A High Order Theory for Linear Thermoelastic Shells: Comparison with Classical Theories

Abstract

A high order theory for linear thermoelasticity and heat conductivity of shells has been developed. The proposed theory is based on expansion of the 3D equations of theory of thermoelasticity and heat conductivity into Fourier series in terms of Legendre polynomials. The first physical quantities that describe thermodynamic state have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby all equations of elasticity and heat conductivity including generalized Hooke's and Fourier's laws have been transformed to the corresponding equations for coefficients of the polynomial expansion. Then in the same way as in the 3-D theories system of differential equations in terms of displacements and boundary conditions for Fourier coefficients has been obtained. First approximation theory is considered in more detail. The obtained equations for the first approximation theory are compared with the corresponding equations for Timoshenko's and Kirchhoff-Love's theories. Special case of plates and cylindrical shell is also considered, and corresponding equations in displacements are presented.

1. Introduction

The development of microelectromechanical and nanoelectromechanical technologies extends the field of application of the classical or nonclassical theories of plates and shells towards the new thin-walled structures. The classical elasticity can be extended to the micro- and nanoscale by implementation of the theory of elasticity taking into account the physical phenomenon that can occur in such structures and devices [1, 2].

Classical theories of beams, rods, plates, and shells are usually related to names of Bernoulli, Euler, Kirchhoff, and Love. These theories are based on well-known physical hypothesis; they are very popular among an engineering community because of their relative simplicity and physical clarity. Numerous books and monographs have been written in the subject among others one can refer to [3–6]. But unfortunately classical theories have some shortcomings and logical contradictions such as their proximity and inaccuracy and as result in some cases not good agreement with results obtained with 3D approach and experiments. Therefore there is demand in developing new more accurate theories.

We can mention at least two approaches to development of the theories of thin-walled structures. One consists in improvement of the classical physical hypothesis and development of more accurate theories. The theory of beamis well-known model that takes into account transversal deformations developed by Timoshenko and extended to the plate theory by Mindlin [7]. This theory was extended and applied to shells of arbitrary geometry in numerous publications and is referred to as Timoshenko’s theory that takes into account in-plane shear deformations and rotation of the elements perpendicular to the middle surface of the shell [8–10].

The second approach consists in expansion of the stress-strain field components into polynomials series in terms of thickness. It was proposed by Cauchy and Poisson and at that time was not popular. Significant extension and development of that method were done by Kil'chevskii [11]. Vekua has used Legendre’s polynomials for the expansion of the equations of elasticity and reduction of the 3-D problem to 2-D one [12]. Such an approach has significant advantages because Legendre’s polynomials are orthogonal, and as a result obtained equations are simpler. This approach was extended and applied to dynamical problems [13], thermoelasticity [14], and composite and laminate shells [15].

The approach developed in [12–15] has been applied to the plates and shells thermoelastic contact problems when mechanical and thermal conditions are changed during deformation in our previous publications [16–29]. The mathematical formulation, differential equations, and contact conditions for the cases of plates and cylindrical shells for the first time have been reported in [16–18]. In more general form with extension to nonstationary processes and calculation of all coefficients of the equations and contact and boundary conditions it was presented in [19, 20]. Then the approach was further developed to contact of plates and shells with rigid bodies through heat conducting layer [21–25], thermoelasticity of the laminated composite materials with possibility of delamination and mechanical and thermal contact in temperature field in [21, 22, 26], the pencil-thin nuclear fuel rods modeling in [22], and functionally graded shells in [28, 29].

Bibliography related to different aspects of the theory and applications of the thin-walled structures contains of several thousands publications for references one can see review papers [30, 31]. For trends and recent development in the shells theory and its applications one can refer to books [10, 32–34].

In this paper, an approach based on expansion of the equations of thermoelasticity and heat conductivity into Fourier series in terms of Legendre polynomials has been developed and applied to high order theory of arbitrary geometry shells. For that purpose we expand functions that describe thermodynamic state of elastic body into Fourier series in terms of Legendre polynomials with respect to thickness and find corresponding relations of thermoelasticity and heat conductivity for Fourier coefficients of those. Then using techniques developed in our previous publications we find system of differential equations and boundary conditions for Fourier coefficients. Case of first approximation (Vekua’s) theory is considered in more detail, and all relations and equations are explicitly presented. Obtained equations for Vekua’s shell theory have been analyzed and compared with classical Timoshenko’s and Kirchhoff-Love's shell theories.

2. 𝟑-D Formulation

Let an elastic body occupy an open 3-D Euclidian space in simply connected bounded domain 𝑉∈𝐑³ with a smooth boundary 𝜕𝑉. We assume that body is homogeneous isotropic shell of arbitrary geometry with 2ℎ thickness, material of which follows linear physical laws of thermoelasticity and heat conductivity. The shell occupies the domain 𝑉=Ω×[−ℎ,ℎ] in Euclidean space. Boundary of the shell can be presented in the form 𝜕𝑉=𝑆∪Ω⁺∪Ω⁻. Here Ω is the middle surface of the shell, 𝜕Ω is its boundary, Ω⁺ and Ω⁻ are the outer sides, and 𝑆=𝜕Ω×[−ℎ,ℎ] is a sheer side.

Thermodynamic state of the body is defined by stress 𝜎_𝑖𝑗(𝐱,𝑡) and strain 𝜀_𝑖𝑗(𝐱,𝑡) tensors and displacements 𝑢_𝑖(𝐱,𝑡), traction 𝑝_𝑖(𝐱,𝑡), and body forces 𝑏_𝑖(𝐱,𝑡) vectors, temperature 𝜃(𝐱,𝑡), and vector of thermal flow 𝑞_𝑖(𝐱,𝑡) and specific strength of the internal heat sources 𝜒(𝐱,𝑡), respectively. These quantities are not independent they are related by equations of linear thermoelasticity and heat conductivity.

We introduce orthogonal system of coordinates 𝑥₁,𝑥₂, and 𝑥₃, such that position vector of arbitrary point is equal to 𝐑(𝑥₁,𝑥₂,𝑥₃)=𝐞_𝑖𝑥_𝑖. Unit orthogonal basic vectors and their derivatives with respect to space coordinates are equal to 𝐞_𝑖=1𝐻_𝑖𝜕𝐑𝜕𝑥_𝑖,𝜕𝐞_𝑖𝜕𝑥_𝑗=Γ^𝑘_𝑖𝑗𝐞_𝑘, (1) where 𝐻_𝑖 are Lame coefficients and Γ^𝑘_𝑖𝑗 are Christoffel symbols. They are calculated by the equations 𝐻_𝑖=𝜕𝐑𝜕𝑥_𝑖=𝜕𝐑𝜕𝑥_𝑖⋅𝜕𝐑𝜕𝑥_𝑖,Γ^𝑘_𝑖𝑗1=−𝐻_𝑖𝜕𝐻_𝑖𝜕𝑥_𝑗𝛿_𝑖𝑘+12𝐻_𝑖𝐻_𝑘×𝛿_𝑗𝑘𝜕𝐻_𝑗𝐻_𝑘𝜕𝑥_𝑖+𝛿_𝑖𝑘𝜕𝐻_𝑖𝐻_𝑘𝜕𝑥_𝑗−𝛿_𝑖𝑗𝜕𝐻_𝑖𝐻_𝑗𝜕𝑥_𝑘. (2) From the last equation it follows that Γ^𝑘_𝑖𝑗=0 for 𝑖≠𝑗≠𝑘 and Γ^𝑘_𝑖𝑖1=−𝐻_𝑘𝜕𝐻_𝑖𝜕𝑥_𝑘,Γ^𝑘_𝑖𝑘1=−𝐻_𝑖𝜕𝐻_𝑘𝜕𝑥_𝑖for𝑖≠𝑘. (3)

In the case if displacements and their gradients are small, the following kinematic Cauchy relations take place: 𝜀_𝑖𝑗=121𝐻_𝑗𝜕𝑢_𝑖𝜕𝑥_𝑗+1𝐻_𝑖𝜕𝑢_𝑗𝜕𝑥_𝑖+Γ^𝑘_𝑖𝑗𝑢_𝑘. (4)

Equations of motion have the form 1𝐻_𝑗𝜕𝜎_𝑖𝑗𝜕𝑥_𝑗+𝜎_𝑖𝑗𝐻_𝑘Γ^𝑘_𝑖𝑘+𝜎_𝑖𝑘𝐻_𝑖Γ^𝑗_𝑘𝑖+𝑏_𝑖=𝜌𝑢̈_𝑖, (5) where upper points are partial derivatives with respect to time 𝑡.

We assume that the stress 𝜎_𝑖𝑗(𝐱) and deformation 𝜀_𝑖𝑗(𝐱) tensors and temperature 𝜃(𝐱,𝑡) are related by equations 𝜎_𝑖𝑗=𝑐_{𝑖𝑗𝑘𝑙}𝜎_𝑘𝑙+𝛽_𝑖𝑗𝜃,𝑐_{𝑖𝑗𝑘𝑙}=𝑐_{𝑗𝑖𝑘𝑙}=𝑐_{𝑘𝑙𝑖𝑗},𝛽_𝑖𝑗=𝛽_𝑗𝑖, (6) where 𝑐_{𝑖𝑗𝑘𝑙} and 𝛽_𝑖𝑗 are elastic modulus and the coefficients of linear thermal expansion. In the isotropic case 𝑐_{𝑖𝑗𝑘𝑙}=𝜆𝛿_𝑖𝑗𝛿_𝑘𝑙+𝜇𝛿_𝑖𝑘𝛿_𝑗𝑙+𝛿_𝑖𝑙𝛿_𝑘𝑗,𝛽_𝑖𝑗=𝛽_𝑇𝛿_𝑖𝑗, (7) where 𝜆 and 𝜇 are the Lame constants, 𝛽_𝑇=(𝜇+3𝜆)𝛼_𝑇, and 𝛼_𝑇 is a coefficient of linear thermal expansion.

Equations (6) and (7) are referred to as generalized Hooke’s law. They were introduced by Duhamel and substantiated using thermodynamic methods by Bio [8, 11].

Heat is distributed in the body according to Fourie’s law: 𝑞_𝑖=𝜆_𝑖𝑗1𝐻_𝑗𝜕𝜃𝜕𝑥_𝑗, (8) where 𝜆_𝑖𝑗 is the tensor of coefficients of thermal conductivity of the body, in the isotropic case 𝜆_𝑖𝑗=𝛿_𝑖𝑗𝜆_𝑇, and 𝜆_𝑇 is the coefficient of thermal conductivity of the body.

For convenience we transform the previous equations taking into account that the position vector 𝐑(𝐱) of any point in domain 𝑉, occupied by material points of shell, may be presented as 𝐑(𝐱)=𝐫𝐱_𝛼+𝑥₃𝐧𝐱_𝛼, (9) where 𝐫(𝐱_𝛼) is the position vector of the points located on the middle surface of shell and 𝐧(𝐱_𝛼) is a unit vector normal to the middle surface.

We consider that 𝐱_𝛼=(𝑥¹,𝑥²) are curvilinear coordinates associated with main curvatures of the middle surface of the shell. In this case 3-D equations (4)–(8) can be simplified taking into account that Lame coefficients and their derivatives have the form 𝐻_𝛼=𝐴_𝛼1+𝑘_𝛼𝑥₃for𝛼=1,2,𝐻₃=1𝜕𝐻_𝛽𝜕𝑥_𝛼=𝜕𝐴_𝛽𝜕𝑥_𝛼1+𝑘_𝛼𝑥₃,𝜕𝐻_𝛽𝜕𝑥₃=𝑘_𝛽𝐴_𝛽,𝜕𝐻₃𝜕𝑥_𝑖=0. (10)

The equations of motion (5) after simplification have the form 𝜕𝐴₂𝜎₁₁𝜕𝑥₁+𝜕𝐴₁𝜎₁₂𝜕𝑥₂+𝐴₁𝐴₂𝜕𝜎₁₃𝜕𝑥₃+𝜎₁₂𝜕𝐴₁𝜕𝑥₂+𝜎₁₃𝐴₁𝐴₂𝑘₁−𝜎₂₂𝜕𝐴₂𝜕𝑥₁+𝐴₁𝐴₂𝑏₁=𝜌𝐴₁𝐴₂𝑢̈₁,𝜕𝐴₂𝜎₂₁𝜕𝑥₁+𝜕𝐴₁𝜎₂₂𝜕𝑥₂+𝐴₁𝐴₂𝜕𝜎₂₃𝜕𝑥₃+𝜎₂₁𝜕𝐴₂𝜕𝑥₁+𝜎₂₃𝐴₁𝐴₂𝑘₂−𝜎₁₁𝜕𝐴₁𝜕𝑥₂+𝐴₁𝐴₂𝑏₂=𝜌𝐴₁𝐴₂𝑢̈₂,𝜕𝐴₂𝜎₃₁𝜕𝑥₁+𝜕𝐴₁𝜎₃₂𝜕𝑥₂+𝜕𝐴₁𝐴₂𝜎₃₃𝜕𝑥₃−𝜎₁₁𝐴₁𝐴₂𝑘₁−𝜎₂₂𝐴₁𝐴₂𝑘₂+𝐴₁𝐴₂𝑏₃=𝜌𝐴₁𝐴₂𝑢̈₃. (11)

The kinematic Cauchy relations (4) have the form 𝜀₁₁=1𝐴₁𝜕𝑢₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₂+𝑘₁𝑢₃,𝜀₂₂=1𝐴₂𝜕𝑢₂𝜕𝑥₂+1𝐴₂𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₁+𝑘₂𝑢₃,𝜀₃₃=𝜕𝑢₃𝜕𝑥₃,𝜀₁₂=1𝐴₂𝜕𝑢₁𝜕𝑥₂−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₂+1𝐴₁𝜕𝑢₂𝜕𝑥₁−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₁,𝜀₁₃=𝜕𝑢₁𝜕𝑥₃−𝑘₁𝑢₁+1𝐴₁𝜕𝑢₃𝜕𝑥₁,𝜀₂₃=𝜕𝑢₂𝜕𝑥₃−𝑘₂𝑢₂+1𝐴₂𝜕𝑢₃𝜕𝑥₂. (12) In (10)–(12), 𝐴_𝛼(𝑥₁,𝑥₂)=(𝜕𝐫(𝑥₁,𝑥₂))/𝜕𝑥_𝛼 are coefficients of the first quadratic form of a surface, and 𝑘_𝛼(𝑥₁,𝑥₂) are its main curvatures. We also take into account that shell is relatively thin, and therefore 1𝐻_𝛼𝜕𝐻_𝛽𝜕𝑥_𝛼=1𝐴_𝛼𝜕𝐴_𝛽𝜕𝑥_𝛼,1+𝑘_𝛼𝑥₃≈1→𝐻_𝛼≈𝐴_𝛼. (13)

Following [8, 11] we can present coupled system of the equations of linear thermoelasticity and heat conductivity in the form 𝐋⋅𝐮+𝛽_𝑇̈[],𝜆∇𝜃+𝐛=𝜌𝐮,∀𝐱∈𝑉,∀𝑡∈ℑ=0,𝑇_𝑇Δ𝜃−𝜒=𝑐_𝜀𝜃̇+𝛽_𝑇𝜃₀(∇⋅𝐮̇). (14)

The differential operator 𝐋=𝐿_𝑖𝑗𝐞_𝑖⊗𝐞_𝑗 for homogeneous isotropic medium has the form 𝐋⋅𝐮=𝜇Δ𝐮+𝜆+𝜇∇(∇⋅𝐮). (15) Other differential operators presented in (14) and (15), respectively, in the system of coordinates introduced here related to middle surface of the shell have the form ∇(𝐮)₁=1𝐴₁𝜕𝑢₁𝜕𝑥₁+1𝐴₂𝜕𝑢₁𝜕𝑥₂+𝜕𝑢₁𝜕𝑥₃,∇(∇⋅𝐮)_𝛼=1𝐴_𝛼𝜕𝜕𝑥_𝛼1𝐴₁𝐴₂𝜕𝐴₂𝑢₁𝜕𝑥₁+𝜕𝐴₁𝑢₂𝜕𝑥₂+𝜕𝑢₃𝜕𝑥₃+𝑘₁+𝑘₂𝑢₃,∇(∇⋅𝐮)₃=𝜕𝜕𝑥₃1𝐴₁𝐴₂𝜕𝐴₂𝑢₁𝜕𝑥₁+𝜕𝐴₁𝑢₂𝜕𝑥₂+𝜕𝑢₃𝜕𝑥₃+𝑘₁+𝑘₂𝑢₃,1∇𝜃=𝐴₁𝜕𝜃𝜕𝑥₁+1𝐴₂𝜕𝜃𝜕𝑥₂+𝜕𝜃𝜕𝑥₃. (16)

For mathematically correct formulation of the coupled problem of thermoelasticity and heat conductivity we have to formulate initial and boundary conditions. Initial conditions consist of assignment of the displacements, velocity, and temperature distribution in the initial moment of time. They can be written in the form 𝑢_𝑖(𝐱,0)=𝑢⁰_𝑖(𝐱),𝜕_𝑡𝑢_𝑖(𝐱,0)=𝑣⁰_𝑖(𝐱),𝜃(𝐱,0)=𝜃⁰(𝐱). (17)

If the body occupied a finite region 𝑉 with the boundary 𝜕𝑉, it is necessary to assign boundary conditions. We consider the mixed boundary conditions in the form 𝑢_𝑖(𝐱,𝑡)=𝑢^𝑏_𝑖(𝐱,𝑡),∀𝐱∈𝜕𝑉_𝑢𝑝,∀𝑡∈ℑ,_𝑖(𝐱,𝑡)=𝜎_𝑖𝑗(𝐱,𝑡)𝑛_𝑗(𝐱)=𝑝^𝑏_𝑖(𝐱,𝑡),∀𝐱∈𝜕𝑉_𝑝𝜃(),∀𝑡∈ℑ,𝐱,𝑡=𝜃^𝑏(𝐱,𝑡),∀𝐱∈𝜕𝑉_𝜃𝑞,∀𝑡∈ℑ,_𝑖(𝐱,𝑡)=𝑞^𝑏_𝑖(𝐱,𝑡),∀𝐱∈𝜕𝑉_𝑞,∀𝑡∈ℑ. (18)

Now we can transform 3-D equations of thermoelasticity and heat conductivity in 2-D equations using Legendre polynomials series expansion.

3. 𝟐-D Formulation

We expand the physical parameters, that describe the thermodynamic state of the body, into the Legendre polynomials series along the coordinate 𝑥₃. Such expansion can be done because any function 𝑓(𝑝), which is defined in domain −1≤𝑝≤1 and satisfies Dirichlet’s conditions (continuous, monotonous, and having finite set of discontinuity points), can be expanded into Legendre’s series according to formulas 𝑓𝑝=^∞_𝑘=0𝑎_𝑘𝑃_𝑘𝑝where𝑎_𝑛=2𝑘+12¹₋₁𝑓𝑝𝑃_𝑘𝑝𝑑𝑝. (19) Any function of more than one independent variable can also be expanded into Legendre’s series with respect to for example, variable 𝑥₃∈[−1,1], but first the new normalized variable 𝜔=𝑥₃/ℎ∈[−1,1] has to be introduced. Taking into account (19) we have 𝑢_𝑖(𝐱,𝑡)=^∞_𝑘=0𝑢^𝑘_𝑖𝐱_𝛼,𝑡𝑃_𝑘(𝜔),𝜎_𝑖𝑗(𝐱,𝑡)=^∞_𝑘=0𝜎^𝑘_𝑖𝑗𝐱_𝛼,𝑡𝑃_𝑘(𝜔),𝜀_𝑖𝑗(𝐱,𝑡)=^∞_𝑘=0𝜀^𝑘_𝑖𝑗𝐱_𝛼,𝑡𝑃_𝑘(𝜔),𝜃(𝐱,𝑡)=^∞_𝑘=0𝜃^𝑘𝐱_𝛼,𝑡𝑃_𝑘(𝜔),𝑢^𝑘_𝑖𝐱_𝛼,𝑡=2𝑘+12ℎ^ℎ_−ℎ𝑢_𝑖𝐱_𝛼,𝑥₃,𝑡𝑃_𝑘(𝜔)𝑑𝑥₃,𝜎^𝑘_𝑖𝑗𝐱_𝛼,𝑡=2𝑘+12ℎ^ℎ_−ℎ𝜎_𝑖𝑗𝐱_𝛼,𝑥₃,𝑡𝑃_𝑘(𝜔)𝑑𝑥₃,𝜀^𝑘_𝑖𝑗𝐱_𝛼,𝑡=2𝑘+12ℎ^ℎ_−ℎ𝜀_𝑖𝑗𝐱_𝛼,𝑥₃,𝑡𝑃_𝑘(𝜔)𝑑𝑥₃,𝜃^𝑘𝐱_𝛼,𝑡=2𝑘+12ℎ^ℎ_−ℎ𝜃𝐱_𝛼,𝑥₃,𝑡𝑃_𝑘(𝜔)𝑑𝑥₃. (20)(21) The following relations take place for the derivatives with respect to time: 2𝑘+12ℎ^ℎ_−ℎ𝜕_𝑡𝑢_𝑖𝐱_𝛼,𝑥₃,𝑡𝑃_𝑘(𝜔)𝑑𝑥₃=𝜕_𝑡𝑢^𝑘_𝑖𝐱_𝛼,𝑡,2𝑘+12ℎ^ℎ_−ℎ𝜕²_𝑡𝑢_𝑖𝐱_𝛼,𝑥₃,𝑡𝑃_𝑘(𝜔)𝑑𝑥₃=𝜕²_𝑡𝑢^𝑘_𝑖𝐱_𝛼,𝑡,2𝑘+12ℎ^ℎ_−ℎ𝜕_𝑡𝜃𝐱_𝛼,𝑥₃,𝑡𝑃_𝑘(𝜔)𝑑𝑥₃=𝜕_𝑡𝜃^𝑘𝐱_𝛼,𝑡, (22) and for the derivatives with respect to coordinates 𝐱_𝛼: 2𝑘+12ℎ^ℎ_−ℎ𝜕𝑢_𝑖𝐱_𝛼,𝑥₃,𝑡𝜕𝑥_𝛼𝑃_𝑘(𝜔)𝑑𝑥₃=𝜕𝑢^𝑘_𝑖𝐱_𝛼,𝑡𝜕𝑥_𝛼,2𝑘+12ℎ^ℎ_−ℎ𝜕𝜎_𝑖𝑗𝐱_𝛼,𝑥₃,𝑡𝜕𝑥_𝛼𝑃_𝑘(𝜔)𝑑𝑥₃=𝜕𝜎^𝑘_𝑖𝑗𝐱_𝛼,𝑡𝜕𝑥_𝛼,2𝑘+12ℎ^ℎ_−ℎ𝜕𝜃𝐱_𝛼,𝑥₃,𝑡𝜕𝑥_𝛼𝑃_𝑘(𝜔)𝑑𝑥₃=𝜕𝑢^𝑘_𝑖𝐱_𝛼,𝑡𝜕𝑥_𝛼, (23) respectively.

Integration of the derivatives with respect to coordinates 𝑥₃ gives us 2𝑘+12ℎ^ℎ_−ℎ𝜕𝑢_𝑖(𝐱,𝑡)𝜕𝑥₃𝑃_𝑘(𝜔)𝑑𝑥₃=𝑢^𝑘_𝑖𝐱_𝛼,𝑡,2𝑘+12ℎ^ℎ_−ℎ𝜕𝜎_𝑖3(𝐱,𝑡)𝜕𝑥₃𝑃_𝑘(𝜔)𝑑𝑥₃=2𝑘+1ℎ𝜎⁺_𝑖3𝐱_𝛼,𝑡−(−1)^𝑘𝜎⁻_𝑖3𝐱_𝛼,𝑡−𝜎^𝑘_𝑖3𝐱_𝛼,𝑡,2𝑘+12ℎ^ℎ_−ℎ𝜕₃𝜃(𝐱,𝑡)𝑃_𝑘(𝜔)𝑑𝑥₃=𝜕₃𝜃𝐱_𝛼,𝑡=𝑄^𝑘₃𝐱_𝛼,𝑡,2𝑘+12ℎ^ℎ_−ℎ𝜕²₃𝜃(𝐱,𝑡)𝑃_𝑘(𝜔)𝑑𝑥₃=2𝑘+1ℎ𝑄⁺₃𝐱_𝛼,𝑡−(−1)^𝑘𝑄⁻₃𝐱_𝛼,𝑡+𝑄^𝑘₃𝐱_𝛼,𝑡, (24) where 𝑢^𝑘_𝑖𝐱_𝛼,𝑡=2𝑘+1ℎ𝑢_𝑖^𝑘+1𝐱_𝛼,𝑡+𝑢_𝑖^𝑘+3𝐱_𝛼,𝜎,𝑡+⋯_𝑘𝑖3𝐱_𝛼,𝑡=𝐴₁𝐴₂2𝑘+1ℎ𝜎^𝑘−1_𝑖3𝐱_𝛼,𝑡+𝜎^𝑘−3_𝑖3𝐱_𝛼,𝑄,𝑡+⋯_𝑘3𝐱_𝛼,𝑡=−𝐴₁𝐴₂2𝑘+1ℎ𝑄₃^𝑘−1𝐱_𝛼,𝑡+𝑄₃^𝑘−3𝐱_𝛼,𝑡+⋯. (25)

Now substituting stress tensor from (20) with considering the equations of motion (11), multiplying obtained relations by 𝑃_𝑘(𝜔), and integrating over interval [−ℎ,ℎ] with respect to 𝑥₃ we obtain 2-D equations of motion in the form 𝜕𝐴₂𝜎^𝑘₁₁𝜕𝑥₁+𝜕𝐴₁𝜎^𝑘₁₂𝜕𝑥₂+𝜎^𝑘₁₂𝜕𝐴₁𝜕𝑥₂+𝜎^𝑘₁₃𝐴₁𝐴₂𝑘₁−𝜎^𝑘₂₂𝜕𝐴₂𝜕𝑥₁−𝜎^𝑘₁₃+𝐴₁𝐴₂𝑓^𝑘₁=𝜌𝐴₁𝐴₂𝜕²_𝑡𝑢^𝑘₁,𝜕𝐴₂𝜎^𝑘₂₁𝜕𝑥₁+𝜕𝐴₁𝜎^𝑘₂₂𝜕𝑥₂+𝜎^𝑘₁₂𝜕𝐴₂𝜕𝑥₁+𝜎^𝑘₂₃𝐴₁𝐴₂𝑘₂−𝜎^𝑘₁₁𝜕𝐴₂𝜕𝑥₁−𝜎^𝑘₂₃+𝐴₁𝐴₂𝑓^𝑘₂=𝜌𝐴₁𝐴₂𝜕²_𝑡𝑢^𝑘₂,𝜕𝐴₂𝜎^𝑘₃₁𝜕𝑥₁+𝜕𝐴₁𝜎^𝑘₃₂𝜕𝑥₂−𝜎^𝑘₁₁𝐴₁𝐴₂𝑘₁−𝜎^𝑘₂₂𝐴₁𝐴₂𝑘₂−𝜎^𝑘₃₃+𝐴₁𝐴₂𝑓^𝑘₃=𝜌𝐴₁𝐴₂𝜕²_𝑡𝑢^𝑘₃, (26) where 𝑓^𝑘_𝑖𝐱_𝛼=𝑏^𝑘_𝑖𝐱_𝛼+2𝑘+1ℎ𝜎⁺_𝑖3𝐱_𝛼−(−1)^𝑘𝜎⁻_𝑖3𝐱_𝛼. (27)

In the same way the 2-D kinematic Cauchy relations can be found: 𝜀^𝑘₁₁=1𝐴₁𝜕𝑢^𝑘₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₂+𝑘₁𝑢^𝑘₃,𝜀^𝑘₂₂=1𝐴₂𝜕𝑢^𝑘₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₁+𝑘₂𝑢^𝑘₃,𝜀^𝑘₁₂=1𝐴₂𝜕𝑢^𝑘₁𝜕𝑥₂−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₂+1𝐴₁𝜕𝑢^𝑘₂𝜕𝑥₁−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₁,𝜀^𝑘₁₃=1𝐴₁𝜕𝑢^𝑘₃𝜕𝑥₁−𝑘₁𝑢^𝑘₁+𝑢^𝑘₁,𝜀^𝑘₂₃=1𝐴₂𝜕𝑢^𝑘₃𝜕𝑥₂−𝑘₂𝑢^𝑘₂+𝑢^𝑘₂,𝜀₃₃=𝑢^𝑘₃. (28) Let us consider generalized Hooke’s law for homogeneous anisotropic body 𝜎^𝑘_𝑖𝑗=𝑐_{𝑖𝑗𝑘𝑙}𝜀^𝑘_𝑘𝑙+𝛽_𝑖𝑗𝜃^𝑘 (29) and for isotropic one 𝜎^𝑘₁₁=𝜆+2𝜇𝜀^𝑘₁₁+𝜆𝜀^𝑘₂₂+𝜀^𝑘₃₃+𝛽_𝑇𝜃^𝑘,𝜎^𝑘₂₂=𝜆+2𝜇𝜀^𝑘₂₂+𝜆𝜀^𝑘₁₁+𝜀^𝑘₃₃+𝛽_𝑇𝜃^𝑘,𝜎^𝑘₃₃=𝜆+2𝜇𝜀^𝑘₂₂+𝜆𝜀^𝑘₁₁+𝜀^𝑘₂₂+𝛽_𝑇𝜃^𝑘,𝜎^𝑘₁₂=𝜇𝜀^𝑘₁₂,𝜎^𝑘₁₃=𝜇𝜀^𝑘₁₃,𝜎^𝑘₂₃=𝜇𝜀^𝑘₂₃. (30) In the previously mentioned relations (26)–(30) the following orthogonality property of the Legendre’s polynomials has been used: ^ℎ_−ℎ𝑃_𝑛(𝜔)𝑃_𝑚(𝜔)𝑑𝑥₃=2ℎ2𝑛+1for𝑛=𝑚0for𝑛≠𝑚. (31)

In order to find 2-D differential equations in the form of displacements we substitute the kinematic Cauchy relations (28) with Hooke’s law for homogeneous body (30). As a result we have 𝜎^𝑘₁₁=1𝜆+2𝜇𝐴₁𝜕𝑢^𝑘₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₂+𝑘₁𝑢^𝑘₃1+𝜆𝐴₂𝜕𝑢^𝑘₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₁+𝑘₂𝑢^𝑘₃+𝑢^𝑘₃+𝛽_𝑇𝜃^𝑘,𝜎^𝑘₂₂=1𝜆+2𝜇𝐴₂𝜕𝑢^𝑘₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₁+𝑘₂𝑢^𝑘₃1+𝜆𝐴₁𝜕𝑢^𝑘₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₂+𝑘₁𝑢^𝑘₃+𝑢^𝑘₃+𝛽_𝑇𝜃^𝑘,𝜎^𝑘₃₃=𝜆+2𝜇𝑢^𝑘₃1+𝜆𝐴₁𝜕𝑢^𝑘₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₂+𝑘₁𝑢^𝑘₃+1𝐴₂𝜕𝑢^𝑘₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₁+𝛽_𝑇𝜃^𝑘,𝜎^𝑘₁₂1=𝜇𝐴₂𝜕𝑢^𝑘₁𝜕𝑥₂−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₂+1𝐴₁𝜕𝑢^𝑘₂𝜕𝑥₁−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₁,𝜎^𝑘₁₃1=𝜇𝐴₁𝜕𝑢^𝑘₃𝜕𝑥₁−𝑘₁𝑢^𝑘₁+𝑢^𝑘₁,𝜎^𝑘₂₃1=𝜇𝐴₂𝜕𝑢^𝑘₃𝜕𝑥₂−𝑘₂𝑢^𝑘₂+𝑢^𝑘₂. (32)

Substitution of these equations in the equations of equilibrium (26) gives us the 2-D equations in displacements in the form ^∞_𝑙=0𝐿^𝑘𝑙_𝑖𝑗𝑢^𝑙_𝑗−𝜎^𝑘_𝑖3+𝐿^𝑘_𝑖𝜃^𝑘+𝐴₁𝐴₂𝑓^𝑘_𝑖=𝜌𝐴₁𝐴₂𝑢̈^𝑘_𝑖,𝜆_𝑇Δ𝜃^𝑘+𝑄^𝑘₃+𝜒^𝑘=𝑐_𝜀𝜃̇^𝑘+𝛽_𝑇𝜃₀∇⋅𝐮̇^𝑘+2𝑘+1ℎ𝑢̇₃^𝑘+1+𝑢̇₃^𝑘+3⋯, (33) where 𝑄^𝑘₃=2𝑘+1𝑄2ℎ⁺³−(−1)^𝑘𝑄⁻₃−1𝐴₁𝐴₂𝑄^𝑘₃+𝑘₁+𝑘₂𝑄^𝑘₃. (34)

Now instead of the finite 3-D system of the differential equations in displacements (14) we have an infinite system of 2-D differential equations for coefficients of the Legendre’s polynomial series expansion. In order to simplify the problem approximate theory has to be developed, and only the finite number of members has to be taken into account in the expansion (20) and in all the previous relations. For example, if we consider 𝑛-order approximate shell theory, only 𝑛+1 members in the expansion (20) are taken into account: 𝑢_𝑖(𝐱,𝑡)=^𝑛_𝑘=0𝑢^𝑘_𝑖𝐱_𝛼,𝑡𝑃_𝑘(𝜔),𝜎_𝑖𝑗(𝐱,𝑡)=^𝑛_𝑘=0𝜎^𝑘_𝑖𝑗𝐱_𝛼,𝑡𝑃_𝑘(𝜔),𝜀_𝑖𝑗(𝐱,𝑡)=^𝑛_𝑘=0𝜀^𝑘_𝑖𝑗𝐱_𝛼,𝑡𝑃_𝑘(𝜔),𝜃(𝐱,𝑡)=^𝑛_𝑘=0𝜃^𝑘𝐱_𝛼,𝑡𝑃_𝑘(𝜔). (35) In this case we consider that 𝑢^𝑘_𝑖=0,𝜎^𝑘_𝑖𝑗=0,𝜀^𝑘_𝑖𝑗=0 and 𝜃^𝑘=0 for 𝑘<0 and for 𝑘>𝑛.

Order of the system of differential equations depends on assumption regarding thickness distribution of the stress-strain parameters of the shell.

4. Vekua’s Shell Equations

In the case if only the first two terms of the Legendre polynomials series are considered in the expansion (20) we have the first approximation shell theory which is usually referred to as Vekua’s shell theory. In this case the thermodynamic parameters, which describe the state of the shell, can be presented in the form 𝜎_𝑖𝑗(𝐱,𝑡)=𝜎⁰_𝑖𝑗𝐱_𝛼,𝑡𝑃₀(𝜔)+𝜎¹_𝑖𝑗𝐱_𝛼,𝑡𝑃₁(𝜔),𝑢_𝑖(𝐱,𝑡)=𝑢⁰_𝑖𝐱_𝛼,𝑡𝑃₀(𝜔)+𝑢¹_𝑖𝐱_𝜈,𝑡𝑃₁(𝜔),𝜀_𝑖𝑗(𝐱,𝑡)=𝜀⁰_𝑖𝑗𝐱_𝛼,𝑡𝑃₀(𝜔)+𝜀¹_𝑖𝑗𝐱_𝛼,𝑡𝑃₁(𝜔),𝜃(𝐱,𝑡)=𝜃⁰𝐱_𝛼,𝑡𝑃₀(𝜔)+𝜃¹𝐱_𝛼,𝑡𝑃₁(𝜔), (36) where coefficients of the expansion are 𝑢⁰_𝑖𝐱_𝛼=1,𝑡2ℎ^ℎ_−ℎ𝑢_𝑖𝐱_𝛼,𝑥₃,𝑡𝑑𝑥₃,𝑢¹_𝑖𝐱_𝛼=3,𝑡2ℎ^ℎ_−ℎ𝑢_𝑖𝐱_𝛼,𝑥₃,𝑡𝑥₃𝑑𝑥₃,𝜀⁰_𝑖𝑗𝐱_𝛼=1,𝑡2ℎ^ℎ_−ℎ𝜀_𝑖𝑗𝐱_𝛼,𝑥₃,𝑡𝑑𝑥₃,𝜀¹_𝑖𝑗𝐱_𝛼=3,𝑡2ℎ^ℎ_−ℎ𝜀_𝑖𝑗𝐱_𝛼,𝑥₃,𝑡𝑥₃𝑑𝑥₃,𝜎⁰_𝑖𝑗𝐱_𝛼=1,𝑡2ℎ^ℎ_−ℎ𝜎_𝑖𝑗𝐱_𝛼,𝑥₃,𝑡𝑑𝑥₃,𝜎¹_𝑖𝑗𝐱_𝛼=3,𝑡2ℎ^ℎ_−ℎ𝜎_𝑖𝑗𝐱_𝛼,𝑥₃,𝑡𝑥₃𝑑𝑥₃,𝜃⁰𝐱_𝛼=1,𝑡2ℎ^ℎ_−ℎ𝜃𝐱_𝛼,𝑥₃,𝑡𝑑𝑥₃,𝜃¹𝐱_𝛼=3,𝑡2ℎ^ℎ_−ℎ𝜃𝐱_𝛼,𝑥₃,𝑡𝑥₃𝑑𝑥₃. (37) The equations of motion (26) in this case have the form 𝜕𝐴₂𝜎⁰₁₁𝜕𝑥₁+𝜕𝐴₁𝜎⁰₁₂𝜕𝑥₂+𝜎⁰₁₂𝜕𝐴₁𝜕𝑥₂+𝜎⁰₁₃𝐴₁𝐴₂𝑘₁−𝜎⁰₂₂𝜕𝐴₂𝜕𝑥₁−𝜎⁰₁₃+𝐴₁𝐴₂𝑓⁰₁=𝜌𝐴₁𝐴₂𝑢̈⁰₁,𝜕𝐴₂𝜎⁰₂₁𝜕𝑥₁+𝜕𝐴₁𝜎⁰₂₂𝜕𝑥₂+𝜎⁰₁₂𝜕𝐴₂𝜕𝑥₁+𝜎⁰₂₃𝐴₁𝐴₂𝑘₂−𝜎⁰₁₁𝜕𝐴₂𝜕𝑥₁−𝜎⁰₂₃+𝐴₁𝐴₂𝑓⁰₂=𝜌𝐴₁𝐴₂𝑢̈⁰₂,𝜕𝐴₂𝜎⁰₃₁𝜕𝑥₁+𝜕𝐴₁𝜎⁰₃₂𝜕𝑥₂−𝜎⁰₁₁𝐴₁𝐴₂𝑘₁−𝜎⁰₂₂𝐴₁𝐴₂𝑘₂−𝜎⁰₃₃+𝐴₁𝐴₂𝑓⁰₃=𝜌𝐴₁𝐴₂𝑢̈⁰₃,𝜕𝐴₂𝜎¹₁₁𝜕𝑥₁+𝜕𝐴₁𝜎¹₁₂𝜕𝑥₂+𝜎¹₁₂𝜕𝐴₁𝜕𝑥₂+𝜎¹₁₃𝐴₁𝐴₂𝑘₁−𝜎¹₂₂𝜕𝐴₂𝜕𝑥₁−𝜎¹₁₃+𝐴₁𝐴₂𝑓¹₁=𝜌𝐴₁𝐴₂𝑢̈¹₁,𝜕𝐴₂𝜎¹₂₁𝜕𝑥₁+𝜕𝐴₁𝜎¹₂₂𝜕𝑥₂+𝜎¹₁₂𝜕𝐴₂𝜕𝑥₁+𝜎¹₂₃𝐴₁𝐴₂𝑘₂−𝜎¹₁₁𝜕𝐴₂𝜕𝑥₁−𝜎¹₂₃+𝐴₁𝐴₂𝑓¹₂=𝜌𝐴₁𝐴₂𝑢̈¹₂,𝜕𝐴₂𝜎¹₃₁𝜕𝑥₁+𝜕𝐴₁𝜎¹₃₂𝜕𝑥₂−𝜎¹₁₁𝐴₁𝐴₂𝑘₁−𝜎¹₂₂𝐴₁𝐴₂𝑘₂−𝜎¹₃₃+𝐴₁𝐴₂𝑓¹₃=𝜌𝐴₁𝐴₂𝑢̈¹₃. (38) The kinematic Cauchy relations (28) have the form 𝜀⁰₁₁=1𝐴₁𝜕𝑢⁰₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢⁰₂+𝑘₁𝑢⁰₃,𝜀⁰₂₂=1𝐴₂𝜕𝑢⁰₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢⁰₁+𝑘₂𝑢⁰₃,𝜀⁰₁₂=1𝐴₂𝜕𝑢⁰₁𝜕𝑥₂−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢⁰₂+1𝐴₁𝜕𝑢⁰₂𝜕𝑥₁−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢⁰₁,𝜀⁰₁₃=1𝐴₁𝜕𝑢⁰₃𝜕𝑥₁−𝑘₁𝑢⁰₁+1ℎ𝑢¹₁,𝜀⁰₂₃=1𝐴₂𝜕𝑢⁰₃𝜕𝑥₂−𝑘₂𝑢⁰₂+1ℎ𝑢¹₂,𝜀⁰₃₃=1ℎ𝑢¹₃,𝜀¹₁₁=1𝐴₁𝜕𝑢¹₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢¹₂+𝑘₁𝑢¹₃,𝜀¹₂₂=1𝐴₂𝜕𝑢¹₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢¹₁+𝑘₂𝑢¹₃,𝜀¹₁₂=1𝐴₂𝜕𝑢¹₁𝜕𝑥₂−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢¹₂+1𝐴₁𝜕𝑢¹₂𝜕𝑥₁−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢¹₁,𝜀¹₁₃=1𝐴₁𝜕𝑢¹₃𝜕𝑥₁−𝑘₁𝑢¹₁,𝜀¹₂₃=1𝐴₂𝜕𝑢¹₃𝜕𝑥₂−𝑘₂𝑢¹₂,𝜀¹₃₃=0. (39) The generalized Hooke’s law for homogeneous isotropic material (30) has the form 𝜎⁰₁₁=𝜆+2𝜇𝜀⁰₁₁+𝜆𝜀⁰₂₂+𝜀⁰₃₃+𝛽_𝑇𝜃⁰,𝜎⁰₂₂=𝜆+2𝜇𝜀⁰₂₂+𝜆𝜀⁰₁₁+𝜀⁰₃₃+𝛽_𝑇𝜃⁰,𝜎⁰₃₃=𝜆+2𝜇𝜀⁰₂₂+𝜆𝜀⁰₁₁+𝜀⁰₂₂+𝛽_𝑇𝜃⁰,𝜎⁰₁₂=𝜇𝜀⁰₁₂,𝜎⁰₁₃=𝜇𝜀⁰₁₃,𝜎⁰₂₃=𝜇𝜀⁰₂₃,𝜎¹₁₁=𝜆+2𝜇𝜀¹₁₁+𝜆𝜀¹₂₂+𝜀¹₃₃+𝛽_𝑇𝜃¹,𝜎¹₂₂=𝜆+2𝜇𝜀¹₂₂+𝜆𝜀¹₁₁+𝜀¹₃₃+𝛽_𝑇𝜃¹,𝜎¹₃₃=𝜆+2𝜇𝜀¹₂₂+𝜆𝜀¹₁₁+𝜀¹₂₂+𝛽_𝑇𝜃¹,𝜎¹₁₂=𝜇𝜀¹₁₂,𝜎¹₁₃=𝜇𝜀¹₁₃,𝜎¹₂₃=𝜇𝜀¹₂₃. (40)

Substitution of the kinematic Cauchy relations (39) with the generalized Hooke’s law (40) and the result of the equations of motion (38) give us the 2-D equations in displacements for Vekua’s shell theory in the form 𝐿⁰⁰_𝑖𝑗𝑢⁰_𝑗+𝐿⁰¹_𝑖𝑗𝑢¹_𝑗+𝐿⁰_𝑖𝜃⁰+𝐴₁𝐴₂𝑏^𝑘_𝑖=𝜌𝐴₁𝐴₂𝑢̈^𝑘_𝑖,𝐿¹⁰_𝑖𝑗𝑢⁰_𝑗+𝐿¹¹_𝑖𝑗𝑢¹_𝑗+𝐿¹_𝑖𝜃¹+𝐴₁𝐴₂𝑏^𝑘_𝑖=𝜌𝐴₁𝐴₂𝑢̈^𝑘_𝑖, (41) and the equation of heat conductivity in the form 𝜆_𝑇Δ𝜃⁰+𝑄⁰₃+𝜒⁰=𝑐_𝜀𝜃̇⁰+𝛽_𝜏𝜃₀∇⋅𝐮̇⁰+1ℎ𝑢̇¹₃,𝜆_𝑇Δ𝜃¹+𝑄¹₃+𝜒¹=𝑐_𝜀𝜃̇¹+𝛽_𝜏𝜃₀∇⋅𝐮̇¹, (42) where 𝑄⁺₃,𝑄⁻₃,𝑄⁰₃, and 𝑄¹₃ depend on thermal conditions on the outer sides Ω⁺ and Ω⁻ of the shell. In the case if temperatures 𝜃⁺ and 𝜃⁻ are prescribed on Ω⁺ and Ω⁻, respectively, they have the form 𝑄⁺₃−𝑄⁻₃=3𝜃4ℎ⁺+𝜃⁻+3𝜃⁰2ℎ,𝑄⁰₃=1𝜃2ℎ⁺−𝜃⁻,𝑄⁺₃+𝑄⁻₃=3𝜃2ℎ⁺−𝜃⁻−5𝜃¹2ℎ,𝑄¹₃=3𝜃2ℎ⁺+𝜃⁻−3𝜃¹.2ℎ (43)

Differential operators that appear in the equations of thermoelasticity (41) and heat conductivity (42) for shells of arbitrary geometry are presented in the Appendix Section.

5. Timoshenko’s Shell Equations

Timoshenko's theory of shells is based on assumptions concerning the value and distribution of the stress-strain state of the shell. Thus, according to static assumptions 𝜎₃₃=0 and according to kinematic assumptions 𝜀₃₃=0. In this theory the thermodynamic state of shells is determined by quantities specified on the middle surface. The stress state is characterized by the normal 𝑛_𝛼𝛼, tangential 𝑛_𝛼𝛽(𝛼≠𝛽), and shear 𝑛_𝛼3 forces, as well as the bending 𝑚_𝛼𝛼 and twisting 𝑚_𝛼𝛽(𝛼≠𝛽) moments. They are defined as follows: 𝑛_𝛼𝛽=^ℎ_−ℎ𝜎_𝛼𝛽𝑑𝑥₃,𝑛_3𝛽=^ℎ_−ℎ𝜎_3𝛽𝑑𝑥₃,𝑚_𝛼𝛽=^ℎ_−ℎ𝜎_𝛼𝛽𝑥₃𝑑𝑥₃. (44) Comparison with (37) gives us the following relation between corresponding parameters in Vekua’s and Timoshenko’s theories: 𝜎⁰_𝑖𝛽∼𝑛_𝑖𝛽2ℎ,𝜎¹_𝛼𝛽∼3𝑚_𝛼𝛽2ℎ². (45) Components 𝜎⁰₃₃ and 𝜎¹₃₃ are not taken into account in Timoshenko's theory of shells. That follows also from the static hypothesis.

Displacements in the Timoshenko's theory of shells are defined by vectors 𝐮(𝐱_𝛼,𝑡) and 𝜸(𝐱_𝛼,𝑡) with components 𝑢_𝑖,𝑖=1,2,3 and 𝛾_𝛼,𝛼=1,2, respectively. They correspond to displacements of the middle surface and rotation of the elements perpendicular to the middle surface in the planes (𝑥_𝛼,𝑥₃). These parameters are related to the coefficients of the displacements expansion in the Vekua’s theory in the following way: 𝑢⁰_𝑖∼𝑢_𝑖,𝑢¹_𝛼∼𝛾_𝛼ℎ. (46) Component 𝑢¹₃ is not taken into account in Timoshenko's theory of shells.

Deformations in Timoshenko’s theory are determined by the relations 𝜀_𝛼𝛽=𝑒_𝛼𝛽+𝜅_𝛼𝛽𝑥₃,𝜀_𝛼3=𝑒_𝛼3. (47) Roughly speaking components 𝑒_𝛼𝛽 correspond to the tension-compression deformation of the middle surface, components 𝑒_𝛼3 to the transversal shear deformation, and components 𝜅_𝛼𝛽 to the bending and twisting middle surface, respectively. The following formulas give us relations with corresponding quantities in Vekua’s theory: 𝜀⁰_𝛼𝑖∼𝑒_𝛼𝑖,𝜀¹_𝛼𝛽∼𝜅_𝛼𝛽. (48) Component 𝜀⁰₃₃ and 𝜀¹₃₃ are not taken into account in the Timoshenko's theory of shells. That follows also from the kinematic hypothesis.

The kinematic Cauchy relations in Timoshenko's theory of shells have the following form: 𝑒₁₁=1𝐴₁𝜕𝑢₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₂+𝑘₁𝑢₃,𝑒₂₂=1𝐴₂𝜕𝑢₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₁+𝑘₂𝑢₃,𝑒₁₂=1𝐴₂𝜕𝑢₁𝜕𝑥₂−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₂+1𝐴₁𝜕𝑢₂𝜕𝑥₁−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₁,𝑒₁₃=1𝐴₁𝜕𝑢₃𝜕𝑥₁−𝑘₁𝑢₁+1ℎ𝛾₁,𝜀⁰₂₃=1𝐴₂𝜕𝑢₃𝜕𝑥₂−𝑘₂𝑢₂+1ℎ𝛾₂,𝜀⁰₃₃𝜅=0,₁₁=1𝐴₁𝜕𝛾₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝛾₂,𝜅₂₂=1𝐴₂𝜕𝛾₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝛾₁,𝜅₁₂=1𝐴₁𝜕𝛾₂𝜕𝑥₁−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝛾₁+𝑘₁𝜔₂,𝜅₂₁=1𝐴₂𝜕𝛾₁𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝛾₂+𝑘₂𝜔₁,𝜔₁=1𝐴₁𝜕𝑢₂𝜕𝑥₁−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₁,𝜔₂=1𝐴₂𝜕𝑢₁𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₂. (49)

The equations of motion have the form 𝜕𝐴₂𝑛₁₁𝜕𝑥₁+𝜕𝐴₁𝑛₁₂𝜕𝑥₂+𝑛₁₂𝜕𝐴₁𝜕𝑥₂−𝑛₂₂𝜕𝐴₂𝜕𝑥₁+𝑛₁₃𝐴₁𝐴₂𝑘₁+𝐴₁𝐴₂𝑓^𝑘₁=𝜌𝐴₁𝐴₂𝑢̈₁,𝜕𝐴₂𝑛₁₂𝜕𝑥₁+𝜕𝐴₁𝑛₂₂𝜕𝑥₂+𝑛₂₁𝜕𝐴₂𝜕𝑥₁−𝑛₁₁𝜕𝐴₂𝜕𝑥₁+𝑛₂₃𝐴₁𝐴₂𝑘₂+𝐴₁𝐴₂𝑓^𝑘₂=𝜌𝐴₁𝐴₂𝑢̈₂,𝜕𝐴₂𝑛₃₁𝜕𝑥₁+𝜕𝐴₁𝑛₃₂𝜕𝑥₂−𝑛₁₁𝐴₁𝐴₂𝑘₁−𝑛₂₂𝐴₁𝐴₂𝑘₂+𝐴₁𝐴₂𝑓^𝑘₃=𝜌𝐴₁𝐴₂𝑢̈₃,𝜕𝐴₂𝑚₁₁𝜕𝑥₁+𝜕𝐴₁𝑚₁₂𝜕𝑥₂+𝑚₁₂𝜕𝐴₁𝜕𝑥₂−𝑚₂₂𝜕𝐴₂𝜕𝑥₁−𝑛₁₃𝐴₁𝐴₂=𝜌𝐴₁𝐴₂ℎ𝛾̈₁,𝜕𝐴₂𝑚₂₁𝜕𝑥₁+𝜕𝐴₁𝑚₂₂𝜕𝑥₂+𝑚₂₁𝜕𝐴₂𝜕𝑥₁−𝑚₁₁𝜕𝐴₂𝜕𝑥₁−𝑛₂₃𝐴₁𝐴₂=𝜌𝐴₁𝐴₂ℎ𝛾̈₂,𝑛₁₂+𝑚₁₂𝑘₁=𝑛₂₁+𝑚₂₁𝑘₂. (50)

The generalized Hooke’s law for Timoshenko's theory of shells has the form 𝑛_𝛼𝛼=2𝐸ℎ1−𝜐²𝑒_𝛼𝛼+𝜐𝑒_𝛽𝛽−(1+𝜐)𝛼_𝑇𝜃⁰,𝑛₁₂=𝐸ℎ𝑒1+𝜐¹²,𝑛_𝛼3=𝐸ℎ𝑒1+𝜐^𝛼3,𝑚_𝛼𝛼=2𝐸ℎ1−𝜐²𝜅_𝛼𝛼+𝜐𝜅_𝛽𝛽−(1+𝜐)𝛼_𝑇𝜃¹.𝑚₁₂=2𝐸ℎ³3()𝑒1+𝜐¹². (51) Such form of the generalized Hooke’s law follows from the static hypothesis according to which 𝜎₃₃=0, and therefore 𝜀₃₃𝜐=−𝜀1−𝜐¹¹+𝜀₂₂. (52)

Substituting kinematic relations (49) with generalized Hooke’s law (51) and considering the result of the equations of motion (50) we obtain the following system of the differential equations in the form of displacements: 𝐿⁰⁰_𝑖𝑗𝑢_𝑗+𝐿⁰¹_𝑖𝛽𝛾_𝛽+𝐿⁰_𝑖𝜃⁰+𝐴₁𝐴₂𝑏_𝑖=𝜌𝐴₁𝐴₂𝑢̈_𝑖,𝐿¹⁰_𝛼𝑗𝑢_𝑗+𝐿¹¹_𝛼𝛽𝛾_𝛽+𝐿¹_𝛼𝜃¹+𝐴₁𝐴₂𝑚_𝛼=𝜌𝐴₁𝐴₂ℎ𝛾̈_𝛼. (53)

Differential operators that appear in (53) for shells of arbitrary geometry are presented in the Appendix Section.

The equations of heat conductivity in Timoshenko's theory of shells have the same form as in the Vekua’s shells theory, that is, defined by (42).

Components of the stress tensor can be calculated from the equations 𝜎_𝛼𝛽(𝐱)=𝑛_𝛼𝛽𝐱_𝛼+2ℎ3𝑚_𝛼𝛽𝐱_𝛼𝑥₃ℎ³,𝜎_𝛼3(𝐱)=𝑛_𝛼3𝐱_𝛼2ℎ,𝜎₃₃(𝐱)=0. (54)

The equations presented here allow us calculate stress-strain state of the shells under static and dynamic action of mechanical and thermal load.

6. Kirchhoff-Love’s Shell Equations

In the classical Kirchhoff-Love's theory of shells in addition to the assumptions of Timoshenko's theory it is assumed that 𝜀_𝛼3=0 and that the angles of rotation of the normal-to-middle surface vector become dependent; they are given by the equations 𝑢¹_𝛼1=−𝐴_𝛼𝜕𝑢⁰₃𝜕𝑥_𝛼+𝑘_𝛼𝑢⁰_𝛼,or𝛾_𝛼1=−𝐴_𝛼𝜕𝑢₃𝜕𝑥_𝛼+𝑘_𝛼𝑢_𝛼. (55)

Substituting (55) with the kinematic equations for Timoshenko's shell theory we obtain kinematic equations for Kirchhoff-Love's shell theory in the form 𝑒₁₁=1𝐴₁𝜕𝑢₁𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₂+𝑘₁𝑢₃,𝑒₂₂=1𝐴₂𝜕𝑢₂𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₁+𝑘₂𝑢₃,𝑒₁₂=1𝐴₂𝜕𝑢₁𝜕𝑥₂−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₂+1𝐴₁𝜕𝑢₂𝜕𝑥₁−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₁,𝜅₁₁=1𝐴₁𝜕𝛾₁𝜕𝑥₁−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂1𝐴₂𝜕𝑢₃𝜕𝑥₂+𝑘₂𝑢₂,𝜅₂₂=1𝐴₂𝜕𝛾₂𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁1𝐴₁𝜕𝑢₃𝜕𝑥₁−𝑘₁𝑢₁,𝜅₁₂=1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂1𝐴₁𝜕𝑢₃𝜕𝑥₁−𝑘₁𝑢₁−1𝐴₁𝜕𝛾₂𝜕𝑥₁1𝐴₂𝜕𝑢₃𝜕𝑥₂−𝑘₂𝑢₂+𝑘₁𝜔₂,𝜅₂₁=1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁1𝐴₂𝜕𝑢₃𝜕𝑥₂−𝑘₂𝑢₂−1𝐴₂𝜕𝜕𝑥₂1𝐴₁𝜕𝑢₃𝜕𝑥₁−𝑘₁𝑢₁+𝑘₂𝜔₁,𝜔₁=1𝐴₁𝜕𝑢₂𝜕𝑥₁−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₁,𝜔₂=1𝐴₂𝜕𝑢₁𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₂. (56) From two last equations of motion for Timoshenko's shell theory (50) we can find 𝑛₁₃=1𝐴₁𝐴₂𝜕𝐴₂𝑚₁₁𝜕𝑥₁+𝜕𝐴₁𝑚₂₁𝜕𝑥₂+𝑚₁₂𝜕𝐴₁𝜕𝑥₂−𝑚₂₂𝜕𝐴₂𝜕𝑥₁,𝑛₂₃=1𝐴₁𝐴₂𝜕𝐴₂𝑚₁₂𝜕𝑥₁+𝜕𝐴₁𝑚₂₂𝜕𝑥₂+𝑚₂₁𝜕𝐴₂𝜕𝑥₁−𝑚₁₁𝜕𝐴₁𝜕𝑥₂. (57) Substituting them with other equations of motion (50) we obtain 𝜕𝐴₂𝑛₁₁𝜕𝑥₁+𝜕𝐴₁𝑛₁₂𝜕𝑥₂+𝑛₁₂𝜕𝐴₁𝜕𝑥₂−𝑛₂₂𝜕𝐴₂𝜕𝑥₁+𝑘₁𝜕𝐴₂𝑚₁₁𝜕𝑥₁+𝜕𝐴₁𝑚₂₁𝜕𝑥₂+𝑚₁₂𝜕𝐴₁𝜕𝑥₂−𝑚₂₂𝜕𝐴₂𝜕𝑥₁+𝐴₁𝐴₂𝑓^𝑘₁=𝜌𝐴₁𝐴₂𝑢̈₁,𝜕𝐴₂𝑛₁₂𝜕𝑥₁+𝜕𝐴₁𝑛₂₂𝜕𝑥₂+𝑛₂₁𝜕𝐴₂𝜕𝑥₁−𝑛₁₁𝜕𝐴₂𝜕𝑥₁+𝑘₂𝜕𝐴₂𝑚₁₂𝜕𝑥₁+𝜕𝐴₁𝑚₂₂𝜕𝑥₂+𝑚₂₁𝜕𝐴₂𝜕𝑥₁−𝑚₁₁𝜕𝐴₁𝜕𝑥₂+𝐴₁𝐴₂𝑓^𝑘₂=𝜌𝐴₁𝐴₂𝑢̈₂,𝜕𝜕𝑥₁1𝐴₁𝜕𝐴₂𝑚₁₁𝜕𝑥₁+𝜕𝐴₁𝑚₂₁𝜕𝑥₂+𝑚₁₂𝜕𝐴₁𝜕𝑥₂−𝑚₂₂𝜕𝐴₂𝜕𝑥₁+𝜕𝜕𝑥₂1𝐴₂𝜕𝐴₂𝑚₁₂𝜕𝑥₁+𝜕𝐴₁𝑚₂₂𝜕𝑥₂+𝑚₂₁𝜕𝐴₂𝜕𝑥₁−𝑚₁₁𝜕𝐴₁𝜕𝑥₂−𝑛₁₁𝐴₁𝐴₂𝑘₁−𝑛₂₂𝐴₁𝐴₂𝑘₂+𝐴₁𝐴₂𝑓^𝑘₃=𝜌𝐴₁𝐴₂𝑢̈₃. (58)

In the same way substituting kinematic relations (56) with generalized Hooke’s law (51) and considering the result of the equations of motion (58) we obtain the following system of the differential equations in the form of displacements for Kirchhoff-Love's shell theory: 𝐿_𝑖𝑗𝑢_𝑗+¹_𝑘=0𝐿^𝑘_𝑖𝜃^𝑘+𝐴₁𝐴₂𝑏_𝑖=𝜌𝐴₁𝐴₂𝑢̈_𝑖. (59)

Differential operators that appear in (59) for shells of arbitrary geometry have more complicated form; therefore they were not presented here.

The equations of heat conductivity in Kirchhoff-Love's theory of shells have the same form as in Vekua’s shells theory, that is, defined by (42).

Components of the stress tensor can be calculated in the same way as for Timoshenko's shell theory using (54).

7. Comparative Study and Applications

Developed here is high order shell theory that gives us the full system of the equations for studying thermodynamical state of the thin-walled structures with high accuracy. We can keep in the polynomial expansion (20) as many terms as it is necessary for approximation of the stress-strain state with necessary accuracy. The only problem that may occur is that, on one hand more members one keeps, the more accurate results he has on the other hand the more members one keeps, the more complicated the system of equations he has to solve. Therefore general suggestion is that one has to keep as many members as it is necessary for getting results with necessary accuracy, but not more. We consider here in more detail first order theory which takes into account two members in the expansion (20) that theory is usually referred to as Vekua’s shell theory. Let us compare Vekua’s shell theory with classical Timoshenko’s and Kirchhoff-Love’s shell theories.

Comparison of Vekua’s, Timoshenko’s, and Kirchhoff-Love’s shell theories has been done in Table 1.

Table 1 

From Table 1 it follows that Vekua’s shell theory considers all components of the stress-strain state in linear approximation. Classical theories are based on physical assumptions regarding stress and strain distribution. For example, according to static hypothesis 𝜎₃₃=0 and according to kinematic hypothesis 𝜀₃₃=0, and therefore components 𝜎⁰₃₃,𝜎¹₃₃, and 𝜀⁰₃₃,𝜀¹₃₃ have no analogy in Timoshenko’s and Kirchhoff-Love’s shell theories. But on the other hand in classical theories 𝜀₃₃ is defined from the Hooke’s law using (52). Also in Kirchhoff-Love’s shell theory 𝜀_𝛼3=0 and in accordance with Hooke’s law 𝜎_𝛼3=0, and therefore 𝑛_𝛼3=0. But in fact 𝜎_𝛼3 and 𝑛_𝛼3 are defined from the equations of equilibrium. Unfortunately classical theories contain some logical contradictions; nevertheless they are very popular and are frequently used in engineering applications.

Systems of differential equations in displacements for Vekua’s (41), Timoshenko’s (53), and Kirchhoff-Love’s (59) theories for the case of arbitrary geometry shells are very complicated because of the structure of the differential operator presented in the Appendix Section. In order to simplify analysis of the structure of those equations, let us consider how they look for plates and cylindrical shells.

7.1. Equations in the Form of Displacements for Plates

All equations that specify thermodynamical state of the plates can be obtained from the general equations. In this case one has to substitute 𝐴₁=𝐴₂=1 and 𝑘₁=𝑘₂=0 and consider all the equations in Cartesian system of coordinates in order to obtain corresponding equations for the plates theory. We present in the following only differential equations in displacements for theories considered here.

For Vekua’s theory differential equations in displacements have the form 𝜇Δ𝑢⁰₁+𝜕𝜆+𝜇𝜕𝑥₁𝜕𝑢⁰₁𝜕𝑥₁+𝜕𝑢⁰₂𝜕𝑥₂+𝜆ℎ𝜕𝑢¹₃𝜕𝑥₁+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₁+𝑓⁰₁=𝜌𝑢̈⁰₁,𝜇Δ𝑢⁰₂+𝜕𝜆+𝜇𝜕𝑥₂𝜕𝑢⁰₁𝜕𝑥₁+𝜕𝑢⁰₂𝜕𝑥₂+𝜆ℎ𝜕𝑢¹₃𝜕𝑥₂+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₂+𝑓⁰₂=𝜌𝑢̈⁰₂,𝜇Δ𝑢⁰₃+𝜇ℎ𝜕𝑢¹₁𝜕𝑥₁+𝜕𝑢¹₂𝜕𝑥₂+𝑓⁰₃=𝜌𝑢̈⁰₃,𝜇Δ𝑢¹₁+𝜕𝜆+𝜇𝜕𝑥₁𝜕𝑢¹₁𝜕𝑥₁+𝜕𝑢¹₂𝜕𝑥₂−3𝜇ℎ𝜕𝑢⁰₃𝜕𝑥₁+1ℎ𝑢¹₁+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₁+𝜕𝜃¹𝜕𝑥₁+𝑓¹₁=𝜌𝑢̈¹₁,𝜇Δ𝑢¹₂+𝜕𝜆+𝜇𝜕𝑥₂𝜕𝑢¹₁𝜕𝑥₁+𝜕𝑢¹₂𝜕𝑥₂−3𝜇ℎ𝜕𝑢⁰₃𝜕𝑥₂+1ℎ𝑢¹₂+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₂+𝜕𝜃¹𝜕𝑥₂+𝑓¹₂=𝜌𝑢̈¹₂,𝜇Δ𝑢¹₃−3𝜆ℎ𝜕𝑢⁰₁𝜕𝑥₁+𝜕𝑢⁰₂𝜕𝑥₂+3𝜆+2𝜇ℎ²𝑢¹₃−3𝛽_𝑇ℎ𝜃⁰+𝑓¹₃=𝜌𝑢̈¹₃. (60)

For Timoshenko’s theory differential equations in displacements have the form 𝜇Δ𝑢₁+𝜕𝜆+𝜇𝜕𝑥₁𝜕𝑢₁𝜕𝑥₁+𝜕𝑢₂𝜕𝑥₂+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₁+𝑓⁰₁=𝜌𝑢̈⁰₁,𝜇Δ𝑢₂+𝜕𝜆+𝜇𝜕𝑥₂𝜕𝑢₁𝜕𝑥₁+𝜕𝑢₂𝜕𝑥₂+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₂+𝑓⁰₂=𝜌𝑢̈⁰₂,𝜇Δ𝑢₃+𝜇𝜕𝛾₁𝜕𝑥₁+𝜕𝛾₂𝜕𝑥₂+𝑓₃=𝜌𝑢̈₃,𝜇Δ𝛾₁+𝜕𝜆+𝜇𝜕𝑥₁𝜕𝛾₁𝜕𝑥₁+𝜕𝛾₂𝜕𝑥₂−3𝜇ℎ𝜕𝑢₃𝜕𝑥₁+𝛾₁+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₁+𝜕𝜃¹𝜕𝑥₁+𝑓¹₁=𝜌𝛾̈₁,𝜇Δ𝛾₂+𝜕𝜆+𝜇𝜕𝑥₂𝜕𝛾₁𝜕𝑥₁+𝜕𝛾₂𝜕𝑥₂−3𝜇ℎ𝜕𝑢₃𝜕𝑥₂+𝛾₂+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₂+𝜕𝜃¹𝜕𝑥₂+𝑓¹₂=𝜌𝛾̈₂. (61)

For Kirchhoff theory the differential equations in displacements have the form 𝜇Δ𝑢₁+𝜕𝜆+𝜇𝜕𝑥₁𝜕𝑢₁𝜕𝑥₁+𝜕𝑢₂𝜕𝑥₂+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₁+𝑓₁=𝜌𝑢̈⁰₁,𝜇Δ𝑢₂+𝜕𝜆+𝜇𝜕𝑥₂𝜕𝑢₁𝜕𝑥₁+𝜕𝑢₂𝜕𝑥₂+𝛽_𝑇𝜕𝜃⁰𝜕𝑥₂+𝑓₂=𝜌𝑢̈⁰₂,𝜇ΔΔ𝑢₃+𝛽_𝑇Δ𝜃¹+𝑓₃=𝜌𝑢̈₃. (62)

System of differential equations for Vekua’s theory (60) is coupled, and all equations have to be solved together. In Timoshenko’s and Kirchhoff’s theories the first two equations are independent and can be solved separately. Also in Timoshenko’s theory equations for the plate deflection 𝑢₃ and angles of rotation 𝛾_𝛼 are coupled (61) whereas in Kirchhoff’s theory we have only one equation for the plate deflection 𝑢₃ of the from (62).

System of differential equations of the heat conductivity is the same for all plate theories considered here and has the form (42), where differential operators have simpler form in Cartesian system of coordinates.

7.2. Equations in the Form of Displacements for Cylindrical Shells

All equations that specify thermodynamical state of the cylindrical shell can be obtained from general equations presented here. In this case one has to introduce cylindrical coordinates in the following way: 𝑥₁=𝑟𝜉,𝑥₂=𝑟𝜗, and substitute 𝐴₁=𝐴₂=𝑟,𝑘₁=0,𝑘₂=𝑟⁻¹. After that all corresponding equations for the cylindrical shell theory can be easily obtained. Differential equations in displacements for cylindrical shells have the same structure as in general case but differential operators are much simpler. We present them here.

For Vekua’s theory differential operators have the form 𝐿⁰⁰₁₁𝜕=𝜇Δ+𝜆+𝜇²𝜕𝜉²,𝐿⁰⁰₁₂=𝜕𝜆+𝜇²,𝐿𝜕𝜉𝜕𝜗₀₀₁₃𝜕=𝜆𝜕𝜉,𝐿⁰¹₁₁=0,𝐿⁰¹₁₂𝐿=0,₀₁₁₃𝑟=𝜆ℎ𝜕𝜕𝜉,𝐿⁰⁰₂₁=𝜕𝜆+𝜇²,𝐿𝜕𝜉𝜕𝜗₀₀₂₂𝜕=𝜇Δ+𝜆+𝜇²𝜕𝜗²𝐿−𝜇,₀₀₂₃=𝜕𝜆+2𝜇𝜕𝜕𝜗+𝜇,𝐿𝜕𝜗₀₁₂₁=0,𝐿⁰¹₂₂=𝜇𝑟ℎ,𝐿⁰¹₂₃𝑟=𝜆ℎ𝜕𝜕𝜗,𝐿⁰⁰₃₁𝜕=−𝜆,𝐿𝜕𝜉₀₀₃₂𝜕=−𝜇−𝜕𝜕𝜗𝜆+2𝜇,𝐿𝜕𝜗₀₀₃₃,𝐿=−𝜇Δ−𝜆+2𝜇₀₁₃₁𝑟=−𝜇ℎ𝜕𝜕𝜉,𝐿⁰¹₃₂𝑟=−𝜇ℎ𝜕,𝐿𝜕𝜗₀₁₃₃𝑟=−𝜆ℎ,𝐿¹⁰₁₁𝐿=0,₁₀₁₂=0,𝐿¹⁰₁₃=−𝜇3𝑟ℎ𝜕,𝐿𝜕𝜉₁₁₁₁=𝐿⁰⁰₁₁−𝜇3𝑟²ℎ²,𝐿¹¹₁₂=𝐿⁰⁰₁₂,𝐿¹¹₁₃=𝐿⁰⁰₁₃,𝐿¹⁰₂₁=0,𝐿¹⁰₂₂=𝜇3𝑟ℎ,𝐿¹⁰₂₃=−𝜇3𝑟ℎ𝜕𝜕𝜗,𝐿¹¹₂₁=𝐿⁰⁰₁₂,𝐿¹¹₂₂=𝐿⁰⁰₂₂−𝜇3𝑟²ℎ²,𝐿¹¹₂₃=𝐿⁰⁰₂₃,𝐿¹⁰₃₁=−𝜆3𝑟ℎ𝜕𝜕𝜉,𝐿¹⁰₃₂𝑟=−𝜆ℎ𝜕,𝐿𝜕𝜗₁₀₃₃=−𝜆3𝑟ℎ,𝐿¹¹₃₁=𝐿⁰⁰₃₁,𝐿¹¹₃₂=𝐿⁰⁰₃₂,𝐿¹¹₃₃=𝐿⁰⁰₃₃−𝜆+2𝜇3𝑟²ℎ². (63)

For Timoshenko’s theory differential operators have the form 𝐿⁰⁰₁₁=𝐷_𝑛𝜕²𝜕𝜉²+1−𝜐2𝜕²𝜕𝜗²,𝐿⁰⁰₁₂=𝐿⁰⁰₂₁=𝐷_𝑛1+𝜐2𝜕²,𝐿𝜕𝜉𝜕𝜗₀₀₁₃=𝐿⁰⁰₃₁=𝐷_𝑛𝜐𝜕,𝐿𝜕𝜉₀₁₁₁=𝐿¹⁰₁₁=𝐿⁰¹₁₂=𝐿¹⁰₁₁𝐿=0,₀₀₂₂=𝐷_𝑛1−𝜐2𝜕²𝜕𝜉²+𝜕²𝜕𝜗²−1−𝜐2,𝐿⁰¹₂₁=𝐿¹⁰₂₁𝐿=0,₀₀₂₃=−𝐿⁰⁰₃₂=𝐷_𝑛𝜕+𝜕𝜉1−𝜐2𝜕,𝐿𝜕𝜗₀₁₂₂=𝐿¹⁰₂₂=𝐸ℎ𝑟,𝐿1+𝜐₀₁₃₁=−𝐿¹⁰₁₃=𝐸ℎ𝑟𝜕1+𝜐,𝐿𝜕𝜉₀₁₃₂=−𝐿¹⁰₂₃=𝐸ℎ𝑟𝜕1+𝜐,𝐿𝜕𝜗₀₀₃₃=𝐷_𝑛𝑟²1−𝜐21Δ−𝑟²,𝐿¹¹₁₂=𝐿¹¹₂₁=𝐷_𝑚1+𝜐2𝜕²,𝐿𝜕𝜉𝜕𝜗₁₁₁₁=𝐷_𝑚𝜕²𝜕𝜉²+1−𝜐2𝜕²𝜕𝜗²,𝐿¹¹₂₂=𝐷_𝑚1−𝜐2𝜕²𝜕𝜉²+𝜕²𝜕𝜗². (64) Here 𝐷_𝑛=2𝐸ℎ/(1−𝜐²) is a stiffness that corresponds to in-plane deformation, and 𝐷_𝑚=2𝐸ℎ³/ (3(1−𝜐²)) is a stiffness that corresponds to bending deformation.

For Kirchhoff-Love's theory differential operators have the form 𝐿₁₁=𝐷_𝑛𝜕²𝜕𝜉²+1−𝜐2𝜕²𝜕𝜗²,𝐿₁₂=𝐿₂₁=𝐷_𝑛1+𝜐2𝜕²,𝐿𝜕𝜉𝜕𝜗₁₃=−𝐿₃₁=𝐷_𝑛𝜐𝜕,𝐿𝜕𝜉₂₂=𝐷_𝑛1−𝜐2𝜕²𝜕𝜉²+𝜕²𝜕𝜗²−1−𝜐2+𝐷_𝑚1𝑟²2()𝜕1−𝜐²𝜕𝜉²+𝜕²𝜕𝜗²,𝐿₂₃=−𝐿₃₂=𝐷_𝑛𝜕𝜕𝜗−𝐷_𝑚1𝑟²()𝜕2−𝜐³𝜕𝜉²+𝜕𝜕𝜗𝜕𝜉³³,𝐿₃₃=−𝐷_𝑛−𝐷_𝑚ΔΔ. (65)

System of differential equations of the heat conductivity is the same for all cylindrical shell theories considered here and has the form (42), where differential operators have simpler form in cylindrical system of coordinates.

8. Conclusions

In this paper a high order theory for homogeneous thermoelastic shells has been developed. The proposed approach is based on expansion of the coupled equations of thermoelasticity and heat conductivity into Fourier series in terms of Legendre polynomials. Starting from the equations of coupled thermoelasticity and heat conductivity, stress and strain tensors, vectors of displacements, traction and body forces, temperature, and vector of thermal flow have been expanded into Fourier series in terms of Legendre polynomials in a thickness coordinate. Thereby all equations of thermoelasticity including generalized Hooke’s law and equations of heat conductivity have been transformed to corresponding equations for Fourier coefficients of Legendre’s polynomial expansion. The system of differential equations in terms of displacements including initial and boundary conditions for Fourier coefficients has been obtained. Special attention has been paid to the case of the first approximation shells theory. All equations of thermoelasticity and heat conductivity for Vekua’s, Timoshenko’s, and Kirchhoff-Love’s shell theories have been obtained. Differential operator for shells of arbitrary geometry, cylindrical shells, and plates has been obtained and presented in the explicit form. Comparison of Vekua’s, Timoshenko’s, and Kirchhoff-Love’s shell theories has been done, and corresponding equations have been analyzed. Obtained equations can be used for thermomechanical analysis of the shells of any specific geometry and also for numerical calculations.

Appendix

For high order shell theory 𝐿^{𝑘(𝑘−𝑖)}₁₁𝑢₁^𝑘−𝑖=𝜇𝐴₁𝐴₂𝑘₁2𝑘+1ℎ𝑢₁^𝑘−𝑖for𝑖<𝑘,odd𝑖−𝜇𝐴₁𝐴₂2𝑘+1ℎ²×((2(𝑘−𝑖)−1)+(2(𝑘−𝑖)−5)+⋯)𝑢₁^𝑘−𝑖𝐿for𝑖<𝑘,even𝑖,_𝑘𝑘11𝑢^𝑘₁=𝜇𝐴₁𝐴₂Δ𝑢^𝑘₁+𝜕𝜆+𝜇𝜕𝑥₁𝐴₂𝐴₁𝜕𝑢^𝑘₁𝜕𝑥₁𝜕−𝜇𝜕𝑥₂𝑢^𝑘₁𝐴₂𝜕𝐴₁𝜕𝑥₂+𝑢^𝑘₁𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₂²+𝐴₁𝐴₂𝑘²₁𝑢^𝑘₁−𝑢𝜆+2𝜇^𝑘1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²1−𝜆𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝑢^𝑘₁𝜕𝑥₁−𝐴₁𝐴₂𝜇2𝑘+1ℎ²×((2𝑘−1)+(2𝑘−5)+⋯)𝑢^𝑘₁,𝐿^{𝑘(𝑘+𝑖)}₁₁𝑢₁^𝑘+𝑖=𝜇𝐴₁𝐴₂𝑘₁2𝑘+1ℎ𝑢₁^𝑘+𝑖for𝑖>𝑘,odd𝑖−𝜇𝐴₁𝐴₂2𝑘+1ℎ²×((2𝑘−1)+(2𝑘−5)+⋯)𝑢₁^𝑘+𝑖𝐿for𝑖>𝑘,even𝑖,_𝑘𝑘12𝑢^𝑘₂=𝜕𝜆+2𝜇𝜕𝑥₁𝑢^𝑘₂𝐴₁𝜕𝐴₁𝜕𝑥₂−1𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝑢^𝑘₂𝜕𝑥₂+𝜕𝜆+𝜇²𝑢^𝑘₂𝜕𝑥₁𝜕𝑥₂𝑢−𝜆^𝑘2𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁+𝜇𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₂𝐴₁𝐴₂+1𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢^𝑘₂𝜕𝑥₁−𝜕𝜕𝑥₂𝑢^𝑘₂𝐴₂𝜕𝐴₂𝜕𝑥₁,𝐿^{𝑘(𝑘−𝑖)}₁₃𝑢₃^𝑘−𝑖=−𝜇𝐴₂2𝑘+1ℎ𝜕𝑢₃^𝑘−𝑖𝜕𝑥₁𝐿for𝑖<𝑘,odd𝑖0for𝑖<𝑘,even𝑖,_𝑘𝑘13𝑢^𝑘₃=𝜕𝐴𝜆+2𝜇¹𝑘₁𝑢^𝑘₃𝜕𝑥₁−𝑘₂𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₃𝜕𝐴+𝜆²𝑘₂𝑢^𝑘₃𝜕𝑥₁−𝑘₁𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₃+𝜇𝐴₂𝑘₁𝜕𝑢^𝑘₃𝜕𝑥₁,𝐿^{𝑘(𝑘+𝑖)}₁₃𝑢₃^𝑘+𝑖=𝜆𝐴₂2𝑘+1ℎ𝜕𝑢₃^𝑘+𝑖𝜕𝑥₁𝐿for𝑖>𝑘,odd𝑖0for𝑖>𝑘,even𝑖,_𝑘𝑘21𝑢^𝑘₁=𝜕𝜆+2𝜇𝜕𝑥₂𝑢^𝑘₁𝐴₂𝜕𝐴₂𝜕𝑥₁−1𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢^𝑘₁𝜕𝑥₁+𝜕𝜆+𝜇²𝑢^𝑘₁𝜕𝑥₁𝜕𝑥₂1−𝜆𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝐴₂𝜕𝑥₁𝑢^𝑘₁1+𝜇𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝑢^𝑘₁𝜕𝑥₂−𝑢^𝑘₁𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝐴₁𝜕𝑥₂−𝜕𝜕𝑥₁𝑢^𝑘₁𝐴₁𝜕𝐴₁𝜕𝑥₂,𝐿^{𝑘(𝑘−𝑖)}₂₂𝑢₁^𝑘−𝑖=𝜇𝐴₁𝐴₂𝑘₂2𝑘+1ℎ𝑢₂^𝑘−𝑖for𝑖<𝑘,odd𝑖−𝜇𝐴₁𝐴₂2𝑘+1ℎ²×((2(𝑘−𝑖)−1)+(2(𝑘−𝑖)−5)+⋯)𝑢₂^𝑘−𝑖𝐿for𝑖<𝑘,even𝑖,_𝑘𝑘22𝑢^𝑘₂=𝜇𝐴₁𝐴₂Δ𝑢^𝑘₂+𝜕𝜆+𝜇𝜕𝑥₂𝐴₁𝐴₂𝜕𝑢^𝑘₂𝜕𝑥₂𝜕−𝜇𝜕𝑥₁𝑢^𝑘₂𝐴₁𝜕𝐴₂𝜕𝑥₁+𝑢^𝑘₂𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₂²+𝐴₁𝐴₂𝑘²₂𝑢^𝑘₂−𝑢𝜆+2𝜇^𝑘2𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²1−𝜆𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝑢^𝑘₂𝜕𝑥₂−𝐴₁𝐴₂𝜇2𝑘+1ℎ²×((2𝑘−1)+(2𝑘−5)+⋯)𝑢^𝑘₂,𝐿^{𝑘(𝑘+𝑖)}₂₂𝑢₂^𝑘+𝑖=𝜇𝐴₁𝐴₂𝑘₂2𝑘+1ℎ𝑢₂^𝑘+𝑖for𝑖<𝑘,odd𝑖−𝜇𝐴₁𝐴₂2𝑘+1ℎ²×((2𝑘−1)+(2𝑘−5)+⋯)𝑢₂^𝑘+𝑖𝐿for𝑖<𝑘,even𝑖,_{𝑘(𝑘−𝑖)23}𝑢₃^𝑘−𝑖=−𝜇𝐴₁2𝑘+1ℎ𝜕𝑢₃^𝑘−𝑖𝜕𝑥₂𝐿for𝑖<𝑘,odd𝑖0for𝑖<𝑘,even𝑖,_𝑘𝑘23𝑢^𝑘₃=𝜕𝐴𝜆+2𝜇²𝑘₂𝑢^𝑘₃𝜕𝑥₂−𝑘₁𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₃𝜕𝐴+𝜆¹𝑘₁𝑢^𝑘₃𝜕𝑥₂−𝑘₂𝜕𝐴₁𝜕𝑥₂𝑢^𝑘₃+𝜇𝐴₁𝑘₂𝜕𝑢^𝑘₃𝜕𝑥₂,𝐿^{𝑘(𝑘+𝑖)}₂₃𝑢₃^𝑘+𝑖=𝜆𝐴₂2𝑘+1ℎ𝜕𝑢₃^𝑘+𝑖𝜕𝑥₂𝐿for𝑖<𝑘,odd𝑖0for𝑖<𝑘,even𝑖,_{𝑘(𝑘−𝑖)31}𝑢₁^𝑘−𝑖=−𝜆2𝑘+1ℎ𝜕𝐴₂𝑢₁^𝑘−𝑖𝜕𝑥₁𝐿for𝑖<𝑘,odd𝑖0for𝑖<𝑘,even𝑖,_𝑘𝑘31𝑢^𝑘₁𝜕𝐴=−𝜇¹𝑘₁𝑢^𝑘₁𝜕𝑥₁−×𝑢𝜆+2𝜇_𝑘1𝑘₂𝜕𝐴₂𝜕𝑥₁+𝑘₁𝐴₂𝜕𝑢^𝑘₁𝜕𝑥₁−𝜆𝑘₁𝑢^𝑘₁𝜕𝐴₂𝜕𝑥₁+𝑘₂𝐴₂𝜕𝑢^𝑘₁𝜕𝑥₁,𝐿^{𝑘(𝑘+𝑖)}₃₁𝑢₁^𝑘+𝑖=𝜇2𝑘+1ℎ𝜕𝐴₂𝑢₁^𝑘+𝑖𝜕𝑥₁𝐿for𝑖<𝑘,odd𝑖0for𝑖<𝑘,even𝑖,_{𝑘(𝑘−𝑖)32}𝑢₂^𝑘−𝑖=−𝜆2𝑘+1ℎ𝜕𝐴₁𝑢₂^𝑘−𝑖𝜕𝑥₂𝐿for𝑖<𝑘,odd𝑖0for𝑖<𝑘,even𝑖,_𝑘𝑘31𝑢^𝑘₁𝜕𝐴=−𝜇²𝑘₂𝑢^𝑘₂𝜕𝑥₂−×𝑢𝜆+2𝜇_𝑘1𝑘₁𝜕𝐴₁𝜕𝑥₂+𝑘₂𝐴₁𝜕𝑢^𝑘₂𝜕𝑥₂−𝜆𝑘₂𝑢^𝑘₂𝜕𝐴₁𝜕𝑥₂+𝑘₁𝐴₁𝜕𝑢^𝑘₂𝜕𝑥₂,𝐿^{𝑘(𝑘+𝑖)}₃₂𝑢₂^𝑘+𝑖=𝜇2𝑘+1ℎ𝜕𝐴₁𝑢₂^𝑘+𝑖𝜕𝑥₂𝐿for𝑖<𝑘,odd𝑖0for𝑖<𝑘,even𝑖,_{𝑘(𝑘−𝑖)33}𝑢₃^𝑘−𝑖=−𝜆𝐴₁𝐴₂𝑘₁+𝑘₂2𝑘+1ℎ𝑢₃^𝑘−𝑖𝐴for𝑖<𝑘,odd𝑖𝜆+2𝜇₁𝐴₂2𝑘+1ℎ²×((2(𝑘−𝑖)−1)+(2(𝑘−𝑖)−5)+⋯)𝑢₃^𝑘−𝑖𝐿for𝑖<𝑘,even𝑖,_𝑘𝑘33𝑢^𝑘₃=𝜇𝐴₁𝐴₂Δ𝑢^𝑘₃−𝜆+2𝜇𝑘²₁+𝑘²₂𝑢^𝑘₃−𝜆+2𝜇𝐴₁𝐴₂2𝑘+1ℎ²×((2𝑘−1)+(2𝑘−5)+⋯)𝑢^𝑘₃−2𝜆𝐴₁𝐴₂𝑘₁𝑘₂𝑢^𝑘₃,𝐿^{𝑘(𝑘+𝑖)}₃₃𝑢₃^𝑘+𝑖=−𝜆𝐴₁𝐴₂𝑘₁+𝑘₂2𝑘+1ℎ𝑢₃^𝑘+𝑖𝐴for𝑖<𝑘,odd𝑖𝜆+2𝜇₁𝐴₂2𝑘+1ℎ²×((2𝑘−1)+(2𝑘−5)+⋯)𝑢₃^𝑘+𝑖𝐿for𝑖<𝑘,even𝑖,_{𝑘(𝑘−𝑖)12}𝑢₂^𝑘−𝑖=0,𝐿^{𝑘(𝑘+𝑖)}₁₂𝑢₂^𝑘+𝑖𝐿=0,_{𝑘(𝑘−𝑖)21}𝑢₁^𝑘−𝑖=0,𝐿^{𝑘(𝑘+𝑖)}₂₁𝑢₁^𝑘+𝑖𝐿=0,_𝑘1𝜃^𝑘=−𝛽_𝜏𝐴₁𝜕𝜃^𝑘−𝜃^𝑘₀𝜕𝑥₁,𝐿^𝑘₂𝜃^𝑘=−𝛽_𝜏𝐴₁𝜕𝜃^𝑘−𝜃^𝑘₀𝜕𝑥₂,𝐿^𝑘₃𝜃^𝑘=𝐴₁𝐴₂𝑘₁+𝑘₂𝛽_𝜏𝜃^𝑘−𝜃^𝑘₀+2𝛽_𝜏𝑄^𝑘₃, (A.1) where 1Δ=𝐴₁𝐴₂𝜕𝜕𝑥₁𝐴₂𝐴₁𝜕𝜕𝑥₁+𝜕𝜕𝑥₂𝐴₁𝐴₂𝜕𝜕𝑥₂,𝜎^𝑘₁₃=𝐴₁𝐴₂2𝑘+1ℎ×𝜇1𝐴₁𝜕𝑢₃^𝑘−1𝜕𝑥₁−𝑘₁𝑢₁^𝑘−1+𝑢₁^𝑘−11+𝜇𝐴₁𝜕𝑢₃^𝑘−3𝜕𝑥₁−𝑘₁𝑢₁^𝑘−3+𝑢₁^𝑘−3,𝜎+⋯_𝑘23=𝐴₁𝐴₂2𝑘+1ℎ×𝜇1𝐴₂𝜕𝑢₃^𝑘−1𝜕𝑥₂−𝑘₂𝑢₂^𝑘−1+𝑢₂^𝑘−11+𝜇𝐴₂𝜕𝑢₃^𝑘−3𝜕𝑥₂−𝑘₂𝑢₂^𝑘−3+𝑢₂^𝑘−3,𝜎+⋯_𝑘33=𝐴₁𝐴₂2𝑘+1ℎ𝜆+2𝜇𝑢₃^𝑘−11+𝜆𝐴₁𝜕𝑢₁^𝑘−1𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₂^𝑘−1+𝑘₁𝑢₃^𝑘−1+1𝐴₂𝜕𝑢₂^𝑘−1𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₁^𝑘−1+𝜎^𝑘−3₃₃,𝑢+⋯_𝑘𝑖=2𝑘+1ℎ𝑢_𝑖^𝑘+1+𝑢_𝑖^𝑘+3+⋯. (A.2) For Vekua’s shell theory 𝐿⁰⁰₁₁𝑢⁰₁=𝜇𝐴₁𝐴₂Δ𝑢⁰₁+𝜕𝜆+𝜇𝜕𝑥₁𝐴₂𝐴₁𝜕𝑢⁰₁𝜕𝑥₁𝜕−𝜇𝜕𝑥₂1𝐴₂𝜕𝐴₁𝜕𝑥₂+1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²+𝐴₁𝐴₂𝑘²₁𝑢⁰₁−1𝜆+2𝜇𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²𝑢⁰₁1−𝜆𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝑢⁰₁𝜕𝑥₁,𝐿⁰⁰₁₂𝑢⁰₂=𝜕𝜆+2𝜇𝜕𝑥₁1𝐴₁𝜕𝐴₁𝜕𝑥₂𝑢⁰₂−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝑢⁰₂𝜕𝑥₂+𝜕𝜆+𝜇²𝑢⁰₂𝜕𝑥₁𝜕𝑥₂1−𝜆𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₂𝑢⁰₂𝜕−𝜇𝜕𝑥₂1𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢⁰₂−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₂𝑢⁰₂−1𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢⁰₂𝜕𝑥₁,𝐿⁰⁰₁₃𝑢⁰₃=𝜕𝜆+2𝜇𝜕𝑥₁𝐴₁𝑘₁𝑢⁰₃−𝑘₂𝜕𝐴₂𝜕𝑥₁𝑢⁰₃𝜕+𝜆𝜕𝑥₁𝐴₂𝑘₂𝑢⁰₃−𝑘₁𝜕𝐴₂𝜕𝑥₁𝑢⁰₃+𝜇𝐴₂𝑘₁𝜕𝑢⁰₃𝜕𝑥₁,𝐿⁰¹₁₁𝑢¹₁=𝜇𝐴₁𝐴₂𝑘₁𝑢¹₁ℎ,𝐿⁰¹₁₂𝑢¹₂𝐿=0,₀₁₁₃𝑢¹₃=𝜆𝐴₂1ℎ𝜕𝑢¹₃𝜕𝑥₁,𝐿⁰⁰₂₁𝑢⁰₁=𝜕𝜆+2𝜇𝜕𝑥₂1𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢⁰₁−1𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢⁰₁𝜕𝑥₁+𝜕𝜆+𝜇²𝑢⁰₁𝜕𝑥₁𝜕𝑥₂1−𝜆𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₂𝑢⁰₁𝜕−𝜇𝜕𝑥₁1𝐴₁𝜕𝐴₁𝜕𝑥₂𝑢⁰₁−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝐴₁𝜕𝑥₂𝑢⁰₁−1𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝑢⁰₁𝜕𝑥₂,𝐿⁰⁰₂₂𝑢⁰₁=𝜇𝐴₁𝐴₂Δ𝑢⁰₂+𝜕𝜆+𝜇𝜕𝑥₂𝐴₁𝐴₂𝜕𝑢⁰₂𝜕𝑥₂𝜕−𝜇𝜕𝑥₁1𝐴₁𝜕𝐴₂𝜕𝑥₁+1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²+𝐴₁𝐴₂𝑘²₂𝑢⁰₂−1𝜆+2𝜇𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²𝑢⁰₂1−𝜆𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝑢⁰₂𝜕𝑥₂,𝐿⁰⁰₂₃𝑢⁰₃=𝜕𝜆+2𝜇𝜕𝑥₂𝐴₂𝑘₂𝑢⁰₃−𝑘₁𝜕𝐴₁𝜕𝑥₂𝑢⁰₃𝜕+𝜆𝜕𝑥₂𝐴₁𝑘₁𝑢⁰₃−𝑘₂𝜕𝐴₁𝜕𝑥₂𝑢⁰₃+𝜇𝐴₁𝑘₂𝜕𝑢⁰₃𝜕𝑥₂,𝐿⁰¹₂₁𝑢¹₁=0,𝐿⁰¹₂₂𝑢¹₂=𝜇𝐴₁𝐴₂𝑘₂𝑢¹₂ℎ,𝐿⁰¹₂₃𝑢¹₃=𝜆𝐴₂1ℎ𝜕𝑢¹₃𝜕𝑥₂,𝐿⁰⁰₃₁𝑢⁰₁𝜕=−𝜇𝜕𝑥₁𝐴₁𝑘₁𝑢⁰₁−×𝑘𝜆+2𝜇₂𝜕𝐴₂𝜕𝑥₁𝑢⁰₁+𝑘₁𝐴₂𝜕𝑢⁰₁𝜕𝑥₁−𝜆𝑘₁𝜕𝐴₂𝜕𝑥₁𝑢⁰₁+𝐴₂𝑘₂𝜕𝑢⁰₁𝜕𝑥₁,𝐿⁰⁰₃₂𝑢⁰₂𝜕=−𝜇𝜕𝑥₂𝐴₂𝑘₂𝑢⁰₂−×𝑘𝜆+2𝜇₁𝜕𝐴₁𝜕𝑥₂𝑢⁰₂+𝑘₂𝐴₁𝜕𝑢⁰₂𝜕𝑥₂−𝜆𝑘₂𝜕𝐴₁𝜕𝑥₂𝑢⁰₂+𝐴₁𝑘₁𝜕𝑢⁰₂𝜕𝑥₂,𝐿⁰⁰₃₃𝑢⁰₃=−𝜇𝐴₁𝐴₂Δ𝑢⁰₃−𝜆+2𝜇𝐴₁𝐴₂𝑘²₁+𝑘²₂𝑢⁰₃−2𝜆𝐴₁𝐴₂𝑘₁𝑘₂𝑢⁰₃,𝐿⁰¹₃₁𝑢¹₁1=−𝜇ℎ𝜕𝐴₂𝑢¹₁𝜕𝑥₁,𝐿⁰¹₃₂𝑢¹₂1=−𝜇ℎ𝜕𝐴₁𝑢¹₂𝜕𝑥₂,𝐿⁰¹₃₃𝑢¹₃−𝜆𝐴₁𝐴₂𝑘₁+𝑘₂1ℎ𝑢¹₃,𝐿¹⁰₁₁𝑢⁰₁=𝜇𝐴₁𝐴₂𝑘₁3𝑢⁰₁ℎ,𝐿¹⁰₁₂𝑢⁰₂𝐿=0,₁₀₁₃𝑢⁰₃=−𝜇𝐴₂3ℎ𝜕𝑢⁰₃𝜕𝑥₁,𝐿¹¹₁₁𝑢¹₁=𝐿⁰⁰₁₁𝑢¹₁−𝜇𝐴₁𝐴₂3𝑢¹₁ℎ²,𝐿¹¹₁₂𝑢¹₂=𝐿⁰⁰₁₂𝑢¹₂,𝐿¹¹₁₃𝑢¹₃=𝐿⁰⁰₁₃𝑢¹₃,𝐿¹⁰₂₁𝑢⁰₁=0,𝐿¹⁰₂₂𝑢⁰₂=𝜇𝐴₁𝐴₂𝑘₂3𝑢⁰₂ℎ,𝐿¹⁰₂₃𝑢⁰₃=−𝜇𝐴₁3ℎ𝜕𝑢⁰₃𝜕𝑥₂,𝐿¹¹₂₁𝑢¹₁=𝐿⁰⁰₁₂𝑢¹₁,𝐿¹¹₂₂𝑢¹₂=𝐿⁰⁰₂₂𝑢¹₂−𝜇𝐴₁𝐴₂3𝑢¹₂ℎ²,𝐿¹¹₂₃𝑢¹₃=𝐿⁰⁰₂₃𝑢¹₃,𝐿¹⁰₃₁𝑢⁰₁3=−𝜆ℎ𝜕𝐴₂𝑢⁰₁𝜕𝑥₁,𝐿¹⁰₃₂𝑢⁰₂1=−𝜆ℎ𝜕𝐴₁𝑢⁰₂𝜕𝑥₂,𝐿¹⁰₃₃𝑢⁰₃=−𝜆𝐴₁𝐴₂𝑘₁+𝑘₂3ℎ𝑢⁰₃,𝐿¹¹₃₁𝑢¹₁=𝐿⁰⁰₃₁𝑢¹₁,𝐿¹¹₃₁𝑢¹₁=𝐿⁰⁰₃₁𝑢¹₁,𝐿¹¹₃₂𝑢¹₂=𝐿⁰⁰₃₂𝑢¹₂,𝐿¹¹₃₃𝑢¹₃=𝐿⁰⁰₃₃𝑢¹₃−𝜆+2𝜇𝐴₁𝐴₂3ℎ²𝑢¹₃. (A.3)

For Timoshenko’s shell theory 𝐿⁰⁰₁₁𝑢₁=𝐷_𝑛𝜕𝜕𝑥₁𝐴₂𝐴₁𝜕𝑢₁𝜕𝑥₁−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²𝑢₁𝜕+𝜐𝜕𝑥₁1𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₁−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝑢₁𝜕𝑥₁+1−𝜐2×𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝜕𝑥₂𝑢₁𝐴₁+𝜕𝜕𝑥₂×𝐴²₁𝐴₂𝜕𝜕𝑥₂𝑢₁𝐴₁−𝐴₁𝐴₂𝑘²₁𝑢₁+𝐷_𝑚𝜕𝜕𝑥₂𝑘₁𝑘₂𝐴₁𝐴₂𝜕𝑢₁𝜕𝑥₂−𝜕𝜕𝑥₂𝑘²₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₁−𝑘₁𝑘₂𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²𝑢₁+𝑘²₂𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝑢₁𝜕𝑥₂,𝐿⁰⁰₁₂𝑢₂=𝐷_𝑛𝜕𝜕𝑥₁1𝐴₁𝜕𝐴₁𝜕𝑥₂𝑢₂−1𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝑢₂𝜕𝑥₂𝜕+𝜐²𝑢₂𝜕𝑥₁𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝑢₂+1−𝜐2𝜕𝜕𝑥₂𝐴₂𝜕𝜕𝑥₁𝑢₂𝐴₂+𝐴₂𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝜕𝑥₁𝑢₂𝐴₂+𝐷_𝑚𝜕𝜕𝑥₂𝑘²₁𝜕𝑢₂𝜕𝑥₁−𝜕𝜕𝑥₂𝑘₁𝑘₂𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₂+𝑘₁𝑘₂𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢₂𝜕𝑥₁−𝑘²₁𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝑢₂,𝐿⁰⁰₁₃𝑢₃=𝐷_𝑛𝜕𝜕𝑥₁𝐴₂𝑘₁+𝜐𝑘₂𝑢₃−𝑘₂+𝜐𝑘₁𝜕𝐴₂𝜕𝑥₁𝑢₃+1−𝜐2𝐴₂𝑘₁𝜕𝑢₃𝜕𝑥₁,𝐿⁰¹₁₁𝛾₁=𝐷_𝑚𝜕𝜕𝑥₂𝐴₁𝑘₁𝐴₂𝜕𝛾₁𝜕𝑥₂−𝜕𝜕𝑥₂𝑘₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝛾₁−𝑘₂𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²𝛾₁+𝑘₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝛾₁𝜕𝑥₂+𝐷_𝑛𝐴₁𝐴₂𝑘₁𝛾₁,𝐿⁰¹₁₂𝛾₂=𝐷_𝑚𝜕𝜕𝑥₂𝑘₁𝜕𝛾₂𝜕𝑥₁−𝜕𝜕𝑥₂𝑘₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝛾₂−𝑘₂𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝛾₂+𝑘₂𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝛾₂𝜕𝑥₁,𝐿⁰⁰₂₁𝑢₁=𝐷_𝑛𝜕𝜕𝑥₂1𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₁−1𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢₁𝜕𝑥₁𝜕+𝜐²𝑢₁𝜕𝑥₁𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝐴₁𝜕𝑥₂𝑢₁+1−𝜐2𝜕𝜕𝑥₁𝐴₁𝜕𝜕𝑥₂𝑢₁𝐴₁+𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝜕𝑥₂𝑢₁𝐴₁+𝐷_𝑚𝜕𝜕𝑥₁𝑘²₂𝜕𝑢₁𝜕𝑥₂−𝜕𝜕𝑥₁𝑘₁𝑘₂𝐴₁𝜕𝐴₁𝜕𝑥₂𝑢₁+𝑘₁𝑘₂𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝑢₁𝜕𝑥₂−𝑘²₁𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝑢₁,𝐿⁰⁰₂₂𝑢₂=𝐷_𝑛𝜕𝜕𝑥₂𝐴₁𝐴₂𝜕𝑢₂𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²𝑢₂𝜕+𝜐𝜕𝑥₂1𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₂−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝑢₂𝜕𝑥₂+1−𝜐2𝐴₂𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝜕𝑥₁𝑢₂𝐴₂+𝜕𝜕𝑥₁×𝐴²₂𝐴₁𝜕𝜕𝑥₁𝑢₂𝐴₂−𝐴₁𝐴₂𝑘²₂𝑢₂+𝐷_𝑚𝜕𝜕𝑥₁𝑘₁𝑘₂𝐴₂𝐴₁𝜕𝑢₂𝜕𝑥₁−𝜕𝜕𝑥₁𝑘²₂𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₂−𝑘₁𝑘₂𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²𝑢₂+𝑘²₁𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝑢₂𝜕𝑥₁,𝐿⁰⁰₂₃𝑢₃=𝐷_𝑛𝜕𝜕𝑥₂𝐴₁𝑘₂+𝜐𝑘₁𝑢₃−𝑘₁+𝜐𝑘₂×𝜕𝐴₁𝜕𝑥₂𝑢₃+1−𝜐2𝐴₁𝑘₂𝜕𝑢₃𝜕𝑥₂,𝐿⁰¹₂₁𝛾₁=𝐷_𝑚𝜕𝜕𝑥₁𝑘₂𝜕𝛾₁𝜕𝑥₂−𝜕𝜕𝑥₁𝑘₂𝐴₁𝜕𝐴₁𝜕𝑥₂𝛾₁−𝑘₁𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝛾₁+𝑘₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝛾₁𝜕𝑥₂,𝐿⁰¹₂₂𝛾₂=𝐷_𝑚𝜕𝜕𝑥₁𝐴₂𝑘₂𝐴₁𝜕𝛾₂𝜕𝑥₁−𝜕𝜕𝑥₁𝑘₂𝐴₁𝜕𝐴₂𝜕𝑥₁𝛾₂−𝑘₁𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²𝛾₂+𝑘₁𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝛾₂𝜕𝑥₁+𝐷_𝑛𝐴₁𝐴₂𝑘₂𝛾₂,𝐿⁰⁰₃₁𝑢₁=𝐷_𝑛𝐴₂𝑘₁+𝜐𝑘₂𝜕𝑢₁𝜕𝑥₁+𝑘₂+𝜐𝑘₁𝜕𝐴₂𝜕𝑥₁𝑢₁+1−𝜐2𝜕𝜕𝑥₁𝐴₂𝑘₁𝑢₁,𝐿⁰⁰₃₂𝑢₂=𝐷_𝑛𝐴₁𝑘₂+𝜐𝑘₁𝜕𝑢₂𝜕𝑥₂+𝑘₁+𝜐𝑘₂𝜕𝐴₁𝜕𝑥₂𝑢₂+1−𝜐2𝜕𝜕𝑥₂𝐴₁𝑘₂𝑢₃,𝐿⁰⁰₃₃𝑢₃=𝐷_𝑛𝐴₁𝐴₂1−𝜐2Δ𝑢₃−𝑘²₁+2𝜐𝑘₁𝑘₂+𝑘²₂𝑢₃,𝐿⁰¹₃₁𝛾₁=𝐷_𝑛1−𝜐2𝜕𝜕𝑥₁𝐴₂𝛾₁,𝐿⁰¹₃₂𝛾₂=𝐷_𝑛1−𝜐2𝜕𝜕𝑥₂𝐴₁𝛾₂,𝐿¹⁰₁₁𝑢₁=𝐷_𝑚(𝜕1−𝜐)𝜕𝑥₂𝐴₁𝑘₂𝐴₂𝜕𝑢₁𝜕𝑥₂−𝜕𝜕𝑥₂𝑘₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝑢₁−𝑘₁𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²𝑢₁+𝑘₂𝐴₂𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢₁𝜕𝑥₂+𝐷_𝑛𝐴₁𝐴₂𝑘₁𝑢₁,𝐿¹⁰₁₂𝑢₂=𝐷_𝑛(𝜕1−𝜐)𝜕𝑥₂𝑘₁𝜕𝑢₂𝜕𝑥₁−𝜕𝜕𝑥₂𝑘₂𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₂−𝑘₂𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝑢₂+𝑘₁𝐴₁𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝑢₂𝜕𝑥₁,𝐿¹⁰₁₃𝑢₃=−𝐷_𝑛1−𝜐2𝐴₂𝜕𝑢₃𝜕𝑥₁,𝐿¹¹₁₁𝛾₁=𝐷_𝑚𝜕𝜕𝑥₁𝐴₂𝐴₁𝜕𝛾₁𝜕𝑥₁−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²𝛾₁𝜕+𝜐𝜕𝑥₁1𝐴₁𝜕𝐴₂𝜕𝑥₁𝛾₁−1𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝛾₁𝜕𝑥₁+1−𝜐2𝜕𝜕𝑥₂𝐴₁𝐴₂𝜕𝛾₁𝜕𝑥₂+1𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝛾₁𝜕𝑥₂−𝜕𝜕𝑥₂1𝐴₂𝜕𝐴₁𝜕𝑥₂𝛾₁−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²𝛾₁+𝐷_𝑛1−𝜐2𝐴₁𝐴₂𝛾₁,𝐿¹¹₁₂𝛾₂=𝐷_𝑚𝜕𝜕𝑥₁1𝐴₁𝜕𝐴₁𝜕𝑥₂𝛾₂−1𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝛾₂𝜕𝑥₂𝜕+𝜐²𝛾₂𝜕𝑥₁𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝛾₂+1−𝜐2𝜕²𝛾₂𝜕𝑥₁𝜕𝑥₂+1𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝛾₂𝜕𝑥₁−𝜕𝜕𝑥₂1𝐴₂𝜕𝐴₂𝜕𝑥₁𝛾₂−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝛾₂,𝐿¹⁰₂₁𝑢₁=𝐷_𝑚1−𝜐2𝜕𝜕𝑥₁𝑘₂𝜕𝑢₁𝜕𝑥₂−𝜕𝜕𝑥₁𝑘₁𝐴₁𝜕𝐴₁𝜕𝑥₂𝑢₁−𝑘₁𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝑢₁+𝑘₂𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝑢₁𝜕𝑥₂,𝐿¹⁰₂₂𝑢₂=𝐷_𝑚1−𝜐2𝜕𝜕𝑥₁𝐴₂𝑘₁𝐴₁𝜕𝑢₂𝜕𝑥₁−𝜕𝜕𝑥₁𝑘₂𝐴₁𝜕𝐴₂𝜕𝑥₁𝑢₂−𝑘₂𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁𝑢₂+𝑘₁𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝑢₂𝜕𝑥₁+𝐷_𝑛1−𝜐2𝐴₁𝐴₂𝑘₂𝑢₂,𝐿¹⁰₂₃𝑢₃=−𝐷_𝑛1−𝜐2𝐴₁𝜕𝑢₃𝜕𝑥₂,𝐿¹¹₂₁𝛾₁=𝐷_𝑚𝜕𝜕𝑥₂1𝐴₂𝜕𝐴₂𝜕𝑥₁𝛾₁−1𝐴₁𝜕𝐴₁𝜕𝑥₂𝜕𝛾₁𝜕𝑥₁𝜕+𝜐²𝛾₁𝜕𝑥₁𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝛾₁+1−𝜐2×𝜕²𝛾₁𝜕𝑥₁𝜕𝑥₂+1𝐴₂𝜕𝐴₂𝜕𝑥₁𝜕𝛾₁𝜕𝑥₂−𝜕𝜕𝑥₁1𝐴₁𝜕𝐴₁𝜕𝑥₂𝛾₁−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝐴₂𝜕𝑥₁𝛾₁,𝐿¹¹₂₂𝛾₂=𝐷_𝑚𝜕𝜕𝑥₂𝐴₁𝐴₂𝜕𝛾₂𝜕𝑥₂−1𝐴₁𝐴₂𝜕𝐴₁𝜕𝑥₂²𝛾₂𝜕+𝜐𝜕𝑥₂1𝐴₂𝜕𝐴₁𝜕𝑥₂𝛾₂−1𝐴₂𝜕𝐴₁𝜕𝑥₂𝜕𝛾₂𝜕𝑥₂+1−𝜐2×𝜕𝜕𝑥₁𝐴₂𝐴₁𝜕𝛾₂𝜕𝑥₁+1𝐴₁𝜕𝐴₂𝜕𝑥₁𝜕𝛾₂𝜕𝑥₁−𝜕𝜕𝑥₁1𝐴₁𝜕𝐴₂𝜕𝑥₁𝛾₂−1𝐴₁𝐴₂𝜕𝐴₂𝜕𝑥₁²𝛾₂+𝐷_𝑛1−𝜐2𝐴₁𝐴₂𝛾₂. (A.4)(A.5)

Acknowledgment

This work was supported by the Committee of Science and Technology of Mexico (CONACYT) through the research Grants (Project no. 000000000101415), which is gratefully acknowledged.