Author’s Accepted Manuscript Investigations on the Compressive Behavior of 3D Random Fibrous Materials at Elevated Temperatures Datao Li, Wei Xia, Wenshan Yu, Qinzhi Fang, Shengping Shen www.elsevier.com/locate/ceri
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S0272-8842(17)30054-8 http://dx.doi.org/10.1016/j.ceramint.2017.01.044 CERI14506
To appear in: Ceramics International Received date: 25 December 2016 Revised date: 7 January 2017 Accepted date: 9 January 2017 Cite this article as: Datao Li, Wei Xia, Wenshan Yu, Qinzhi Fang and Shengping Shen, Investigations on the Compressive Behavior of 3D Random Fibrous Materials at Elevated Temperatures, Ceramics International, http://dx.doi.org/10.1016/j.ceramint.2017.01.044 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Investigations on the Compressive Behavior of 3D Random Fibrous Materials at Elevated Temperatures Datao Li, Wei Xia, Wenshan Yu, Qinzhi Fang, Shengping Shen*
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China *
Corresponding author.
[email protected]
Abstract
By means of the experimental method, micromechanical model and Finite Element Method (FEM), this paper studied the compressive behaviors of the three-dimensional random fibrous (3D RF) material in the through-the-thickness (TTT) and in-plane (IP) directions at elevated temperatures. The compressive experiments showed that the fracture strength and Young’s modulus of the 3D RF material in the TTT and IP directions decrease as increasing temperature. The specimens fracture through breaking the fibers under the bending deformation, while almost all the bonding zones keep intact. A simple micromechanical model and a FEM model are developed to simulate the mechanical properties of the 3D RF material. The micromechanical model ignores the randomness of the fibers, while in the FEM model special attention is drawn to the influence of the morphological characteristic. Numerical results from the micromechanical model and FEM model agree well with the observations from the compressive experiments.
Keywords
Random fibrous materials (RF materials); Elevated temperature; Compressive strength; Micromechanical model; FEM model
1. Introduction
Due to the high porosity, low weight, high specific surface area and excellent thermal properties [1-6] inorganic random fibrous (RF) materials with three-dimensional (3D) bonded networks have been extensively used in a variety of fields, such as building, filtration and thermal insulation of space shuttle [3, 4, 7, 8]. Inorganic fibers in 3D RF materials are bonded by high temperature binder, lapped and connected with each other, forming a stable 3D fibrous networks [9]. The mechanical properties of 3D RF materials are very important during their use, for instance, the performance of the thermal insulators is decreased when the bonded fiber networks are destroyed under high compression [10]. Numerous studies have been investigated the mechanical properties of the 3D RF materials through experimental method at room temperature. Such as the compressive strength of 3D RF materials in the in-plane (IP) direction is stronger than that in the through-the-thickness (TTT) direction and increases as increasing the strain rate. However, the modulus of 3D RF materials is somewhat insensitive to the strain rate [11]. Compressive experiments of lightweight ceramic ablators (a kind of 3D RF material) indicate discontinuous breaking of bonds between the fibers and buckling of fibers [12, 13]. The compressive analysis of the two kinds of alumina based fibrous ceramics (two kinds of 3D RF materials) are conducted to investigate the effect of the kind of high temperature binder and their contents on physical and mechanical properties of the sample. For the alumina fibrous ceramics, the compressive strength increases with the increase of silica sol concentration due to the increasing amount of the silica
phase [9]. Contrary to the monolithic ceramics which exhibit a brittle fracture, 3D RF ceramics present a dissipative damage tolerant behavior [14-17]. The compressive strength of 3D RF materials depends on the sintering temperature, in the sintering process the suitable additive is chosen to lower the sintering temperature [18]. A smaller number of studies focus on mechanical behaviors of the 3D RF materials at elevated temperatures, such as high-temperature toughness behaviors of FRCI-12 (a kind of 3D RF material with 78 wt% of the amorphous silica fibers, plus 22 wt% of the aluminoborosilicate fibers and the density is 192 kg/m³) and HTP-6 (a kind of 3D RF material and the density is 96 kg/m³), the strong behavioral change of two kinds of 3D RF materials at elevated temperatures is linked to the glass-softening temperature of the multiphase structure characteristic [5]. Moreover, the 3D RF materials show a much high degree of elasticity because of the bending of the fibers. The rebound resilience decreases with the increase of the temperature [19].
Besides the efforts on the experiments of 3D RF materials, also some research has been performed on developing theoretical models and FEM models to investigate the mechanical properties of 3D RF materials at room temperature. Such as, a number of theoretical models based on affine approximations of fiber networks have been proposed to study the mechanical properties of 3D RF materials [20-22], in these works the fiber in bulk material is considered as an end-loaded cantilever beam under the influence of an applied force. The derived theoretical model shows the Young’s modulus of the 3D RF material is related to the fiber modulus, fiber volume fraction, and fiber length and diameter. The fibrous randomness is not shown in the theoretical model. In addition, Green et al. [23] presented a simplicity micromechanical model for 3D RF ceramic bodies to
investigate the mechanical properties. It is found that properties such as the elastic modulus, fracture toughness, and strength depend on the fiber spacing.
Compared to theoretical models, Finite Element Method (FEM) modeling encourages us to set up the relationship between macro-properties and fibrous randomness of RF materials [24-27]. For instance, by using FEM modeling many efforts have been invested to evaluate the mechanical properties of RF materials including the onset and propagation of damage in fiber and bonding points and the effects of orientation distribution function in low-density RF materials [28]. 2D FEM model based simulations have been applied to investigate the elastic-plastic behavior of porous metal fiber-sintered sheets and flexible fiber-fiber bonding [29-32]. A further advance in FEM modeling was made by Lu et al. [4]. They established a 3D FEM model to explore compressive performance of 3D RF materials. In their works beam element was used to model silica, mullite fibers and the bonding materials between the bonded fibers, which compose the RF material. The morphology of 3D RF materials observed in the experiments was successfully used in the model. Their simulated strength and modulus were in good agreement with the experimental results at room temperature. Moreover, the compressive behaviors (the effective Young’s modulus and compressive strength) of two types of porous silicon nitride ceramics (kinds of 3D RF materials) with different porosities were also well simulated using 3D FEM models at room temperature [33].
So far, numerous efforts have been invested into the study of 3D RF materials, most of studies were only performed at the room temperature. 3D RF materials exhibit many intrinsic properties which make them promising high-temperature structural materials [19]. However, we found that investigations on mechanical behaviors of 3D RF materials at elevated temperatures were still
under-researched. In our previous study [34] we investigated the tensile behaviors of the 3D RF material at elevated temperatures using experimental method and a developed 3D FEM model. Numerical results from the 3D modeling are in good agreement with the experimental observations. Facing to tensile behaviors of the 3D RF material, we may think about a question, i.e., how do the compressive behaviors of 3D RF materials change at elevated temperatures? In this study, we will attempt to answer it using experimental method, micromechanical model and FEM modeling. This study is organized as follows: we performed the compressive test of the 3D RF material in two directions, i.e., TTT (fifteen specimens) and IP (fifteen specimens), at the elevated temperatures ranging from 299 K to 1273 K (five temperatures: 299 K, 573 K, 773 K, 1073 K and 1273 K). The compressive response of the 3D RF material in the two directions, as well as the compressive strength and Young’s modulus at different temperatures are obtained. To reveal the temperature dependence of material properties, then a micromechanical model and a 3D FEM model are adopted in this study. Finally, we compare the results obtained from experiments, theoretical models and FEM models. It is found that they are in agreement with each other.
2. Experiment
2.1. Materials
In the present work the 3D RF material was sintered by using substantial high purity silica fibers (purity ≥ 99.95%) and few mullite fibers. The detailed process for sintering of the RF material can be found in Ref. [3, 4, 34]. The average density and porosity of the 3D RF material are 0.28 g/cm³ and 87%, respectively. The density of the 3D RF material is defined as its mass divided by its bulk volume, and determined by its exterior dimensions. The average length and diameter of
fibers in the 3D RF materials are about 0.6 mm and 0.1 mm. In the TTT direction, as shown in Fig. 1a, the fibers distribute randomly and have high porosity. In the IP direction, as shown in Fig. 1b, the fibers have preferred orientations and low porosity. These morphological characteristic affects the mechanical properties of the 3D RF materials in the two directions.
Fig.1 The scanning electron microscope (SEM) images of the 3D RF material in the (a) TTT and (b) IP directions
2.2. Experiment devices
Using a universal test machine (Z005, Zwick Roell, Germany) and a 1473 K high temperature mechanics test system (GW 1200, Changchun Fangrui Technology CO. LTD, China), the compressive tests for specimens are conducted at the loading rate of 1 mm / min through the displacement control mode. Before the compressive tests are performed, the test temperature in the furnace is held 20 minutes (except room temperature (299 K)). During the testing, we use a high temperature furnace extensometer (Model 3448, Epsilon Technology Corp, USA) to measure the displacement of the specimens.
2.3. Compression specimen and fixture
Two groups of compression specimens are fabricated from 3D RF bulk materials in the TTT and IP directions with cylinder-shaped. The diameter and height of the cylinder-shaped specimen are 20 and 30 mm (height-dimension ratio of the specimen is 1.5), respectively. The specimen and fixture assembled in the furnace are shown in Fig. 2.
Fig. 2 Specimen in furnace
2.4. Experimental results
During the compressive testing, the random distribution of fibers in the 3D RF material imparts the variations in the mechanical response of 3D RF material. Figs. 3a and 4a show only some typical stress-strain curves in the TTT and IP directions, respectively, at the different temperatures. In the two figures, the associated stress-strain curve is calculated based on initial geometric specimen dimensions, i.e., the stress is calculated as force divided by initial circular area of the specimen and strain by dividing displacement with initial height of the specimen. For the TTT direction, the curves before 773 K can be divided into three stages: a linear stage, a plateau
stage and a compaction stage. In the linear stage, compressive stress increases linearly with applied compressive strain. The maximum linear strain (compressive strength defined as elastic limit) is about 4.6% which is equivalent to the ceramic matrix composites reinforced with continuous fibers (about 3-9%) [19, 35, 36]. The curves experienced a plateau stage at the elastic limit up to an applied strain of about 10%. The events recorded on the stress-strain curves may be related to failures of fibers or necks. At the last stage, the compressive stress increased because of a gradual densification of the structure (see Fig. 3b). The fracture fragments of the fibers and the glassy phase at the compaction stage stacked together [19]. From 1073 K to 1273 K, there are only two stages in the stress-strain curves: a linear stage and a compaction stage. The elastic limit is not clear and the softening phenomenon is markedly observed. Especially at 1273 K, the linear stage is short. The densified structure induces a significant increase of the compressive stress. For the IP direction, from the room temperature to 1273 K, the fracture of specimens is typical brittle. There is only a linear stage. For example, at room temperature the compressive stress increases linearly with applied compressive strain. When the stress reaches the peak, the specimen fractures brittlely as shown in Figs. 4a and b.
Fig. 3 (a) Stress-strain curves of the TTT direction at different temperatures and (b) compaction of the specimen at room temperature
Fig. 4 (a) Stress-strain curves of the IP direction at different temperatures and (b) brittle fracture of the specimen at room temperature
Further inspection of stress-strain curves reveals that the fracture strength of 3D RF material in the TTT and IP direction doesn’t decrease as increasing the temperature. For the two directions the strength variation with temperature has almost the same trend (Fig. 5). Moreover, the fracture strength of the IP direction is almost three times greater than that of the TTT direction. The
differences can be attributed to the different distribution of fibers in the two directions. In the IP direction there are more fibers to bear the load than those in the TTT direction. Therefore the fracture strength of the IP direction is greater than that of the TTT direction. When the temperature varies from room temperature to 1073 K, the strength fluctuates but does not decrease. At 1273 K the strength decreases about 52% and 26% compared with the strength at room temperature in the TTT and IP directions, respectively. The temperature-dependent modulus (Fig. 6) of the IP direction decreases linearly with the temperature, while in the TTT direction the modulus has some increase from room temperature to 573 K beyond which the modulus decreases.
Fig. 5 Compressive strength of the TTT and IP directions at different temperatures. Each point represents the average strength of three specimens. The error bar for each point is the standard deviation.
Fig. 6 Compressive modulus of the TTT and IP directions at different temperatures. Each point represents the average modulus of three specimens. The error bar for each point is the standard deviation.
3. Micromechanical model
In order to analyze the mechanical properties of 3D RF materials theoretically, a micromechanical model is developed based on Ref. [23]. It is assumed that the 3D RF material can be modelled by an orthotropic network of fibers (Fig. 7). Fig. 7 shows one possible until cell which varies in the model. The random fibers are considered aligned parallel to the axis. In the model, the orthogonal fibers are jointed at the nodes of the network. It is expected that elastic properties of the model will be determined by the fiber spacings ( Di , i
D1
D2
1, 2,3 ). For the 3D RF material, we expect:
D3
where D1 , D2 and D3 are the fiber spacings in the x, y and z directions, respectively.
(1)
Fig. 7 Micromechanical model of the 3D RF material [23] It is reasonable to assume that Young’s modulus in the IP and TTT directions depend on the area fraction of fibers aligned in the corresponding directions, i.e.
E2
R2 E f / ( D2 )2 , E3
R2 E f / ( D3 )2
(2)
where E f is the modulus of the fiber at room temperature, R is the fiber diameter. By assuming the spacings ( Di , i
1, 2,3 ) are independent of temperature, the temperature-dependent modulus of
the 3D RF material can be written as:
E2 (T )
R2 E f (T ) / ( D2 )2 , E3 (T )
R2 E f (T ) / ( D3 )2
(3)
where T is absolute temperature. Based on the temperature-dependent strength model developed by Li and Fang [37] and Eq.(3), the temperature-dependent strength of the 3D RF material in the IP and TTT directions can be obtained as: 1/2
1/2
T T E f (T ) E f (T ) 1 1 0 2 (T )= 1 Tm C p (T )dT 2， 3 (T )= 1 Tm C p (T )dT 30 E E f C p (T )dT 0 f C p (T )dT 0 0 0 (4)
0 where 2
0 and 3 are the fracture strength of specimen in the IP and TTT directions at
absolute temperature zero. Tm is the melting temperature. C p (T ) is the specific heat capacity for constant pressure, and is fitted as:
C p (T ) A1 A2 103T A3 103T 2 A4 106 T 2 A5 108 T 3
(5)
All coefficients in Eq. (5) can be found in Ref. [34]. The subscripts 2 and 3 of the temperature-dependent strength 2 (T ) and 3 (T ) represent the IP and TTT directions (y and z directions), respectively.
4. FEM model
4.1. Modeling method
To perform FEM modelling of the 3D RF material in this study, a 3D FEM model is adopted in this study. The detailed modeling and calculating processes are shown in the Appendix. All the simulations are achieved by using ANSYS software. In the FEM model, Beam 188 element is used to model high purity silica fibers. There are about 57100 nodes and 60315 elements (average nodes and elements of four FEM models) in the FEM model. All fibers in the model are considered to be isotropic materials, then there are only two independent material parameters in their constitutive relationships. In addition, fibers are fragile, thus the maximum principal stress criterion is used to predict the onset of damages [3]. In the present analysis, we just consider the temperature-dependent elastic modulus and fracture strength, while the Poisson’s ratio [4] is independent of the temperature. During the testing, the thermal expansion of the fibers occurs as increasing the temperature. The coefficient of linear thermal expansion of silica fibers is
0.5 106 K1 [38] and assumed to keep unchanged from room temperature to 1273 K. By doing so,
we are able to simulate the mechanical behaviors of the 3D RF material at elevated temperatures. A FEM model of the representative volume element (RVE) created by using the procedure in the Appendix is shown in Fig. 8, and the compressive loadings are applied in the TTT and IP directions that align with z and y directions. In the simulation, the constitutive equation of fibers is:
=ET ( f T ) , T =T -Tro where Tro is the room temperature, f is the coefficient of linear thermal expansion of fibers.
(6)
Fig. 8 The FEM model of the RVE: (a) fiber distribution in the TTT direction, (b) fiber distribution in the IP direction. The cyan solids represent silica fibers.
4.2. Mechanical properties of silica fibers
The elastic modulus and fracture strength of the high purity silica fibers at room temperature are 78 GPa and 3.6 GPa [39] respectively. In addition, the temperature-dependent modulus of silica fibers can be found in Ref. [40] and can be fitted by Wachtman’s equation [41]: ET E0 BT exp(
T0 3579 ) 78 0.341T exp( ) T T
(7)
where E0 is the elastic modulus at absolute zero temperature, B and T0 are fitting constants of the material. In this study, E0 , B and T0 are 78 GPa, 0.341 and 3579 K, respectively. The temperature-dependent strength of the silica fibers is given by Ref. [37]:
E th (T )= T E 0
1/2
T 1 0 1 0 C p (T )dT th Tm C p (T )dT 0
(8)
0 where th is the fracture strength of silica fibers at absolute zero. Based on Eqs. (7) and (8), the
elastic modulus and fracture strength of silica fibers at different temperatures are shown in Fig. 9.
Fig. 9 Fracture strength and elastic modulus [40] of silica fiber at different temperatures
5. Simulations and discussions
5.1. Compressive strength of the TTT direction
The compressive deformation of the specimen is mainly attributed to the elastic bending of fibers which can be proved by the simulated results from the FEM model. For instance, the compressive displacement (0.045 mm) was applied in the -z direction (see red arrows in Fig. 10a) of the FEM model, the principal stress contours are shown in Fig. 10a. We can find most zones of the model are in low stress levels (from 0 to 392.257 MPa). In order to determine the location of the maximum principal stress, a local FEM model (Fig. 10b) is sliced from the red rectangle in Fig. 10a. The maximum principal stress (3530.31 MPa) locates near the red arrow in Fig. 10b. The principal stress contours of the local structure with two fibers and a bonding zone show the maximum stress appears on the fiber and the fiber bears bending deformation (see Fig. 10c). Our earlier study also shows that the specimens fracture through breaking the fibers, while almost all the bonding zones keep intact when the specimens are applied tensile displacement [34].
Fig. 10 Principal stress contours (the strain is about 0.05) of (a) the RVE model, (b) the sliced model from the red rectangle in (a), and (c) the local structure at room temperature.
To evaluate the simulated results, the experimental results are compared with the results from the micromechanical model and the FEM model. Fig. 11 exhibits the compressive strength from experimental results, FEM model and micromechanical model in the TTT direction at elevated temperatures. In this figure it is clearly found that the strength from the micromechanical model agrees well with experimental results at room temperature and 1273 K. Between 573 K and 1073 K, the experimental results are higher than the results of micromechanical model. Compared to the micromechanical model, the simulated results from the FEM model agree well with the experimental results between room temperature to 1073 K. At 1273 K, the simulated results are higher than the experimental results.
Fig. 11 Compressive strength of the TTT direction. Simulated strength from the FEM model is the average data from four models. The error bar for each point is the standard deviation.
5.2. Compressive modulus of the TTT direction
The compressive modulus of the TTT direction increases from the room temperature to 573K beyond which the modulus continuously decreases (Fig.12). Using the micromechanical model and the FEM model, the simulated modulus agreed with the experimental results from room temperature
to 1073 K. At 1273K the simulated modulus is higher than the experimental one. This may be attributed to the softening and viscous flow of the bonding zone between the fibers.
Fig. 12 Compressive modulus of the TTT direction. Simulated modulus from the FEM model is the average of four models. The error bar for each point is the standard deviation.
5.3. Compressive strength of the IP direction
In the IP direction, the compressive displacement (0.0195 mm) is applied in the -y direction of the FEM model, the principal stress contours are shown in Fig. 13a. The maximum principal stress (3722.72 MPa) is located at a local model (Fig. 13b) sliced from the red rectangle in Fig. 13a. We can find the maximum principal stress at the location of the red arrow (Fig. 13b). The principal stress contours of the local structure with three fibers and two bonding zone shows that the maximum stress (3372.72 MPa) appears on the fiber and the fiber also bears the bending deformation (see Fig. 13c).
Fig. 13 Principal stress contours (the strain is about 0.025) of (a) the RVE model, (b) the sliced model from the red rectangle in (a), and (c) the local structure at room temperature.
The sintering temperature plays an important role in the 3D RF material. When the sintering temperature decreased to room temperature, this process was often accompanied by a 3-5% volume change and might generate thermal stress, resulting in cracking of the bonding zone, which might affect the compressive strength of the specimens [14, 42], such as fluctuations of the experimental strength at elevated temperatures ranging from room temperature to 1273 K (see Figs. 11 and 14). The strength of the IP direction from the micromechanical model agrees with the experimental results at room temperature and 773 K and decreases linearly with the temperature (Fig. 14). At room temperature, 773 K and 1273 K, the simulated strength from the FEM model fits well with the experimental results (Fig. 14).
Fig. 14 Compressive strength of the IP direction. Simulated strength from the FEM model is the average of four models. The error bar for each point is the standard deviation.
5.4. Compressive modulus of the IP direction
Fig.15 exhibits the compressive modulus from experiments and simulations of the IP direction at different temperatures. In the figure it can be clearly found that the modulus from the micromechanical model and FEM models fitted well with the experimental results, especially at
room temperature, 573 K and 1073 K. In addition, the simulated modulus from the micromechanical model and FEM model has the same trend.
Fig. 15 Compressive modulus of the IP direction. Simulated modulus from the FEM model is the average of four models. The error bar for each point is the standard deviation.
6. Conclusions
In this study, using the experimental method, micromechanical model and FEM simulation, we demonstrate the mechanical behaviors (compressive strength and Young’s modulus) in the TTT and IP directions of the 3D RF materials at elevated temperatures. The conclusions are summarized as follows:
(1) The experimental results show the compressive strength of the 3D RF material in the TTT and IP directions almost has the same trend from room temperature to 1273 K. At 1273 K the compressive strength decreases about 52% and 26% compared with the compressive strength at room temperature in the TTT and IP directions, respectively. The elastic modulus in the IP direction decreases linearly from the room temperature to 1273 K. However, in the TTT direction the
modulus increases firstly from room temperature to 573 K, then decreases rapidly from 573 K to 1273 K. The bending deformation of fibers plays a decisive role in the fracture process of the specimens.
(2) A simplified micromechanical model is developed to analyze the temperature-dependent strength and modulus of 3D RF materials. Using fiber spacings obtained from the elastic modulus of specimens at room temperature, and temperature-dependent modulus of fibers which compose the 3D RF material, the micromechanical model can predict the mechanical behaviors of the 3D RF material at elevated temperatures.
(3) Based on the critical model parameters obtained through the detailed SEM photo analyses, a 3D FEM model, which reveals the meso-structure of the 3D RF material and incorporates the temperature effects, is used to simulate the compressive performance in ANSYS software. The simulated results agreed well with the experimental results. The failure mechanism of fibers in the specimens can be verified in the FEM model.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11632014, 11372238, 11302161, 11302162 and 11502191), the Chang Jiang Scholar program and China Postdoctoral Science Foundation (Grant No: 2015M580836).
Appendix
Here, we make a brief introduction of the FEM modeling. Based on the morphological characteristic of the 3D RF material in Fig. 1, we assume that the orientation of each fiber can be
determined by two angles and shown in Fig. 16. Two angles and are taken from a Gaussian distribution function G(0, ) with the mean value of zero and the standard deviation and a random function R(0, 2 ) , respectively. Two endpoints of the fiber in the representative volume element (RVE) (see Fig. 16) are K0 ( x0 , y0 , z0 ) and K1 ( x1 , y1 , z1 ) .
Fig. 16 Position of a fiber in RVE In the present study, the standard deviation in Gaussian distribution function G(0, ) is determined to be 21° from the experimental observation. The dimensions of the RVE model are
1.5Ly 1.3Ly 1.5Ly , where Ly is the size in the IP direction and equals fiber length L . Then, coordinates of two endpoints of a fiber in the RVE can be expressed as:
x0 R(0,1.5L) y0 R(0,1.3L) z R(0,1.5L) 0 x1 x0 Lcos( )cos( ) y1 y0 Lcos( ) sin( ) z z Lsin( ) 0 1
(9)
(10)
where R(0,1.5L) and R(0,1.3L) are the random functions in the range of 0 to 1.5L and 1.3L , respectively. The flowchart for modeling and calculation is shown in Fig. 17. The porosity P, fiber diameter D, fiber length L and the standard deviation are known parameters in the program. Then, the amount of fibers N is calculated via [3]:
N
f
4V (1 P) D2 L
(11)
where V is the volume of the RVE, f is a proportionality coefficient and can be changed in the modeling process. Then all the fibers are generated whether they are in or out of the RVE. The periodic boundary conditions (PBCs) are applied to RVE model. The implement of PBCs are described in the Ref. [4]. In the analysis, we make some simplifications, such as the variation of the fiber length and the fiber flexure are neglected, and all fibers are assumed to have uniform length L (0.6 mm) and diameter D (0.01 mm).
Fig. 17 Flowchart of algorithm for FEM model setup and calculating
In the RVE two fibers are bonded together when the mutual distance of two fibers evaluated using the common perpendicular line is between D and 2D. Such a common perpendicular line will be treated as a new fiber by which two fibers are connected [3, 4]. To avoid two fibers overlapping, one of fiber needs to be removed in the model if mutual distance of two fibers is less than
1.0 105 mm . The above procedure may not guarantee the entirely connected network of all fibers. Here, we propose an algorithm to choose the longest fibers group Lmax in order to efficiently exclude those fibers not bonded together. The choosing process could be described as:
Lmax Max( LGi ) , i 1, 2...n,
(12)
where Gi presents a local fibers group containing the ith fiber, LGi is the total fiber length of Gi .
Max( LGi ) presents a function for returning the longest fiber group. n is the maximum fiber
number in the RVE. The longest fibers group Lmax in the RVE is obtained using this approach while other fibers not falling into the longest fiber group will be removed. Then we re-calculate the true porosity of the RVE and judge the true porosity if it meets the desired porosity (the desired porosity P is 87%) within the deviation 0.1% [4]. If all the above conditions are fulfilled, the model then could be used to perform the compressive simulation. Otherwise, the coefficient f needs to be adjusted by adding or subtracting a constant from the absolute difference between the true porosity and the desired porosity, and the program will repeat the above procedure (see Fig. 17). The initial value of the proportionality coefficient f is 1.12.
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