Solution of longitudinal shear problems of physically nonlinear bodies with properties depending on the type of stress state

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Abstract

Deformation properties of structural materials, rocks, composite materials, etc. depend on the type of external action, and the degree of this dependence is determined by the structural features of the materials. These materials are characterized by a relationship between volume and shear deformation. Deformation curves are nonlinear even at small deformations. This paper presents constitutive relations describing the nonlinear behavior of these materials under small deformations. It is shown that classical hypotheses of anti-plane shear cannot be applied. The problem of anti-plane shear of a long prismatic body with a square cross-section containing a round through hole in the plane of the cross-section is solved numerically. It is shown that under shear loading conditions, the body is characterized by a triaxial stress state and a change in volume.

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About the authors

E. V. Lomakin

Lomonosov Moscow State University

Author for correspondence.
Email: evlomakin@yandex.ru

Corresponding Member of the RAS

Russian Federation, Moscow

O. P. Korolkova

Lomonosov Moscow State University; Research Institute of Mechanics of Lomonosov Moscow State University

Email: ol.shendrigina@mail.ru
Russian Federation, Moscow; Moscow

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Supplementary files

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2. Fig. 1. Equivalent stress-strain curves for gray cast iron SCh 15–32 at different values of the parameter ξ: ⅓ (1), 0.232 (2), 0 (3), –0.064 (4), –0.126 (5), –⅓ (6).

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3. Fig. 2. Section of the body under consideration under longitudinal shear conditions.

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4. Fig. 3. Distribution of deformations (a) and stresses (b) along the straight line x₂ = 0.

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5. Fig. 4. Distribution of deformations (a) and stresses (b) along the straight line x₁ = 0.

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6. Fig. 5. Volumetric deformation (a) and hydrostatic pressure (b) at x₁² + x₂² = a²

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