Geodesic
Numerical investigation of hyperelastic solids with contact interaction
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 4, pp. 644-656 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is devoted to the construction of a computational algorithm for the investigation of hyperelastic solids with the contact interaction. In the framework of the previously developed algorithm for the investigation of large deformations of three-dimensional solids, the solutions of contact problems based on the equation of the principle of virtual work in velocity term have been considered. Contact interaction has been modeled on the basis of the “master-slave” approach. The closest point projection procedure has been used to find the contact area. A contact functional has been built on the basis of the principle of virtual work in velocity term within the penalty method. The linearization of the kinematic relations and contact functional are based on the capacity on the possible velocities of penetration. The solution of the nonlinear system of equations has been obtained using the method of step loading. The numerical implementation is based on the finite element method.
Keywords: finite deformations, contact interaction, penalty method. 
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A. I. Abdrakhmanova; L. U. Sultanov. Numerical investigation of hyperelastic solids with contact interaction. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 4, pp. 644-656. http://geodesic.mathdoc.fr/item/UZKU_2018_160_4_a3/

[1] Wriggers P., Computational contact mechanics, John Wiley Sons Ltd, 2002, 464 pp.

[2] Wriggers P., “Finite element algorithms for contact problems”, Archives of Computational Methods in Engineering, 2:4 (1995), 1–49 | DOI | MR

[3] Konyukhov A., Izi R., Introduction to computational contact mechanics: a geometrical approach, John Wiley Sons Ltd, 2015, 302 pp. | MR | Zbl

[4] Konyukhov A., Geometrically exact theory for contact interactions, KIT Scientific Publ., Karlsruhe, 2011, xix+516 pp.

[5] Konyukhov A., Schweizerhof K., Computational Contact Mechanics – Geometrically Exact Theory for Arbitrary Shaped Bodies, Springer, Heidelberg–New York–Dordrecht–London, 2012, 443 pp. | MR

[6] Laursen T. A., Computational Contact and Impact Mechanics, Springer, Berlin–Heidelberg, 2002, xv+454 pp. | MR | Zbl

[7] Puso M. A., Laursen T. A., Solberg J., “A segment-to-segment mortar contact method for quadratic elements and large deformations”, Comput. Methods Appl. Mech. Eng., 197:6–8 (2008), 555–566 | DOI | MR | Zbl

[8] Yang B., Laursen T. A., Meng X., “Two dimensional mortar contact methods for large deformation frictional sliding”, Int. J. Numer. Methods Eng., 62:9 (2005), 1183–1225 | DOI | MR | Zbl

[9] Burago N. G., Zhuravlev A. B., Nikitin I. S., “Analysis of stress state of GTE contact system “disk-blade””, Vychisl. Mekh. Sploshnykh Sred, 4:2 (2011), 5–16 (In Russian)

[10] Badriev I. B., Makarov M. V., Paimushin V. N., “Contact statement of mechanical problems of reinforced on a contour sandwich plates with transversally-soft core”, Russ. Math., 61:1 (2017), 69–75 | DOI | MR | Zbl

[11] Badriev I. B., Paimushin V. N., “Refined models of contact interaction of a thin plate with positioned on both sides deformable foundations”, Lobachevskii J. Math., 38:5 (2017), 779–793 | DOI | MR | Zbl

[12] Al-Dojayli M., Meguid S. A., “Accurate modeling of contact using cubic splines”, Finite Elements in Analysis and Design, 38:4 (2002), 337–352 | DOI | Zbl

[13] Laursen T. A., “Convected description in large deformation frictional contact problems”, Int. J. Solids Struct., 31:5 (1994), 669–681 | DOI | MR | Zbl

[14] Parisch H., Lubbing Ch., “A formulation of arbitrarily shaped surface elements for three dimensional large deformation contact with friction”, Int. J. Numer. Methods Eng., 40:18 (1997), 3359–3383 | 3.0.CO;2-5 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[15] Puso M. A., Laursen T. A., “A 3D contact smoothing method using Gregory patches”, Int. J. Numer. Methods Eng., 54:8 (2002), 1161–1194 | DOI | MR | Zbl

[16] Yang B., Laursen T. A., Meng X., “Two dimensional mortar contact methods for large deformation frictional sliding”, Int. J. Numer. Methods Eng., 62:9 (2005), 1183–1225 | DOI | MR | Zbl

[17] Izi R., Konyukhov A., Schweizerhof K., “Large penetration algorithm for 3D frictionless contact problems based on a covariant form”, Comput. Methods Appl. Mech. Eng., 217–220 (2012), 186–196 | DOI | MR | Zbl

[18] Simo J. S., Laursen T. A., “An augmented lagrangian treatment of contact problems involving friction”, Computers and Structures, 42:1 (1992), 97–116 | DOI | MR | Zbl

[19] Bonet J., Wood R. D., Nonlinear continuum mechanics for finite element analysis, Cambridge Univ. Press, Cambridge, 1997, 268 pp. | MR | Zbl

[20] Oden J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, 1972, 442 pp. | Zbl

[21] Sultanov L. U., “Analysis of large elastic-plastic deformations: Integration algorithm and numerical examples”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 159, no. 4, 2017, 509–517 (In Russian)

[22] Davydov R. L., Sultanov L. U., “Numerical algorithm for investigating large elasto-plastic deformations”, J. Eng. Phys. Thermophy., 88:5 (2015), 1280–1288 | DOI

[23] Abdrakhmanova A. I., Sultanov L. U., “Numerical modelling of deformation of hyperelastic incompressible solids”, Mater. Phys. Mech., 26:1 (2016), 30–32

[24] Davydov R. L., Sultanov L. U., “Numerical algorithm of solving the problem of large elastic-plastic deformation by FEM”, Vestn. Permsk. Nats. Issled. Politekh. Univ.: Mekh., 2013, no. 1, 81–93 (In Russian) | DOI

[25] Abdrakhmanova A. I., Sultanov L. U., “Numerical investigation of nonlinear deformations with contact interaction”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 160, no. 3, 2018, 423–434 (In Russian) | MR

[26] Golovanov A. I., Sultanov L. U., “Postbuckling state analysis of three-dimensional bodies taking into account finite strains”, Russ. Aeronaut., 51:4 (2008), 362–368 | DOI

[27] Bathe K.-J., Finite element procedures, Prentice-Hall, Englewood Cliffs, N. J., 1996, xiv+1037 pp.

[28] Zienkiewicz O. C., Taylor R. L., The finite element method, v. 1, 2, McGraw-Hill, London–N. Y., 1989 | MR