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Objective Stress Rates and Residual Strains in Stress Cycles

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The main objective of the present work is to study the residual deformation effects of Truesdell's widely-used hypoelastic model for four well-known objective stress rates under a four-phase stress cycle associated with axial tension and/or torsion of thin-walled cylindrical tubes. Here, two kinds of models based upon the Cauchy stress and the Kirchhoff stress will be treated. The reduced systems of differential equations of these rate constitutive equations are derived and studied for Jaumann, Green-Naghdi, logarithmic and Truesdell stress rates, separately. Analytical solutions in some cases and numerical solutions in all cases are obtained using these reduced systems. Comparisons between the residual deformations are made for different cases. It may be seen that only the logarithmic stress rate results in no residual deformation. In particular, results indicate that Green-Naghdi rate would generate unexpected residual deformation effect that is essentially different from that resulting from Jaumann rate.

On the other hand, it is realized that this study accomplishes an alternative, direct proof for the non-integrability problem of Truesdell's hypoelastic rate equation with classical stress rates. This problem has been first treated successfully by Simo and Pister in 1984 using Bernstein's integrability conditions. However, such treatment needs to cope with a coupled system of nonlinear partial differential equations in Cauchy stress. Here, a different idea is used. It is noted that every integrable hypoelastic equation is just an equivalent rate form of an elastic equation and hence should produce no residual deformations under every possible stress cycle. Accordingly, a hypoelastic model with a stress rate has to be non-integrable, whenever a stress cycle can be found under which this model generates residual deformation. According to this idea of reductio ad absurdum, a well-designed stress cycle is introduced and the corresponding residual deformations are calculated. Unlike the treatment of Bernstein's integrability conditions, it may be a simple and straightforward matter to calculate the final deformations for a given stress cycle. This has been done in this study for several well-known stress rates.

Affiliations: 1: Institute of Numerical and Applied Mathematics, Georg-August-University of Göttingen, Lotzestrasse 16-18, D-37083 Göttingen, Germany;, Email: xiazhang@math.uni-goettingen.de

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