The global and local conditions of uniqueness and the criteria excluding a possibility of bifurcation of the equilibrium state for small strains are derived. The conditions and criteria are derived analyzing the problem of uniqueness of solution of the basic incremental boundary problem of coupled generalized thermo-elasto-plasticity. This paper is a continuation of some previous works by the author, but contains new derivation of the global and local criteria excluding a possibility of bifurcation of the equilibrium state for a comparison body dependent on statically admissible fields of stress velocity. All the thermal elastoplastic coupling effects, non-associated laws of plastic flow and influence of plastic strains on thermoplastic properties of a body were taken into account in this work. Thus, the mathematical problem considered here is not a self-conjugated problem. The paper contains four Appendices A, B, C and D where the local necessery and sufficient conditions of uniqueness have been derived.
Part I. Derivation of basic equations and relationships
The paper contains two parts. In the first part, basic relationships were derived and some problems connected with stability loss during hydraulic forming of round metallic drawpieces with liquid pressure were discussed. The aim of the considerations is to test drawability of sheets by estimation of acceptable values of plastic strains and the corresponding heights of spherical shells. The analysis was based on some selected real conditions of stability loss. The influence of the coefficient of material hardening for the drawpiece, the coefficient of normal anisotropy and coefficients of plane anisotropy on acceptable values of plastic strains and heights of the formed drawpieces corresponding to the given condition of stability loss was also tested.
According to the postulate concerning a local change of the “actual active radius” with a bending angle in the bend zone, a generalized model of strain during metal tube bending was derived. The tubes should be subjected to bending at tube bending machines by the method of wrapping at the rotating template and with the use of a lubricated steel mandrel. The model is represented by three components of strain in the analytic form, including displacement of the neutral axis. Generalization of the model during bending metal tubes at the tube bending machines as compared with the existing papers (Śloderbach, 1999; Śloderbach and Rechul, 2000) consists in including the neutral axis displacement and possibility of determination of strains at each point along the thickness of the wall of the bent tube in the bending and bend zone. The derived scheme of strain satisfies initial and boundary kinematic conditions of the bending process, conditions of continuity and inseparability of strains. The obtained analytic expressions can be classified as acceptable from the kinematic point of view
The generalized strain scheme in bending metal tubes at bending machines with the use of a mandrel presented in Śloderbach (1999; 2002; 20131,2; 2014) satisfies initial and boundary kinematic conditions of bending, conditions of continuity and inseparability of strains. This paper introduces three formal simplifications gradually imposed into forms of principal components of the generalized strain model giving suitable simplifications of the 1st, 2nd and 3rd types. Such mathematical simplifications cause that the obtained strain fields do not satisfy the condition of consistency of displacements and strain continuity. The simplified methods determine safer values of the wall thickness than those from the generalized continuous strain scheme. The condition of plastic incompressibility was used for the derivation of an expression for distribution of wall thickness of the bent elbow in the layers subjected to tension and compression for three examples of discontinuous kinematic strain fields.
In this paper, the field equations of the generalized coupled thermoplasticity theory are derived using the postulates of classical thermodynamics of irreversible processses. Using the Legendre transformations two new thermodynamics potentials P and S depending upon internal thermodynamic forces Π are introduced. The most general form for all the thermodynamics potentials are assumed instead of the usually used additive form. Due to this assumption, it is possible to describe all the effects of thermomechanical couples and also the elastic-plastic coupling effects observed in such materials as rocks, soils, concretes and in some metalic materials. In this paper not only the usual postulate of existence of a dissipation qupotential (the Gyarmati postulate) is used to derive the velocity equation. The plastic flow constitutive equations have the character of non-associated flow laws even when the Gyarmati postulate is assumed. In general formulation, the plastic strain rate tensor is normal to the surface of the generalized function of plastic flow defined in the the space of internal thermodynamic forces Π but is not normal to the yield surface. However, in general formulation and after the use the Gyarmati postulate, the direction of the sum of the plastic strain rate tensor and the coupled elastic strain rate tensor is normal to the yield surface.
In the paper a procedure and methodology of acceptable calculations of powers, displacements (of amplitudes) and velocities of pulses according to ISO standards depending on the rotational speeds (frequencies) of rotor during attempts of dynamic breaking down taking into account effectiveness of vibration isolation were described. The proposed method is regarding one-sidedly hung rotors appearing in such appliances as rotors of fans, blowers, pumps and others. As permissible parameters of pulses values which are given in international standards PN-ISO-10816-1:1998P and PN-ISO-7919-1:2001 were assumed. Calculations were compared in the form of tables and they depicted with relevant graphs which have the character of useful nomograms. Also, recommendations for the design of the foundations and guidelines related to dimensions of the foundation, the weight and the way of fixing the device were given. In the end also closing remarks and conclusions were given.
The relations to calculate the maximum value of strains in processes of bending tubes on benders, in stretched layers of tubes, are presented in this work on the basis of the EU-Directive concerning production of pressure equipment. It has been shown that for large deformations that occur during bending of the pipes on knees, logarithmic strain measures (real) and relative strain measures give different values of strain but equal wall thicknesses in the bending zone. Logarithmic measures are frequently used in engineering practice and are valid for large and small deformations. Reverse expressions were also derived to calculate the required initial wall thickness of the tube to be bent, in order to obtain the desired wall thickness of the knee after bending.
In this study, the influence of thermomechanical coupling effect - the effect of thermal expansion due to dissipation of the energy of plastic deformation, with and without taking into account the stored energy of plastic deformation (SEPD) for the distribution of stresses, strains, temperature, the applied pressure and the residual stresses is examined. The residual stresses remain in a thick-walled tube (a cylindrical thick-walled tank) after removing the internal pressure in the process of purely elastic unloading. The analysis is made on the example of an analitycal solution for a thick-walled tube subjected to a quasistatically increasing internal pressure for the case of adiabatic processes (without heat flow). Since the loading with internal pressure is quasi-static (monotonic), then neglecting the process of heat flow can lead to some different results in calculated stresses, deformations, temperature, internal pressure and residual stresses. The calculations for isothermal type of processes of deformations (without heat or ideal cooling) are also performed for the estimation of these differences. The results calculated for the process with heat flow should be intermediate between the values obtained for isothermal and adiabatic processes.
The paper presents an aproximate analytic method for determination of the stored energy of plastic deformation during cold bending of metal tubes at bending machines. Calculations were performed for outer points of the tube layers subjected to tension and compression (the points of maximum strains). The percentage of stored energy related to the plastic strain work was determined and the results were presented in graphs. The influence and importance of the stored energy of plastic deformation on the service life of pipeline bends are discussed.