Бас редактор Байжуманов М. К
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| [4, 5]. The use of the modified theory of shell dynamics, which takes into account the inertia of rotation and
transverse shift of a normal element, is due to the fact that polymer and composite materials, widely used in modern technology, are characterized by weak resistance to shear deformation, which are not taken into consideration by the classical theory of shells, and within the framework of this approach take nonzero values. Key words: two-dimensional thermoviscoelastic waves, stability of a difference scheme, convergence of a solution of a difference problem, indenter, deformation, stress tensor. Introduction Let us consider a thin shell, the middle surface of which is formed by rotation of a smooth curve s R around the axis Oz (Fig. 1 ). The radius-vector , s r of an arbitrary point on the median surface is set as follows Figure 1 s z s R s R s r , , cos , . (1) Directing vectors of the orthogonal local coordinate system in the point are entered as follows: ISSN 1607-2774 Вестник Государственного университета имени Шакарима города Семей № 4(92) 2020 117 ; , sin , cos z R R s r r s ; 0 , sin , cos R R r r (2) ; ; ds s dz z ds s dR R . s R R Using these vectors, we define the components of the metric tensor ; 1 : , , 2 2 z R r r g r g s s ss , ; 0 ; 2 2 R g g g g R g s s (3) as well as Lamé coefficients . 1 , 1 : 1 R H H g H s The normal vector n will be set as ; , sin , cos 1 R z z r r g n s ; , sin , cos R z z s n n s ; 0 , cos , sin z z n n (4) ; 2 2 ds s R d R ; 2 2 ds s z d z and the components of the second metric tensor b will be calculated as : r n b b R z z R r n b s s ss or, considering the relations between ; 1 2 R z , 1 2 R R R z get ; 1 1 0 2 s ss R R R b ; 0 2 R R z R r n b (5) . 0 s s b b Here , 1 2 0 R R R s 2 0 1 R R R - radii of curvature in directions s and respectively. Mixed components of the second metric tensor are calculated as follows: ; 1 0 s s s R b ; 1 0 R b . 0 s s b b (6) Christophele's characters in the selected coordinate system are as follows ; 0 s s s s ss s ss r r r r r . ; R R r r R R r s s s (7) Now let us proceed to the finding of the basic ratios of the shell dynamics. The determining equations in tensor form for the shell of arbitrary curvature are given in [6]. Let us write the equations of motion, Hooke's law and Cauchy's relations, keeping only their linear part. Motion equation: ; p Q b N h ISSN 1607-2774 Семей қаласының Шәкәрім атындағы мемлекеттік университетінің хабаршысы № 4(92)2020 118 ; q N b Q w h (8) ; q N b Q w h Here, the second derivative in time t is marked by two dots above the letters. For recording the expressions in the right part (8), the rule of summing up by two repetitive indexes is used. Symbols , denote variables , , , , w s − the counter-variant components of the displacement vector and the normal angle of rotation, respectively. M Q N , , − counter-variant components of force and moment tensors; m q p , , − intensities of forces (tangential and transverse), as well as moments distributed on the shell surface. − covariant derivative, − density of material, h − thickness of the shell, 12 3 h I Hook's Law: ; h N ; I M (9) ; 3 3 2 w hC k Q ; 3333 33 33 C C C C The coefficients C are Hook's elastic coefficient matrix: . 2 , 1 ; 3 , 2 , 1 , , , s l k j i C kl ijkl ij 6 5 2 k − the shear coefficient in the S.P.Timoshenko theory. The tensors included in the right parts (9) are determined from Cauchy ratios: ; 2 1 l l ; 2 1 ; 2 1 w r ; w b l (10) ; . b w w The representation of relations (8) − (10) in terms of physical components of displacements, forces and moments is illustrated by the example of the first equation in (8) at s . By definition of the covariant derivative from the invariant tensor is obtained as follows l l l l N N x N N or at s RN R N R R N s N N N N N N s N N ss s ss s s s s s s s s s ss ss s ss s 3 2 Here we used formulas (7). Further from (6) we have . 1 0 s s s s s s s Q R Q b Q b Q b Then the first equation in (8) takes the form . 1 0 s s s ss s ss s p Q R N R R N R R N s N h (11a) Similar calculations lead to the following system of equations: ISSN 1607-2774 Вестник Государственного университета имени Шакарима города Семей № 4(92) 2020 119 ; 1 3 0 p Q R N R R N s N h s s ; 1 0 2 0 q N R R N R Q R R Q s Q w h s ss s s s (11b) ; s s ss s ss s m Q M R R M R R M s жүктеу/скачать 9,41 Mb. Достарыңызбен бөлісу:
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