# Isothermal Pneumo-Forming of Hemispherical Parts
## Made Out of Anisotropic Materials In Short-Term
## Creep Mode

S.N. Larin, V.I. Platonov, G.A. Nuzhdin

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S.N. Larin, V.I. Platonov, G.A. Nuzhdin. Isothermal Pneumo-Forming of Hemispherical Parts Made Out of Anisotropic Materials In Short-Term Creep Mode. Mechanics, Materials Science & Engineering Journal, 2016, 10.13140/RG.2.1.3033.6408. hal-01307352

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# Isothermal Pneumo-Forming of Hemispherical Parts Made Out of Anisotropic Materials In Short-Term Creep Mode

S.N. Larin¹, V.I. Platonov¹, Nuzhdin G.A.²

1 – Tula State University, Tula, Russia

2 – quality assurance company “Konsersium”, Moscow, Russia

<figure><img src="image_2.png" /></figure>

DOI 10.13140/RG.2.1.3033.6408

**Keywords:** anisotropy, mathematical model, high-strength materials, domed parts, pneumo-forming, creep, pressure, stress, thickness, damage rate, failure.

**ABSTRACT.** Provided here are results of theoretical and experimental research of strained and stressed state, force modes, geometrical sizes for the blanks, and limit possibilities of deformation during isothermal blow molding of hemispheric parts of anisotropic material in creeping mode. Determined is the effect for the researched parameters of the studied deformation process, produced by anisotropy of mechanical properties, loading conditions and blank's geometric dimensions. Comparison of the theoretical and experimental data regarding the relative blank thickness in the blank dome and base points, and of data regarding the relative height of the blank, point to their satisfactory agreement (up to 10 percent). Recommendations have been developed regarding calculation of scientifically-based technological parameters for operations of isothermal straining of semi-spherical components made out of highly strong anisotropic materials in the mode of short-time creeping. The recommendations were used during development of technological processes of manufacture -- in the mode of short-time creeping and out of highly strong anisotropic materials --. of semispherical components conforming to the operational technical requirements. The technological processes provide for increasing specific strength by 1,5 – 1,7 times, for decreasing the mass by 1,5 times, for reducing labor content by 2-3 times, and for growth of capacity factor – from 0,3 to 0,9.

**Introduction.** Spherical sheet domes make bodies for tanks for fuel and liquid nitrogen, which are used in aviation and space technologies. The conventional methods of their manufacture are in the mode of multistage squeeze drawing with transitional thermal treatment rounds, or else it is hammer stamping in subpress dies, these being highly labor-consuming operations. Isothermal deformation of domed parts with gas of sheet high-strength aluminum and titanium alloys has considerable advantage over the traditional methods of processing, and it is quite promising for its industrial use [1-5].

Materials that are subjected to processes of plastic straining, possess, as a rule, anisotropy of mechanical properties. Anisotropy of mechanical properties may produce both positive or negative effect upon stable procedure of technological processes of plastic deformation under various thermomechanical modes [1-19].

**Equations.** Let us consider straining of a round sheet blank with radius $R_0$ and with thickness $h_0$ by free bulging in the mode of viscous flow of material as influenced by excessive gas pressure

$$p = p_0 + a_p t^{np}$$

into a spherical matrix (Fig. 1). Here $p_0, a_p, n_p$ – are pressurizing constants.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Diagram for calculation of strained condition of middle surface of blank in meridian plane</figcaption></figure>

The blank is fixed along the outer contour. The blank material is taken as transversal-isotropic with anisotropy ratio R ; the stressed state of the dome is plane, i.e., the stress is perpendicular to the plane of the sheet and equals zero (σz = 0). We consider straining in the meridian plane of the dome as straining a membrane. Because of the symmetry of the material’s mechanical properties, relative to the blank axis and the character of action of the external forces, -- the strain stresses and speeds, that are meridian, circumstantial and normal to the middle surface of the blank, are the major ones.

The middle surface of the blank remains a part of the spherical surface at any stage of straining. Taking place in any meridian section of the dome is radial flow of material relative to the new center at every stage of straining.

According to the adopted allowance, the radiuses of curvature of the meridian section ρm of the middle surface and of the slashing of the dome by the taper surface that is perpendicular to the meridian arc, ρt, are found according to the following formula.

$$
\rho_m = \rho_t = \rho = \frac{H^2 + R_0^2}{2H}, \qquad (1)
$$

where H – is the height of the dome at a given moment of straining.

Since the trajectories of the points of the middle surface are orthogonal at this moment to the profile
in formation, straining speeds in the pole of the meridian surface (point “c” ) shall be determined in
the meridian sections as

$$
\xi_{tc}^{c} = \frac{2H\dot{H}}{H^2 + R_0^2}; \quad \xi_{mc}^{c} = \frac{2H\dot{H}}{H^2 + R_0^2}; \quad \xi_{zc}^{c} = \frac{\dot{h}}{h}, \qquad (2)
$$

where $\dot{H} = dH/dt$ ; $\dot{h} = dh/dt$ .
The blank is fixed in the contour (point “k”), i.e., the strain speed along the contour equals zero $\xi_{tk}^c = 0$, and according to the associated flow law, we have

$$ \xi_{tk}^c = 0; \sigma_{tk} = \frac{R\sigma_{mk}}{1+R}; \xi_{zk}^c = -\xi_{mk}^c, \quad (3) $$

where R – is the normal anisotropy factor for viscous flow of material.

Further in the text we make no limitations about changing the shell thickness along the circumference arc in the meridian section. In this case the speed for the strain in the meridian $\xi_m^c$, circumferential direction $\xi_t^c$ and the thickness strain $\xi_z^c$ of the shell is found after the following formulas accordingly

$$ \xi_m^c = \left( \frac{\sin \theta}{\theta \sin \alpha} - ctg \alpha \right) \dot{\alpha}; \quad \xi_t^c = \left( \frac{\cos \theta}{\sin \alpha} - ctg \alpha \right) \dot{\alpha}; \quad \xi_z^c = \frac{\dot{h}}{h}. \quad (4) $$

where $\theta$ – is here actual angle between the blank’s vertical axis of symmetry and radius-vector determining the position of point in the section of middle surface by diagonal plane; $\dot{\alpha} = d\alpha/dt$.

It was assumed during the straining of the shell that there takes place in the meridian plane during every stage of straining a radial flow of point of middle surface, that is relative to the new center at moment $t+dt$, i.e., in the direction of $\theta+d\theta$.

Connection between angle $\alpha$ and straining time $t$, when functional connection is assumed $H = H(t)$, is found as follows

$$ \alpha = 2arctg \frac{H(t)}{R_0}. \quad (5) $$

The thickness of the meridian plane dome of the shell ($\theta=0$) is found from the following formula

$$ h = h_0 / \left( 1 + \frac{H^2(t)}{R_0^2} \right)^2. \quad (6) $$

The changing thickness of the shell from straining time $t$ at the spot of its fixation ($\theta=\alpha$) is found after the following formula

$$ h = h_0 \frac{H(t)}{R_0 \left( 1 + \frac{H^2(t)}{R_0^2} \right) \operatorname{arctg} \frac{H}{R_0}}. \quad (7) $$

Cutting elements from the membrane by meridian planes and cone surfaces in the vicinity of the point under consideration, and assuming that the stresses are evenly distributed throughout the thickness of
the element, we write down the equation of equilibrium of momentless shell loaded with uniform pressure p, as follows:

$$ \frac{\sigma_m}{\rho_m} + \frac{\sigma_t}{\rho_t} = \frac{p}{h}; \sigma_{mx} = \frac{p\rho_t}{2h}. \qquad (8) $$

Solving them together and taking account of $\rho_m = \rho_t$, we find

$$ \sigma_m = \sigma_t = \frac{p\rho}{2h}. \qquad (9) $$

The equivalent speed of straining $\xi_e^c$ and equivalent stress $\sigma_e$ in the apex of the dome (point “c”) and in the point of fixation of the shell over the entire circuit (point “k”) are computed respectively for anisotropic material after the following formulas:

$$ \xi_{ec}^c = \frac{2}{\sqrt{3}} \sqrt{2+R} \xi_{mc}^c; \quad \sigma_{ec}^c = \frac{\sqrt{3}}{2\sqrt{2+R}} \sigma_{mc}; \qquad (10) $$

$$ \xi_{ek}^c = \left\{ \frac{2(2+R)(R+1)}{3(2R+1)} \right\}^{1/2} \xi_{mk}; \quad \sigma_{ek} = \left\{ \frac{3}{2} \frac{2R+1}{(2+R)(R+1)} \right\}^{1/2} \sigma_{mk}. \qquad (11) $$

Let us consider a case of slow isothermal straining of a shell made out of material for which hold constitutive equations of the energy theory of creep and creep damage [4]

$$ \xi_e^c = \frac{B(\sigma_e / \sigma_{e0})^n}{(1 - \omega_A^c)^m}; \quad \dot{\omega}_A^c = \frac{\sigma_e \xi_e^c}{A_{np}^c}, \qquad (12) $$

where B, n, m – are material invariables depending on the test temperature;

$\omega_A^c$ – is the material damage rate for viscous deformation in the energy failure models;

$A_{np}^c$ – is specific fracture work for mode of viscous flow;

$\dot{\omega}_A^c = d\omega_A^c/dt$; $\xi_e^c$ and $\sigma_e$ – are equivalent straining speed and strain;

$\sigma_{e0}$ – is equivalent stress dividing the viscous and viscoplastic flows of material.

The value of the specific fracture work $A_{np}^c$ for viscous flow of anisotropic material is found from

$$ A_{np}^c = D(b_0 + b_1 \cos \alpha + b_2 \cos \beta + b_3 \cos \gamma), $$
where $D, b_0, b_1, b_2, b_3$ – are material constants;

$\sigma = (\sigma_1 + \sigma_2 + \sigma_3)/3$ – is mean stress; $\sigma_1, \sigma_2$ and $\sigma_3$ – are primary stresses;

$\alpha, \beta, \gamma$ – are orientation angles of the first main stress axis $\sigma_1$ as relative to the main axes of anisotropy x, y and z respectively.

Since the value of pressure $p$ at every moment of straining is distributed evenly over the surface of the shell, we shall be finding its value in the apex of the shell dome (point “c”).

By substituting into the first material state equation (12) its constituent values $\sigma_e$ and $\xi_e^c$, found after formulas (10), and taking account of formulas (1), (4), (9), we receive

$$p^n dt = \frac{\sigma_{e0}^n (1 - \omega_{Ac}^c)^m 2^{2n+2} (2+R)^{\frac{n+1}{2}} H^{n+1} h^n dH}{3^{\frac{n+1}{2}} B (H^2 + R_0^2)^{n+1}} \quad (13)$$

Shell thickness $h$ is found after formula (6).

We now find the value of the damage accumulation $\omega_{Ac}^c$. By substituting into the second state equation (12) formulas (10), account being taken of (1), (4) and (9), we obtain

$$\dot{\omega}_{Ac}^c = \frac{p \left(1 + \frac{H^2}{R_0^2}\right)^2}{h_0 A_{np}^c} \dot{H} \quad (14)$$

This equation is fitfully used if the pressurizing provides for $p = const$.

If we substitute the first state equation into the second one, we shall have a different type of equation for finding the damage rate

$$\dot{\omega}_{Ac}^c = \frac{\sigma_{e0} (1 - \omega_{Ac}^c)^{m/n} (\xi_{ec}^c)^n}{A_{np}^c B^{1/n}} \quad (15)$$

This equation can be comfortably used if $\xi_{ec}^c = \xi_{el}^c = const$. In the latter case integration of equation (15) results in a formula of the following type:
$$ \omega_{Ac}^{c} = 1 - \left[ 1 - \frac{n-m}{n} \frac{\left(\frac{\xi_{e1}^{c}}{n}\right)^{\frac{n+1}{n}} \sigma_{e0} t}{A_{np}^{c} B^{1/n}} \right]^{\frac{n}{n-m}} \quad (16) $$

Destruction time $t_*$ is found from assumption $\omega_{Ac}^c = 1$

$$ t_* = \frac{A_{np}^c B^{1/n} n}{\sigma_{e0} (n-m) \left(\frac{\xi_{e1}^c}{n}\right)^{\frac{n+1}{n}}}. \quad (17) $$

Pressure $p$, which is necessary for creating the straining conditions, shall be computed after the following formula

$$ p(t) = \frac{\sigma_{e0} (1 - \omega_{Ac}^c)^{m/n} 2^2 (2 + R)^{1/2} Hh}{\sqrt{3} B^{1/n} (H_0^2 + R_0^2)} \left(\frac{\xi_{e1}^c}{n}\right)^{1/n}. \quad (18) $$

Functional connection $\omega_A^c = \omega_A^c(t)$ is found as per formula (16), while $H = H(t)$ can be found from the following equation

$$ t = \frac{2\sqrt{2+R} \ln \frac{H^2+R_0^2}{H_0^2+R_0^2}}{\sqrt{3} \xi_{e1}^c}. \quad (19) $$

Limiting dome height $H_*$ can be found by equation (19) with $t = t_*$.

Function setting $H = H(t)$ allows us to find $\omega_A^c = \omega_A^c(t)$ from formulas (15) or (16), whereas function $p = p(t)$ is calculated after formula (13).

The same procedure is used for finding the stressed and strained conditions of the blank in the point of fixation of the shell (point “k”), and also found are the principal equations and correlations for the solving of the problem with the assumption that the behavior of the material is subject to equations of the kinetic theory of creep and creep damage under the known law of time-pressure $p = p(t)$ and under permanent equivalent strain rate in the blank dome $\xi_{e1}$.

There have been developed the calculation algorithm for the force and strain parameters of the studied technological process, and the computer software.

**Discussion of results.** Evaluated were the stressed and strained conditions, the material flow kinematics, the force modes and frontiers of the studied process of strain – as relating to accumulated microscopic damages, and depending on anisotropy of the mechanical properties of the source material, on the stress sequence, and on the geometric dimensions of the blank and of the ready item.
Calculation was performed for titanium alloy LT31 under $T = 860^{\circ}C$, the behavior of which is subject to the energy theory of creep and creep damage, and for titanium alloy 4AL-3MO-1V under $T = 950^{\circ}C$, the behavior of which is subject to the kinetic theory of creep and creep damage. Mechanical characteristics of these materials during deformation under conditions of viscous flow of material are shown in the work [3, 4].

The curves for changing values of gas pressure $p$, for relative values of blank thickness in the dome $\bar{h}_c = h_c/h_0$ and at the place of its attachment $\bar{h}_k = h_k/h_0$, for the height of the dome-shaped blank $\bar{H}' = H/R_0$, for strain time $t$ for titanium alloy LT31 ($T = 860^{\circ}C$) under permanent value of the equivalent strain rate in the blank dome $\xi_{e1}$ --are shown in Fig. 2. Experimental data are shown here with dots.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Changes of $p$, $\bar{H}'$ and $\bar{h}$ in studied points of blank as depending on $t$ of titanium alloy LT31 ($R_0 = 300$; $\xi_{e1} = 0,002 \ 1/c$)</figcaption></figure>

It follows from analysis of the calculation results and graphic dependencies that, as there grows to a certain limit the strain time $t$, there takes place a steep increase in the relative height of the blank $\bar{H}'$ and reduction in the relative thickness of the blank in the dome $\bar{h}_c$ and in the place of its fixation $\bar{h}_k$. Further increase in the time of straining $t$ results in a smooth changing of the values under study. At time moment $t$, close to the destruction of the blank, there takes place a sudden change of the relative values of $\bar{H}'$, $\bar{h}_c$ and $\bar{h}_k$. This is linked to the intensive accumulation of micro-damages at the concluding stage of the process.

It has been found that change of the relative thickness in the blank dome proceeds in a more intensive manner as versus changing the relative thickness at the place of its fixation. As the $t$ straining time grows, that difference grows too and may reach as much as 50%.

It was shown that for the purpose of ensuring a permanent equivalent speed of strain in the blank dome, the law of changing pressure $p$ during time of straining $t$ has a complicated character. At the initial moment of deformation we observe a sudden growth of pressure $p$, since there takes place a substantial change of half-sphere radius $\rho_m$. Further extension of the straining time $t$ is accompanied with reduction in the magnitude of the gas pressure $p$.
Comparison of theoretical and experimental data for the relative thickness in the dome of blank $\bar{h}_c$ and at the place of its fixation $\bar{h}_k$, as well as the data for relative height of blank $\bar{H}'$ points to their satisfactory coincidence (up to 10%).

It is defined, that destruction of the blank during isothermal staining takes place in blank dome where there happens the maximum thinning of the blank.

Dependence of change in destruction time $t_*$, of relative height $\bar{H}_*$ and of thickness in the blank dome $\bar{h}_*$ at the moment of destruction as determined after the magnitude of accumulated micro-damages with $\omega_A^c = 1$, -- on the magnitude of the standing equivalent speed of strain in the blank dome $\xi_{e1}$ and on anisotropy ratio $R$ is shown in Fig.3 and Fig.4 respectively. It is shown that increasing the parameters of stress sequence $a_p, n_p$ and increasing the magnitude of the permanent equivalent speed of strain in the blank dome $\xi_{e1}$, result in decreasing destruction time $t_*$ and blank relative height $\bar{H}'$, and result also in increasing the relative thickness in the blank dome $\bar{h}_*$.

It follows from the curves' analysis (Fig. 4) that normal anisotropy factor $R$ produces a substantial effect upon the magnitude of the destruction time and upon the relative values $\bar{H}'$, $\bar{h}_*$. As there grows the anisotropy factor $R$, the relative value $\bar{h}_*$ grows steeply, while destruction time $t_*$ and blank relative height $\bar{H}'$, abruptly drop in magnitude. It was found, that failure to take account of anisotropy of blank's mechanical properties during analysis of the process of isothermal deformation of a spherical shell results in a measurement error in the evaluation of destruction time $t_*$ in the order of 35%, and this results in an error of 15% in the evaluation of the relative height $\bar{H}'$ and of the thickness in the blank dome $\bar{h}_*$ during the destruction moment.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Dependence on $\xi_e$ of changing $t_*$ and $\bar{H}'$, $\bar{h}_*$ in blank dome, titanium alloy LT31 ($\bar{R}_0$ = 300)</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Dependence on R of changing $t_*$ and $\bar{H}'$, $\bar{h}_*$ (kinetic theory; $\bar{R}_0 = 300$; $p_0 = 0,013 \text{ МПа}$; $a_p = 4 \cdot 10^{-3} \text{ МПа}/c^{n_p}$; $n_p = 0,6$)</figcaption></figure>

Calculation analysis shows that the limitations of deformation under conditions of viscous flow of material whose behavior obeys the kinetic theory of creep and creep damage (alloy 4AL-3MO-1V), do not depend on the loading conditions of the blank. Shown is the substantive dependence of destruction time $t_*$ on loading parameters $a_p, n_p$ and on the magnitude of the permanent equivalent strain rate $\xi_{e1}$. Increasing the loading parameter $a_p$ from $0,2 \cdot 10^{-3} \text{ МПа}/c^{n_p}$ to $1,4 \cdot 10^{-3} \text{ МПа}/c^{n_p}$, and increasing $n_p$ from 0,5 to 0,8 under fixed other parameters, results in the reduction of the destruction time $t_*$ by 1,8 times.

There has been shown substantive effect of the geometric dimensions of the blank upon the values of the destruction time $t_*$. It was found that increasing the relative size of the blank radius $\bar{R}_0 = R_0 / h_0$ from 200 to 800 results in reduction of the destruction time by more than 4 times over.

**Summary.** There has been carried out experimental research of operations of isothermal straining of hemispherical parts of high-strength anisotropic materials in the mode of short-time creeping. It was experimentally shown that in case of direct one-stage molding, there forms significant unevenness in the width of the wall along the generatrix from the flange radius to the dome center ($h/h_0$). In case of shells of alloys LT31 it was on an average 0,4; and was 0,33 for shells made out of alloy AA5052. In order to lessen the unevenness in the wall thickness, it is recommended to use two-stage stamping with an insert (reversal). Comparison of theoretical and experimental data on relative thickness in the blank dome and in the basic points, as well as comparison of the blank's relative heights, points to their satisfactory coincidence (up to 10%).

Recommendations have been developed for calculating scientifically-based technological parameters for operations of isothermal straining of semispherical components made out of highly strong anisotropic materials in the mode of short-time creeping. The technological processes are based on observing the proper sequence of operations with the provided blanks in one operating position: heating, vacuuming – forming – thermal stabilization – cooling.

The above recommendations were used for developing technological processes of manufacturing semispherical components meeting the operational technological requirements, the components were made out of highly strong anisotropic materials in the mode of short-time creeping. Samples of dome-
shaped critical-task components made out of titanium alloys 4AL-3MO-1V, manufactured by isothermal stamping in the mode of short-time creeping, are shown in Fig. 5.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Samples of dome-shaped components made out of titanium alloys 4AL-3MO-1V (a) and BT23 (b)</figcaption></figure>

The described technological processes provide for: greater specific strength – 1,5 ... 1,7 times; weight reduction - 1,5 times; lower labor consumption - 2...3 times; greater capacity factor (from/to) - 0,3/0,9.

This research was carried out as per basic part of state task №2014/227 for scientific research, issued by the Ministry of Education and Science of the Russian Federation for 2014-2020, and as per grant RFFI № 14-08-00066 a, № 16-48-710016 and 16-08-00020.

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