Abstract
In research and industry, the in-plane torsion test is applied to investigate the material behaviour at large plastic strains: a sheet is clamped in two concentric circles, the boundaries are twisted against each other applying a torque, and simple shear of the material arises. This deformation is analysed within the scope of finite elasto-plasticity. An additive decomposition of the Almansi strain tensor is derived, valid as an approximation for arbitrary large plastic strains and sufficiently small elastic strains and rotations. Constitutive assumptions are the von Mises yield criterion, an associative flow rule, isotropic hardening, and a physically linear elasticity relation. The incremental formulation of the elasticity relation applies covariant Oldroyd derivatives of the stress and the strain tensors. The assumptions combined with equilibrium conditions lead to evolution equations for the distribution of stresses and accumulated plastic strain. The nonzero circumferential stress must be determined from the equilibrium condition because no deformation is present in tangential direction. As a result, a differential-algebraic-equation (DAE) system is derived, consisting of three ordinary differential equations combined with one algebraic side condition. As an example material, properties of a dual phase steel DP600 are analysed numerically at an accumulated plastic strain of 3.0. Radial normal stresses of 3.1% and tangential normal stresses of 1.0% of the shear stresses are determined. The influence of the additional normal stresses on the determination of the flow curve is 0.024%, which is negligibly small in comparison with other experimental influences and measurement accuracies affecting the experimental flow curve determination.
Highlights
The experimental identification of material parameters and functions is one key aspect of every numerical process analysis
Remark: There is a positive circumferential stress, but no corresponding strain. This is no contradiction because the non-vanishing stress σφφ is a pure consequence of the equilibrium condition and has nothing to do with any constitutive law
The maximum deviation to the classical Eq (8) is 0.024%, which is negligibly small in comparison with other experimental influences and measurement accuracies influencing the experimental flow curve determination
Summary
The experimental identification of material parameters and functions is one key aspect of every numerical process analysis. The in-plane torsion test is the only test known enabling ideal simple shear deformation experimentally This test has been first applied by Marciniak [8] for the characterisation of the flow curve and the Bauschinger effect of copper. Very large equivalent plastic strains up to 3.0 can be achieved in the in-plane torsion test [17], one basic discussion focusses on the stress and strain states: by use of the FEM, Sowerby et al [18] showed that there are radial and tangential normal stresses and strains for elastic and plastic deformations Their results indicate that neglecting the additional stresses and strains does not produce a significant error in the evaluation of the test. Analytical solutions for the in-plane torsion test for small elastic and large plastic deformations are considered to estimate the error by the traditional method of evaluation
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