Geodesic
Stress and strain fields in a plate of stress state dependent material properties
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 4, pp. 458-466 
E. V. Lomakin; O. P. Shchendrigina. Stress and strain fields in a plate of stress state dependent material properties. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 18 (2018) no. 4, pp. 458-466. http://geodesic.mathdoc.fr/item/ISU_2018_18_4_a8/
@article{ISU_2018_18_4_a8,
     author = {E. V. Lomakin and O. P. Shchendrigina},
     title = {Stress and strain fields in a plate of stress state dependent material properties},
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     pages = {458--466},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2018_18_4_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The paper analyzes the properties of the constitutive relations proposed to describe the behavior of materials whose deformation diagrams depend on the type of external forces. In this case, various forms of nonlinearity arise, related to the dependence of the properties of materials on the type of the stressed state, the nonlinearity of the deformation diagrams, and the relationship between the shear and volume deformation processes. The influence of these forms of nonlinearity on the distribution of stresses and strains in a circular plate under different boundary conditions is investigated. The stress and strain fields were calculated for structural graphite and they are compared with the results of calculations for a classical physically nonlinear material whose properties are invariant to the form of external forces. The conditions imposed on the parameters of material functions that ensure the uniqueness of the solution of boundary value problems are established.

[1] Ambartsumyan S. A., Elasticity Theory of Different Modulus, Translated by Wu R., China Railway Publishing House, Beijing, 1986

[2] Jones R. M., “Buckling of circular cylindrical shells with different moduli in tension and compression”, AIAA J., 9:1 (1971), 53–61 | DOI | Zbl

[3] Bert C. W., “Models for fibrous composites with different properties in tension and compression”, J. Eng. Mater. Technol., 99:4 (1977), 344–349 | DOI

[4] Vijayakumar K., Rao K. P., “Stress-strain relations for composites with different stiffnesses in tension and compression”, Comput. Mech., 2:3 (1987), 167–175 | DOI | Zbl

[5] Ye Z., Yu H., Yao W., “A new elasticity and finite element formulation for different Young's modulus when tension and compression loadings”, J. of Shanghai Univ., 5:2 (2001), 89–92 | DOI | MR | Zbl

[6] Patel B. P., Khan K., Nath Y., “A new constitutive model for bimodular laminated structures: Application to free vibrations of conical/cylindrical panels”, Composite Structures, 110 (2014), 183–191 | DOI

[7] Timoshenko S., Strength of Materials, v. II, Advanced Theory and Problems, D. Van Nostrand Company, Inc., Lancaster, PA, 1941, 510 pp. | MR

[8] Lomakin E. V., “Difference in the modules of composite materials”, Mech. Compos. Mater., 17:1 (1981), 18–24 | DOI

[9] Khan K., Patel B. P., Nath Y., “Vibration analysis of bimodulus laminated cylindrical panels”, J. Sound Vibrations, 321:1–2 (2009), 166–183 | DOI

[10] Khan K., Patel B. P., Nath Y., “Free and forced vibration characteristics of bimodular composite laminated circular cylindrical shells”, Composite Structures, 126 (2015), 386–397 | DOI

[11] Du Z., Guo X., “Variational principles and the related bounding theorems for bi-modulus materials”, J. Mech. Phys. Solids, 73 (2014), 83–211 | DOI | MR

[12] Querin O. M., Victoria M., Marti P., “Topology optimization of truss-like continua with different material properties in tension and compression”, Struct. Multidisc. Optim., 42:1 (2010), 25–32 | DOI

[13] Liu S., Qiao H., “Topology optimization of continuum structures with different tensile and compressive properties in bridge layout design”, Struct. Multidisc. Optim., 43:3 (2011), 369–380 | DOI

[14] Lomakin E. V., “Constitutive relations of deformation theory for dilatant media”, Mech. Solids, 26:6 (1991), 66–75

[15] Belyakova T. A., Lomakin E. V. Elastoplastic deformation of a dilatant medium subjected to a plane stress state near a crack tip, Mech. Solids, 39:1 (2004), 81–87

[16] Novozhilov V. V., “On the physical meaning of stress invariants used in the theory of plasticity”, PMM, 16:5 (1952), 615–619 (in Russian) | Zbl