Multiscale heat conduction and fractal oxidation behaviors of needle-punched carbon/carbon composites

Meng Han; Chuwei Zhou; Vadim V. Silberschmidt; Qinsheng Bi

doi:10.1515/secm-2022-0174

Open Access Published by De Gruyter December 31, 2022

Multiscale heat conduction and fractal oxidation behaviors of needle-punched carbon/carbon composites

Meng Han , Chuwei Zhou , Vadim V. Silberschmidt and Qinsheng Bi

From the journal Science and Engineering of Composite Materials

https://doi.org/10.1515/secm-2022-0174

Abstract

Needle-punched (NP) carbon/carbon composites (CCCs) are widely used in structures re-entering the atmosphere of aerospace, thanks to their excellent mechanical properties. They are easily oxidized at high temperatures with atmospheric oxygen. The oxidation behavior is influenced by the process of heat conduction. In this study, longitudinal and transverse heat conduction in micro- and mesoscale models of CCCs was investigated. It was established that the heat transfer interface of a fiber bundle demonstrated peak-like morphology, while the punching structures of composites slowed down the process of heat conduction. Oxidation behavior of NP CCCs was predicted with a fractional Brownian motion strategy. It was found that the oxidized fiber bundles formed “bamboo shoots” morphology due to different oxidation rates of the matrix and fibers. Stochastic oxidation behavior was successfully described with this fractal strategy.

Keywords: heat conduction; fractal strategy; oxidation behavior; carbon/carbon composite

1 Introduction

Carbon/carbon composites (CCCs) are widely used for the atmospheric re-entry of thermal protections systems (TPSs) [1] because of their excellent thermo-chemical and mechanical performances. With the development of multiple-use recyclable devices, researchers pay increasing attentions to the oxidation behavior and resistance of CCCs, since they experience high temperatures in the process of penetrating the atmosphere. In this harsh service environment with high temperature, they react with oxygen, producing gaseous carbon oxides [2], and suffer from the mechanical erosion caused by heat flux.

Experimental studies at high temperature in oxygen atmosphere is a common method to investigate the oxidation behaviors. Huang et al. [3] experimentally researched the anti-oxidation performance and microstructures of SiC-coated CCCs with the thermodynamic analysis at 1,500°C. It was found that Al-modified SiC coating provided better thermal shock performance than Al₂O₃-modified SiC one. Wang et al. [4] proposed a three-dimensional high-thermal-conductivity TPS for CCCs with a directional heat transfer function. Guo et al. [5] investigated the oxidation mechanisms and non-isothermal oxidation kinetics of graphite in air with scanning electron microscopy (SEM) and thermogravimetry (TG). The obtained results demonstrated that the oxidation rate had three stages: (i) rapid increase, (ii) steady state and (iii) the sharp decrease. Deng et al. [6] researched the effect of oxidation on the stress of first matrix cracking and proposed a criterion for interface debonding. Kim et al. [7] analyzed and used the oxidation stabilization of CCCs to improve their mechanical strengths. Mu et al. [8] characterized the chemical composition and mechanical properties of CCCs modified with synthetized single-crystalline hafnium carbide nanowires added into carbon fiber preforms, and as a result, flexural strength of CCCs was improved by 79.3%. Qin et al. [9] investigated the oxidation kinetics of four-direction CCCs in H₂O–Ar atmosphere at temperatures ranging from 600 to 1,400°C.

Theoretical and numerical research works on the oxidation behaviors of CCCs are also important. Vignoles et al. [10] proposed an approach of ablation by sublimation, and implemented a Brownian motion method to simulate efficiently the process of mass transfer [11]. Lachaud et al. [12] proposed a strategy to predict behaviors of a composite and its components, and provided a method for the identification of a combination with convection and reactions [13]: they modeled the evolution of surface roughness and effective reactivity [14]. Xie et al. [15] investigated the oxidation behavior and microstructure evolution of specimens with Lu₂O₃-SiC-HfB₂ ceramic coating at 1,700°C with an experimental study and first-principles simulations, implemented with density functional theory (DFT) methods. Dang et al. [16] studied a laser ablation process, expressed in terms of the enthalpy of each phase, with a finite element method (FEM); processes of thermal conduction, heat radiation, oxidation and sublimation were considered in these simulations. Zhang [17] investigated the microstructure and thermal stresses in the coating after ablation, and analyzed the effect of thickness and thermal stresses on ablation resistance of HfC coating, combining heat flux ablation tests and finite element analysis. The study indicated that the coating thickness could effectively release the thermal stress generated during the ablation process.

Various studies demonstrated the nontrivial geometry of the oxidation domains. Fractal geometry is a method to study the irregular curves with self-similar configurations [18]. A parameter describing this self-similarity is a fractal dimension. In oxidation of solid particles, a process developing from an outer surface to inner layers, the volume of changing particles and the shape of a process zone remain the same, hence, it can be approximated with a self-similar process.

In this study, fractal strategies are employed to predict the oxidation behavior of NP CCCs, considering the mass transfer at short times together with a long-time concentration decay. Longitudinal and transverse heat conduction in a fiber bundle is investigated, in order to analyze its oxidation mechanism. Multiscale oxidation behavior of NP CCCs and rough surfaces of oxidized fiber bundles are predicted employing a fractional Brownian motion (FBM) strategy.

2 Mesoscopic structure of NP CCCs

Carbon fiber preforms were orthogonally placed at 0/90°, using unidirectional 12 K T700 carbon laminas, with short-cut fiber felt cloths sandwiched between them [19] (Figure 1). Then, they were punched employing a specially designed panel with needles and heat-treated at 2,000°C for 3 h. It was followed by densification with isothermal chemical vapor infiltration (CVI) at 1,000°C under 1 kPa pressure, with the natural gas as pyrocarbon precursor, and H₂ as the carrier and diluent gas.

Figure 1

Structures of NP C/Cs [19].

A 3D needle-punched technology could balance the cost and performance, it effectively reduces the manufacturing cost providing good interlayer performance. It introduces parts of horizontal (in XY plane) short-cut fibers of the felts into the neighboring layers, forming conical clusters of short fibers along the thickness direction (Z direction). After punching orthogonally twice, a region of 24 mm × 24 mm was considered as minimum period distribution pattern of holes, as shown in Figure 2.

Figure 2

Minimum period distribution of needle holes.

3 Theoretical model

3.1 Standard diffusion model

Carbon oxidation is considered as a mass transfer process. Oxygen transport limits the surface recession rate for both fiber and matrix phases [14], and the diffusion coefficient influences the effective oxidation rate.

The second Fick’s law of diffusion equation has the following form:

(1) ∂ C ∂ t = D ∂ 2 C ∂ x 2 ,

and the boundary condition was expressed as follows:

(2) ∂ C ∂ x ∣ x = 0 = 0 ,

where C is the linear concentration; x is the spatial coordinate; D is the effective diffusion coefficient and t is the time.

At the initial time, the concentration was constant on the oxidized surface of the sample:

(3) C ( x , t ) ∣ t = 0 = C 0 ( x ) = const , 0 ≤ x ≤ L ,

where L is the length of the sample along the oxidation direction.

The following formalism [20] was defined with the Green’s function G for equation (1) in order to get the solution:

(4) G ( x , t , ε ) = 1 4 π D t exp − ( x − ε ) 2 4 D t + exp − ( x + ε ) 2 4 D t ,

where ε is the unitary mass convolution.

The solution of a standard diffusion equation (equation (1)) by integrating the relation with respect to ε can be given as follows [21]:

(5) C ( x , t ) = C 0 2 erf L − x 4 D t + erf L + x 4 D t ,

where “erf” is the abbreviation of error function.

The concentration evolution with time on the permeable boundary ( x = L ) of the pellet was obtained.

(6) C ( L , t ) = C 0 2 erf L D t .

The Taylor series expansion was applied, and neglecting the high-order terms, the mass transfer description at short times was obtained as follows:

(7) C s ( L , t ) = C 0 L π D t .

The long-time concentration decay on the permeable boundary was given [21] as follows:

(8) C l ( L , t ) = C 0 A exp − π 2 L 2 D t .

where A is the coefficient that depends on the initial and boundary conditions.

3.2 FBM model

Fractals and multifractals [18,22] are self-similar and self-affine sets, which can be found in many structures and evolution of complex systems. Thousands of solid carbon particles are stochastically oxidized as gaseous CO and CO₂, so a relationship between the oxidation behavior and fractal evolution of its front is the focus of this study.

FBM is widely used in analysis of stochastic processes [23]. It exhibits a long-range property dependence and self-similarity. The FBM with Hurst parameter H ∈ ( 0 , 1 ) is a zero mean Gaussian process B H = { B t H , t ≥ 0 } with the following covariance function:

(9) R H ( s , t ) = 1 2 ( s 2 H + t 2 H − ∣ t − s ∣ 2 H ) , ( s , t ≥ 0 ) .

An FBM model was employed [21,24] to describe the mass transfer in oxidation. Since diffusion is a random process, the effective diffusion coefficient ( D , in equation (1)) is determined by a long-time coherence effect of an irregular fluctuation force. This coefficient is described as follows:

(10) D ( t ) = K β F ( β ) t β − 1 ,

where β is a parameter determined by the dynamic oxidation process.

So, the diffusion equation (1) can be rewritten as follows:

(11) ∂ C ∂ t = K β F ( β ) t β − 1 ∂ 2 C ∂ x 2 ,

When F ( β ) = β , equation (11) is simplified as follows:

(12) ∂ C ∂ t = K β β t β − 1 ∂ 2 C ∂ x 2 ,

where K β is the FBM diffusion coefficient. Equation (12) is the Fokker–Planck equation, describing the FBM.

Accounting for a probability density function, the Green’s function for the FBM diffusion equation ( G β ) with the boundary condition (2) is given by [21]

(13) G β ( x , t , ε ) = 1 4 π K β t β exp − ( x − ε ) 2 4 K β t β + exp − ( x + ε ) 2 4 K β t β .

The short-time ( C β s ) and the long-time ( C β l ) solutions were derived:

(14) C β s ( L , t ) = C 0 L π K β t β ,

(15) C β l ( L , t ) = C 0 A exp − π 2 K β t β 4 L 2 .

4 Simulation of oxidation and mechanical behaviors

4.1 Multiscale heat conductions

The oxidation process is mainly influenced by the temperature for sufficient supply of oxygen [25,26,27]. Microscale longitudinal and transverse heat conduction and oxidation behaviors of fiber bundles were simulated. The results obtained for the longitudinal direction of the fiber bundle (Figure 3(a)), demonstrated that the heat transfer interface had a peak, with the temperature near boundary higher than that in the specimen’s interior on the same plane during heat transfer. Similar results were obtained for the transverse direction of the fiber bundle (Figure 3(b)). The isothermal contour plots of longitudinal and transverse heat transfers showed peak morphology.

Figure 3

Heat conduction of fiber bundle: (a) longitudinal and (b) transverse.

The density of a material is an effective indicator of its internal interconnection state. At higher material density, a better-connected state is observed, with fewer defects and easier heat conduction, so the thermal conductivity is higher. On the other hand, if the density of the material is lower, many structural defects hinder the heat conduction channels, so the thermal conductivity is lower. Due to the CVI process, the density of fiber bundles of preform is higher than that of the pyrocarbon matrix, so the heat transfer shows the peak morphology.

To study the effect of needle punching, a mesoscale representative volume element (RVE) of NP CCCs was developed for analysis of heat conduction (Figure 4). The dimensions of RVE were 24 mm × 24 mm × 1 mm; it was layered with two orthogonal layers. Meshed circular cylindrical areas represented the punched short-cut fiber felts that enhanced the interlaminar properties. The meshing of the punched areas was refined.

Figure 4

RVE of NP CCCs: (a) geometrical model and (b) mesh model.

Heat conduction in composite was investigated numerically. The thermal conductivity parameters of the components [28,29] are listed in Table 1.

Table 1

Thermal conductivity parameters of components

	Description	Temperature (K)	Value (W m⁻¹ K⁻¹)
Fiber	Thermal conductivity	1,000	80
		1,100	82
		1,200	83
	Thermal diffusivity	1,000	70
		1,100	65
		1,200	60
Pyrocarbon	Thermal conductivity	1,000	78
		1,100	80
		1,200	81
	Thermal diffusivity	1,000	68
		1,100	63
		1,200	58

Mesoscale heat conduction along the horizontal and vertical directions was investigated. Along the horizontal direction, heat flux was transferred from a high-temperature side to inner areas. Due to different thermodynamic properties of the orthogonal layers and punched areas, the heat transfer interface was not straight. Inside the punched holes, the punched short-cut felts formed vertical structures, so the horizontal heat transfer was influenced by the punched areas as shown in Figure 5.

Figure 5

Simulation results for horizontal heat conduction.

The vertical heat conduction of RVE was simulated (Figure 6) with the heat source applied on the top surface. It was found that the vertical heat conduction, influenced by the layers and inherent defects, resulted in the isotherms of heat conduction changed by the layer interfaces and the punched areas.

Figure 6

Simulation results for vertical heat conduction: (a) 5 s and (b) 10 s.

The heat conductivity coefficient is an indicator of oxidation and ablation. The heat flux was transferred into inner parts of the sample from the environment, and the amount of heat inside the sample increased, starting the oxidation reaction in the positions with high initial temperature.

Heating curves for different points (Figure 7a) at distances of 0.1 mm (point A), 1.1 mm (point B), 2.1 mm (point C) and 4.1 mm (point D) from the heating end were plotted (Figure 7b). In 20 s, temperature of point A was raised with a two-stage trend, it increased quickly at first, followed by a final linear growth, while temperature in point B raised nonlinearly but more gently. Temperature in point C increased approximately linearly, while temperature in point D grew with the lowest rate. At the 20th second, the temperatures of points A, B, C and D were 918, 659.7, 535.9 and 353 K, respectively.

Figure 7

Temperature–time curves of monitoring points: (a) monitoring points and (b) temperature–time curves for points in (a).

4.2 Oxidation behavior simulated with FBM strategy

Oxidation behavior of the studied component was predicted with the FBM strategy [30], which is defined as

(12) B H ( t ) = 1 Γ H + 1 2 ∫ 0 t ( t − s ) H − 1 2 d B ( s ) ,

where Γ is the gamma function, and B is the Brownian motion process [31,32].

Since oxidation is a stochastic process in the reaction regions, the mass loss ratios of fiber bundles, matrix and composite were calculated with the FBM strategy. The calculated curves of mass loss evolution with time are plotted in Figure 8. Apparently, the oxidation process in the components demonstrated the approximate linear growth. In the initial 24 min, their mass loss ratios increased fast, followed by a constant rate growth in the matrix. The mass loss of fiber bundles showed a periodic linear growth, since fiber bundles are transversely isotropic, the transverse oxidation reaction had a periodic character due to the complete oxidation of fibers one by one.

Figure 8

Mass loss ratios of components and composite calculated with FBM strategy.

The oxidation experiment of the composite at 550°C was performed. Three samples were oxidized for 1 h in the muffle furnace, and they were weighed every 15 min. The results demonstrated that the mass loss ratio of CCCs increased linearly, with the final magnitude of 1.8%. Apparently the theoretical and numerical results agreed well with the experimental data (Figure 8).

Rough surfaces of oxidized fiber bundles were simulated with the FBM strategy, the results of these simulations are presented in Figure 9(a). They showed that oxidized fiber bundles formed bamboo shoots morphology, because the oxidation rate of the matrix was higher than that of the fibers. The roughness of oxidized fiber bundles was 1.2 × 10⁻³ mm.

Figure 9

Morphologies of the oxidized fiber bundles: (a) simulation and (b) experiment.

The micro-morphology of the oxidized NP CCC was observed with SEM (Figure 9b). When oxidation reached the steady state, the pyrocarbon matrix between the fibers was oxidized first due to the higher porosity and oxygen permeability. So, the oxidized fibers presented the obvious bamboo shoots shape more slowly. As the oxidation continued, the formed shoots were gradually corroded.

5 Conclusion

The observed heat conduction interface in NP CCCs is wavy since the punching technology changed the component structures. Temperature of the composite was raised with a two-stage trend first. The macroscopic oxidation behavior of NP CCCs was investigated numerically with the fractal strategy. The magnitudes of mass losses were predicted for the fiber bundles, the matrix and the composite, with the oxidation process of components demonstrating an approximately linear growth. When the oxidation reached the steady state, the pyrocarbon matrix between the fibers was oxidized first due to the higher porosity and oxygen permeability, so the oxidized fibers exhibited the obvious bamboo shoots shape more slowly.

Funding information: This work is partially supported by the National Natural Science Foundation of China (Grant no. 12102152) and the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (Grant no. MCMS-E-0221Y02).
Conflict of interest: Authors state no conflict of interest.
Data availability statement: The datasets generated and analyzed in the current study are available from the corresponding author on reasonable request.

References

[1] Manocha LM, Fitzer E. Carbon reinforcement and C/C composites. Berlin: Springer; 1998.10.1007/978-3-642-58745-0Search in Google Scholar

[2] Luthra KL. Oxidation of carbon/carbon composites-a theoretical analysis. Carbon. 1998;26:217–24.10.1016/0008-6223(88)90040-1Search in Google Scholar

[3] Huang JG, Guo LJ, Li KJ, Yan NN, Zhou L, Li YY. Microstructures and oxidation behaviors of Al-modified and Al2O3 modified SiC coatings on carbon/carbon composites via pack cementation. Ceram Int. 2021;47:8105–12.10.1016/j.ceramint.2020.11.165Search in Google Scholar

[4] Wang H, Ji RT, Qu F, Bai JQ, Fu QG, Li HJ. Three-dimensional pore-scale study of the directional heat transfer in a high thermal conductivity carbon/carbon composite protection system. Aerosp Sci Technol. 2021;112:106609.10.1016/j.ast.2021.106609Search in Google Scholar

[5] Guo WM, Xiao HN, Zhang GJ. Kinetics and mechanisms of non-isothermal oxidation of graphite in air. Corros Sci. 2008;50:2007–11.10.1016/j.corsci.2008.04.017Search in Google Scholar

[6] Deng Y, Li WG, Ma JZ, Li Y. Thermal-mechanical-oxidation coupled first matrix cracking stess model for fiber reinforced ceramic-matrix composites. J Eur Ceram Soc. 2021;41(7):4016–24.10.1016/j.jeurceramsoc.2021.02.033Search in Google Scholar

[7] Kim J, Jo A, Choi Y, Lee K, Lm J, Bai B. Improving the mechanical strength of carbon-carbon composites by oxidative stabilization. J Mater Res Technol. 2020;9(6):16513–21.10.1016/j.jmrt.2020.11.064Search in Google Scholar

[8] Mu JR, Shi XH, Zhang HR, Yang L, Han X. Microstructures and enhanced flexural properties of single-crystalline HfC nanowires in situ modified carbon/carbon composites by electrophoresis-thermal evaporation using CNTs as the template. Ceram Int. 2021;47(3):3063–9.10.1016/j.ceramint.2020.09.142Search in Google Scholar

[9] Qin F, Peng LN, He GQ, Li J, Yan Y. Oxidation kinetics and mechanisms of carbon/carbon composites and their components in water vapour at high temperatures. Corros Sci. 2015;90:340–6.10.1016/j.corsci.2014.10.027Search in Google Scholar

[10] Vignoles GL, Lachaud J, Aspa Y, Goyheneche JM. Ablation of carbon-based materials: Multiscale roughness modelling. Compos Sci Technol. 2009;69:1470–7.10.1016/j.compscitech.2008.09.019Search in Google Scholar

[11] Lachaud J, Vignoles GL. A Brownian motion technique to simulate gasification and its application to C/C composite ablation. Comput Mater Sci. 2009;44(4):1034–41.10.1016/j.commatsci.2008.07.015Search in Google Scholar

[12] Lachaud J, Aspa Y, Vignoles GL. Analytical modeling of the steady state ablation of a 3D C/C composite. Int J Heat Mass Transf. 2008;51:2614–27.10.1016/j.ijheatmasstransfer.2008.01.008Search in Google Scholar

[13] Lachaud J, Bertrand N, Vignoles GL, Bourget G, Rebillat F, Weisbecker P. A theoretical/experimental approach to the intrinsic oxidation reactivities of C/C composites and of their components. Carbon. 2007;45:2768–3776.10.1016/j.carbon.2007.09.034Search in Google Scholar

[14] Lachaud J, Aspa Y, Vignoles GL. Analytical modeling of the transient ablation of a 3D C/C composite. Int J Heat Mass Transf. 2017;115:1150–65.10.1016/j.ijheatmasstransfer.2017.06.130Search in Google Scholar

[15] Xie W, Fu QG, Cheng CY, Li J, Wang HH, Li K. Experimental and first-principles simulation study on oxidation behavior at 1700°C of Lu2O3-SiC-HfB2 ternary coating for SiC coated carbon/carbon composites. Ceram Int. 2022;48:8088–96.10.1016/j.ceramint.2021.12.010Search in Google Scholar

[16] Dang X, Yin X, Fan X, Ma Y, Wang J, Ju P, et al. Microstructural evolution of carbon fiber reinforced SiC-based matrix composites during laser ablation process. J Mater Sci Technol. 2019;35:2919–25.10.1016/j.jmst.2019.04.042Search in Google Scholar

[17] Zhang J, Zhang YL, Fu YQ, Hu D, Meng JC, Li T. Ablation behavior of HfC coating with different thickness for carbon/carbon composites at ultra-high temperature. J Eur Ceram Soc. 2021;41:1769–78.10.1016/j.jeurceramsoc.2020.10.055Search in Google Scholar

[18] Mandelbrot BB. The fractal geometry of nature. San Francisco, USA: Freeman; 1982.Search in Google Scholar

[19] Han M, Zhou CW, Bi QS. Residual mechanical properties of needle-punched carbon/carbon composites after oxidation. Compos Commun. 2021;28:100966.10.1016/j.coco.2021.100966Search in Google Scholar

[20] Jeon J-H, Chechkin AV, Metzler R. First passage behavior of multi-dimensional fractional Brownian motion and application to reaction phenomena. First-Passage Phenomena and Their Application. World Scientific; 2014. p. 175–202.10.1142/9789814590297_0008Search in Google Scholar

[21] Zhokh A, Strizhak P. Macroscale modeling the methanol anomalous transport in the porous pellet using the time-fractional diffusion and fractional Brownian motion: A model comparison. Commun Nonlinear Sci Numer Simul. 2019;79:104922.10.1016/j.cnsns.2019.104922Search in Google Scholar

[22] Evertsz JG, Mandelbrot BB. Multifractal measures. In: Peitgen HO, Jurgens H, Saupe D, editors. Chaos and fractals: New frontiers of science. New York: Springer-Verlag; 1992. 921–54.Search in Google Scholar

[23] Shen GJ, Xiang J, Wu JL. Averaging principle for distribution dependent stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. J Differ Equ. 2022;321:381–414.10.1016/j.jde.2022.03.015Search in Google Scholar

[24] Wang KG, Lung CW. Long-time correlation effects and fractal Brownian motion. Phys Lett A. 1990;151:52–56.10.1016/0375-9601(90)90175-NSearch in Google Scholar

[25] Akhtar M, Khajuria A, Sahu JK, Swaminathan J, Kumar R, Bedi R, et al. Phase transformations and numerical modelling in simulated HAZ of nanostructured P91B steel for high temperature applications. Appl Nanosci. 2018;8:1669–85.10.1007/s13204-018-0854-1Search in Google Scholar

[26] Akhtar M, Khajuria A. The synergistic effects among crystal orientations, creep parameters, local strain, macro-microdeformation, and polycrystals’ hardness of boron alloyed P91 steels. Steel Res Int. 2022;93:2100819.10.1002/srin.202100819Search in Google Scholar

[27] Khajuria A, Akhtar M, Bedi R. A novel approach to envisage effects of boron in P91 steels through Gleeble weld-HAZ simulation and impression-creep. J Strain Anal Eng Des. 2022;8:1–15.10.1177/03093247211061943Search in Google Scholar

[28] Pardere C, Batsale JG, Goyheneche JM, Pailler R, Dilhaire S. Thermal properties of carbon fibers at very high temperature. Carbon. 2009;47:737–43.10.1016/j.carbon.2008.11.015Search in Google Scholar

[29] Sessim M, Shi L, Phillpot SR, Tonks MR. Phase-field modeling of carbon fiber oxidation coupled with heat conduction. Comput Mater Sci. 2022;204:111156.10.1016/j.commatsci.2021.111156Search in Google Scholar

[30] Chen CC, Daponte JS, Fox MD. Fractal feature analysis and classification in medical imaging. IEEE Trans Med Imaging. 1989;8(2):133–42.10.1109/42.24861Search in Google Scholar PubMed

[31] Heydari MH, Mahmoudi MR, Shakiba A, Avazzadeh Z. Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion. Commun Nonlinear Sci Numer Simul. 2018;64:98–121.10.1016/j.cnsns.2018.04.018Search in Google Scholar

[32] Heydari MH, Avazzadeh Z, Mahmoudi MR. Chebyshev cardinal wavelets for nonlinear stochastic differential equations driven with variable-order fractional Brownian motion. Chaos Solitons Fractals. 2019;124:105–24.10.1016/j.chaos.2019.04.040Search in Google Scholar

Received: 2022-07-25

Revised: 2022-11-15

Accepted: 2022-11-28

Published Online: 2022-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Multiscale heat conduction and fractal oxidation behaviors of needle-punched carbon/carbon composites

Abstract

1 Introduction

2 Mesoscopic structure of NP CCCs

3 Theoretical model

3.1 Standard diffusion model

3.2 FBM model

4 Simulation of oxidation and mechanical behaviors

4.1 Multiscale heat conductions

4.2 Oxidation behavior simulated with FBM strategy

5 Conclusion

References

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