A generalized criterion of plastic instabilities and its application to creep damage and superplastic flow

Acta metall, mater. Vol. 43, No. 11, pp. 4093-4100, 1995

~

Pergamon 0956-7151(95)00091-7

ElsevierScienceLtd Copyright © 1995Acta MetallurgicaInc. Printed in Great Britain.All rights reserved 0956-7151/95$9.50+ 0.00

A GENERALIZED CRITERION OF PLASTIC INSTABILITIES A N D ITS APPLICATION TO CREEP DAMAGE A N D SUPERPLASTIC FLOW

PETERH.AHNER European Commission, Institute for Advanced Materials, Joint Research Centre, 1-21020 Ispra (Va), Italy (Received 24 October 1994; in revised form 5 February 1995)

Abstract--A novel criterion to distinguish between stable (homogeneous) and unstable (localized) plastic flow in tensile specimens is proposed which is based on a consideration of the structural evolution on characteristic time-dependent intrinsic or extrinsic length scales. Macroscopic localization is predicted to occur if fluctuations grow with respect to these length scales. Previous instability criteria by Hart, Estrin and Kubin, and Molinari are recovered as special cases of the present generalized criterion. Moreover, it allows one to distinguish between homogeneous and heterogeneous nucleation of slip. Two applications are discussed: (i) the peculiar necking resistance during superplastic flow is shown to depend not only on a high value of the strain-rate sensitivity but also on a large amount of deformation accommodated by grain boundary sliding;(ii) the instability criterion is applied to creep damage described in terms of a model of the Kachanov-Rabotnov type. This allows for a determination of the critical damage parameter at failure.

1.

INTRODUCTION

What constitutes the failure of a structural component? From the phenomenological point of view, this question is addressed by distinguishing three major reasons of failure [1]: (i) failure due to excessive elastic deformation; (ii) failure due to excessive plastic deformation; and (iii) failure by fracture. With respect to the theoretical understanding, knowledge is decreasing in the order (i)-(iii). While problems of the first type (e.g. buckling occuring when the Euler critical load is exceeded) were solved a long time ago, the second type of problem is still the subject of recent investigations and controversial conclusions. As to fracture, theoretical understanding is even less comprehensive, as various phenomena combine: crack tip plasticity, fracture mode, embrittlement etc. [2, 31. The present paper addresses failure due to excessive plastic deformation by the derivation of a novel criterion of plastic instability. As a clear definition of what constitutes unstable flow is not available, some general remarks are in order. Mathematically speaking, a system is called stable if trajectories corresponding to slightly different initial conditions remain close to each other for all later times (stability in the Lyapunov sense [4]). This may not be the case if the constitutive equations exhibit singular points such as nodes or focuses associated with bifurcations, since then the qualitative behaviour of the system depends sensitively on the initial conditions.

However, the macroscopic constitutive equations used for the description of the structural evolution and plastic deformation during a tensile test do not, in most cases, possess singular points but are analytic in the entire phase plane. In this case, a well-known theorem states a continuous dependence of the solutions on the initial conditions. Hence, instabilities in the above-mentioned sense do not occur. Nevertheless, the problems related to material failure show that the instabilities occur in a wider, more phenomenological sense. As a clue to understanding this apparent discrepancy, one must realize that macroscopic constitutive equations are incomplete. An improved description should rather account for the fluctuations that are inevitably present on various scales. Fluctuations on the microscopic scale are due to the discreteness of defects, in particular, dislocations. On a mesoscopic scale, geometrically necessary fluctuations of stress and strain rate arise from the long-range interactions during dislocation glide. A crystal undergoing plastic deformation can thus be considered as an effective fluctuating medium [5]. Additional sources of fluctuations may be related to the granular structure, texture etc. Fluctuations provide additional degrees of freedom that can give rise to plastic instability and failure, as soon as they become macroscopically manifest. The central question to be addressed in this paper reads: "Can we arrive at a physically sound instability

4093

4094

HAHNER: CRITERION FOR PLASTIC INSTABILITIES

criterion based on macroscopic constitutive equations, even when a proper micro- or mesoscopic description is not available?" The organization of this work is as follows. In Section 2, ambiguities and conceptual inadequacies with existing instability criteria are outlined. This leads us to the formulation of a generalized instability criterion for a tensile test by making additional assumptions on the microstructural evolution and the corresponding length scales. Instability is then defined as localization occurring with respect to these length scales (Section 3). This is compared with former results in Section 4. In Section 5, the generalized instability criterion is applied to creep damage and superplasticity. Conclusions are drawn in Section 6. 2. EXISTING CRITERIA OF PLASTIC INSTABILITY The various criteria proposed in the literature differ with respect to the very definition of what is considered an indication of unstable plastic flow in a tensile test. According to Hart [6, 7], plastic flow is assumed unstable if some inhomogeneity 6A < 0 of the cross-sectional area A (either a mechanical imperfection or a machining defect [7]) becomes more pronounced in the course of time t, i.e. ~dSA < 0.

(1)

Provided that the infinitesimal fluctuation 6A obeys the same evolution law as A, this is equivalent to 3" < 0

(2)

where a dot denotes differentiation with respect to time. Although equation (1) represents a clear definition of instability, the choice of the extensive variable A as stability controlling parameter is astonishing, because instability is generally expected to be an intrinsic phenomenon that does not depend on specimen size. Estrin and Kubin [8] have circumvented this shortcoming by monitoring plastic instability by the plastic strain e (intensive quantity). Their criterion may be written as g,&>O

or

t'>O.

It has been stated that there is no reason to prefer one criterion over the other [7], although equations (1), (3) and (4) yield obviously different results. While these differences are small as far as a softening instability (type h instability [8]) is concerned, they may be crucial for the question whether or not plastic flow becomes unstable at all, if stability is governed by the strain-rate sensitivity (cf. Sections 4 and 5.1). Hence, ambiguities with respect to the choice of the stability controlling variable should be removed in a satisfactory stability analysis. There is actually another source of arbitrariness. Comparing equations (2)-(4), one notices that instability implies in each case that plastic flow is accelerated in some sense. From the practical point of view, however, it is not necessarily an acceleration but a localization of flow that acts as a precursor of failure. This is illustrated in Fig. 1, where two different sequences of deformation states are schematically depicted. Intuitively, one would associate stable plastic flow with the case (a), while (b) is obviously unstable, although )i < 0 or ( > ~2/E may be fulfilled in either case. This is closely related to the difference between "homogeneous nucleation"[case (a)] and "heterogeneous nucleation" of slip [case (b)] and stresses the importance of the length scales with respect to which localization is assumed detrimental (cf. also Fig. 3). Depending on the specific situation considered, the relevant length scales may be intrinsic (e.g. grain size, cross-slip height, dislocation cell size, slip band distance etc.) or extrinsic (e.g. specimen diameter). We shall come back to this point in Sections 4 and 5. It is concluded that the stability problem can principally not be addressed in terms of macroscopic constitutive equations alone. In general, one must rather refer to some mesoscopic approach accounting for spatio-temporal fluctuations and couplings. Since

(~)

(b)

[±

(3)

Finally, Molinari [9] proposed to consider the relative growth of a perturbation.i" Instability is then related to O,

>0

or

~'i>--. £

(4)

According to this supposition, instability is confined to a regime where strain fluctuations grow faster with time than the macroscopic strain. tActually, Molinari being interested in adiabatic instabilities in simple shear, considered a temperature perturbation. Here we transfer his analysis to strain perturbations.

tl t2

tl

t3

t3

t2

Fig. 1. On the relationship between strain localization and plastic instabiliy. (a) Sequence of tensile deformation states at times tt-t 3 representing stable flow; (b) the corresponding situation for unstable flow.

H,~HNER: CRITERION FOR PLASTIC INSTABILITIES

Figure 2 illustrates this in a schematic way. Curve 1 represents the initial fluctuation intensity in dependence of the length scale L over which fluctuations are averaged (coarse graining). While in the case (2a) fluctuations decay with respect to a reference scale L* (stable case), they become macroscopically manifest in the case (2b) (unstable case). For a 3 D fluctuation analysis, we introduce the quantity 6q~gby which the actual values of q5 deviate from the averages performed over some surrounding volume V to be specified below

/$¢~ mesoscopic macroscopic -_ ,, _, ~ ( ( 2 ( 1 )

t= 0 (2a) t > 0, stable b) t > 0, unstable

log L*

log L

Fig. 2. Length scale dependence of the variations of an averaged internal state variable ~L (schematic). Plastic flow is unstable if fluctuations grow with respect to a characteristic length scale L* and become macroscopically manifest [case (2b)l. this may be difficult in practice, we shall pursue a different approach. It consists of considering macroscopic constitutive equations backed up by additional assumptions on the microstructural evolution and the corresponding length scales. This allows one to derive a generalized instability criterion based on localization with respect to specific length scales. It will be shown later (Section 4) that the three previously mentioned instability criteria are included as special cases.

Consider a set of state variables, {¢ }, relevant for the temporal evolution during a tensile test. {~b} may enclose structural variables (such as the various defect densities), the plastic strain and, if deformation is adiabatic, temperature. In any case, ~ should be intensive variables, in order to base the consideration on physical arguments independent of specimen dimensions. On the microscopic and mesoscopic scales, the variables th always exhibit spatio-temporal fluctuations. For instance, mesoscopic fluctuations of the plastic shear strain rate ~ are caused by the long-range dislocation interactions [5] (~i.,) s

6dpv(Y¢, t) = qS(Yc, t) -- - ~ 1

(5)

where ( . . . ) denotes a temporal expectation value effected during the glide motion of a representative dislocation, S is the strain-rate sensitivity, and zi~t and z e~ are the resolved internal back stress and the effective shear stress, respectively. From equation (5) it is seen that strain-rate fluctuations are appreciable, because (z int) >> S usually holds. Although this intrinsic noise does.not inevitably lead to an instability, it may serve as a seed for macroscopic localization. Wether or not plastic flow becomes macroscopically unstable depends on the evolution of the fluctuations with respect to some reference length scale L*.

fV(0 d3x'q~(~:"

t)

def

= tk(~, t) - qSv(3C,t).

(6)

Note that V is allowed to evolve with time and does not generally coincide with a material volume. The temporal evolution of 6q~v consists of three contributions

dt6cP v = c3tbd?v - - ~

d3x'~ '. [~b(~', t)b(Sc, t)]

+V

d3x'q~(Sc', t)

(7)

where the first term on the r.h.s. at&by = 0,~b - -~

3. DERIVATION OF A GENERALIZED INSTABILITY CRITERION

(6~,2) (9)2

4095

d3x'c~tq~(3c', t)

(8)

corresponds to the local evolution of 4, itself, while the second and third terms describe the evolution of the boundary and the normalization constant, respectively. The velocity field b is related to the expansion/contraction of the volume V

f/=

I d3x'V"b(Sc" t). (9) dv By application of Gauss' theorem, equation (7) is transformed into a surface integral over the boundary t3Vof V d, fdpv = O,6dpv---~

dA'. b(Sc', t)6dpv(K , t).(10)

V

So far, no assumption was made on the nature of V. We are free to choose V = V* such that V* contains the fundamental (i.e. the longest wavelength) mode of the spatio-temporal deviations 3q~v.. This is achieved by setting ~ *r .t - re? dA~t "v(x,t)cSdpv.(X,t)= V*

l?*~c~z.(Sc, t).

(11)

According to this definition, the flux of 6qSr, into V* [i.e. the l.h.s, of equation (11)] is directly related to the rate of growth of V* so that, provided that V* is characteristic of the fundamental fluctuation at time t, it will remain so at later times. In 1D, where equation (11) is reduced to - ½[aCL.(x - L * / 2 ) + &~L.(x + L*/2)1 = a ~ . ( x ) (12)

4096

H/~HNER: CRITERION FOR PLASTIC INSTABILITIES

this is obviously fulfilled for 5~bL.(x)~ sin(21tx /L *). Equations (10) and (11) give us the final result d,5~bv. = O,6dpv,+ ~

6dpv,.

(13)

As V* was adapted to the spatial extension and the temporal evolution of the fundamental mode of the fluetuations f4~v,, growth with time of f o r , is proposed to correspond to unstable plastic flow due to macroscopic localization. This leads us to the following eriterion of plastic instability

O,fdpv, + ~ 3~bv. > 0.

(14)

This constitutes a criterion of relative instability, since deviations are assumed detrimental only if they grow with respect to some reference volume V* such that they become macroscopically manifest. As it stands, equation (14) is not yet of much help. In addition, a biased guess on the physical significance of V* is necessary, in order to relate it to the characteristic scale of incompatibility stresses, microscopic flaws, slip bands etc. in a material. In the following sections it will be shown by means of various examples how this can be achieved with regard to specific modes of deformation.

4. C O M P A R I S O N

WITH PREVIOUS CRITERIA

As a first demonstration of what is new with equation (14), we compare it to the existing instability criteria outlined in Section 2. To be specific, we consider a tensile test of a material that obeys an incremental visco-plastic law of the form

d a = h d E + m ~ dg

(15)

E

where a and E denote the tensile flow stress and plastic strain, respectively• The momentary material response is characterized by the strain-hardening coefficient aa

h =~

(16)

deformation by glide, & = -5A/A, equation (1) may be rewritten as

A

d,& + ] &

> 0.

(19)

This is obviously equivalent to equation (18) if V* = const. A.

(20)

Within the framework of the present theory, Hart's criterion is thus applicable if variations of an intrinsic length scale are not important and fluctuations are detrimental only if they extend over the entire specimen cross-section A. This is supposed to be realistic in the special case of localized necking of a polycrystalline material• Applied to the flow rule of equation (15), Hart's criterion predicts instability for [6] h

- + m < 1.

(21)

O"

This was used to explain the high necking resistance of superplastic materials where h/a ,~ 1. According to equation (21), superplastic flow is then expected for m ~> 1, while m ~>0.3 proved sufficient in many experiments [10]. In Section 5 we shall come back to this point assigning to V* a significance different from equation (20). In a similar way, the Estrin-Kubin criterion (3) can be reconsidered• From a comparison of equations (3) and (18), it is seen to imply V* = const.

(22)

One may either attribute this to the fact that this criterion does not account for time-dependent length scales or, more convincingly, that V* is identified with the gauge volume that does not change during plastic deformation. The latter case may be related to the diffuse necking of a polycrystalline material. The fundamental difference between the instability criteria by Hart, and Estrin and Kubin becomes evident when equation (3) is applied to the flow rule (15). This gives h

- < 1

(23)

O"

and the strain-rate exponent (assumed positive)

Taking 4~v*= E as the macroscopic structural variable, equation (14) predicts instability due to strain localization for

as a condition of instability [cf. equation (21)] which is actually identical with the well-known Considrre criterion [11]. Since this is independent of the strainrate sensitivity, the phenomenon of superplasticity remains unexplained• Let us finally consider Molinari's criterion, equation (4), which can be expressed as

d,& + ~* fie > 0.

a,& - - ~ 5 E > 0 .

Olne I

m= ~ .

07)

(18)

(24)

V ~

£

First, this is to be compared with Hart's criterion [6]. As the specimen volume is invariant under plastic

When compared to the present equation (18), this is seen to represent a criterion of relative instability with

H/~HNER:

CRITERION FOR PLASTIC INSTABILITIES

4097

Table 1. Comparisonof the present instabilitycriterion,equation (18), with previousones by Hart [6], Estrin and Kubin [8],and Molinari I91 Hart

Estrin and Kubin

Molinari

Instability criterion

~9,fie + (,4 IA )& > 0

~,& > 0

9 , & - (i le )& > 0

Application to flow rule (15) Characteristic volume Physical realization

(h/a) + m < 1 V* = const.A Localized necking in polycrystals

(h/a) < 1 V* = const. Diffuse necking in polycrystals Heterogeneous slip nucleation

(h/a) + (re~e)< 1 V* = const, e ]

/////

q

(~)

(b)

I homogeneous nucleation

heterogeneous nucleation

Fig. 3. Difference between homogeneous (a) and heterogeneous (b) nucleation of slip during single glide. While the situation (a) is stable (contraction of the characteristic length scale defined by the slip band distance), situation (b) corresponds to a plastic instability (Liiders-like slip with constant slip band distance).

Homogeneous slip nucleation

The preceding discussion which is summarized in Table 1 shows that the present instability criterion (18) includes the previously established ones as special cases. This is why equation (14) is considered a generalized instability criterion that is versatile to be applied to various situations, provided that physically sound assumptions on the nature of the characteristic volume V* can be made. In the following section, these ideas will be used to arrive at new conclusions concerning superplasticity and creep damage. 5. APPLICATIONS

5.1. Superplasticity respect to an intrinsic length scale that is contracting with strain: V* = const, e - l .

(25)

As a physical realization, one may envisage the homogeneous nucleation of slip bands in a singleglide oriented crystal.]" In this case, the relevant length scale is identified with the average slip-band distance that decreases in inverse proportion to the macroscopic strain, provided that slip bands carry approximately equal amounts of slip. This corresponds to a particularly stable situation, as can be seen from applying the instability criterion (24) to equation (15) h

m

- + - - < 1. (7 £

(26)

At the onset of yielding, e ,~ l, homogeneous slip band nucleation has a stabilizing effect in strain-rate sensitive materials with m > 0 [Fig. 3(a)]. A completely different situation is met with the heterogeneous nucleation of slip, i.e. the propagation of a Lfiders band [Fig. 3(b)], when the intrinsic length scale corresponds to the cross-slip height [12]. As this can be considered constant within the yield regime (i.e. 17"= 0), the Estrin-Kubin criterion, equation (23), is appropriate and a softening instability is predicted for h < a (see also Fig. 1).

tSince slip bands extend over the whole specimen crosssection, A does not influence the characteristic volume V*. This is in contrast to the polycrystalline case, cf. equation (20).

Various fine-grained alloys deformed at elevated temperatures are able to accommodate large neckfree plastic strains up to 2000%. Recent reviews on this phenomenon termed superplastic flow are presented in Refs [10, 13, 14]. It is generally accepted that superplastic flow occurs to a large extent ( > 50%) by grain boundary sliding (GBS), while only a minor contribution is due to dislocation slip. Several theories on the microscopic mechanisms of GBS have been developed that are able to explain the comparatively large strain-rate exponents (0.3 < m < 1) and the absence of strain hardening (h/a ~ 1) observed under superplastic conditions. However, the phenomenological relation between GBS and a large value of m on the one hand, and the high necking resistance and large failure strain on the other hand is not fully understood. This question is to be addressed in terms of the generalized instability criterion (18). It was pointed out in the preceding section that superplasticity (absence of instability) can principally not be understood in terms of Estrins criterion, whereas Hart's criterion predicts superplastic behaviour for m/> 1. As superplastic flow is already observed for m > 0.3 [10], it is tempting to attribute this quantitative discrepancy to an inappropriate definition of the relevant characteristic volume V*. To clarify this point, we adopt the geometrical model of GBS controlled deformation proposed by Ashby and Verrall [15]. Figure 4 gives a simplified idea of the elementary deformation step assumed to occur during superplastic flow. The corresponding elemental plastic strain was estimated to be e0 ~ 0.5.

(27)

4098

H,~HNER:

CRITERION FOR PLASTIC INSTABILITIES

t

Fig. 4. On the basic deformation mechanism in grain boundary sliding involving an elemental strain E0 (after Ashby and Verrall [15]). It is important to notice that this mechanism: (i) does not change the average shape of grains (no dislocation slip considered); and (ii) gives rise to trajectories of neighbouring grains that are perpendicular to each other (as indicated by the arrows in Fig. 4). Having in mind a sequence of these elementary events, deformation is seen to involve a "stirring" of grains. Due to this stirring mode of deformation, intergranular correlations of stress and strain decay within a characteristic interval of strain on the order of E0. We believe that the stirring effect is responsible for the exceptional necking resistance of superplastic materials, as it impedes the emergence of macroscopic fluctuations and strain localization. The stirring effect is readily accounted for in terms of an exponentially decaying representative volume V*. Instead of equation (20) corresponding to Hart's criterion, one may thus write

V* = Ad expI-tl ~ 1

~ 1+

fl

l

> 1.

(30)

F o r h/e 4. 1, superplastic flow is thus related to a Strain-rate exponent E0

As the assumptions made for its derivation are not very restrictive, the instability criterion (14) is also applicable to plastic instabilities in a wider sense. We may, therefore, use it for studying creep damage, where failure occurs by the nucleation and coalescence of microvoids or -cracks. To this end, we adopt a phenomenoiogical model of the KachanovRabotnov (KR) type [16, 17] extended in a way to account for strain hardening and geometrical softening, and consider the coupled dynamics of the plastic strain E and a damage variable D defined as

m > - ~ + E o"

(31)

"['Withoutloss of generality for the following results of linear stability analysis, linear strain hardening is assumed.

0 ~< D ~< 1

(32)

where R and PD denote the linear extension and number density of microvoids/-cracks, respectively, and c is a numerical coefficient introduced to account for the geometry of these flaws. The K R model then reads d,E = B 1 (acxt(E) -- 0"int(6))n (1 - D ) "

'

(33)

(~oxt)v

diD Bz (~ ~ -ff)~,

(29)

immediately follows. By using this in the instability criterion (18) and applying it to the flow rule (15), we expect absence of necking for - + m

5.2. Creep damage

D = cR3pD, (28)

where d is the grain size and r/is the fraction of plastic strain accommodated by GBS (0 < q < 1). According to this ansatz, strain fluctuations are considered detrimental if they cover a monogranular layer extending over the whole specimen cross-section A and grow fast enough to outweigh the stirring effect. From equation (28) V* -

With e0 = 0.5 and r / = 1 this gives approx, m > 0.3 in favourable agreement with experimental findings. It is noteworthy that, according to this result, superplasticity is not only a consequence of a sufficiently high strain-rate sensitivity, but also depends on the strain proportion r/ accommodated by GBS. This may be related to the enormous scatter observed in the correlation between strain-rate exponents and elongations to fracture (failure strains may remain small, although m is comparatively large) [13]. Failure is then due to damage accumulation by internal cavitation. This is explained in terms of a small value of r/ meaning that GBS and, hence, the beneficial influence of stirring have little effect. Ideal superplasticity is rather supposed to occur when the accommodation of stress concentrations at grain boundary precipitates and triple points is such that dislocation movement is not the rate-controlling process. GBS is then very effective giving rise to ~/ values close to unity [14].

=

(34)

The coefficients B~ and B2 and the exponents n, # and v are positive parameters to be fitted to the experimental creep damage curves. The strain dependence of the external tensile stress o"ext is due to the geometrical softening under a constant applied load

F0 F0

°'°xt(e) = Tooexp e

(35)

where A 0 is the specimen cross-section prior to deformation. The internal stress flint increases due to strain hardeningt flint = o.~nt + he.

(36)

H~HNER:

CRITERION FOR PLASTIC INSTABILITIES

Although phenomenological in nature, equation (33) may be interpreted as a Norton law that is corrected for the stress increase due to the formation of internal stress-free volume. According to equation (34), this is assumed to accelerate damage accumulation. It is pointed out that equations (33) and (34) do not account for static recovery. The constant strain rate during secondary creep is rather ascribed to a balance between strain hardening on the one hand, and geometrical softening and damage accumulation on the other hand. Equations similar to (33) and (34) are frequently used to fit experimental creep damage curves. Although this works quite well, there is some lack of understanding with respect to the values of strain and damage at failure. This may be seen as follows. At first glance, one would assign failure to D = 1, where the strain and damage rates diverge. In practice, however, this value is much too high. On the other hand, application of the Estrin-Kubin instability criterion (3) to equations (33) and (34) does not yield new insights, as a reasonable critical damage parameter D c does not derive. It is thus interesting to see what the generalized instability criterion (14) predicts when it is applied to equations (33) and (34)• As a first step, one has to make a physical assumption on the representative volume V*. In the present case, it is natural to identify V* with the average flawless volume between the microdefects: V* ~ p/1. Assuming in addition that damage accumulation is due to the nucleation rather than g r o w t h of flaws, one has

~?*

15

V* -

D"

(37)

According to equation (14), the condition of marginal stability reads: 8~ - D_ 8~ = 0, D

(38)

8/) - D 8D = 0 D

(39)

~16~: -k- C~26D = 0,

(40)

~36E q'- g48D = 0

(41)

4099

Marginal stability is thus predicted for ~1~4 -

~2~3 = 0

(46)

which gives after some rearrangements 1 --

D (# + 1)D = vn h fir ext - 1 m

/) .

(47)

Dd

Note that this result depends implicity on the unknown coefficients B1 and B2 v i a / ) N . Equation (47) may be simplified by assuming that the onset of critical damage accumulation occurs during secondary creep (stage II of the strain vs time curve) = En = const,

and

h > o-ext.

(48)

Moreover, one knows from experimental observation [18] that damage accumulates rapidly once D/> D c has been reached, i.e.

/5

--)')" ~ll " D~

(49)

These simplifications lead us to the following result for the critical damage parameter Dc -

1

#+1

(50)

which can be obtained directly by setting cq = 0 in equation (41) (damage controlled failure). We have thus derived a value 0 < D c < 1, at which damage starts to accumulate in a critical way by rapidly filling the flawless volume between microvoids/-cracks. This constitutes a useful criterion for the onset of creep failure and is to be compared with findings on an AMCR 0033 steel, where failure is due to the coalescence of intergranular microcracks [18]. Here, an exponent/~ ~ 10 derives from the experimental creep curves. According to the theoretical result (50), failure is predicted for D c ~ 0.09 which compares well with the combined results of non-destructive ultrasonic techniques and post-mortem damage analysis giving Dc = 0.085 [18].

or equivalently

where equations (33) and (34) have been used, and the coefficients ~l to ~4 are defined as follows ct1 =

1 --

-- B2 (1 -- D)"~'

~2 = n l -- D'

(42) (43)

(~ eX')v ~3 = vB2 (1 -- D)~'

(aext)v ( cq=B2~

# "I-D

(44)

1).

(45)

6. CONCLUSIONS Proceeding from some general considerations on what constitutes unstable plastic flow, and obvious discrepancies in previously established instability criteria by Hart [6], Estrin and Kubin [8] and Molinari [9], a novel criterion of "relative instability" was developed. Its peculiarity consists in accounting for the temporal evolution of intrinsic (microstructural) or extrinsic (dimensional) length scales. Instability is then defined to become macroscopically evident (localization), if fluctuations grow with respect to these reference length scales. As a comparison with the former instability criteria revealed, the present result is more general,

4100

HXHNER:

CRITERION FOR PLASTIC INSTABILITIES

inasmuch as those are recovered as special cases if particular assumptions on the relevant length scales are made. This led to a re-interpretation of, and a removal of ambiguities in, the former criteria. Application of the generalized instability criterion to superplasticity showed the high necking resistance to result not only from a high value of the strain-rate exponent (m > 0.3), but also from the geometry of the particular deformation mechanism: grain boundary sliding impedes the development of macroscopic strain fluctuations (stirring effect). As another application, creep failure by microvoid/ -crack coalescence was considered in terms of a model of the Kachanov-Rabotnov type [17]. The critical damage parameter was determined and compared with experimental findings. In summary, it can be stated that the instability criterion developed here may serve as a useful base for assessing plastic instabilities both from the qualitative and quantitative point of view.

Acknowledgement--I am grateful to H. Stamm for directing my interest to the problems of creep damage which initiated the present work.

REFERENCES

I. G. E. Dieter, Mechanical Metallurgy. McGraw-Hill, London (1988). 2. R. Thomson, Mater. Sci. Engng A176, 1 (1988), and other papers in that volume. 3. P. Hfihner and H. Stature, Acta metall, mater. In press. 4. H. Haken, Advanced Synergetics. Springer, Berlin (1987). 5. P. H~hner, submitted to Acta metall, mater. 6. E. W. Hart, Acta metall. 15, 351 (1967). 7. I.-H. Lin, J. P. Hirth and E. W. Hart, Acta metall. 29, 819 (1981). 8. Y. Estrin and L. P. Kubin, Res. Mechanica. 23, 197 (1988). 9. A. Molinari, J. Mdc. Thdor. AppL 4, 659 (1985). 10. A. H. Chokshi, A. K. Mukherjee and T. G. Langdon, Mater. Sci. Engng RI0, 237 (1994). 11. A. Considrre, Ann. Ponts Chauss. (Set. 6). 9, 574 (1885). 12. P. H~ihner, AppL Phys. A~, 49 (1994). 13. O. D. Sherby and J. Wadsworth, Prog. Mater. Sci. 33, 169 (1989). 14. T. G. Langdon, Mater. Sci. Engng A174, 225 (1994). 15. M. F. Ashby and R. A. Verrall, Acta metall. 21, 149 (1973). 16. L. M. Kaehanov, Izv. Akad. Nauk. S. S. R., Otd. Tekh. Nauk. 8, 26 (1958). 17. Y. N. Rabotnov, Proc. XII Congr. of Applied Mechanics, p~ 342. Springer, Berlin (1968). i8. U. von Estorff and H. Stamm, Proc. 5th Int. Conf. on Creep of Materials, p. 287. ASM International Materials Park, Ohio (1992).