Synergetic approach to work hardening of metals

Materials Science and Engineering,

Al37

119

(1991) 119-123

Synergetic approach to work hardening of metals A. FranGk C?KLISemiconductors,

NVZ- V2, 14002 Prague (Czechoslovakia)

J. Kratochvil Institute of Physics, Czechoslovak Academy of Sciences, 18040 Prague 8 (Czechoslovakia)

M. SaxlovH and R. SedlEek Faculty of Mathematics and Physics, Charles Unicersity, 12116 Prague (Czechoslovakia)

Abstract Work hardening is understood as a transient stage that starts at the onset of plasticity and approaches steady state. The process of hardening, which is a consequence of generation of dislocations, their clustering and annihilation, is described in terms of a simple synergetic model of the dislocation population. The population is spontaneously arranged into a characteristic dislocation pattern of high and low dislocation density regions, where the wavelength and profile of the pattern change during deformation. In the steady state all newly generated dislocations are annihilated in the high density regions. The work-hardening process can be understood as the tendency of a deformed metal to minimize its internal energy.

1. Introduction Typically the phenomenon of work hardening is manifested in a stress-strain diagram of a tensile test. An idealized stress-strain curve is shown in Fig. 1, where ais the stress and E is the plastic strain. In Fig. 1, two characteristic features are emphasized: the formation of the dislocation patterns (spontaneous structuralization) that underlies work hardening and the tendency to reach a steady state through dynamic recovery. The most significant difference between the idealized stress-strain diagram and a real one is the existence of strain localization. If localization reaches macroscopic proportions, it may cause failure and interrupt the hardening process. This possibility is indicated in Fig. 1 schematically by the broken line. Within the synergetic approach the problem of strain location has been outlined in previous papers [l-3]; the present paper concerns the idealized stress-strain curve. It is generally accepted that dislocation interactions cause dislocation accumulation during straining, and dislocation rearrangements during work hardening and dynamic recovery. A wide range of structural and macroscopic observations 0921-5093/91/$3.50

indicate, however, that the detailed nature of dislocation interactions is of less importance than the spatial arrangement of dislocations in characteristic patterns such as tangles or cells. These observations have been respected as a basic fact in recent work-hardening studies by others [4-71. Our synergetic model accepts the same point of view but, unlike the previous attempts, it gives a reason for the dislocation pattern formation, dislocation rearrangement during straining, and the approach to steady state.

steady-state

/

structuralization

E

Fig. 1. Idealizedstress-strain diagram. 8 Elsevier Sequoia/Printedin The Netherlands

120

The synergetic approach to plastic properties of materials has been developed in several recent papers [8-14]. The main stream of this research has been directed towards the formation of dislocation patterns as predicted from a linear stability analysis of the system of equations describing the dislocation population. However, the stressstrain diagram seems to be the result of nonlinear effects that cannot be described by linear analysis. This will be shown in the present paper using a one-dimensional model. Despite its simplicity the model qualitatively simulates the main features of the idealized stress-strain diagram. The model includes the initial linear workhardening and the subsequent transition to dynamic recovery with a decreasing workhardening coefficient; both are consequences of the generation of dislocations, their clustering and annihilation. The model proposed in this paper demonstrates that a uniform distribution of stored dislocations is unstable and dislocation tangles are spontaneously formed. The tangles serve as storage facilities for redundant dislocations. When the density of dislocations stored within tangles reaches a critical level, the mechanism of massive annihilation starts to operate. As a consequence the tangles begin to be rebuilt into cell walls and the dislocation pattern is changed. The walls serve mainly as sinks for dislocations generated in the cell interiors. There is a tendency to form as many walls as needed to annihilate all the newly generated dislocations. The increase in the number of walls (i.e. the decrease in cell size) means a higher proportion of the hard components (walls) in the material. Hence, the stress required for further deformation increases. The stress levels off in the steady state, where a sufficient number of walls has been created to annihilate all subsequent dislocations generated during straining.

quantity of the model is the stored dislocation density p. The proposed one-dimensional model is assumed to be rate independent and elastic strain is neglected. The more general two- and threedimensional theories of the early stage of dislocation pattern formation were presented in refs. 9-1 1. The results were based on the linearized theory. An advantage of the present one-dimensional model is that non-linear effects, which control the idealized stress-strain curve, can be directly evaluated. The first basic equation of the model is the balance law for the stored dislocation density p(x, e), where x is the coordinate and the plastic strain e serves as the evolution parameter: 0p De

(1)

Equation ( 1 ) means that the rate of accumulation of stored dislocations per unit strain must be equal to their net flux - OJ/Ox plus the net rate of generation (or annihilation) of stored dislocations per unit strain. The physical meaning of p requires that p >/0. From dislocation dynamics the flux J was derived in the form [9] ~T

J=D -Ox

(2)

where D is positive ("uphill diffusion") and generally depends on p and v. For the purpose of this paper, D is assumed to be a positive constant. ~:(x,e) represents the local stress. The macroscopic stress o can be identified with a spatial average of ~-, i.e. o = (r). The dependence of the rate r on the density p is considered in the form a

2. Model To obtain a sufficiently simple and mathematically tractable theory, the dislocation population is idealized. It is assumed that it consists of glide dislocations and stored dislocations. The glide dislocations carry plastic deformation while the stored dislocations, mainly in the form of dislocation dipoles and multipoles, hinder the glide. The glide dislocations will enter the theory only implicitly through the plastic strain e. The principal

0 Ox J + r

r=

I P ~
-b

(3)

P>Pc

where Pc, a and b are positive constants. The relations (3) mean that, as long as the density p is less than the critical value Pc, the stored dislocations are generated at the rate a. However, in places where p exceeds Pc, the dislocations become annihilated at the rate - b (for relevant dislocation mechanism see refs. 6, 15 and 16).

121

The second basic equation proposed in ref. 9 relates the stress r and the stored dislocation density: co

r(x,e) = r , , + f M(~)p(x+~,e)d~

(4)

co

where r 0 is the yield stress and M is the "influence" function of hardening. Equation (4) expresses the fact that, on the micrometre scale, work hardening is a non-local effect represented in (4) by the integral term. According to the more detailed analysis given in ref. 9, the integral comprises both the so-called passing stress and the bowing stress. M(~) is expected to be a positive symmetric function which decreases with increasing I and approaches zero for I -~ co. It should be noted that, for uniform p, eqn. (4) is reduced to a standard linear hardening r = to+Hop, H,,= fM(~)d~. In our model, eqn. (4) will be employed in the approximate form

r(x,e)= zo+ Hc~p(x,e)+ Hl

02p(X, 6) Ox 2

(5)

where 2H I = f M ( ~ ) ~ 2 d~. Equation (5) follows from (4), if p(x, e) is expanded in a Taylor series, used in (4) and restricted to the first two non-zero terms (owing to the symmetry fM (~)~ d~ = 0). Inserting (2), (3), (5) into (1) we obtain the following equation for the model:

_ OP + D H o 06

Ox 2

HI

a

a' l

p = ae + fi exp(ikx + we)

(7)

where /5 is the infinitesimal amplitude, k is the wavevector and ~o represents the amplification factor which determines the rate of growth or decay of the perturbation. Using (7) and (6) we get the relation

w = Dk2(Ho- H 1k 2)

(s)

As there is always a certain range of k for which w is positive, the perturbation wave grows exponentially, and according to model (6) the uniform distribution of stored dislocations is unstable. As oJ occurs in an exponential, it is convenient to ignore the growth of all waves but those near the fastest-growing wave. Hence, in the first approximation the wave with maximum w determines the wavelength of the dislocation pattern in the initial stage of straining. The amplification factor w reaches the maximum value for k = +(Ho/2H1) I/2. This means that the most probable wavelength of the "tangle" structure is ~'b = 2~(2HI/Ho) I/2 and the dominant initial stored dislocation density distribution is

7x

{ p<~pc if

-b

tion of dislocation tangles can be deduced from the linear stability analysis of eqn. (6). Initially the density p is close to zero, i.e. much below the critical value Pc, i.e. r = a. When investigating the linear stability of (6), we are looking for the fastest-growing perturbation in the form of an infinitesimal wave superposed on the homogeneous solution p = ae. The perturbed solution is assumed in the form

(6) P >Pc

The reason for the drastic simplification employed in (6) is that very useful analytical estimates of a solution of (6) can be obtained, while the most essential features of the mechanism of work hardening and dynamic recovery are preserved. 3. Formation of dislocation tangles, cells and the steady state

Two analytical estimates of the solution of (6) are possible. First, the early stage of the forma-

[[g°'l/2x}]exp(D~Hle) (9) For increasing e the density p given by (9) becomes always in some regions negative, i.e. it violates the condition p >/0. A preliminary nonlinear evaluation of (6) that respects p >/0 predicts a coarsening of dislocation distribution (9). Secondly, an analytical solution of the steady state of (6) can be found. In the steady state, ap/ 0 e = 0 and (6) becomes an ordinary differential equation. The solution ps(X) of such an equation that possesses three continuous derivatives can

122

be written for x >10 in the form

the differential equation (6) numerically indicate that the steady state solution (10) is an attractor of (6). In terms of the stress-strain diagram this means that the stress approaches or oscillates around the steady state value o s. From the physical point of view such behaviour is quite realistic. It is observed in situations where the tendency to localization is suppressed [17, 18].

I(,,ol } i(.0/1, }

p~(x)=cl +c2x + c 3 s i n [ ~ ]

"~ C4 COS [ k ~ l l ]

X

× ~ (-1)h(x-xn n = l

x

+2DHo+2D--~o

)((

2

x - x , ) -~

F f/H \1/2 lcoslt )

2H1

14o

17)

4. Summary

,10,

where x, belongs to the sequence of x, 0 ~ xn. An analogous formula for p~(x) can be derived for x ~<0. A "tangle" structure solution (9) (broken line) and a steady state "cellular" solution for c 1 = c 2 = c3 = c4 = 0 (full line) are compared in Fig. 2. The wavelength of the "cellular" structure is ~.c = x~ +2 - x,; the peaks of (10) simulate the cell walls. For H 0 e H~ the contribution of the terms in the square brackets in (10) is small and the dislocation density profile of the "cellular" structure is represented by parts of parabolae; then )lc = 2(2DHopca)l/2(a + b)/ab and the "cell wall" width w=2(2DHoPca)l/2/b; hence, w is smaller for higher annihilation rate b. The stress-strain diagram of the model can be evaluated as the spatial average cr(e)=(r(x,e)), where r is given by (5). If we use for p(x, e) the "tangle" structure solution (9), we get the initial part of the diagram o = r 0 + Hoae. The steady state value o s follows from spatial averaging of (5), where (10) is used for p(x,E). We get o~ = r 0 + HoP¢(2a + b)/3b + H 1a/DH o. Preliminary mathematical analysis and attempts to solve 9 X, 9~

W

k / _-t_

~/I/

x

Fig. 2. "Tangle" structure ( - - - ) and steady state "cellular" structure ( ).

The results of the previous sections might contribute to a deeper understanding of the physical nature of work hardening. The results are based on the simple synergetic model of collective behaviour of dislocations that underlies hardening. The model described by eqn. (6) is the combination of two ingredients: (i) balance law (1) for stored dislocations; (ii) dependence (4) of the local stress r on the stored dislocation density p. It was shown that in the idealized case, where strain localization is disregarded, the workhardening process has three basic stages. (a) Immediately at the outset of plastic deformation the characteristic pattern of low and high dislocation density regions ("tangle" structure) is spontaneously formed. The widths of the regions are comparable. (b) The rearrangement of the "tangle" structure into the "cellular" structure is triggered by massive annihilation of stored dislocations in the high density regions. During this process the wavelength and profile of the pattern are changed (Fig. 2). The low density regions (cell interiors) may become much wider than the high density regions (cell walls). The cell walls are thinner for a higher dislocation annihilation rate. (c) With increasing strain the process approaches (or oscillates around) the steady state, in which the generation and annihilation of dislocations are balanced. For effective annihilation a sufficient number of cell walls must be formed. It should be noted that the transition from the initial stage of work hardening with a high hardening coefficient to the steady state, where there is no work hardening, can be understood as a tendency to minimize the internal energy of a deformed solid. In all stages of plastic straining, many more dislocations are produced than is needed to carry the deformation. The redundant dislocations are stored within the material and hinder further glide. To continue straining, the stress must be increased, i.e. the internal energy

123

increases. The cooperative phenomena in the dislocation population are directed towards liquidating the excess dislocations. The places of liquidation are mostly cell walls. When the walls are built in a sufficient number, all the newly generated dislocations are annihilated, steady state is reached, the stress levels off, and the internal energy ceases to increase.

References 1 J. Kratochvfl, J. Czech. Phys. B, 38(1988) 421. 2 J. Kratochvfl, in P. Lukfi~ and J. Pol~ik (eds.), Basic Mechanism of Fatigue of Metals, Academia, Prague; Elsevier, Amsterdam, 1988, p. 15. 3 J. Kratochvfl, J. Mech. Behav. Mater., ( 1991 ) in the press. 4 E.W. Hart, J. Eng. Mater. Technol., 98(1976) 193. 5 H. Mughrabi, Mater. Sci. Eng., 85(1987) 15. 6 U. E Koks, in Dislocations and Properties of Real Ma-

terials, Metals Society, London, 1984. 7 E R. N. Nabarro, Acta Metall., 37(1989) 1521. 8 J. Kratochvfl and S. Libovick3~, Scr. Metall., 20 (1986) 1625. 9 J. Kratochvfl, Rev. Phys. Appl., 23(1988) 419. 10 J. Kratochvfl, Scr. Metall. Mater., 24 (1990) 891. 11 A. Fran6k, R. Kalus and J. Kratochvfl, Philos. Mag., in the press. 12 E. C. Aifantis, in L. Kubin and G. Martin (eds.), Nonlinear Phenomena in Materials Science, Trans Tech, Aedermannsdorf, 1988, p. 397. 13 E. C. Aifantis, in A. S. Khan and M. Tokuda (eds.), Advances in Plasticity 1989, Pergamon, Oxford, 1989, p. 537. 14 D. Walgraef and E. C. Aifantis, in G. J. Weng (ed.), Micromechanics and lnhomogeneity, Toshio Mura Anniversary, Springer, Berlin, 1990, p. 511. 15 D.J. Quesnel and J. S. Tsou, Scr. Metall., 14 (1980) 935. 16 E B. Prinz and A. S. Argon, Acta Metall., 30 (1984) 1021. 17 M. Bo6ek and J. H. Choi, Res. Mech., ( 1991 ) in the press. 18 T. Sakai and J. J. Jonas, Acta Metall., 32 (1984) 189.