Finite element modeling of thermal stresses in aerospace structures from polymer composite materials

. The conjugate problem of aerodynamics and heat and mass transfer in heat-shielding structures made of ablating polymer composite materials is considered. To determine the heat fluxes in the shock layer, the chemical composition of the atmosphere is taken into account. A mathematical formulation of the joint problem is formulated and an algorithm for numerical solution is proposed. An example of a numerical solution of the problem for the reentry spacecraft Stardust is presented. It is shown that due to the high temperatures of aerodynamic heating of a structure made of a polymer composite material, an intense internal generation of gas appears in the structure of the material, which can lead to the destruction of the material


Introduction
In the process of entry into the Earth's atmosphere of the returning spacecraft, high heat fluxes are formed in the shock layer and non-equilibrium chemical reactions occur, which leads to high temperatures on the surface of the spacecraft itself, and, as a result, to the need for its thermal protection [1][2][3][4].The most commonly used method of protection is passive, which consists in the use of ablative composite materials.Ablative thermal protection is based on physical and chemical processes occurring in the bulk of the material, with significant energy absorption.The advantages of the passive method of thermal protection include: cost, ease of use, reliability, no restrictions on the maximum heat flow and a large selection of materials due to the development of organic chemistry.In [5][6][7], a mathematical model of the processes of internal heat and mass transfer in polymeric heat-shielding materials during non-stationary heating was developed, taking into account the processes of thermal destruction, gas formation and filtration of gases in the pores of the material.
Since to determine the thermal loads it is necessary to solve the problem of aerodynamics, the adjoint problem will be two systems of equations: the Navier-Stokes equations, supplemented by the equations of chemical kinetics, describing the change in the chemical composition of the gas; equations of internal heat and mass transfer in a heat-shielding structure.

Navier-Stokes system and equations of chemical kinetics
Let us consider the oncoming flow as a mixture of 23.3% oxygen O2 and 76.7% nitrogen N2.Let us assume that the gas flowing around the body consists of seven components (O, N, NO, NO + , O2, N2, e -), and as possible chemical reactions occurring in the shock layer at high temperatures, we consider the following: Here M means any of the 6 considered components that are catalysts, e -electronic component.We will associate the index 1, 6 i with components O, N, NO, NO + , O2, N2 respectively.
For the considered chemical reactions, the conditions of conservation of the atomic composition and the condition of quasi-neutrality of the mixture take place:

Equations of internal heat and mass transfer
Next, consider a heat-shielding structure made of a polymer composite material.Under the action of high temperatures, usually above 300 °C, internal physicochemical transformations occur in polymer composite materials -thermal destruction processes that lead to irreversible changes in the structure and loss of mass of the material.The starting material consists of two phases: a filler (fiber or reinforcing particles) and a polymer matrix.When heated in the process of thermal destruction, the polymer matrix decomposes into two new phases: coke (solid pyrolytic residue) and gas, which is in the pores and is filtered through them to the surface of the material.Thus, the original composite material is considered as a 4-phase structure [5][6][7][8].The heat and mass transfer equations (5) are closed by the constitutive relations that relate the vector functions and to the temperature and pressure gradients using the Fourier and Darcy laws, as well as the Arrhenius relation for the mass rate of thermal destruction and the Mendeleev-Clapeyron equation ( 4): q Λ -Fourier's law; (7) where 0 J -pre-exponential factor, A E -activation energy of thermal destruction processes, Λ -thermal conductivity tensor, K -gas permeability tensor of the composite.
In addition, for a composite, as for a multiphase system, the mixture ratios and the normalization condition are satisfied (the index refers to the reinforcing filler): , , 1.

Boundary conditions for the adjoint problem
Consider the boundary and initial conditions for the adjoint problem (1) -(9).For the system of equations of gas dynamics (1), it is necessary to add boundary conditions: 3) on the symmetry surface the following conditions are set: 0, 0, 0, 0, 0 the initial conditions for system (1) have the form: i p p y y It is necessary to add the initial and boundary conditions to the system of heat and mass transfer equations ( 5)-( 9): 1) on the inner surface of the composite, we set the adiabaticity and tightness of the gaseous products of thermal destruction: 0, 0 5) the initial conditions look like: To the listed boundary conditions, it is necessary to add the relations on the surface common for the gas and the composite w , with the help of which the problems of external aerodynamics and internal heat and mass transfer in a composite material are connected.These ratios look like: the no-slip condition for the molecules of the incoming gas flow: .
n -surface normal vector; c q -density of convective heat flux to the surface; BL q - heat flux density removed from the surface due to the release of gaseous pyrolysis products from the composite; RW q -the density of the heat flux removed from the surface due to its own radiation by the heated surface into the environment; ( ) where wg and wscomposite gas temperature at the surface w respectively; mechanical impact on the composite material from the oncoming gas flow

Numerical methods
To solve the gas dynamics system the finite volume method is used and for the heat and mass transfer system the finite element method is used [11,13].

Modelling of material properties anisotropy
For thermal protection, consider fabric orthotropic materials.To correctly take into account the properties in different directions, it is necessary to specify the direction of laying the fibers of the fabric.To do this, we introduce a local coordinate system in each finite element (Fig. 1).

Results and discussion
The numerical solution of the problem was carried out for the reentry spacecraft Sturdust [4].The results were obtained for an altitude of 60 km.and speed of movement 5000 m/s.On figures 2-4 show the density distribution fields, the concentration of atomic nitrogen, and the temperature distribution in the shock layer.It can be seen that the temperature in the shock layer exceeds 8000 K.When using the perfect gas model, the temperature value is 1.5-2The design of the spacecraft was supposed to be two-layer.Phenolic fiberglass (outer area) and epoxy fiberglass (inner area) were chosen as heat-shielding materials.On figures 5 and 6 show the distribution of temperature and volume concentration of the gas phase in the heat-shielding layer for a heating time of 210 sec.The most heat-loaded region is the critical bluntness, where the temperature reached 1946 K.It can be seen that the phenolic matrix decomposes under the influence of high temperatures, while the epoxy matrix remains in the low temperature region.

Conclusion
A mathematical formulation of the conjugate problem of high-velocity aerothermodynamics of a chemically reacting gas flow and internal heat and mass transfer in the heat-shielding structure of the reentry spacecraft from ablation composites is considered.Numerical finitevolume and finite-element methods for solving the joint problem is proposed.An example of numerical simulation of the aerothermodynamics of a gas flow is given, taking into account chemical reactions and internal heat and mass transfer for the return space apparatus Stardust.The importance of the chemical composition is shown atmosphere to determine the thermal effects on the surface apparatus design.
at constant volume and pressure [J/(kg K)]; temperature [K]; p mixture pressure [Pa]; 0 8.31 R universal gas constant [J/(mol K)the gas mixture and the i-th component of the gas mixture [kg/mol]; i i y mass concentration of the i-th component; i D -diffusion coefficients [m 2 /s]; i imass rate of formation of chemical components [kg/(m 3 •s)]; molecules for the Lighthill model [J/kg]; 0 k h -enthalpy of formation of the component [J/kg] (Table 1 concentrations [mol/kg]. the notation is introduced here: b , gvolume concentrations of the polymer and gas phases; b , gdensity of the original polymer matrix and gas phase [kg/m 3 ]; g c -heat capacity of a gas at constant volume [J/(kg K)], and c -density and heat capacity of the composite, q -heat flow vector [W/m 2 ], -composite temperature [K]; g v -gas phase velocity [m/s]; 0 especific heat of thermal degradation of the matrix [J/kg]; J -mass rate of thermal degradation of the matrix [kg/m 3 s]; Г -matrix gasification coefficient.
boundary), boundary conditions are not specified; ) boundary condition for the concentration of components (O and N atoms, NO molecules) on the non-catalytic surface of a solid wall 0 the recombination of charged particles, the conditions of an ideal catalytic surface composite surface temperature; wintegral emissivity of the composite surface; SB -Stefan-Boltzmann constant;heat transfer coefficient between the surface and the external incoming gas flow; p c , g c -specific heat capacities of the external gas flow and the gas phase of the composite, respectively;temperature continuity condition at the interface w : pressure of the external gas flow on the surface.

Fig. 1 .
Fig. 1.Visualization of basis vectors to account for material anisotropy properties.

Fig. 3 .
Fig. 3. Distribution of atomic nitrogen concentrations in the shock layer.

Fig. 6 .
Fig. 6.Distribution of the volume concentration of the gas phase.
Therefore, taking into account the chemical composition of the atmosphere under such conditions is necessary.