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.

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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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