Stress in Rotating Disks and Cylinders

Source: arXiv


The solution of the classic problem of stress in a rotating elastic disk or cylinder, as solved in standard texts on elasticity theory, has two features: dynamical equations are used that are valid only in an inertial frame of reference, and quadratic terms are dropped in displacement gradient in the definition of the strain. I show that, in an inertial frame of reference where the dynamical equations are valid, it is incorrect to drop the quadratic terms because they are as large as the linear terms that are kept. I provide an alternate formulation of the problem by transforming the dynamical equations to a corotating frame of reference of the disk/cylinder, where dropping the quadratic terms in displacement gradient is justified. The analysis shows that the classic textbook derivation of stress and strain must be interpreted as being carried out in the corotating frame of the medium.

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    • "Furthermore, this derivation makes the role of the reference (unstrained) configuration more clear in the definition of the strain tensors. Clarifying this role is of importance for applying finite deformation theory to prestresssed materials, which are capable of withstanding higher-stress applications, such as in rotating machinery [7] [16]. Finally, the derivation presented here allows the generalization of the definition of strain tensors to the realm where general relativity applies [17] [18]. "
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    ABSTRACT: A coordinate independent derivation of the Eulerian and Lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a three-point tensor. The derivation explicitly shows that the Eulerian and Lagrangian strain tensors are two-point tensors, each a function of both the spatial and material coordinates. The Eulerian strain is a two-point tensor that transforms as a second rank tensor under transformation of spatial coordinates and transforms as a scalar under transformation of the material coordinates. The Lagrangian strain is a two-point tensor that transforms as scalar under transformation of spatial coordinates and transforms as a second rank tensor under transformation of the material coordinates. These transformation properties are needed when transforming the strain tensors from one frame of reference to another moving frame.
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    ABSTRACT: Many recent papers have questioned Irving and Kirkwood's atomistic expression for stress. In Irving and Kirkwood's approach both interatomic forces and atomic velocities contribute to stress. It is the velocity-dependent part that has been disputed. To help clarify this situation we investigate (i) a fluid in a gravitational field and (ii) a steadily rotating solid. For both problems we choose conditions where the two stress contributions, potential and kinetic, are significant. The analytic force-balance solutions of both these problems agree very well with a smooth-particle interpretation of the atomistic Irving-Kirkwood stress tensor.
    Physical Review E 04/2009; 79(3 Pt 2):036709. DOI:10.1103/PhysRevE.79.036709 · 2.29 Impact Factor
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    ABSTRACT: We have obtained an exact solution of the plane problem for a porous tube rotating around its fixed axis of symmetry. The material of the tube is saturated with an ideal compressible liquid. Within the framework of the Bowen theory, for constructing the initial relations, we take into account the interconnections between stress tensors, strain tensors, relative density of the liquid filler, and varying porosity of the material. We present relations for the radial displacements, components of the stress tensor and densities in the skeleton and liquid filler, and also the porosity and mass concentration of the mixture. We have carried out a numerical analysis of the radial distributions of these characteristics for solid and hollow cylinders made of sandstone and saturated with kerosene. The critical numbers of revolution for tubes of saturated and dry materials have been found.
    Journal of Mathematical Sciences 05/2010; 167(2):182-196. DOI:10.1007/s10958-010-9914-0


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