Closed set of the uniqueness conditions and bifurcation criteria in generalized coupled thermoplasticity for small deformations

Abstract

This paper reports the results of a study into global and local conditions of uniqueness and the criteria excluding the possibility of bifurcation of the equilibrium state for small strains. The conditions and criteria are derived on the basis of an analysis of the problem of uniqueness of a solution involving the basic incremental boundary problem of coupled generalized thermo-elasto-plasticity. This work forms a follow-up of previous research (Śloderbach in Bifurcations criteria for equilibrium states in generalized thermoplasticity, IFTR Reports, 1980, Arch Mech 3(35):337-349, 351-367, 1983), but contains a new derivation of global and local criteria excluding a possibility of bifurcation of an equilibrium state regarding a comparison body dependent on the admissible fields of stress rate. The thermal elasto-plastic 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.Keywords Bifurcation of the equilibrium state · Conditions and criteria of uniqueness · Boundary-value problem · Generalized coupled thermo-elasto-plasticity · Comparison bodies IntroductionThe incremental boundary-value problem of generalized coupled thermoplasticity is formulated in this paper. This is followed by an interpretation of the uniqueness conditions for the solution of that problem. The necessary and sufficient local uniqueness conditions are deduced together with the global sufficient uniqueness condition. A similar incremental boundary-value problem of coupled generalized thermoplasticity was investigated and discussed [1,3]. In this paper, necessary and sufficient local and global conditions of uniqueness of solution of an incremental boundary-value problem of coupled generalized thermoplasticity for the case of small displacements gradients (small strains) are derived. Uniqueness conditions for the generalized coupled thermoplasticity [1][2][3] and suitable comparison bodies [1][2][3][4][5][6][7] are identified for this purpose. The derived local and global uniqueness conditions are suitable necessary and sufficient conditions excluding occurrence of the bifurcation of equilibrium state in coupled generalized thermoplasticity and for suitable comparison bodies (also in isothermal loading processes).Early papers by Mróz [8,9] defined local conditions of uniqueness for solving an incremental boundary problem for the case of non-associated laws of plastic flow, for isothermal processes and small strains. A similar local condition was obtained by Hueckel and Maier in [10,11]. In their analysis, the stability of the material is defined by means of a condition which states that the half of the product of the stress rate tensor needs to be a positive value. The reported study was confined to the case of the isothermal theory of plasticity (with Communicated by Andreas Öchsner. Z.Śloderbach (B)Faculty of Applications of Chemistry and Mechani...

On the Uniqueness Conditions and Bifurcation Criteria in Coupled Thermo-Elasto-Plasticity

“…The global conditions and local conditions of uniqueness are derived. The uniqueness conditions for the generalized coupled thermoplastic body and suitable comparison bodies (Mróz and Raniecki,[4], [5]; Raniecki, [6]; Śloderbach, [1], [3], [7]) were derived. The derived global and local uniqueness conditions are suitable necessary and sufficient conditions excluding the occurrence of the bifurcation state of equilibrium in coupled generalized thermoplasticity and also in isothermal loading processes.…”

“…The derived global and local uniqueness conditions are suitable necessary and sufficient conditions excluding the occurrence of the bifurcation state of equilibrium in coupled generalized thermoplasticity and also in isothermal loading processes. We prove that the local uniqueness conditions for the generalized thermo-elastoplastic body and the comparison bodies are same, but the methods of calculating the bifurcation state (using the global sufficient uniqueness condition) for the case of comparison bodies are less complicated (there exists a linear dependence between the stress rate and strain rate), see (Mróz and Raniecki,[4], [5]; Raniecki, [6]; Śloderbach, [1], [3], [7]). Thus, introduction of such comparison bodies seems to be advisable.…”

“…In a generalized case, the constitutive equations of coupled thermoplasticity are non-associated laws of plastic flow, even if Gyarmati's (Gyarmati, [9]) postulate is assumed (see Śloderbach, [1], [3], [7], [8]). They also include the effects of thermomechanical coupling effects and take into account the elastic-plastic conjugation.…”

“…If the critical loads are exceeded, bifurcation of the equilibrium state is possible. In comparison to the previous papers about the uniqueness of solution and bifurcation of equilibrium states (Śloderbach, [1], [3], [7]; Mróz and Raniecki, [4], [5]; Raniecki, [6];), this paper presents a derived new comparison body dependent on statically admissible fields of stress rate. For such a body a sufficient local and global uniqueness conditions were derived, developing problems discussed in the above mentioned papers.…”

“…However, it is necessary to determine velocity fields of strain and displacement as well as temperatures for the input values of stress and divergence of a vector of the flux of heat exchanged by the surface through the elementary area [see the problems b 1 and b 2 considered in this paper and in Śloderbach [1], [3], [7].…”