Abstract

The axisymmetric torsion of a layer of incompressible Mooney–Rivlin material placed between two rigid coaxial cylindrical surfaces is considered. It is assumed that the ends of the cylinder are fixed to avoid axial displacement and the cross-sections orthogonal to the axis of symmetry do not distort during deformation. In this case, the material points move along arcs of circles with an angular velocity which in the general case may depend on both the axial and the radial coordinates. Closed-form solution of the equilibrium equation is obtained. The well-known Rivlin solution is a particular case of this solution. Like in the classical solution, the stress tensor is a function of only the radial coordinate and the twist angle of the material points is a linear function of the axial coordinate. It is shown that torsion that is symmetric about a plane orthogonal to the axis of rotation can be realized only within the framework of the Rivlin solution. The obtained exact solution can be realized in the presence of friction on cylindrical surfaces. The initial-boundary-value problem described by this solution is considered.

中文翻译：

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门尼-里夫林固体的圆形剪切扭转

摘要

考虑了放置在两个刚性同轴圆柱表面之间的不可压缩Mooney-Rivlin材料层的轴对称扭转。假定圆柱体的端部是固定的，以避免轴向位移，并且在变形过程中正交于对称轴的横截面不会变形。在这种情况下，物质点以一定的角速度沿着圆弧移动，角速度通常取决于轴向和径向坐标。得到平衡方程的闭式解。众所周知的Rivlin解决方案就是这种解决方案的特例。像经典解决方案一样，应力张量仅是径向坐标的函数，而材料点的扭转角是轴向坐标的线性函数。结果表明，仅在Rivlin解决方案的框架内才能实现围绕正交于旋转轴的平面对称的扭转。所获得的精确解决方案可以在圆柱表面上存在摩擦的情况下实现。考虑该解决方案描述的初边值问题。