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2024. 20(1). 27-39
Строительная механика инженерных конструкций и сооружений Structural Mechanics of Engineering Constructions and Buildings
HTTP://JOURNALS.RUDN.RU/STRUCTURAL-MECHANICS
ТЕОРИЯ ПЛАСТИЧНОСТИ THEORY OF PLASTICITY
DOI: 10.22363/1815-5235-2024-20-1-27-39 UDC 539.3
EDN: XNRJTY Research article / Научная статья
Mixed FEM for Shells of Revolution Based on Flow Theory and its Modifications
Rumia Z. Kiseleva1 H, Natalia A. Kirsanova2 , Anatoliy P. Nikolaev1 , Yuriy V. Klochkov1 , Vitaliy V. Ryabukha1
1 Volgograd State Agrarian University, Volgograd, Russia
2 Financial University under the Government of the Russian Federation, Moscow, Russia O rumia1970@yandex.ru
Article history
Received: September 21, 2023 Revised: December 3, 2023 Accepted: December 17, 2023
Conflicts of interest
The authors declare that there is no conflict of interest.
Authors' contribution
Undivided co-authorship.
Abstract. For describing elastoplastic deformation, three versions of constitutive equations are used. The first version employs the governing equations of the flow theory. In the second version, elastic strain increments are defined the same way as in the flow theory, and the plastic strain increments are expressed in terms of stress increments using the condition of their proportionality to the components of the incremental stress deviator tensor. In the third version, the constitutive equations for a load step were obtained without using the hypothesis of separating strains into the elastic and plastic parts. To obtain them, the condition of proportionality of the components of the incremental strain deviator tensor to the components of the incremental stress deviator tensor was applied. The equations are implemented using a hybrid prismatic finite element with a triangular base. A sample calculation shows the advantage of the third version of the constitutive equations.
Keywords: shell of revolution, physical nonlinearity, prismatic finite element, mixed functional, implementation of mixed FEM
For citation
Kiseleva R.Z., Kirsanova N.A., Nikolaev A.P., Klochkov Yu.V., Ryabukha V.V. Mixed FEM for shells of revolution based on flow theory and its modifications. Structural Mechanics of Engineering Constructions and Buildings. 2024;20(1):27-39. http://doi.org/10.22363/1815-5235-2024-20-l-27-39
Rumia Z. Kiseleva, Candidate of Technical Sciences, Associate Professor of the Department of Applied Geodesy, Environmental Management and Water Management, Volgograd State Agrarian University, Volgograd, Russia; ORCID: 0000-0002-3047-5256; E-mail: rumia1970@yandex.ru Natalia A. Kirsanova, Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematics, Financial University under the Government of the Russian Federation, Moscow, Russia; ORCID: 0000-0003-3496-2008; E-mail: nagureeve@fa.ru
Anatoliy P. Nikolayev, Doctor of Technical Sciences, Professor of the Department of Mechanics, Volgograd State Agrarian University, Volgograd, Russia; ORCID: 0000-0002-7098-5998; E-mail: anpetr40@yandex.ru
Yuriy V. Klochkov, Doctor of Technical Sciences, Professor, Head of the Department of Higher Mathematics, Volgograd State Agrarian University, Volgograd, Russia; ORCID: 0000-0002-1027-1811; E-mail: klotchkov@bk.ru
Vitaliy V. Ryabukha, Postgraduate student of the Department of Mechanics, Volgograd State Agrarian University, Volgograd, Russia; ORCID: 00000002-7394-8885; E-mail: vitalik30090@mail.ru
© Kiseleva R.Z.. Kirsanova N.A.. Nikolaev A.P.. Klochkov Yu.V.. Ryabukha V.V.. 2024
(DCS) I ' ls work is licensed under a Creative Commons Attribution 4.0 International License kSHas https://creativecommons.0rg/licenses/by-nc/4.o/legalcode
Смешанная формулировка МКЭ в расчете оболочек вращения на основе теории течения и ее модификаций
Р.З. Киселева1 н, H.A. Кирсанова2 , А.П. Николаев1 , Ю.В. Клочков1 , В.В. Рябуха1
1 Волгоградский государственный аграрный университет, Волгоград, Россия
2 Финансовый университет при Правительстве Российской Федерации, Москва, Россия И rumia1970@yandex.ru
История статьи
Поступила в редакцию: 21 сентября 2023 г.
Доработана: 3 декабря 2023 г.
Принята к публикации: 17 декабря 2023 г.
Заявление о конфликте интересов
Авторы заявляют об отсутствии конфликта интересов.
Вклад авторов
Нераздельное соавторство.
Аннотация. Для учета упругопластического деформирования используются физические уравнения в трех вариантах. В первом варианте применяются определяющие уравнения теории течения, во втором варианте физических уравнений приращения упругих деформаций определяются, как и в теории течения, а приращения пластических деформаций выражаются через приращения напряжений с использованием условия их пропорциональности компонентам девиатора приращений напряжений, в третьем варианте физические уравнения на шаге нагружения получены без гипотезы о разделении деформаций на упругие и пластические части. Для их получения использовано условие пропорциональности компонент девиаторов приращений деформаций компонентам девиаторов приращений напряжений. Реализация уравнений выполнена с использованием гибридного призматического конечного элемента с треугольным основанием, на конкретном примере показано преимущество третьего варианта физических уравнений. Ключевые слова: оболочка вращения, физическая нелинейность, призматический конечный элемент, смешанный функционал, реализация смешанного МКЭ
Для цитирования
Kiseleva R.Z., Kirsanova N.A., Nikolaev A.P., Klochkov Yu.V., Ryabukha V.V. Mixed FEM for shells of revolution based on flow theory and its modifications // Строительная механика инженерных конструкций и сооружений. 2024. Т. 20. № 1. С. 27-39. http://doi.org/10.22363/1815-5235-2024-20-1-27-39
1. Introduction
For the majority of deformable materials, Hook's law is only valid at loading levels, at which the stresses do not exceed the yield stress of the material. Usually, plastic deformations emerge in stress concentration zones already at insignificant levels of loading. Hence, structural analysis with account of elastoplastic deformation zones is an important engineering problem.
Two elastoplastic deformation theories are most commonly used for solid bodies: flow plasticity theory and the theory of incremental elastoplastic deformation1 [1-3].
Displacement-based finite element method (FEM) has been widely used for elastoplastic deformation analysis2 [4-7]. This method was applied to thermoplastic and contact problems of continuum mechanics [8-12].
Киселева Румия Зайдуллаевна, кандидат технических наук, доцент кафедры прикладной геодезии, природообустройства и водопользования, Волгоградский государственный аграрный университет, Волгоград, Россия; ORCID: 0000-0002-3047-5256; E-mail: rumia1970@yandex.ru Кирсанова Наталья Анатольевна, доктор физико-математических наук, профессор департамента математики, Финансовый университет при Правительстве Российской Федерации, Москва, Россия; ORCID: 0000-0003-3496-2008; E-mail: nagureeve@fa.ru
Николаев Анатолий Петрович, доктор технических наук, профессор кафедры механики, Волгоградский государственный аграрный университет, Волгоград, Россия; ORCID: 0000-0002-7098-5998; E-mail: anpetr40@yandex.ru
Клочков Юрий Васильевич, доктор технических наук, профессор, заведующий кафедрой высшей математики, Волгоградский государственный аграрный университет, Волгоград, Россия; ORCID: 0000-0002-1027-1811; E-mail: klotchkov@bk.ru
Рябуха Виталий Васильевич, аспирант кафедры механики, Волгоградский государственный аграрный университет, Волгоград, Россия; ORCID: 0000-0002-7394-8885; E-mail: vitalik30090@mail.ru
1 Malinin M.M. Applied Theory of Plasticity and Creep. Moscow: Engineering; 1975. (In Russ.); Samul V.I. Fundamentals of the theory of elasticity and plasticity: a textbook for university students. 2nd ed., revised. Moscow: Vysshaya shkola Publ.; 1982. (In Russ.); Samul V.I. Fundamentals of the theory of elasticity and plasticity: a textbook for university students. 2nd ed., revised. Moscow: Vysshaya shkola Publ.; 1982. (In Russ.)
2 Levin V.A. Nonlinear Computational Mechanics of Strength. Models and methods. Moscow: Fizmatlit. Publ.; 2015. (In Russ.)
FEM was also effectively employed in finite strain cases of elastoplastic deformation processes [13-16]. Mixed finite element method has been extensively applied to elastoplastic deformation problems [17-21].
In this study, a prismatic finite element with triangular bases has been developed in mixed FEM formulation. Three versions of governing equations are used as constitutive relations. The first version uses the flow theory equations. The second version employs the governing equations obtained from the authors' hypothesis, that the components of the incremental plastic strain tensor are proportional to the components of the combined stress deviator tensor.
The third version does not separate strains into the elastic and plastic parts. For determining the relationships between the strain increments and the stress increments, the condition of proportionality between the components of the incremental strain deviator tensor to the components of the incremental stress deviator tensor was used.
2. Methods 2.1. Shell Geometry
Arbitrary point M0 of the shell, which is located at distance t from the middle surface, is defined by the following position vector:
R0t = R0 + ta0, (1)
where R0 = xi + r sin 9 j + r cos 9 k is the position vector of the corresponding point M0 of the middle surface; r is the radius of curvature of the middle surface point; i, j, k are the unit vectors of the Cartesian coordinate system; x,9 are the axial and angular coordinates of point M0; a0 = a° x a0 is the normal line to the
middle surface at pointM0; a° x 5° are the unit basis vectors at point M0.
The basis vectors of arbitrary point Mot are determined by differentiating position vector (1):
g=R0t; g = Rr; g0 = R=a0, (2)
and by following [17], the matrix expressions of the derivatives of the basis vectors of an arbitrary point in the basis of this point are formed:
{g^M^}; {fc^I»]^j; {fc?}=№}, (3)
3x1 3x3 3x1 3x1 3x3 3x1 3x1 3x3 3x1
where {g,0}T = ggj g,03 }; {g% }T = {g0r 19 g0r29 g0r39}; {g,0}T = {g,n g,02 g,03 };
1x3 1x3 1x3
{g0 jT = {j0 g2g3 } are the row matrices of the derivatives of the basis vectors ofM0t.
Under gradually applied load, the incremental displacement vector at a load step is represented by components in the basis of point M0t:
'V = Av1 g0 + Av2 g0 + Av3 g° = {g0 }T (Avj, (4)
1x3 3x1
where {'v}T = {av1 Av2 Av3 } is the row matrix of displacements of point M01. 1x3
The derivatives of the displacement vector are also expressed in terms of the basis vectors of point Mot:
\\J A - 0 , z-2-o ,3-0 AV,S = /jgj + fx g2 + fi g3;
AV,r9 = /igi0 + /2g2 + /3g3; (5)
= figO + /32g2 + /3g0,
where
/ = Av,xs +Av1m11 + Av2m21 + Av3m31; — /33 = Av,3 +Av713 + Av2l23 + Av3l33; mj, nj,ly are the elements of matrices [m], [n] and [/].
Under specified load, an arbitrary point of the shell will di splace to po sition M, which i s dete rmine d by position vector
Rt = R0t +AV. (6)
The strain increments for a load step are governed by relations [3] in a geometrically linear definition
^=2 (g0 •Av,J+g0 -Av,i *. (7) Considering (5), strains (7) can be expressed in matrix form as
#AS}=[l]{Av}, (8)
6x1 6x3 3x1
where {As}^ = {Asss Asqq Astt 2Assq 2Asst 2AsQt } is the row matrix of strain increments; %L ] is the matrix
1x6
of differentiation operators.
2.2. Relations of Flow Plasticity Theory Full strain increments As j are combinations of elastic strains As j and plastic strains AsP .
Asj =Asj +Asf . (9)
The relationships between the elastic strain increments and the stress increments are defined by expressions3
'sj = E [(1 -v)A.y-vAac5j], (10)
E
where E is the material Young's modulus; v is the Poisson's ratio; A.c is the mean value of the normal stress increments; 8 j is the Kronecker delta.
In the flow theory, the plastic strain increments are defined by relations4
3 Samul V.I. Fundamentals o/ the theory o/ elasticity and plasticity: a textbook for university students. 2nd ed., revised. Moscow: Vysshaya shkola; 1982. (In Russ.); Demidov S.P. Theory of elasticity. Moscow: Vysshaya shkola; 1979. (In Russ.); Demidov S.P. Theory of elasticity. M.: Vysshaya shkola; 1979. (In Russ.)
4 Malinin M.M. Applied Theory o/Plasticity and Creep. Moscow: Engineering; 1975. (In Russ.)
Asp = k ij-.Ду ), (11)
where k is the coefficient of proportionality, which is defined according to6 expression
k =
3
6 1 1 3
2a
. Ek Ен 2
A.i. (12)
Here: .i is the stress intensity; EH is the modulus of the initial segment of the stress-strain intensity diagram; EK is the tangent modulus at the considered point on the stress-strain intensity diagram;
A. =-— A.mn is the stress intensity increment.
mn
By combining (10) and (11) and taking into account (12), the matrix expression for the constitutive equations of the flow theory is formed:
{Ae} = {C!11} {Ao}. (13)
6x1 6x6 6x1
The second version of the post-yield constitutive equations uses the hypothesis of proportionality between the components of the incremental plastic strain tensor and the components of the incremental stress deviator tensor:
'sf = 81 (Aa j-'.cSj). (14)
Proportionality coefficient 81 is defined according to5 expression
(15)
3
8,=2
1 13
5 Ek Ен 2
By combining (10) and (14), the matrix expression for the second version of constitutive relations is obtained:
{As} = [C2n]{Ag) . (16)
"2
6x1 6x6 6x1
The third version of constitutive relations is based on the hypothesis of proportionality between the incremental strain deviator tensor and the incremental stress deviator tensor components:
Asj - 8 j Asc = 8 2 ('. j - -ij A.c ), (17)
where = — —-!- = , and the volumetric strain increment is determined as in the case of elastic 2 2 Ao,. 2 Ek
1- 2/
deformation, Asc = A.c-.
E
Based on (17), the third version of the constitutive relations is formed:
{'-} = [Qn ]{'o} . (18)
5 Malinin M.M. Applied Theory of Plasticity and Creep. Moscow: Engineering; 1975. (In Russ.)
2.3. Finite Element Stiffness Matrix
A prismatic finite element with triangular bases is considered. The nodal unknowns are the displacement and stress increments. Coordinates s, 0, t of an arbitrary point of the shell are defined in terms of nodal coordinates using linear functions n, Z with ranges 0 n < 1; -1 < Z <1,
A = {/ ((,n, Z)} {}, (19)
1x6 6x1
where {y } = /a' Aj Ak Am An Ap } is the row of nodal coordinate s, 0 or t;
{ (,„, c) = {(i - 5 - n ; 5 Izi: „ Idz; (1 - 5 - „)!!?: 5 l±i: „ ^
By using linear approximating functions (19), the interpolation expressions for Av components and the components of the incremental stress tensor are formed:
/Av}= [a] /Avy}; {Ao} = [s] /Aay}, (20)
3x1 3x18 1gx1 6x1 6x36 36x1
where {Avy} = {Av1' Av1 jAv1kAv2iAv2jAv2k... Av3mAv3nAv3p} is the row-matrix of the nodal displace-
1x18
ment increments;
{Ao} = {Ao ss Ao00Aott Ao s0Ao st Ao0t} is the row of stress increments at a point;
1x6
{Aoy } = -j/Aossy } {Ao00y } - {Ao0ty} j is the row of stress increments at the nodes of the finite
1x36 I 1x6 1x6 1x6 J
element.
Considering (20), strain increments (8) can be represented in matrix form:
{Ae}= [l] {Av}= [l] [a] {Avy}= [5] {Avy}. (21)
6x1 6x3 3x1 6x3 3x18 i8xi 6x18
The nonlinear mixed functional for a load step, obtained in [17], is expressed as 0 = J[A*}r [L][Av]dV-2J[A*}r [C*][Aa}dV-
V 1x6 6x3 3x1 2 V 1x6 6x6 3x1
-2J[Av} [ Aq}dS-J{Av} [q}dS + J[&} [Af}dV; (// = 1,2,3).
2 S 1x3 3x1 S 1x3 3x1 V 1x6 6x1
(22)
Taking into account matrix relations (18) and (21), functional (22) for the prismatic finite element becomes
Ф iK ГЛ5 & [ в ] ^^ {'vy}- 2 К }T J[ s f№ ][ s ] ^^ К}
1x36 V 36"6 6x18 18x1 2 1x36 V 36"6 6x6 6x36 36x1
2{'Vy}T J[Af {'q}dS-{'Vy}T J[A]T {q}dS + {'Vy}T J[B]T {#}dV.
2 1x18 S 18x3 3x1 1X18 s 18x1 3x1 1x18 V 18x6 6x1
(23)
By varying functional (23) with respect to nodal unknowns {a.y ^ and {'Vy ^, the following systems of equations are obtained:
1x36
1x18
J^TFI Q] К)- H] К }= 0; I Q]T {'.}-{!/, j-{R}= 0,
7fCT yf 36x18 18x1 36x36 36x1 7\'Vy/
18x36 36x1
18x1
18x1
(24)
where Q] = J [S]T [в] dV; [H] = J [S]T ^ ] [S] dV; {'/ }= J [a]t {'q)ds;
36x18 V36x6 6x18 36x36 V36x6 6x6 6x36 1gx1 S 18x3 3x1
{Я} = J[A]T {q}dS - J[B]T {#}dV is the Raphson residual.
18x1 S 18x3 3x1 V 18x6 6x1
Systems (24) can be combined into one [K ]{Zy }={Fy}
54x54 54x1 54x1
(25)
where [K & =
54x54
-[H ] Q]
36x36 36x18
Q]T [0]
18x36 18x18
— is the matrix of the stress-strain state of the hybrid finite element at a load
step; {zy ^ = j {'ay }T {'vy }T D — is the vector of nodal unknowns; {Fy } = {0 }T : {afq } + {r }T
1x54 F 1x36 1x18 J is the vector of nodal loads with residuals.
у
1x 54
1x 36
1x18
1x18
3.1. Sample Calculation 1
The shell of revolution depicted in Figure 1 with the middle surface in the shape of a truncated ellipsoid was analyzed. The following input values were specified: a = 0.21 m; e = 0.15 m; h = 0.01 m; ¡k = 0.2 m; E = 2* 105 MPa; v = 0.3. The height of truncation of the elliptical shell is
zk =
V
12
1 = 0.15' a
V
0 202
1--— = 0.0457 m.
0.212
The stress-strain curve for the shell material was assumed to be in the form of Figure 2, where oT = 200 MPa is the yield stress of the material; eT = 0.001 is the yield strain; -k = 0.02 is the final strain; ok = 400 MPa is the final stress.
Figure 1. Truncated elliptical shell Source: made by R.Z. Kiseleva
The stress-strain intensity curve was constructed using formulas6
a, =
1
s, =■
V2 V2
(011 0 022 )2 +(022 0 033 )2 +(033 0 011 )2
1
V2
[o2 + 0 + a2 ] = a;
(s11 S 22 * + ( S22 S33 * + ( S33 S11 *
S
(s + vs)2 + 0 + (0vs 0 s)2
= 2 (1 + v )
2
3
3
3
Figure 2. Stress-strain curve of the elliptical shell material Source: made by R.Z. Kiseleva
Figure 3. Stress-strain intensity curve of the shell material Source: made by R.Z. Kiseleva
Values of the parameters of the stress-strain intensity curve: .iT =.T = 200 MPa is the stress intensity at yield point;
2 2
siT = 3(1 + v*sT = 3(1 + 0.3)- 0.001= 0.866667* 10-3 is the strain intensity at yield point; 2 2
,ik = 3 (1 + v),k = 3 (1 + 0.3* - 0.01 = 0.866667* 10-2 is the final strain intensity; . iK = . k = 300 MPa is the final stress intensity.
6 Malinin M.M. Applied Theory of Plasticity and Creep. Moscow: Engineering; 1975. (In Russ.)
The stress-strain intensity curve is assumed to be defined by function o, = f (e,) in the form of a parabola
o, = ae2 + be, + c (when s, > s,T),
where a = -6612835.5282 MPa; b = 242231.47902 MPa; c = 1795.0330258 MPa.
The presented shell of revolution was analyzed for the case of elastic deformation (q = 18.0 MPa). The normal stress values at the fixed support are presented in Table 1, where the first column contains the number of discretization nodes of the shell along its axis (NM) and along its thickness (NT).
The other columns contain normal stresses of the internal fibers along the axis ^a"11j and circumference (a(nt). For the external fibers, these variables are denotes as (o™1) and (o 9f ) respectively.
Table 1
Numerical values of normal stresses at the fixed support
NM x NT int . п , MPa . S , MPa #ext, MPa 11 Ом ' MPa
20x3 116.484 210.103 117.640 203.671
40x5 116.324 209.843 118.267 203.857
30x7 116.234 209.766 118.396 203.834
The results presented in Table 1 demonstrate convergence of the computational process with respect to normal stresses of the shell at the fixed support.
3.2. Sample Calculation 2
The analysis of the shell from the previous section was performed under internal pressure q = 27.65 MPa. The specified load value was achieved in 16 steps and in 32 steps, and the results of the analysis using the three versions of constitutive equations were found to be virtually identical.
The values of meridional stresses .ss and circumferential stresses Gqq after 32 load steps are presented in
Table 2. The stress values were calculated along the shell thickness h in the left section using the third version of the constitutive equations.
Table 2
Numerical values of meridional and circumferential stresses after 32 load steps along the shell thickness h in the left section
Oss, MPa 163.8 170.7 175.4 181.8 186.9 193.2 205.4
oee, MPa 323.2 318.9 314.4 313.1 309.0 306.9 302.8
h, m 0 0.00166 0.0033 0.005 0.0066 0.00833 0.01
The of results from Table 2 are used to plot the distributions of meridional stresses (Figure 4) and circumferential stresses (Figure 5).
In order to control the accuracy of computation of meridional stresses, the check of Ox = 0 is performed. The check gives an acceptable discrepancy in the values of the resultant external and internal forces:
Q tQ
5= Qmt x100% = 2.4%,
Qext
where Qext is the resultant external force; Qint is the resultant internal force.
500 1000 1500 2000 2500
Figure 4. Distribution of meridional stresses . ss along the section thickness Source: made by R.Z. Kiseleva
/'cm
3000 3050 3100 3150 3200 3250
Figure 5. Distribution of circumferential stresses aee S o u r c e : made by R.Z. Kiseleva
As seen from Figure 5, the circumferential stresses exceed the elastic limit significantly. Table 3 provides the values of meridional and circumferential stresses in the external fibers along the meridian arc length.
Table 3
Numerical values of meridional and circumferential stresses in external fibers along the meridian arc length
Meridian arc length S, m
airess 0.005 0.025 0.04 0.06 0.081 0.102 0.124 0.147 0.172 0.185 0.2 0.237
. , MPa 164.2 165.0 164.6 163.1 161.0 157.1 152.9 150.0 142.2 134.9 121.0 0.6
.00 , MPa 302.9 302.7 301.0 295.6 288.6 279.5 267.7 254.5 249.5 242.7 244.0 251.1
The results from Table 3 were used to plot the distributions of meridional stresses stresses Gqq (Figure 6).
and circumferential
Figure 6. Distributions of meridional and circumferential stresses in external fibers along the meridian arc length S o u r c e : made by R.Z. Kiseleva
ss
The values of the meridional stresses in the end section are almost zero, which complies with the loading condition. The circumferential stresses vary insignificantly along the meridian.
The analysis of the results in Tables 1 -3 indicates correctness of the developed algorithm and shows adequate convergence of the computational process.
4. Conclusion
The 3D stress-strain state of a shell is studied without using the straight-normal hypothesis for elastoplastic deformation.
1. The constitutive relations beyond the elastic limit are implemented in three versions.
The first version uses the relationships of the flow theory.
The second version employs the governing equations, where the authors' hypothesis is used for determining the plastic strain increments. The hypothesis assumes that the components of the incremental plastic strain tensor are proportional to the components of the incremental stress deviator tensor.
The third version of equations is based on the hypothesis of proportionality between the components of the incremental strain deviator tensor and the components of the incremental stress deviator tensor without separating the strains into elastic and plastic.
2. The analysis of the shell is performed using mixed FEM. For this purpose, the authors developed a 6-node solid prismatic finite element with triangular bases. The nodal unknowns are the displacement vector components and the nodal stress tensor components. The target variables are approximated by the nodal unknowns using bilinear shape functions.
3. The presented study shows that all three versions of the governing equations for plastic deformation produce identical results. The analysis of the constitutive equations shows that the most physically reasonable version is the third one. This version does not separate the strain increments into elastic and plastic parts, and is based on the hypothesis of proportionality between the components of the incremental strain deviator tensor to the components of the incremental stress deviator tensor.
The proposed governing equations, without the strain separation, correspond to the physical meaning of the process of deformation and have great potential for analyzing reservoirs, submersibles and other engineering structures containing shells of revolution.
References
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