Clarification of the plastic deformations zone borders for the fuel tank diaphragm

Толық мәтін

Introduction

The fuel tanks of liquid-propellant low-thrust rocket engines operating in zero-gravity conditions are subject to high demands to ensure guaranteed separation of the liquid and gas phases. This is necessary for repeated starting of engines in the absence of gravity [1–3].

To ensure normal supply of the liquid fuel component from the tank to the low-thrust liquid rocket engine, a guaranteed separation of the liquid and gaseous phases is necessary, which is only possible with mechanical phase separation. Using axisymmetric metal eversible diaphragm separators [3–5] is efficient (Fig. 1). The deformation of the separator during the eversion process occurs in a small volume of the torus rolling zone moving along the generatrix and it has a clearly defined plastic character. Under the influence of distributed pressure as a result of the movement of the plastic zone, the deformed middle part of the separator is in an elastic deformed state and moves along the axis of the tank [3–6].

 

Рис. 1. Эскиз металлической диафрагмы-разделителя: а – в составе однокомпонентного сферического бака; б – с основными геометрическими параметрами

Fig. 1. Drawing of a metal diaphragm separator: а – as a part of a one-component spherical tank; b – with basic geometric parameters

 

For an ideally plastic body, plastic flow is determined by the finite combination of loads. The loading path, initial stress and deformation are not taken into account. To determine the main parameters of the plastic deformation process, the extremal principle for an ideally plastic body is used, which characterizes the minimum properties of the actual velocity field based on finite plastic deformations [3–8]:

$\underset{F}{\int }{X}_n{V}_{n}dF\le {\text{τ}}_s\underset{V}{\int }{H}^{\text{'}}dV,$ (1)

where X_n – surface load; V_n – surface movement speed; F – surface moving as a result of deformation; τ_s – material shear yield strength; H’ – kinematically possible intensity of strain rates; V – plastic zone volume.

Mathematical dependencies compiled on an experimentally substantiated physical model of plastic deformation of thin-walled shells with an arbitrary shape of the generatrix do not have sufficient accuracy and correspondence to the experimental results. These discrepancies are explained by the symmetrical boundaries of the plastic deformation zone accepted without sufficient justification and the type of surface considered as a torus. To calculate the parameters of plastic deformation of shell sections during their design, time- and cost-consuming refinement coefficients are used, determined experimentally for each shape of the separator generatrix and depending primarily on the angle φ [4; 5]. We could provide an example of formulae to calculate the eversion pressure and the radius of the torus zone for certain types of generatrix shapes in the eversion section [4; 5]:

$P=\frac{4{\text{τ}}_{s}s}{R_C^2\sin \phi }\cdot \sqrt{R_{C}s\left(\sin \phi -\phi \cos \phi \right)}\cdot \frac{1}{1-k\cdot \phi },$ (2)

$r=\mathrm{0,5}\sqrt{\frac{x_C\cdot s}{\sin \phi -\phi \cdot \cos \phi }}\cdot \left(1-k\cdot \phi \right),$ (3)

or

$r=\mathrm{0,5}\sqrt{\frac{x_C\cdot s}{\sin \phi -\phi \cdot \cos \phi }}\cdot \left(1-k\cdot \phi \right)\cdot \left(1-k_1\frac{R_C-A}{R_C}\right).$ (4)

To improve the quality of design when using any deformation-energy approach, it is necessary to clarify the boundaries of the zone of plastic deformation of the shell at various stages of everting the separator.

The problem statement

We propose to use finite element modeling of an elastic-plastic separator with subsequent study of the stress-strain state of the torus and the adjacent section of the shell as a method for solving the problem. To realize this, simulating the diaphragm loading scheme is necessary (Fig. 2).

 

Рис. 2. Схема нагружения диафрагмы

Fig. 2. Diaphragm loading diagram

 

A freely eversible diaphragm is investigated, that is, it is assumed that the inverted part of the tank walls does not touch. From the inside, the diaphragm is affected by the eversion pressure difference “p”. The area where the diaphragm is attached to the tank frame is pinched.

Numerical simulation parameters

To conduct the study, the MSC NASTRAN computer modeling package is used. When performing numerical studies in such software products, the basis of the calculation is the correct application and consideration of the loads acting on the body, as well as the creation of a high-quality finite element model. Developing the latter is always a difficult and time-consuming task.

The results of finite element modeling directly depend on the correct choice of the type of finite elements (FE) and the high-quality construction of a finite element mesh based on them. In the research [9] (Fig. 3), two types of FEs were used to solve a similar problem – Brick and Tetra. With their assistance, a model of a quarter diaphragm was obtained. However, the resulting model has a number of disadvantages. Firstly, it is the required amount of memory. The model occupies about 8 GB of disk space, this demands a significant consuming time and computer resources to carry out calculations and process the results. Secondly, even with such a volume of the model, it is not possible to divide the model section in the torus zone into small enough finite elements, about 1/20 of the diaphragm thickness, to study the distribution of plastic deformation zones more precisely.

 

Рис. 3. Модель диафрагмы на основе КЭ Brick и Tetra: а – четверть диафрагмы; б – зона соединения КЭ Brick и Tetra

Fig. 3. Diaphragm model based on Brick and Tetra FE: а – a quarter of the diaphragm; b – connection zone between the Brick and Tetra FEs

 

Since the problem posed is axisymmetric, it is advisable to use FEs of the Axisymmetric type from the section of volume FEs (Volume Elements) [10–12]. An axisymmetric FE is a ring-shaped element with a triangular or quadrangular cross-section. This type of FE is constructed in the XZ plane of the base coordinate system, where Z is the axis of rotation of the body. When constructing a model, only half of the section of the body of rotation is constructed, and the model should not intersect the Z axis. For the mesh, quadrangular FEs of the Quads type were selected, which have better convergence compared to Tri elements [10–14].

To construct a finite element model, we used the algorithm described in [15; 16]. Half of the diaphragm section is immediately created by finite elements without creating a geometric model. The finite elements in the torus and adjacent zone are specified more densely (1/20 s) than in the spherical section (1/4 s). The number of elements is chosen so that the resulting elements are as close to square as possible. The material in the calculations is aluminum AD-1M (E = 0.7×10⁵ MPa, µ = 0.27, σ₀₂ = 50 MPa, σ_в = 80 MPa, ε_в = 0.35, isotropic, elastic-plastic). Figure 4, a, b shows the material parameters in MSC NASTRAN. Graph 13 specified nonlinear properties. Stress vs Strain is specified at three points (0;0), (σ₀₂/E; σ₀₂), (ε_в; σ_в) (Fig. 4, c).

 

Рис. 4. Задание свойств материала в MSC NASTRAN: а – общие свойства; б – свойства пластичности; в – график напряжение-деформация для сплава АД-1М

Fig. 4. Setting material properties in MSC NASTRAN: а – general properties; b – plasticity properties; c – stress-strain graph for the AD-1M alloy

 

To perform calculation, nonlinear static analysis “22...Advanced Nonlinear Static” with large displacement options was used [10–12]. The load in the calculation is specified as a function of time (number of steps) (Fig. 5) and gradually increases from 0 to 100% per 100 time steps. The solver settings specify 100 steps with a time step of 1. The maximum pressure value was set as 0.1 MPa. In this case, the solver gradually approaches and stops at a certain load (Fig. 6). This appeared to be a good solution, since with a previously unknown eversion pressure, we obtain a value that is of the same order of magnitude as the experimental pressure. With this method of setting parameters, the pressure value is obtained as a percentage equal to the time step at the final moment of calculation from the maximum specified pressure.

 

Рис. 5. График приложения нагрузки

Fig. 5. Load application graph

 

Рис. 6. График приложения нагрузки в процессе расчета

Fig. 6. Graph of load application during the calculation process

 

Calculation results

For the research, models were constructed and calculations were performed according to the above scheme for separators with an internal radius R 100, 200, 350, 500 mm, thickness s 1.0; 1.5; 2.0 mm, at angles j 90, 80, 70, 60, 50, 40, 30°. The radius r was determined automatically based on the known dependence (3). The calculation results allow to identify some regularities. As an example, we could demonstrate the calculation result for a separator with parameters R = 200 mm, s = 2 mm, φ = 80° (Fig. 7, a). We will display stresses only that exceed the yield strength. The calculation results for the remaining separators look similar.

 

Рис. 7. Результат расчета: а – распределение напряжений, превышающих предел текучести, в районе торовой области; б – схема обозначений новых параметров зоны пластического деформирования

Fig. 7. Calculation result: а – distribution of stresses exceeding the yield strength in the torus region; b – designation scheme for new parameters for the plastic deformation zone

 

Stress analysis shows that the area of plastic deformation does not reach the boundary of the torus zone at the outer edge of the separator (point H´) and exceeds it at the inner boundary (point B´) (Fig. 7, b). To take into account the obtained results, it is proposed to introduce new refined angles of the plastic deformation zone ψ⁺ from the side of point H´ and ψ^- from the side of point B´. This choice of angles does not depend on the shape of the central part of the separator and will allow them to be used in the future for diaphragms not only with a spherical central part. Considering the fact that R >> 1, part of the diaphragm in the section B-B´, regardless of the shape of this part, can be replaced by a conical one with a segment BB´ on the middle meridian. We summarize the results of measurements of new angles in the table 1–4.

 

Table 1. Calculation result for a separator with R = 100 mm

φ,°	ψ+,°			ψ-,°
	s, мм
	1,0	1,5	2,0	1,0	1,5	2,0
90	80	82	85	139	140	141
80	75	75	76	126	127	130
70	69	70	70	116	118	118
60	60	60	60	102	105	107
50	50	50	50 (к)	91	95	102 (к)
40	40	40 (к)	40 (к)	80	95 (к)	88 (к)
30	30 (к)	30 (к)	30 (к)	68 (к)	70 (к)	76 (к)

 

Table 2. Calculation result for a separator with R = 200 mm

φ,°	ψ+,°			ψ-,°
	s, мм
	1,0	1,5	2,0	1,0	1,5	2,0
90	80	80	80	139	141	137
80	74	74	74	128	129	126
70	67	67	68	117	117	114
60	60	60	60	105	106	102
50	50	50	50	92	95	89
40	40	40 (к)	40	79	80 (к)	77
30	30 (к)	30 (к)	30 (к)	75 (к)	65 (к)	56 (к)

 

Table 3. Calculation result for a separator with R = 350 мм

φ,°	ψ+,°			ψ-,°
	s, мм
	1,0	1,5	2,0	1,0	1,5	2,0
90	78	78	80	137	138	140
80	72	73	75	126	128	127
70	67	65	70	116	118	116
60	60	60	60	104	106	106
50	50	50	50	92	92	104
40	40	40	40	75	77	94
30	30	30 (к)	30	61	65 (к)	77

 

Table 4. Calculation result for a separator with R = 500 мм

φ,°	ψ+,°			ψ-,°
	s, мм
	1,0	1,5	2,0	1,0	1,5	2,0
90	75	78	78	135	139	140
80	71	72	75	127	128	129
70	65	68	66	116	116	118
60	60	60	60	104	103	107
50	50	50	50	89	87	93
40	40	40	40	74	75	76
30	30	30	30	60	61	78

 

It is worth mentioning that not in all cases, the calculation according to the scheme described above was successful immediately, especially it is specific at angles φ < 40°. This can be explained by the fact that when approaching the polar zone, the radius r of the torus zone becomes commensurate with the radius R of the central part of the separator. In this case, eversion of the diaphragm occurs with a significant influence of elastic stresses in the non-inverted area according to the “slow clap” pattern. This resulted in the solver being able to pass the pop and continue loading. In these cases, it was necessary to manually limit the maximum pressure to obtain a graph of the form presented in Fig. 6. The angle values in these cases are marked in the table 1–4 as (к). On the other side, conditions were identified under which the use of the energy principle equation (1) is inappropriate due to the described phenomenon. These conditions can be expressed as follows: (R/r ≤ 6.5) or (6.5 < R/r < 10 and R/Xc < 0.8), where Xc is the X coordinate of the center of radius r.

It can be noted that parameters such as R and s do not significantly affect the refined angles of the plastic deformation zone. Additional studies of changes in radius r also did not reveal a strong effect on these angles.

We could display the values from the table 1–4 on the coordinate plane as a function ψ = f(φ) (Fig. 8, points 1, 2). The minimum and maximum values within one angle j differ by no more than 20°. Such a gap will not greatly affect the result of the preliminary design of the diaphragms, therefore, for each angle j we will find the average value of the ψ⁺ and ψ^– parameters and combine them. As a result, we obtain a graph of the dependence of the refined angles of the plastic deformation zone on the angle of the torus zone (Fig. 8, curves 3, 4).

Рис. 8. Зависимость уточненных углов зоны пластических деформаций от угла торовой зоны: 1 – ψ^-= f(φ) по табл. 1–4; 2 – ψ⁺ = f(φ) по табл. 1–4; 3 – уточненная функция ψ^-= f(φ) по усредненным значениям; 4 – уточненная функция ψ⁺ = f(φ) по усредненным значениям

Fig. 8. Dependence of refined angles of the plastic deformation zone from the corner of the torus zone: 1 – ψ^– = f(φ) according to table 1–4; 2 – ψ⁺ = f(φ) according to table 1–4; 3 – refined function ψ^– = f(φ) based on averaged values; 4 – refined function ψ⁺ = f(φ) based on averaged values

 

Conclusion

The results obtained clarify the boundaries of the plastic deformation region of the fuel tank separator diaphragm. This allows to adjust the previously developed algorithm for calculating the main parameters of the eversion process [4] and eliminate the process of identifying these parameters based on the results of experiments in future design.

We have obtained averaged dependencies of the new angles of the plastic deformation region, independent on the parameters R, s, r of the diaphragm.

Conditions have been identified under which the use of the energy principle equation is inappropriate due to the presence of membrane stresses. This condition can be expressed as (R/r ≤ 6.5) or (6.5 < R/r < 10 and R/Xc < 0.8).

The refined angles of the plastic deformation zone should be used when solving the basic equation of the energy principle (1). To perform this, when solving the integral on the right side, the volume of the plastic deformation zone must be taken not at angle 2φ, but at angle (ψ^– + ψ⁺).

Авторлар туралы

Dmitriy Klimovskiy

Reshetnev Siberian State University of Science and Technology

Хат алмасуға жауапты Автор.
Email: Klinsky92@yandex.ru

assistant professor of a department of Aircraft Engines

Ресей, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037

Viktor Zhuravlev

Reshetnev Siberian State University of Science and Technology

Email: vz@sibsau.ru

Cand. Sc., professor of a department of Aircraft Engines

Ресей, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037

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