Isothermal Pneumatc Molding of Dome-Shaped Parts of Anisotropic Material in Short-Time Creeping Mode

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Modern Applied Science; Vol. 9, No. 8; 2015

ISSN 1913-1844 E-ISSN 1913-1852
Published by Canadian Center of Science and Education

# Isothermal Pneumatc Molding of Dome-Shaped Parts of Anisotropic Material in Short-Time Creeping Mode

Sergey Sergeevich Yakovlev 1, Sergey Nikolaevich Larin 1 & Valeriy Ivanovich Platonov 1

1 Tula state university, Tula, Russian Federation

Correspondence: Sergey Nikolaevich Larin, Tula state university, Tula, Russian Federation.

Received: October 16, 2014  Accepted: November 1, 2014  Online Published: June 29, 2015

doi:10.5539/mas.v9n8p13   URL: http://dx.doi.org/10.5539/mas.v9n8p13

Abstract

Studying isothermal straining of dome-shaped shells with the aim of evaluating kinematics of the material flow,
stress and strained conditions, force conditions, geometrical dimensions of shells and possibility frontier of
straining. The theoretical study of the isothermal straining processes has been performed on the basis of the
theory of short-time creeping of anisotropic material. There has been revealed the influence of technological
parameters, conditions of loading, and geometry of working tools on the kinematics of material flow produced
by friction conditions on the blank and tool contact surfaces, stress and strain states, force conditions, possibility
frontiers of form-shaping of operations of isothermal deforming of dome-shaped shells made out of
high-strength anisotropic materials. Conclusions. The equations obtained can be used for theoretical analysis of
operations of isothermal straining of dome-shaped shells in the mode of short-time creeping.

Keywords: anisotropy, dome-shaped shells, high-strength materials, stress, strain, isothermal straining, viscosity,
damaging, destruction

1. Introduction

Dome-shaped parts have found their wide usage in various industries. The traditional methods of their
production by means of pressing are rather labor-consuming and problematic for securing requisite geometric
precision, due to residual stress, which causes deformation of the contour and hence an extensive scope of fitting
and component-adjusting operations to adjust the parts to the requisite dimensions.Residual stress is largely
caused by the inbuilt anisotropy of the mechanical properties in the sheet under deformation and by the uneven
character of the deformations. Isothermal changing of the shape of dome-shaped parts with gas of highly strong
aluminum and titan alloys is considerably advantageous versus the conventional processing methods, and is quite
promising for industrial use [1-6].

Sheet material, subjected to deforming processes, as a rule, possesses certain anisotropy of mechanical properties,
determined by the material grade, by technological modes of its production, which can produce a positive or a
negative effect upon stable procedure of technological processes of metal treatment by pressure under various
temperature-speed modes of deformation [5-15].

Single- and multilayered cellular structures (waffle-grid structure) are used for the shells of the product vessels.
At a high load-bearing capacity, these structures possess relatively light weight and strength balance while
loading. The existing technological processes of manufacturing cellular structures include machining operations
(milling) and electroerosion. Technological principles of a hot slow-speed forming through the excess gas
pressure can be used in the production of cellular structures out of aluminum and titanium alloys.

2. Methodology

The theoretical study of processes of isothermal straining is carried out on the basis of the theory of short-time
creeping of anisotropic material. The object of study was stressing of an anisotropic material under conditions of
short-time creeping. Short-time creeping shall be assumed as a slow straining under conditions of viscous or
ductile flow. The elastic components of strain are ignored [1]. The analysis of the stress and strain conditions of
the blank, of the force conditions of isothermal pneumatic molding of dome-shaped parts has been performed
numerically by the method of finite-difference mutual relations, use being made of computers, for joint solution
of differential balance equations, equations of state and kinematic correlations under postulated initial and
boundary conditions [10]. The possibility frontiers of deforming have been set on the basis of use of

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phenomenological criteria of destruction (stress and strain) of anisotropic material associated with accumulation
of microdamages.

Main part. We have considered straining of a round sheet blank with radius _R_ 0 and thickness _h_ 0 by free
bulging in the mode of viscous flow of material when exposed to the excess gas pressure _p_ = _p_ 0 + _a_ _pt_ _n p_ into a

spherical matrix (Fig. 1). Here _p_ 0 , _a_ _p_, _np_ are stressing constants.

Figure 1. Scheme to the calculation of strained state of the blank middle surface in the meridian plane

The blank is fixed along the outer contour. The blank material is taken as transversally isotropic with anisotropy
factor _R_ ; the stressed state of the shell is two-dimensional, i.e., the stress perpendicular to the sheet plane
equals zero ([sigma] _z_ =0). We consider stressing in the meridian plane of the shell as a membrane. Due to the
symmetry of the mechanical properties of the material versus the blank axis, and due to the nature of the external
forces behavior, stresses and velocities of straining, which are meridian, circumferential and normal to the
median surface of the blank, are the chief ones.

The median surface of the blank remains a part of the spherical surface at every stage of the straining. A radial
flow of material relative to the new center at every stage of the straining takes place at any meridian section of
the shell.

According to the adopted allowances, the curvature radii of the meridian section [rho] _m_ of the median surface,
and those of the shell section by the conical surface perpendicular to the meridian arch, [rho] _t_ are equal, i.e.,

ρ _m_ = ρ _t_ = ρ= _H_ 22+ _HR_ 02 (1)

where _H_ - is the height of the dome at the given time instant of straining.

It is assumed that the stresses are distributed evenly over the thickness of the element cut out of the membrane
by meridian and conical surfaces in the vicinity of the point under consideration. As distinct from the available
solutions, we make no limitation as to changing the thickness of the shell along the circumference arch in the
meridian section.

The straining velocities in the meridian [ksi] _c_

_m_ , circumferential direction [ksi] _c_

_t_ and along the thickness [ksi] _c_

The straining velocities in the meridian [ksi] _c_ _m_ , circumferential direction [ksi] _c_ _t_ and along the thickness [ksi] _c_ _z_ of

the shell are determined accordingly by the following formulas

(2)

ξ _mc_ =  sinθ - _ctg_ α  α 
 θsinα 

ξ _tc_ =  cosθ− _ctg_ α  α 
 sinα 

sin

sin

α  ; ξ _tc_ =  cos

 sin



α  ;



ξ _zc_ = _hh_

_c_
_z_

Here [teta] – is the actual angle between the blank vertical line of symmetry and the radius-vector determining

the position of the point in the median surface section by a diagonal plane; α  = _d_ α _dt_ ; _h_  = _dh_ _dt_

It was assumed for the shell strain that at every step of straining there takes place a radial flow of the median

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surface point in the meridian plane relative to the new center at time instant _t_ + _dt_, i.e., in direction

[teta]+ _d_ [teta].

The shell thickness in the dome of the middle surface of the shell ([teta]=0) is found by the following expression

 _H_ 2 
_h_ = _h_ 0  1+ _R_ 02  (3)

Change of the shell thickness depending on straining time _t_ at the point of fixation ([teta=alfa]) is found by the
formula

_H_
_h_ = _h_ 0

0  _H_ 2

 _H_ 2  _H_

_R_ 0  1+ 2 

 _R_ 0  _arctg_ _R_ 0

 _H_ 2 
 1+ _R_ 02 

(4)

0  1+ 2 

 _R_ 0  _arctg_ _R_ 0

The values of the meridian [sigma] _m_ and tangential [sigma] _t_ stresses in the process of deforming with [rho _m_ =rho _t_ ],
are found by the formula [10]:

_p_

_p_ ρ

σ _m_ = σ _t_ = 2 _h_ (5)

_m_ _t_ 2

The equivalent velocity of straining [ksi] _c_ _e_ and stress [sigma] _e_ at the dome apex (point “с”) and at the fixation

point of the shell in the contour (point “k”) are calculated for anisotropic material respectively after formulas [5,
6]:

ξ _ecc_ = 23 2 + _R_ ξ _mcc_

2
2
3

ξ _ecc_ = 2 2 + _R_ ξ _mcc_ ; σ _ecc_ = 3

σ _ecc_ = σ _mc_

2 2 + _R_

2 2

;            (6)

1/2
2 (2 + _R_ )( _R_ +1) 



3 2 _R_ +1 

1/2

ξ _ekc_ =  2 (2 + _R_ )( _R_ +1)  ξ
 3 2 _R_ +1 

(2 + _R_ )( _R_ +1)

(7)

1/2

_ekc_ =  2 (2 + _R_ )( _R_ +1)  ξ _mk_

 3 2 _R_ +1  1/2

; σ _ek_ =   σ _mk_

 2 (2 + _R_ )( _R_ +1) 

There has been considered a slow isothermal straining of a shell made out of a material, for which work the
equations of state for the energy theory of creeping and damaging [6]:

### ξ ec = B (σ e σ e 0 ) n / ( 1−ω Ac ) m ;  ω  Ac =σ e ξ ec / Anрc,          (8)

Where _B_, _n_, _m_ - are material constants depending on the test temperature; [omega] _c_ _A_ – is damageability of

material for viscous deformation after energy fracture model; _Aпрс_ - specific fracture work for viscous flow of

material; ω  _Ac_ = _d_ ω _Ac_ / _d t_ ; [ksi] _c_ _e_ and [sigma] _e_ – are equivalent velocities of straining and stress; [sigma] _e0_ – is

equivalent stress dividing the viscous and ductile flows of material.
Since the pressure value _p_ is distributed evenly at every moment of deformation over the surface of the shell,
its value was found at the apex of the shell dome (point “ _с_ ”).

By substituting in the first equation of the material state (8) its values of [sigma] _e_ and [ksi] _c_ _e_ , are to be found

after formulas (6) and (7), taking account of correlations (3), (4), (5), we obtain

_n_ +1
### n σ en 0 (1−ω Acc ) m 22 n + 2 (2 + R ) 2 H n +1 hn

_n_ +1
### 0 (1−ω Acc ) m 22 n + 2 (2 + R ) 2 H n +1

+1

### (1−ω Acc ) m 22 n + 2 (2 + R )

### pndt = σ en 0 (1−ω Acc ) n +122 n + 2 (2 + R ) H n +1 hndH

### 1−ω Acc ) 22 n + 2 (2

(9)

### σ en 0 (1−ω Acc ) 22 n + 2 (2 +

= _n_ +1

### σ en 0 (1−ω c

_n_ + 2 2 _n_ +1

_n_ +1

2 2 _n_

_n_ +1
### 2 B ( H 2 + R 02 ) n +1

+1

2 2 _n_ +1

### B ( H 2 + R

### ( H 2 + R 02 )

Shell thickness _h_ is found after formula (3).

The value of the accumulated damage is found after:

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2
2 

 _H_ 2 

ω  _Acc_ = _p_  1+ _Rc_ 02  _H_  (10)

 _H_ 2 

ω  _Acc_ = _p_  1 _h_ +0 _ARnpc_ 02  _H_ 

2
0

_c_ 
_Ac_ = _h_ 0 _Anpc_

Where _H_  = _dH_ _dt_ .

Destruction time _t_ ∗ is found from [omega] _c_

_Ac_ _=1_ .

Carried out in a similar manner was the study of stress and strain states of a blank in the apex of the dome shell
(point “ _с_ ”) and at the point of the shell fixation (point “k”), as well as received were basic equations and
proportions for solving the set task, with the assumption that material behavior is subject to the equations of the
kinetic theory of creeping and damaging under the well-known law of pressure and time dependence _p_ = _p_ ( _t_ )
and under permanent equivalent velocity of straining in the blank dome [ksi] _e1_
There has been developed the algorithm for calculation of force and straining parameters of the analyzed
technological process and the software.

There have been evaluated the stress and strained states, the kinematic of the material flow, force conditions and
the possibility frontier of straining process under investigation as relating to accumulation of microdamages, as
depending on the anisotropy of the mechanical properties of the basic material, depending on the law of loading,
and on the geometric dimensions of the blank and the product.

Calculation was made for titanium alloys ВТ6С at _T_ = 860  _C_, which behavior is described by the energy

theory of creeping and damaging, and for titanium alloy ВТ14 at _T_ = 950  _C_, which behavior is subject to the
kinetic theory of creeping and damaging. The mechanical characteristics of these materials under straining in
conditions of viscous flow of material are shown in the works [5, 6].
The characteristic curves for changing values of gas pressure _p_, relative values of blank thickness in the dome

_hс_ = _hс_ _h_ 0 and at the place of its fixation _hk_ = _hk_ _h_ 0, dependence of the height of the dome-shaped blank

_H_ = _H_ / _R_ 0 on straining time instant _t_ for titanium alloy ВТ6С ( _T_ = 860  _C_ ) at a permanent equivalent
straining velocity in the blank dome [ksi] _e1_ are shown in Fig. 2. The experimental data are shown here with dots.

Figure 2. Dependences of _p_, _H_ and _h_ changing in the discussed points of a blank from _t_ for titanium

alloy ВТ6С ( _R_ 0 = 300 ; ξ _e_ 1 = 0,002 1 _c_ )

It follows from an analysis of the results of calculations and characteristic curves that as straining time _t_ grows

to a certain limit, there takes place a steep increase in the blank relative height in dome _h_ _с_ and in its place of

fixation _h_ _k_ . Further growth of straining time _t_ results in a smooth change of the analyzed values. At time

instant _t_, which is close to the blank destruction, there takes place an abrupt change of the relative values _H_,

_h_ _с_ и _h_ _k_ . This is caused by an intensive growth of accumulated microdamages at the concluding stage of the
process.

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It has been established that the changing of the relative thickness of the blank’s dome _h_ _с_ takes place in a more

intensive manner as versus the changing of its relative thickness at its place of fixation _h_ _k_ . The difference grows
together with straining time _t_ and may reach 50 %.
It has been shown that for securing a stable equivalent speed of deformation in the blank dome, the law of
changing pressure _p_ during straining time _t_ is of a complex nature. During the initial moment of the
shape-changing we witness a steep change in the pressure _p_, since there takes place a substantial change of the
radius of the half-sphere [rho] _m_ . Subsequent growth of the straining time _t_ is accompanied by a decrease in the
magnitude of the gas pressure _p_ .

Comparing theoretical and experimental data on the relative thickness in the blank dome _h_ _с_ and at the place of

its fixation _h_ _k_ , as well as on the relative height of the blank _H_ , shows their satisfactory coincidence (up to 10 %)

[6].
It has been established that the destruction of the blank during the isothermal straining takes place in the dome of
the part, where happens the maximum thinning-out of the blank.
Figure 3 shows the changed time of destruction _t_ - , the changed relative height _H_ ∗ , and the changed thickness

of the blank dome _h_ ∗ at the moment of destruction, as determined by the magnitude of the accumulated
microdamages with [omega] c A =1, -- as having functional connections to the value of the permanent equivalent

velocity of straining in the blank dome [ksi] _e1_ . It is shown that increased parameters of the law of loading _a_ _р_,

_n_ _р_ and increased magnitude of the value of the permanent equivalent straining velocity in the blank dome

[ksi] _e1_ result in a shorter time of destruction _t_ - and in a smaller relative height _H_ ∗ , as well as in an increased

relative thickness of the blank dome _h_ ∗ .

Figure 3. Dependences of changing _t_ ∗ and _H_ ∗, _h_ ∗ in the blank dome as relative to [ksi] _e_ for titanium alloy

ВТ6С ( _R_ 0 = 300 )

It is determined that normal anisotropy factor _R_ influences considerably the magnitude of the destruction time
and the relative values _H_ ∗, _h_ ∗ . The calculations have been made at _R_ 0 = 300 ; _p_ 0 = 0,013 _МПа_ ;

_a_ _p_ = 4 ⋅10−3 _МПа_ _сnp_ ; _n_ _p_ = 0,6 (the kinetic theory). As there grows the anisotropy factor _R_, the relative

value of _h_ ∗ steeply increases, while destruction time _t_ - and the relative blank height fall abruptly. It has been
established that the failure of considering the anisotropy of the mechanical properties of the blank when
analyzing the process of isothermal changing of the spherical shell shape – makes us err in destruction time _t_
by about 35 %, while in the relative height _H_ ∗ and in the thickness of the blank dome _h_ ∗ at the moment of
destruction – we err by 15 %.
Analysis of the calculation results shows that the limit possibilities of shape-changing in the mode of viscous

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flow of material, which conduct is subject to the kinetic theory of creeping and damaging (alloy ВТ14), do not
depend on the conditions of the blank loading. Shown is the substantial dependence of destruction time _t_ - on
loading parameters _a_ _р_, _nр_ and on the value of the permanent equivalent straining velocity [ksi] _e1_ . Increasing

the loading parameters _a_ _р_ from 0.2 ⋅10−3 _МПа_ _сnp_ to 1.4 ⋅10−3 _МПа_ _сnp_, and those of _nр_ from 0.5 to 0.8

with unchanged other parameters, results in reduction of the destruction time _t_ - by 1.8 times.
There has been shown a substantial influence of the geometric dimensions of the blank upon the destruction time
magnitude _t_ - . It has been established that an increase in the relative size of the blank radius _R_ 0 = _R_ 0 / _h_ 0 from
200 to 800 results in a shorter destruction time by more than 4 times.
3. Discussion
Isothermal gas forming of the domed components made of high-strength aluminum and titanium sheet alloys has
significant advantages over traditional methods of treatment and is very promising when it is used within the
industrial sector. It is less time-consuming and problematic in terms of providing the necessary geometric
accuracy due to the presence of residual stresses; it does not cause edge deformation and does not require a large
volume of thereof associated mechanical and finishing works on parts fitting to the specified dimensions [11 20].

The suggested mathematic model for the isothermal straining of domed shells made of anisotropic high-strength
materials in the mode of creeping, allows us to evaluate the influence of technological parameters, loading
conditions, geometry of the working tool, friction conditions on the contact surface of the workpiece and tool on
the kinematics of material flow, stress and strain states, force conditions, possibility frontier of forming the
process in question, taking into account the anisotropy of mechanical properties, which distinguishes this work
out of many other alike studies. (Authors of article offer mathematical model of isothermal deformation of
dome-shaped covers by superfluous pressure of gas from anisotropic high-strength materials in a creep mode.
The given model allows to estimate kinematics of a current of the material, the intense and deformed conditions
of a cover, power modes, the geometrical sizes of made dome-shaped covers and limiting possibilities of
deformation. The offered mathematical model can be used as base by working out of technological processes of
isothermal deformation of dome-shaped covers).

4. Conclusions

Result analysis of the considered process allows drawing the following conclusions:

On the basis of the developed mathematical model the analysis of isothermal deformation of dome-shaped
covers with a view of an estimation of kinematics of a current of the material strained and deformed conditions,
power modes, the geometrical sizes of covers and limiting possibilities of deformation, in which result has been
carried out:

- We have established the influence of technological parameters, the loading conditions, tools geometry,
conditions of friction on the contact surfaces of the blank and the tool as affecting the kinematics of the material
flow, the stress and strain states, force conditions, the possibility frontier in forming the operations of isothermal
straining of dome shells made out of highly strong anisotropic materials.

- It has been shown that for the purpose of providing for permanent equivalent straining velocities in the blank
dome, the law of pressure change during straining has a complicated character. It has been determined that
changing the relative thickness of the blank dome happens in a more intensive manner by comparison to
changing the relative thickness at the main points. As the straining time is increased, the difference may grow to
reach 50 %.

- It has been established that in the process of isothermal straining of domed blanks, the destruction of the blank
occurs in the dome of the blank where there happens a maximal thinning-out of the blank. Increase of the
parameters of loading and equivalent straining velocity results in reduction of the relative limit height of the
blank and of the destruction time for materials, which conduct is described by the energy theory of creeping and
damaging. The possibility frontier of shape-forming during isothermal straining of anisotropic materials, which
behavior is described by the energy theory of creeping and damaging, do not depend on the blank loading
conditions.

- It has been established that lack of consideration for anisotropy of the blank mechanical properties when
analyzing the process of isothermal straining of dome-shaped shells, gives us an error in evaluating the
destruction time of about 45%, and of 30% for the relative height and thickness in the blank dome at the moment
of destruction.

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Acknowledgement

This work has been performed within the frameworks of the basic part of the State Task #2014/227 for
performing research and development works of the Ministry of Education and Science of the Russian Federation
for the years 2014-2020 and of the RFFI grant # 14-08-00066 а.

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