Geodesic
Consistent equations of nonlinear rectilinear laminated bars theory in quadratic approximation
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 75-87

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Two versions of one-dimensional equilibrium equations for rectilinear laminated bars on basis of S. P. Timoshenko's model subject to transversal compression for each layer and describing geometrical nonlinear deformation by arbitrary displacements and small strain have been derived. The equations are based on the earlier proposed consistent theory of elasticity relations, the usage of which does not lead to spurious bifurcation solutions. The first version corresponds to contact problem statement, when contact stresses are introduced in the coupling points of layers as unknown parameters. The second version corresponds to preliminary satisfaction to the kinematic coupling conditions of layers with respect to displacements.
Keywords: rectilinear bar, laminated structure, geometrically nonlinearity, arbitrary displacements, small strain, S. P. Timoshenko's model, contact stresses, kinematic coupling conditions. 
@article{UZKU_2017_159_1_a6,
     author = {V. N. Paimushin and S. A. Kholmogorov},
     title = {Consistent equations of nonlinear rectilinear laminated bars theory in quadratic approximation},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {75--87},
     publisher = {mathdoc},
     volume = {159},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a6/}
}
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V. N. Paimushin; S. A. Kholmogorov. Consistent equations of nonlinear rectilinear laminated bars theory in quadratic approximation. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 75-87. http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a6/