Dynamic strain aging and plastic instabilities

J. Mech. Phys. Solids. Vol. 43, No. 5, pp. 671-700,

0022~50%(95)00010-O

DYNAMIC

STRAIN

AGING SINISA

Harvard

University,

(Received

Division

AND PLASTIC

1995

Elsevier Science Ltd Printed in Great Britain 0022-5096/95 $9.50 + 0.00

Pergamon

INSTABILITIES

DJ. MESAROVIC

of Applied

Sciences, Cambridge,

MA 02138, U.S.A.

10 June 1994; in retCsed,form 23 January 1995)

ABSTRACT A constitutive model proposed by McCormick [(1988) Theory of flow localization due to dynamic strain ageing. Acta. MetaN. 36,3061-30671 based on dislocation-solute interaction and describing dynamic strain aging behavior, is analyzed for the simple loading case of uniaxial tension. The model is rate dependent and includes a time-varying state variable, representing the local concentration of the impurity atoms at dislocations. Stability of the system and its post-instability behavior are considered. The methods used include analytical and numerical stability and bifurcation analysis with a numerical continuation technique. Yield point behavior and serrated yielding are found to result for well defined intervals of temperature and strain rate. Serrated yielding emerges as a branch of periodic solutions of the relaxation oscillation type, similar to frictional stick-slip. The distinction between the temporal and spatial (loss of homogeneity of strain) instability is emphasized. It is found that a critical machine stiffness exists above which a purely temporal instability cannot occur. The results are compared to the available experimental data.

1.

INTRODUCTION

This paper is concerned with stability of mechanical response based on a constitutive model for dynamic strain aging. Strain aging in alloys is associated with the time dependent segregation of mobile solute to temporarily arrested dislocations, which partially impedes dislocation motion. Two types of instabilities are associated with strain aging : Luders fronts and the Portevin-LeChatelier (PLC) effect. The Luders front, in a tensile specimen, is a delineation between plastically deformed and undeformed material (Lomer, 1952). It appears at one end of the specimen and propagates with typically constant velocity, if the cross-head velocity is kept constant, towards the other end (Butler, 1962). The nominal stress-strain curve appears flat during the propagation. However, the localization is preceded by yield point behavior; after reaching a peak, the flow stress quickly drops to a lower value. The PLC effect, on the other hand, is seen either as a sequence of shear bands appearing sequentially with sometimes regular spacing, or as a set of propagating bands with a source at one end of the specimen (Chihab et al., 1987). The resulting nominal stress-strain curve is serrated ; stress and plastic strain rate oscillate. During the oscillations, the average stress may remain constant or increase, either steadily or in steps corresponding to the crossing of the specimen length by the bands, indicating continuous strain hardening. These phenomena may affect manufacturing processes, as well as the fracture properties of a material. Delafosse et al. (1993) recently found that dynamic strain 671

672

S. D. MESAROVIC

aging affects the fracture resistance of Al-Li alloy. Their tests, in the range of temperatures and strain rates where dynamic strain aging is important, showed localization in the form of the PLC bands near the crack tip and strong temperature and loading rate dependence of the tearing modulus. Transition from Luders front to PLC response, while changing temperature or strain rate, has been observed (Ardley and Cottrell, 1953 ; Nadai, 1950; Brindley et al., 1962; Sleeswyk, 1958), which indicates that these are related phenomena, stemming from similar, but continuously varying, constitutive properties. Further, the yield point is known to reoccur after some rest period, indicating that a time dependent process, i.e. strain aging, is responsible for this phenomenon. In the present work, it is shown that a dynamic strain aging constitutive model (McCormick, 1988) describes both oscillatory behavior and yield point phenomenon. The following, important distinction should be noted. Another type of yield point behavior and Luders fronts may occur in low dislocation density single crystals (Johnston, 1962), which is due to the rapid multiplication of dislocations and their spatial propagation by the cross-slip mechanism (Johnston and Gilman, 1960). This type of behavior is not repeatable after unloading (Johnston, 1962), unless of course, the specimen is annealed. Luders fronts and the PLC effect have been observed in a variety of fee and bee, substitutional and interstitial alloys in particular ranges of temperatures and strain rates, as well as in non-metals such as silicon (Mahajan et al., 1979). Recently, the PLC effect has been observed in ordered intermetallics (Brzeski et al., 1993) suggesting a disordering near dislocations. For a recent review of experimental observations see Neuhauser and Hampel(1993) and McCormick et al. (1993) ; the former concentrates on single crystals. Cuddy and Leslie (1972) demonstrated that the PLC effect occurs in Fe-3% Ni and Fe-1.5% Si within a region of temperature-strain rate space bounded from below and above. Here, it is shown that McCormick’s (1988) constitutive model describes this dependence. Once dynamic strain aging is accepted as a model, the physical interpretation of these bounds is rather simple. Consider a stage of deformation with abundant mobile and forest dislocations. Dislocation glide is assumed to be grossly discontinuous with a long waiting time at obstacles and a very short flight time between them. The yield stress is determined by the strength of obstacles. This strength, in turn, depends on the local concentration of diffused impurity atoms in the cores of temporarily arrested dislocations and in their neighborhoods. This diffusion process is very complex and time dependent. The average waiting time of dislocations at obstacles is the time available for diffusion. It is inversely proportional to the strain rate. Qualitatively, high strain rates or low temperatures result in a long characteristic diffusion time and short waiting time, so that, in the limit, dislocations glide through a practically nominal concentration of solute. Low strain rates or high temperatures give a short characteristic diffusion time and long waiting time, so that in the limit, dislocations glide by breaking off from equilibrium segregated (hence, often practically saturated) cores and atmospheres. This scenario excludes the possibility of solute atoms being dragged by a moving dislocation. Intermediate strain rates and temperatures lead to the competition between two processes : (a) The nature of thermal activation is such that the flow strength increases with increasing strain rate.

Dynamic

(b) An increase in strain weakens the obstacle,

strain aging and plastic instabilities

rate shortens the time available for diffusion leading to a decrease in flow strength.

673

and thus

When the second effect dominates, the steady state strain rate sensitivity, do/d In tilrteadys,ate,becomes negative ; here E denotes plastic strain. Transient behavior in the positive strain rate jump experiment is always characterized by positive stress jump (positive instantaneous strain rate sensitivity). The negative steady state SRS is substantiated experimentally by Estrin and Kubin (1989) for Al-5% Mg [see also Nadai (1950) for iron], while the strain rate step experiments by McCormick and Estrin (1989) and Ling and McCormick (1990) on the same material demonstrate the positive instantaneous SRS. Estrin and Kubin (1989) and Ling and McCormick (1990) obtained large negative SRS for Al-5% Mg of the order of - 1.5 MPa. Cottrell and Bilby (1949) developed the theory of strain aging in iron based on the segregation of carbon atoms to temporarily arrested dislocations, in the limit of dilute concentrations. While their theory predicts the negative strain rate sensitivity of yield stress, it does not include the saturation of solute atoms at dislocations and would, if included in the present model, predict large negative strain rate sensitivity at low strain rates, contrary to experiments. On the phenomenological side, Penning (1972) assumed an intermediate range of strain rates where the strain rate sensitivity is negative to explain the PLC effect. Van den Beukel (1975) demonstrated that the strain aging model which includes saturation behavior would indeed result in a negative SRS region, as shown in Fig. l(a). The Cottrell-Bilby aging kinetics were modified by Louat (1981) to include saturation. He simply included the fact that the probability of the solute atom coming to a core site is proportional to the fraction of available core sites. The local concentration C varies between the nominal value, C,,. and the equilibrium concentration, C,, with the characteristic diffusion time r,, : C-C0

=(C,-C,)*(l-exp{-(C,/C,)*(t/t,)‘!’)).

Based on this, McCormick (1988) developed a constitutive model exhibiting negative steady state SRS and positive instantaneous SRS [Fig. l(b)]. McCormick (1988) and Estrin and McCormick (1991) also demonstrated the oscillatory behavior predicted in the tensile test with this model. Recent observations by Springer and Schwink (1991) support an exponent of l/3, instead of 2/3, in the above formula. The physical basis for such a difference is thought to be pipe diffusion along dislocation lines. Ling et al. (1993) performed numerical time integration, with the l/3 exponent, to demonstrate the similarity of predicted oscillations to the experimental results. In the present work, the full bifurcation analysis is done and temperature-strain rate bounds established. The model used is similar to McCormick’s; the concentration of solute atoms is described through a state variable, varying as the one above. It is shown that the value of the exponent has no qualitative effect on the behavior of the constitutive model, but the quantitative differences are important for accurate modeling. Data assembled by Nadai (1950) for mild steel at 200°C demonstrate that while the upper yield point has positive SRS, the yield stress after the yield point (as well as the ultimate stress) has a region of mild, but clearly defined, negative SRS. This is also predicted by the present model. While the present model is based on physical processes which occur on an atomic/dislocation scale, the formulation is macroscopic and

614

-8

7 co A

-14 -16 -18

-6

-4

-2

0

2

4

6

8

10

178 177 3 B ;

176 175 174 173 0

0.0004

0.0008

0.0012

0.0016

0.002

E Fig. 1. (a) Strain rate dependence of the yield stress for CI= l/3,2/3 and 1. (b) Response to the strain rate jump : instantaneous vs steady state strain rate sensitivity. a = l/3, T = 300 K ; parameters from Table 1, jump lo-’ + 10d4 s-‘,

phenomenological, so that the number of independent parameters is minimized. The state variable formulation attempts to achieve some generality and to include a variety of materials and microscopic aging mechanisms. Estrin et al. (1991) performed a load controlled tensile experiment with two strain gauges. By plotting the readings from the two gauges, one versus the other, they concluded that the temporal oscillations precede the spatial localization. The stability analysis in Section 3 predicts simultaneous occurrence of both types of instabilities, for a load controlled experiment. For a cross-head velocity controlled experiment,

Dynamic strain aging and plastic instabilities

675

the loss of homogeneity occurs before the temporal instability (oscillatory behavior) sets in. However, an interpretation of the results of Estrin et al. (1991) consistent with the stability results is possible. This explanation is presented in Section 3. The uniaxial model used here cannot predict the mode of non-homogeneous deformation. A threedimensional continuum analysis, analogous to the Hutchinson and Miles (1974) analysis of necking in rate independent materials, is needed.

2.

CONSTITUTIVE

MODEL FOR DYNAMIC STRAIN AGING

The following is a slightly modified McCormick (1988) constitutive model for dynamic strain aging. Assuming that obstacle controlled glide is the dominant mechanism for plastic deformation for the temperatures and strain rates in question, one can regard plastic straining as a thermally activated process (Kocks et al., 1975) : i = i. - exp { - AG/kT}

(2.1)

where & is a reference strain rate, which clearly depends on the current strain, i.e. current dislocation densities, and stress. However, for strong obstacles, the exponential varies strongly with stress so that pre-exponential can be regarded as constant. Frost and Ashby (1982) find that the value B0= lo6 ss’ fits experimental data for both fee and bee metals. In (2. l), T is the temperature, k. the Boltzman constant, and AG is the free enthalpy (Gibbs free energy) of activation, referred to as activation energy in the sequel. It is the average energy barrier encountered by dislocations at obstacles to be overcome by thermal fluctuations. The activation energy can be considered as a function of the excess stress, ~7- 6, and of the phenomenological state variable, C, representing the local measure of the concentration of solute atoms in the dislocation cores and in the surrounding atmosphere : AG = AG(o-6,

C).

Here, CJis the applied stress and 6 is the average flow stress at 0 K, for a given nominal solute concentration and dislocation density. For example, Frost and Ashby (1982) find the value 6 = 183 MPa appropriate for (Al). The temperature dependence of AG is considered weak compared to that of l/T and is neglected. The state variable C must increase with the average waiting time of dislocations at obstacles t, and approach its equilibrium value at infinite average waiting time. Following Louat’s (1981) theoretical development, and the Springer and Schwink ( I99 1) experiments (see Introduction), the non-dimensional state variable, varying from 0 to 1 with the characteristic diffusion time td is defined as C = 1 -exp { -(ta/fd)‘}.

(2.2)

The effect of the nominal concentration is lumped into 8. One may actually think of C as a non-dimensional activation energy variation with time. The values l/3,2/3 and 1 for c1will be used for numerical study and the predictions will be compared. When the plastic strain rate is constant, the average waiting time has its steady state value, t,,,,. For the case of discrete obstacles, the free flight time between obstacles

S. D. MESAROVIC

616

is negligible compared to the average waiting time, so that the Orowan equation, relating plastic strain rate to dislocation densities and velocity, may be written as

where P,,, and pi are the densities of mobile and immobile dislocations, L is the mean free path between obstacles, b is the length of the Burgers vector, and R is the strain produced by all mobile dislocations moving to the next obstacle on their path. For example, for Pm = pi = 10” m-‘, Q = 10p4. Clearly, R varies with dislocation densities and thus, with accumulated plastic strain. Models for such variation can be found in Kubin et al. (1990) Kubin and Estrin (1992) and Balik and Lukac (1993). The main result is that the PLC effect can occur only for some intervals of accumulated plastic strain. Here, this variation is not considered. Nevertheless, the resulting stability condition in Section 3 includes this effect through the current value of 0. Thus, the last result states that the steady state average waiting time is inversely proportional to the plastic strain rate (Kubin and Estrin, 1990). Upon sudden change in strain rate, the average waiting time relaxes to its new steady state. The simplest phenomenological model for this relaxation is (McCormick, 1988) i,

= )tass - ta; t a,ss

sz c3,ss=

T

E

(2.3)

In the limit of vanishing strain rate, the average waiting time should approach the real time, i, -+ 1. Hence, the characteristic relaxation time [denominator in (2.3)] must be equal to the new steady state average waiting time. If the strain rate is continuously varying, the steady state waiting time acts as a “moving target” for the average waiting time. Linearizing the activation energy dependence on the excess stress and on the concentration variable, one defines

The stress variation during PLC and yield point phenomena is small compared to the total stress. The linearization of activation energy with respect to stress is thus plausible. The linearization of activation energy with respect to C is phenomenological. Since the activation energy decreases with the applied stress, the instantaneous strain rate sensitivity S must be positive (cf. “apparent activation area” of Kocks et al., 1975). These definitions imply the temperature dependence of H and S. The phenomenological diffusion time, fd, is solute mobility (and hence temperature) dependent. Thus, See T,

H cc l/T,

td a exp (l/T}.

This temperature dependence of H neglects the possible temperature dependence of the activation energy at saturation. At very high temperatures, this assumption is violated ; the equilibrium concentration becomes strongly temperature dependent

Dynamic strain aging and plastic instabilities

677

(Hirth and Lothe, 1982). The plastic strain (rate)-stress relation can be cast into the following equivalent forms :

i: = C,*exp

0 - B(E) p--H-C S

(2.4)

where h = d&/d&is the strain hardening. Equations (2.2), (2.3) and (2.4), together with the elastic stress-strain relations, fully define a one-dimensional constitutive law with one state variable. Here, one has a choice of state variable, since (2.2) defines one-to-one correspondence between C and t,. Under the condition of constant plastic strain rate, the constitutive equations can be integrated to give flss =~(s)+S*[ln$+H~(l-exp{-(&I})].

(2.5)

The steady state strain rate sensitivity S,, is then given by S,, =&=

S[l-H~$$exp{-(&I}].

(2.6)

Clearly, S,, can take negative values for intermediate strain rates. The stress-strain rate dependence (2.6) is shown in Fig. l(a) for the three values of exponent a. Parameters for this plot are deduced from the Ling and McCormick (1990) experimental results for strain rate sensitivity in Al-Mg-Si alloy at room temperature. Using three values for S,, and the value for S, all for a given plastic strain, one can fit three independent coefficients in (2.6) : H, c( and n/t,. For the accumulated plastic strain of 0.02, and strain rates of 1.1, 5.5 and 11.0 x lop4 s-l, Ling and McCormick find, from the rate jump experiments, S,, = - 1.25, - 1.Oand -0.75 MPa, and S = 1.O MPa. Since only three discrete values of c1are considered, the fitting procedure is as follows. Two experimental values are used to compute Hand n/t,, for the three values of x, from (2.6). The prediction for the third value of S,, is then computed and compared to the experimental one. It is found that CI= l/3 fits the experimental data better than the other two values. Table 1 shows the values of CY and fd for Q = 10-4. The response of the material to a jump in strain rate is depicted in Fig. l(b). An instantaneous jump in stress is observed which is proportional to S; the stress then relaxes to a steady state value which differs from the one before jump by the amount proportional to S,,. This is, of course, accompanied by continuous strain hardening. The behavior in Fig. l(a,b) is confirmed by experiments (Estrin and Kubin, 1989, McCormick and Estrin, 1989, Ling and McCormick, 1990). In the jump experiment, the characteristic relaxation time t,,,, of (2.3) is inversely proportional to the new strain rate, i.e. equal to Q/e. This brings about the alternative interpretation of Q as a characteristic relaxation strain. However, this point of view

678

S. D. MESAROVIC Table 1. Parameters for the constitutive

model based on experimental results of Ling and McCormick (1990), Figs 2 and 3, for Al-0.4% Mg4.2% Si at 293 K. For accumulated strain of 0.02 and strain rates of 1.1 x 1OP4 S -‘, 5.5 x 1O-4 s-’ and 11.0 x 10e4 sP’, the values of S,, are - 1.25 MPa, - 1.0 MPa and -0.75 MPa (the first S,, is extrapolatedfrom the values for accumulated strain from 0.0 to 0.017. Fit from (2.6) for three values

213 1

300 K 10.5935- T 300 K 9.6437.T

is limited to specific experiments. In general, strain aging is a time dependent process ; it occurs without any straining, e.g. in a stress free specimen, after unloading. Figure 2(a) shows the loci of the points in temperature-strain rate space, where steady state strain rate sensitivity changes sign, for the three values of a. The points inside the curves have negative steady state SRS. This constitutive model, as well as features of the experimental response, bears resemblance to the rate and state dependent friction models (Dieterich, 198 1 ; Ruina, 1983 ; Rice and Ruina, 1983). There, the friction stress, slip, velocity and characteristic slip weakening length are analogous to the stress, strain, strain rate and characteristic relaxation strain in the present work. Relaxation of the state variable, analogous to t,, is governed by laws similar to (2.3). In the early models, the steady state velocity dependence (analogous to S,,) was always negative. Rice and Ruina (1983) analyzed the stability and bifurcations of a spring-slider system where slider friction is governed by such a law. They showed that there is a critical spring stiffness below which the steady sliding becomes unstable. An analogous condition is derived here. Since the steady state velocity sensitivity is always negative, the quasi-static system has no stabilizing mechanism and instability results in an unbounded motion. In the dynamic case, inertia serves as a stabilizing factor and sustained oscillatory behavior becomes possible. Recently, Weeks (1993) implemented a phenomenological model in which steady state velocity sensitivity becomes zero or positive at high velocities. Gu et al. (1984) performed a full non-linear analysis of the spring-slider system, with slider friction governed by rate and state dependent laws with one or more state variables. They found that multi-variable laws lead to chaotic behavior. The constitutive law has no limit at vanishing strain rate. This is the consequence of unidirectional activation considerations (2.1) ; the full thermal activation analysis

Dynamic

strain aging and plastic instabilities

619

6

-6

I

0

2

4

,7--m

,

J

6

8

10

6

8

10

300K

6t

0

2

4 300K

Fig. 2. (a) Regions Hopf bifurcation

of negative strain rate sensitivity (inside the curves) for N = l/3, 2/3 and 1. (b) Loci of points (purely temporal bifurcation), x = 113, 2/3 and 1. Parameters from Table I

would include the probability of reversal, which would, at vanishing stress, cancel out the forward term. Further, at very low and very high stresses the dislocation glide may not be discontinuous; the glide time may be more important than the waiting time (Nabarro, 1991).

680

S. D. MESAROVIC

Another problem with this type of formulation was recently noted by Perrin et al. (1994) in the context of the rate and state dependent friction model. The state variable relaxation law (2.3) does not describe the actual behavior close to i: = 0 and large t,. The relaxation towards the steady state is then unrealistically slow. A simple three-dimensional generalization of the constitutive law is to interpret (2.4) as a relation between equivalent stress and strain rate measures, and to introduce the equivalent accumulated strain as a new state variable : &=

,s

s-dt

0

where sii are deviatoric components of the stress tensor, and the indicial notation (summation over repeated indices) is used. The incompressible isotropic constitutive law is then (2.7) Based on a concept of constrained equilibrium states, Kestin and Rice (1970) and Rice (1971) showed that for a material which yields according to the Schmid-like law (i.e. the plastic flow on each slip system, in every grain of polycrystal, depends on the stress state only through the resolved shear stress on that system) a flow potential exists, such that the components of the strain rate tensor can be represented as derivatives, with respect to corresponding stress components, of the flow potential. Equation (2.7) is derivable from such a potential as

where XI

-_=e=i: aa

o . exp

Clearly, other state variables may be introduced, such as mobile and immobile dislocation densities. Indeed, the appearance of PLC is conditioned by accumulated dislocation densities. The evolution laws for those are discussed in, e.g. Kubin and Estrin (1992) and Balik and Lukac (1993). The variation of these densities with strain is slow. Here, interest lies in “fast” instabilities, dominated by rate sensitivity variations. Dislocation densities can then be treated as constants. Nevertheless, the required dislocation densities are included in the resulting stability condition through the current values of R and h.

3.

STABILITY AND BIFURCATIONS

IN UNIAXIAL

TENSION

A uniaxial quasi-static tension experiment with controlled cross-head velocity V is considered next. The machine stiffness is k (force per unit displacement). The total

Dynamic

strain aging and plastic instabilities

681

stretch ratio tr can be viewed as the elastic stretch superposed to the permanent (plastic) stretch :

Consistent with the earlier notation, the plastic stretch is without subscript, while the total stretch has subscript T; lo, 1, and IT are initial, plastic and total length. Constitutive relation (2.4) is assumed to relate the true (Cauchy) stress and logarithmic strain (rate) : cT = lnt,

= ln<,,+lnt

= ~,,+a = a/E+e.

The nominal stress, IZ,is related to the true stress by tT*n = n*exp {&+2va/E}.

CT =

(3.1)

In a cross-head velocity controlled test the following condition applies :

(3.2) where E and v are Young’s modulus and Poisson ratio, respectively, and A0 is the initial cross-sectional area. Together with the constitutive relations in the form fl- B(E) s-H*[l-exp{-(kJ}]}

c: = d,*exp i i, =

l--i.,,

(3.3)

and an appropriate strain hardening law, represented by B(E), this defines the time evolution of the three-dimensional system (E(x, t), t,(x, t), CJ(X,t)) for a given overall deformation rate R = V/l,,. The system described by (3.2) and (3.3) has no steady state solution, but only a continuously evolving one. Further, it may have spatially non-homogeneous solutions. The stability and bifurcations of the system are first analyzed using the linear perturbation theory. In Section 4, the numerical continuation technique is used to trace the branches of periodic solutions of the simplified spatially homogeneous system having a steady state solution. For a rigorous mathematical background, one may consult Guckenheimer and Holmes (1983) or, for a simpler application-oriented approach, Seydel(l988). To determine the stability of a spatially homogeneous solution, one introduces an infinitesimal perturbation (d&(x, t), &,(x, t), 60(x, t)) to the initially homogeneous solution (s(t), t,(t), a(t)). The uniaxial equilibrium implies that the nominal stress variation is independent of x, so that 6n = &z(t). Introducing the perturbation in (3.3) and (3.2), taking the gradients of the resulting equations, and taking the reference state to instantaneously coincide with the current one,

682

S. D. MESAROVIC

(3.4)

where (.),.Yis used to denote the spatial derivative and I- = I(t,(t))

= H~(t,(f)l@

exp { - (ta(t)/td)“}

is introduced for convenience, so that in the steady state limit : T(t,,,,) = 1 -S,,(C)/,% Equation (3.4) describes the evolution of the gradients of (6&(x, t), 6&(x, t), 60(x, t)) at every point x along the bar. It is assumed that the fundamental solution (E(t), t,(t), a(t)) varies slowly so that for the initial behavior of an infinitesimal perturbation can be computed by taking the coefficients in (3.4) to be constant. Assuming the usual separable form of perturbation hE(X,

t)

=

u(x)

se”’

S&(x,

t)

=

b(x)

se”

&7(x,

t)

=

c(x)

-ei’

(3.5)

and eliminating the stress gradient variation from (3.4), one obtains the linear eigenvalue problem with the characteristic equation det

-s(h-a)/S-2 st,(h-a)/Si2

-d-/t,

1 0,

-r)/n-A =

-e(i

and solutions

1s=

&[PsI!IJiGi1

(3.6)

where pL,= - [h-o+S(l --I-)/Q] and o, = 4S(h-o)/R; the subscript S is used to denote spatial instability, i.e. loss of homogeneity. The homogeneous solution is unstable if Re {A,}IS . non-negative. This occurs if ps > 0, or h-o

< -S-(1

-I-/Q

(3.7)

or, if the pre-bifurcation state is close to the steady state, h-a d - &/il This condition was derived by Estrin and Kubin (1991) for the special case of load controlled experiment. It bears emphasis that the instability condition (3.7) applies to any boundary conditions; the spatial and temporal parts of the perturbation decouple, so that the resulting stability condition is independent of a particular spatial variation of the perturbation. However, an additional condition is imposed to the initial perturbation by the “machine equation” (3.2) :

Dynamic strain aging and plastic instabilities

-5

-2

4

1 In

7

683

10

&

10-4sec-’ Fig. 3. Variation of real and imaginary parts of characteristic exponents, characterizing the spatial and temporal instabilities, with the steady state strain rate.

a(x) dx + 6n,l,

(3.8)

6n, is the initial value of &z(t). Therefore, the instability can take place under a variety of loading and extension conditions ; characteristic exponents are independent of the deformation mode. The most unstable mode can only be determined by the full three-dimensional analysis. Loss of homogeneity under no variation in load may occur even if no load control is applied ; (3.8) then implies that the variation in total plastic extension should vanish, too. This cannot be concluded from (3.8) for the load controlled experiment; the machine stiffness k is zero, so that the average plastic extension cannot be determined from (3.8). It should also be noted that the above analysis does not include the homogeneous perturbation as a special case ; (3.4) would then be identically satisfied and no eigenvalue problem would emerge. At bifurcation, ,u~ = 0, the characteristic exponents are pure imaginary. Therefore, the perturbed solution starts to diverge in an oscillatory manner. However, the characteristic exponents remain imaginary for a very small interval of strain rates. Figure 3, to be discussed later, shows the values of real and imaginary parts of the “spatial” characteristic exponent, as a function of strain rate, assuming that the fundamental solution is a steady state. In Section 5, a typical path to instability, in the beginning of a uniaxial tension test, is analyzed. There, it is shown by numerical examples that the characteristic exponents quickly become real after entering the instability region. Indeed, the stable oscillatory behavior emerges as an interplay between rate sensitive plastic deformation and elasticity of the whole system which where

684

S. D. MESAROVIC

allows for repeated load peaks and drops. In what follows, the possibility of a homogeneous oscillatory behavior is investigated with the hope that it will provide an understanding of a real experiment where loss of homogeneity and oscillatory behavior combine. It is important to note the following three points : (a) The first approximation in the analysis is the assumption of a one-dimensional stress state. In other words, the only localization allowed is of a long wavelength necking type. The experimentally observed shear localization is not one-dimensional. Further, even the uniaxial, necking type localization may require a three-dimensional approach. A three-dimensional analysis, by Hutchinson and Miles (1974), of necking in axially loaded bars with a simpler, rate independent constitutive model, shows that there is a difference, albeit small, in the instability conditions derived from the one- and three-dimensional analyses. In other words, the unperturbed homogeneous one-dimensional state does not completely determine stability ; a mode of the bifurcated solution provides a small contribution, dependent on the ratio of bar radius to length. (b) Stability condition (3.7) depends on current values of the unperturbed variables. Due to the evolving nature of the unperturbed solution, the utility of the stability condition (3.7) depends on the following condition : The time required for a change in unperturbed evolving state sufficient to cause a change in inequality (3.7) must be much greater than the characteristic perturbation decay time (the inverse of the characteristic exponent). In other words, the evolution of the unperturbed state must be slow and the variation of the characteristic exponents fast. Numerical studies reported in the next section confirm the above assumption ; the characteristic exponents change abruptly near the instability point. (c) Stability condition (3.7) applies not only to the constant crosshead velocity test but to any prescribed cross-head velocity variation which does not violate the assumption in (b). To analyze the stability of the homogeneously evolving system to the purely temporal perturbation, (3.2) is replaced with its time derivative : (3.9) Now, (3.9) and (3.3) define the time evolution of the system (e(t), t,(t), n(t)). The linear stability analysis of such a system, with respect to infinitesimal spatially homogeneous perturbations, is algebraically cumbersome and is shown in the Appendix. The system (3.9), (3.3) is truly three-dimensional [as opposed to (3.4)], implying that there should be two conditions for all the eigenvalues to have the same sign. Incidentally, the numerical distinction between the two conditions is negligible (Appendix). The instability condition for the system (3.9), (3.3) is K+h-a

< -s-(1-I-)/R,

(3.10)

where l/K = l/E+ A,/(k - lo), so that K is the equivalent elastic stiffness of the specimen and the machine. For the rigid machine, K = E. Again, if the pre-bifurcation state is close to the steady state the instability condition can be expressed as: K+h--a

< -&/Cl.

Dynamic strain aging and plastic instabilities

685

The nature of temporal instability is analyzed next. For this purpose, the system is approximated by neglecting changes in the slowly varying variable, strain. This also involves neglecting the difference between variations in nominal and true stress. (The difference in pre-bifurcation values is eliminated by simply taking the reference state to instantaneously coincide with the current one.) The system will then have a steady state solution and a full bifurcation analysis can be performed. The solutions of such reduced a system should then be interpreted as “moving targets” for the original system, as the slow variation with strain proceeds. The analysis of the reduced system is instructive and is pursued in some detail below. For the homogeneously deforming system, the “machine equation” (3.9) takes the form (3.11)

& = K(R-8).

After differentiating the first of equations (3.3) and eliminating the stress rate by virtue of (3.11), one obtains the system of two first order differential equations in variables B and t, E

KR

c:

s

i, =

K+h-a

H*cr .~_~.exp{-(t)“}.[;-~].(!$-’

s

td

l-i-t,.

(3.12)

Since the slow variations with strain are neglected, h and a are considered constants during the steady state. However, (3.12) also admit an oscillatory solution. During the oscillations, the variation in a is not slow but is small compared to its average value and to h, and certainly negligible compared to K. The steady state solution (with possible simplification for h <
KR K+h-a

Introducing non-dimensional

n R t, = t,,,, = F z $. ss

z R;

variables and non-dimensional E

x, =-;

Ess

ta

x2=t;

a.ss

(3.13)

time,

i=+ d.Sb

(3.12) takes the form I,

=

a-(x,

12 = l-x,

-x;)-H**b.

‘X2

{-b*x”z}.(l-x,.X2)‘X,

*x;-’ (3.14)

where, for the purpose of numerical studies in Section 4, the following notation is introduced :

S. D. MESAROVIC

686

and

H* = H*cr.

The steady state solution and steady state strain rate sensitivity are then X 1,s --

1;

xl,ss = 1

S,, = S*[l-H**b*exp(-b)].

(3.15)

A linear stability analysis gives the characteristic exponents

where pt = - [K+h--o+ S(l -I)/fl] and o, = 4S(K+h-o)/R; the subscript t denotes purely temporal instability. The only difference between the characteristic exponents of the spatial and temporal instabilities is the K term. Its effect is that the characteristic exponents of the temporal instability remain complex for a much wider interval of strain rates. The instability occurs when the real part of the characteristic exponent becomes positive. The values of /1cross the imaginary axis with a non-zero imaginary part (the argument of the square root is negative when the first term approaches zero). The bifurcation is, therefore, a Hopf bifurcation, and the emerging solution is a limit cycle. Note that the function b. e -b has the maximum value l/e, so that the bifurcation is possible only if

Therefore, for a given material composition, temperature and accumulated dislocation densities (hardening stage), there is a critical equivalent stiffness K above which no temporal bifurcation is possible. However, K cannot be greater than E, so that, for a sufficiently high S and small R, the critical K may not be achievable. Given a sufficiently low K, bifurcation occurs when the condition (3.10) is satisfied. This temporal instability is completely analogous to the Rice and Ruina (1983) results for a spring-slider system with rate and state dependent friction. For the parameters K+ h - cr = lo4 MPa, R = 10m4,S = 1 MPa x (T/300 K), and other values shown in Table 1, convenient non-dimensional parameters are : a=--

300 K T

and

f=ln

R 10-4

s-l

.

The locus of Hopf bifurcation points in a-fspace is shown in Fig. 2(b) for the three values of CI.Outside these curves, the steady state is stable. Upon crossing the curve, the steady state becomes unstable, and a limit cycle emerges. Comparing Fig. 2(a) with 2(b), it is clear that the temporal instability occurs well within the negative S,,

Dynamic

strain aging and plastic instabilities

687

range. This has also been concluded by Ling and McCormick (1990). The turning points of the curves in Fig. 2(a,b) should be taken with some reservation, due to the earlier mentioned high temperature effect on the solute saturation at dislocations. For K = 0, i.e. for the load controlled experiment, instability conditions (3.10) and (3.7) coincide; both types of instability occur simultaneously. The rate independent case is obtained in the limit of vanishing SRS. In the rate independent limit, S = 0, the instability condition (3.7) reduces to the well known Considere condition for necking. As an aside, note that an additional information is obtained when Considere condition is derived as a limiting case of rate dependent analysis. The usual rate independent one-dimensional stability analysis proves only the possibility of a nonhomogeneous increment, i.e. the bifurcation ; the present approach confirms that the homogeneous solution becomes unstable when the Considere condition is satisfied. System (3.14) is two-dimensional and autonomous (i.e. the governing equations have no explicit time dependence). Therefore, the only type of oscillatory behavior expected is a periodic one. Both conditions (3.7) and (3.10) implicitly include the effect of accumulated strain and dislocation densities, through the strain hardening modulus h and the relaxation strain s1. The evolution of Q with accumulated strain is non-monotonic (Kubin et al., 1990, Kubin and Estrin, 1992, Balik and Lukac, 1993). For a hardening high enough to avoid Considere necking, both instability conditions require sufficiently small Q. The result is that there are intervals of accumulated strain where the instability conditions, (3.7) and (3.10), cannot be satisfied even if the strain rate sensitivity is negative. For example, if h-o z GPa, and S,, z MPa, the value of R required for instability (3.7) is of the order 0.001 or less. More about this effect can be found in the above references. Characteristic exponents determine the initial behavior of the perturbed system, i.e. while the perturbation is still small. In Section 5, it is shown that, in the neighborhood of instability, the fundamental (unperturbed) solution varies rapidly; due to the negative strain rate sensitivity, the system moves through the bifurcation point quickly towards the target strain rate imposed by the cross-head velocity. The behavior of the system “in the large” is then determined by the state corresponding to the target strain rate, and not by the state near the bifurcation point. It is, therefore, instructive to examine the variation of characteristic exponents as the fundamental solution moves through the range of strain rates. Figure 3 shows the variation of real and imaginary parts of “spatial” and “temporal” exponents with strain rate. These are computed assuming that the steady state conditions are valid for each strain rate. In Section 5, similar calculations are discussed for the non-steady state fundamental solution. The steady state SRS is shown for reference. The spatial instability occurs close to the zero SRS. The imaginary part of the spatial exponent exists only briefly. This implies that, unless the target strain rate is close to the bifurcation point, oscillatory behavior will not have sufficient time to develop; upon entering the region of real spatial exponent, the monotonic necking sets in. In contrast, the temporal exponent is always complex and large, which implies relatively fast oscillations. Estrin et al. (1991) compared the readings from two symmetrically positioned strain gauges (symmetry axis being normal to the specimen axis), during the load controlled

688

S. D. MESAROVIC 6.0 ,

h-J-10w4set-’ (b)

5.0

unstable limit cycles unstablesteady states

I

I

-1.5 In

I R

I - 1.o

lo4 set-’

Fig. 4. Bifurcation diagrams for system (3.14). Parameters from Table 1, T = 300 K. (a) a = l/3, c( = 2/3, and a = 1 ; (b) detail of (a) for 61= l/3.

experiment, and concluded that the temporal instability precedes the loss of homogeneity. The present analysis predicts that, for the load controlled experiment, the two instabilities coincide. However, the spatial instability starts as diffuse necking, and, it seems reasonable to assume, symmetric. The symmetrically positioned strain gauges would not detect symmetric necking. A sharp shear band localization is expected to develop later. This is a common observation in rate independent materials (Rice, 1976). Therefore, what appears to be a purely temporal instability may actually be an oscillating symmetric necking.

4.

CONTINUATION

AND STABILITY ANALYSIS OF PERIODIC OSCILLATIONS

A numerical continuation of (3.14) with bifurcation and stability analyses was performed using the program AUTO: Software for Continuation and Bifurcation Problems of ODES (Doedel, 1986). Figure 4(a) shows the bifurcation diagrams for

Dynamic

strain aging and plastic instabilities

689

the three values of c(. The limit cycles are represented by their strain rate amplitudes. These are the cuts through the temperatureestrain rate space in Fig. 3 at T = 300 K ; a magnified detail is shown on Fig. 4(b). Going from the left stable steady state, at the first Hopf bifurcation point, a branch of unstable limit cycles emerges and bends backwards only to gain stability at the turning point. For the strain rates between the turning point and the Hopf bifurcation, three solutions exist : one stable steady state and two limit cycles, one stable and the other unstable. This means that, in this region. for a sufficiently large perturbation from the steady state, the system can converge to the stable limit cycle. Figure 5(a))(f) shows different views of a high amplitude limit cycle, close to the middle of the region with two solutions, and a transient from the perturbed unstable steady state. Figure 5(e) compares the steady state stress-strain rate relation with the same relation during oscillations. In Fig. 5(f), an appropriately normalized strain rate, average waiting time and stress are plotted in parallel. During most of the cycle, the stress is close to the maximum. There is a large delay between the minimum strain rate and the maximum average waiting time. The burst in strain rate is then followed by a drop in average waiting time and stress. Figure 6 shows behavior similar to that in Fig. 5(e) but now in the region with three solutions. Here, a finite perturbation must be introduced to escape the attraction of a stable steady state. Convergence to the limit cycle is very slow. This is a region of imperfection sensitivity. In any uniaxial test, the end conditions at grips introduce a finite perturbation. The start of the localization instability is always observed there.

5.

YIELD

POINT

BEHAVIOR

AND PATH

TO INSTABILITY

There are at least two scenarios for the onset of instability (3.7). First, the system may be evolving at approximately constant strain rate. The instability than occurs when the stress becomes sufficiently high and hardening modulus sufficiently low, i.e. its occurrence is due to the changes in accumulated strain (or dislocation densities) ; changes in R also fall into this scenario. The more common case is the instability at the beginning of the uniaxial tension test, when the plastic strain rate changes from zero to some finite value determined by the imposed cross-head velocity. The basic mechanism can already be seen on Fig. l(b) which shows the response to the rate-step experiment. The beginning of any uniaxial test is very similar to such an experiment; the initial total strain rate is zero, and jumps to a finite value as the cross-head begins to move. The step in plastic strain rate is not as sharp as the one in Fig. 1(b) ; the jump in the cross-head velocity is first accommodated by mostly elastic deformation, and later, as stress increases, by mostly plastic deformation. While the present constitutive model breaks down at vanishing stress and strain rate, the experiment can be approximated by a large step in crosshead velocity such that the nominal strain rate R,as defined in (3.9) varies by many orders of magnitude. The final value of plastic strain rate is approximately equal to R, which determines the type of behavior. The plastic strain rate has to cross all the values between the initial and the final one, and so must the steady state SRS (see Fig. 1). As the plastic strain rate varies, the state variable is lagging behind its steady

S. D. MESAROVIC

690

(a)

5

-5

-I

I I

(b)

a’

I I t I

10

0

’ t 20

’ I I

’

I I

30

40 time [sec.]

50

60

70

80

2 c”“,““,““,““,““I”“I”“~~‘~‘J

c 1 0 -1

h L

-3 -4

1

0

10

20

30

40

50

60

70

80

time [sec.] Fig. 5. Time histories of the system perturbed from the unstable steady state R = IO-“SC'. a = l/3, T = 300 K, Parameters from Table 1. (a) Strain rate; (b) average waiting time ; (c) stress ; (d) phase space (G, t,) ; (e) strain rate-stress superposed to the steady state relation ; (f) detail of the history of (a), (b) and (c).

Dynamic

691

strain aging and plastic instabilities

A 0

20

30

50 40 time [sec.]

60

80

2 1 0 -1 -2 -3 -4 -5 -5

0

Fig. 5. (c) and (d)

5

692

S. D. MESAROVIC

0

T 3 T

-12

‘b

-14

&

-16

stable limit cycle 1

-18

-1 F

(f)

-20 79

81

83

85

Fig. 5. (e) and (f).

87

89

Dynamic

strain aging and plastic instabilities

693

-8 -9 0

-10

a a + ‘b

I VI

-11

b

-12 -13 -14 -3 Fig. 6. Response

-2.5

-2

-1.5 Int 1odsec-’

-1

-0.5

to a large perturbation in a region with three solutions: ln(R/lOP 10m4. c( = l/3, T = 300 K, parameters from Table I.

0

s-- ‘) = - 1.37 x

state value. Three possibilities are considered here: (a) the final strain rate is in the negative SRS region but lower than needed for temporal instability, (b) the final strain rate is in the region of temporal instability, and (c) the final strain rate is in the negative SRS region, but higher than needed for temporal instability, i.e. the system must cross the region of temporal instability. All three cases are within the region of spatial instability. Numerical integration of (3.11) and (3.3) with enforced spatial homogeneity was performed with the power law hardening (b/g,,) =(E/E,)‘.‘~ for several values of N between 2 and 15. Figure 7(a) shows the results for N = 10, cU = 183 MPa, E,,= 0.02, and initial strain of 0.02, so that the initial hardening is h = 918 MPa. Two values of equivalent stiffness are considered: rigid machine, K = E = 66 GPa and the soft machine, K = 10 GPa. It should be noted that the yield point behavior exhibited for the cases of rigid machine does not represent instability ; it is the fundamental evolving solution, the stability of which was analyzed in Section 3. The spatial homogeneity is enforced and the temporal instability does not occur for such a high equivalent stiffness. As expected, for Rfinal= 10e4 SK’ [case (b)] and soft machine, the typical PLC serrations occur. Other hardening laws give essentially the same behavior. Not surprisingly, extremely high hardening can obliterate the effects of rate sensitivity. With increasing stiffness, K, and hardening, h, the upper yield points in Fig. 7 become sharper; the transition from elastic to plastic strain rate occurs earlier, so that the state variable relaxes faster towards the new steady state. The amplitude of serrations decreases with increasing K. At initial stages of instability, the perturbation is still small and the characteristic exponents provide some information about the nature of instability. Figure 7(b)

S. D. MESAROVIC

694

(a)

186

182

178

174

170

166 -0.002

0

0.002

0.004

0.006

0.008

0.01 E 184 182 180 178 7 176 174

si 13

172 170 -15

168

‘. CI= l/3, T = 300 K, Fig. 7. Beginning of the tensile test. System (3.1 l), (3.3) initial condition: R = 10m6 sparameters from Table 1, (a) Stress-strain curves, (b) evolution of real and imaginary parts of characteristic exponents for Rhna, = lo-’ SC’.

Dynamic strain aging and plastic instabilities

695

shows the variation of characteristic exponents for Rfinal = 10e3 s-‘. (In these calculations, spatial homogeneity is enforced. The characteristic exponent for the temporal instability is computed assuming K = 10 GPa, while the equations are integrated for a rigid machine. This is necessary in order to avoid the oscillatory instability; characteristic exponents computed from the oscillating state have no meaning. Differences in fundamental solutions computed for the hard and soft machine are small.) The spatial characteristic exponent is complex only briefly, while the characteristic exponent for the temporal instability retains its imaginary part. For the enforced spatial homogeneity, the oscillations and stress drops are possible only for a soft machine. In a real experiment, loss of homogeneity is expected to occur before the temporal instability. Successive stress drops can then be accomplished either by successive localizations along the specimen or by discontinuous propagation of Luders front. If this is not possible, then the localization is expected to give a continuous propagation of Luders front. The model also predicts a positive SRS of the upper yield point. This is consistent with the data collected by Nadai (1950) for mild steel at 200°C which show small stress drop at low strain rates, serrated yielding for the intermediate strain rates, between the regions of yield point behavior, and the upper yield point with positive SRS. The corresponding temperature dependence is consistent : at intermediate temperatures serrations occur, at high temperatures only the yield point is visible, while at low temperatures the yield point is followed by a stress plateau. The results of Sleeswyk (1958) for ingot iron at low temperatures are consistent with this. Mahajan rt al. (1979) find that yield point behavior in silicon results in Luders fronts at 8OO”C, but not at 900°C and 1000°C.

6.

CONCLUSIONS

AND DISCUSSION

The constitutive model for dynamic strain aging describes both the yield point behavior and serrated yielding (PLC). The temperature and strain rate bounds for these phenomena are derived. The oscillatory behavior is of the relaxation oscillation type, analogous to frictional stick-slip, with abrupt changes followed by periods of relative tranquility. There is a detailed mathematical analogy between the present work and the rate and state dependent friction analysis of Rice and Ruina (1983) and Gu rt al. (1984). The yield point behavior is a property of the fundamental solution. It occurs even if spatial homogeneity is enforced. Without this enforcement an instability, such as Luders front or serrated yielding, is expected. Serrated yielding emerges as an interplay between rate dependence, state variable evolution and elastic properties of the system. In agreement with experimental observations, the upper yield point shows a positive strain rate dependence, while the “lower yield point” shows a negative one. The exponent CI, which determines the relaxation of the state variable, is not of crucial importance for qualitative behavior, but accurate, quantitative modeling requires that CIbe determined. The oscillatory behavior requires a mechanism for stress drops. Without localization, this kind of behavior is severely restricted by the machine stiffness; high

696

S. D. MESAROVIC

equivalent elastic stiffness prevents serrated yielding and sharpens the upper yield point. Further, when serrated yielding does occur, increasing equivalent stiffness lowers the amplitude of serrations. Saitou et al. (1988) demonstrated very systematic and clear effects of the machine stiffness on the amplitude of serrations (in accord with the present model) on 7075-Al alloy. In the Van den Brink et al. (1977) experiments on AI-Cu alloy the machine stiffness influenced the type of serrations and their onset. However, Chihab’s (1986) results on Al-Mg alloys indicate at most marginal dependence of the onset of serrations on the machine stiffness. Chihab’s results also contradict Ling and McCormick’s (1991) findings on a similar material, in that the instability in his experiments seems to initiate at the very beginning of the negative SRS range. However, in all experiments, localization begins at the grips, where the stress and strain rate states are different than in the middle of the specimen. In order to determine the conditions for : (1) PLC as a sequence of shear localizations, (2) PLC as discontinuous Luders front propagation, and (3) continuous Luders front propagation, two continuum mechanics problems require further analysis : (a) shear band localization, and (b) deformation front propagation, in a strain aging material. The shear localization is often observed to be a limiting mode of instability (Rice, 1976). A more general stability analysis of the rate dependent material with a time varying state variable(s) is required if one is to understand the effects of strain aging on fracture properties, or localization under more complex conditions which may occur during material processing. Shear localization in rate independent materials is fairly well understood. The general framework is reviewed by Rice (1976). No such theoretical framework exists for strain aging materials. Given that the instability is observed to initiate at the grips, and that this region is subjected to the range of stress-strain rate states, it is not inconceivable that the appearance of instability at the gauge length is governed by the propagation condition and that its onset, in the grip region, is possible for a wide range of imposed boundary conditions. The propagation of the instability and the associated length scale are still awaiting a full continuum analysis. Experimental evidence of the dependence of a bandwidth on specimen dimensions (e.g. Van den Brink et al., 1977) suggests that the length scale is determined on the macroscopic continuum level. A finite deformation, three-dimensional analysis, analogous to the Hutchinson and Neale (1983) analysis of neck propagation in polymers, is needed. Recently, McCormick and Ling (1994) analyzed a one-dimensional continuum model with Bridgman-type corrections for triaxiality at localizations. They were able to simulate the propagation of the PLC band and the variation of the bandwidth with the specimen geometry.

ACKNOWLEDGEMENTS The author acknowledges guidance and suggestions by J. R. Rice. In particular, his suggestions and preliminary analysis were essential for the author’s understanding of the onset of non-homogeneous deformation. The comments by Y. Estrin, J. W. Hutchinson and L. P. Kubin are also appreciated. The work was supported by the Office of Naval Research, Mechanics Division (grant N00014-90-J-1379) and by a Harvard University Graduate Fellowship.

Dynamic

strain aging and plastic instabilities

697

REFERENCES Ardley, G. W. and Cottrell, A. H. (1953) Dislocation theory of yielding and strain ageing of iron. Proc. Royal Sot. A219,328-341. Balik, J. and Lukac, P. (1993) Portevin-LeChatelier effect in Al-3 Mg conditioned by strain rate and strain. Acta Metall. 41, 1447-1454. Brindley, B. J., Corderoy, D. J. H. and Honeycombe, R. W. K. (1962) Yield points and Luders bands in single crystal of copper-base alloys. Acta Metall. 10, 1043-1050. Brzeski, J. M., Hack, J. E., Darolia, R. and Field, R. D. (1993) Strain ageing embrittlement of the ordered intermetallic compound NiAl. Mater. Sci. Engng A170, 1 l-18. Butler. J. F. (1962) Luders front propagation in low carbon steels. J. Mech. Phys. Solids 10, 3 13--334. Chihab, K. (1986) Etudes des instabilites de la deformation plastique associees a l’effet Portevin-LeChatelier dans les alliages aluminium-magnesium. Doctoral dissertation. l’universite de Poitier. Chihab, K., Estrin Y., Kubin, L. P. and Vergnol, J. (1987) The kinetics of the Portevin-Le Chatelier bands in Al-Sat%Mg alloy. Scripta Metall. 21, 203-208. Cottrell, A. H. and Bilby, B. A. (1949) Dislocation theory of yielding and strain ageing of iron. Proc. Phys. Sot. LXII(I-A), 4962. Cuddy, L. J. and Leslie, W. C. (1972) Some aspects of serrated yielding in substitutional solid solutions of iron. Acta Metall. 20, 1157-l 167. Delafosse, D., Lapasset, G. and Kubin, L. P. (1993) Dynamic strain ageing and crack propagation in the 2091 AI-Ti Alloy. Scripta Metall. 29, 1379-l 384. Dieterich, J. H. (1981), Constitutive properties of faults with simulated gauge. In Mechanica/ Behavior oJ‘Crusta1 Rocks. (ed. Carter et al.), pp. 103-120. AGU, Washington D.C. Doedel, E. (1986) AUTO: Software ,for Continuation and Bifurcation Problems in 0rdinar-l. D[l/&ential Equations. California Institute of Technology, California. Estrin, Y. and Kubin, L. P. (1989) Collective dislocation behaviour in dilute alloys and the Portevin-LeChatelier effect. J. Mech. Behall. Mater. 2, 255-292. Estrin, Y. and Kubin, L. P. (1991) Plastic instabilities: phenomenology and theory. Mater. Sci. Engng A137, 125-134. Estrin, Y. and McCormick, P. G. (1991) Modelling the transient behaviour of dynamic strain ageing materials. Acta Metall. 39, 2977-2983. Estrin. Y., Ling, C. P. and McCormick, P. G. (1991) Localization of plastic flow: spatial vs. temporal instabilities. Acta Metall. 39, 2943-2949. Frost, H. J. and Ashby, M. F. (1982) Deformation-Mechanism Maps. Pergamon Press, New York. Gu, J.-C., Rice, J. R., Ruina, A. L. and Tse, S. T. (1984) Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction. .I. Mech. Phys. Solids 32, 167-196. Guckenheimer, J. and Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems und Bifurcations of Vector Fields. Springer-Verlag, New York. Hirth, J. P. and Lothe, J. (1982) Theory of Dislocations, 2nd Edn. John Wiley & Sons. New York. Hutchinson, J. W. and Miles, J. P. (1974) Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension. J. Mech. Phys. Solids 22, 61-71. Hutchinson, J. W. and Neale, K. W. (1983) Neck Propagation. J. Mech. Phys. Solids 31,405-426. Johnston, W. G. (1962) Yield points and delay times in single crystals. J. Appl. Phys. 33,271& 2730. Johnston, W. G. and Gilman, J. J. (1960) Dislocation multiplication in lithium fluoride crystals. J. Appl. Ph_vs.31, 632-643. Kestin, J. and Rice, J. R. (1970) Paradoxes in the application of thermodynamics to strained solids. In Critical Review qf Thermodynamics (ed. Stuart et al.), p. 275. Mono Book Corp., Baltimore. Kocks, U. F., Argon, A. S. and Ashby, M. F. (1975) Thermodynamics and kinetics of slip. Pro~qr.Mater. Sci. 19.

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Kubin, L. P. and Estrin, Y. (1990) Evolution of dislocation densities and the critical condition for the Portevin-LeChatelier effect. Acta Metall. 38, 697-708. Kubin, L. P. and Estrin, Y. (1992) The critical conditions for jerky flow. Phys. Stat. Sol. (b) 172, 173-185. Kubin, L. P., Estrin, Y. and Canova, G. (1990) Dislocation patterns and plastic instabilities. In Patterns, Defects and Material Instabilities (ed. Walgraef and Ghoniem), pp. 277-301. Ling, C. P. and McCormick, P. G. (1990) Strain rate sensitivity and transient behaviour in AlMg-Si alloy. Acta Metall. 38,2631-2635. Ling, C. P., McCormick, P. G. and Estrin, Y. (1993) A load perturbation method of examining dynamic strain ageing. Acta Metall. 41, 3323-3330. Lomer, W. M. (1952) The yield phenomenon in polycrystalline mild steel. J. Mech. Phys. Solids 1, 64-73. Louat, N. (1981) On the theory of the Portevin-LeChatelier effect. Scripta Metall. 15, 11671170. Mahajan, S., Brasen, D. and Haasen, P. (1979) Luders bands in deformed silicon crystals. Acta Metall. 27, 1165-l 173. McCormick, P. G. (1988) Theory of flow localization due to dynamic strain ageing. Acta Metall. 36, 3061-3067. McCormick, P. G. and Estrin, Y. (1989) Transient flow behaviour associated with dynamic strain ageing. Scripta Metall. 23, 1231-1234. McCormick, P. G. and Ling, C. P. (1994) Numerical modelling of the Portevin-LeChatelier effect. Submitted to Acta Metall. McCormick, P. G., Vendekasan, S. and Ling, C. P. (1993) Propagative instabilities: An Experimental view. Scripta Metall. 29, 1159-l 164. Nabarro, F. R. N. (1991) The dependence of activation enthalpy on stress in shear processes. Acta Metall. 39, 187-189. Nadai, A. (1950) Theory of Flow and Fracture of Solids. McGraw-Hill, New York. Neuhauser, H. and Hampel, A. (1993) Observations of Luders bands in single crystals. Scripta Metall. 29, 1151-l 158. Penning, P. (1972) Mathematics of the Portevin-LeChatelier effect. Acta Metall. 20, 1169-l 175. Perrin, G., Rice, J. R. and Zheng, G. (1994) Self-healing pulse on a frictional surface. Submitted to J. Mech. Phys. Solids. Rice, J. R. (1971) Inelastic constitutive relations for solids : An internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19,433-455. Rice, J. R. (1976) The localization of plastic deformation. In Theoretical and Applied Mechanics, Proc. of the 14th II/TAM Congress (ed. Koiter, W. T.), pp. 207-220. North Holland Publishing Co. Rice, J. R. and Ruina, A. L. (1983) Stability of steady frictional slipping. J. Appl. Mech. 105, 343-349. Ruina, A. L. (1983) Slip instability and state variable friction law. J. Geophys. Res. 88(B12), 10,359-10,370. Saitou, K., Otoguro, Y. and Kihara, J. (1988) Effect of the stiffness of the testing system on the discontinuous deformation of 7075-Al Alloy. J. Japan Inst. Metals 52, 1238-1245. Seydel, R. (1988) From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis. Elsevier, New York. Sleeswyk, A. W. (1958) Slow strain-hardening of ingot iron. Acta Metall. 6, 598-603. on dynamic strain ageing in Springer, F. and Schwink, Ch. (1991) Q uantitative investigations polycrystalline CuMn alloys. Scripta Metall. 25, 2739-2744. Van den Beukel, A. (1975) Theory of the effect of dynamic strain ageing on mechanical properties. Phys. Stat. Sol. (a) 30, 197-206. Van den Brink, S. H., Van den Beukel, A. and McCormick, P. G. (1977) The influence of specimen geometry and machine stiffness on the Portevin-LeChatelier effect in AI-Cu Alloy. Phys. Stat. Sol. (a) 41, 513-521. Weeks, J. D. (1993) Constitutive laws for high-velocity frictional sliding and their influence on stress drop during unstable slip. J. Geophys. Res. 98(BlO), 17,637-17,648.

Dynamic

APPENDIX

: LINEAR

699

strain aging and plastic instabilities

STABILITY

SOLUTION

ANALYSIS

OF THE

SYSTEM

OF THE

EVOLVING

(3.1 l), (3.3)

The system (3.11), (3.3) is perturbed homogeneously by (S&(t), at,(t), &z(t)). Taking the reference state to instantaneously coincide with the current one, and neglecting n/E compared to 1, and ri/E compared to 8, and setting e”clz 1, the variation in total stretch becomes 6kr(t) = c.6&(1)+ 2+(l)+

+(r)+&(1).

(Al)

After applying the variational operator to (3.11) and (3.3) and after some algebraic manipulation, one obtains the linear system of first order ODES with variables (6~. 6t,, 6n) :

ir

I(h -n) S &(h-n)

c

s

t*

it,

C(1 -r) R

SCI

22

where r(t,(t)) = Hcc(t,(t)/t,)“exp { - (fa(t)/fd)‘}. The eigenvalues of the matrix in (A2) are determined

The necessary and sufficient Hurwitz theorem) is

condition

from the characteristic

for all the roots to have a negative

equation

real part (Routh-

K*S2 A,, = ---a0 sz

a, *a, -&

= (K+h-n).

K+h-n+

S-(1-r) n

cl

+K

1

K*S*T --20. R

(A3)

The quantities denoted with a caret (^) are the original coefficients normalized by positive numbers. In (A3) the additional assumption E >> S is made. This seems reasonable since S is of order 1 MPa, while E is many GPas. The second condition of (A3) is always satisfied. The difference between the first and the third is negligible. This can be seen in the following way : The instability occurs when the quantity T, which (if one assumes that the evolving system is well approximated by the steady state one) equals

becomes sufficiently large, that is, when the steady state strain rate sensitivity becomes sufficiently negative. To distinguish between the two conditions in (A3), one looks for the interval of r, such that the first condition is satisfied and the third is not. This results in

S. D. MESAROVIC

700 R-(Kfh-n) S

+

K+h-n

< r < Q*(K+h-n)

K(l +R)+h-n

’

’

S

+1.

The interval is maximized when h = n : Al-=

1-p

1 1+n

x n.

Since Q is very small compared to the variation in I- required to go from the positive rate sensitivity to the sufficiently negative one, which would violate the first condition of (A3), one can neglect the third condition.