A theory for small deformation analysis of growing bodies with an application to the winding of magnetic tape packs
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A general theory is proposed for the analysis of small deformations in growing bodies. The growing nature of these bodies is due to reasons that are not related to the main deformation process, as for example is the case in solidifying or melting bodies and in magnetic tapes during winding. We assume that the growing boundary is completely described in time. The strain tensor in such growing bodies does not satisfy the compatibility conditions. However, the strain rate tensor satisfies the compatibility condition and as such a hypoelastic constitutive model is proposed as an appropriate elastic model and the whole deformation problem is casted in the form of an initial/boundary value problem. The obtained solution satisfies continuity of deformation and it can easily be extended to account for rate dependent phenomena.
As an application of this theory, a two-dimensional axially symmetric model and its finite element implementation for computing the induced stresses and deformation in a magnetic tape pack during a winding operation, are presented. The effects on the stresses of a tape pack due to a non-uniform across the tape width applied winding tension, are examined and presented.
KeywordsConstitutive Model Compatibility Condition Induce Stress Small Deformation Elastic Model
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- Brown, C. B., Goodman, L. E.: Gravitational stresses in accreted bodies. Proc. R. Soc. London Ser.A276, 571–576 (1963).Google Scholar
- Trincher, V. K.: Formulation of the problem of determining the stress-strain state of a growing body. Mech. Solids19, 119–125 (1984).Google Scholar
- Bykovtsev, G. I., Lukanov, A. S.: Some problems of the theory of solidifying and incrementing media. Mech. Solids20, 112–115 (1985).Google Scholar
- Arutyunyan, N. Kh., Drozdov, A. D., Naumov, V. E.: Mechanics of growing visco-elasto-plastic bodies. Moscow: Nauka Science Publishers 1987.Google Scholar
- Arutyunyan, N. Kh., Manzhirov, A. V.: Contact problems in the mechanics of bodies with accretion. J. Appl. Math. Mech.53, 117–128 (1989).Google Scholar
- Heinlein, M., Mukherjee, S., Richmond, O.: A boundary element method analysis of temperature fields and stresses during solidification. Acta Mech.59, 58–81 (1986).Google Scholar
- Bazant, Z. P.: Viscoelasticity of solidifying porous materials. J. Eng. Mech. Div. Proc. ASCE103, 1049–1067 (1977).Google Scholar
- Bykovtsev, G. I., Lukanov, A. S.: Deformation of a wedge in the process of accretion. Mech. Solids25, 190–193 (1990).Google Scholar
- Altmann, H. C.: Formulas for computing the stresses in center-wound rolls. J. Tech. Ass. Paper Pulp Ind. (TAPPI)51, 176–179 (1968).Google Scholar
- Yogoda, H. P.: Resolution of a core problem in wound rolls. ASME J. Appl. Mech.47, 847–854 (1980).Google Scholar
- Yogoda, H. P.: Generalized formulas for stresses in wound rolls. J. Tech. Ass. Paper Pulp Ind. (TAPPI)64, 91–93 (1981).Google Scholar
- Pfeiffer, J. D.: Prediction of roll defects from roll structure formulas. J. Tech. Ass. Paper Pulp Ind. (TAPPI)62, 83–88 (1979).Google Scholar
- Hakiel, Z.: Non linear model for wound roll stresses. J. Tech. Ass. Paper Pulp Ind. (TAPPI)70, 113–117 (1987).Google Scholar
- Willet, M. S., Poesch, W. L.: Determining the stress distributions in wound rolls of magnetic tape using a non-linear finite difference approach. ASME J. Appl. Mech.55, 365–371 (1988).Google Scholar
- Zabaras, N., Ruan, Y., Richmond, O.: On the calculation of deformations and stresses during axially symmetric solidification. ASME J. Appl. Mech.58, 865–871 (1991).Google Scholar
- Zabaras, N., Liu, S., Koppuzha, J., Donaldson, E.: A hypoelastic model for computing stresses in center-wound rolls of magnetic tape. ASME J. Appl. Mech.61, 290–295 (1994).Google Scholar