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    High Temperature Ceramic Matrix Composites 8
    High Temperature Ceramic Matrix Composites 8
    High Temperature Ceramic Matrix Composites 8
    Ebook1,495 pages12 hours

    High Temperature Ceramic Matrix Composites 8

    By Dongliang Jiang

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    This proceedings contains 78 papers from the 8th International Conference on High Temperature Ceramic Matrix Composites, held September 22-26, 2013 in Xi'an, Shaanxi, China. Chapters include:
    • Ceramic Genome, Computational Modeling, and Design
    • Advanced Ceramic Fibers, Interfaces, and Interphases
    • Nanocomposite Materials and Systems
    • Polymer Derived Ceramics and Composites
    • Fiber Reinforced Ceramic MatrixComposites
    • Carbon-Carbon Composites: Materials, Systems, and Applications
    • Ultra High Temperature Ceramics and MAX Phase Materials
    • Thermal and Environmental Barrier Coatings
    LanguageEnglish
    PublisherWiley
    Release dateJun 9, 2014
    ISBN9781118933008
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      High Temperature Ceramic Matrix Composites 8 - Litong Zhang

      DESIGN OF NEW GRADIENT CEMENTED CARBIDES AND HARD COATINGS THROUGH CERAMIC GENOME

      Weibin Zhanga, Yong Dua,*, Li Chena, Yingbiao Penga, Peng Zhoua, Weimin Chena, Kaiming Chenga, Lijun Zhanga, Wen Xieb, Guanghua Wenb, and Shequan Wangb

      a State Key Lab of Powder Metallurgy, Central South University, Hunan, 410083, China

      b Zhuzhou cemented carbide cutting tools limited company, Zhuzhou, Hunan, 412007, China

      ABSTRACT

      The concept of ceramic genome is briefly introduced, and its combination with CALPHAD (CALculation of PHAse Diagrams) method is a powerful tool for materials process optimization and alloy design. The quality of CALPHAD-type calculations is strongly dependent on the quality of the thermodynamic and diffusivity databases. The development of thermodynamic and diffusivity databases for cemented carbides is described. Several gradient cemented carbides sintered under vacuum and various partial pressures of N2 have been studied via experiment and simulation. Examples of ceramic genome applications in design and manufacture for different kinds of cemented carbides are shown using the databases and comparing where possible against experimental data, thereby validating their accuracies. Metastable Ti-Al-N coatings have been well acknowledged as protective layer for industrial applications due to their excellent mechanical, chemical and thermal properties. Here, we study the effect of Zr addition on structure and thermal properties of Ti1-xAlxN based coatings under the guidance of ab initio calculations. The preparation of Ti1-x-zAlxZrzN by magnetron sputtering verifies the suggested cubic (NaCl-type) structure for x below 0.6–0.7 and z ≤ 0.4. Alloying with Zr also promotes the formation of cubic domains but retards the formation of stable wurtzite AlN during thermal annealing.

      INTRODUCTION

      Cemented carbides have long been used in applications such as cutting, grinding, and drilling¹. Cemented carbides² are hard and tough tool materials consisting of micrometer-sized tungsten carbide embedded in a metal binder phase, usually rich in Co. Cubic carbides or carbonitrides based on Ta, Ti, and Nb are often added in cemented carbides to increase the resistance to plastic deformation or as gradient formers¹. Some grain growth inhibitors such as Cr and V may also be added in small amounts. In order to increase cutting performance of the cemented carbide inserts, the wear surface are usually coated with a thin layer of hard material³. Due to the high deposition temperature and a large difference in thermal expansion coefficients between the coating and substrate, cracks would be introduced into the coating unavoidably⁴. And the formed cracks might easily propagate into the substrate to cause failure when coating tools are employed in metal machining⁵. In order to prevent crack propagation from the coating into the substrate, a gradient layer, which is free of cubic phases and enriched in binder phase, is introduced between coating and substrate⁵.

      In the past decades, cemented carbides were mainly developed through a large degree of testing. However, there are numerous factors influencing the microstructure and properties of cemented carbides, such as alloy composition, sintering temperature, time and partial pressure and so on. These factors can only be varied in an infinite number of ways through experimental method. In order to shorten development time, reduce the cost and improve outcome, the concept of materials genome has been proposed. Computation, experimentation, and database are identified as three major components of materials genome. The ceramic genome can describe the interaction of the various process conditions, which presents the opportunity to design and produce new kinds of cemented carbides more efficiently. CALPHAD (CALculation of PHAse Diagrams) method is a powerful tool to establish the database of various properties in the ceramic genome. Computational thermodynamics, using, e.g. the Thermo-Calc and DICTRA packages, has shown to be a powerful tool for processing advanced materials in cemented carbides, which is more efficient on composition and process parameters optimization compared with expensive and time consuming experimental methods. With the development of thermodynamic and diffusivity databases, it is possible to make technical calculations on commercial products which are multicomponent alloys. On the basis of thermodynamic database, thermodynamic calculations can give an easy access to what phases form at different temperatures and alloy concentrations during the manufacture process. By combining CSUTDCC1 and CSUDDCC1 databases, DICTRA⁶ permits simulations of the gradient process, which is a major advance in the understanding of the gradient zone formation in the cemented carbides.

      Ti-Al-N hard coatings with cubic (c) NaCl structure, where Al substitutes for Ti in the TiN-based structure (i.e., c-Ti1-xAlxN), are widely used in cutting tools because of their high hardness and wear resistance as well as good thermal properties⁷. Because of the spinodal decomposition of c-Ti1-xAlxN into nano-size cubic Ti-rich and Al-rich domains at elevated temperatures, the age-hardening abilities of Ti-Al-N coatings can improve the mechanical properties of coatings⁸. Due to solid solution strengthening, alloying Zr to Ti-Al-N coating can improve the hardness. Unfortunately, a huge amount of work is needed to find appropriate elements by using experimental method, while first-principles calculations on the investigation of structural and mechanical properties can reduce the workload effectively and provide the reasonable explanation for the experimental observation.

      This paper is devoted to 1) describe the development of the thermodynamic and diffusion databases in cemented carbides, 2) design experiments to investigate the gradient zone formation under different sintering environments, 3) validate the accuracy and reliability of the presently established databases in ceramic genome by comparing the experimental and simulation results, 4)investigate the structural and thermal properties of Ti-Al-Zr-N hard coatings by first-principles calculations and experiment, and 5) present the schematical ceramic genome strategy for the development of new cemented carbides and hard coating.

      ESTABLISHMENT OF THE CERAMIC GENOME

      Development of Thermodynamic and Diffusivity Databases

      The development of thermodynamic and diffusivity databases in cemented carbides has started from the major elements in gradient cemented carbides C-Co-Cr-W-Ta-Ti-Nb-N. The usage of ceramic genome in industry will necessarily require that thermodynamic and diffusivity databases provide data that is not only of high quality, but relevant to industrially complex materials.

      Developed using the CALPHAD approach, the thermodynamic database, CSUTDCC1, is based on critical evaluations of binary, ternary and even higher order systems which enable making predictions for multicomponent systems and alloys of industrial importance. Six important phases in cemented carbides, e.g. liquid, WC, carbides or carbonitrides, binder (Co), and eta (M6C or M12C) phases, are the main focus during the modeling. A large number of additional phases are also included in CSUTDCC1, because many of the binary and ternary systems in CSUTDCC1 have been assessed over their entire composition and temperature ranges. It is important to have reliable descriptions for all the stable phases in the system when developing new alloys and exploring new composition ranges. The thermodynamic models describe the thermodynamic properties of various types of phases depending on the crystallography, order-disorder transitions, and magnetic properties of the phases. With parameters stored in database, many different models⁹, including the substitution solution model, sublattice model, order-disorder model, have been adopted for the phases in cemented carbide systems. The thermodynamic models for Gibbs energy of a phase can be represented by a general equation:

      (1)

      Here refGmθ represents the Gibbs energy of the pure elements of the phase and idGmθ represents the contribution due to the ideal mixing. The term EGmθ represents the excess energy and magnGmθ the magnetic contribution.

      In contrast to extensive efforts on the establishment of thermodynamic database for multicomponent cemented carbides, diffusivities in the multicomponent cemented carbides have received limited investigations both experimentally and theoretically. In a multicomponent system, a large number of diffusivities need to be evaluated, making a database very complex. A superior alternative is to model atomic mobility instead. In this way, the number of the stored parameters in the database is substantially reduced and the parameters are independent. A detailed description for the atomic mobility is given by Andersson and Ågren¹⁰. The atomic mobility for an element B, MB, can be expressed as

      (2)

      where R is the gas constant, T the temperature, MB⁰ a frequency factor and QB the activation enthalpy. Both MB⁰ and QB are in general dependent on composition and temperature.

      The simulation of gradient zone formation is based on the model for long-range diffusion occurring in a continuous matrix with dispersed phases. Due to the presence of dispersed phases (WC, carbides and carbonitrides), the diffusion is reduced in the matrix (the liquid binder phase)¹¹. A so-called labyrinth factor λ(f), where f is the volume fraction of the matrix, was introduced to reduce the diffusion coefficient matrix.

      (3)

      First-principles Calculations of Hard Coatings

      First-principles calculation is one of the theoretical methods to study the microstructure and properties of materials. This method is based on the density functional theory (DFT) with the local density approximation (LDA)¹² or generalized gradient approximation (GGA)¹³. In the present work, first-principles calculations are performed by Vienna ab-initio simulation package (VASP)¹⁴. The electron-ion interactions are described by the projector augmented wave (PAW) method¹⁵, and the exchange-correlation is depicted by GGA and LDA. The energy cutoff of the wave functions is taken as 1.3 times higher than the default values in the psudopotentials. The Monkhorst-Pack scheme¹⁶ of k-points sampling and the linear tetrahedron method including Blöchl corrections¹⁷ are adopted for the integration in the Brillouin zone. The total number of k-points is at least 10,000 per reciprocal atom for all the calculations. The convergence criterion for electronic self-consistency is 10−6 eV per unit cell.

      The quasiharmonic approach is adopted to evaluate the finite-temperature Helmholtz energy as a function of volume V and temperature T¹⁸

      (4)

      where E(V) is the static energy at 0 K without the zero-point vibrational energy, Fvib(V,T) is the vibrational contribution to Helmholtz energy with the input of phonon density of state (DOS), and Fele(V,T) is the thermal electronic contribution to the free energy, which can be calculated by Mermin statistics¹⁹ with input of electronic DOS from first-principles directly.

      EXPERIMENTAL

      The alloys were prepared from a powder mixture of WC, (Ti,W)C, Ti(C,N), and metallic Co powder provided by Zhuzhou cemented carbide cutting tools limited company. The composition of the sintered material is given in Table 1. After milling and drying, the powders were pressed into cutting tool inserts. Samples were dewaxed and sintered under different nitrogen partial pressures (0, 20 and 40 mbar) at 1723 K for 1 h. After sintering the samples were cut, embedded in resin and polished. SEM (Nova NanoSEM 230, USA) was employed to investigate the microstructure of the gradient zone, and EPMA (JXA-8230, JEPL, Japan) was used to determine the concentration profiles of the elements.

      Table I. Chemical composition of the investigated cemented carbides (wt%)

      Ti1-x-zAlxZrzN films were deposited onto several substrates by unbalanced magnetron sputteringin a mixed Ar+N2 (both of 99.999% purity) glow discharge. DSC with TGA was performed in a Netzch-STA 409C from room temperature (RT) to 1500 °C with a heating rate of 20 K/min in flowing He (99.9% purity, 20 sccm flow rate) The chemical compositions of the films were determined using energy dispersive X-ray analysis (EDX) with an Oxford Instruments INCA EDX. Phase identification and structural investigations of the layers in their as deposited state and after thermal treatment with the DSC equipment in He or synthetic air were conducted by XRD with CuKα radiation using a Brucker D8 diffractometer in Bragg/Brentano mode. DFT calculations were performed using the VASP code.

      RESULTS AND DISCUSSION

      Verification and Application of the Databases

      It is generally known that the graphite and eta (M6C or M12C) phases are unexpected phases and the carbon content in cemented carbides should be carefully controlled to avoid the formation of these phases. With the aid of thermodynamic calculations, it is easy to see how to control the carbon content and how the carbon content affects the choice of sintering temperature when developing a new alloy. Figure 1(a) shows a calculated phase equilibria closing to the sintering region of an alloy with the composition of C-W-9Co-15Ti-10Ta-2Nb-0.1N (wt.%). As can be seen, the carbon content have to be carefully located in a narrow range about 0.2 wt.% in order to avoid the appearance of unwanted phases. Figure 1(b) presents a similar calculation by adding 2 wt.%, of Cr. From Fig. 1(b), it can be seen that the melting point of binder phase is decreased substantially by Cr addition and the existence of the preferable fcc_Co + M(C, N)x + WC equilibrium is broadened. On the basis of CSUTDCC1, a similar calculation can be performed on alloys with any composition, which will be a useful guidance for developing new alloys.

      Figure 1. Calculated phase equilibria closing to the sintering region of alloys with the composition of (a) C-W-9Co-15Ti-10Ta-2Nb-0.1N (wt.%) and (b) C-W-2Cr-9Co-15Ti-10Ta-2Nb-0.1N (wt.%)

      Study of Gradient Zone Formation in Cemented Carbides

      Figure 2 shows SEM micrograph of the cross section of alloys sintered under different nitrogen partial pressures (0, 20 and 40 mbar) at 1450 °C for 1 h. It is obvious that the near-surface of the alloy has formed the gradient zone which is enriched in binder phase and depleted in cubic carbides. Comparing the micrographs of the cross section of the cemented carbides sintered under different low nitrogen gas pressure shows that a decreasing thickness of gradient layer with increasing nitrogen gas pressure.

      Figure 2. SEM micrograph of the cross section of alloys sintered under different nitrogen partial pressures (0, 20 and 40 mbar) at 1450 °C for 1 h.

      By combining the presently established thermodynamic and diffusivity databases, DICTRA software has been used to simulate the formation of the gradient zone. Figures 3(a)–(b) illustrate the simulated elemental concentration profiles for Co and Ti in alloys after sintering for 1 h at 1450 °C under different nitrogen partial pressures (0, 20 and 40 mbar), compared with the measured data. This result indicates that the content of Ti is free in the near-surface zone and enrich inside the surface zone. At the near-surface zone, the content of Co increases sharply and reached a maximum value. Beneath the near-surface zone, a decrease of Co is observed, which leads to the minimum value. Above this minimum value, the content of Co increases slowly to its bulk value. The calculated thickness of the gradient layer decreases with the increasing of nitrogen partial pressure, which shows the similar diffusion behavior as the experimental results. As can be seen in Figs. 3(a)–(b), the presently obtained thermodynamic and diffusion databases can reasonably reproduce most of the experimental concentration profiles.

      Figure 3. Concentration profile for (a) Co and (b) Ti in alloys: measurement (symbols) and calculation (curve).

      Ti-Al-Zr-N Hard Coatings

      Elemental analysis by EDX reveals that our Ti1-x-zAlxZrzN films are stoichiometric with N/metal ratios of 1±0.1 and compositions of Ti0.48Al0.52N, Ti0.40Al0.55Zr0.05N, Ti0.39Al0.51Zr0.10N, Ti0.36Al0.47Zr0.17N, and Ti0.34Al0.37Zr0.29N, respectively. XRD investigations in Fig. 4 reveal a single phase cubic structure, which is in agreement with ab intio calculations. Figure 5a presents the energy of formation (Ef) of the cubic and wurtzite Ti1-x-zAlxZrzN alloys with constant z = 0, 0.05 and 0.1 as a function of the AlN mole fraction x. The data suggest a transition from cubic to wurtzite structure Ti1-x-zAlxZrzN at x ~0.72, 0.70, and 0.68 for a ZrN mole fraction z of 0, 0.05, and 0.10, respectively. Since the compositional steps given by the supercell sizes are different for the cubic (1/18) and for the wurtzite (1/16) alloys, to evaluate the maximum solubility of AlN in the cubic phase we proceeded as follows. First, we fitted each set of data with constant ZrN mole fraction with a quadratic polynomial (a0 + a1·x + a2·x²). For c-Ti1-x-zAlxZrzN, the AlN mole fraction x was varied 19 times for z = 0, 12 times for z = 0.055, and 10 times for z = 0.111. The calculations of w-Ti1-x-zAlxZrzN were obtained with 7, 6, and 5 variations in x for z = 0, 0.0625, and 0.125, respectively, in the composition range x = 0.5-1. Subsequently, for each phase (i.e. cubic or wurtzite) we fitted individually the coefficients (i.e. a0, a1, a2) of their quadratic polynomial for the three different ZrN mole fractions, z, with a linear expression in the ZrN contents. This way, two polynomial fits (one for the cubic and one for the wurtzite modification) as functions of x (AlN mole fraction) and z (ZrN mole fraction) were obtained. In the last step, we used these fits to estimate the cross-over between the formation energies of the cubic and wurtzite phases at fixed ZrN mole fractions (and thus to estimate the influence of Zr on the maximum AlN mole fraction in the cubic Ti1-x-zAlxZrzN).

      Figure 4. XRD patterns of as deposited powdered Ti1-x-zAlxZrzN thin films.

      Figure 5. (a) Energy of formation (Ef) for c-Ti0.33Al0.39Zr0.28N and cubic and wurtzite phases Ti1-x-zAlxZrzN with z = 0, 0.05 and 0.1 as functions of AlN mole fraction. (b) Overall chemical compositions of our Ti1-x-zAlxZrzN films, in the as deposited state, plotted within the TiN–AlN–ZrN quasi-ternary phase diagram. The solid line indicates the transition between preferred cubic and wurtzite phases

      Ab initio obtained mixing enthalpies (from the binary constituents c-TiN, c-ZrN, and w-AlN) for c-Ti0.48Al0.52N, c-Ti0.4Al0.55Zr0.05N, c-Ti0.39Al0.51Zr0.1N, and c-Ti0.34Al0.37Zr0.29N are 104, 116, 119, and 120 meV/at, respectively, and hence increase with increasing ZrN content. When using a cubic solid solution between TiN and ZrN, i.e. c-Ti1-yZryN (with y = z/(1-x)) as a constituent next to w-AlN, we obtain mixing enthalpies of 104, 98, 87, and 72 meV/at for z = 0, 0.05, 0.10, and 0.29, respectively. The latter reference and the comparison of the mixing enthalpies with the DSC experiments, which exhibit an overall exothermic contribution of 192, 232, 227, and 140 W·K/g for the coatings c-Ti0.48Al0.52N, c-Ti0.4Al0.55Zr0.05N, c-Ti0.39Al0.51Zr0.1N, and c-Ti0.34Al0.37Zr0.29N, suggests that there is no separation into the constituents c-TiN, c-ZrN, and w-AlN but into c-Ti1-yZryN and w-AlN. This is verified by XRD analysis of samples annealed to various temperatures using the DSC equipment with the same setup, atmosphere, heating and cooling rates.

      Figure 6 shows the structural evolution during annealing of our Ti0.48Al0.52N (a) Ti0.4Al0.55Zr0.05N (b) and Ti0.39Al0.51Zr0.1N (c) films by means of XRD patterns after annealing to 700, 850, 1100, 1200, and 1500 °C. The Zr-free Ti0.48Al0.52N film exhibits a small shift of the XRD reflexes during annealing to 700 °C as compared with the as deposited state, see Fig. 6a, suggesting only minute structural changes like recovery and relaxation which contribute to the exothermic DSC feature in this temperature range. The XRD patterns of the Zr-containing films Ti0.4Al0.55Zr0.05N and Ti0.39Al0.51Zr0.1N annealed to 700 °C reveal also a shift in the peak position to higher diffraction angles but also an increase in peak broadening, see Figs. 6b and c. The latter is an indication for a reduction in grain size and/or an increase in microstresses which can result from the onset of a decomposition process. This can better be seen after annealing at 850 °C, where the XRD reflexes exhibit on both sides (lower and higher diffraction angles) an increase in intensity and width, suggesting the formation of Al-depleted and Al-enriched domains. After annealing at 1100 °C, a pronounced shoulder-formation on both sides of the ‘matrix’ XRD peak can be seen clearly. These shoulders indicate the formation of TiN- and AlN-rich cubic domains for Ti0.48Al0.52N, and TiN-, ZrN- and AlN-rich cubic domains for the Zr-containing films. While the Zr-free film, Ti0.48Al0.52N, exhibits the formation of w-AlN already after annealing at 1100 °C, no w-AlN formation can be detected for the Zr-containing films (compare Figs. 6a, b, and c), indicating that Zr effectively retards the formation of w-AlN. After annealing at 1200 °C, almost no intensity at the XRD peak positions of as deposited films can be detected, indicating close-to-complete decomposition of the original supersaturated matrix. The Ti0.48Al0.52N film is mainly composed of TiN- and AlN-rich cubic phases and w-AlN, whereas it is still hard to detect any w-AlN for the Zr containing films, Ti0.4Al0.55Zr0.05N and Ti0.39Al0.51Zr0.1N, which at this stage compose mainly of ZrN-, TiN- and AlN-rich cubic phases.

      Fig. 6 XRD patterns after annealing in vacuum to temperatures Ta up to 1500 °C of (a) Ti0.48Al0.52N, (b) Ti0.40Al0.55Zr0.05N, and (c) Ti0.39Al0.51Zr0.10N.

      SCHEMATICAL CERAMIC GENOME

      Based on the ceramic genome strategy, several brands of cemented carbides have been developed in the present work, as shown in Fig. 7. The path toward ceramic genome implementation in cemented carbides design and manufacture can be expressed in Fig. 8. Firstly, alloy composition and process parameters are designed via CALPHAD or First-principles calculations. Secondly, the gradient zone formation, microstructure and mechanical properties can be predicted based on the accurate databases. Thirdly, a series of corresponding cemented carbides and hard coatings are prepared under the guidance of the previous steps. After that, the microstructure and mechanical properties of the cemented carbides and hard coatings are investigated experimentally, thereby validating the accuracy of the calculations/simulations. Process routes are optimized and finally chosen for industrial production of cemented carbides with excellent or special performances.

      Figure 7. Designed and manufactured industrial cemented carbides with the integration of ICME.

      Figure 8. A schematical ceramic genome strategy to develop new gradient carbides and hard coatings.

      CONCLUSION

      The ceramic genome is a powerful tool for materials process optimization and alloy design. Thermodynamic database and diffusivity database for cemented carbides have been developed through a combination of experimental, theoretical and assessment work. Gradient cemented carbides WC-Ti(C,N)-Co, sintered under various partial pressures of N2 have been prepared and investigated by means of SEM and EPMA techniques. Good agreements between simulated and measured results indicate the powerful ability of the presently established databases in cemented carbides design and process optimization The structure and thermal properties are investigated using experimental methods combined with first-principles calculations. Based on the results presented, we can conclude that the coating Ti0.40Al0.55Zr0.05N which contains only 5% Zr at the metal sublattice exhibits the best mechanical and thermal properties. A schematical ceramic genome strategy to develop new gradient carbides and hard coatings is presented.

      ACKNOWLEDGEMENT

      The financial support from Creative Research Group of National Natural Science Foundation of China (Grant No. 51021063) and Zhuzhou cemented carbide cutting tools limited company of China is acknowledged.

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      ¹⁹Y. Wang, Z. K. Liu, and L. Q. Chen, Thermodynamic properties of Al, Ni, NiAl, and Ni3Al from first-principles calculations, Acta. Mater, 52, 2665–71 (2004).

      ²⁰C. K. Gan, Y. P. Feng, and D. J. Srolovitz, First-principles calculation of the thermodynamics of InxGa1-xN alloys: Effect of lattice vibrations, Phys. Rev. B, 73, 235214–21 (2006).

      THE EFFECTS OF NESTING AND STACKING SEQUENCE ON THE STRUCTURAL AND GAS TRANSPORT PROPERTIES OF PLAIN WOVEN COMPOSITES DURING CHEMICAL VAPOR INFILTRATION PROCESS

      Kang Guan, Laifei Cheng, Qingfeng Zeng, Yunfang Liu, Haitao Ren, Litong Zhang

      Science and Technology on Thermostructure Composite Materials Laboratory, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, P. R. China.

      ABSTRACT

      In this study, we numerically investigate the effects of nesting and stacking sequence on the structural and gas transport properties of plain woven composites during chemical vapor infiltration process. Our results show that the structural properties of these composites are strongly affected by nesting and slightly influenced by the stacking sequence, while the gas transport properties are strongly affected by nesting and stacking sequence.

      INTRODUCTION

      Chemical vapor infiltration (CVI) has been considered as one important method for producing high-quality fiber reinforced ceramic- or carbon-matrix composites over the half century [1, 2]. One typical feature of this technique is, the structural and gas transport properties of the preform during the infiltration will change significantly. Thus, the precise knowledge of the evolution of these properties is essential and helpful for design and optimization the whole CVI process.

      In general, the most accurate value of structural and gas transport properties are obtained by direct experiments [3, 4]. However, experimental studies on this problem are expensive and sometimes unfeasible. In contrast, numerical approaches allow researchers to model any complex architecture and study the effects of different geometry factors. Until now, the most efficient way to study this problem is to reconstruct the realistic fibrous structure through X-ray digital data, simulate the pore evolution and calculate the corresponding structural and gas transport properties [5, 6].

      This paper is the continuation of the study to the evolution of structural and gas transport properties of plain woven composites during the CVI process. In the previous paper [7], we investigated 0°/90° woven composites by considering the nesting (the average thickness of one layer is less than two times the tow thickness) exists between the adjacent layers. In this work, we investigate the effects of nesting and stacking sequence on the evolution of structural and gas transport properties of these composites during CVI process.

      THE MACRO-PORE MODEL

      In our previous study, we used three parameters, e(the aspect ratio of the tow cross section), kL(the ratio of the gap width to the tow width between two adjacent tows in the longitudinal direction), and m(tow shape factor) to characterize the geometry of the composite. Because m has been found to almost not affect the evolution of structural and gas transport properties, m is fixed as 0.3 in this study. For the case of considering the nesting, the layers are randomly shifts along the in-plane direction, then the thickness of the layer is estimated by Ito’s model [8]. For the case of ignoring the nesting, the thickness of the layer is fixed and equal to twice the tow thickness. The difference between both structures is illustrated in Figure 1. By varying e and kL, considering no nesting exists and rotating each layer relative to the corresponding lower layer, the macro-pore structures with three different stacking sequences (0°/90°, 0°/45° and 0°/30°60°) are constructed. The details of the numerical method can be found in our previous articles[7, 9, 10] and will not be repeated here. An example of the simulated macro-pores evolution is visualized in Figure 2.

      Figure 1. Illustration of the geometric models (a) without nesting and (b) with nesting.

      Figure 2. An example of simulated macro-pore structure of 0°/30°/60° woven composites with four layers at the (a) initial, (b) intermediate and (c) final stages of the CVI.

      RESULTS AND DISCUSSION

      Structural Properties of Macro-pores

      For simulating and optimizing CVI technique, the indispensable problem is knowing the relationship of the specific area, Sv and porosity, p. For partially densified woven composites, we had found that the structural properties for macro-pores can also be approximated by the partially overlapping fiber structure, in which the initial radius of this imaginary fiber, r0 can be considered as a length scale associate with the preform:

      (1a) equation

      where t and b are the adjustable parameter and the half thickness of the tow, respectively. Then the radius of the coated fiber r can be expressed as follows:

      (1b) equation

      where km is the ratio of the thickness of deposited matrix to the half thickness of the tow.

      Then the relationship of Sv and p for macro-pores of woven composites can be expressed as

      (2a) equation

      where r0 and δ are the initial radius of the fiber and the ratio of the radius of the coated fiber to the initial radius of the fiber (δ = r/r0), pa is the accessible porosity and estimated by the following functions [9]:

      (2b)

      equation

      Here pc is the critical porosity, which means the isolated pores are gradually present when p is less than pc; pr is the residual porosity, which means all pores are isolated and the densification is terminated when is p equal to pr.

      In Figures 3, 4 and 5, the estimated pc, pr and t values are compared for different woven composites. The comparison shows that: First, ignoring or considering the nesting will give contradictory trends: pc values for the case of ignoring the nesting effect scatter across the wide ranges of 0.15–0.30, while pr values fluctuate between 0.04–0.12; almost all t values for the case of ignoring the nesting effect is less than the ones for considering the nesting effect, which implies the nesting effect is beneficial to the densification, because a larger t value means faster densification rates. Second, these values for 0°/90°, 0°/45° and 0°/30°/60° layup can be considered as approximately the same. Thus we may conclude that the structural properties of woven composites are mainly controlled by the extent of nesting, while they are not very sensitive to the stacking sequence.

      Figure 3. The variation of estimated pc values with kL for plain woven composites in the (a) longitudinal and (b) through-thickness directions.

      Figure 4 The variation of estimated pr values with kL for plain woven composites in the (a) longitudinal and (b) through-thickness directions.

      Figure 5. The variation of estimated t values with p⁰ for plain woven composites.

      Gas Transport Properties of Macro-pores

      For simulating and optimizing the CVI technique, another indispensable problem is knowing the relationship of the effective diffusivity, De for reactant gas and porosity. Generally, De is estimated by the tortuosity, τ:

      (3a) equation

      where D0 is the gas diffusivity in the void. τ is usually expressed by incorporating the percolation effect [6]:

      (3b) equation

      where τmin is the tortuosity for the case that the porosity is unity. pr is the percolation porosity (residual porosity). f is the parameter that characterizes the increasing magnitude of tortuosity over the range of solid volume fraction from dilute to percolation concentration. In this study, we only consider the effective diffusivity of woven composites in the ordinary regime. If the preforms are applied a pressure gradient to enhance infiltration (e.g. FCVI), another gas transport property of the preform—permeability, K must be known. Our recent work [10] had shown that the permeability for woven composites can be estimated well by Gebart’s model:

      (4) equation

      where C1 and C2 are the adjustable parameters; is the average pore radius of the porous medium, and is estimated as = 2p/Sv. In this study, Eqs. (3) and (4) are found to give a sufficient fit to calculated diffusivities and permeabilities for all woven composites, thus the effects of nesting and layup sequence on the diffusivities and permeabilities can be estimated by substituting the corresponding fitting parameters in Eqs. (3) and (4).

      Given the same tow thickness, the effects of nesting on the gas transport properties are shown in Figure 6. Here the subscripts nest and 90 denote the gas transport properties of 0°/90° woven composites considering the nesting and without considering the nesting, respectively. In Figure 6, a different trend is found for τnest/τ90 and Knest/K90 with p. In general, τ is not so sensitive to the nesting; when p is not very close to the percolation porosity, τnest/τ values for longitudinal and through-thickness directions are between 0.9–1.1 and 1–1.5, respectively. On the contrary, nesting has more influence on K; Knest/K90 values for the longitudinal and through-thickness directions can change rapidly between the ranges of 0.8–4 and 0.8–3.5, respectively. The main reason for these is permeability is strongly affected by the average radius of the pore structure, which may vary due to the nesting.

      Figure 6. The variation of (a) τnest/τ90, (b) Knest/K90 with p for plain woven composites.

      Given the same tow thickness, the effects of the stacking sequence on the gas transport properties are shown in Figures 7 and 8. Here the subscripts 45 and 30 denote the gas transport properties of 0°/45° and 0°/30°/60° woven composites without considering the nesting, respectively. It is clearly shown that the stacking sequence has more influence on permeability than on tortuosity. τ45/τ90 or τ30/τ90 values for longitudinal and through-thickness directions are between 0.8–1.2 and 1.1–1.5, respectively. K45/K90 or K30/K90 values for longitudinal and through-thickness directions are between 0.2–0.8 and 0.8–2.3, respectively. Since the stacking sequence plays little effect on the structural properties of woven composites (see previous section), the large variation of permeability is attributed to the large variation of local pore radius for gas flow.

      Figure 7. The variation of (a) τ45/τ90, (b) K45/K90 with p for plain woven composites.

      Figure 8. The variation of (a) τ30/τ90, (b) K30/K90 with p for plain woven composites.

      In most situations, the actual values of e and kL. are between 4–8 and 0.2–0.3, respectively, thus we use following function to estimate the effects of the stacking sequence on the gas transport properties of plain woven composites:

      For in-plane direction, De30 ≈ De45 ≈ 0.9 De90, K30 ≈ K45 ≈ 0.6 K90.

      For through-thickness direction, De30 ≈ De45 ≈ 1.1De90, K30 ≈ K90, K45 ≈ 1.1K90.

      CONCLUSION

      In this study, the effects of the nesting and stacking sequence on the structural and gas transport properties of plain woven composites are investigated. Our study shows that gas transport properties are more sensitive to the nesting and stacking sequence than the structural properties, especially in the longitudinal direction. But our geometric model considering the nesting is constructed by Ito’s analytical expressions [8], which is only an approximation method. In addition, our model is based on the assumption that the layers are randomly stacked, e(the aspect ratio of the tow cross section) and kL(the ratio of the gap width to the tow width between two adjacent tows in the longitudinal direction) are not varied. Actually, these assumptions do not perfectly coincide with the reality. Anyhow our study provides more precise knowledge of the evolution of structural and gas transport properties for woven composites during CVI process, and thus is useful for design and optimization of CVI technique.

      ACKNOWLEDGEMENTS

      The authors acknowledge the financial support from the Natural Science Foundation of China (Grant No. 51032006). We also thank Northwestern Polytechnical University High Performance Computing Center for the allocation of computing time on their machines.

      REFERENCES

      [1] Besmann TM, Sheldon BW, Lowden RA, Stinton DP. Vapor-phase fabrication and properties of continuous-filament ceramic composites. Science. 1991;253(5024):1104–1109.

      [2] Naslain RR. SiC-Matrix Composites: Nonbrittle Ceramics for Thermo-Structural Application. International Journal of Applied Ceramic Technology. 2005;2(2):75–84.

      [3] Starr TL, Hablutzel N. Measurement of gas transport through fiber preforms and densified composites for chemical vapor infiltration. J Am Ceram Soc. 1998;81(5):1298–1304.

      [4] Lee S-B, Stock SR, Butts MD, Starr TL, Breunig TM, Kinney JH. Pore geometry in woven fiber structures: 0°/90° plain-weave cloth layup preform. J Mater Res. 1998; 13(05): 1209–1217.

      [5] Coindreau O, Vignoles GL. Assessment of geometrical and transport properties of a fibrous C/C composite preform using x-ray computerized micro-tomography: Part I. Image acquisition and geometrical properties. J Mater Res. 2005;20(9):2328–2339.

      [6] Vignoles GL, Coindreau O, Ahmadi A, Bernard D. Assessment of geometrical and transport properties of a fibrous C/C composite preform as digitized by x-ray computerized microtomography: Part II. Heat and gas transport properties. J Mater Res. 2007;22(6):1537–1550.

      [7] Guan K, Cheng L, Zeng Q, Zhang L, Deng J, Li K, et al. Modeling of pore structure evolution between bundles of plain woven fabrics during chemical vapor infiltration process: the Influence of preform geometry. J Am Ceram Soc. 2013;96(1):51–61.

      [8] Ito M, Chou T-W. An analytical and experimental study of strength and failure behavior of plain weave composites. J Compos Mater. 1998;32(1):2–30.

      [9] Guan K, Cheng L, Zeng Q, Feng Z-Q, Zhang L, Li H, et al. Modeling of pore structure evolution within the fiber bundle during chemical vapor infiltration process. Chem Eng Sci. 2011;66(23):5852–5861.

      [10] Guan K, Cheng L, Zeng Q, Li H, Liu S, Li J, et al. Prediction of Permeability for Chemical Vapor Infiltration. J Am Ceram Soc. 2013;96(8):2445–2453.

      AN EFFICIENT APPROACH TO DETERMINE THE EFFECTIVE PROPERTIES OF RANDOM HETEROGENEOUS MATERIALS

      Yatao Wu and Yufeng Nie

      Department of Applied Mathematics, Northwestern Polytechnical University Xi’an, Shaanxi, China

      ABSTRACT

      Many studies of effective properties of random heterogeneous materials focus on determination of the size of representative volume element (RVE), but few discuss acceleration of the convergence rates of the apparent material properties as the sizes of volume elements increase. In this study, the convergence of the apparent thermo-mechanical properties of aluminum/alumina random heterogeneous materials is investigated, and Richardson extrapolation technique is introduced to accelerate the convergence rates of apparent material properties. It is found that this approach efficiently predicts the effective material properties with significantly reduced computational effort. Meanwhile, the homogenized material properties are compared with theoretical bounds, i.e., the Voigt-Reuss bounds and the Hashin-Shtrikman bounds.

      INTRODUCTION

      Composites with complex microstructures are widely used in modern engineering. To deal with difficulties of direct simulation of the structure with multiscale characteristics, homogenized material model, which reflects the effects of microscale heterogeneities, has been used in a macroscale analysis.¹ For a statistically homogeneous material, a representative volume element (RVE) is always defined at the micro-scale, and effective properties are computed after solving special partial differential equations (PDEs) with uniform boundary conditions.² An RVE should be sufficiently large such that it contains enough microscale features and leads to deterministic effective property. Determination of the size of RVEs for random heterogeneous materials attracts a continuous interest of the scientific community.³–⁵ To this end, PDEs are solved within larger and larger volume elements, until the effective property with satisfactory accuracy is gained. Actually this procedure is much too time-consuming and high-demanding for computer memory, especially for three-dimensional complex microstructures with high contrast ratio of constituent properties.

      The effective properties determined within not large enough volume elements, which are called apparent⁶ properties, will converge to the true effective property defined in an infinite domain.⁷ Few researches discuss acceleration of the convergence rate of the apparent property. Gloria added a zero-order term into original PDEs to improve the convergence rate of the apparent property under periodic or a certain stochastic assumption on coefficients.⁸ In this work, under general statistically homogeneity assumption, we investigate the convergence rate of the apparent property numerically, and then utilize Richardson extrapolation technique to improve its rate of convergence. The method is applied to predict the effective thermo-mechanical properties of aluminum/alumina random heterogeneous materials. Numerical results demonstrate that homogenization combined with extrapolation efficiently determines the effective properties within some smaller volume elements, and it is no more necessary to solve PDEs within larger RVEs. A great deal of computation time and computer memory can be saved.

      HOMOGENIZATION

      Theoretically, the RVE should be an infinite domain such that the homogenized property is deterministic. However, from a computational perspective, the homogenization procedure is performed in finite domains. Some PDEs with uniform boundary conditions defined in a volume element are solved firstly, and apparent property can then be computed. The apparent property is indeterminate and dependent on the size of volume element and boundary condition. Moreover, the apparent property relies on the microstructural realization.

      To obtain effective thermal conductivity, following PDE (1) with one type of boundary conditions (2) or (3) are solved

      (1) equation

      (2) equation

      (3) equation

      Temperature T and thermal conductivity coefficients κ are functions of position and realization ω. For simplicity, let YL = [0, L]×[0, L] be a square volume element. (2) and (3) are uniform temperature gradient (UTG) and uniform heat flux (UHF) boundary’ condition, respectively. n denotes the outward normal vector to ∂YL. ξ and η are arbitrary constant vectors.

      The apparent thermal conductivity can be computed by the relations of volume average of relevant variables

      (4) equation

      Analogous with thermal problem, homogenization of linear elastic problem solves PDEs with uniform strain or uniform stress boundary condition. For conciseness it will be omitted here to retain the focus on the key points of the present study. The reader is encouraged to reference Kanit et al. for more details.⁹

      Apparent thermal conductivity κLH and apparent elastic moduli CLH are functions of realization. Hence the ensemble averages of the apparent properties are adopted in a macroscale analysis, after solving PDEs in the volume elements of same size for a sufficient number of realizations.

      EXTRAPOLATION

      For periodic microstructure and a specific random microstructure of checker-board type, research shows that the apparent properties converge with first order accuracy O(1 / L).¹⁰ In our numerical examples, convergence rates of apparent thermo-mechanical properties for a more morphologically realistic random heterogeneous microstructure are numerically investigated, and it shows first order rates of convergence. In order to deal with the difficulties of prediction of effective properties by solving PDEs in a sufficient large RVE, we introduce Richardson extrapolation technique to the homogenization procedure. For convenience, let pLH represents apparent properties E{κij,LH(ω)} or E{CHijkl,L(ω)}, where E{·} denotes the ensemble average. As L goes to infinity, pHL converges to its limit pH with

      (5) equation

      When combine pHL and pH2L, we have

      (6) equation

      Obviously extrapolation sequence RL improves the convergence rate to a higher order β. It will show that the approximations computed by extrapolation of apparent properties with volume elements of size L and 2L estimate the effective properties not only better than themselves, but also better than the apparent properties within larger volume elements (e.g. size of 4L in our examples).

      NUMERICAL RESULTS

      Vel et al. proposed a method to create morphologically realistic microstructures with computer. The microstructures closely resemble actual micrographs of random heterogeneous materials manufactured by techniques including plasma spraying and powder processing.¹¹ We can generate random microstructures of varying volume fractions using the method. Figure 1 shows some simulated microstructures of different sizes for volume fraction V1 = V2 = 0.5.

      Figure 1. Some volume elements of dimensionless sizes L=1, 2, 4 & 8.

      In this work, convergence rates of apparent properties for random heterogeneous materials by homogenization are estimated. As an example, we compute the effective thermo-mechanical properties of Al/Al2O3, random heterogeneous materials. The constituent material properties are listed in Table 1. In Table 2, we list some convergence rates of apparent thermal conductivities with both uniform boundary conditions for a variety of volume fractions. Obviously the rates of convergence are close to 1.

      Table 1. Properties of the constituent materials.

      Table 2. Convergence rates of apparent thermal conductivities of Al/Al2O3.

      We are then encouraged to utilize Richardson extrapolation technique to obtain more accurate results. As can be seen in Figure 2, the properties computed by extrapolation approximate the true effective properties much better than the apparent properties computed by homogenization. In fact the results by extrapolation of apparent properties with volume elements of size L and 2L estimate the effective properties not only better than themselves, but also better than the apparent properties within volume elements of size 4L. Figure 3 compares apparent thermal conductivities computed by homogenization and their extrapolation for the entire range of volume fractions. Richardson extrapolation technique produces sharper upper and lower bounds of true effective properties than homogenization. To verify our numerical results, Voigt-Reuss bounds and Hashin-Shtrikman bounds are also compared. It is observed that the homogenized properties lie within the two bounds.

      Figure 2. Convergence of apparent thermal conductivities as the volume element size increases and Comparison of apparent thermal conductivities computed by homogenization and their extrapolation.

      Figure 3. Comparison of extrapolation, homogenization, Hashin-Shtrikman bounds and Voigt-Reuss bounds for apparent thermal conductivities as a function volume fraction.

      The same work has been down for effective mechanical properties of Al/Al2O3 composites. And similar results have been shown. Here we just present the comparison of extrapolation, homogenization and theoretical bounds for apparent Young’s modulus and shear modulus, see Figure 4 and Figure 5.

      Figure 4. Comparison of extrapolation, homogenization, Hashin-Shtrikman bounds and Voigt-Reuss bounds for apparent Young’s modulus as a function volume fraction.

      Figure 5. Comparison of extrapolation, homogenization, Hashin-Shtrikman bounds and Voigt-Reuss bounds for apparent shear modulus as a function volume fraction.

      CONCLUSION

      Richardson extrapolation technique is introduced into homogenization method to deal with the difficulties of determining effective properties of random heterogeneous materials in large RVEs. First order convergence rates of apparent properties are shown. And Richardson extrapolation technique improves the convergence rates. Since much better estimations of effective properties can be computed within some smaller volume elements rather than larger RVEs, it will save a great deal of computation time and computer memory.

      ACKNOWLEDGEMENT

      This work is supported by the National Natural Science Foundation of China (No. 90916027, No. 11071196).

      REFERENCES

      ¹T. I. Zohdi, P. Wriggers, An Introduction to Computational Micromechanics. Lecture Notes in Applied and Computational Mechanics, Springer-Verlag Berlin Heidelberg, 2005.

      ²N. Charalambakis, Homogenization Techniques and Micromechanics. A Survey and Perspectives, Appl Mech Rev, 63 (3), (2010).

      ³Z. H. Shan, A. M. Gokhale. Representative volume element for non-uniform micro-structure. Comp Mater Sci, 24 (3), 361–379 (2002).

      ⁴M. Ostoja-Starzewski, X. Du, Z. F. Khisaeva, and W. Li, Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures, Int J Multiscale Com, 5 (2), 73–82 (2007).

      ⁵C. Pelissou, J. Baccou, Y. Monerie, and F. Perales, Determination of the size of the representative volume element for random quasi-brittle composites, J Mech Phys Solids, 46 (14–15), 2842–2855 (2009).

      ⁶C. Huet, Application of variational concepts to size effects in elastic heterogeneous bodies, J Mech Phys Solids, 38 (6), 813–841 (1990).

      ⁷A. Bourgeat, A. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann I H Poincare-Pr, 40 (2), 153–165 (2004).

      ⁸A. Gloria, Reduction of the resonance error - part 1: approximation of homogenized coefficients, Math Mod Meth Appl S, 21 (8), 1601–1630 (2011).

      ⁹T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin, Determination of the size of the

      representative volume element for random composites: statistical and numerical approach, Int J Solids Struct, 40 (13–14), 3647–3679 (2003).

      ¹⁰X. Y. Yue, W. N. E. The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size, J Comput Phys, 222 (2), 556–572 (2007).

      ¹¹S. S. Vel, A. J. Goupee, Multiscale thermoelastic analysis of random heterogeneous materials Part I: Microstructure characterization and homogenization of material properties, Comp Mater Sci, 48(1), 22–38 (2010).

      CONTRIBUTION OF IMAGE PROCESSING TECHNIQUES TO THE SIMULATION OF CHEMICAL VAPOR INFILTRATION OF SiC IN CMCs

      G. L. Vignolesa, C. Chapoulliéa,b, W. Rosa,b, C. Mulata,b, G. Couégnatc, C. Germainb, J.-P. Da Costab M. Cataldid, C. Descampsd

      aUniversity of Bordeaux, LCTS, 3 allée de la Boétie, F-33600 Pessac, France

      bUniversity of Bordeaux, IMS, 351 Cours de la Libération, F-33405 Talence, France

      cCNRS, LCTS, 3 allée de la Boétie, F-33600 Pessac, France

      dSAFRAN Herakles, Les 5 Chemins, F-33187 Le Haillan, France

      ABSTRACT

      Woven-reinforcement CMCs are promising materials in aeronautic industry. Their matrix is often prepared by Chemical Vapor Infiltration. To increase their competitiveness with respect to more classical materials, their fiber architecture and matrix processing conditions need more control and optimization. This paper summarizes recent developments of computer methods aimed at providing design tools in this context, paying special attention to image processing techniques and image-based modeling. A first code extracts morphological features from X-ray CMT images of raw woven fabrics. An algorithm for segmentation and labeling produces the yarns envelopes; then, fiber labeling is done at a higher resolution. Based on the analysis results, a synthesis technique allows numerical reconstitutions of yarns as filament bundles. We present an object dynamics method placing fiber sections in a yarn section as repelling disks. Another synthesis is the macro-wire dynamics approach in which a yarn is divided in a small number of large filaments whose mechanics are simulated in detail. Once realistic 3D images of woven architectures are obtained, numerical simulations of matrix infiltration take place. They are either based on region dilation tools or on realistic simulations of gas diffusion and deposition; the latter have been implemented as random-walk algorithms at two scales.

      INTRODUCTION

      Ceramic-Matrix Composites (CMCs) are promising candidates for, among other applications, civil jet engine hot parts¹. Their behavior in actual conditions of use has been characterized experimentally, showing excellent lifetimes². They are constituted of a woven/interlocked 3D arrangement of SiC fiber tows, infiltrated by a matrix, possibly multilayered in the case of self-healing CMCs³.

      The large-scale industrial production of such materials will be economically possible if the fabrication expenses are cut down, while guaranteeing sufficient life duration. In this aim, optimization of the material by varying its organization parameters, i.e. weaving patterns, layered matrix design (number, thicknesses, etc…) is a key issue. Unfortunately, the production of samples is very lengthy and expensive. Therefore, a modeling approach looks attractive: a prediction of the variation of the material properties when changing some design parameter would be fast and inexpensive provided it is reliable enough.

      The most important points for reliability are the accuracy in the description of the materials architecture on the one hand, and of the physical, mechanical and chemical phenomena on the other hand. To address this, all simulations presented here are in strong relationship with image analysis and synthesis, in order to provide a sound basis for experimental validation with respect to actual material samples. The materials structure is observed and the resulting information is utilized either directly or indirectly for simulations of the materials life. A class of realistic methods is to perform direct simulations inside accurate 3D representations of the materials, such as those produced by ultra-sound or tomographic investigations⁴; however, the question of the representativeness of the scanned sample is not addressed in this way, unless the procedure is repeated many times on many samples. On the other hand, it is always possible to start from a priori representations of the materials geometry, that should include as many parameters as possible to account for their non-ideality, and these parameters have to be identified from the analysis of a large enough number of sample images. Yet, direct simulations are of interest since they allow capturing effects that could be missed in an a priori approach of the material. In addition to these elements, since the material architecture is organized upon several length scales, multi-scale analysis and change-of-scale methods have to be performed. In the following we will present the methods that we have developed and discuss some results.

      OVERALL STRATEGY

      Keeping in mind the idea of handling simultaneously actual and virtual representations of the material, the strategy is articulated as shown in figure 1.

      Figure 1. Scheme of the overall two-scale analysis and modeling strategy

      The starting point is the acquisition of morphological information by 2D and 3D imaging. Images are available at low and high resolutions, i.e. at fiber-scale and yarn-scale. 3D CMT scans are of course a choice source of data, but micrographs taken in 2D are very useful too.

      From this raw material, two processing routes are possible⁵. The first one (plain lines) is to perform pattern recognition, recover the objects (yarns at large scale and fibers at a smaller scale), then to produce statistical data from them for further image synthesis or retain them as a 2D or 3D mesh for physicochemical and mechanical computations. The other way (dashed lines) is to extract directly image statistics without any pattern recognition and go to image synthesis only with this type of data as constraints.

      The production of CMC models by image synthesis usually involves two steps: the first one is the production of the reinforcement architecture, and the second one addresses the insertion of the matrix. Image dilation or mesh dilation are simple but sometimes efficient techniques; more realistic matrix infiltration models are also available. They may be run either on synthesized images (plain lines) or directly on the original CMT scans (dotted lines).

      Once the full CMC architecture has been produced, it is possible to enter the domain of mechanical, thermal, and thermo-mechanical simulations, to get an insight into the potential forces and weaknesses of the material under mechanical and thermal loads.

      IMAGE ANALYSIS TOOLS

      A typical starting point is illustrated by figure 2a. it is a high-resolution tomographic scan of a 2D woven CMC before matrix reinforcement, embedded in epoxy resin. 3D image data have been obtained by Synchrotron X-ray computerized tomography at a resolution of 1.377 μm per voxel, at the ID19 line of ESRF⁶. The data of interest represent a CMC sample of size 1.4×0.99×1.4 mm³, i.e. 1041×721×1024 voxels. The resolution and size of these tomographic images allow identifying simultaneously yarns at coarse scale and fibers at fine scale. Images show a good contrast between fibers and resin. The cylindrical shape of fibers is clearly visible. Their radius is about 5 pixels.

      Figure 2. Summary of the image analysis approach for the fibrous reinforcement. a) Example of high-resolution X-ray CMT slice. One particular yarn has been highlighted in red. b) Typical result of the yarn extraction procedure. c) Typical result of the fiber extraction procedure. d) Fiber diameter statistics. e) Density map in a particular yarn.

      Two segmentation algorithms sharing the same basic principles but working at different scales have been developed⁷. The first one allows extraction of yarn envelopes while the second one identifies individual fibers. Once the yarn and fibers are identified, a characterization stage provides parameters such as fiber diameter distribution, diameter variations along fibers, fiber orientation distribution, and intra-yarn density.

      The yarn-scale algorithm starts with the production of two masks separating respectively warp yarns and weft yarns. To obtain them, three primary masks have to be calculated first. One is obtained by basic intensity grey level thresholding, separating ceramic and resin. The other two result from the calculation of local 3D orientation vectors using the structure tensor⁸ and of their classification into weft and warp. Two coarse masks are thus obtained. The weft and warp coarse masks are then multiplied with the first all-yarn mask and, with a few morphological operations (dilation and erosion to smooth the envelope and fill porosity) fine weft and warp masks are ready to use.

      The second step addresses individual yarn identification in two passes: isolated chunk identification and yarn pursuit. Isolated chunks are yarn sections without any contact with neighboring yarns. An area-based criterion is used to decide whether the yarn is in contact with another one or not. This criterion also allows elimination of yarns intersected by data block borders. The isolated chunks are then fed into the pursuit algorithm. They are grown inside the mask following 3D orientation vectors. After spreading labels inside entire yarns, a post-treatment is applied to paste together chunks belonging to the same yarn. This final step results in the identification of all individual yarns. Figure 2b illustrates typical results, where weft yarns only are made visible.

      The fiber detection algorithm is similar but operates at a finer scale. The first step is the detection of individual fiber centers, i.e. voxels belonging to fiber axes⁹. Specific differential geometry tools¹⁰,¹¹ have been used. The second step is the identification and labeling of fiber segments, done by associating neighboring individual axes voxels and by operating a geodesic dilation inside the fiber binary mask. This step results in a 3D block were fibers are split into small cylindrical segments. The last step is, as for the yarns, a pursuit algorithm. Indeed, after center detection, labels are not the same along a fiber. Interruptions can also occur due to interruptions of the center extraction procedure itself – often in high-density parts of the yarn. The pursuit algorithm consists in using local 3D orientation vectors to follow fibers and associating labels along them. It allows unambiguous labeling of the fibers: every fiber has a single label and there is no spreading of this label into neighboring fibers, as can be seen in Figure 2c.

      Once the yarns and fibers have been extracted, a large quantity of morphological parameters becomes available. We have focused on the following ones:

      - Fiber diameter statistics: all fiber sections in perpendicular slices s are computed, and the diameter inferred as d = 2(s/π)¹/². The cumulative information is displayed as in Figure 2d; diameter variations along a single fiber have also been computed, showing modest variations on the test sample data.

      - Fiber orientation statistics: By the nature of the extraction algorithms, orientations of the individual fibers and of their local yarn are simultaneously available; therefore it is straightforward to obtain two angles for each fiber: a pitch and a yaw angle, giving respective inclinations with respect to the horizontal and vertical planes of a yarn. This type of statistics gives very interesting insights into the mechanical history of the fibrous yarns: narrowly dispersed orientation statistics indicate a high tensile stress state;

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