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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