16. Simulated windows (C–G).
Table 5 max Summary of rmax 1 , r1 , r1 =r1 in matrix of various windows and digital image
Window-C Window-D Window-E Window-F Window-G Digital image
rmax (MPa) 1
r1 (MPa)
224.3 228.98 227.51 224.37 230.03 228.46
109.64 109.02 108.83 108.99 109.14 109.24
rmax 1 =r1 2.05 2.10 2.08 2.06 2.10 2.09
principal stress in the matrix is in good agreement with the corresponding distribution for the largearea digital image, and (iii) the micro-stress distribution does not vary significantly with location, or for different realizations, or for different type of boundary conditions (traction versus displacement control). Therefore, it is tempting to regard such a window as an RVE for micro-mechanical simula-
Fig. 17. Comparison of the fraction of matrix with high maximum principal stresses of simulated windows C–G and digital image of CMC.
tions. However, there is yet another factor (that arises due to non-uniform spatial fiber distribution) that needs to be examined before arriving at such a conclusion. All the earlier two-dimensional RVE studies on the fiber composites have been performed on the materials that are locally and globally transverse isotropic, for example, transverse loading behavior of an aligned continuous fiber ceramic composite having uniform random spatial distribution of fibers [18]. However, although the present material is locally transverse isotropic, it is not globally transverse isotropic due to layered nature of the fiber-rich and -poor regions present in the microstructure (Fig. 2), which is typical of many real composites. In such materials, the local stress (and
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Fig. 18. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) of window-C under traction and displacement controlled boundary condition. (a) Full scale, (b) tail part.
strain) distributions (but not the modulus) are expected to be a function of loading direction. Therefore, it is of interest to determine, if the size (of type-C window) that yields a statistically similar local stress distribution for different window locations, type of boundary conditions, and realizations of the models, varies with the loading direction or not. Fig. 22 shows micro-stress distribution plot for the two simulations on the same window of type C but with different loading directions (X and Y as shown in Fig. 13). The applied load (118 MPa), the boundary conditions, and all other simulation parameters are the same as before. Recall that in all the earlier simulations discussed the simulated load has been applied
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along X-direction (Fig. 13). The distribution of the maximum principal stresses (r1 ) in these two cases as shown Fig. 22 indicates that the micromechanical response (maximum principal stress) of the composite is quite sensitive to loading direction, as expected. Therefore, it is of interest to determine if the optimal size of type-C window is significantly different when the load is applied along Y-direction. To test this, all the previous simulations were repeated with Y-direction as the loading direction. It was observed that, the local stress distributions did not vary with the location of the window, or for different realizations of the model, or with the type of boundary conditions, etc. Therefore, it is concluded that the optimal size of the type-C window does not depend on the loading direction. Although the local stress distribution varies with the loading directions, for a given loading direction the local stress distribution does not depend on window location, type of boundary conditions, etc. It is now shown that (i) micro-structure of window-C is statistically similar to that of the large-area digital image, and it does not vary (in a statistical sense) with the location or for different realizations of the model, (ii) the distribution of local maximum principal stress in the matrix of the window-C is in good agreement with the corresponding distribution for the large-area digital image, (iii) the micro-stress distribution of type-C window does not vary significantly with location, or for different realizations, or for different type of boundary conditions (traction versus displacement control), and (iv) the micro-mechanical response of type-C window is unique for a given loading condition. It is concluded that the type-C window can be regarded as RVE for simulation of local stress (and strain) distributions and damage initiation, although the simulated local stress distribution strongly depends on the loading direction. In this contribution, a methodology has been developed to arrive at a sufficiently small microstructural window that can be regarded as RVE for the simulation of micro-mechanical response that can be used as an input to simulate damage initiation. Note that all earlier RVE studies were only concerned with arriving at a micro-structural window that yield the same elastic properties (Young’s
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Fig. 19. Contour plot of the maximum principle stress r1 (MPa) of matrix of various type-C windows.
modulus) as that of a macro-size specimen, and were not concerned with local stress distributions and damage initiation. Earlier RVE studies ignored the window-to-window statistical variations in the size distribution and spatial arrangement of features. All the earlier studies involved RVEs for uniform random micro-structures and did not account for local stress distributions. Such RVE models are not useful for real composites having non-uniform spatial arrangement of the fibers, nor are they useful for simulation of damage initiation in composites having uniform random or nonuniform distribution of fibers. It is realized that residual stresses caused by manufacturing processing (i.e., cooling down from processing), interfacial properties of fiber/matrix interfaces, and
presence of processing defects in the composites may also influence the local stress/strain distribution [19], and consequently, the size of the RVE. If experimental data on these attributes is available, then the present methodology can be used to arrive at sufficiently small RVE size by including these attributes in the computer simulated micro-structure model. Note that the RVE for the present non-uniform micro-structure (i.e., window-C) contains about 250 fibers, whereas large-area digital image of the CMC (Fig. 2) contains about 2000 fibers. Therefore, it is significantly more efficient to use type-C window for FE simulations of macro-size specimens. An important utility of RVE size is for FEbased simulation of macro-mechanical response of
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Fig. 20. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) of type-C window at three typical locations. (a) Full scale, (b) tail part.
Fig. 21. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) between window-C and C0 . (a) Full scale, (b) tail part.
components, where a homogenized continuum in assumed. For such simulations, the FE size must be larger than the RVE for the material, so that each FE has the same constitutive behavior as a macrosize specimen of the composite. For the present composite the absolute RVE size (for fibers having 14 lm average size) is 0.1 mm2 . Therefore, for simulation of macro-mechanical response of a component the element size should be larger than 0.1 mm2 area. It is important to point out that FE simulations are scale independent. Therefore, the ratio of the absolute size of the RVE and average length scale of micro-structure (in this case average fiber size) should be the same for micro-structures that differ only in scale. Therefore, for microstructures that differ from the present composite
Table 6 max Summary of rmax 1 , r1 , r1 =r1 in matrix of various type-C windows Simulated window
rmax (MPa) 1
r1 (MPa)
rmax 1 =r1
Window-C (i) Window-C (ii) Window-C (iii) Window-C0 (i)
224.3 223.3 223.9 225.9
109.64 109.41 109.36 110.63
2.05 2.04 2.05 2.04
micro-structure only in scale and has say 28 lm average fiber size, the absolute RVE area should be about 0:4–0:5 mm2 (about 0:7 mm 0:7 mm), which is not so small. In general, for a given microstructure, the size of RVE is expected to vary with the variations in the constitutive equations of the
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4. Summary and conclusions
Fig. 22. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) of window-C at different loading direction. (a) Full scale, (b) tail part.
phases involved. For example, in the present case, the local stress (or strain) distributions are strong function of elastic modulii of the fibers and the matrix, and consequently, the size of the RVE is expected to depend on the ratio of the elastic modulii of fibers and matrix. The RVE model can be used for FE-based simulation of mechanical response of real complex micro-structures. Further, one can then change sizes and number density of fibers (or any other attributes) in the micro-structural model to reflect the changes in the processing conditions and raw materials. Then FE-based simulations on the RVE model can be repeated on the new structure to gauge the effect of changes in the micro-structure (and therefore, changes in the processing conditions) on the mechanical response of the material.
A methodology has been developed to arrive at a sufficiently small micro-structural window that can be regarded as RVE of a non-uniform microstructure of a composite containing a range of fiber sizes, and fiber-rich and -poor regions at the length scale of about 100 lm. The methodology involves quantitative characterization of microstructure, micro-structure modeling, FE-based simulations on computer simulated micro-structural widows of different sizes containing 60–2000 fibers, and FE simulations on large-area highresolution digital image of the composite microstructure containing about 2000 fibers. The RVE has the micro-structure that is statistically similar to that of the composite material having fiber-rich and -poor regions, its Young’s modulus is very close to that of the composite, and has local stress distribution that is comparable to that in the real composite under similar loading conditions. Therefore, such an RVE is useful for realistic simulations of damage initiation, based on which parametric studies can be performed by repeating FE-based simulations on the RVE model with different sizes and number density of fibers to reflect the changes in the micro-structure (and therefore, in the processing conditions) on the mechanical response of the materials. Acknowledgements This research work is a part of NSF sponsored project DMR-9816618. Dr. B. MacDonald is the project monitor. The financial support of NSF is gratefully acknowledged. References [1] R. Hill, J. Mech. Phys. Solids 11 (1963) 357–372. [2] W.J. Drugan, J.R. Willis, J. Mech. Phys. Solids 44 (1996) 497–524. [3] H.J. B€ ohm, F.G. Rammerstorfer, F.D. Fischer, T. Siegmund, J. Eng. Mater. Technol., Trans. ASME 116 (1994) 268–273. [4] H.E. Deve, Metall. Mater. Trans. A 30 (1999) 2513–2522. [5] C. Zweben, Composites 25 (1994) 451–454. [6] M.R. Wisnom, Compos. Struct. 18 (1991) 47–63.
Z. Shan, A.M. Gokhale / Computational Materials Science 24 (2002) 361–379 [7] G. Zhou, G. Davis, Composites 26 (1995) 587–596. [8] M.F. Crowther, M.S. Starkey, Compos. Sci. Technol. 31 (1988) 87–95. [9] S. Yang, A. Tewari, A.M. Gokhale, Acta Mater. 45 (1997) 3059–3069. [10] K.K. Chawala, Ceramic Matrix Composite, Chapman and Hall, New York, 1993. [11] P. Louis, A.M. Gokhale, Metall. Trans. 26A (1995) 1449– 1456. [12] P. Louis, A.M. Gokhale, Acta Metall. Mater. 44 (1996) 1519–1528. [13] S. Yang, A. Tewari, A.M. Gokhale, in: G.F. Vander Voort, J.J. Friel (Eds.), Development in Materials Char-
[14] [15] [16] [17] [18] [19]
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acterization Technologies, ASM International, Materials Park, Ohio, 1996, pp. 33–43. J.R. Brockenbrough, S. Suresh, H.A. Wienecke, Acta Metall. et Mater. 39 (1991) 735–752. J. Wulf, T.H. Steinkopff, H.F. Fischmeister, Acta Mater. 44 (1996) 1765–1779. K.K. Chawala, in: Composite Materials, Springer, New York, 1998, p. 99. S. Yang, A.M. Gokhale, Z. Shan, Acta Mater. 48 (2000) 2307–2322. M. Jiang, Ph.D. thesis, Georgia Tech., 2000. T. Nakamura, S. Suresh, Acta Metall. Mater. 41 (1993) 1665–1681.
Representative volume element for non-uniform micro-structure Zhaohui Shan, Arun M. Gokhale
*
School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, USA Received 9 July 2001; received in revised form 28 September 2001; accepted 28 September 2001
Abstract A representative volume element (RVE) of micro-structure is an important input for computational-mechanicsbased simulations of micro-mechanical response of heterogeneous materials such as composites. In this contribution, a methodology has been developed to arrive at a sufficiently small micro-structural window that can be regarded as a RVE of a non-uniform micro-structure of a ceramic matrix composite (CMC) containing a range of fiber sizes, and fiber-rich and -poor regions at the length scale of about 100 lm. The RVE contains about 250 fibers of 14 lm diameter average size. The absolute size of the RVE is 0.1 mm2 . The methodology involves a unique combination of quantitative characterization of geometry and spatial arrangement of micro-structural features using stereological and image analysis techniques, development of computer simulated micro-structure model that is statistically similar to the real micro-structure, finite element (FE)-based simulations of micro-mechanical response on computer-simulated microstructural widows of different sizes containing 60–2000 fibers, and FE-based simulations on large-area high-resolution digital image of the composite micro-structure containing about 2000 fibers. The RVE has the micro-structure that is statistically similar to that of the CMC having fiber-rich and -poor regions, its Young’s modulus is very close to that of the composite, and has local stress distribution that is comparable to that in the real composite under similar loading conditions. Therefore, such an RVE is useful for realistic simulations of damage initiation. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Composite; Micro-structure; FEM; RVE; Stereology; Simulations
1. Introduction The concept of representative volume element (RVE) is important in the study of mechanical
* Corresponding author. Tel.: +1-404-894-2887; fax: +1-404894-9140. E-mail address: [email protected] (A.M. Gokhale).
response of heterogeneous materials. Mathematically, RVE is the infinite length scale limit relative to the micro-scale (or the length scale of a single heterogeneity), where the material appears uniform, and therefore, the continuum concepts are applicable. RVE is useful for modeling the effect of nano- and meso-scale heterogeneities on the overall mechanical response of the macro-size components/specimens. Different research workers have defined RVE in different ways. The first formal
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 2 5 7 - 9
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definition of RVE was given by Hill [1], according to which, RVE (i) must be structurally entirely typical of whole micro-structure on average, and (ii) must contain a sufficiently large number of micro-structural heterogeneities for apparent overall moduli to be effectively independent of the surface values of traction and displacement, as long as these values are ‘macroscopically uniform’. Drugan and Willis [2] have defined RVE as the smallest volume element of a material for which the usual spatially constant ‘overall modulus’, i.e., macroscopic constitutive representation, is sufficiently accurate to represent overall constitutive response. According to Drugan and Willis [2], a micro-structural volume element that yields modulus value within 5% of the global bulk modulus can be regarded as an RVE for practical applications. Therefore, such a volume element is useful to simulate global mechanical response of material using finite element (FE)-based simulations. However, majority of FE-based simulations are performed on a micro-structural window of arbitrary small size, with an assumption that such micro-structural window is an RVE, which may be reasonable only if the micro-structure consists of periodic arrays of micro-structural features of mono-size. Unfortunately, micro-structures of many materials (including composites) do not satisfy this requirement. For example, Fig. 1 shows micro-structure of a ceramic matrix composite (CMC) that has non-uniform spatial distribution of fibers (observe the fiber-rich and -poor regions). Further, a distribution of fiber sizes exists in this micro-structure. In the earlier studies, on the basis
Fig. 1. Low magnification micrograph of the glass CMC shows the alternating fiber-rich and -poor regions in the composite micro-structure.
of computer-simulated micro-structures, it has been shown that the spatial arrangement of fibers or particles has a strong effect on the mechanical responses of the composites [3,4]. These simulations have been performed on small micro-structural windows that may not be descriptive of the corresponding RVEs. Consequently, the simulated mechanical response obtained from such small windows of micro-structures may not be representative of the overall mechanical response of the material. The problem persists even in composites having uniformly distributed mono-size fibers as well, because the simulated properties of small micro-structural windows are of stochastic nature, and depend on the window size (scale effects), especially at low length scales [5–8]. On the other hand, from computational point of view, it is not practical to perform FE-based simulation on very large micro-structural windows containing very large number of heterogeneities (for example, say millions of fibers). Therefore, it is of interest to determine a sufficiently small micro-structural window that can serve as an RVE for simulation of micro-mechanical response of a heterogeneous micro-structure that may or may not be spatially homogenous or uniform random. In this context, it is essential to point out that most of the previous studies on RVE concern the simulation of global bulk properties such as modulus; very little attention has been paid to simulation of micro-stress (and strain) distributions that are statistically representative of the corresponding distributions in macro-size specimens. For simulation of damage initiation and evolution in heterogeneous materials such as composites it is of interest to know the micro-stress (and strain) distributions at the length scale of fiber size. Therefore, it is of interest to develop a methodology to arrive at sufficiently small micro-structural window size (which can be termed as an RVE in the present context) that has the bulk properties (such as modulus), microstructure, and the micro-stress (and strain) distributions that are statistically representative of those in the macro-size specimens. Such an RVE should be useful (i) to simulate mechanical response of large components using mesh size approximately equal to or larger than RVE size, and (ii) for representative parametric studies on the effect of
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variations in the micro-structure on the micromechanical response, using micro-structural window of RVE size. It is the purpose of this contribution to develop a methodology to arrive at a sufficiently small window size that can serve as an RVE. The technique involves a combination of digital image processing, quantitative micro-structure characterization, computer-simulated microstructure models, and FE-based simulations on large-area high-resolution digital images of microstructure, and computer-simulated micro-structures. In an earlier contribution [9], a methodology has been developed to arrive at detailed microstructure model for composites having nonuniform spatial distribution of fibers consisting of fiber-rich and -poor regions, and a range of fiber sizes. The methodology yields a computer-simulated micro-structure that is statistically similar to the micro-structure of the composite. In this contribution, this computer-simulated micro-structure model is used to arrive at a sufficiently small micro-structural window size that has the microstructure, modulus, and the micro-stress (and strain) distributions that are statistically representative of a macro-scale specimen having the same micro-structure. The next section of the paper gives a brief description of processing of the composite and development of the geometric model for its micro-structure. The subsequent sections describe the simulated window and the methodology to arrive at sufficiently small microstructure window size that has the geometrical attributes, the bulk properties (such as modulus) and the micro-stress (and strain) distributions that are statistically representative of those in the macro-size specimens.
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[10]. Fig. 1 is a low magnification micrograph of the composite. Observe that the spatial arrangement of the fibers is not uniform random. The micro-structure contains fiber-rich and -poor regions due to which the number density of fibers is spatially non-uniform. Fig. 2 is a digital image montage of the micro-structure created through pixel-by-pixel matching and pasting of 120 contiguous micro-structural fields grabbed at the magnification of 500. The image montage has been digitally compressed for the presentation in Fig. 2. The montage contains about 2000 fibers; it has a resolution of 0.5 lm, and covers an area of 1 mm2 on the transverse metallographic plane of the composite specimen. Therefore, it is a large-area high-resolution micro-structural image of the size that approaches a macro-size specimen. It is extremely difficult to experimentally measure the local stress (and strain) distributions around the fibers at the micron length scale, therefore, for the development of the methodology, the micromechanical response and local stress (and strain) distributions resulting from FE-based simulations on this image montage are regarded as representative of a macro-size specimen of the composite. It is of interest to determine if there exists a
2. Background The material used in this study is a glass CMC (Nicalon/MAS-5), supplied by Corning Incorporated. It consists of MAS cordierite (2MgO– 2Al2 O3 –5SiO2 ) matrix and continuous aligned Nicalon (SiC) fibers of 14 lm average diameter having volume fraction of 0.35. The composite is manufactured by slurry-infiltration-type process
Fig. 2. Actual micro-structure of a glass CMC with unidirectional aligned SiC fibers in a glass ceramic matrix (MAS).
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sufficiently small micro-structural window having local stress (and strain) distributions, modulus and micro-structure, that are statistically similar to those of the large-area high-resolution montage. For this purpose, a computer-simulated microstructure model has been developed that has the fiber size distribution, volume fraction, number density, nearest neighbor distributions, and radial distribution function of fibers statistically similar those in the composite [9]. The model has been developed by using experimentally measured data on nearest neighbor distribution, radial distribution function, size distribution, number density, and volume fraction of fiber in the composite obtained via extensive digital image analysis [9]. Fig. 3 is the computer-simulated micro-structure of the composite obtained from such a model. The following sections describe FE-based simulations on the large-area high-resolution image montage, and the various windows from the computer simulated micro-structure model. These simulations have been subsequently utilized to arrive at a sufficiently small window of the geometric microstructure model that has local stress (and strain) distributions, modulus and micro-structure, that
are statistically similar to those of the large-area high-resolution montage.
3. Representative volume element study of the composite with non-uniformly distributed fibers In the present context, a micro-structural window is regarded as an RVE provided (i) microstructure, global mechanical properties such as modulus, and micro-mechanical response of the window are statistically similar to those of a macrosize specimen of the same material, (ii) the microstructure, simulated global mechanical properties, and micro-stress (and strain) distributions of the window do not vary with the location of the window, or with the nature of boundary conditions (for example, displacement control versus traction control), (iii) different realizations of the simulated micro-structure window have statistically similar micro-structure and micro-mechanical response, and (iv) the micro-mechanical response of the window is unique for a given loading direction, although different loading directions may yield different micro-mechanical response due to nonuniform nature of the micro-structure. It is extremely difficult to experimentally measure local stress (and strain) distributions in a real macro-size specimen, at the length scale of fiber size (14 lm). Therefore, as mentioned earlier, in this study, these local stress distributions have been obtained by performing FE-simulations on a large-area highresolution digital image of a seamless micro-structural montage of about 1 mm2 area having about 2000 fibers, but having an image resolution of about 0.5 lm [9,11–13]. 3.1. Comparison of the micro-structural geometry of a large window from simulated micro-structure and the large-area micro-structural digital image
Fig. 3. Simulated micro-structural geometric model of the CMC.
The micro-structural attributes of the large-area high-resolution image in Fig. 2 image are representative of the CMC, and therefore, they are compared with the corresponding attributes of simulated micro-structural windows of different sizes. Fig. 3 is a simulated micro-structure containing approximately the same number of fibers
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as digital image in Fig. 2. Fig. 4 compares fiber size distribution, and the first and second nearest
Fig. 4. Fiber size distribution, the first and second nearest neighbor function of fibers in the digital image and the simulated window at the same size.
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neighbor distance distributions of fibers in the simulated structure in Fig. 3 and the digital image in Fig. 2. Observe that all the three distributions (size as well as spatial) of the two micro-structures are in an excellent agreement. Therefore, simulated window (Fig. 3) has a micro-structure that is representative of the CMC. It is now of interest to determine, if smaller windows from the simulated micro-structures also have micro-structures that are statistically similar to that of the digital image of CMC (Fig. 2). In the present composite, there are fiber-rich and -poor regions. Therefore, a useful window of this micro-structure must have this typical fiber distribution pattern. The windows A– G in Fig. 5 show the windows of different sizes from the simulated micro-structure. Window-G is
Fig. 5. Different size simulated windows in the study of microstructural RVE.
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of the biggest size of Fig. 3, which contains about 2000 fibers. Window-A is of the smallest size, and it consists of one layer of fiber-rich and -poor region. This window contains about 60 fibers. Table 1 lists the fiber volume fraction, Young’s modulus. The volume fraction of the fibers in the microstructure model is measured by using image analysis. The Young’s modulus is calculated from the volume fraction of the fibers in the specific microstructural window and the Young’s modulii of the fibers and matrix using a linear interpolation method. Fig. 6 compares the fiber size distribution in windows A, B, C, G and the digital image of CMC (Fig. 2). Notice that fiber volume fraction, Young’s modulus of all the windows in Fig. 5 are very close to that of the composite, and fiber size distributions of these windows are in reasonable agreement with digital image of the micro-structure of the composite. Figs. 7 and 8 show the first and second nearest neighbor distance distributions of fiber centers in windows A and B. Observe that although the fiber volume fraction and fiber size distributions are in reasonable agreement, the first and second nearest neighbor distributions for windows A and B exhibit lot of noise, which is due to the edge effects and window size effects (these windows do not have sufficient number of fibers for stable statistical distributions). Therefore, stable statistical distributions can only be obtained by averaging over many realizations of such small windows. In other words, the micro-structure of such small windows (A and B) is expected to randomly vary from one location to another, and from one realization to another. It follows that windows A and B cannot be treated as RVE of the micro-structure. Fig. 9 compares and first and second nearest neighbor distance distributions of fiber centers in window-C and the large-area highresolution digital image of CMC (Fig. 2). The data in Table 1, Fig. 6, and Fig. 9 show that the fiber volume fraction, fiber size distribution and the first
Fig. 6. Size distribution of fibers in different simulated windows and digital image of CMC.
and the second nearest neighbor distributions for the window-C are in good agreement with the corresponding distributions for the digital image of the CMC (Fig. 2). Therefore, simulated window-C is a sufficiently small window that can serve as a micro-structural RVE for the CMC microstructure, provided the micro-structural attributes of this window do not vary significantly from one realization to another and one location to another. This window contains about 250 fibers (a factor of 8 less than the number of fibers in the digital image in Fig. 2. Due to the stochastic nature of micro-structure of the composite, there are no two micro-structural windows that can have exactly the same spatial arrangement of fibers. The fiber distribution pattern varies with the locations of the selected window. An RVE is required to have statistically equivalent micro-structural attributes regardless of its location. In the present composite, there are three characteristic fiber distribution patterns associated with windows of type C, which are shown
Table 1 Fiber volume fraction and Young’s modulus of the simulated windows and the composite Simulated window
Window-A
Window-B
Window-C
Window-G
Digital image
Volume fraction of fibers (%) Young’s modulus (GPa)
34.5 117.95
34.8 118.28
34.7 118.17
34.9 118.39
35.0 118.5
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Fig. 8. The first and second nearest neighbor function of fibers in window-B. Fig. 7. The first and second nearest neighbor function of fibers in window-A.
in Fig. 10: (i) a window that starts and ends at the boundary of fiber-rich and -poor region; (ii) a window that starts and ends at the middle fiberpoor region; and (iii) a window that starts and ends at the middle fiber-rich region. If it can be shown that type-C windows in these locations have statistically similar micro-structure, then it can be concluded that the micro-structural geometry of window-C is independent of its location (which is one of the requirements for it to be an RVE). The simulated window of type C at location (i) has been proven having the statistically equivalent micro-structural attributes of the composite in the early of this part (Table 1, Figs. 6 and 9), Table 2 and Fig. 11 make a similar comparison for type-C windows at locations (i), (ii) and (iii). Table 2,
Figs. 9 and 11 clearly reveal that the fiber volume fraction, fiber size distributions as well as the first and the second nearest neighbor distributions for type-C windows at the three locations are in good agreement of the corresponding attributes with each other, also with the large-area digital image (Fig. 2). Therefore, it can be concluded that, in the statistical sense, the micro-structure of type-C window does not vary with its location, and it is representative of the non-uniform micro-structure of the CMC. Table 2 and Fig. 12 show a similar comparison for two type-C windows obtained from two different realizations (simulations) ((a) and (d) in Fig. 10) of the computer simulated micro-structure model. From inspection of Table 2 and Fig. 12, it can be concluded that different realizations lead to statistically similar micro-structures in window-C.
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C are statistically similar to those in the large-area digital image of the composite (Fig. 2). 3.2. Finite element-based simulations of micromechanical response of the composite The following assumptions, loading conditions, boundary conditions, and constitutive properties are used for all the FE-based simulations of the present work.
Fig. 9. The first and second nearest neighbor function of fibers in window-C and digital image.
The volume fraction, the size distribution, and the first and second nearest neighbor distribution functions of fibers in the simulated type-C windows at different locations or obtained from different realizations are statistically similar and agree well with corresponding attributes of the large-area digital image of the micro-structure (Fig. 2). These results demonstrate that the simulated window at the size scale of window-C (containing two layers of fiber-rich and -poor region) is sufficiently large to have the statistically equivalent micro-structural attributes as those in the composite, and it can be regarded as a micro-structural RVE of the composite. It remains to be shown if the local stress and strain distributions in window-
1. It is assumed that both the fibers and the matrix are perfectly elastic, and there is a perfect bond between them. 2. The constitutive properties of the fibers and the matrix are given in Table 3. 3. Micro-structure model is embedded in a homogeneous medium to minimize the effects of the finite boundaries [14,15]. The elastic modulus of the homogeneous medium is equal to the over all elastic modulus of the corresponding composite micro-structure window embedded in that homogeneous medium. The volume fraction of the fibers in the micro-structure model is measured by using image analysis. The elastic modulus of the homogeneous medium is calculated from the volume fraction of the fibers in the specific micro-structural window embedded in that homogenous medium and the elastic modulii of the fibers and matrix using a linear interpolation method. In-plane isotropy is assumed for the homogenous medium. 4. In each simulation, a traction boundary condition, which contains a constant uniaxial tensile stress of 118 MPa in a transverse direction along X-axis (Fig. 13) is applied on the boundary of the homogeneous medium. 118 MPa is 80% of the fracture stress of the matrix [16]. The boundary conditions and the FE mesh of one simulation are also shown in Fig. 13. Note that the simulated micro-structural window is embedded in the homogeneous medium, the area of which is about three times that of the simulated micro-structural window. 5. The generalized plane strain elements are more suitable for handling the longitudinal constraints; therefore CGPE5 elements are utilized in the present simulations rather than CPE3
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Fig. 10. Type C windows at various locations (a–c), and from different realization (d).
Table 2 Volume fraction of fibers in the various type-C windows Simulated window
Window-C (i)
Window-C (ii)
Window-C (iii)
Window-C0 (i)
Volume fraction of fibers (%)
34.7
34.5
34.8
34.6
elements. For simulations on windows containing one to six layers of fiber-rich and -poor regions, about three to ten thousands CGPE5 elements of the same mesh density are required. Our early study about element mesh density [17] showed that such mesh density is sufficiently fine to avoid any artifacts. All the
simulations are performed with ABAQUS software package. 3.3. Results of the FE-based simulations Due to brittle nature of the matrix and assumed perfect interface, the damage initiation can
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Fig. 11. Fiber size distribution, the first and second nearest neighbor function of fibers in the type-C window at three typical locations.
be assumed to occur in the matrix, and controlled by distribution of the local maximum principal
Fig. 12. Fiber size distribution, the first and second nearest neighbor function of fibers of type-C windows from different realizations of the same model.
stress in the matrix. Therefore, in the present FE simulations, the distribution of local maximum principal stress is of interest. In this section, the FE
Z. Shan, A.M. Gokhale / Computational Materials Science 24 (2002) 361–379 Table 3 Mechanical properties of Nicalon (SiC) fiber, MAS-5 matrix and the composite
E (GPa) m
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Table 4 max Summary of rmax 1 , r1 , r1 =r1 in matrix of window-C and digital image
Fiber (Nicalon (SiC))
Matrix (MAS-5)
Simulated window
(MPa) rmax 1
r1 (MPa)
rmax 1 =r1
190 0.29
80 0.3
Window-C Digital image
224.3 228.46
109.64 109.24
2.05 2.09
Fig. 13. FE mesh overlaid on the simulated window and boundary conditions. The model is embedded in an effective media. (a) A window (b) FE mesh and boundary conditions.
simulations are performed on micro-structural windows of type C (or larger) as such windows can be regarded as micro-structural RVE of the material. Table 4 gives the values of the maximum
principal stress (r1 ), the average maximum principal stress (r1 ), and the ratio of the r1 =r1 in the matrix of the simulated window-C and the large-area digital image of the micro-structure. Table 4 reveals that the average maximum principal stresses are almost identical and the highest maximum principal stress are slightly different in these two windows. However, only one tiny element with higher stress may not cause the final failure of the composite, the extent of matrix having high local stress and the connectivity of the high stress regions are other important factors to cause the failure of composite. Therefore, the comparison of fraction of matrix experiencing high maximum principal stress (r1 ) in each window is more meaningful. The fracture strength of the matrix of this composite depends on the processing conditions, and has an average value about 140 MPa [16]. Therefore, the fraction of the matrix with maximum principal stresses higher than 140 MPa is a critical attribute of significant interest for analyzing the damage initiation in the composite. Fig. 14a compares the distribution of the maximum principal stresses in the matrix of window-C and large-area digital image, where Y-axis is the area fraction of the matrix with maximum principal stress higher than a given value. Inspection of Fig. 14a reveals excellent agreement between the patterns of local maximum principal stresses in the two simulations. Fig. 14b shows the fraction of matrix experiencing a stress higher than a given value for stresses higher than 140 MPa. These plots for the two simulations essentially represent the tail ends of the corresponding plots in Fig. 14a, and again it is observed that for any given stress level the area fractions of matrix with the stress higher than that value are within 5% of each other. Therefore, it can be concluded that the distribution of local maximum principal stress in windowC is representative of the large-area digital image
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Fig. 14. Comparison of the fraction of matrix with high maximum principal stresses of simulated window-C and digital image of CMC.
(Fig. 2). Fig. 15 shows the contour plot of the maximum principal stress in the matrix of window-C and large-area digital image. Note that higher maximum principal stresses are present in the regions where fibers are clustered and very closely spaced. These sites are the weak areas of the composite, where the cracks are likely to initiate. Observe that the elements experiencing higher stresses are more or less linked together, which is deleterious to the fracture resistance. These critical elements may form micro-cracks first and cause the final failure of the composite. It is of interest to determine if increasing the window size leads to significantly better agreement between the local stress distributions of the simulated micro-structural window and largearea digital image. Fig. 16 shows the simulated windows C–G, from computer simulated micro-
structure of CMC, which contains two to six layers of fiber-rich and -poor regions, respectively. Note that windows D–G are larger than window-C. Table 5 gives the values of the maximum principal stress (r1 ), the average maximum principal stress (r1 ), and the ratio of the r1 =r1 in the matrix of the five simulated windows and large-area digital image of CMC, showing that both the average maximum principal stress and the highest maximum principal stress does not vary significantly with scale of the simulated window, and are close to those in the matrix of the large-area digital image of CMC. The maximum principal stress distributions in the matrix of the five simulated windows and large-area digital image of CMC are compared in Fig. 17 with the focus on the higher stress part (Fig. 17b). The fraction of matrix experiencing a stress higher than a given value is similar for all windows. Therefore, window-C can be regarded as an RVE for FE simulations of damage initiation, provided the stress distributions are comparable for displacement control and traction control simulations, and they do not vary with the location of the window, or from one realization of the model to another realization of the same model, are unique for a given loading direction. Fig. 18 compares the distribution of the maximum principal stresses ðr1 ) with displacement and traction controlled boundary conditions. These data show that the differences in the distribution of the maximum principal stresses ðr1 Þ for these two conditions are very small at any stress level. The total strain energy of the matrix with the displacement and traction controlled boundary condition are also investigated, and they are 5:179 109 and 5:152 109 J/m3 , respectively. Fig. 19 shows the contour plots of the maximum principal stress (r1 ) in the matrix of type-C windows at three different typical locations in Fig. 10. Fig. 20 compares the distribution of the maximum principal stresses in the matrix for the three type-C windows. The results show that distributions of the maximum principal stresses in the matrix for the type-C windows at three locations are almost identical in the entire stress range. The contour plots of the maximum principal stress (r1 ) in the matrix of another type-C window obtained
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Fig. 15. Contour plot of the maximum principal stresses r1 (MPa) in a simulated window-C and larger area digital image.
from a different computer simulation (realization) of the same model is shown in Fig. 19(d). Fig. 21 compares the distribution of the maximum principal stresses in the matrix of the two windows from different realizations at similar location (Fig. 10(a) and (d)). Observe that the difference between the stress distributions for these two simulations is very small in the entire stress range. Table 6 also lists the values of the maximum principal stress (r1 ), the average maximum principal stress (r1 ), and the ratio of the r1 =r1 in the matrix of the type-
C windows at the three typical locations and from different realizations of the same model, showing that the average and highest maximum principal stress does not vary significantly among these typeC windows. It is now shown type-C is a window of sufficiently small size (250 fibers) for which (i) microstructure is statistically similar to that of large-area digital image, and it does not vary (in statistical sense) with the location or different realizations of the model, (ii) the distribution of local maximum
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Fig. 16. Simulated windows (C–G).
Table 5 max Summary of rmax 1 , r1 , r1 =r1 in matrix of various windows and digital image
Window-C Window-D Window-E Window-F Window-G Digital image
rmax (MPa) 1
r1 (MPa)
224.3 228.98 227.51 224.37 230.03 228.46
109.64 109.02 108.83 108.99 109.14 109.24
rmax 1 =r1 2.05 2.10 2.08 2.06 2.10 2.09
principal stress in the matrix is in good agreement with the corresponding distribution for the largearea digital image, and (iii) the micro-stress distribution does not vary significantly with location, or for different realizations, or for different type of boundary conditions (traction versus displacement control). Therefore, it is tempting to regard such a window as an RVE for micro-mechanical simula-
Fig. 17. Comparison of the fraction of matrix with high maximum principal stresses of simulated windows C–G and digital image of CMC.
tions. However, there is yet another factor (that arises due to non-uniform spatial fiber distribution) that needs to be examined before arriving at such a conclusion. All the earlier two-dimensional RVE studies on the fiber composites have been performed on the materials that are locally and globally transverse isotropic, for example, transverse loading behavior of an aligned continuous fiber ceramic composite having uniform random spatial distribution of fibers [18]. However, although the present material is locally transverse isotropic, it is not globally transverse isotropic due to layered nature of the fiber-rich and -poor regions present in the microstructure (Fig. 2), which is typical of many real composites. In such materials, the local stress (and
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Fig. 18. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) of window-C under traction and displacement controlled boundary condition. (a) Full scale, (b) tail part.
strain) distributions (but not the modulus) are expected to be a function of loading direction. Therefore, it is of interest to determine, if the size (of type-C window) that yields a statistically similar local stress distribution for different window locations, type of boundary conditions, and realizations of the models, varies with the loading direction or not. Fig. 22 shows micro-stress distribution plot for the two simulations on the same window of type C but with different loading directions (X and Y as shown in Fig. 13). The applied load (118 MPa), the boundary conditions, and all other simulation parameters are the same as before. Recall that in all the earlier simulations discussed the simulated load has been applied
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along X-direction (Fig. 13). The distribution of the maximum principal stresses (r1 ) in these two cases as shown Fig. 22 indicates that the micromechanical response (maximum principal stress) of the composite is quite sensitive to loading direction, as expected. Therefore, it is of interest to determine if the optimal size of type-C window is significantly different when the load is applied along Y-direction. To test this, all the previous simulations were repeated with Y-direction as the loading direction. It was observed that, the local stress distributions did not vary with the location of the window, or for different realizations of the model, or with the type of boundary conditions, etc. Therefore, it is concluded that the optimal size of the type-C window does not depend on the loading direction. Although the local stress distribution varies with the loading directions, for a given loading direction the local stress distribution does not depend on window location, type of boundary conditions, etc. It is now shown that (i) micro-structure of window-C is statistically similar to that of the large-area digital image, and it does not vary (in a statistical sense) with the location or for different realizations of the model, (ii) the distribution of local maximum principal stress in the matrix of the window-C is in good agreement with the corresponding distribution for the large-area digital image, (iii) the micro-stress distribution of type-C window does not vary significantly with location, or for different realizations, or for different type of boundary conditions (traction versus displacement control), and (iv) the micro-mechanical response of type-C window is unique for a given loading condition. It is concluded that the type-C window can be regarded as RVE for simulation of local stress (and strain) distributions and damage initiation, although the simulated local stress distribution strongly depends on the loading direction. In this contribution, a methodology has been developed to arrive at a sufficiently small microstructural window that can be regarded as RVE for the simulation of micro-mechanical response that can be used as an input to simulate damage initiation. Note that all earlier RVE studies were only concerned with arriving at a micro-structural window that yield the same elastic properties (Young’s
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Fig. 19. Contour plot of the maximum principle stress r1 (MPa) of matrix of various type-C windows.
modulus) as that of a macro-size specimen, and were not concerned with local stress distributions and damage initiation. Earlier RVE studies ignored the window-to-window statistical variations in the size distribution and spatial arrangement of features. All the earlier studies involved RVEs for uniform random micro-structures and did not account for local stress distributions. Such RVE models are not useful for real composites having non-uniform spatial arrangement of the fibers, nor are they useful for simulation of damage initiation in composites having uniform random or nonuniform distribution of fibers. It is realized that residual stresses caused by manufacturing processing (i.e., cooling down from processing), interfacial properties of fiber/matrix interfaces, and
presence of processing defects in the composites may also influence the local stress/strain distribution [19], and consequently, the size of the RVE. If experimental data on these attributes is available, then the present methodology can be used to arrive at sufficiently small RVE size by including these attributes in the computer simulated micro-structure model. Note that the RVE for the present non-uniform micro-structure (i.e., window-C) contains about 250 fibers, whereas large-area digital image of the CMC (Fig. 2) contains about 2000 fibers. Therefore, it is significantly more efficient to use type-C window for FE simulations of macro-size specimens. An important utility of RVE size is for FEbased simulation of macro-mechanical response of
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Fig. 20. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) of type-C window at three typical locations. (a) Full scale, (b) tail part.
Fig. 21. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) between window-C and C0 . (a) Full scale, (b) tail part.
components, where a homogenized continuum in assumed. For such simulations, the FE size must be larger than the RVE for the material, so that each FE has the same constitutive behavior as a macrosize specimen of the composite. For the present composite the absolute RVE size (for fibers having 14 lm average size) is 0.1 mm2 . Therefore, for simulation of macro-mechanical response of a component the element size should be larger than 0.1 mm2 area. It is important to point out that FE simulations are scale independent. Therefore, the ratio of the absolute size of the RVE and average length scale of micro-structure (in this case average fiber size) should be the same for micro-structures that differ only in scale. Therefore, for microstructures that differ from the present composite
Table 6 max Summary of rmax 1 , r1 , r1 =r1 in matrix of various type-C windows Simulated window
rmax (MPa) 1
r1 (MPa)
rmax 1 =r1
Window-C (i) Window-C (ii) Window-C (iii) Window-C0 (i)
224.3 223.3 223.9 225.9
109.64 109.41 109.36 110.63
2.05 2.04 2.05 2.04
micro-structure only in scale and has say 28 lm average fiber size, the absolute RVE area should be about 0:4–0:5 mm2 (about 0:7 mm 0:7 mm), which is not so small. In general, for a given microstructure, the size of RVE is expected to vary with the variations in the constitutive equations of the
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4. Summary and conclusions
Fig. 22. Comparison of the fraction of matrix with high maximum principal stresses r1 (MPa) of window-C at different loading direction. (a) Full scale, (b) tail part.
phases involved. For example, in the present case, the local stress (or strain) distributions are strong function of elastic modulii of the fibers and the matrix, and consequently, the size of the RVE is expected to depend on the ratio of the elastic modulii of fibers and matrix. The RVE model can be used for FE-based simulation of mechanical response of real complex micro-structures. Further, one can then change sizes and number density of fibers (or any other attributes) in the micro-structural model to reflect the changes in the processing conditions and raw materials. Then FE-based simulations on the RVE model can be repeated on the new structure to gauge the effect of changes in the micro-structure (and therefore, changes in the processing conditions) on the mechanical response of the material.
A methodology has been developed to arrive at a sufficiently small micro-structural window that can be regarded as RVE of a non-uniform microstructure of a composite containing a range of fiber sizes, and fiber-rich and -poor regions at the length scale of about 100 lm. The methodology involves quantitative characterization of microstructure, micro-structure modeling, FE-based simulations on computer simulated micro-structural widows of different sizes containing 60–2000 fibers, and FE simulations on large-area highresolution digital image of the composite microstructure containing about 2000 fibers. The RVE has the micro-structure that is statistically similar to that of the composite material having fiber-rich and -poor regions, its Young’s modulus is very close to that of the composite, and has local stress distribution that is comparable to that in the real composite under similar loading conditions. Therefore, such an RVE is useful for realistic simulations of damage initiation, based on which parametric studies can be performed by repeating FE-based simulations on the RVE model with different sizes and number density of fibers to reflect the changes in the micro-structure (and therefore, in the processing conditions) on the mechanical response of the materials. Acknowledgements This research work is a part of NSF sponsored project DMR-9816618. Dr. B. MacDonald is the project monitor. The financial support of NSF is gratefully acknowledged. References [1] R. Hill, J. Mech. Phys. Solids 11 (1963) 357–372. [2] W.J. Drugan, J.R. Willis, J. Mech. Phys. Solids 44 (1996) 497–524. [3] H.J. B€ ohm, F.G. Rammerstorfer, F.D. Fischer, T. Siegmund, J. Eng. Mater. Technol., Trans. ASME 116 (1994) 268–273. [4] H.E. Deve, Metall. Mater. Trans. A 30 (1999) 2513–2522. [5] C. Zweben, Composites 25 (1994) 451–454. [6] M.R. Wisnom, Compos. Struct. 18 (1991) 47–63.
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