STRESS-STRAIN STATE OF THICK-WALLED ANISOTROPIC CYLINDRICAL SHELLS UNDER THERMAL POWER LOAD, PROTECTED BY THE FUNCTIONALLY GRADED MATERIAL

In the article the stress-strain state of thick-walled structurally anisotropic composite cylindrical shells under thermal power load, which are protected by a functionally graded material, are analysed. Based on the interrelations of the spatial theory of elasticity, a system of inhomogeneous differential equations in three-dimensional formulation, which describes the stress-strain state of thick-walled anisotropic cylindrical shells, was obtained. To reduce the dimensionality of this system, the Bubnov-Galerkin analytical method was used. Thus, the obtained one-dimensional system of twelve equations of normal Cauchy form was implemented using the numerical method of discrete orthogonalization. To represent the possibilities of the proposed approach, there were used stress-strain states of two, four and five-layered anisotropic cylindrical shells of fibrous composites, protected from temperature by a layer of transversely isotropic functionally graded material.


Introduction
Thin-walled structures made of composite materials are widely used in a variety of elements of up-to-date equipment. For example, the aerospace and rocket industries require the use of shells made of lightweight, high-strength composite materials. Unfortunately, traditional composite materials are not always able to be used in high temperatures, because their load-bearing capacity can be significantly reduced. Heat-resistant ceramics can be used to protect thin-walled composites from temperatures, but it is well known that this material has brittle properties and does not bend and twist.
Relatively recently, a new class of composite materials known as functionally graded materials (FGMs) has emerged [16]. Typical FGMs is an inhomogeneous composite made of different phases of material components (usually ceramics and metal). FGMs ceramic components are able to withstand high-temperature environments due to the better heat resistance characteristics, and metal components provide higher mechanical properties and reduce the possibility of destruction. Thus, the use of FGMs can help to protect the shell structure from the effects of variable temperature fields, which will allow the structure to absorb the load without reducing its strength, for instance.
At present, a sufficiently detailed analysis of the stress-strain state of thinwalled and thick-walled cylindrical shells of both conventional composites and FGMs in the calculations of two-dimensional systems under the thermal power load [1,3,12,15,16,17] is made. In this paper, the change of the characteristics of the stress-strain state in the thickness of the structure is modeled by hypotheses of varying degrees of accuracy. It is generally known that to calculate the stress-strain state of thick-walled cylindrical shells it should be applied an approach [2,6,7,10,11,13], based on the use of equations of the spatial theory of elasticity and which allows you to correctly analyze changes of parameters such as stress-strain state of the construction by the thickness.
The authors propose an approach to the establishment of the stress-strain state of a thick layered anisotropic cylindrical shell made of a fibrous composite, which is made at an angle to the generatrix, and a layer of FGMs. It is also necessary to take into account the effect of anisotropy caused by the discrepancy between the directions of reinforcement and the shell axes ( Fig. 1) [1,2,3,6,7,10,11,13,14], and to assess the impact of temperature on such a combined structure thickness.
In this paper a three-dimensional theory of elasticity is used to solve the problem of the stressstrain state of a thickwalled anisotropic cylindrical shell made of fibrous composites [5]. The obtained solutions can serve as references in the calculations of stressstrain states of thinwalled structures of more complex geometry established, for example, when using the finite element method.

Formulation of the problem
Linear equilibrium equations in non-axisymmetric stress-strain state for each i-th layer are as follows [5]: where i r (i=1,2) -radius of the cylinder, which does not depend on the coordinates z and  ; i -vector projections of specific volume forces on the directions tangent to the coordinate lines r , z ,  .
The relationship between the components of deformation and displacement will take the form: In the equations (3, 4) / i kl a and i kl a -are mechanical constants of the i-th layer of orthotropic material and material with one plane of elastic symmetry, the relationship between which is established in [4].

Research methodology
The relation of the generalized Hooke's law for a material with one plane of elastic symmetry (4) takes the form [2], which is used in the solution of the system (1): In this system of equations and in (5) (2) we obtain for each i-th layer a complete system of differential equations in partial derivatives, in which we take into account that the shell is deformed according to the axial symmetry: If the temperature field is constant to the cylinder surface, the system (6) will be rewritten:  ;   12  66  36  22  26  36   11  16  36  12  26  23  22  22  23  11  12  23  12  16  13   22  12  13  11  11  13  33  3 The solution of the system (7) must correspond to the conditions on the side surfaces at (9) and the conditions of hard contact of the layers: There is a diaphragm, which is absolutely rigid in its plane and flexible, at the edges of the cylinder in conditions (9). In (8) 1 ( ) r q z , 2 ( ) r q z -an internal and external pressure is distributed on the side surfaces of the shell, respectively.
To solve the three-dimensional problem (7) and (8-9), we use the Bubnov-Galerkin methodology. According to it, we decompose all functions into trigonometric series on the coordinate along the cylinder generatrix z , so that they would satisfy the boundary conditions (9) where pk i y , , After some mathematical transformations and separation of variables in equations (7) using the ratios (11), we obtain for each i-th layer a system of differential equations of the twelfth order in the normal Cauchy form  (7), i f -the components of the stress-strain state related to the temperature in the system (7) and are determined as: 11  12  23  12  16  13  22  12  13  11  11  13  33  10 Implementation of the obtained one-dimensional problem on the stressstrain state of a thick-walled cylinder was carried out using the numerical method of discrete orthogonalization [3]. After solving the system (12) taking into account the boundary conditions (8), ratios (11) were used for the transition from the obtained functions to the components of the stress-strain state.

Results of the numerical calculations and their analysis
The object of the study is a cylindrical shell made of layers of fibrous boroplastic material and a layer of functionally graded material [16] under the distributed external pressure and external constant temperature. Silicone nitride was chosen as the ceramic component of FGMs, and titanium (Ti-6Al-4V) was chosen as the metal component. The temperature distribution along thickness of the cylinder was determined according to [9,16]. Physical and mechanical properties of the functionally graded component of the cylinder dependent on the temperature were determined in the tables 1-4 [12] and according to the dependences [18]:     Common characteristics of the functionally graded materials were determined in accordance with [8]: coordinate of the cylinder inner surface, N -volume fraction of mixed materials [16]. A cylindrical shell with the following geometric parameters was considered ( fig. 1): L=1.2 m, 1 r =0.57 m, 2 r =0.63 cm, h=0.06 m. Three variants of a thick-walled cylindrical shell were calculated. The first one took into the consideration only power load, and then combined power and thermal load. In this case, the temperature field changed only according to the thickness of the layer of functionally graded material.
In the first variant of this structure, it was assumed that the cylinder consists of two layers: the inner 1 r =0.57 m, 11 r =0. Ti-6Al-4V( 02 r ) -silicone nitride ( 2 r )). All presented variants of shell structures were under pressure load that was distributed on the external surface , where q 0 =100 MPa. In the case of joint power and thermal load, the temperature field changed only by the thickness of the functionally graded component of the material from Т=293 0 К (20 0 С), when 02 r = 0.6 m, to Т=373 0 К (100 0 С), when 2 r =0.63m for all three variants of the cylindrical shell, and distributed by thickness according to the law presented in [9] Fig . 2-4 show the change of stress-strain components of the shells with one layer of boroplastic, that is curve 1 (solid), with two and four layers, i.e curve 2 (dashed) and curve 3 (dotted), respectively. The calculation results are given by the coordinate r for the cross-section that is on the middle of generatrix, i.e, when z = 0.5L. The analysis results of calculations in Fig. 2-4 describe the stress-strain state of the layered thick-walled anisotropic composite shells and show that the variable temperature field of the functionally graded component of the shell material within the studied temperatures ranges from Т=293 0 К (20 0 С), when r=0.6 m to Т=373 0 К (100 0 С), when r=0.63 m and does not significantly affect the stress-strain state of a thick-walled anisotropic cylinder. However, it should be noted that variable temperature field and distributed lateral pressure ( fig. 2b) effects that the values of normal stresses are slightly, up to 5%, higher compared with those, which are under power load only ( fig. 2a).
The graphs presented in figs. 3a and 3b indicate that the cross-laying of fibrous material allows reducing the value of the tangential stresses rz  compared to one layered of the composite. This is especially evident in the example of the thickness distribution of the tangential stresses   r in figs. 4a and 4b, which appear only in anisotropic material. There are anisotropic components of the stress-strain state   r for one composite layer in the transversely isotropic FGMs. However, for cross-ply composites having two and four layers the effect is almost absent.

Conclusions
In the article the numerical calculations of the stress-strain state of a thickwalled cylindrical anisotropic shell of fibrous composite material that is protected by a functionally graded material under power and thermal load, were conducted based on three-dimensional theory of elasticity. Variants of increasing the number of layers stacked so that their axes of orthotropy of fiber composites do not coincide with the axes of the cylinder coordinate system, creating the effect of a material with one plane of elastic symmetry, are analyzed. It is taken into account that the physical and mechanical properties of FGMs are dependent on temperature. It was found that the stress-strain state of a thick-walled anisotropic cylinder does not change significantly due to the action of high temperature, which is the effect of using FGMs to protect the cylindrical shell.  In the article the stress-strain state of thick-walled structurally anisotropic composite cylindrical shells under thermal power load, which are protected by a functionally graded material, are analysed. Based on the interrelations of the spatial theory of elasticity, a system of inhomogeneous differential equations in three-dimensional formulation, which describes the stress-strain state of thick-walled anisotropic cylindrical shells, was obtained. To reduce the dimensionality of this system, the Bubnov-Galerkin analytical method was used. Thus, the obtained one-dimensional system of twelve equations of normal Cauchy form was implemented using the numerical method of discrete orthogonalization. To represent the possibilities of the proposed approach, there were used stress-strain states of two, four and five-layered anisotropic cylindrical shells of fibrous composites, protected from temperature by a layer of transversely isotropic functionally graded material. Tabl. 4. Fig. 1