Effect of Preliminary Mechanical and Thermal Treatment on the ...

Materials Science, Vol. 38, No. 6, 2002

EFFECT OF PRELIMINARY MECHANICAL AND THERMAL TREATMENT ON THE UNSTEADY CREEP OF METALS A. K. Rusynko

UDC 539.376

The synthetic theory of plasticity and creep is generalized for the description of unsteady creep of metals preceded by their mechanical and thermal treatment in the form of plastic prestraining at room temperature followed by annealing. This type of treatment decreases the level, rate, and duration of unsteady creep.

As one of the methods aimed at the improvement of heat-resistant properties of materials, one can mention their mechanical and thermal treatment (MTT) including, in particular, the following sequence of operations: — a specimen is stretched at room temperature to a certain level of plastic strains εx0 ; — annealing: the unloaded specimen is held at temperature T1 for time t1 and the level of strains ε x0 does not exceed a value above which the process of heating to temperature T1 causes active recrystallization. The data of numerical experiments [1–3] demonstrate that the procedure of preliminary MTT strongly affects the parameters of unsteady creep for nickel, copper, and steels. Thus, according to the results of subsequent testing for creep, the preliminary MTT decreases the initial level of plastic strains, strain rate, and duration of the unsteady section of the diagram (see Fig. 1). As εx0 increases (for the same time of annealing t1 and temperature T1 ), the indicated parameters of unsteady creep decrease. For some value of ε x0 in the diagram of creep (Fig. 1, curve 3), the unsteady section completely disappears and the process of deformation runs with a constant rate from the very beginning. As the degree of cold-work hardening increases further, we observe the formation of a section of inverse creep, which is not studied in the present work. The difference observed in the behavior of specimens is explained by specific features of their structure formed as a result of preliminary stretching and annealing. In the course of plastic deformation, intact relatively perfect crystals split into fragments whose sizes and mutual orientation depend on the level of strains and temperature. The boundaries of the fragments are characterized by the presence of dislocation pileups forming a three-dimensional network of subboundaries. The process of heating of a metal is accompanied by the redistribution of dislocations and some of them annihilate [4]. The indicated processes decrease the total elastic energy of a crystal. As a result, we observe the formation of a stable three-dimensional network of dislocations whose walls separate the subgrains. The outlined process of formation of substructures is called polygonization. The preliminarily created polygonized structure decreases the intensity of (rough and smooth) sliding in the course of creep [3]. This decreases the rate of unsteady creep. Moreover, this substructure decreases the difference between the local and mean stresses and, hence, the time of relaxation of elastic distortions of the crystal lattice caused by active loading. Hence, the stage of unsteady creep terminates faster than in the absence of MTT as a result of the attainment of an equilibrium block structure. “L’vivs’ka Politekhnika” National University, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 38, No. 6, pp. 51–57, November – December, 2002. Original article submitted February 20, 2001. 824

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© 2002

Plenum Publishing Corporation

E FFECT OF PRELIMINARY MECHANICAL

AND

T HERMAL TREATMENT ON THE UNSTEADY CREEP OF METALS

825

Fig. 1. Qualitative diagrams of creep for specimens with different degrees of prestraining: ε (x30) > ε (x20) > ε (x10) , ε (x10) = 0. After prestraining, specimens 2 and 3 were annealed at temperature T1 for time t1 . The superscripts correspond to the numbers of the specimens and curves and ti* ( i = 1, 2) are the durations of unsteady creep for specimens 2 and 3, respectively. The dotted lines correspond to the conditions of steady creep.

In what follows, we give an analytic description of the diagrams of creep (Fig. 1) within the framework of the synthetic theory of plasticity and creep [5]. In the Il’yushin three-dimensional deviator subspace, the component of the vector of irreversible strains e1 under the conditions of uniaxial tension is given by the formula [5] e1 =

2π r

∫ ∫ ϕ sin β cos β cos λ dβ dλ ,

e2 = 0,

e3 = 0,

(1)

λ β

where ϕ is the strain intensity determined from the differential equation [7] dψ = r dϕ – K ψ dt,

(2)

ψ is the intensity of defects, dt is an increment of time, r is a material constant, and K is a function of the homological temperature Θ and stresses σx : K = K1 exp( K2 Θ)

(

2 σx 3

)

K3

,

K1 , K2 , K3 = const.

(3)

In the case of plastic deformation, the intensity of defects ψ is given by the formula [5] 2 2 σS . 3

(4)

2 2 σ P − [ I N (t )]2 , 3

(5)

ψ (H ) = H2 − At the same time, in the case of unsteady creep, we have ψ (H ) = H2 −

where H is the distance from the origin of coordinates to a plane tangential to the loading surface. If the endr r r r point of the loading vector S lies in a certain plane, then the distance to this plane H = S ⋅ N , where N is the unit normal to the plane whose orientation is specified by the angles λ and β.

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A. K. RUSYNKO

The intensity of defects ψ is their averaged continuous measure in a homogeneous continuum (for the actual solid body, these are dislocations, vacancies, interstitial atoms, etc.). The term “defects in a plane with norr mal N ” denotes defects from the sliding system of the microvolume corresponding to this plane. r In the process of loading, the endpoint of the vector S shifts tangential planes. This corresponds to irreversible deformation. The plasticity (creep) surface is chosen in the form [8] 2 2 σy , 3

S12 + S22 + S32 =

(6)

where σy is either the plasticity limit σS (if we study the case of plastic strains) or the creep limit σP (if we stur dy the case of creep strains). In the case of uniaxial tension, only one component of the vector S is not equal to zero, namely, 2 σx . 3

S1 = Therefore [5],

H ( β, λ ) = S1 sin β cos λ =

2 σ x sin β cos λ. 3

(7)

As follows from relations (4), (5), and (7), the quantity H and, hence, the intensity of defects ψ increase with the level of stresses. Thus, the quantity H characterizes the degree of strain-hardening of the material. This is explained by the fact that the density of structural defects in the crystal lattice of the material increases with the level of stresses, which complicates the process of plastic deformation. Therefore, to attain higher levels of plastic strains, it is necessary to increase the level of stresses. The inhomogeneity integral IN is a measure of local peak stresses, i.e., stresses of the third kind responsible for local elastic distortions of the crystal lattice (interlacing of dislocations, formation of complex irregular dislocation networks, etc.). The difference between the local and mean stresses increases with the loading rate. Therefore, we introduce the integral IN as follows [7]: t

IN ( t ) = Iν sin β cos λ = B ∫ 0

r dS exp [ − p(t − τ)] dτ sin β cos λ, dt

(8)

where B and p are material constants depending on the testing temperature: B = B1 + B2 Θ,

p = p1 + p2 Θ,

pi , Bi = const

( i = 1, 2 ).

(9)

Under active loading at a constant rate k for time t2, we get (for time t2 , the level of stresses becomes equal to σx = k t2 ): IN ( t = t2 ) = Iν ( t = t2 ) sin β cos λ =

Bk (1 – exp (– p t2 )) sin β cos λ. p

(10)

At the same time, if the specimen is held under a constant stress σ x ( t > t2 ) independent of time, then we find


AND

IN ( t ) =


Bk (exp ( p t2 ) – 1) exp ( – p t ) sin β cos λ. p

827

(11)

Relation (10) characterizes the accumulated elastic distortions of the crystal lattice after active loading. Relation (11) describes the process of thermally activated relaxation of these distortions (disentanglement of entangled dislocations and formation of a polygonized structure). In this case, preliminarily anchored dislocations become mobile under the action of temperature fluctuations and, thus, promote the process of unsteady creep. The constants B and p in relation (8) determine the intensity and kinetics of unsteady creep [7]. Thus, the smaller the constant B, the lower the level of strains and strain rate. At the same time, the larger the constant p, the smaller the size of the nonlinear section in the diagram of creep. This type of behavior of the constants B and p characterizes the increase in the initial level of plastic strains [7]. For IN (t ) = 0, i.e., when the factor exp (– p t) approaches zero, the stage of unsteady creep is terminated. In the process of plastic deformation, we have d t = 0 and relation (2) takes the form ϕ =

ψ , r

(12)

where ψ is given by relation (4). If, in the case of steady creep, we have IN ( t ) = 0, then d ψ = 0 and relation (2) takes the form r d ϕ = K ψ dt

or

ϕ =

K ψt . r

(13)

The analysis of the diagrams of creep for different materials shows that the accumulation of strains in their linear sections runs slower than in the unsteady sections. Therefore, in describing the process of unsteady creep, we can omit the term K ψ dt in relation (2), i.e., return to relation (12). The time dependence of strains is described by the term containing the inhomogeneity integral IN ( t ) on the right-hand side of relation (5). Thus, within the framework of the synthetic theory, the same relation (2) describes the cases of plastic deformation and unsteady (or steady) creep. We now substitute the function ψ given by relation (5) in formula (12) and insert the expression for ϕ obtained as a result in relation (1). The angle λ in relation (1) varies within the limits 0 ≤ λ ≤ λ 1 . The parameter λ 1 is determined from the condition ψ = 0 [5] as follows: cos λ1 ( β, λ ) =

σP σ 2x −

3 ( Iν (t ))2 sin 2 β 2

.

For the angle β, we have β 1 ≤ β ≤ π / 2, where β1 can be found from the condition λ1 = 0 [5]: sin β1 ( t ) =

σP . 3 2 2 σ x − ( Iν (t )) 2

(14)

As a result of integration, we obtain the component of the vector of unsteady creep e1 ( t ) (without MTT): e1 ( t ) = a Φ ( sin β1 ( t ) ),

(15)

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A. K. RUSYNKO

where Φ (ξ) =

1 ξ2

 1 + 1 − ξ2  2 2 2 4 − ξ − ξ − ξ + ξ ln 2 1 5 1 3  , ξ  

a =

πσ 2P , 9r

and σP is the creep limit of the material at temperature T2 . For t = t2 , we get the initial level of plastic strains and, for t > t2 , the levels of creep strains in the process of unsteady creep. If we remove the load after plastic prestraining ( ε x0 ) and, thus, stop the process of growth of irreversible strains ( d ϕ = 0), then relation (2) takes the form dψ = – K ψ dt

(16)

and describes the relaxation of defects after unloading (thermally activated annihilation of dislocations of the opposite signs, annihilation of vacancies on dislocations and subgrain boundaries, mutual annihilation of vacancies and interstitial atoms, etc.). We now generalize the synthetic theory to the case of unsteady creep after MTT. Recall that the dislocation structure formed in the course of preliminary MTT restricts the processes accompanying subsequent steady creep. In order to describe this fact qualitatively, we replace relation (5) by the formula ψM ( H ) = H 2 − HT2 − [ I˜N (t )]2 ,

(17)

where HT is the distance to the planes after MTT characterizing the degree of hardening of the material in the course of preliminary MTT. In fact, the defects accumulated in the process of cold-work hardening that do not annihilate in the course of annealing create a stable polygonal network. This substructure complicates the processes running in the material in the case of unsteady creep as compared with the case of creep without preliminary MTT. In order to study the specific features of the kinetics of unsteady creep after MTT (the decrease in its level, rate, and duration), we change the constants B and p in the inhomogeneity integral: 2 2 BT = B − B3  HT max − σ P  = B1 + B2Θ − B3  HT max − σP  ,     3 3 (18) 2 2 σ P  = p1 + p2Θ + p3  HT max − σP  , pM = p + p3  HT max −     3 3 where HT max is the maximum value of HT regarded as a function of the angles β and λ and B3 and p 3 are material constants. Since BM < B and pM > p, in view of the comments to formulas (10) and (11) made above, we conclude that the new constants BM and p M qualitatively describe the decrease in the rate and duration of unsteady creep preceded by MTT. The increase in the initial level of plastic strains caused by the conditions BM < B and pM > p is compensated by the negative term HT in relation (17). In addition, we replace relation (3) by the formula K3 , KM = f ( Θ, Hmax ) = K1 exp ( K2Θ) Hmax

(19)

where Hmax is the maximum distance to the planes at given temperature for the entire loading history. The quantity Hmax is determined from relation (7) with β = π / 2 and λ = 0:


AND


Hmax =

829

2 σ x max , 3

where σx max is the maximum level of stresses at given temperature for the entire loading history. If the procedure of preliminary MTT was not performed, i.e., sphere (6) was not transformed, then HT2 = HT2 max =

2 2 σP 3

and relations (17)–(19) coincide with relations (4), (9), and (3), respectively. The quantity KM is equal to K if the level of stresses remains constant as a function of time. By using relation (16) with KM given by relation (19), we can describe the relaxation of defects in the absence of loading [formula (3) gives K = 0 when the level of stresses is equal to zero]. The data about preliminary cold-work hardening is preserved indirectly via the value of H max because the higher the attained level of plastic strains, the longer the loading vector and, thus, the larger the distance to the planes located at its endpoint. The quantity HT is determined by analyzing the procedure of cold plastic prestraining with subsequent annealing. We substitute the quantity ϕ given by formula (12) and the function ψ given by formula (4) in relation (1). The angle λ in relation (1) varies within the limits 0 ≤ λ ≤ λ1 0 . The quantity λ 1 0 is determined from the condition ψ = 0 [5] as follows: cos λ1 0 ( β ) =

σS . σ x 0 sin β

(20)

For the angle β, we have β 1 0 ≤ β ≤ π / 2, where β1 0 is determined from the condition λ1 0 = 0 [5]: sin λ1 0 =

σS , σ x0

(21)

where σx0 is the stress of plastic prestraining and σ S is the plasticity limit of the material at room temperature. Integrating relation (1) within the limits specified by relations (20) and (21), we get the level of plastic prestraining e10 in the form e10 = a0 Φ (sin β1 0 ),

where

a0 =

πσ 2S . 9r

(22)

In the process of annealing of unloaded specimens, the defects of the crystal lattice accumulated during plastic prestraining undergo relaxation described by the differential equation (16): 2 ψ = ψ0 exp ( – KM t ) ⇒ H =

{

}

2 [(σ x0 sin β cos λ )2 − σ 2S ] exp (− K M t ) + σ 2S , 3

(23)

where ψ0 is the intensity of defects accumulated in the course of preliminary cold-work hardening. Relation (23) describes the return of planes to sphere (6). To take into account the effect of preliminary treatment on the parameters of creep, we consider the reverse motion of planes to the creep surface. The identical decrease in H in all directions (for all values of λ and β ) caused by the replacement of σS by σP in relation (23) means that the angles λ1 0 and β1 0 remain constant.

830

A. K. RUSYNKO

The squared distance to the planes after annealing is determined from relation (23) with t = t1 as follows:

HT2

[

]

 [(σ x sin β cos λ )2 − σ 2S ] e − K M t1 + σ 2P , 0 2 =  3 2  σP ,

β10 ≤ β ≤ π / 2 ,

0 < λ < λ10 ,

− π / 2 < β < β10 ,

λ10 < λ ,

(24)

where K3 2 KM = K1 exp( K2 Θ1 )  σ x0   3 

and Θ1 is the homological temperature of annealing. Substituting HT2 from relation (24) in expression (17), we get the following formula for the intensity of defects:  (σ x sin β cos λ )2 − [(σ x 0 sin β cos λ )2 − σ 2S ] e − K M t1 − σ 2P − 3 [ I˜N (t )]2 , β10 ≤ β ≤ π / 2, 0 < λ < λ10 ,  2 2 ψM ( t ) =  3  (σ sin β cos λ )2 − σ 2 − 3 [ I˜ (t )]2 , β < β < β , λ < λ < λ , P N 1M 10 10 1M  x 2 (25) where λ1M is determined from the condition ψM = 0 : cos λ1M ( β, t ) =

σP 2

σx −

3 ˜ ( Iν (t ))2 sin 2 β 2

and β1M is determined from the condition λ1M = 0: σP . 3 2 σ x − ( I˜ν (t ))2 2

sin β1M ( t ) =

(26)

We now find the level of strains e1M ( t ) attained as a result of unsteady creep preceded by MTT. Substituting the quantity ϕ from relation (12) and the function ψ given by formula (25), in relation (1), we get 2π e1M ( t ) = r

=

λ1 M π / 2

∫ ∫ 0

ψ M (t ) sin β cos β cos λ dλ d β

β1 M

λ 1 M 4π   3r  ∫0 

π/2

∫

β1 M

(σ sin β cos λ )2 − σ 2 − 3 [ I˜ (t )]2  sin β cos β cos λ dλ dβ   x P ν 2  

– exp (− K M t1 )

λ 10 π / 2

∫ ∫ {(σ x 0

β10

0

}

}

sin β cos λ )2 − σ 2S sin β cos β cos λ d λ d β .

(27)


AND


831

Fig. 2. Curves of creep plotted for nickel specimens at σ x = 25 MPa and T2 = 700°C: (1) without preliminary MTT, (2) prestraining (1)

to ε x = 1.8% at 20°C followed by annealing at T1 = 800°C for 1 h. The solid lines are the experimental curves taken from 0

[3] and the dotted lines are plotted according to the results of numerical calculations.

We perform integration in (27) and take into account the fact that, according to relations (4), (20), and (21), the second integral specifies the level of plastic prestraining e10 . As a result, we obtain e1M ( t ) = a Φ (sin β1M ( t )) – e10 exp ( – KM t1 ).

(28)

Let us now analyze this formula. The argument of the function Φ contains the integral I˜ν with constants BM and pM such that BM < B and pM > p according to relations (9) and (18), and the difference between the constants B, p and B M, p M determines the difference between ordinary unsteady creep and unsteady creep with preliminary MTT. The analysis of the arguments of the function Φ given by relations (26) and (14) shows that the investigated curve in the diagram of creep lies above the curve of unsteady creep without preliminary MTT. However, this contradiction is removed by the negative term e10 exp (– KM t1) in relation (28) which shifts the entire curve e1M ( t ) downward, below the curve e1 ( t ). For some values of the constants B and p , the integral I˜ vanishes, which means that the intensity of M

M

ν

defects is invariant as a function of time. Thus, relation (2) turns into relation (13) which describes steady creep. This corresponds to the absence of the section of unsteady creep in the diagram of creep (see Fig. 1, curve 3). If e10 = 0, then HT = 0 and, according to (18), BM = B and pM = p. This means that relation (29) coincides with relation (15) which describes unsteady creep without preliminary MTT. By using relation (28), we plotted the curves of unsteady creep for nickel (Fig. 2) with σ S = 100 MPa and σP = 10.2 MPa [9]. To do this, it was necessary to determine the material constants r, Ki , Bi , and pi ( i = 1, 2, 3 ). The constant r was selected analytically by describing the experimental plastic tensile stress–strain diagram of nickel (see [10]) with the help of relation (22). The constants Ki ( i = 1, 2, 3 ) were chosen by constructing the analytic description of the experimental dependences of the steady creep rate in nickel on temperature and stresses according to the relations presented in [10]. The constants B1 , B2 and p1 , p2 were chosen by describing the experimental curves of unsteady creep without preliminary MTT [11] for different loading rates and temperatures with the help of relations (14) and (15).

832

A. K. RUSYNKO

Since the synthetic theory describes plastic deformation and both sections of the diagram of creep by the same determining equation, the constants of the model must be present in all relations. Therefore, we choose the constants B3 and p3 as free for variation in the computed curves of unsteady creep with preliminary MTT: B3 = 3.241 ⋅ 10– 4 (MPa)– 1 and p3 = 4.2 ⋅ 10– 7 (MPa ⋅ sec)– 1. For the transition from the component of the vector e10 to the component of the strain tensor ε x0 , we use the formula ε x0 =

2 / 3 e10 [6].

CONCLUSIONS We generalized the synthetic theory to the case of unsteady creep of metals with preliminary MTT, including plastic prestraining at room temperature and annealing of cold-worked specimens. A simple formula was deduced for the evaluation of the level of strains after unsteady creep preceded by MTT. Since the agreement of the numerical results with the experimental data is quite good, this relation can be used for the prediction of the parameters of unsteady creep in materials depending on the degree of preliminary cold-work hardening in the first stage of MTT. REFERENCES 1. I. I. Ivanova, Mechanical and Thermal Treatment as a Method for Improving the Heat Resistance of Metals and Alloys [in Russian], Sci.-Techn. Society of Machine Building, Public University, Moscow (1964). 2. G. Ya. Kozyrskii, P. N. Okrainets, and G. A. Petrunin, “Specific features of the structures of nickel with elevated creep resistance,” Vopr. Fiz. Met. Metalloved., No. 20, 42–51 (1964). 3. V. M. Rozenberg, “Influence of substructures on the creep of nickel,” Fiz. Met. Metalloved., 6, No. 11, 899–909 (1961). 4. I. I. Novikov, Theory of Thermal Treatment of Metals [in Russian], Metallurgiya, Moscow (1974). 5. Ya. F. Andrusik and K. N. Rusinko, “Plastic deformation of strain-hardening materials loaded in a three-dimensional subspace of the five-dimensional deviator space,” Mekh. Tverd. Tela, No. 2, 78–83 (1993). 6. A. A. Il’yushin, Plasticity [in Russian], Izd. Akad Nauk SSSR, Moscow (1963). 7. K. N. Rusinko, Specific Features of Inelastic Deformation of Solid Bodies [in Russian], Vyshcha Shkola, Lviv (1986). 8. A. K. Rusinko, “Creep with temperature hardening,” Fiz.-Khim. Mekh. Mater., 33, No. 6, 71–74 (1997). 9. N. I. Kornilov, Nickel and Its Alloys [in Russian], Izd. Akad Nauk SSSR, Moscow (1958). 10. A. K. Rusynko, “Analytic dependence of the steady creep rate in metals on the degree of plastic prestraining,” Probl. Prochn., No. 4, 91–102 (2002). 11. M. G. Lozinskii, Structure and Properties of Metals and Alloys at High Temperatures [in Russian], Gos. Nauch.-Tekh. Izd. Lit. Chern. Tsvet. Metallurg., Moscow (1964).