Behaviour of a non-linear mesh cylindrical shell as an element of mems and nems

. A new mathematical model of a non-linear mesh cylindrical shell behaviour in the temperature field under normal distributed load is constructed. The construction of the mathematical model takes into account the Kirchhoff-Love kinematic model and the Duhamel-Neumann hypothesis. The scaling effect is taken into account by the modified moment theory of elasticity. It is assumed that the displacement and rotation fields are not independent. Geometric nonlinearity is taken into account according to T. von Karman's theory. The equations of motion of the smooth shell, boundary and initial conditions are derived from the variational Ostrogradsky-Hamiltonian principle. The lattice structure of the shell was modelled by the continuum theory of G. I. Pshenichny. This allowed us to replace the regular rib system by a continuous layer. The equilibrium conditions of a rectangular element were used to write down the relations connecting stresses occurring in an equivalent smooth shell with stresses in the ribs. The Lagrange multiplier method has been used to determine the physical relationships for the mesh shell. By means of the method of establishment the study of features of the shell's plasticity has been carried out and the frequencies of natural linear vibrations depending on the mesh geometry have been obtained.


Introduction
Scientific and technological progress and the development of modern technology are inexorably leading to nano-and microelectromechanical systems (NEMS and MEMS) becoming firmly established in all areas of human life and activity.Cylindrical micro-and nano-shells, including mesh ones, are among the typical sensing elements of MEMS and NEMS often operating in wide temperature ranges.Therefore, the need to study the behavior of such mechanical objects as cylindrical micro-and nano-shells under various external influences does not weaken [1][2][3][4][5][6].Many authors rely on mathematical models based on linear theory; however, experimental data indicate the need to take nonlinearity into account when modelling the behaviour of the objects in question [7].Cylindrical micro-and nano-shells as elements of electronics often operate in a wide range of temperatures.The analysis of the behaviour of these mechanical objects under the action of temperature, including the solution of problems about natural vibrations, are devoted to [8][9][10].There are not many works devoted to the analysis of the behavior of cylindrical dimensional-dependent shells.In [11], a continuum model of a mesh shell formed by two families of flexible nonlinear elastic fibers that keep their orthogonality during the deformation process was proposed.An analysis of the behaviour of dimensional-dependent geometrically nonlinear mesh shells of the Kirchhoff-Love's model with different mesh geometries without considering temperature effects is given in [12].
The purpose of this work is to investigate the particularities of the stress-strain state of meshed cylindrical shells with allowance for the scale effect, different geometry of the mesh structure of the shell and the stationary temperature field.

Mathematical model
The subject of the study is a cylindrical mesh shell occupying ℝ 3 area 0 ; 0 2 ; 2 2 Fig1.Introduce the coordinate system as follows: coordinate z is measured from the median surface of the shell along its outer normal, the axis  is directed along the formative, the axis  is directed along the circumference of the cylinder formed by the centre surface of the shell Fig. 2. The shell material is isotropichomogeneous, obeying Hooke's law.The geometric nonlinearity is taken into account according to the Theodor von Karman model [13].The motion equations, boundary and initial conditions are obtained from the Ostrogradsky -Hamilton's variational principle [14][15] by using the Kirchhoff -Love's kinematic hypotheses.
The non-zero components of the strain tensor will then take the form: The scale effect is taken into account based on a modified moment theory [16].That is, in addition to the conventional stress field, moment stresses are also considered.It is assumed that the displacement and rotation fields are not independent.The components of the torsional bending tensor are written as follows: 22 2 11 2, 4 ,, u v w -directional moves ,   and deflection respectively, R -shell radius.
Following the accepted hypotheses, considering the Duhamel-Neumann hypothesis that thermal deformations are a superposition of elastic deformations and thermal expansions and based on Yang theory [16], we write the defining relations for the shell material in the form: The equations of motion of a continuous micropolar shell, taking into account temperature and force influences, boundary and initial conditions are obtained from the Ostrogradsky-Hamiltonian energy principle [14][15]: Here K -kinetic energy, U -potential energy, W -the work of external forces.Given a modified moment theory, the potential energy U in an elastic body, is written in the form:


-variation of external normal distributed load operation q and dissipation, respectively.Here ε -dissipation factor,  -shell material density.By performing a variation, collecting the coefficients at the same variations, we obtain the equations of motion of a smooth micropolar cylindrical shell considering temperature and force influences.
Suppose that the shell consists of n families of frequently spaced edges of the same material.Let's replace the regular system of edges by a continuous layer using the continuum model of Pshenichny G.I. [17].Figure 2 shows the median surface of the shell consisting of one family of ribs referenced to the orthogonal curvilinear coordinate system z  .Denote arising in the equivalent smooth shell, associated with the stresses in the ribs, comprising the angles j  pivotally  will be of the form: To determine the physical relationships of the mesh shell, consider the Lagrange multiplier method.When constructing the functional, an expression for the potential energy of deformation expressed in terms of stresses and moments of higher orders is used.Additional conditions of static equivalence of the original mesh shell and the equivalent continuum shell are introduced by means of Lagrange multipliers (deformation for stresses and curvature tensor components for moments of higher orders).It is necessary for the constructed functional to have a stationary value -0 L which will allow to write down additional conditions of static equivalence of the original mesh shell and its equivalent continuous shell: .
Introduce the notation: , 0, 4. sk We write down the expressions for the classical forces and moments as well as the forces due to moment stresses of the cylindrical smooth shell equivalent to the original mesh shell (the upper index shows the consideration of the mesh structure): .
In this case, the equations of motion of the cylindrical shell element with respect to its mesh structure will be:  ,   .
Let's add boundary conditions to the equations of motion of the shell element and zero initial conditions: In order to construct a mathematical model of the behaviour of a mesh micropolar cylindrical shell under temperature influences, let us add the stationary three-dimensional heat conduction equation to the equations of motion: with boundary conditions of the first kind:

Numerical experiment
The method of establishment [18,19] analyses the effect of the number of rib families and temperature influences on the stress-strain state of the shell in question.The method of establishment has a high accuracy as it can be referred to iterative methods, here each time step is a new approximation to the exact solution of the problem.Following the method of establishment, solving a system of partial equations reduces to solving a Cauchy problem for a system of ordinary differential equations.a range of values of the normal time constant load parameter i q , we get a sequence of deflections i w (or other components of the displacement vector ) for an arbitrary, in our case central, point of the shell.On the basis of these data the dependence   wq and the stress- strain state of the system is investigated.The essence of the method is schematically shown in figure 3. The partial differential problem was reduced to the Cauchy problem by the finite difference method of second order accuracy.The Cauchy problem was solved by the Runge-Kutta method of the fourth order.
To substantiate the validity of the results obtained by the method used, a comparison with the results published in [20] has been carried out.Fig. 4  .The epiure constructed without taking scale effects into account according to the classical theory is fully consistent with the calculations of the authors of [20].An increase in the dimensionally dependent parameter leads to an increase in the shell stiffness.Figure 5 shows moment and stress diagrams along the length of the shell, depending on the number of rib families in the mesh structure of the shell(case1 -12 45 The values of the moments closer to the centre of the shell (away from the edge) do not differ much, but it can be seen that the values of the moments are greater in the case of fewer families of ribs in the mesh.A change in the mesh geometry has a more significant effect on the normal forces (Figure 6).An increase in the number of rib families in the mesh results in a change in the shape of the normal force epure along the length of the shell.The values of the normal force decrease as the number of rib families in the mesh increases.
The numerical experiment showed that in the calculation of the stress-strain state of the shells in question, the temperature effects should be taken into account in the mathematical model.The graph (Fig. 7) shows that the values of normal forces obtained without taking into account temperature effects in the model and at 300 K (room temperature) are different.An increase in temperature leads to an increase in the values of the normal forces.The natural frequencies of the shell depending on the number of rib families in the mesh at room temperature (300 K) have also been calculated.It is interesting to note that the natural frequency of the shell increases significantly when a rib family with an angle of 90 degrees is added to the mesh (Table 1).

Conclusion
Based on the modified moment theory, a mathematical model of a mesh cylindrical nanoshell in the temperature field was constructed.The mathematical model was verified by comparing the resultsobtained with those of other authors.The problem of natural linear oscillations of the shell depending on the geometry of its mesh was solved.It has been shown that the presence of families of ribs with 90-degree angles in the mesh significantly increases the natural frequency.The peculiarities of the stress-strain state of the shells depending on the mesh geometry and the stationary temperature field are studied by the method of embedding.A pronounced marginal effect is observed in the distribution of moments along the length of the shell.The distribution of normal force along the length of the shell strongly depends on the geometry of the mesh.
Designations for classical and non-classical forces and moments: the symmetrical moment tensor of higher order, E -Young's modulus,  -Poisson's ratio of the plate material, l -additional independent material length parameter related to the torsion bending tensor, h-shell thickness,   ,, z  -a known function of the absolute shell temperature, t  -material coefficient of thermal expansion.
j a -fin spacing j-th of the family, j rib width j-th of the family, j angle between the axis  and the axis of the ribs j-th of the family.In such a case, the stresses E3S Web of Conferences 431, 05006 (2023) ITSE-2023 https://doi.org/10.1051/e3sconf/202343105006

Fig. 5 .
Fig.5.Dependence of M  on mesh geometryA change in the mesh geometry has a more significant effect on the normal forces (Figure6).An increase in the number of rib families in the mesh results in a change in the shape of the normal force epure along the length of the shell.The values of the normal force decrease as the number of rib families in the mesh increases.

Fig. 6 .
Fig.6.Dependence of N  on mesh geometryFigure 7  shows the dependence of the normal force along the length of a shell consisting of two families of ribs 1245 ,  135

Fig. 7 .
Fig. 7. Dependence of N  on on temperature Solving the Cauchy problem at 45 ,  135