Geodesic
Scale-dependent deformation model of a layered rectangle
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 4, pp. 48-57 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the problem of deformation of a layered rectangle, the lower side of which is rigidly clamped, a distributed normal load acts on the upper side, and the sides are in conditions of sliding termination. One-parameter gradient theory of elasticity is used to take into account the scale effects. The boundary conditions on the side faces allow the use of the method of separation of variables. The displacements and mechanical loads were decomposed into Fourier series. To find the harmonics of displacements, we have a system of two differential equations of the fourth order. The solution of the system of differential equations is based on the introduction of the elastic potential of displacements. The unknown integration constants are found by satisfying the boundary conditions and the conjugation conditions written in the displacement harmonics. Based on specific examples, the calculations of the horizontal and vertical distribution of displacements, couple and total stresses of a layered rectangle are carried out. The difference between the distributions of displacements and stresses found on the basis of solutions to the problem in the classical formulation and in the gradient formulation is shown. It was found that the total stresses experience a small jump on the conjugation line, due to the fact that, according to the gradient theory of elasticity, not the total stresses, but the components of the load vectors, should be continuous on the conjugation line. A significant influence of an increase in the scale parameter on changes in the values of displacements, total and couple stresses was revealed. 
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A. O. Vatulyan; S. A. Nesterov. Scale-dependent deformation model of a layered rectangle. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 4, pp. 48-57. http://geodesic.mathdoc.fr/item/VMJ_2022_24_4_a3/

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