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

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.


Introduction
The 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 Mechanics, Opole University of Technology, Luboszycka 7, 45-036 Opole, Poland E-mail: z.sloderbach@po.opole.pl no thermomechanical couplings), the elastic-plastic coupling effects and non-associated laws of plastic flow being preserved. The minimum principle (a principle of stability) for incremental, isothermal elasto-plasticity with non-associated flow laws for small deformation was derived in a paper by Maier [12]. In [4,5] Mróz and Raniecki derived local necessary and sufficient conditions of uniqueness in coupled thermoplasticity for associated laws of plastic flow, without the elastic-plastic coupling effects and for small strains. The obtained local conditions were not optimal (i.e. they were not minimal). The procedure of optimization was presented in [1,3,13,14]. In Ref. [15], Raniecki and Sawczuk derived the equations of field and constitutive equations for a body of coupled thermoplasticity with associated laws of plastic flow. In Ref. [16] they applied the local conditions of uniqueness for considerations of the influence of selected thermomechanical couplings on unstable behaviour of materials and selected machine components subjected to variable thermomechanical fields. The studies into global conditions of uniqueness, stability and criteria of bifurcation for elasto-plastic bodies for the case of large strains and the associated laws of plastic flow and isothermal processes were presented by Hill, see for example [17][18][19][20]. Uniqueness conditions for the case of non-associated laws of plastic flow and large strains were derived by Raniecki and Raniecki and Bruhns [13,14]. In these papers [14,15], local and global conditions were derived with regard to a comparison body depending on kinematically admissible strain rate fields. In addition, local and global conditions of uniqueness for a comparison body dependent on statically admissible stress rate fields for a case of non-isothermal processes, large strains and non-associated laws of plastic flow were deduced in [21]. An attempt at the derivation of the local condition of uniqueness for a certain non-compressible elasto-plastic comparison body and for case of large strains was made in [22], but the study did not offer satisfactory results.
The current paper contains a summary and synthesis of this author's papers and also includes the results obtained by the earlier authors. This paper presents a closed set of uniqueness conditions and bifurcation criteria in generalized coupled thermoplasticity [1][2][3] for the case of small deformations. The paper analyses and includes a study regarding the influence of selected thermomechanical couplings on the uniqueness of the solution of boundary problems. A discussion is included regarding non-associated laws of plastic flow, and elastic and plastic behaviour (elastic-plastic coupling effects). Such a requirement poses a greater problem than the ones stated previously. The problem is not a mathematical self-adjoint. In comparison with the results given in [1,3], local and global conditions of uniqueness are identified with regard to a comparison body in relation to the kinematically admissible stress rate fields. Thus, a set of conditions of uniqueness and bifurcation criteria could be presented in a closed form. In contrast to the earlier papers focusing on uniqueness conditions and bifurcations of equilibrium states [4][5][6][7], this paper deals with non-associated laws of plastic flow accounting for elastic-plastic coupling effects and relations are established for a comparison body dependent on kinematically admissible stress rate fields. In addition, influence of the chosen thermomechanical couplings and influence of non-associated laws of plastic flow on some non-typical cases of behaviour of bodies under influence of thermomechanical loadings provided by the theory are specified and described.
This paper demonstrates that the local uniqueness conditions for the generalized thermo-elasto-plastic body and for the comparison bodies are the same. The methods of calculating the bifurcation state (using the global sufficient uniqueness condition) for the case of comparison bodies are less complicated than for the generalized thermo-elasto-plastic body, as there is a linear dependence between stress rate and strain rate, see [1][2][3][4][5][6][7] and the discussion in Sects. 4.2.2 and 4.2.4 of this paper. Thus, the use of such comparison bodies seems to be advisable.
In a generalized case, constitutive equations of coupled thermoplasticity take the form of non-associated laws of plastic flow, even if Gyarmati's [23] postulate is assumed (see [1][2][3]). They also include the effects of thermomechanical coupling and take into account the elastic-plastic conjugation. It means that they can be applied for the description with regard to not only metallic bodies, but also porous materials, sintered powders, rocks and soils. The paper also contains a description of special cases of the local conditions of uniqueness for more specific body models. In such less general models, constitutive functions occurring under conditions of uniqueness can take simpler form.
The obtained conditions of uniqueness seem to be important from both mathematical and practical point of view. They can offer a device useful in the estimation of critical loads. If the critical loads are exceeded, bifurcation of the equilibrium state is possible [1][2][3][4][5][6][7].
The incremental boundary-value problem of coupled generalized thermo-elasto-plasticity is formulated in this paper. In order to analyse the uniqueness of the boundary problem solution, it is assumed that a thermodynamic state of the body at a given instant of plastic deformation is known. It is necessary to determine rate fields of strain or stress as well as temperatures appropriately for set values of stress or strain rate and the divergence of the vector of heat flux exchanged through the surface in the elementary area (see the problems b 1 and b 2 considered in this paper and also in [1,3]). Similar incremental boundary problems of coupled thermo-elasto-plasticity for small and large deformations were tested and discussed in the literature, see e.g. [1][2][3][4][5][6][7][8][9]13,14,21].

Symbols and abbreviations
c ε and c σ Specific heat capacity measured at constant elastic strain and stresses in Reference temperature corresponding to the TRS-it may be, for example, the ambient temperature, Abbreviation for "thermodynamic reference state", where T = T 0 , K = 0 (see Greek symbols) and ε e = 0, x D Set of dissipative (mechanical) thermodynamic flows,ẋ D = {ε p , q,K }, X D Set of dissipative (mechanical) thermodynamic forces, Yield stress in uniaxial tension, Y 0 Initial yield stress, for ε p = 0, Y 1 Yield stress in uniaxial tension as dependent on (π, κ, T ) ; Greek symbols α Symmetric tensor of thermal expansion coefficients, such that αδ i j = const, δ s Amount of entropy generated within a unit volume over a unit time and referred to a given material particle, ε Tensor of total deformations, ε = ε e + ε p , ε e and ε p Tensor of small elastic and plastic deformations, e p Deviatoric part of plastic deformation tensors ε p , e p ≡ devε p , K Pair of internal parameters K ⇔ {κ (M) ; κ (N ) } M=1 m,N =1 n , κ and κ Symmetric second-rank tensor and scalar internal parameter, respectively, Λ Plasticity multiplier, μ and λ Lamé elastic constants, ν Poisson ratio, Π Pair of internal thermodynamic forces associated with a pair of internal parameters Heat flux exchange with the neighbourhood per unit time across an unit area in [J/(m 2 s)], ρ 0 and ρ Body density in a thermodynamic reference state and in an actual one, respectively, σ Cauchy stress tensor, Effective deviator of stress, σ 0 Yield stress value obtained in the uniaxial tension test for ε p = 0, Tensors will be printed in a bold typeface. The summation convention is assumed along with the following detailed notation , where x j − coordinates of a material particle, If Z denotes pairs of tensors of the fourth and the second order, then Z ⇔ {Z mnkl ; Z mn }, and if M is the tensor of the fourth order, then the operation MZ is a pair of tensors of the fourth and the second order defined as follows If Π and K denote pairs of tensors of the second and the zeroth order, then the operation Π · K produces a scalar, cf. [1,2,21,45,47] Π · K = π : κ + πκ = π i j κ i j + πκ.
If the function F is relative to Π and K , then F(Π ) = F(π, π) and F(K ) = F(κ, κ). The derivatives of function F with respect to a pair Π and K are defined as follows The differentials of function F with respect to the pairs K and Π of tensors of the second and the zeroth order produce the following form The differential of function F with respect to the pair K of tensors of the second and the zeroth order, produces a sum, cf. [1,2,45,47], such that If α is a second order tensor, then the operation α • (MZ) produces a pair of tensors of the second and the zeroth order If Z denotes pairs of tensors of the fourth and the second order and K is a pair of tensors of the second and the zeroth order, then the operation Z * K produces a pair of tensors of the second order such that [1,2,45,47] Z * K ⇔ Z i jmn κ mn ; Z i j κ .

Fundamental assumptions and equations
A homogeneous physical body of unit mass is being considered. When the thermodynamic state of each particle of the body is the same at any moment of the process, the process is called homogeneous. In the case of such processes, the quantity divq occurring in the equations for temperature should be understood as the rate of global heat exchange between the physical body and the environment, and ρ 0 as a reverse of the total volume of the body.
Let us assume that the local thermodynamic state is described by the following parameters of state [1,2,7,15,45]: ε e -tensor of elastic strain, s-specific entropy (per unit mass), Differential of the internal energy depending on internal parameters of the state {s, ε e , K } has a form [1,2,7,15,45,47] dU (s, ε e , The local approach to the principle of conservation of energy is as follows [1,2,7,15] The equation of local entropy balance per unit volume of the body has the form [1,2,7,15,45] The local formulation of the second law of thermodynamics is given by the inequality The entropy production can be evaluated by solving a set of three Eqs.
where D expresses dissipation of mechanical energy per unit time and volume The set of forces X D involved in (2.5), (2.6) is named as a set of dissipation forces or a set of thermodynamic impulses and a set of thermodynamic flow ratesẋ D joined with the set of forces X D are the following

Thermostatic identities and properties of an thermo-elasto-plastic body
The fundamental physical quantities describing the thermostatic properties of solids are defined as follows [1,2,45,47] where Z is the vector of pairs of tensors of the fourth and second order representing the isothermal variation of elastic deformation due to the internal processes accompanying plastic deformation in state Y T σ K . Physically, it means a change of the Young's modulus caused by plastic deformations. N is the vector of pairs of tensors of the fourth and second order representing isothermal variation of the state of stress due to internal processes accompanying plastic deformation too in the state Y T ε K . The quantities (2.9)-(2.12) are not independent. They satisfy the set of the following identities resulting from the existence of thermodynamic potentials, see e.g. [1,2,7,15,45,47]) 14) If the thermodynamic potentials being not expressed in an additive form but in the most general form [1,2,45,47] we have the following additional identities of thermostatic couplings, which will be used in a further part of the paper and The relations (2.18), (2.20) and (2.21) are complex identities and are sometimes called thermodynamic identities of the second order. Physical interpretation is applied with regard to the identities expressing thermostatic couplings (2.19). A variation in internal forces Π due to the acting elastic strain results in the process of material hardening (softening), and variation in internal forces as a result of stress is connected with the variation in the elasticity modulus as a result of variability in internal parameters, see (2.19) and (2.11) 3 .
Eliminatingṡ from the Eqs. (2.3), (2.22) 2 and making use of the Eqs. (2.9)-(2.12) and (2.23) 2 yields two equations for the temperature rate and of elastic strain rate [1,2,46,47]. (2.25) Closed system of thermostatic couplings in the area of thermo-elasto-plastic interactions is presented in Fig. 1. A similar but incomplete (not closed) schemes of couplings are presented in papers [1,6,15] The diagram shows 15 possible coupled connections between two independent parameters of state. Particular types of thermodynamic, thermomechanical and mechanical (elastic-plastic) couplings are denoted as the Greek letter γ with a suitable subscript or with subscript and dash. It was assumed that couplings with the upward (vertical or at an angle) and horizontal (to the right) arrows were expressed by the symbol γ i (i = 1, 2, . . . , 15) (with no dash). The couplings with downward arrows (vertical or at an angle) and horizontal (to the left) arrows were denoted by the symbol with subscript and dashγ i (i = 1, 2, . . . , 15). Dimensionless coefficients γ i andγ i have no physical meaning. The ideas presented in, see e.g. [1][2][3][4][5][6][7]15,16,[45][46][47] and others, have been introduced for simplification of interpretation of different terms of effects of couplings occurring in the equations. They can be also useful for some simplifications in general constitutive equations. They are a kind of numbers which take a value 1 (when any of the effects of couplings shown in Fig. 1 is taken into  Fig. 1 is neglected. For example, when γ 4 = 0 is substituted, thermal expansion is neglected, if γ 11 = 0, a change of internal forces caused by a change of elastic strains is neglected, whenγ 6 = 0, heat of elastic deformations is not taken into account, and in the case ofγ 9 = 0, influence of internal changes on elastic strains is omitted. If we are concentrated only on internal changes in the material caused by plastic strains and occurring as plastic hardening, i.e. the effects denoted by numbers 9-12 in Fig. 1, they can be also called the effects of elastic-plastic coupling [1,2,10,11,[45][46][47]. Symbol γ 0 shows including dissipation heat which is not a thermostatic thermal effect, and it is not specified in the diagram in Fig. 1. Theoretically, there are 30 physical kinds of thermodynamic, thermomechanical and mechanical (elasticplastic) couplings. Not all of them, however, have their physical interpretation and physical explanation, and not all of them have been observed. They are a result of assumption of a general (non-additive) form for thermodynamic potentials and formal derivation of total differentials for the chosen dependent parameters of state, depending on the assumed systems of independent parameters of state [1,2,45,47].
Let us observe that the elastic strain rate involved in (2.25) 2 can be written as follows [1,2,46,47] ε e =ε eI +ε eI I , (2.26) ε eI = γ 2 Lσ + γ 4 αṪ , ε eI I =γ 9 ZK , (2.27) whereε eI is termed the "uncoupled" part of the elastic strain rate, andε eI I is the "coupled" part of the elastic strain rate connected with the internal processes accompanying the plastic strain. Such a separation of the tensorε e into two parts was adopted and interpreted for the case of isothermal processes in papers [10,11] and for non-isothermal processes in [1,2,45,47].

Rate equations: plastic flow equations
Equations of plastic flow are the followinġ where b-function describing evolution for internal parametersK . Here, is a generalized function of plastic flow determined in the space of thermodynamic forces X d = {σ, −Π } such that (F 1 = 0) determines the flow area in this space. The expression for an equation of evolution for the vector of pairsK in form of (2.28) 2 is a result of application of idea of the preparation space [1,2,45,47,48]. Using additional experimental data, in that space we can determine the equations of evolution for internal parameters K or dissipation forces Π and determine their initial value.
If Π or K is replaced by a suitable equation of state [1,2,46,47], then we will obtain the plastic flow function and suitable flow conditions F in the stress space, written as The factor Λ in (2.28) can be eliminated by making use of the "association condition" [1,2,7,15,46,47].
is what is termed the strain-hardening function. By assuming the classical condition for plastic loading (ε p = 0), if and only if ∂ F ∂σ :σ + ∂ F ∂ TṪ ≥ 0, we find, by virtue of (2.31) 2 , that h ≥ 0. Equation (2.28) 1 can be expressed in the forṁ
Condition (2.40) should be interpreted as a limitation for the functions occurring in the group of constitutive equations. If (H = 0), from the theory of thermo-elasto-plasticity it results that an instantaneous change of stresses and temperature is possible when the body element is not being deformed (ε = 0) and it does not exchange heat with the environment (q 0 = 0). It means theoretically that if (H = 0 andε = 0, q 0 = 0 theṅ σ = 0 andṪ = 0). However, such phenomena do not take place in real physical bodies [1].
If (h 1 = 0), then in an adiabatic process (q 0 = 0) the body behaves similarly as the perfectly plastic body (i.e. there is no hardening). This means that momentary adiabatic plastic flow (ε p = 0) is possible under constant stresses (σ = 0). Such a phenomenon can occur in the range of large deformations; therefore, for small deformations we assume that constitutive functions satisfy the condition stated in (2.39).
It is worthwhile to observe that in the case of metallic materials the satisfaction of the condition (2.39) implies, in general, fulfilment of the condition (2.40) (cf. [1,3,4,6,7,16,21,47]) Let us assume that the conditions (2.39) and (2.40) are both satisfied. The solutions of the incremental problems (b 1 ) and (b 2 ) can be expressed in the following forms: The symbol L a denotes the tensor of adiabatic elasticity. Let us observe that the second right-hand term of (2.44a) is not equal to the plastic strain rate, but may be considered as representing the adiabatic plastic strain rate. The tensorK is asymmetric,K i jmn =K mni j . A lack of symmetry is caused by not only thermal expansion accompanying power dissipation of the plastic strain and a change of the yield point together with increase of temperature resulting from the piezoelectric effect, but also effects of elastic-plastic coupling and assumption of non-associated laws of plastic flow. It makes difficult a proof of the theorem concerning uniqueness of solution of the incremental boundary-value problem in the case of heterogeneous processes and in a consequence formulation of suitable criteria of bifurcation. The equations for the thermodynamic flow rates can also be expressed in terms ofσ and q, They have the formK Taking into considerations the Gyarmati postulate and the resulting condition (cf. [1,3,47]), the relation (2.47) takes the form (2.48)

Formulation of the incremental boundary-value problem
If the condition (2.39) is satisfied, the set of Eqs.
where v is the vector of velocity of particles, b m is the body force, constitutes a set of fundamental field equations of generalized coupled thermo-elasto-plasticity. Together with the boundary conditions and the initial conditions, it may be used as a basis for analysis of many problems of generalized coupled thermoelasto-plasticity, both static and dynamic [1,3,6,7,47,48].
where n is a unit vector normal to S, directed towards the outside of D [1,3,4,7,16,47]. Our task is to find the set of functions (σ,ε, v) defined inD and the functionṪ defined in D, which satisfy, in the region D, Eqs. (2.37)-(2.40) and (3.1), expression q = 1 ρ 0 c σ divq and the rate equations of equilibrium Let us observe that, knowing the functions (3.2), we can determine q at every point of the region D, directly from the q = ϕ q ∇T, Y T K , where ∇T = gradT and Y T K = {T, K }, by differentiating T and q with respect to the coordinate variables x i , where i = 1, 2, 3.
In the coupled generalized thermo-elasto-plasticity, the formulated incremental boundary problem plays the same role as a suitable boundary incremental problem in isothermal theory of plasticity. Namely, if its solution is ambiguous, then a solution of a general problem where the history of variation of surface forces, velocity and temperature on the surface of the considered body are given is ambiguous, too [1,4,6,7,16,47].

Discussion of uniqueness conditions
Tests of uniqueness of the solution of the incremental boundary problem presented in Sect. 3 belongs to the most important problems shown in this paper. Such tests and the obtained results can be the basis for formulation of two criteria allowing to estimate the critical thermodynamic state after exceeding of which bifurcation of the equilibrium state is possible. These criteria are also two sufficient conditions (local and global condition) of uniqueness of solution of the incremental boundary-value problem.
The local condition can be easily applied in practice because it is directly expressed by constitutive functions and material constants. However, it gives less accurate estimations of the critical state. The global condition gives better estimations of critical states, but its application is more difficult because it requires searching kinematically acceptable velocity fields for which the functional J (see point 4.2) reaches zero.

Local uniqueness condition
The following theorem is proved in author's papers [1,3,47] Theorem 1 If the inequality The inequality (4.1) is the sufficient local uniqueness condition. Each thermodynamic state, for which the condition (4.1) is satisfied, is secure against bifurcation. Since in the course of a deformation process of the body the value of the strain-hardening function (the modulus) decreases, in general, therefore the value of h * 1 may be treated as an upper estimation of the unknown critical value h corresponding to the critical state.
Some particular cases of the expression (4.1) have already been mentioned in the literature. A similar condition was obtained in [10,11] in their analysis of the stability of material defined as a condition of half the product of the stress rate tensor and the strain rate tensor being positive. Their study was confined to the case of the isothermal theory of plasticity (with no thermomechanical couplings), the elastic plastic coupling effects and non-associated laws of plastic flow being preserved. An expression of this type was also obtained in [1,[3][4][5]7,13,14,21,47] for the case of isothermal uncoupled and non-isothermal coupled theory of thermoplasticity with a non-associated law of plastic flow.

Global condition for a thermo-elasto-plastic body dependent on kinematically admissible strain rate fields
Let us assume that there exist two sets of functions σ,ε,Ṫ , v and σ * ,ε * ,Ṫ * , v * which are solutions of the incremental boundary-value problem of generalized coupled thermoplasticity, which was formulated in Sect. 3. Then, the following equality must be satisfied due to the fact that both solutions satisfy the same boundary conditions (3.3), in the case of Gauss-Ostrogradski theorem. Let us denote by J the integrand in the expression (4.3), which depends on (ε andε * ), for an elastic-plastic body, as follows where ε =ε −ε * andσ * =σ ε * , and j 1 = j 1 (ε) and j * 1 = j 1 ε * are defined by the Eqs. (2.50a) and (2.50b). The quantities (ε andε * ) and (σ * andε * ) are interrelated by Eq. (2.49a), which can be rewritten in a more compact form as followṡ Since the expression (4.3) with zero at the right side provides existence of two sets of functions σ,ε,Ṫ , v and σ * ,ε * ,Ṫ * , v * , which are a solution of the formulated incremental boundary-value problem, so positivity of the expression (4.3), i.e. Λ * > 0 [1,3,6,7,13,14,21,47] and (4.4) will be a condition excluding occurrence of the bifurcation state. Inequality Λ * > 0 is a sufficient global condition of uniqueness and a global criterion excluding occurrence of the bifurcation state.

Global condition for a comparison body dependent on kinematically admissible strain rate fields
Let us introduce the following function J ', depending on (ε andε * ) J ε,ε * = ε: where x 2 is a scalar quantity. The expression J is a comparison body function and represents a one-parameter family of expressions of J , with respect to the parameter x 2 .
The functions J and J depend in addition to the variables (ε andε * ) on the thermodynamic state of the body (3.2).
As it results from comparison of the expression (4.7) and (4.4), it presents a certain linear dependence between ( σ and ε). Differentiating J in relation to ε, we obtain a linear dependence between σ and ε, which does not occur in expression (4.4), because these dependences are nonlinear.

Lemma 1
It will be shown that if the same thermodynamic state is prescribed for J and J ', then for each pair (ε andε * ) the following inequality holds J ε,ε * , j 1 , j * 1 − J ε,ε * ≥ 0. (4.8) Let us introduce the following notations for the function J ε,ε * , j 1 , j * 1 : Then, by evaluating the difference (4.8) for all the possible four cases (4.9), we obtain, by virtue of (4.4) and (4.5) and (4.7), that Proof The integrands in (4.12) are The validity of the sufficient global uniqueness criterion (4.12), being a safer criterion excluding the state of bifurcation, follows directly from the inequalities Λ * > 0 and (4.8) and (4.10).
The integral condition (4.12) is, in particular form, of essential practical importance. If for a prescribed state {T, σ, K } it is impossible to find such a field v that the sum of integrals at the left-hand side of the expression (4.12) is zero, we are assured that this state is secure against bifurcation.
The idea of deriving such criterion was conceived as early as in Hill's works [17][18][19][20] for elastic-plastic bodies under large strain, for the isothermal incremental boundary-value problem. For the incremental boundaryvalue problem in coupled thermoplasticity in the case of associated laws of plastic flow and for small deformations such criterion has been derived by Mróz and Raniecki [4][5][6][7], for the case of non-associated laws of plastic flow byŚloderbach [1,3,47] and for large deformations by Raniecki and Bruhns [13,14] andŚloderbach and Pajak [21]. Another sufficient global uniqueness criterion for incremental problems of isothermal plasticity of elastic-plastic bodies with non-associated laws of plastic flow has been given by Hueckel and Maier [10][11][12]. In paper [12] the author introduced an idea of two-point asymmetric scalar function linearly dependent on the Green function for a linearly elastic body. Application of the criterion in practice is difficult because the Green function for bodies of an arbitrary shape is usually unknown.
It is shown in [1,3,13,14,47] that the sufficient local uniqueness condition following from the requirement that the intergrand J should be definite positive is the same as for an generalized thermo-elasto-plastic body Eqs. (2.44a) and (2.49a) or (4.5) provided that the parameter x 2 takes its optimum form (4.14) A procedure for obtaining the optimum parameter x 2 0 is also discussed in [1,3,13,14,47]. For the parameter x 2 0 , the sufficient local uniqueness condition becomes the optimum (strongest) condition for the entire one-parameter family of sufficient uniqueness conditions. Now, by substituting the optimum value of the parameter (4.14) into the expressions (4.7) and (4.12) we shall obtain the optimum (strongest) integrand which will be denoted by the symbol J 0 and optimum form of the bifurcation criterion,
At presentσ * andε * are connected with a suitable constitutive equation, see [1,3,46] written aṡ where g = F 1,σ +γ 9 ξ Z * b + γ 4 m σ α , d 1 = γ 4 qα, Like in the case of kinematically admissible strain rate field, the expression (4.3) with the sign zero at the right side allows for existence of two sets of functions σ,ε,Ṫ , v and σ * ,ε * ,Ṫ * , v * , being a solution of the formulated incremental boundary problem. Thus, the positive definition of the expression (4.3) is a condition excluding occurrence of the bifurcation state, i.e. Λ * > 0 [1,3,6,7,13,14,21,47]. The positive definition of the expression (4.17) is a result of the positive definition of (4.13), too. In this case, the inequality (4.13) is a sufficient global condition of uniqueness of solution of the incremental boundary problem for a reference body and a global criterion excluding occurrence of the bifurcation state for a case of kinematically admissible stress rate fields.

Global condition for a comparison body dependent on statically admissible stress rate fields
The function I dependent onσ andσ * is introduced in the following way I σ,σ * = σ : where y 2 is a scalar parameter. The above expression expresses a one-parameter series of expressions I related to the parameter y 2 . The functions I and I depend not only on independent variablesσ andσ * , but on the thermodynamic state as well (3.2).
In coupled generalized thermo-elasto-plasticity, an idea of the reference body dependent on statically acceptable stress rate fields was introduced in the author's papers [21,22,47]. As in the case of kinematically admissible strain rate fields, from comparison of the expression (4.20) with (4.17) it appears that it is a certain linear dependence between ( σ and ε). Differentiating I in relations to σ, we obtain a certain linear dependence between ε and σ, which does not occur in the expression (4.17), because in Eq. (4.17) those dependences are not linear.
Next, calculating the difference (4.21) for all the possible above four cases from the expressions (4.17), (4.20) and (4.22), we obtain where γ g = g: σ and γ f =F σ : σ and γ f = A σ − A * σ and also A σ =F σ :σ + z 1 and A * σ =F σ :σ * + z 1 . From the set of inequalities (4.24), it appears that the inequality (4.21) is true. Using the inequality Λ * > 0, see the expression (4.3), and the inequalities (4.21) and (4.24), we can formulate (like in the item 4.2.2) the following sufficient condition of uniqueness of a solution of the incremental boundary-value problem for the reference body dependent on statically admissible stress rate fields, which is a safer criterion excluding occurrence of the bifurcation state.
Theorem 3 Let us assume h 1 > 0 in each point of the body x ∈ D P in its part where plastic deformations take place, i.e. where D P = {x: F = 0}. If for each statically admissible stress velocity fieldσ (or the field of stress rate difference σ), which disappears on a surface part (a body boundary) S t , the following inequality is satisfied The integral condition (4.26) presented in this form is very important from a practical point of view. Namely, if for a given thermodynamical state {T, σ, K} it is not possible to find such statically admissible stress velocity fieldσ, for which a sum of integrals occurring at the left side of the expression is equal to zero, then we must be sure that such a state is safe from the point of view of possibility of bifurcation state occurrence.
In Ref. [21] for the case of large deformations and in [47] for the case of small deformations, it is shown that the sufficient local condition of uniqueness resulting from the requirement of a positively defined integrand I ' is the same as for a case of a thermo-elasto-plastic body (2.44a) and (4.18) when the parameter y 2 takes the following optimum form (4.28) For the parameter y 2 0 , the local condition of uniqueness becomes the optimum (safest) condition from all the set of conditions. Substituting a value of (4.28) into the expressions (4.20) and (4.26), we obtain the optimum element of integration I 0 and the optimum form of the bifurcation criterion as Now we can state that in the case of the comparison body expressed by Eqs. (4.7), (4.13) or as J 1 (4.8) 1 dependent on kinematically admissible strain rate fields at the boundary transition g * =F * σ and (x 2 = 1), we obtain a body of coupled generalized thermoplasticity determined by (4.5). From the expressions (4.20) or (4.27) for the reference body dependent on statically acceptable stress velocity fields, it appears that substituting at the boundary (g =F σ ) and the value (y 2 = 1) we obtain the expression I 4 like for the thermoelastic body. Thus, reference bodies are not obtained by their mutual inversion like in the case of a thermo-elasto-plastic bodies; they are independently derived so as to satisfy the inequalities (4.8), (4.10) and (4.12) for a body dependent on kinematically acceptable strain fields and inequalities (4.21), (4.24) and (4.26) for the reference body dependent on statically acceptable stress velocity fields.

Local uniqueness conditions for special cases of bodies
In the constitutive equations of coupled generalized thermoplasticity (2.24) and (2.26) and in the comparison bodies as well in derived local conditions o uniqueness, see expression (4.1) the functions g,F σ , M a and m σ , occur. Their forms will be different in the case of less general models of bodies. In such cases, the functions will be simpler.
In such a case, the functions g,F σ and m σ take the following forms: Function M (a) is the same as in Sect. 2.
Here, the generalized function of plastic flow F 1 (plastic potential) does not depend on a vector of internal pairs of dissipation forces Π , which are dependent on the stress state (see [1,3,21,[45][46][47]). Moreover, all the effects of the thermomechanical couplings and effects of the elastic-plastic couplings are kept.

Case 2
The case not including effect of the elastic plastic coupling. Then where m σ = 1 ρ 0 c σ (σ : and the function M (a) remains the same as previously.
In this case, also all the effects of thermomechanical couplings and the non-associated laws of plastic flow are still valid. A model of such a body can be useful, for example for analysis of adiabatic and non-isothermal processes of location of plastic strains and non-isothermal, adiabatic or quasi-adiabatic processes of plastic deformation and forming of metals. Such local adiabatic instabilities can occur during some machining processes, for example while milling or turning, and they cause vibrations and irregularities of the machined surface or other negative effects.

Case 4
The case of constitutive approximate equations.
Case 5 Isothermal theory of plasticity concerning non-associated plastic flow with elastic-plastic coupling and with no thermal coupling. Then Such a model of the elastic-plastic body is often applied for description of porous materials, sinters, rocks and soils [10][11][12]. In this model, influence of plastic strains on elastic properties of the body is included. Conditions of uniqueness for this model were studied by Mróz [8,9]. In Ref. [9] Mróz also derived the sufficient local condition of uniqueness for compressible and isotropic elastic-plastic bodies.
The presented chosen cases of elastic-plastic bodies and the corresponding global and local conditions of uniqueness of a solution of the incremental boundary problem are not all possible models of bodies resulting from the generalized model of coupled thermoplasticity derived in [1][2][3]21,[45][46][47]. The presented cases of body models 1 ÷ 6 are more or less similar to standard models of elastic-plastic bodies, discussed previously in the literature.

Conclusions
6. In the paper, only general expressions were derived for constitutive functions of coupled generalized thermoplasticity. They occur in both necessary and sufficient global and local conditions of uniqueness of solution of the formulated incremental boundary-value problem. During further investigations, the constitutive functions of coupled generalized thermoplasticity should be specified within mechanics of continuous media according to experimental results. 7. In the paper, it is assumed that gradients of displacements and strain rates are small. For simplicity purposes, it is assumed that all the mathematical operations and all the description are realized in the Cartesian coordinate system. In some last years, many papers describing some chosen kinds of thermomechanical couplings including large deformations were published, see e.g. . 8. The future tests concerning uniqueness of the solutions of incremental boundary problems can be realized also for viscoplastic materials. Hence, for the case of non-associated laws of plastic flow, the generalized function of plastic flow F 1 (plastic potential) should be replaced by the dynamic function of flow dependent on F 1 , see [48][49][50]. In the case of the associated laws of plastic flow (then F 1 = F and F 1,σ = F σ ), the function of plastic flow F should be replaced by the dynamic function dependent on F. This problem remains open to further research.
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