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A higher order theory for shells, plates and rods
Author’s Accepted Manuscript A higher order theory for shells, plates and rods V.V. Zozulya
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S0020-7403(15)00314-8 http://dx.doi.org/10.1016/j.ijmecsci.2015.08.025 MS3090
To appear in: International Journal of Mechanical Sciences Received date: 11 June 2015 Revised date: 26 August 2015 Accepted date: 28 August 2015 Cite this article as: V.V. Zozulya, A higher order theory for shells, plates and r o d s , International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2015.08.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A higher order theory for shells, plates and rods. V.V. Zozulya Centro de Investigacion Cientifica de Yucatan, A.C., Calle 43, No 130, Colonia: Chuburna de Hidalgo C.P. 97200, Merida, Yucatan, Mexico. [email protected]
Abstract. In this paper a new theory for shells, plates and rods has been developed. The proposed theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. The case of axially symmetric shell has been considered in more details. As a special case of the general approach the equations of the first and second approximations have been developed for circular plates, curvilinear rods and cylindrical shells. Numerical calculations have been done numerically using the finite element method (FEM) along with the Comsol Multiphysics, Matlab and Mathematica software.
Keywords: Shell, plate, rod, Legendre polynomial, FEM. 1. Introduction Classical theories of beams, rods, plates and shells are usually related with Bernoulli, Euler, Kirchhoff and Love. These theories are based on well-known physical hypothesis; they are very popular among the engineering community because of their relative simplicity and physical clarity. Numerous books and monographs have been written on the subject, among others one can refer to [1-4]. In the same time classical theories have some shortcomings such as its proximity and inaccuracy and as result in some cases are observed not good agreement with practice. For this reason new and more accurate theories have been developed. We can mention several approaches to development of the theories of thin-walled structures. One consists in improvement of the classical physical hypothesis and the development of more accurate theories. In the beams theory is a well-known model that takes into account transversal deformations developed by Timoshenko and was then extended to the plate theory by Mindlin [5]. This theory was extended and applied to shells in numerous publications and is referred to as the theory that takes into account in-plane shear deformations [6-8]. So-called direct approach has been developed and applied to plates, shells and rods analysis in [9, 10]. Another approach consists in the expansion of the stress-strain field components into polynomials series in terms of thickness. It was proposed by Cauchy and Poisson and that a time was not popular. Significant extensions and developments of that method have been done by Kil'chevskii [11]. Vekua has used Legendre’s polynomials for the expansion of the equations of elasticity and the reduction of the 3-D problem to the 2-D one [12]. Such an approach has significant advantage since Legendre’s polynomials are orthogonal and as result the developed equations are simple. This approach was extended and applied to dynamical problems [13] and thermoelasticity [14], composite and laminate shells [15], the theory of classical and micropolar elasticity [16].
The approach developed in [11-16] has been applied to the plates and shells thermoelastic contact problems when mechanical and thermal conditions are changed during deformation. Details of the approach, equations and contact conditions have been reported for the first time in [17-19]. In [20, 21] it was extended to no stationary processes, presented in general form all coefficients of the corresponding equations with initial, boundary and contact condition. Then the higher order theory was further developed to the contact of plates and shells with rigid bodies though a heat conducting layer [22-26], thermoelastic the laminated composite materials with the possibility of delamination and thermoelastic contact in the temperature field in [22, 27, 28], the pencil-thin nuclear fuel rods modeling in [29], functionally graded shells in [30-32] and for electrostatically actuated MEMS in [33]. An analysis of the higher order shells theories of thermoelasticity and comparison with classical theories has been done in [34]. Different aspects of the theory and applications of the thin-walled structures can be found in thousands publications, for references one can see review papers [35-40] and for trend and resent development books [41-44]. In this paper, based on the expansion of the equations of elasticity into Fourier series in terms of Legendre polynomials a higher order theory of shells, plate and rods shell has been developed. More specifically, we expanded functions that describe stress-strain relations into Fourier series in terms of Legendre polynomials and found corresponding relations of elasticity for Fourier coefficients of expansions of the stress and strain. Then using the techniques developed in our previous publications we found a system of differential equations and boundary conditions for Fourier coefficients. The case of axially symmetric shells is considered in more detail and all relations explicitly presented. From these equations as a special case there can be easily derived higher order equations for beams, rods, circular axisymmetric plated, cylindrical and spherical shells. Cases of the first and second approximations have been considered in more detail. Numerical examples are presented. Comparison with the theory of elasticity and the classical theories has been done. The analysis of the numerical results shows a good coincidence of the displacements and stresses obtained by using the higher order theory and elasticity solution for even relatively thick structural elements.
2. 3-D formulation As it was mentioned above in this study we further develop an approach based on the expansion of the equations of elasticity into Fourier series in terms of Legendre polynomials and apply it to construct higher order moles of shells, plates and rods. Therefore we start our consideration with a 3-D formulation of the problem. We consider a linear elastic body occupying an open in 3-D Euclidian space simply connected bounded domain V R3 with a smooth boundary V . The elastic body is supposed to be a homogeneous isotropic shell of arbitrary geometry with 2h thickness and occupies the domain V [h, h] in Euclidean space. Here is the middle surface of the shell, is its boundary. The boundary of the body can be presented in the form + V S , where and are the outer sides and S [h, h] is a sheer side. In order to examine the stress-strain state of the elastic body we introduce stress ij and ij strain tensors and displacements ui , traction p i , and body forces b i vectors. Covariant and mixed components of the related tensors and vectors can be found in a standard way by raising and lowering indexed using metric tensors g ij , g ij and g ij respectively. These quantities are related by equations of linear elasticity. In the case if displacements and their gradients are small Cauchy relations take place 1 1 (1) ij (i ui j ui ) ( i ui j ui ) ijk uk 2 2
Here i are covariant derivatives, ijk are the Christoffel symbols, i xi are partial derivatives with respect to the space variables xi . Equations of equilibrium have the form j ij bi 0,
j ij j ij ijk kj jkj ik
(2)
Stress and strain tensors are related by generalized Hooke’s law
ij cij kl kl , cij kl g ij g kl ( g il g jk g ik g jl )
(3)
Here and are Lame constants, 0 and , are Lame constants. Throughout this paper we use the Einstein summation convention. For convenience we transform the above equations of elasticity taking into account that the radius vector R(x) of any point in domain V, occupied by material points of shell may be presented as
R(x) r(x ) x3n(x )
(4)
where r(x ) is the radius vector of the points located on the middle surface of shell, n(x ) is a unit vector normal to the middle surface. We consider that x ( x1 , x2 ) are curvilinear coordinates associated with the main curvatures of the middle surface of the shell which is commonly used in application. In this case 3-D equations of elasticity (1) - (3) significantly simplify making it unnecessary to distinguish co- and contra-variant components of tensors and vectors. Therefore from here to the end of the paper lower index will be used for components of vectors and tensors and upper index for components of the coefficients of the Legendre’s polynomial series expansion. The equations of equilibrium have the form ( A2 11 ) ( A1 12 ) A A A1 A2 13 12 1 13 A1 A2 k1 22 2 A1 A2b1 0, x1 x2 x3 x2 x1 ( A2 21 ) ( A1 22 ) A A A1 A2 23 21 2 23 A1 A2 k2 11 1 A1 A2b2 0, x1 x2 x3 x1 x2
(5)
( A2 31 ) ( A1 32 ) ( A1 A2 33 ) 11 A1 A2 k1 22 A1 A2 k2 A1 A2b3 0. x1 x2 x3
Here A ( x1 , x2 ) r ( x1 , x2 ) r ( x1 , x2 ) are coefficients of the first quadratic form of a surface, r
r is a derivative the radius vector of the points located on the middle surface of shell, x
k ( x1 , x2 ) are it main curvatures. The Cauchy relations have the form
11
1 u1 1 A1 1 u2 1 A2 u2 k1u3 , 22 u1 k2u3 , A1 x1 A1 A2 x2 A2 x2 A2 A1 x1
u3 1 u 1 A2 1 u2 1 A1 , 12 1 u2 u1 x3 A2 x2 A1 x1 A1 x1 A2 x2 u u 1 u3 1 u3 13 1 k1u1 , 23 2 k2u2 x3 A1 x1 x3 A2 x2
33
(6)
The generalized Hooke’s law (3) in this case has the form
ij cijkl kl ,
cijkl ij kl ( ik jl il kj ),
(7)
where ij is a Kronecker’s symbol. Substituting the Cauchy relations (1) in the Hooke’s law (3) and obtained relations in the equations (2) we obtain the differential equations of equilibrium in the form of displacements which may be presented in the form L u b 0,
x V
(8)
The differential operator L Lij ei e j for homogeneous anisotropic and isotropic medium has the form
Lij cikjl k l and Lij ij k k ( )i j
(9)
respectively. If the problem is defined in an infinite region, then additional conditions at the infinity in the form have to be satisfied u j (x) O(r 1 ), ij (x) O(r 2 ), for r .
(10)
If the body occupied a finite region V with the boundary V , it is necessary to establish boundary conditions. We consider the mixed boundary conditions in the form ui (x) i (x) , x Vu , pi (x) ij (x)n j (x) Pij [u j (x)] i (x) , x Vp
(11)
The differential operator Pij : u j pi is called stress operator. It transforms the displacements into the tractions. For homogeneous anisotropic and isotropic medium they have the forms
Pij cikjl nk l
and
Pij ni j ij n n j i
(12)
respectively. Here ni are components of the outward normal vector, n ni i is a derivative in direction of the vector n(x) normal to the surface V .
3. 2-D formulation In order to transform a 3-D problem into 2-D one let us expand the parameters that describe stress-strain of the cylindrical shell in the Legendre polynomials series along the coordinate x3
ui x uik x Pk ( ) , uik x k 0
2k 1 ui x , x3 Pk ( )dx3 , 2h h h
ij x ijk x Pk ( ) , ijk x k 0
ij x ijk x Pk ( ) , ijk x k 0
where
x3 . h
2k 1 ij x , x3 Pk ( )dx3 , 2h h h
2k 1 ij x , x3 Pk ( )dx3 . 2h h h
(13)
For the derivatives with respect to coordinates x the following relations take place h uik x 2k 1 ui x P dx k 3 2h h x x h ijk x 2k 1 ij x P dx k 3 2h h x x
(14)
Integration of the derivatives with respect to coordinates x3 gives us h 2k 1 ui x Pk dx3 uik x 2h h x3
2k 1 i 3 x 2k 1 k Pk dx3 i 3 x 1 i3 x ik3 x 2h h x3 h h
(15)
where 2k 1 k 1 ui x uik 3 x , h 2k 1 k 1 ik3 x A1 A2 i3 x ik33 x h uik x
(16)
Now substituting stress tensor from (13) into the equations of equilibrium (5) , multiplying obtained relations by Pk ( ) and integrating over interval [h, h] with respect to x2 we obtain 2-D equations of equilibrium in the form A2 11k x1
k A2 21
x1
A2 31k x1
A1 12k x2
k A1 22
x2
A1 32k x2
12k
A1 k A2 13k A1 A2 k1 22 13k A1 A2 f1k 0 , x2 x1
12k
A2 A k k 23 A1 A2 k2 11k 2 23 A1 A2 f 2k 0 , x1 x1
(17)
k 11k A1 A2 k1 22 A1 A2 k2 33k A1 A2 f 3k 0 .
where fi k x bik x
2k 1 i3 x (1)k i3 x h
(18)
In the same way can be found the 2-D Cauchy relations
11k
1 u1k 1 A1 k 1 u2k 1 A2 k k u2 k1u3k , 22 u1 k2u3k , A1 x1 A1 A2 x2 A2 x2 A1 A2 x1
1 u1k 1 A2 k 1 u2k 1 A1 k u2 u1 , A2 x2 A1 x1 A1 x1 A2 x2 1 u3k 1 u3k k 13k k1u1k u1k , 23 k2u2k u2k , 33 u3k . A1 x1 A2 x2
12k
(19)
The Hooke’s law (7) after Legendre’s polynomial series expansion (13) has the form: for homogeneous anisotropic body
ijk cijlm lmk
(20)
and for isotropic one
11k 2 11k 22k 33k , 22k 2 22k 11k 33k , 33k 2 22k 11k 22k , 12k 12k , 13k 13k , 23k 23k ,
(21)
In the above relations (17) - (21) orthogonality of the Legendre’s polynomials has been used. In order to find 2-D differential equations in form of displacements we substitute Cauchy relations into Hooke’s law for homogeneous body 1 u1k 1 u2k 1 A1 k 1 A2 k u2 k1u3k u1 k2u3k u3k , A1 x1 A1 A2 x2 A2 x2 A1 A2 x1
11k 2
1 u2k 1 u1k 1 A2 k 1 A1 k u1 u2 k1u3k u3k , A2 x2 A1 A2 x1 A1 x1 A1 A2 x2
2k2 2
1 u1k 1 u2k 1 A1 k 1 A2 k u2 k1u3k u1 , A2 x2 A1 A2 x1 A1 x1 A1 A2 x2
33k 2 u3k 1 u k
1 A
1 u k
1 A
(22)
2 k 1 k 12k 1 u2 2 u1 , A2 x2 A1 x1 A1 x1 A2 x2
1 u3k 1 u3k k1u1k u1k , 2k3 k2u2k u2k , A1 x1 A2 x2
13k
Substitution of these equations in the equations of equilibrium (17) gives us the 2-D equations in the form of displacements
2
A2 u1k 1 A1 k 1 u2k 1 A2 k k u A k u u1 k2u3k u3k 2 2 1 3 x1 A1 x1 A1 x2 x1 A2 x2 A1 A2 x1
A1 u1k 1 A2 k u2k 1 A1 k u2 u1 x2 A2 x2 A1 x1 x1 A2 x2 1 u k 1 A2 k 1 u2k 1 A1 k A1 1 u2 u1 A2 x2 A1 x1 A1 x1 A2 x2 x2
1 u3k 1 u2k 1 A2 k k1u1k u1k A1 A2 k1 2 u1 k2u3k A1 x1 A2 x2 A1 A2 x1 1 u1k A 1 A1 k u2 k1u3k u3k 2 13k A1 A2 f1k 0 , A1 x1 A1 A2 x2 x1
u1k 1 A2 k A2 u2k 1 A1 k u2 u1 x1 x2 A1 x1 A1 x1 A2 x2 A u k 1 A2 k 1 u1k 1 A1 k u1 A1k2u3k u2 k1u3k u3k 2 1 2 x2 A2 x2 A2 x1 x2 A1 x1 A1 A2 x2
1 u1k 1 A2 k 1 u2k 1 A1 k A2 u2 u1 A2 x2 A1 x1 A1 x1 A2 x2 x1
(23)
1 u3k 1 u1k 1 A1 k k2u2k u2k A1 A2 k2 2 u2 k1u3k A2 x2 A1 x1 A1 A2 x2
1 u2k A 1 A2 k k u1 k2u3k u3k 2 23 A1 A2 f 2k 0 , A x A A x x 1 2 1 2 2 1
A2 u3k A1 u3k k k A k u A u A1k2u2k A1 u2k 2 1 1 2 1 x1 A1 x1 x2 A2 x2 1 u1k 1 u2k 1 A1 k 1 A2 k u2 k1u3k u1 k2u3k u3k A1 A2 k1 A1 x1 A1 A2 x2 A2 x2 A1 A2 x1
2
1 u2k 1 u1k 1 A2 k 1 A1 k u1 k2u3k u2 k1u3k u3k A1 A2 k2 A2 x2 A1 A2 x1 A1 x1 A1 A2 x2
2
33k A1 A2 f 3k 0 . As result of the performed transformations instead of a finite 3-D system of the differential equations in displacements (8) we have a infinite system of 2-D differential equations, which are related to coefficients of the Legendre’s polynomial series expansion. This system of equations is not suitable for solution of the problems of the shell, plate theory. An approximate theory, which takes into account only a finite number of members in the expansion (13) has to be developed. The number of members in the expansion (13) and order of the system of differential equations (23) depends on the demanding accuracy of the stress-strain parameters calculation.
4.
Axially symmetric shells
In order to simplify situation we consider a 1-D case when all functions that describe stress-strain state of the shell depend only of one coordinate x1 . Such situation takes place for example on the case of axially symmetric deformation of the shell. Stress and strain tensors and displacements vector have the following form
11
0
13
11
0
ij 0 22 0 , ij 0 22 31 0 33 31 0
13
u1 0 , ui 0 33 u3
(24)
All relations obtained in the previous section significantly simplify because the components of the stress-strain state have the simple form (24) and do not depend on coordinate x2 . The Equations of equilibrium (17) now have the form
A2 11k x1
k 13k A1 A2 k1 22
A2 31k x1
A2 13k A1 A2 f1k 0 x1
(25)
k 11k A1 A2 k1 22 A1 A2 k2 33k A1 A2 f 2k 0
where
2k 1 i 3 x1 (1)k i3 x1 , h 2 k 1 ik3 x1 A1 A2 ik31 x1 ik33 x1 h fi k x1 bik x1
(26)
The Cauchy relations (19) have the form
11k
1 u1k 1 A2 k k k1u3k , 22 u1 k2u3k , A1 x1 A1 A2 x1
1 u3k u , k1u1k u1k . A1 x1 k 33
k 3
(27)
k 13
where uik x1
2k 1 k 1 ui x1 uik 3 x1 h
(28)
We consider isotropic material and Hooke’s law (21) in this case has the form
11k 2 11k 22k 33k , 22k 2 22k 11k 33k , 33k 2 33k 11k 22k ,
(29)
13k 13k . Substituting Cauchy relations (27) into Hooke’s law (29) instead of (22) we obtain more simple relations 1 u1k 1 A2 k k1u3k u1 k2u3k u3k , A1 x1 A1 A2 x1
11k 2
1 A2 k 1 u1k u1 k2u3k k1u3k u3k , A1 A2 x1 A1 x1
2k2 2
1 u1k 1 A2 k 2 u k1u3k u1 k2u3k , A1 A2 x1 A1 x1 k 33
(30)
k 3
1 u3k k1u1k u1k . A1 x1
13k
Substitution of these equations in the equations of equilibrium (25) gives us the 1-D equations in displacements in the form
2
1 u3k A2 u1k 1 A2 k k k k A k u u k u u A A k k1u1k u1k 2 1 3 1 2 3 3 1 2 1 x1 A1 x1 x1 A1 A2 x1 A1 x1
1 A2 k A 1 u1k A 2 u1 k2u3k 2 k1u3k u3k 2 13k A1 A2b1 0, A1 A2 x1 x1 A1 x1 x1
(31)
1 u 1 A2 k A2 u A2 k1u1k A2 u1k 2 A1 A2 k1 k1u3k k2 u1 k2u3k x1 A1 x1 A1 A2 x1 A1 x1 k 3
k 1
1 A2 k 1 u1k A1 A2 k1 u1 k2u3k u3k A1 A2 k2 k1u3k u3k 33k A1 A2b2 0 A1 A2 x1 A1 x1
This system of the differential equations is also infinite dimensional, and finite dimensional reduction is needed in order to construct approximate equations which can be solved analytically or numerically. It is important to mention that from this system of equations as particular case equations for rods, plates and shells of the different shape can be obtained. It is important to clarify here that in the case of plates and shells one has to use Lame constants in the traditional form E E , (32) , (1 )(1 2 ) 2(1 ) that correspond to 3-D elasticity. In the case of the rods and beams in order to take into account Poisson’s effect one has to use the modified Lame constants in the form 2 E E , , (33) 2 1 2 1 that correspond to plane stresses state in 2-D elasticity. 4.1. Axially symmetric circular plate If
in
all
of the above equations we assume that A1 1, A2 x1 , k1 k2 0 then equations of axisymmetric plates will be obtained. The equations of equilibrium (25) have the form 11k 11k 2k2 13k f1k 0, x1 x1 3k1 31k 33k f 2k 0, x1 x1
(34)
where fi k x1 bik x1
2k 1 i 3 x1 (1)k i3 x1 , h
2k 1 k 1 x1 i3 x1 ik33 x1 . h k i3
The Cauchy relations (27) have the form
(35)
11k
u k u1k uk , 22 1 , 33 u3k , 13k 3 u1k . x1 x1 x1
(36)
Hooke’s law has the same form us (29) and the equations in displacements have the form
2
u3 2u1k 1 u1k 1 2 2 2 u1k 13k f1k 0 2 x1 x1 x1 x1 x1 k
u1 2u k 1 u3k 1 21 u1k 33k f 2k 0 x1 x1 x1 x1 x1 k
(37)
This system of 1-D differential equations for coefficients of the Legendre’s polynomial series expansion contains infinite number of equations. In order to simplify the problem we consider approximate theories which contain only finite number of equations. To do that, a finite number of members in the expansion (13) has to be taken into account. The order of the approximation depends on the number of members in the series expansion (13). For the first approximation only the first two members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1 and the system of differential equations (37) contain only four equations of the form
2
2u10 1 u10 u31 1 2 2 2 u10 f10 0 2 x1 x1 x1 h x1 x1
2u30 1 u30 u11 1 u1 f 20 0 2 x1 x1 x1 h x1 x1h
(38)
2
u 3 u 1 u 1 3 2 2 2 u11 2 u11 f11 0 x h x1 x1 x1 x1 h 2 1 1 2 1
0 3
1 1
2u11 3 u10 1 u31 3 u10 3 2 2 u31 f 21 0 2 x1 h x1 x1 x1 h x1 h
For the second approximation only the first three members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1,2 and the system of differential equations (37) contain only six equations of the form
2
2
2u10 1 u10 u31 1 2 2 2 u10 f10 0 x12 x1 x1 h x1 x1
2u30 u30 u11 1 u1 f 20 0 x12 x1 x1 h x1 x1h
2u11 3 u30 1 u11 3 u32 1 3 2 2 2 u11 2 u11 f11 0 2 x1 h x1 x1 x1 h x1 x1 h
2 u1 3 u10 1 u31 3 u12 3 u0 3 3 2 21 1 2 2 u31 u1 f 21 0 x1 h x1 x1 x1 h x1 h x1 h x1h
(39)
2
2u12 5 u31 1 u12 1 15 2 2 2 u12 2 u12 f12 0 2 x1 h x1 x1 x1 x1 h
2u12 5 u11 1 u32 5 u11 15 2 2 u32 f 22 0 2 x1 h x1 x1 x1 h x1 h
The systems of differential equations (38) and (39) together with corresponding boundary conditions can be solved and the stress-strain state of axisymmetric circular plate investigated in frame of the first and second order approximation respectively. 4.2. Curvilinear rods and arks The equations (25) - (31) can be used to investigate of the stress-strain state of curvilinear rods and arks. Let us consider in those equations A2 1, k2 0 , as result we obtain equations for ark of arbitrary shape. The equations of equilibrium (25) in this case have the form 11k 13k A1k1 13k A1 f1k 0 , x1 31k 1k1 A1k1 33k A1 f3k 0 , x1
(40)
where fi k x1 bik x1
2k 1 i 3 x1 (1)k i3 x1 , h
2k 1 k 1 x1 A1 i 3 x1 ik33 x1 . h
(41)
k i3
The Cauchy relations (27) have the form
11k
1 u1k 1 u3k k1u3k , 13k k1u1k u1k , 33k u3k . A1 x1 A1 x1
(42)
Hooke’s law has the same form us (29) and the equations in displacements have the form
2
1 u2k 1 u1k k k k u u A k k1u1k u1k 13k A1 f1k 0 , 1 3 3 1 1 x1 A1 x1 x1 A1 x1
1 u1k 1 u3k k1u1k u1k 2 A1k1 k1u3k A1k1 u3k 33k A1 f 3k 0 , x1 A1 x1 A1 x1
(43)
where
13k x1
2k 1 1 u3k 1 1 u3k 3 k1u1k 1 u1k 1 k1u1k 3 u1k 3 , h A1 x1 A1 x1
1 u1k 1 2k 1 1 u1k 3 k 1 k 3 x1 k1u3k 1 k1u3k 3 2 u3 u3 h A1 x1 A1 x1 k 33
(44)
This system of 1-D differential equations for coefficients of the Legendre’s polynomial series expansion contains an infinite number of equations. In the same way as in the case of axisymmetric
plate in order to simplify the situation a finite number of members in the expansion (13) has to be taken into account. For the first approximation only first two members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1 and the system of differential equations (37) contain only four equations of the form
2
1 u30 1 u10 1 1 1 0 k u u A k k1u10 u11 A1 f10 0 , 1 3 3 1 1 x1 A1 x1 h x1 h A1 x1
1 u10 1 u30 1 1 1 0 k u u 2 A k k1u30 A1k1 u31 A1 f 30 0 , 1 1 1 1 1 x1 A1 x1 h h A1 x1
2
1 u31 1 u11 1 k u A k k1u11 1 3 1 1 x1 A1 x1 A1 x1
(45)
3 1 u30 1 A1 k1u10 u11 +A1 f11 0 , h A1 x1 h
1 u11 1 u31 k1u11 2 A1k1 k1u31 x1 A1 x1 A1 x1 3 1 u10 1 A1 k1u30 2 u31 A1 f 31 0 . h A1 x1 h
For the second approximation only the first three members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1,2 and the system of differential equations (37) contain only six equations of the form
2
1 u30 1 u10 1 1 1 0 k u u A k k1u10 u11 A1 f10 0 , 1 3 3 1 1 x1 A1 x1 h x1 h A1 x1
1 u10 1 u30 1 1 1 0 k u u 2 A k k1u30 A1k1 u31 A1 f 30 0 , 1 1 1 1 1 x1 A1 x1 h h A1 x1
2
1 u31 1 u11 3 2 1 k u u A k k1u11 1 3 3 1 1 x1 A1 x1 h x1 A1 x1 3 1 u30 1 A1 k1u10 u11 A1 f11 0 , h A1 x1 h
1 u11 1 u31 3 u12 3 1 k u 2 A k k1u31 A1k1 u32 1 1 1 1 x1 A1 x1 h x1 h A1 x1 3 1 u10 1 A1 k1u30 2 u31 A1 f 31 0 , h A1 x1 h
2
1 u32 1 u12 5 1 u31 3 2 2 k u A k k u A k1u11 u12 A1 f12 0 , 1 3 1 1 1 1 1 x1 A1 x1 h A1 x1 h A1 x1
1 u12 15 1 u32 5 1 u11 k1u12 2 A1 k1 k1u32 2 u32 A1 k1u31 A1 f 31 0 . x1 A1 x1 h A1 x1 h A1 x1
The systems of differential equations (45) and (46) together with corresponding boundary conditions can be solved and the stress-strain state of axisymmetric circular plate investigated in frame of the first and second order approximation respectively. 4.3. Axially symmetric cylindrical shell If in the equations (25) - (31) we suppose A1 1, A2 R, k1 0, k2 k 1/ R as result we obtain equations for investigation of the stress-strain state of axisymmetric cylindrical shell. The equations of equilibrium (25) in this case have the form 11k 13k f1k 0, x1 31k 2k2 33k f 2k 0, x1 R
(47)
where fi k x1 bik x1
2k 1 i 2 x1 (1)k i2 x1 , h
2k 1 k 1 x1 i 3 x1 ik33 x1 . h k i3
(48)
(46)
The Cauchy relations (27) have the form
11k
u k u1k k u3k k , 22 , 33 u3k , 13k 3 u1k . x1 R x1
(49)
Hooke’s law has the same form us (29) and the equations in displacements have the form
2
u3 2u1k u3k 13k f1k 0 2 x1 R x1 x1 k
u1 2u k u1k uk 23 2 32 u3k 33k f 2k 0 x1 R x1 x1 R R k
(50)
where
13k
1 u3k 3 2k 1 1 u3k 1 k 1 u u1k 3 , 1 h R x1 R x1
u k 1 u k 3 2k 1 33k x1 2 R u3k 1 u3k 3 1 u3k 1 1 u3k 3 h x1 x1
(51)
The situation here is the same as in above cases. The system (50) of 1-D differential equations for coefficients of the Legendre’s polynomial series an expansion contains infinite number of equations. In the same way as in the previous cases in order to simplify the situation a finite number of members in the expansion (13) have to be taken into account. For the first approximation only the first two members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1 and the system of differential equations (37) contain only four equations of the form
2
2u10 u30 u31 f10 0 x12 R x1 h x1
2u30 u10 u11 u30 1 2 2 2 u3 f 20 0 x1 R x1 h x1 R Rh
(52)
2
u 3 1 u u 3 1 u1 f11 0 x h R x1 R x1 h 2 2 1 1 2 1
0 3
1 3
2u31 3 u10 u11 3 0 1 3R u3 2 2 2 u31 f 21 0 2 x1 h x1 R x1 h h R
For the second approximation only the first three members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1,2 and the system of differential equations (37) contain only six equations of the form
2
2u10 u30 u31 f10 0 2 x1 R x1 h x1
2u30 u10 u11 u30 1 2 u3 f 20 0 2 2 x1 R x1 h x1 R Rh
2
2u11 3 u30 u31 3 u32 3 1 u1 f11 0 x12 h x1 R x1 h x1 h 2
2u1 3 u10 u11 3 u12 3 0 3 3 2 1 23 u3 2 2 2 u31 u3 f 21 0 x1 h x1 R x1 h x1 Rh h Rh R
2
(53)
2u12 5 u31 u32 15 2 2 u1 f12 0 2 x1 h x1 R x1 h
2u32 5 u11 u12 5 1 1 15 u3 2 2 2 u32 f 22 0 2 x1 h x1 R x1 Rh h R
The differential equations obtained here can be used to investigate the stress-strain state thinwalled structures like shells, plates, rods. Boundary-value problems are defined by a system of differential equations with the corresponding boundary.
5.
Numerical results and discussion
In order to check the validity, accuracy and efficiency of the proposed models we consider the beam and the axisymmetric cylindrical shell with build-in ends subjected to uniform load for the cases first and second order approximation theories. We compare the obtained solutions with the solution obtained using the equations of elasticity and the classical theories. The corresponding boundary-value problems have been solved using finite element method (FEM). Calculations have been done using commercial softwares MATLAB, Mathematica 10 and Comsol Multiphysics. Corresponding differential equations have been transformed to the form suitable for Comsol software input
C
2u( x1 ) u( x1 ) B A u( x1 ) f ( x1 ) 2 x1 x1
(54)
For details of FEM implementation using MATLAB, Mathematica 10.1 and Comsol Multiphysics one can refer to the corresponding tutorials. 5.1. Uniformly loaded beam The calculations have been done for the first and second order theories using differential equations (45) and (46) respectively. In order to adapt these equations for the case beam one has to take A1 1 and k1 0 . Boundary conditions for the case of the build-in ends have the form
uik (0) uik ( L) 0
(55)
Differential equations (45) and (46) have been transformed to the form (54), the corresponding matrixes for the first and second order approximation theories are presented in Appendix. In order to simplify the calculations and analysis of the obtained results dimetionless coordinates x x 1 1 and 3 3 have been introduced. For numerical analysis we used the following L h mechanical and geometrical parameters: Young’s modulus is E 1 Pa and for Poisson’s ratio is 0.3 , length of the beam is L 1 and its thickness is h 0.1L , respectively. The results of the calculations are presented on Fig. 1-6. Fig. 1 presents the verticals displasements of the beam calculated for the first and second order approximation theories in
comparison with the ones obtained using 2-D equations of the theory of elasticity and classical Euler-Bernully theory using MATLAB and Comsol Multiphysics.
Fig.1. Normalized vertical displacements. One can see a very good agreement of the results obtained using the proposed theories and those obtained using the theory of elasticity and the classical theory. It is important to mention that the second approximation theory gives results that better coincide with those obtained for the theory of elasticity. Also one has to take into account that all results presented in Fig.1 were numerically obtained using FEM and above mentioned software. Fig. 2 and Fig. 3 show the distribution of the Legendre polynomials coefficients for the displacements versus the normalized length for the first and second approximation theories respectively. These coefficients are FEM solutions of the systems of differential equations (38) and (46) respectively. They are calculated with MATLAB and represent direct output from that software. The displacements have been calculated using Legendre polynomials coefficients for the displacements and representations (13) for the first and second approximation theories with k 0,1 and k 0,1, 2 respectively. For the stress calculation the same representations have been used, but first using Legendre polynomials coefficients for stress have been calculated using Cauchy relations (42) and Hook’s law (29) for the first and second approximation theories respectively. These calculations cannot be done directly in Comsol Multiphysics, therefore a special MATLAB computer program has been created for this purpose. In that program the derivatives of functions uik ( x1 ) calculated in Comsol Multiphysics have been used. Fig. 4 and Fig. 5 show displacements and stresses distribution versus normalized length and thickness for the first and second approximation theories respectively calculated in MATLAB. The presented results show that the displacements are qualitatively and quantitatively distributed in the same way, but stresses differ qualitatively and quantitatively. More accurate results give us second approximation theory.
Fig.2. Legendre polynomials coefficients for the displacements. First approximation.
Fig.3. Legendre polynomials coefficients for the displacements. Second approximation.
Fig.4. Displacements and stresses versus normalized length and thickness. First approximation.
Fig.5. Displacements and stresses versus normalized length and thickness. Second approximation.
Fig.6. Normalized displacements and stresses in the middle point versus normalized thickness. In Fig. 6 calculated for first and second order approximation theories normalized vertical displacements u2 and normalized normal stresses 22 in the middle point of the beam are presented against normalized thickness h / L and compared with ones calculated using equations of 2-D elasticity. The calculations have been done using the FEM software embedded in Mathematica 10.1 and the results in Fig. 6 are a direct output from the Mathematica 10.1 notebook. The diagrams presented in Fig. 6 show that for all considered normalized thickness coincidence of the results obtained using proposed here approach and the theory of elasticity is very good. It means that the proposed theories may be successfully used for relatively thick beams, where classical theories give inaccurate results.
5.1. Uniformly loaded axially symmetric cylindrical shell Calculations have been done for the first and second order theories using differential equations (52) and (53) respectively. Boundary conditions for the case of the build-in ends of the shell have the form (55). Differential equations (52) and (53) have been transformed to the form (54), corresponding matrixes for the first and second order approximation theories are presented in the Appendix. In order to simplify the calculations and analysis of the obtained results dimetionless coordinates x x 1 1 and 3 3 have been introduced. For a numerical analysis we used the following L h mechanical and geometrical parameters: Young’s modulus is E 1 Pa and for Poisson’s ratio is 0.3 , length of the shell is L 1 , its radius is R 0.25L and its thickness is h 0.1R , respectively. The results of the calculations are presented in Fig. 7-12. Fig. 6 presents the verticals displasements of the shell calculated for the first and second order approximation theories compared with ones obtained using the equations of the theory of axisymmetric elasticity and classical Kirkgoff-Love theory. One can see very a good agreement of the results obtained using the proposed theories and those obtained using the theory of elasticity and classical theory. It is important to mention that the second approximation theory gives results that better coincide with those obtained for the theory of elasticity. Also one has to take into account that the results presented in Fig.7 were obtained numerically using MATLAB and Comsol Multiphysics. Fig. 8 and Fig. 9 show the distribution of the Legendre polynomials coefficients for the displacements versus the normalized length for the first and second approximation theories respectively. These coefficients are FEM solutions of the systems of differential equations (52) and (53) respectively. They are calculated with MATLAB and represent direct output from that software.
Fig.7. Normalized radial displacements. The displacements have been calculated using Legendre polynomials coefficients for the displacements and representations (13) for the first and second approximation theories with k 0,1 and k 0,1, 2 respectively. For stress calculation the same representations have been used, but first
using Legendre polynomials coefficients for stress have been calculated using Cauchy relations (49) and Hook’s law (29) for the first and second approximation theories respectively. These calculations cannot be done directly in Comsol Multiphysics, therefore a special MATLAB computer program have been created for this purpose. In that program the derivatives of functions uik ( x1 ) calculated in Comsol Multiphysics have been used. Fig. 9 and Fig. 10 show displacements and stresses distribution versus normalized length and thickness for the first and second approximation theories respectively calculated in MATLAB. The presented results show that the displacements are qualitatively and quantitatively distributed in the same way, but stresses differ qualitatively and quantitatively. More accurate results give us a second approximation theory.
Fig.8. Legendre polynomials coefficients for the displacements. First approximation.
Fig.9. Legendre polynomials coefficients for the displacements. Second approximation. In order to compare the displacements and stresses calculated by the proposed method for first and second approximation theories with those obtained using equations of elasticity for different values of the normalized thickness we consider the relatively long ( L 10R ) uniformly loaded cylindrical shell. For this specific case we will use an analytical elasticity solution for long asymmetrically loaded long cylinder. Displacements and stresses can be presented by the equations [41]
11
2 p ( R h) 2 p ( R h) 2 p ( R h) 2 ( R h ) 2 , 22 , 4 Rh 4 Rh 4 Rh( R x2 ) 2
33
p ( R h) 2 p ( R h) 2 ( R h) 2 4 Rh 4 Rh( R x2 ) 2
u2
p(1 )( R h) 2 p(1 )( R h) 2 ( R h) 2 ( R x2 ) 4 ERh 4 EeRh( R x2 )
(56)
Here 11 , 22 and 33 are normal stresses in axial, radial and circumferential directions respectively, u2 is a radial displacements.
Fig.10. Displacements and stresses versus normalized length and thickness. First approximation.
Fig.11. Displacements and stresses versus normalized length and thickness. Second approximation.
Fig.12. Normalized radial displacements and circumferential stresses versus normalized thickness for inner side of the cylindrical shell. In Fig. 12 the normalized radial displacements and the circumferential stresses in inner sides of the shell for first and second order approximation theories are calculated and presented against normalized thickness h / L . Then they are compared with the ones that are calculated using the equations (56) of the axisymmetric elasticity. The calculations have been done in Mathematica 10.1, for first and second order approximation using the FEM software embedded and for elasticity solution directly, using the equations (56). The results presented in Fig. 12 are a direct output from the Mathematica 10.1 notebook. The diagrams presented in Fig. 12 show that for all considered normalized thickness coincidence of the results obtained using the approach that has been proposed here and the theory of elasticity is very good. It means that the proposed theories may be successfully used for a relatively thick cylinder, where classical theories give inaccurate results.
6.
Conclusions
In this paper a higher order theory for homogeneous elastic shells, plates and rods has been developed. The proposed approach is based on the expansion of the equations of elasticity into Fourier series in terms of Legendre polynomials. Starting from the equations of elasticity, stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Fourier series in terms of Legendre polynomials in a thickness coordinate. Thereby all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. The system of differential equations in terms of displacements and boundary conditions for Fourier coefficients has been obtained. Special attention has been paid to the case of axially symmetric shells. For the special case of circular plates, curvilinear rods and cylindrical shells the first and second approximations theories have been considered in more detail. All necessary equations and their coefficients have been written explicitly and the corresponding boundary-value problems have been formulated. For a numerical solution of the formulated problems FEM has been applied and commercial software Matlab, Mathematica 10 and Comsol Multiphysics have been used. For validation of the proposed theories, a comparison of the numerical calculations with the results obtained using equations of the theory of elasticity and classical theories of Euler-Bernoulli for beams and Kirchhoff-Love for shell have been done. The presented calculations show a good agreement of the results obtained using proposed theories with the theory of elasticity and the classical theories. The second order approximation theory gives better coincidence with theory of elasticity than the first one and it is important to mention that the proposed theories give more accurate results then the classical theories. They can be recommended for calculation of the stresses and displacements in relatively thick beams, rods, plated and shells, where classical theories give inaccurate results.
Acknowledgments This work was supported by the Committee of Science and Technology of Mexico (CONACYT) by the Research Grants (Project No. 101415), which is gratefully acknowledged.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
20. 21. 22. 23.
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Appendix. Matrixes and vectors are presented in the equation (54). 1. Beams 1.1. First approximation beam theory
2 C
0 0 0
0
0 0 0 2 0 0
0 0 ,B 0
0
0
0
0
0 3 h
0
0
h
3 h
0
0
0
0 0 0 0
h
, 0
A 0 0 3 h2 0 0
0
1.2. Second approximation beam theory
2 0 0 0 0 0
C
0 0 0 2 0 0 0 0 0 0
0
0
0
0
0 0 0
0 0 0 0 0 2 0 0
0
h
0
0
0
0
0
3 h
0
0
0
0
0
0
0
0
5 h
0
0
0 0 0 0 0
0
h
3 h
0 B
0
0 0
3 h , 0 0
5 h
0
0
0
0
0
,
0
0 0 0 2
3 h2
0 0 0 0
0 0
0 0
3 h2
0 0
0 0
0 0
0
0
0
0
0
15 h2
0
2
A 0 0
0
0 0
0
0
0 0
0
0
3 h2
0
2
15 h2
2. Axially symmetric cylindrical shell 2.1. First approximation shell theory 0
2 C
0 0 0
0 0 0 2 0 0
0
0 2 0 R2 A
0 0
3 Rh
0
0
R
0
3 R h2 0
h
3 h
3 h
0
0
0
0
R
0 0 R ,B 0 0
0
0
Rh 0
1 3R 2 2 R h
2.2. Second approximation shell theory
C
2
0
0 0 0 0 0
0 0 0 0
0 0 2 0 0 0
0 0 0
0 0 0 0 0 2 0 0
0 0 0 0 0
,
h 0
R 0
,
0
0
0 R
0
0
0
0
0
5 h
0
2
0
R
0
R
5 h 0
0
0
0
0
3 h
3 h 0
0
0
Rh
R
3 1 2 2 2 h R
0
0
0
0
0
0
0
5 Rh
0
0
0
0
R
0
3 Rh
0
0 0
0
0
0
0
,
0
3 h2
0 A
3 h
0
2
0
h
3 h
0
0
h
0
R
0 B
R
0
3 Rh
15 h2
0
0
1 15 2 2 2 h R
We develop a higher order theory of elasticity for linearly elastic shells, plates, roods and beams. We use for that Fourier series in terms of Legendre’s polynomials expansion For numerical solution the FEM implemented in the software COMSOL Multiphysics, MATLAB and Mathematica. The displacements, stress distribution in the asymmetrical cylindrical shell and beam have been studied Comparison with elasticity and classical shell and beam solution has been done.
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PII: DOI: Reference:
S0020-7403(15)00314-8 http://dx.doi.org/10.1016/j.ijmecsci.2015.08.025 MS3090
To appear in: International Journal of Mechanical Sciences Received date: 11 June 2015 Revised date: 26 August 2015 Accepted date: 28 August 2015 Cite this article as: V.V. Zozulya, A higher order theory for shells, plates and r o d s , International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2015.08.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A higher order theory for shells, plates and rods. V.V. Zozulya Centro de Investigacion Cientifica de Yucatan, A.C., Calle 43, No 130, Colonia: Chuburna de Hidalgo C.P. 97200, Merida, Yucatan, Mexico. [email protected]
Abstract. In this paper a new theory for shells, plates and rods has been developed. The proposed theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. The case of axially symmetric shell has been considered in more details. As a special case of the general approach the equations of the first and second approximations have been developed for circular plates, curvilinear rods and cylindrical shells. Numerical calculations have been done numerically using the finite element method (FEM) along with the Comsol Multiphysics, Matlab and Mathematica software.
Keywords: Shell, plate, rod, Legendre polynomial, FEM. 1. Introduction Classical theories of beams, rods, plates and shells are usually related with Bernoulli, Euler, Kirchhoff and Love. These theories are based on well-known physical hypothesis; they are very popular among the engineering community because of their relative simplicity and physical clarity. Numerous books and monographs have been written on the subject, among others one can refer to [1-4]. In the same time classical theories have some shortcomings such as its proximity and inaccuracy and as result in some cases are observed not good agreement with practice. For this reason new and more accurate theories have been developed. We can mention several approaches to development of the theories of thin-walled structures. One consists in improvement of the classical physical hypothesis and the development of more accurate theories. In the beams theory is a well-known model that takes into account transversal deformations developed by Timoshenko and was then extended to the plate theory by Mindlin [5]. This theory was extended and applied to shells in numerous publications and is referred to as the theory that takes into account in-plane shear deformations [6-8]. So-called direct approach has been developed and applied to plates, shells and rods analysis in [9, 10]. Another approach consists in the expansion of the stress-strain field components into polynomials series in terms of thickness. It was proposed by Cauchy and Poisson and that a time was not popular. Significant extensions and developments of that method have been done by Kil'chevskii [11]. Vekua has used Legendre’s polynomials for the expansion of the equations of elasticity and the reduction of the 3-D problem to the 2-D one [12]. Such an approach has significant advantage since Legendre’s polynomials are orthogonal and as result the developed equations are simple. This approach was extended and applied to dynamical problems [13] and thermoelasticity [14], composite and laminate shells [15], the theory of classical and micropolar elasticity [16].
The approach developed in [11-16] has been applied to the plates and shells thermoelastic contact problems when mechanical and thermal conditions are changed during deformation. Details of the approach, equations and contact conditions have been reported for the first time in [17-19]. In [20, 21] it was extended to no stationary processes, presented in general form all coefficients of the corresponding equations with initial, boundary and contact condition. Then the higher order theory was further developed to the contact of plates and shells with rigid bodies though a heat conducting layer [22-26], thermoelastic the laminated composite materials with the possibility of delamination and thermoelastic contact in the temperature field in [22, 27, 28], the pencil-thin nuclear fuel rods modeling in [29], functionally graded shells in [30-32] and for electrostatically actuated MEMS in [33]. An analysis of the higher order shells theories of thermoelasticity and comparison with classical theories has been done in [34]. Different aspects of the theory and applications of the thin-walled structures can be found in thousands publications, for references one can see review papers [35-40] and for trend and resent development books [41-44]. In this paper, based on the expansion of the equations of elasticity into Fourier series in terms of Legendre polynomials a higher order theory of shells, plate and rods shell has been developed. More specifically, we expanded functions that describe stress-strain relations into Fourier series in terms of Legendre polynomials and found corresponding relations of elasticity for Fourier coefficients of expansions of the stress and strain. Then using the techniques developed in our previous publications we found a system of differential equations and boundary conditions for Fourier coefficients. The case of axially symmetric shells is considered in more detail and all relations explicitly presented. From these equations as a special case there can be easily derived higher order equations for beams, rods, circular axisymmetric plated, cylindrical and spherical shells. Cases of the first and second approximations have been considered in more detail. Numerical examples are presented. Comparison with the theory of elasticity and the classical theories has been done. The analysis of the numerical results shows a good coincidence of the displacements and stresses obtained by using the higher order theory and elasticity solution for even relatively thick structural elements.
2. 3-D formulation As it was mentioned above in this study we further develop an approach based on the expansion of the equations of elasticity into Fourier series in terms of Legendre polynomials and apply it to construct higher order moles of shells, plates and rods. Therefore we start our consideration with a 3-D formulation of the problem. We consider a linear elastic body occupying an open in 3-D Euclidian space simply connected bounded domain V R3 with a smooth boundary V . The elastic body is supposed to be a homogeneous isotropic shell of arbitrary geometry with 2h thickness and occupies the domain V [h, h] in Euclidean space. Here is the middle surface of the shell, is its boundary. The boundary of the body can be presented in the form + V S , where and are the outer sides and S [h, h] is a sheer side. In order to examine the stress-strain state of the elastic body we introduce stress ij and ij strain tensors and displacements ui , traction p i , and body forces b i vectors. Covariant and mixed components of the related tensors and vectors can be found in a standard way by raising and lowering indexed using metric tensors g ij , g ij and g ij respectively. These quantities are related by equations of linear elasticity. In the case if displacements and their gradients are small Cauchy relations take place 1 1 (1) ij (i ui j ui ) ( i ui j ui ) ijk uk 2 2
Here i are covariant derivatives, ijk are the Christoffel symbols, i xi are partial derivatives with respect to the space variables xi . Equations of equilibrium have the form j ij bi 0,
j ij j ij ijk kj jkj ik
(2)
Stress and strain tensors are related by generalized Hooke’s law
ij cij kl kl , cij kl g ij g kl ( g il g jk g ik g jl )
(3)
Here and are Lame constants, 0 and , are Lame constants. Throughout this paper we use the Einstein summation convention. For convenience we transform the above equations of elasticity taking into account that the radius vector R(x) of any point in domain V, occupied by material points of shell may be presented as
R(x) r(x ) x3n(x )
(4)
where r(x ) is the radius vector of the points located on the middle surface of shell, n(x ) is a unit vector normal to the middle surface. We consider that x ( x1 , x2 ) are curvilinear coordinates associated with the main curvatures of the middle surface of the shell which is commonly used in application. In this case 3-D equations of elasticity (1) - (3) significantly simplify making it unnecessary to distinguish co- and contra-variant components of tensors and vectors. Therefore from here to the end of the paper lower index will be used for components of vectors and tensors and upper index for components of the coefficients of the Legendre’s polynomial series expansion. The equations of equilibrium have the form ( A2 11 ) ( A1 12 ) A A A1 A2 13 12 1 13 A1 A2 k1 22 2 A1 A2b1 0, x1 x2 x3 x2 x1 ( A2 21 ) ( A1 22 ) A A A1 A2 23 21 2 23 A1 A2 k2 11 1 A1 A2b2 0, x1 x2 x3 x1 x2
(5)
( A2 31 ) ( A1 32 ) ( A1 A2 33 ) 11 A1 A2 k1 22 A1 A2 k2 A1 A2b3 0. x1 x2 x3
Here A ( x1 , x2 ) r ( x1 , x2 ) r ( x1 , x2 ) are coefficients of the first quadratic form of a surface, r
r is a derivative the radius vector of the points located on the middle surface of shell, x
k ( x1 , x2 ) are it main curvatures. The Cauchy relations have the form
11
1 u1 1 A1 1 u2 1 A2 u2 k1u3 , 22 u1 k2u3 , A1 x1 A1 A2 x2 A2 x2 A2 A1 x1
u3 1 u 1 A2 1 u2 1 A1 , 12 1 u2 u1 x3 A2 x2 A1 x1 A1 x1 A2 x2 u u 1 u3 1 u3 13 1 k1u1 , 23 2 k2u2 x3 A1 x1 x3 A2 x2
33
(6)
The generalized Hooke’s law (3) in this case has the form
ij cijkl kl ,
cijkl ij kl ( ik jl il kj ),
(7)
where ij is a Kronecker’s symbol. Substituting the Cauchy relations (1) in the Hooke’s law (3) and obtained relations in the equations (2) we obtain the differential equations of equilibrium in the form of displacements which may be presented in the form L u b 0,
x V
(8)
The differential operator L Lij ei e j for homogeneous anisotropic and isotropic medium has the form
Lij cikjl k l and Lij ij k k ( )i j
(9)
respectively. If the problem is defined in an infinite region, then additional conditions at the infinity in the form have to be satisfied u j (x) O(r 1 ), ij (x) O(r 2 ), for r .
(10)
If the body occupied a finite region V with the boundary V , it is necessary to establish boundary conditions. We consider the mixed boundary conditions in the form ui (x) i (x) , x Vu , pi (x) ij (x)n j (x) Pij [u j (x)] i (x) , x Vp
(11)
The differential operator Pij : u j pi is called stress operator. It transforms the displacements into the tractions. For homogeneous anisotropic and isotropic medium they have the forms
Pij cikjl nk l
and
Pij ni j ij n n j i
(12)
respectively. Here ni are components of the outward normal vector, n ni i is a derivative in direction of the vector n(x) normal to the surface V .
3. 2-D formulation In order to transform a 3-D problem into 2-D one let us expand the parameters that describe stress-strain of the cylindrical shell in the Legendre polynomials series along the coordinate x3
ui x uik x Pk ( ) , uik x k 0
2k 1 ui x , x3 Pk ( )dx3 , 2h h h
ij x ijk x Pk ( ) , ijk x k 0
ij x ijk x Pk ( ) , ijk x k 0
where
x3 . h
2k 1 ij x , x3 Pk ( )dx3 , 2h h h
2k 1 ij x , x3 Pk ( )dx3 . 2h h h
(13)
For the derivatives with respect to coordinates x the following relations take place h uik x 2k 1 ui x P dx k 3 2h h x x h ijk x 2k 1 ij x P dx k 3 2h h x x
(14)
Integration of the derivatives with respect to coordinates x3 gives us h 2k 1 ui x Pk dx3 uik x 2h h x3
2k 1 i 3 x 2k 1 k Pk dx3 i 3 x 1 i3 x ik3 x 2h h x3 h h
(15)
where 2k 1 k 1 ui x uik 3 x , h 2k 1 k 1 ik3 x A1 A2 i3 x ik33 x h uik x
(16)
Now substituting stress tensor from (13) into the equations of equilibrium (5) , multiplying obtained relations by Pk ( ) and integrating over interval [h, h] with respect to x2 we obtain 2-D equations of equilibrium in the form A2 11k x1
k A2 21
x1
A2 31k x1
A1 12k x2
k A1 22
x2
A1 32k x2
12k
A1 k A2 13k A1 A2 k1 22 13k A1 A2 f1k 0 , x2 x1
12k
A2 A k k 23 A1 A2 k2 11k 2 23 A1 A2 f 2k 0 , x1 x1
(17)
k 11k A1 A2 k1 22 A1 A2 k2 33k A1 A2 f 3k 0 .
where fi k x bik x
2k 1 i3 x (1)k i3 x h
(18)
In the same way can be found the 2-D Cauchy relations
11k
1 u1k 1 A1 k 1 u2k 1 A2 k k u2 k1u3k , 22 u1 k2u3k , A1 x1 A1 A2 x2 A2 x2 A1 A2 x1
1 u1k 1 A2 k 1 u2k 1 A1 k u2 u1 , A2 x2 A1 x1 A1 x1 A2 x2 1 u3k 1 u3k k 13k k1u1k u1k , 23 k2u2k u2k , 33 u3k . A1 x1 A2 x2
12k
(19)
The Hooke’s law (7) after Legendre’s polynomial series expansion (13) has the form: for homogeneous anisotropic body
ijk cijlm lmk
(20)
and for isotropic one
11k 2 11k 22k 33k , 22k 2 22k 11k 33k , 33k 2 22k 11k 22k , 12k 12k , 13k 13k , 23k 23k ,
(21)
In the above relations (17) - (21) orthogonality of the Legendre’s polynomials has been used. In order to find 2-D differential equations in form of displacements we substitute Cauchy relations into Hooke’s law for homogeneous body 1 u1k 1 u2k 1 A1 k 1 A2 k u2 k1u3k u1 k2u3k u3k , A1 x1 A1 A2 x2 A2 x2 A1 A2 x1
11k 2
1 u2k 1 u1k 1 A2 k 1 A1 k u1 u2 k1u3k u3k , A2 x2 A1 A2 x1 A1 x1 A1 A2 x2
2k2 2
1 u1k 1 u2k 1 A1 k 1 A2 k u2 k1u3k u1 , A2 x2 A1 A2 x1 A1 x1 A1 A2 x2
33k 2 u3k 1 u k
1 A
1 u k
1 A
(22)
2 k 1 k 12k 1 u2 2 u1 , A2 x2 A1 x1 A1 x1 A2 x2
1 u3k 1 u3k k1u1k u1k , 2k3 k2u2k u2k , A1 x1 A2 x2
13k
Substitution of these equations in the equations of equilibrium (17) gives us the 2-D equations in the form of displacements
2
A2 u1k 1 A1 k 1 u2k 1 A2 k k u A k u u1 k2u3k u3k 2 2 1 3 x1 A1 x1 A1 x2 x1 A2 x2 A1 A2 x1
A1 u1k 1 A2 k u2k 1 A1 k u2 u1 x2 A2 x2 A1 x1 x1 A2 x2 1 u k 1 A2 k 1 u2k 1 A1 k A1 1 u2 u1 A2 x2 A1 x1 A1 x1 A2 x2 x2
1 u3k 1 u2k 1 A2 k k1u1k u1k A1 A2 k1 2 u1 k2u3k A1 x1 A2 x2 A1 A2 x1 1 u1k A 1 A1 k u2 k1u3k u3k 2 13k A1 A2 f1k 0 , A1 x1 A1 A2 x2 x1
u1k 1 A2 k A2 u2k 1 A1 k u2 u1 x1 x2 A1 x1 A1 x1 A2 x2 A u k 1 A2 k 1 u1k 1 A1 k u1 A1k2u3k u2 k1u3k u3k 2 1 2 x2 A2 x2 A2 x1 x2 A1 x1 A1 A2 x2
1 u1k 1 A2 k 1 u2k 1 A1 k A2 u2 u1 A2 x2 A1 x1 A1 x1 A2 x2 x1
(23)
1 u3k 1 u1k 1 A1 k k2u2k u2k A1 A2 k2 2 u2 k1u3k A2 x2 A1 x1 A1 A2 x2
1 u2k A 1 A2 k k u1 k2u3k u3k 2 23 A1 A2 f 2k 0 , A x A A x x 1 2 1 2 2 1
A2 u3k A1 u3k k k A k u A u A1k2u2k A1 u2k 2 1 1 2 1 x1 A1 x1 x2 A2 x2 1 u1k 1 u2k 1 A1 k 1 A2 k u2 k1u3k u1 k2u3k u3k A1 A2 k1 A1 x1 A1 A2 x2 A2 x2 A1 A2 x1
2
1 u2k 1 u1k 1 A2 k 1 A1 k u1 k2u3k u2 k1u3k u3k A1 A2 k2 A2 x2 A1 A2 x1 A1 x1 A1 A2 x2
2
33k A1 A2 f 3k 0 . As result of the performed transformations instead of a finite 3-D system of the differential equations in displacements (8) we have a infinite system of 2-D differential equations, which are related to coefficients of the Legendre’s polynomial series expansion. This system of equations is not suitable for solution of the problems of the shell, plate theory. An approximate theory, which takes into account only a finite number of members in the expansion (13) has to be developed. The number of members in the expansion (13) and order of the system of differential equations (23) depends on the demanding accuracy of the stress-strain parameters calculation.
4.
Axially symmetric shells
In order to simplify situation we consider a 1-D case when all functions that describe stress-strain state of the shell depend only of one coordinate x1 . Such situation takes place for example on the case of axially symmetric deformation of the shell. Stress and strain tensors and displacements vector have the following form
11
0
13
11
0
ij 0 22 0 , ij 0 22 31 0 33 31 0
13
u1 0 , ui 0 33 u3
(24)
All relations obtained in the previous section significantly simplify because the components of the stress-strain state have the simple form (24) and do not depend on coordinate x2 . The Equations of equilibrium (17) now have the form
A2 11k x1
k 13k A1 A2 k1 22
A2 31k x1
A2 13k A1 A2 f1k 0 x1
(25)
k 11k A1 A2 k1 22 A1 A2 k2 33k A1 A2 f 2k 0
where
2k 1 i 3 x1 (1)k i3 x1 , h 2 k 1 ik3 x1 A1 A2 ik31 x1 ik33 x1 h fi k x1 bik x1
(26)
The Cauchy relations (19) have the form
11k
1 u1k 1 A2 k k k1u3k , 22 u1 k2u3k , A1 x1 A1 A2 x1
1 u3k u , k1u1k u1k . A1 x1 k 33
k 3
(27)
k 13
where uik x1
2k 1 k 1 ui x1 uik 3 x1 h
(28)
We consider isotropic material and Hooke’s law (21) in this case has the form
11k 2 11k 22k 33k , 22k 2 22k 11k 33k , 33k 2 33k 11k 22k ,
(29)
13k 13k . Substituting Cauchy relations (27) into Hooke’s law (29) instead of (22) we obtain more simple relations 1 u1k 1 A2 k k1u3k u1 k2u3k u3k , A1 x1 A1 A2 x1
11k 2
1 A2 k 1 u1k u1 k2u3k k1u3k u3k , A1 A2 x1 A1 x1
2k2 2
1 u1k 1 A2 k 2 u k1u3k u1 k2u3k , A1 A2 x1 A1 x1 k 33
(30)
k 3
1 u3k k1u1k u1k . A1 x1
13k
Substitution of these equations in the equations of equilibrium (25) gives us the 1-D equations in displacements in the form
2
1 u3k A2 u1k 1 A2 k k k k A k u u k u u A A k k1u1k u1k 2 1 3 1 2 3 3 1 2 1 x1 A1 x1 x1 A1 A2 x1 A1 x1
1 A2 k A 1 u1k A 2 u1 k2u3k 2 k1u3k u3k 2 13k A1 A2b1 0, A1 A2 x1 x1 A1 x1 x1
(31)
1 u 1 A2 k A2 u A2 k1u1k A2 u1k 2 A1 A2 k1 k1u3k k2 u1 k2u3k x1 A1 x1 A1 A2 x1 A1 x1 k 3
k 1
1 A2 k 1 u1k A1 A2 k1 u1 k2u3k u3k A1 A2 k2 k1u3k u3k 33k A1 A2b2 0 A1 A2 x1 A1 x1
This system of the differential equations is also infinite dimensional, and finite dimensional reduction is needed in order to construct approximate equations which can be solved analytically or numerically. It is important to mention that from this system of equations as particular case equations for rods, plates and shells of the different shape can be obtained. It is important to clarify here that in the case of plates and shells one has to use Lame constants in the traditional form E E , (32) , (1 )(1 2 ) 2(1 ) that correspond to 3-D elasticity. In the case of the rods and beams in order to take into account Poisson’s effect one has to use the modified Lame constants in the form 2 E E , , (33) 2 1 2 1 that correspond to plane stresses state in 2-D elasticity. 4.1. Axially symmetric circular plate If
in
all
of the above equations we assume that A1 1, A2 x1 , k1 k2 0 then equations of axisymmetric plates will be obtained. The equations of equilibrium (25) have the form 11k 11k 2k2 13k f1k 0, x1 x1 3k1 31k 33k f 2k 0, x1 x1
(34)
where fi k x1 bik x1
2k 1 i 3 x1 (1)k i3 x1 , h
2k 1 k 1 x1 i3 x1 ik33 x1 . h k i3
The Cauchy relations (27) have the form
(35)
11k
u k u1k uk , 22 1 , 33 u3k , 13k 3 u1k . x1 x1 x1
(36)
Hooke’s law has the same form us (29) and the equations in displacements have the form
2
u3 2u1k 1 u1k 1 2 2 2 u1k 13k f1k 0 2 x1 x1 x1 x1 x1 k
u1 2u k 1 u3k 1 21 u1k 33k f 2k 0 x1 x1 x1 x1 x1 k
(37)
This system of 1-D differential equations for coefficients of the Legendre’s polynomial series expansion contains infinite number of equations. In order to simplify the problem we consider approximate theories which contain only finite number of equations. To do that, a finite number of members in the expansion (13) has to be taken into account. The order of the approximation depends on the number of members in the series expansion (13). For the first approximation only the first two members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1 and the system of differential equations (37) contain only four equations of the form
2
2u10 1 u10 u31 1 2 2 2 u10 f10 0 2 x1 x1 x1 h x1 x1
2u30 1 u30 u11 1 u1 f 20 0 2 x1 x1 x1 h x1 x1h
(38)
2
u 3 u 1 u 1 3 2 2 2 u11 2 u11 f11 0 x h x1 x1 x1 x1 h 2 1 1 2 1
0 3
1 1
2u11 3 u10 1 u31 3 u10 3 2 2 u31 f 21 0 2 x1 h x1 x1 x1 h x1 h
For the second approximation only the first three members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1,2 and the system of differential equations (37) contain only six equations of the form
2
2
2u10 1 u10 u31 1 2 2 2 u10 f10 0 x12 x1 x1 h x1 x1
2u30 u30 u11 1 u1 f 20 0 x12 x1 x1 h x1 x1h
2u11 3 u30 1 u11 3 u32 1 3 2 2 2 u11 2 u11 f11 0 2 x1 h x1 x1 x1 h x1 x1 h
2 u1 3 u10 1 u31 3 u12 3 u0 3 3 2 21 1 2 2 u31 u1 f 21 0 x1 h x1 x1 x1 h x1 h x1 h x1h
(39)
2
2u12 5 u31 1 u12 1 15 2 2 2 u12 2 u12 f12 0 2 x1 h x1 x1 x1 x1 h
2u12 5 u11 1 u32 5 u11 15 2 2 u32 f 22 0 2 x1 h x1 x1 x1 h x1 h
The systems of differential equations (38) and (39) together with corresponding boundary conditions can be solved and the stress-strain state of axisymmetric circular plate investigated in frame of the first and second order approximation respectively. 4.2. Curvilinear rods and arks The equations (25) - (31) can be used to investigate of the stress-strain state of curvilinear rods and arks. Let us consider in those equations A2 1, k2 0 , as result we obtain equations for ark of arbitrary shape. The equations of equilibrium (25) in this case have the form 11k 13k A1k1 13k A1 f1k 0 , x1 31k 1k1 A1k1 33k A1 f3k 0 , x1
(40)
where fi k x1 bik x1
2k 1 i 3 x1 (1)k i3 x1 , h
2k 1 k 1 x1 A1 i 3 x1 ik33 x1 . h
(41)
k i3
The Cauchy relations (27) have the form
11k
1 u1k 1 u3k k1u3k , 13k k1u1k u1k , 33k u3k . A1 x1 A1 x1
(42)
Hooke’s law has the same form us (29) and the equations in displacements have the form
2
1 u2k 1 u1k k k k u u A k k1u1k u1k 13k A1 f1k 0 , 1 3 3 1 1 x1 A1 x1 x1 A1 x1
1 u1k 1 u3k k1u1k u1k 2 A1k1 k1u3k A1k1 u3k 33k A1 f 3k 0 , x1 A1 x1 A1 x1
(43)
where
13k x1
2k 1 1 u3k 1 1 u3k 3 k1u1k 1 u1k 1 k1u1k 3 u1k 3 , h A1 x1 A1 x1
1 u1k 1 2k 1 1 u1k 3 k 1 k 3 x1 k1u3k 1 k1u3k 3 2 u3 u3 h A1 x1 A1 x1 k 33
(44)
This system of 1-D differential equations for coefficients of the Legendre’s polynomial series expansion contains an infinite number of equations. In the same way as in the case of axisymmetric
plate in order to simplify the situation a finite number of members in the expansion (13) has to be taken into account. For the first approximation only first two members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1 and the system of differential equations (37) contain only four equations of the form
2
1 u30 1 u10 1 1 1 0 k u u A k k1u10 u11 A1 f10 0 , 1 3 3 1 1 x1 A1 x1 h x1 h A1 x1
1 u10 1 u30 1 1 1 0 k u u 2 A k k1u30 A1k1 u31 A1 f 30 0 , 1 1 1 1 1 x1 A1 x1 h h A1 x1
2
1 u31 1 u11 1 k u A k k1u11 1 3 1 1 x1 A1 x1 A1 x1
(45)
3 1 u30 1 A1 k1u10 u11 +A1 f11 0 , h A1 x1 h
1 u11 1 u31 k1u11 2 A1k1 k1u31 x1 A1 x1 A1 x1 3 1 u10 1 A1 k1u30 2 u31 A1 f 31 0 . h A1 x1 h
For the second approximation only the first three members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1,2 and the system of differential equations (37) contain only six equations of the form
2
1 u30 1 u10 1 1 1 0 k u u A k k1u10 u11 A1 f10 0 , 1 3 3 1 1 x1 A1 x1 h x1 h A1 x1
1 u10 1 u30 1 1 1 0 k u u 2 A k k1u30 A1k1 u31 A1 f 30 0 , 1 1 1 1 1 x1 A1 x1 h h A1 x1
2
1 u31 1 u11 3 2 1 k u u A k k1u11 1 3 3 1 1 x1 A1 x1 h x1 A1 x1 3 1 u30 1 A1 k1u10 u11 A1 f11 0 , h A1 x1 h
1 u11 1 u31 3 u12 3 1 k u 2 A k k1u31 A1k1 u32 1 1 1 1 x1 A1 x1 h x1 h A1 x1 3 1 u10 1 A1 k1u30 2 u31 A1 f 31 0 , h A1 x1 h
2
1 u32 1 u12 5 1 u31 3 2 2 k u A k k u A k1u11 u12 A1 f12 0 , 1 3 1 1 1 1 1 x1 A1 x1 h A1 x1 h A1 x1
1 u12 15 1 u32 5 1 u11 k1u12 2 A1 k1 k1u32 2 u32 A1 k1u31 A1 f 31 0 . x1 A1 x1 h A1 x1 h A1 x1
The systems of differential equations (45) and (46) together with corresponding boundary conditions can be solved and the stress-strain state of axisymmetric circular plate investigated in frame of the first and second order approximation respectively. 4.3. Axially symmetric cylindrical shell If in the equations (25) - (31) we suppose A1 1, A2 R, k1 0, k2 k 1/ R as result we obtain equations for investigation of the stress-strain state of axisymmetric cylindrical shell. The equations of equilibrium (25) in this case have the form 11k 13k f1k 0, x1 31k 2k2 33k f 2k 0, x1 R
(47)
where fi k x1 bik x1
2k 1 i 2 x1 (1)k i2 x1 , h
2k 1 k 1 x1 i 3 x1 ik33 x1 . h k i3
(48)
(46)
The Cauchy relations (27) have the form
11k
u k u1k k u3k k , 22 , 33 u3k , 13k 3 u1k . x1 R x1
(49)
Hooke’s law has the same form us (29) and the equations in displacements have the form
2
u3 2u1k u3k 13k f1k 0 2 x1 R x1 x1 k
u1 2u k u1k uk 23 2 32 u3k 33k f 2k 0 x1 R x1 x1 R R k
(50)
where
13k
1 u3k 3 2k 1 1 u3k 1 k 1 u u1k 3 , 1 h R x1 R x1
u k 1 u k 3 2k 1 33k x1 2 R u3k 1 u3k 3 1 u3k 1 1 u3k 3 h x1 x1
(51)
The situation here is the same as in above cases. The system (50) of 1-D differential equations for coefficients of the Legendre’s polynomial series an expansion contains infinite number of equations. In the same way as in the previous cases in order to simplify the situation a finite number of members in the expansion (13) have to be taken into account. For the first approximation only the first two members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1 and the system of differential equations (37) contain only four equations of the form
2
2u10 u30 u31 f10 0 x12 R x1 h x1
2u30 u10 u11 u30 1 2 2 2 u3 f 20 0 x1 R x1 h x1 R Rh
(52)
2
u 3 1 u u 3 1 u1 f11 0 x h R x1 R x1 h 2 2 1 1 2 1
0 3
1 3
2u31 3 u10 u11 3 0 1 3R u3 2 2 2 u31 f 21 0 2 x1 h x1 R x1 h h R
For the second approximation only the first three members of the Legendre polynomials series expansions (13) have to be taken into account. In this case n 0,1,2 and the system of differential equations (37) contain only six equations of the form
2
2u10 u30 u31 f10 0 2 x1 R x1 h x1
2u30 u10 u11 u30 1 2 u3 f 20 0 2 2 x1 R x1 h x1 R Rh
2
2u11 3 u30 u31 3 u32 3 1 u1 f11 0 x12 h x1 R x1 h x1 h 2
2u1 3 u10 u11 3 u12 3 0 3 3 2 1 23 u3 2 2 2 u31 u3 f 21 0 x1 h x1 R x1 h x1 Rh h Rh R
2
(53)
2u12 5 u31 u32 15 2 2 u1 f12 0 2 x1 h x1 R x1 h
2u32 5 u11 u12 5 1 1 15 u3 2 2 2 u32 f 22 0 2 x1 h x1 R x1 Rh h R
The differential equations obtained here can be used to investigate the stress-strain state thinwalled structures like shells, plates, rods. Boundary-value problems are defined by a system of differential equations with the corresponding boundary.
5.
Numerical results and discussion
In order to check the validity, accuracy and efficiency of the proposed models we consider the beam and the axisymmetric cylindrical shell with build-in ends subjected to uniform load for the cases first and second order approximation theories. We compare the obtained solutions with the solution obtained using the equations of elasticity and the classical theories. The corresponding boundary-value problems have been solved using finite element method (FEM). Calculations have been done using commercial softwares MATLAB, Mathematica 10 and Comsol Multiphysics. Corresponding differential equations have been transformed to the form suitable for Comsol software input
C
2u( x1 ) u( x1 ) B A u( x1 ) f ( x1 ) 2 x1 x1
(54)
For details of FEM implementation using MATLAB, Mathematica 10.1 and Comsol Multiphysics one can refer to the corresponding tutorials. 5.1. Uniformly loaded beam The calculations have been done for the first and second order theories using differential equations (45) and (46) respectively. In order to adapt these equations for the case beam one has to take A1 1 and k1 0 . Boundary conditions for the case of the build-in ends have the form
uik (0) uik ( L) 0
(55)
Differential equations (45) and (46) have been transformed to the form (54), the corresponding matrixes for the first and second order approximation theories are presented in Appendix. In order to simplify the calculations and analysis of the obtained results dimetionless coordinates x x 1 1 and 3 3 have been introduced. For numerical analysis we used the following L h mechanical and geometrical parameters: Young’s modulus is E 1 Pa and for Poisson’s ratio is 0.3 , length of the beam is L 1 and its thickness is h 0.1L , respectively. The results of the calculations are presented on Fig. 1-6. Fig. 1 presents the verticals displasements of the beam calculated for the first and second order approximation theories in
comparison with the ones obtained using 2-D equations of the theory of elasticity and classical Euler-Bernully theory using MATLAB and Comsol Multiphysics.
Fig.1. Normalized vertical displacements. One can see a very good agreement of the results obtained using the proposed theories and those obtained using the theory of elasticity and the classical theory. It is important to mention that the second approximation theory gives results that better coincide with those obtained for the theory of elasticity. Also one has to take into account that all results presented in Fig.1 were numerically obtained using FEM and above mentioned software. Fig. 2 and Fig. 3 show the distribution of the Legendre polynomials coefficients for the displacements versus the normalized length for the first and second approximation theories respectively. These coefficients are FEM solutions of the systems of differential equations (38) and (46) respectively. They are calculated with MATLAB and represent direct output from that software. The displacements have been calculated using Legendre polynomials coefficients for the displacements and representations (13) for the first and second approximation theories with k 0,1 and k 0,1, 2 respectively. For the stress calculation the same representations have been used, but first using Legendre polynomials coefficients for stress have been calculated using Cauchy relations (42) and Hook’s law (29) for the first and second approximation theories respectively. These calculations cannot be done directly in Comsol Multiphysics, therefore a special MATLAB computer program has been created for this purpose. In that program the derivatives of functions uik ( x1 ) calculated in Comsol Multiphysics have been used. Fig. 4 and Fig. 5 show displacements and stresses distribution versus normalized length and thickness for the first and second approximation theories respectively calculated in MATLAB. The presented results show that the displacements are qualitatively and quantitatively distributed in the same way, but stresses differ qualitatively and quantitatively. More accurate results give us second approximation theory.
Fig.2. Legendre polynomials coefficients for the displacements. First approximation.
Fig.3. Legendre polynomials coefficients for the displacements. Second approximation.
Fig.4. Displacements and stresses versus normalized length and thickness. First approximation.
Fig.5. Displacements and stresses versus normalized length and thickness. Second approximation.
Fig.6. Normalized displacements and stresses in the middle point versus normalized thickness. In Fig. 6 calculated for first and second order approximation theories normalized vertical displacements u2 and normalized normal stresses 22 in the middle point of the beam are presented against normalized thickness h / L and compared with ones calculated using equations of 2-D elasticity. The calculations have been done using the FEM software embedded in Mathematica 10.1 and the results in Fig. 6 are a direct output from the Mathematica 10.1 notebook. The diagrams presented in Fig. 6 show that for all considered normalized thickness coincidence of the results obtained using proposed here approach and the theory of elasticity is very good. It means that the proposed theories may be successfully used for relatively thick beams, where classical theories give inaccurate results.
5.1. Uniformly loaded axially symmetric cylindrical shell Calculations have been done for the first and second order theories using differential equations (52) and (53) respectively. Boundary conditions for the case of the build-in ends of the shell have the form (55). Differential equations (52) and (53) have been transformed to the form (54), corresponding matrixes for the first and second order approximation theories are presented in the Appendix. In order to simplify the calculations and analysis of the obtained results dimetionless coordinates x x 1 1 and 3 3 have been introduced. For a numerical analysis we used the following L h mechanical and geometrical parameters: Young’s modulus is E 1 Pa and for Poisson’s ratio is 0.3 , length of the shell is L 1 , its radius is R 0.25L and its thickness is h 0.1R , respectively. The results of the calculations are presented in Fig. 7-12. Fig. 6 presents the verticals displasements of the shell calculated for the first and second order approximation theories compared with ones obtained using the equations of the theory of axisymmetric elasticity and classical Kirkgoff-Love theory. One can see very a good agreement of the results obtained using the proposed theories and those obtained using the theory of elasticity and classical theory. It is important to mention that the second approximation theory gives results that better coincide with those obtained for the theory of elasticity. Also one has to take into account that the results presented in Fig.7 were obtained numerically using MATLAB and Comsol Multiphysics. Fig. 8 and Fig. 9 show the distribution of the Legendre polynomials coefficients for the displacements versus the normalized length for the first and second approximation theories respectively. These coefficients are FEM solutions of the systems of differential equations (52) and (53) respectively. They are calculated with MATLAB and represent direct output from that software.
Fig.7. Normalized radial displacements. The displacements have been calculated using Legendre polynomials coefficients for the displacements and representations (13) for the first and second approximation theories with k 0,1 and k 0,1, 2 respectively. For stress calculation the same representations have been used, but first
using Legendre polynomials coefficients for stress have been calculated using Cauchy relations (49) and Hook’s law (29) for the first and second approximation theories respectively. These calculations cannot be done directly in Comsol Multiphysics, therefore a special MATLAB computer program have been created for this purpose. In that program the derivatives of functions uik ( x1 ) calculated in Comsol Multiphysics have been used. Fig. 9 and Fig. 10 show displacements and stresses distribution versus normalized length and thickness for the first and second approximation theories respectively calculated in MATLAB. The presented results show that the displacements are qualitatively and quantitatively distributed in the same way, but stresses differ qualitatively and quantitatively. More accurate results give us a second approximation theory.
Fig.8. Legendre polynomials coefficients for the displacements. First approximation.
Fig.9. Legendre polynomials coefficients for the displacements. Second approximation. In order to compare the displacements and stresses calculated by the proposed method for first and second approximation theories with those obtained using equations of elasticity for different values of the normalized thickness we consider the relatively long ( L 10R ) uniformly loaded cylindrical shell. For this specific case we will use an analytical elasticity solution for long asymmetrically loaded long cylinder. Displacements and stresses can be presented by the equations [41]
11
2 p ( R h) 2 p ( R h) 2 p ( R h) 2 ( R h ) 2 , 22 , 4 Rh 4 Rh 4 Rh( R x2 ) 2
33
p ( R h) 2 p ( R h) 2 ( R h) 2 4 Rh 4 Rh( R x2 ) 2
u2
p(1 )( R h) 2 p(1 )( R h) 2 ( R h) 2 ( R x2 ) 4 ERh 4 EeRh( R x2 )
(56)
Here 11 , 22 and 33 are normal stresses in axial, radial and circumferential directions respectively, u2 is a radial displacements.
Fig.10. Displacements and stresses versus normalized length and thickness. First approximation.
Fig.11. Displacements and stresses versus normalized length and thickness. Second approximation.
Fig.12. Normalized radial displacements and circumferential stresses versus normalized thickness for inner side of the cylindrical shell. In Fig. 12 the normalized radial displacements and the circumferential stresses in inner sides of the shell for first and second order approximation theories are calculated and presented against normalized thickness h / L . Then they are compared with the ones that are calculated using the equations (56) of the axisymmetric elasticity. The calculations have been done in Mathematica 10.1, for first and second order approximation using the FEM software embedded and for elasticity solution directly, using the equations (56). The results presented in Fig. 12 are a direct output from the Mathematica 10.1 notebook. The diagrams presented in Fig. 12 show that for all considered normalized thickness coincidence of the results obtained using the approach that has been proposed here and the theory of elasticity is very good. It means that the proposed theories may be successfully used for a relatively thick cylinder, where classical theories give inaccurate results.
6.
Conclusions
In this paper a higher order theory for homogeneous elastic shells, plates and rods has been developed. The proposed approach is based on the expansion of the equations of elasticity into Fourier series in terms of Legendre polynomials. Starting from the equations of elasticity, stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Fourier series in terms of Legendre polynomials in a thickness coordinate. Thereby all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. The system of differential equations in terms of displacements and boundary conditions for Fourier coefficients has been obtained. Special attention has been paid to the case of axially symmetric shells. For the special case of circular plates, curvilinear rods and cylindrical shells the first and second approximations theories have been considered in more detail. All necessary equations and their coefficients have been written explicitly and the corresponding boundary-value problems have been formulated. For a numerical solution of the formulated problems FEM has been applied and commercial software Matlab, Mathematica 10 and Comsol Multiphysics have been used. For validation of the proposed theories, a comparison of the numerical calculations with the results obtained using equations of the theory of elasticity and classical theories of Euler-Bernoulli for beams and Kirchhoff-Love for shell have been done. The presented calculations show a good agreement of the results obtained using proposed theories with the theory of elasticity and the classical theories. The second order approximation theory gives better coincidence with theory of elasticity than the first one and it is important to mention that the proposed theories give more accurate results then the classical theories. They can be recommended for calculation of the stresses and displacements in relatively thick beams, rods, plated and shells, where classical theories give inaccurate results.
Acknowledgments This work was supported by the Committee of Science and Technology of Mexico (CONACYT) by the Research Grants (Project No. 101415), which is gratefully acknowledged.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
20. 21. 22. 23.
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Appendix. Matrixes and vectors are presented in the equation (54). 1. Beams 1.1. First approximation beam theory
2 C
0 0 0
0
0 0 0 2 0 0
0 0 ,B 0
0
0
0
0
0 3 h
0
0
h
3 h
0
0
0
0 0 0 0
h
, 0
A 0 0 3 h2 0 0
0
1.2. Second approximation beam theory
2 0 0 0 0 0
C
0 0 0 2 0 0 0 0 0 0
0
0
0
0
0 0 0
0 0 0 0 0 2 0 0
0
h
0
0
0
0
0
3 h
0
0
0
0
0
0
0
0
5 h
0
0
0 0 0 0 0
0
h
3 h
0 B
0
0 0
3 h , 0 0
5 h
0
0
0
0
0
,
0
0 0 0 2
3 h2
0 0 0 0
0 0
0 0
3 h2
0 0
0 0
0 0
0
0
0
0
0
15 h2
0
2
A 0 0
0
0 0
0
0
0 0
0
0
3 h2
0
2
15 h2
2. Axially symmetric cylindrical shell 2.1. First approximation shell theory 0
2 C
0 0 0
0 0 0 2 0 0
0
0 2 0 R2 A
0 0
3 Rh
0
0
R
0
3 R h2 0
h
3 h
3 h
0
0
0
0
R
0 0 R ,B 0 0
0
0
Rh 0
1 3R 2 2 R h
2.2. Second approximation shell theory
C
2
0
0 0 0 0 0
0 0 0 0
0 0 2 0 0 0
0 0 0
0 0 0 0 0 2 0 0
0 0 0 0 0
,
h 0
R 0
,
0
0
0 R
0
0
0
0
0
5 h
0
2
0
R
0
R
5 h 0
0
0
0
0
3 h
3 h 0
0
0
Rh
R
3 1 2 2 2 h R
0
0
0
0
0
0
0
5 Rh
0
0
0
0
R
0
3 Rh
0
0 0
0
0
0
0
,
0
3 h2
0 A
3 h
0
2
0
h
3 h
0
0
h
0
R
0 B
R
0
3 Rh
15 h2
0
0
1 15 2 2 2 h R
We develop a higher order theory of elasticity for linearly elastic shells, plates, roods and beams. We use for that Fourier series in terms of Legendre’s polynomials expansion For numerical solution the FEM implemented in the software COMSOL Multiphysics, MATLAB and Mathematica. The displacements, stress distribution in the asymmetrical cylindrical shell and beam have been studied Comparison with elasticity and classical shell and beam solution has been done.