Probabilistic reliability analysis of carbon-carbon composite nozzle exit cone with uncertain parameters

We describe in this paper a methodology to perform the probabilistic and reliabilitybased design of a novel carbon/carbon rocket nozzle subjected to operational thermal and mechanical loads. In this methodology the nozzle is represented by a multiphysics finite element model capable of predicting the temperature and stress fields of the exit cone. The analysis shows * Professor, Science and Technology on Advanced Composites in Special Environment Laboratory, Harbin Institute of Technology, No.2 Yikuang Street, Harbin 150080, China. † Ph.D. Student, Science and Technology on Advanced Composites in Special Environment Laboratory, Harbin Institute of Technology, No.2 Yikuang Street, Harbin 150080, China. ‡ Senior engineer, STAR UAV Systems co., Ltd. No.2999 Westport Avenue, Shuangliu District, Chengdu, China. ※ Ph.D. Dynamics Group, Department of Mechanical Engineering, Imperial College London, SW7 2AZ, London, UK. § Professor, Bristol Composites Institute (ACCIS), University of Bristol, BS8 1TR Bristol, UK. ☆ Associate professor, Science and Technology on Advanced Composites in Special Environment Laboratory, Harbin Institute of Technology, No.2 Yikuang Street, Harbin 150080, China. that the most likely failure modes of the exit cone are related to compressive loading along the axial and hoop directions, and interlaminar shear. The probabilistic models used in this methodology account for the uncertainty of the material properties by using uniform and normal distributions and different variances. The reliability analysis is performed by using surface response methods. A global sensitivity analysis is also carried out using polynomial expansion chaos surface response models. A particular novelty of analysis is the use of Sobol indices to rank the importance of the single uncertain parameters in the models. The methodology provides a high level of confidence and robustness in determining that the axial thermal conductivity of the carbon/carbon material is the most critical material property to affect the three main failure modes, while the coefficient of the thermal expansion and the heat capacity play a very marginal role.

that the most likely failure modes of the exit cone are related to compressive loading along the axial and hoop directions, and interlaminar shear. The probabilistic models used in this methodology account for the uncertainty of the material properties by using uniform and normal distributions and different variances. The reliability analysis is performed by using surface response methods. A global sensitivity analysis is also carried out using polynomial expansion chaos surface response models. A particular novelty of analysis is the use of Sobol indices to rank the importance of the single uncertain parameters in the models. The methodology provides a high level of confidence and robustness in determining that the axial thermal conductivity of the carbon/carbon material is the most critical material property to affect the three main failure modes, while the coefficient of the thermal expansion and the heat capacity play a very marginal role. conventional materials, the carbon/carbon version involves a simplified design that produces a reduction in weight and internal ablation, and also features improved operational reliability.
The operational environment of a solid rocket nozzle is very complex and challenging, with temperatures usually exceeding 3000°C and the presence of high pressure fields. The internal part of the nozzle is also subjected to sharp heat flux gradients and erosion. The short time burn generates some significant temperature gradients, and the complex thermodynamic environment generated by the presence of high pressures presents a critical challenge to the structural design of the rocket nozzle. The C/C nozzle exit cone structure is one of the most critical components in the SRM system, leading to potential failures during use.
Several efforts have been devoted to understanding the behavior of C/C composites at high temperatures [1][2][3][4][5]. Li et al [1] and Peng et al [2] investigated the microstructure and the ablation mechanism of C/C composites with scanning electron microscopy. Other research groups have focused on the effect of the ablative environment on the ablation behavior of C/C composites and have obtained valuable performance data. For example, Liu et al [3] studied the ablation characteristics of a C/C composite in a lab-scale solid rocket motor under a flux of combustion products containing a high content of particulate alumina. Zaman et al [4] reported on the residual mechanical and thermophysical properties of C/C composites repeatedly ablated using 3000 °C oxyacetylene flame. From the numerical standpoint, Vignoles et al., [5] identified an efficient method to simulate the ablation of C/C composites by considering the bulk transport of reactants and heterogeneous mass transfer conditions. In recent years, several groups have also started to consider the ablation behavior of ceramic based nozzles doped with ceramic composition (e.g. ZrB2, ZrC, etc.) for enhanced performance at high temperatures [6,7]. These works have built on the know-how accumulated on studying traditional C/C nozzles Other significant activities related to C/C nozzle design have considered various aspects of heat transfer [8][9][10]. The analysis of the reliability of C/C composite nozzle exit cone structures is however scarcely represented in open literature [11,12]. The reasons behind this lack of information are mainly related to the significant costs involved in the development and testing of these composites, and also confidentiality issues related to intellectual property ownership [13]. The resources involved in obtaining large amounts of experimental data may be prohibitive, and that justifies the use of reliability-based models to design solid rocket motor components. The results from system reliability analyses calculated on small data samples are affected by significant errors. Therefore numerical models have been widely used to simulate the thermal-structural response of the nozzle and integrate the reliability-based approach [13][14][15][16][17]. Morozov and de la Beaujardiere [13] developed a finite element method to investigate the dynamic thermo-structural response of a composite rocket nozzle throat. Goyal et al [16] developed reliable reduced size models based on 2D plane strain assumptions for SRM structural analysis. Turchi et al [17] suggested a numerical approach to describe a carbon-phenolic SRM nozzle model and then investigated the role of the most important uncertainty parameters affecting the design. Heller et al [11,12] pioneered a methodology for reliability analysis of C/C composites, and analyzed the stress state of a cylindrical structure consisting of multiple layers of C/C composite under thermal and pressure shock by assuming elasticity within the structure. The reliability of the composite configuration was also calculated. Bozkaya et al [18,19] and Akpan and Wong [20] developed some useful and efficient methods for the sensitivity analysis and reliability calculations [21] based on surface response methods (SRM) and Monte Carlo simulation techniques.
In this paper we describe a probabilistic design methodology to assess, at the initial design stage, the reliability of carbon/carbon rocket nozzles under thermal and mechanical load conditions. The proposed finite element methodology allows the estimation of the reliability and probability using a combination of surface response methods and sensitivity analysis based on Sobol's approach with Polynomial Chaos expansions for the surface responses. A particular advantage of this approach also consists in the reduction of the computational resources required to perform the overall analysis.
The paper is organized as follows. In the first section we introduce a numerical multiphysics (thermal and structural) finite element analysis methodology to design a C/C exit cone with uncertain parameters, and present a method to estimate the probability of structural failure by combining the finite element model with response surface methods. In the second part of the paper the reliability metrics of the exit cone are calculated. Finally, the extent and the influence of different failure modes on structural reliability are analyzed and discussed. The global sensitivity of the uncertain parameters to the failure modes is also evaluated through a Sobol analysis.
2 Thermal stress and failure mode analysis of the C/C exit cone

Model of the rocket nozzle
The nozzle is represented as an axisymmetric structure ( Figure 1). The nozzle consists of a throat insert, an exit cone, a back wall, the inlet section and a metal flange. The exit cone is made from 2D needle-punched felt reinforced C/C composites [22,23]. Carbon-felt reinforced C/C composites constitute the throat, and carbon cloth phenolic makes its inlet. The back wall insulation is made of silica cloth phenolic insulation materials, and high-strength alloy steel is  Tables 1, 2 and 3. The parameters k, c, α, υ, E, ρ represent the thermal conductivities, heat capacities, thermal expansion coefficients, Poisson's ratios, elastic moduli and densities, respectively. The subscripts 1, 2 and 3 indicate the principal orthotropic directions of the materials (i.e., the radial, tangential and axial directions in a cylindrical coordinate system) respectively.   The thermal domain (i.e., the transient heat transfer) representing the nozzle is modeled as axisymmetric because of the rotational symmetry of the nozzle geometry. The gas flow generates nearly 6MPa pressure [13] and wall shear stresses on the inner wall of the nozzle.
Because the magnitude of the wall shear stress is very small [22], its influence on the nozzle performance can be therefore neglected. The pressure generated by the gas flow generates the main mechanical load. The heat transfers between the high-temperature gas and the inner wall generally can be classified into convective, radiant and conductive. The forced convective heat is the most significant factor for the heat transfer [23][24][25][26].
To simplify the model we impose the following assumptions: (1) only convective heat transfers between the flow and the nozzle are considered on the inner wall of the nozzle; radiation and conduction are ignored [23][24][25][26]. The mechanical erosion of the material and the ablation of the inner nozzle wall are also neglected [27,28]. (2) Only convective heat transfer between the air and the outer wall of the nozzle is considered, and the radiation heat loss is neglected [29][30][31][32]. (3) The flow field parameters (pressure and temperature) do not change for a SRM in steady-state operations (4) The thermally induced erosion of the insulation material is ignored [27,28].
The temperature profile around the nozzle, the gas pressure and the convective heat transfer coefficient vary with the axial distance ( Fig. 2. (a) and (b)) [32].

Results of the numerical analysis
The overall structural temperature reaches its peak at the end of the time domain simulation. At the entrance of the throat the highest temperature is 3191 °C, and 2091 °C at the exit cone. The temperature of the external alloy steel is less than 30 °C at the end of the simulation.

Radial stress of the exit cone
The radial stress field of the exit cone is shown in Fig. 4(a). The maximum radial tensile stress occurs at point A, which is located at the sharp contact corner of the exit cone and the throat.
During the thermal loading stress concentrations appear at this corner. The maximum radial tensile stress is 9.8MPa, which is significantly lower than the value of the material tensile strength (160MPa). While, the stress concentration at sharp corners does not represent the correct stress magnitude, it must be larger than the actual stress, so he radial tensile stress should not cause failure of the exit cone. The maximum radial compressive stress occurs at point B, with the compressive stress being mainly generated by the thermal expansion of the throat. The maximum radial compressive stress is 60.2MPa, which is again lower than the value of the material compressive strength (140MPa). The radial compressive stress therefore should not damage the exit cone.

Axial stress at the exit cone
The axial stress field of the exit cone is shown in Fig. 4(b). The maximum axial tensile stress occurs at point C, which is located at the contact corner of the exit cone and the back wall. The maximum axial tensile stress is 28MPa, lower than the value of the material tensile strength.
The maximum axial compressive stress is 75.5MPa after 5.

Hoop stress of the exit cone
The distribution of the hoop stresses is shown in Fig. 4(c).

Interlaminar shear stress of the exit cone
The interlaminar shear stress of the exit cone is shown in Fig. 4.

Reliability analysis of the exit cone
The structure of the nozzle is complex. The high combustion temperature and the internal gas flow velocity create an extremely harsh environment for the nozzle materials. Moreover, size scale effects given by the discrete nature of the C/C microstructure bring further uncertainties in determining the overall mechanical performance and directly affect the reliability of the nozzle structure.

Quantization of parameter uncertainty
In the probability model, the uncertainty comes from the discretization of the uncertainty from the variability of the material properties. The material properties are usually measured through experiments, and the uncertainty includes not only the natural dispersion of the properties, but also the testing errors. In this paper, the value of the structural strength at failure is also non-deterministic. For simplicity we assume that the material properties follow a normal distribution, and the strength has a uniform distribution. In order to facilitate the calculation, seven main material parameters are normalized as shown in Table 4, and the probability distribution types of material properties are also specified.

Computational process
From the above analysis it is evident that there are three main factors leading to the failure of exit cone: axial compression, hoop compression and interlaminar shear failures. The maximum stress failure criterion is used to evaluate the reliability of the exit cone. Failure of the exit cone is assumed to occur if the stress is greater than the strength along specific directions. The maximum stress failure criterion does not consider interaction between the failure modes, but it is a conservative approximation of failure especially for combined tensile/shear/compressive loading.
Among the most common structural reliability analysis methods, response surfaces and Monte Carlo simulations are well developed and benchmarked. In this study we combine a surface response method (SRM) with finite elements, as well as Monte Carlo simulations to analyze the reliability. We first obtain the structural performance functions for the different failure modes by response surface approximation. To this end, 300 sample sets used to obtain the surface response are derived by optimization using Latin Hypercube sampling of the uncertainty space. Each set is computed independently by fitting higher order polynomials. We finally obtain a surface response for the three types of failure modes by higher order polynomial fit, and then the response surface is used for further reliability analysis (see the flowchart in Figure 6).

Reliability based on probabilistic theory
The structural performance function of the exit cone can be represented by a two-dimensional function that relates response and capability. If R represents the capability -i.e. strength -and S is the response (stress), it is possible to denote the performance function as: (3) For the three failure modes concerned in this study the corresponding limit state functions can be presented as follows: The subsequent reliability analyses from Monte Carlo simulations are performed based on these SRMs in order to reduce the computational expense. The reliability index , failure probability , and the standard deviation and coefficient of failure probability are calculated using a Monte Carlo method with 5,000 and 10,000 Monte Carlo simulations respectively (Table 5).

The results of the failure probability analysis
The reliability results for the three failure modes are listed in Table 6. The results show that the axial compression failure is the most likely cause of the structural failure. Failure can also occur by interlaminar shear, while hoop compression is the least likely of the three to cause failure.
Variance based sensitivity analysis, namely Sobol analysis, is a form of global sensitivity analysis. It decomposes the variance of the output of the model or system into fractions which can be attributed to inputs or sets of inputs [34]. It would be very useful to identify the most significant input uncertain parameters that contribute to the variation of the different types of stress associated to these three failure modes. In this case study, the Sobol analysis method is used in this case to carry out the global sensitivity analysis [35]. of the given input factor is measured by a term defined as the sensitivity index, which is the fractional contribution to the output variance due to the uncertainties in the inputs. The Sobol sensitivity index can be expressed as: 7* = 1 −~*/ ( ) (11) 4 is the first order Sobol sensitivity index, 46 denotes as the second order term and 7* denotes as the total Sobol sensitivity index corresponding to the ith model parameter. The resulting Sobol's sensitivity indices therefore rely on not only the input parameter distribution, but also the contribution of the input parameter in the mathematical model. The first order Sobol indices 4 are used to quantify the separate effect on the failure modes of each input parameter, while the total Sobol indices 7* quantify the total effect of the input parameter, which include both the first order effect and also its interactions with the other input variables [36].
The use of Polynomial Chaos Expansion (PCE) techniques is widely recognized as a quick way to directly compute the Sobol sensitivity indices. PCE is therefore employed in this case study.
Since the distribution of the input parameters (i.e., the thermal expansion coefficient, heat capacity, radial thermal conductivity, axial thermal conductivity, axial elastic moduli, radial elastic moduli and density) are assumed to have normal distributions, the PCE surface response is generated based on the Gaussian probabilistic distribution by using Hermite orthogonal polynomials. A linear enumeration strategy is employed to construct the multivariate orthonormal basis. The multivariate orthonormal basis is then truncated by using a fixed strategy to build the complete basis with respect to a maximal degree of four [37]. A nonintrusive method based on least squares minimization is used to optimize the PCE coefficients using 200 sampling points generated by the FE model. With this PCE response surface model, the Sobol's indices can then be computed directly without further Monte Carlo simulations. Figure 7 shows the first order and total Sobol's indices of these seven uncertain parameters for the three failure modes separately. The result shows that the indices of these uncertain parameters share very similar trends related to the axial compressive failure (Figure 7 (a)), hoop compression failure (Figure 7 (b)) and shear failure (Figure 7 (c)). Both types of indices clearly show that the axial thermal conductivity has the highest impact on the variation of the outputs (First order: 28%, 36% and 36%; Total: 60%, 70% and 70%), despite the coefficient of variation (CoV) of its uncertain distribution shown in Table 4 is the smallest among the seven input variables. The radial thermal conductivity comes second in terms of impact. However, in terms of total contributions, the second largest influence comes from the elastic moduli for the axial compression failure (25%). For the hoop compression and shear failures, the radial elastic moduli and the density equally take the second leading contributions by about 21%. This suggests that the coupling contribution from the elastic moduli and density is more significant than the one from the thermal conductivity and the elastic moduli along the radial direction.
The first order Sobol's indices clearly show that the thermal expansion coefficient and heat capacity provide negligible contributions to all the three failure modes. These two parameters could be therefore excluded in future uncertainty analysis. directions is relatively small. If we ignore local stress concentration in the corners of the exit cone, the most likely failure region is in the contact area between the exit cone and the throat induced by the axial compressive stress. Also, another critical area is represented by the front region of the exit cone exposed to the combustion chamber, which is characterized by the presence of hoop compressive stresses. The third possible failure mode is the interlaminar shear failure, which is located at the region of the exit cone far away to the throat.
A global sensitivity analysis also was carried out to identify the most influential input parameters on these three failure modes. A Sobol analysis was used to rank the impacts of the input parameters based on the PCE response surface model. The results show that the axial thermal conductivity has the highest contribution on all the three modes. The contributions from the thermal expansion coefficient and heat capacity are negligible. The conclusions from this probabilistic analysis are quite useful to streamline and optimize the outcome of the initial design stage for these C/C composite nozzles. The importance of the axial thermal conductivity in this type of structure may lead to a design of C/C composites and their stacking sequences that favor a strong unidirectional thermal conductivity response for the nozzle configuration. In a similar manner, the design and materials selection of these types of nozzles can be streamlined to prevent the axial compression failure, which is the most important failure mode of the system.
Compressive failure in C/C composites can be tailored by fibers architectures or needle-punch techniques [38], and these approaches could be used to increase the reliability of these carbonbased nozzle designs.