Mathematical modeling of the stress-strain state in surface hardened thin-walled tubes with regard to the residual shear stresses

We suggest the phenomenological mathematical model of the stress-strain state reconstruction in the surface hardened thin-walled tube with an inner diameter 45 mm and outer diameter 51.5 mm made of steel EI961 and treated by diamond smoothing of the outer surface. It is shown that if all stress components depend on radius only, then the components are in the cylindrical coordinate system. The experimental research is made for the samples which were softened under two load modes (radial force) of the diamond ball attachment of 200 and 300 N value. The experimental values of residual stresses , , and in the surface layer are obtained by the ring and strip method using the layer-by-layer electrochemical pickling of the hardened layer. The experimentally measured values of the strip beam deflection, split ring angular opening and axial displacement of cut edges relative to each other are used for this purpose. The hardening anisotropy parameter which relates the axial and circumferential components of plastic strain is included in the mathematical model. To solve the formulated problems we use the hypotheses of plastic incompressibility of the material, the absence of secondary plastic deformations of the material in the surface layer compression area and the hypotheses of flat sections and straight radii. We present the method for solving the stress-strain state reconstruction boundary value problems, which allows obtaining the missing component and all residual plastic strain components. The validation of the computational data obtained by mathematical modelling for adequacy to the experimental data for the two modes of hardening is made. There is a close agreement between the computational and experimental data. The numerical values for the hardening anisotropy parameter are given. By using this parameter we are able to theoretically describe the observable experimental layering of axial and circumferential stresses in depth of the hardened layer. It is theoretically and experimentally established that the absolute values of maximum shear stresses is an order of magnitude smaller than the absolute values of maximum normal stresses. We also discuss the questions of the effect of shear stresses on high-cycle fatigue and creep of the hardened thin-walled tubes. The main results of the research are illustrated by the tabular data and corresponding diagrams of the residual stresses distribution in depth of the hardened layer.