_{2}Ceramics Ablation Behavior

_{2}Ceramics Ablation Behavior

_{2}Ceramics Ablation Behavior

The ablation of ultra-high-temperature ceramics (UTHCs) is a complex physicochemical process including mechanical behavior, temperature effect, and chemical reactions. In order to realize the structural optimization and functional design of ultra-high temperature ceramics, a coupled thermo-chemo-mechanical bond-based peridynamics (PD) model is proposed based on the ZrB_{2} ceramics oxidation kinetics model and coupled thermo-mechanical bond-based peridynamics. Compared with the traditional coupled thermo-mechanical model, the proposed model considers the influence of chemical reaction process on the ablation resistance of ceramic materials. In order to verify the reliability of the proposed model, the thermo-mechanical coupling model, damage model and oxidation kinetic model are established respectively to investigate the applicability of the proposed model proposed in dealing with thermo-mechanical coupling, crack propagation, and chemical reaction, and the results show that the model is reliable. Finally, the coupled thermo-mechanical model and coupled thermo-chemo-mechanical model are used to simulate the crack propagation process of the plate under the thermal shock load, and the results show that the oxide layer plays a good role in preventing heat transfer and protecting the internal materials. Based on the PD fully coupled thermo-mechanical model, this paper innovatively introduces the oxidation kinetic model to analyze the influence of parameter changes caused by oxide layer growth and chemical growth strain on the thermal protection ability of ceramics. The proposed model provides an effective simulation technology for the structural design of UTHCs.

_{2}ceramics

Ultra-high-temperature ceramics (UHTCs) refer to a class of ceramic materials that can maintain physical and chemical stability under high-temperature and oxygen atmospheres. They are mainly multi-component composite ceramics composed of borides, carbides, nitrides and other transition metal compounds based on hafnium, zirconium, and molybdenum, such as ZrB_{2}, HfB_{2}, TaC, HfC, ZrC, and HfN [

The research on UHTCs began as early as 1960s [_{2}-and HfB_{2}-based ceramics render good anti-oxidation and ablation properties, and achieve long-term stability under ultra-high temperature, showing promise as non-ablative ultra-high-temperature heat protection materials. In 2007, Han et al. [_{2}-SiC composite under the oxyacetylene flame. At greater than 2200°C, the mass loss rate and linear oxidation rate of the composite after 10 min were found to be 0.23 mg/s and 0.66 μm/s, respectively. Also, a dense ZrO_{2} layer was formed on the oxidized surface, and no obvious cracks and denudation were observed. Then, in 2011, Wang et al. [_{2} layer when ZrC was exposed to a high-temperature oxygen environment. When the temperature is increased to 2800°C, the solid ZrO_{2} transforms into a molten state and covers the material surface. The formation of a dense ZrO_{2} layer prevents further ablation of C/C-ZrC composites. In 2020, Qing et al. [_{2}-SiC multiphase ceramic rods. The results demonstrated that the ablation mechanism of Zr-Si-B-C multiphase ceramics in the temperature range of 25°C to 3000°C generates B_{2}O_{3} film in the low-temperature zone, SiO_{2} film in the medium-temperature zone and ZrO_{2} film in the high-temperature zone. This Zr-Si-B-C multiphase ceramic ensures that the material will not be seriously ablated in the low-and high-temperature regimes. Based on these studies, it can be found that a liquid or solid oxide layer is formed on the surface of ablative UHTCs under a high-temperature aerobic environment, which can effectively inhibit or prevent the further reaction between external oxygen and ceramic matrix to reduce the amount of ablation.

At present, the ablation resistance mechanism of ZrB_{2} ceramic materials has been clearly understood, but the numerical theoretical research on ablation has not been fully studied. Considering the complexity of the actual service environment, a numerical model needs to be established to describe various complex physical and chemical reactions during the ablation process of UHTCs, and this numerical model can be used to study the factors affecting the ablation process, as well as the influence behavior and mathematical quantitative function of each factor against ablation, so as to predict the ablation rate, temperature distribution, structural deformation and denudation damage. These works are helpful to the structural optimization and functional design of UHTCs. In 2005, Fahrenholtz [_{2}, ZrO_{2} and B_{2}O_{3}. The proposed model can also predict the mass fraction of liquid B_{2}O_{3} remaining in porous solid ZrO_{2}. The model was experimentally verified by Talmy et al. [_{2}-SiC UTHCs and proposed the reaction sequence, oxide composition and formation rate during the oxidation process. Based on these studies, Parthasarathy et al. [_{2}, HfB_{2} and TiB_{2}. Obviously, the ultra-high-temperature ablation of ceramics needs to comprehensively consider the interaction of temperature field, structural deformation and chemical reactions. Zhou et al. [_{2} and proposed a thermal force coupling model, which could describe the multi-field coupling behavior of ZrB_{2} during high-temperature oxidation and predict the stress state of ceramic matrix and oxide layer during ablation process. Similarly, Wang et al. [_{2}-SiC ceramics.

Though these studies can describe the changes in chemical reactions during the ablation process of UHTCs, the ultra-high-temperature ablation of ceramics is also accompanied by some discontinuous problems, such as thermal response, material damage, boundary movement and structural failure. Obviously, the previous studies did not consider these discontinuities. However, the peridynamics (PD) render unique advantages in dealing with discontinuities. The peridynamic theory was first proposed by Silling as a nonlocal theory [

The thermo-chemo-mechanical coupling needs to comprehensively consider the complex interactions between temperature field, chemical reaction and structural displacement. In recent years, based on the general thermodynamic principle and combined with the experimental data, scholars have studied the thermal force coupling of continuous solid materials [

Based on single-phase ZrB_{2} ceramic, a novel thermo-chemo-mechanical coupling equation is established by combining the bond-based peridynamics (BB-PD) coupled thermo-mechanical with the high-temperature oxidation kinetics theory of single-phase ZrB_{2} ceramics in this paper. In _{2} ceramics, and the bond-based peridynamic thermo-chemo-mechanical coupling model. Then, we present the initial boundary conditions of the model and damage criteria. Based on the coupled thermo-chemo-mechanical model (_{2} ablation, the calculation results of the fully coupled thermo-mechanical model and the coupled thermo-chemo-mechanical model are compared and analyzed. Finally,

The relevant experimental results [_{2} ceramics, the following chemical reactions occur during the ablation process:

The abovementioned reaction process is described in _{2} in the oxygen environment begins to become obvious and ZrB_{2} oxidizes to produce solid particles of ZrO_{2} and liquid B_{2}O_{3}. The former serves as a skeleton and the latter serves as a filling and cover. One should note that the oxygen needs to diffuse through the oxide film composed of ZrO_{2} and B_{2}O_{3} in order to react with the underlying matrix material. The oxygen transport rate in the liquid film is very low, which drastically decreases the ablation rate. This is a typical inert oxidation phenomenon. The purpose of protecting the material is achieved by the slight oxidation of the material. When the temperature is higher than 1273 K, the evaporation of B_{2}O_{3} on the surface becomes significant and, with the increase of temperature, B_{2}O_{3} gradually shrinks into the pores of ZrO_{2}. When the temperature is higher than 2073 K, the volatilization rate of B_{2}O_{3} becomes greater than the generation rate and almost only porous ZrO_{2} remains in the oxide layer.

In this section, based on the oxidation process (_{2} ceramics is established (_{2} in the intermediate temperature range (1273–2073 K) in a high-temperature aerobic environment.

As shown in

_{2} density, kg/m^{3}; _{2} molar mass, kg/mol; _{2} oxide layer; _{2} concentration at 2 and 3 reaction interfaces, respectively.

Similarly, _{2}O_{3}, _{2}O_{3} concentration at the 2 and 3 reaction interfaces, respectively. The B_{2}O_{3} concentration in the environment can be taken as 0, like _{2}O_{3} concentration at the 2-2 interface can be calculated by the following equation [

_{2}O_{3} layer thickness (

For the calculation of

The peridynamic theory was first proposed by Silling et al. [

In the bond-based peridynamic theory, the force response function is the force density vector. Considering the effect of temperature change on deformation, the force density vector can be expressed as:

Combined with

Herein,

Combined with

Based on Fourier’s law of heat transfer, the bond-based PD thermal conduction equation can be expressed as:

_{v} refers to the specific heat capacity of the substance,

Introducing the temperature change term

In summary, _{2} ceramics. The proposed model is a simple thermo-chemo-mechanical coupling. The chemical reaction only affects the deformation of the oxide layer, while the effect of the chemical reaction is not considered for the unreacted matrix. However, it should be noted that the model can perform fully coupled thermoelastic analysis. In this model, the structural deformation and temperature field affect each other. The detailed numerical realization method is described in

In bond-based peridynamic thermal diffusion theory, an initial temperature is set for all material points. If the boundary temperature distribution

For the heat flow boundary, it is necessary to convert the heat flow into the heat source density and apply the heat source density to the material points of the boundary. Assuming that the heat flux at the boundary

If there is a temperature difference between the material surface (

The boundary conditions in dynamics’ problem usually include displacement boundary, velocity boundary, acceleration boundary and force boundary. In peridynamic theory, displacement constraints, velocity constraints and acceleration constraints are represented by vector

When distributed pressure

It is assumed that the high-temperature oxidation of ZrB_{2} only takes place at the interface under the thermal load and only the initial conditions of chemical reaction need to be considered. In the oxidation kinetics model of

In the solution of peridynamics, the displacement of each material point and elongation between each pair of material points are calculated. A time scalar function

When

Combined with the

The incremental form of the oxide layer thickness calculation using

It is important to note that the solution of the oxidation kinetics model belongs to a semi-independent process, which requires the distribution of temperature field. The bond-based PD coupled thermo-chemo-mechanical equation in this paper is based on the equations of motion and thermal conduction, and the kinetic equation of oxidation is used as an auxiliary. The calculation results of

In addition, the material parameters of the newly generated oxide layer during the ablation process are quite different from the original substrate material. As shown in

For the numerical solution of a classical fully-coupled thermo-mechanical equation, there are two generic methods, i.e., a single-step algorithm and an interleaved algorithm. In the single-step algorithm, also called the ensemble algorithm, time steps are simultaneously applied to the whole system of equations and multiple unknown variables are solved simultaneously. If the time integral of a single-step algorithm is implicit, absolute stability can usually be achieved, but it leads to a massive solution system. The interleaved algorithm partitions the coupled equations, solving the temperature and displacement fields with different time display algorithms. The interleaved algorithm can achieve a stable solution only under certain conditions.

Herein, according to the characteristics of the discrete form of the bond-based peridynamic equation and considering the particularity of the coupled thermo-chemo-mechanical model, an interleaved algorithm is used for the solution of coupled thermo-mechanical equations. The system is automatically divided according to the displacement field and temperature field, where the equation of motion is used for solving the displacement field and the equation of heat conduction is used for solving the temperature field. The solution of both equations is calculated by explicit time integration. Also, the computational equation of oxidation kinetics is relatively simple and can be solved directly.

The numerical calculations of the coupled thermo-chemo-mechanical model during the ablation of ultrahigh-temperature ceramics based on BB-PD mainly includes the following steps:

Defining the array, initializing variables and generating the physical numerical model;

Searching each material point within the neighborhood particles;

Initializing time-dependent function and surface correction factor;

Starting the first-time step cycle and calculating the temperature field;

Starting oxidation kinetics calculation and calculating the oxide layer film thickness produced due to the chemical reaction;

Starting the calculations of displacement field and reassigning material parameters to the material points of the oxide layer;

Judging whether the calculations are terminated and, if no, proceeding to Step 4 to start the next time step cycle;

Ending the cycle and obtaining the output results at each time step.

The time integral of PD heat conduction equation is calculated by forward difference method, which is conditionally stable. In order to prevent the divergence of numerical solutions, it is necessary to give a stability condition to limit the time step of thermal diffusion. Referring to the existing literature, similar to the method used by Silling et al. [

In this section, first, the correctness and accuracy of the bond-based peridynamic coupled thermo-chemo-mechanical numerical model are verified through three benchmark numerical examples: the fully coupled thermo-mechanical analysis of two-dimensional flat plate under temperature load; crack propagation simulations of the ceramic plate under cold shock, and evolution of oxide layer thickness of ZrB_{2} ceramic under high-temperature environments. In the fully coupled thermo-mechanical analysis of a two-dimensional flat plate under temperature load, the influence of neighborhood size and particle size on PD calculation results is investigated by comparing the calculation results under different neighborhood sizes and particle sizes with the analytical solution and the results of ABAQUS. The chemical reaction and coupled thermo-mechanical parts are verified separately because there is no corresponding ablation experimental research. Hence, the calculation results of the ZrB_{2} oxidation kinetics model can only be compared with the relevant high-temperature oxidation experiments of ZrB_{2}. Accordingly, the coupled thermo-mechanical part can be verified by commercial software or an analytical method. Finally, considering the growth and evolution of damage and oxide layer, the coupled thermo-chemo-mechanical model and fully-coupled thermo-mechanical model are compared and analyzed.

As shown in

^{3}) |
_{v} (J/(kg·K)) |
_{IC} (MPa·m^{1/2}) |
_{T} (W/(m·K)) |
||
---|---|---|---|---|---|

1 | 1 | 0.03 | 1 | 1 | 1 |

In peridynamic theory, the material point only interacts with other material points in the neighborhood of this point, so the size of horizon will affect the numerical calculation results of PD. In order to investigate the influence of horizon size on numerical calculation, this section fixes the distance between material points as

In addition to the influence of neighborhood range on the calculation results of bond-based peridynamic, the BB-PD model is also highly sensitive to particle spacing, i.e., the grid size. In this section, three different grid spacing will be selected in turn, i.e.,

In summary, when the horizon size is 3 times the particle spacing, the calculation results of the BB-PD model are accurate enough; when the particle spacing is smaller, the displacement and temperature calculation results are closer to the analytical solution and the calculation results of commercial software. The above results also prove that the bond-based peridynamic coupled thermo-chemo-mechanical numerical model given in this paper can be used to deal with the thermo-mechanical coupling response.

In this section, the crack growth process of the ceramic plate under cold shock is simulated and compared with the ceramic quenching experiment of Li et al. [

The initial temperature of the ceramic plate is 773 K. The water temperature is 293 K and the coefficient of convection heat exchange is 70,000 W/(m^{2}·K).

^{3}) |
_{v} (J/(kg·K)) |
_{IC} (MPa·m^{1/2}) |
_{T} (W/(m·K)) |
||
---|---|---|---|---|---|

370 | 3980 | 7.5 × 10^{−6} |
880 | 2.998 | 31 |

In this section, the calculation model in _{2} at different temperatures, and predict the growth and evolution of oxide layer thickness at different temperatures. The oxide layer thickness can be calculated by _{2} oxidation kinetics, the density of ZrO_{2} ceramics is taken as ^{3}.

When ZrB_{2} is oxidized in air for 5 h, the porosity (_{2} oxidation kinetics model are consistent with the experimental results, confirming that the proposed oxidation kinetics model can be used to simulate the growth in oxide thickness of ZrB_{2} ceramics at high temperatures.

In this section, the bond-based PD coupled thermo-chemo-mechanical model is used to simulate the dynamic response of ZrB_{2} ceramic plates under thermal shock. The boundary conditions of the ceramic plate are the same as those of _{2} ceramic plate are ^{2}

The distance between material points is _{2} matrix and ZrO_{2} oxide layer. In order to analyze the influence of chemical reactions on the ablation process, the dynamic response of ZrB_{2} ceramic plate under fully coupled thermo-mechanical conditions is simulated.

Parameters | ZrB_{2} [ |
ZrO_{2} [ |
---|---|---|

Material density ^{3}) |
6085 | 5600 |

Young’s modulus |
489 | 117 |

Poisson’s ratio λ | 0.333 | 0.333 |

Specific heat capacity _{v} (J/(kg·K)) |
48.2 | 615.6 |

Thermal conductivity coefficient _{T} (W/(m·K)) |
60 | 2.063 |

Thermal expansion coefficient |
5.9 × 10^{−6} |
7.11 × 10^{−6} |

Fracture toughness _{IC} (MPa·m^{1/2}) |
4.52 × 10^{6} |
5 × 10^{6} |

_{2} ceramic plate, and compare with the temperature distribution results on the vertical centerline, obtained from both calculation methods. As shown in _{2} ceramic plate under thermo-chemo-mechanic coupling is lower than the thermo-mechanical coupling. According to _{2} ceramic plate. It can be found that the ZrO_{2} oxide layer effectively resists the heat flow and inhibits temperature transfer on the ceramic plate to a certain extent, playing a certain protective role for the ZrB_{2} ceramic matrix.

From the above analysis, it can be seen that the evolution of oxide layer caused by chemical reaction during the ceramic ablation can indeed affect the thermal protection performance and damage development of ZrB_{2} ceramics, and the BB-PD coupled thermo-chemo-mechanical model based on ZrB_{2} ceramic ablation can deal with the problems of damage, fracture and temperature effect during the ZrB_{2} ceramic ablation to a certain extent.

Based on the bond-based peridynamic coupled thermo-mechanical theory and high-temperature oxidation kinetics model, a simple coupled thermo-chemo-mechanical model is established to simulate the ultrahigh-temperature ablation of ceramics. Compared with the coupled thermo-mechanical model, the coupled thermo-chemo-mechanical model considers the oxide layer generated by chemical reaction during ablation and the influence of chemical growth strain caused by the oxide layer on the bond basis force response function of PD. Finally, the influence of oxide layer on the ablation protection of ultra-high-temperature ceramics is analyzed in detail. The key results can be summarized as:

The BB-PD coupled thermo-chemo-mechanical model based on the ablation of ultra-high-temperature ceramics is proposed, which can describe the heat conduction and damage process of ZrB_{2} ceramics under high-temperature ablation conditions. It is proved that this numerical model can be used to describe the high-temperature ablation process of ZrB_{2} ceramics, as verified by the cold impact damage experiment and evolution of oxidation layer thickness.

The BB-PD coupled thermo-chemo-mechanical model is also used to simulate the high-temperature ablation process of ceramics. One should note that the results are significantly different from the calculation results of the BB-PD coupled thermo-mechanical, indicating that the presence of a protective oxide layer plays a key role in protecting the internal materials and preventing heat transfer.

In general, the bond-based peridynamic coupled thermo-chemo-mechanical model can describe the high-temperature ablation process of ZrB_{2} ceramics to a certain extent, revealing the changes in temperature distribution and damage behavior during the ceramic ablation.

_{2}-SiC composites at 2200°C

_{2}volatility diagram

_{2}/ZrC/SiC system prepared by reactive processing.

_{2}-SiC oxidation: Formation of a SiC-depleted region

_{2}

_{2}, HfB

_{2}and TiB

_{2}

_{2}-SiC