# Investigation of Temperature Deformation of the IES Electrodes Based on the Continuum Thermo-Mechanical Calculation Model

### Aerospace propulsion engineering

### Аuthors

^{*},

^{1}

^{*},

^{2}

^{*}

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

2. ,

*e-mail: riame@sokol.ru

### Abstract

A continuum thermo-mechanical model for densely perforated electrodes of the ion-extraction system (IES) was developed. It is based on representing the emissive, accelerating and decelerating electrodes as structurally orthotropic sloping thin-walled spherical (profiled in general case) panels. An uneven heating of the electrodes along the radius and through their thickness was considered as the basic loading factor. As a consequence, the coefficients of electrode filling with materials, which are simultaneously the coefficients of constructive orthotropy in the considered model, were directly taken into account in the initial equations of thermoelasticity.Relations between the geometric sizes of round perforation holes, regularly arranged according to a triangular scheme, and the coefficients of electrode filling with materials were calculated by a special algorithm, which gave an opportunity to properly consider the thermal impact on the IES as a whole. Acceptance of such thermo-mechanical model of electrodes resulted in a diagram of dependence of the constructive orthotropy coefficients on the ratio of hole diameter to the distance between them.

For the direct calculation of strain state of spherical electrodes at their uneven axisymmetric heating, there was used the mathematical apparatus of matrix boundary integral and integrodifferential equations represented in convenient dimensionless form. Solution of such equations obtained in the geometrically nonlinear statement of the problem on the bending of sloping spherical panel was based on iterative methods in combination with process of step-by-step temperature loading of considered electrodes.

It is advisable to use the above algorithm for solving the obtained integral equations at critical stages of checking calculations on durability and rigidity. When conducting multivariate research calculations, it is better and more effective to make preliminary calculations by simplified algorithm.

It is the first approximation of the iterative process for solving the appropriate integral equations not in the numerical but in the analytical form. The obtained resulting nonlinear algebraic equation of the third degree relative to the dimensionless additional deflection is solved by using the corresponding software package, such as MathCAD.

In accordance with the above simplified algorithm of calculation of additional deflections of sloping spherical panels, there was conducted a multivariate study of the deformation process of emissive electrodes made of molybdenum and its alloys, titanium-niobium alloy of invar-class, as well as of a carbon-carbon composite material. For the accelerating electrode there was considered a molybdenum alloy and carbon-carbon composite material. The base diameter of the ion beam was taken to be equal to 160 mm and 450 mm.

The developed thermo-mechanical model can be extended to a wider range of problems relating to the geometry of electrodes and the materials used.

### Keywords:

ion thruster, ion-optical system, perforation, deformation, constructive orthotropy, thermal load, thermal deformation, thermo-mechanical model### References

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