Transformation geometry under simple shear

A.M. Dumansky^a*, M.Y. Rusin^b**, V.I. Nepovinnykh^b***

^a Mechanical Engineering Research Institute, Russian Academy of Sciences, Moscow, 101990 Russia

^b AO ORPE Technologiya, Obninsk, 249031 Russia

E-mail: *aldumans@yandex.ru, **mrusin@technologya.ru, ***nepvi@mail.ru

Received January 17, 2018

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Abstract

Transformation geometry under simple shear has been analyzed. For this purpose, the mathematical apparatus of the nonlinear theory of elasticity has been used. The sequence of transformations under simple shear has been investigated. Comparison with polar decomposition has been performed. The regions of compression and extension in the deformation ellipse have been determined. It has been shown that a simple shift can be represented in the form of successive actions of rotation, deformation and rotation.

Keywords: simple shear, deformation ellipse, tensor of deformation gradient, Cauchy–Green tensor, polar decomposition.

Figure Captions

Fig. 1. Deformation on simple shear scheme.

Fig. 2. Circle transformation under simple shear.

Fig. 3. Areas of tension and compression in strain-ellipse.

Fig. 4. Cross-section transformation with the help of polar decomposition.

Fig. 5. Cross-section transformation with the help of polar decomposition.

Fig. 6. Cross-section transformation with the help of rotation, tension and rotation.

Fig. 7. Circle transformation with the help of rotation, tension and rotation.

References

1. Dumansky A.M., Tairova L.P. Time-dependent behavior of carbon fibre reinforced laminates. Proc. 2nd Int. Conf. on Advanced Composite Materials and Technologies for Aerospace Applications, ACMTAA-2012. Wrexham, North Wales, UK, 2012, pp. 75–79.

2. Dumansky A.M., Tairova L.P. Analysis of nonlinear mechanical behavior of carbon fibre reinforced polymer laminates.Proc. 3rd Int. Conf. on Advanced Composite Materials and Technologies for Aerospace Applications, ACMTAA-13. Wrexham, North Wales, UK, 2013, pp.  66–69.

3. Dumanskii A.M, Nepovinnykh V.I., Rusin M.Yu., and Terehin A.V. Estimating the limit state of sealants in aircraft structures. Polym. Sci., Ser. D, 2014, vol. 7, no. 3, pp. 201–207. doi: 10.1134/S1995421214030046.

4. Tatarniov O.V., Rusin M.Y., Vorob'ev S.B., Prasolov A.N. Modeling of elastic behavior of organic silicon sealants under finite strain. Klei, Germetiki, Tekhnol., 2008, no. 2, pp. 20–23. (In Russian)

5. Nepovinnykh V.I., Rusin M.Yu., Dumanskii A.M. General rules of formation of an ordered relief on a sealant fracture surface under shear loading. Polym. Sci., Ser. D, 2016, vol. 9, no. 2. pp. 145–150. doi: 10.1134/S1995421216020167.

6. Kolodyazhnyi S.Yu. Strukturno-kinematicheskaya evolyutsiya yugo-vostochnoi chasti baltiiskogo shchita v paleoprotorozoe [Paleoproterozoic Structural Kinematic Evolution of the Southeastern Part of the Baltic Shield]. Moscow, GEOS, 2006. 334 p. (In Russian)

7. Gzovsky M.V. Osnovy tektonofiziki [Principles of Tectonophysics]. Moscow, Nauka, 1975. 536 p. (In Russian)

8. Yakovlev F.L. Strain analysis of shear zones in terms of types of secondary fractures: Problem statement. Geodin. Tektonofiz., 2011, vol. 2, no. 1, pp. 68–82. (In Russian)

9. Hart-Smith L.J. The design of adhesively bonded joints. In: Dillard D.A., Pocius A.V. (Eds.) The Mechanics of Adhesion. Amsterdam, Elsevier, 2002, pp. 746–777.

10. Truesdell C. A First Course in Rational Continuum Mechanics. Baltimore, Md., 1972. 592 p.

11. Ogden R.W. Elements of the theory of finite elasticity. In: Fu Y.B., Ogden R.W. (Eds.) Nonlinear Elasticity. Theory and Applications. Cambridge, Cambridge Univ. Press, 2001, pp. 1–57.

12. Boulanger Ph., Hayes M. Shear. In: Fu Y.B., Ogden R.W. (Eds.) Nonlinear Elasticity. Theory and Applications. Cambridge, Cambridge Univ. Press, 2001. pp. 201–232.

13. Golovanov A.I. Kinematics of finite elastoplastic deformations. Russ. Math., 2010, vol. 54, no. 7, pp. 12–25. doi: 10.3103/S1066369X10070029.

14. Mase G. Theory and Problems of Continuum Mechanics. New York, McGraw Hill Company, 1970. 320 p.

15. Godunov S.K., Romensky E.I. Elementy mekhaniki sploshnykh sred i zakony sokhraneniya [Elements of Continuum Mechanics and Conservation Laws]. Novosibirsk, Nauchn. Kn., 1998. 267 p. (In Russian)

16. Destrade M., Murphy J.G., Saccomandi G. Simple shear is not so simple. Int. J. Non-Linear Mech., 2010, vol. 47, pp. 210–214.

For citation: Dumansky A.M., Rusin M.Y., Nepovinnykh V.I. Transformation geometry under simple shear. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 1, pp. 196–206. (In Russian)

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