Chaos, Solirons & Fractolr Vol. 3, NO. 3, pp. 269-277. 1993 Pnnted in Great Britain

0960~0779/93$6.00 + .oO Pergamon Press Ltd

Sneaky Plasticity and Mesoscopic Fractals

B. BERNSTEIN and M. KARAMOLENGOS

Department of Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

T. ERBER*

Department of Physics, University of California, Los Angeles, CA 90024, USA

(Received for publication 2 February 1993)

Abstract-Precise measurement of plastic deformations in many materials show step-like variations or the Portevin-Le Chatelier effect. This serrated yielding is presumed to be due to intermittent slipping and locking of microstructural elements. The conventional macroscopic description of plasticity can be extended to include these processes by introducing singular functions whose integrals and derivatives are not constrained by the fundamental theorem of calculus. As an illustrative example we consider a rigid-perfectly plastic body whose energy includes a singular component that is strictly monotonic, fractal, and whose derivative vanishes almost everywhere.

1. INTRODUCTION

We present here a novel approach to plasticity that provides a connection between macroscopic phenomenology and microstructural processes. The essential idea is to represent variables such as stress, work, and internal energy by means of functions whose integrals and derivatives are not constrained by the fundamental theorem of calculus. There are several characteristics of plasticity that suggest a description in terms of functions whose variation is not completely smooth: (1) The most conspicuous effect of this type is serrated yielding, or the Portevin-Le Chatelier effect [l-3]. This refers to the step-like variations that appear in the stress-strain curves of many single or polycrystalline materials such as aluminum or steels during plastic deformations. These abrupt shifts, which occur on time scales of the order of 10m6 s, are presumed to be due to intermittent locking and slipping of microstructural elements such as grains or crystallites. With scanning tunneling microscope techniques this step-like response can be followed down to strains as small as 1o-8 [4]. (2) 0 n a still finer scale the existence of sudden microstructural rearrangements during plastic deformations can be detected by acoustic emission techniques. Although the individual acoustic pulses dissipate sixteen orders of magnitude less energy than typical stress-strain cycles, the cumulative effect of the damage associated with these signals is significant on the macroscopic level in contributing to fatigue. (3) When stresses are applied to plastic materials the energy of mechanical work is generally balanced by changes in the elastic strain energy, dissipated in heat, stored in the semi-permanent ‘set’ of plastic deformations-i.e. the stored energy of cold work [5]-and expended in the generation of microscopic damage. In the particular case of cyclic stress-strain hysteresis in metals, most

*On leave from the Departments of Physics and Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA.

269

270 B. BERNS.I.EIN et al

of the energy (2 95%) is usually dissipated in heat; a minute fraction (9 lo-‘) is expended in damage or dislocation formation; and a small portion (s 1%) is stored in lattice deformations that, in principle, can be released by restoration processes [5]. Since these energy transfers occur at various meso- and microscopic levels it is not possible to characterize them in terms of a single set of classical thermodynamic variables. In particular, since plastic deformations by cold work are neither reversible nor steady-state processes they cannot be described by Onsager’s relations [6]. Moreover, the fact that sequences of plastic states cannot be linked by any kind of idealized reversible process-or. equivalently, that they are completely ‘surrounded’ by irreversibility-implies that their structural order cannot be measured by classical entropy [7].

The mixed contributions of meso- and microscopic processes to plastic deformations suggests that their cumulative effects can be described by a set of functions of varying degrees of smoothness. Specifically, let UT(eZ) - UT(el) denote the total energy transfer to a plastic material during a strain increment l 2 - l 1. Suppose further that the energy transfer can be split into a smoothly varying part (U,,) and a part that represents the intermittent jumps that occur during serrated yielding or acoustic emission (U,). Then clearly

UT(E2) - UT(El) =

~sm(E2) - ~srII(~l) + Urv(E2> - UdEl) 3 (1)

E2 - El E2 - El E2 - El

where the usual connection between force and energy follows from the idealization that U,, is differentiable, i.e.

where ( 0) macr represents the effective macroscopic stress applied to the material. The physical interpretation of

lim ~rv(E2) - U,v(El)

(3) EZ_E, E2 - El

is more complicated. If the significant variations of U,(E) are indeed concentrated at a denumerable set of values of E where microstructural rearrangements occur, then it is plausible to assume that U,(E) is differentiable almost everywhere, and, furthermore, that wherever the derivative exists

au, - = 0. ac (4)

The non-vanishing energy transfer that does occur due to the cumulative effects of microstructural rearrangements can then be described by the following construction: Suppose that the significant rearrangements occur in a narrow set of strain increments centered at the values E,, i = 1, 2, . . ., N. In the simplest approximation, this corresponds to embedding each E, symmetrically in an interval I,, of width RAE, i.e. Z, = [E; - AE, E, + he]. Then the total energy transfer mediated by U, during the strain change E, -+ Ed is approximately given by

$J,,(t, + Ae) - Urv(~, - AE)I = Urv(ElV) - U,(E1) (5) I=1

if CT,.,, varies monotonically throughout the interval Ed - cl. The idealization that these energy transfers-or microscopic ‘latent heats’-occur principally in very narrow strain

Sneaky plasticity 271

intervals is equivalent to the condition

@ = 2NAe << e,,, - El. (6)

In mathematical terms, a function that has a finite total fluctuation in an arbitrarily small interval is not absolutely continuous. Clearly (5) and (6) are reasonably close physical approximations to these conditions. Accordingly, in the ensuing development we shall assume that U, is not an absolutely continuous function of strain. This is the essential property that removes the constraints of the fundamental theorem of calculus. Specifically, if U, is not absolutely continuous then it cannot be represented as the Lebesgue integral of any other function. Consequently, even if U, has a derivative almost everywhere, as in (4), and this derivative in turn is Lebesgue integrable, then the resulting function is not U,,, i.e.

= 0 # U,(EN) - U,(El). (7)

It is precisely this loosening of the connection between average and cumulative changes that allows U,, and U, to describe energy transfers at different meso- and microstructural levels. In particular, equations (l), (2), and (4) reflect the observation that measurements of macroscopic stress during fatigue cycles generally cannot distinguish the small portion of the work that ultimately appears in material damage, i.e.

(o)maer - z = $. (8)

Only very refined checks of the energy balance in mechanical [5] and magnetic [8,9] hysteresis systems can detect these discrepancies.

The assumption that U, is not absolutely continuous does not clash with the vanishing of its derivative and is also compatible with strictly monotonic behavior-for instance, the steady increase of damage. There is a large inventory of singular functions, some of which can be computed numerically, that share all of these properties [lo]. In the following Sections we will illustrate how a particular function of this type-which also happens to be a fractal-can be adapted to describing some aspects of plastic behavior.

2. THE SNEAK FUNCTION

The ‘sneak’ is a singular function that prescribes the optimum betting strategy for a desperate gambler [ll, 121. By identifying monetary gain with energy transfer it is also possible to give the sneak a heuristic physical interpretation. In simplest form, the domain of the sneak function @(x; p) is taken to be 0 G x d 1, 0 < p < 1, where x is a normalized variable corresponding either to money or energy, and p is a parameter that denotes gambling odds or damage increments. The ‘bold play’ strategy is equivalent to the system of functional equations.

$5; P = PG(Xi P> i i 1+x

$(y; P) = P + (1 - PM@; PI

with the boundary conditions

Pb)

4@; p) = 0; 40; P> = 1. (9c)

272 B. BERNSTEIN et al.

The unique continuous solution of (9a-c) can then be represented in the form [lo, 13]

(loa)

where the nk are integers that appear in the (infinite) binary expansion of x, namely

x = k$C, &- O C fl,) < n, < n2 < . . . . (lob)

If p = l/2, it can be verified directly from (9a-c) or (lOa, b) that

@(x; l/2) = x. (11)

However, for all other values of p E (0, l), 4 is a singular function of x with the following properties:

(9 (ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

4 is uniformly continuous but not absolutely continuous for all 0 s x c 1. @ is strictly increasing for all 0 S x S 1.

_ = 0 for all 0 c x d 1, except for a set of Lebesgue measure zero. EJX

At each dyadic rational the right hand derivative is 0 and the left hand derivative is +m. Hence the dyadic values of x lie in a set of Lebesgue measure zero where there is no derivative at all.

34) z IS a Riemann integrable function for which

I I

ds wr, P) = o

0 a? ’ p f l/2, (12)

in the interval 0 c x < 1, although, of course, C$ itself is different from zero for O<xSl. @(x; p) is self-similar or fractal. This is easily verified by setting @(x; p) = y, and introducing a mapping li of the X-Y plane that contracts the unit square, li: (x, y) + (x/2, py). According to (9a) the net result is simply that the value of 4 at x/2 is equal to that at x but compressed by a factor p. It can be shown that the Hausdorff or fractal dimension of @ is 1 ([ 121 and private communication).

W(x; P> > 0 for all 0 <x < 3P 1. 0 < p < 1.

Approximate numerical representations of the sneak function can be obtained from equations (10a) and (lob). The resulting graphs for p = 0.0625, 0.125, and 0.25 are shown in Figs l(a), l(b), and l(c) respectively. In the special case p = 0.5. equation (11) shows that the total arc length of @ is V’2. For all other values of p E (0, l), the fractal nature of @ implies that the arc length has the constant value

2 [12].

3. A RIGID-PERFECTLY PLASTIC BODY

We shall treat here a simple, idealized one-dimensional situation involving stress s and strain E. Let us first assume that there is a yield stress k > 0, such that if (sl always remains below k, then F remains equal to zero. In the following discussion we shall assume that the

Sneaky plasticity 273

direction of s is positive and that E 2 0. Furthermore, when the applied stress s reaches the yield stress k, we shall assume that the stored energy is given by

u = k[e - @(e; P)I (13)

where @(E, p), p f l/2 is the sneak function defined in (lOa, b), and

au

S=z’ (14)

Consequently the stress remains at k whenever it is defined, which-in view of property (iii)-is almost everywhere. Should s fall below the yield value k, the strain remains

THE GRAPH OF THE SNEAK FUNCTION FOR P YkUE.O 0625

1

021

01:

o! ‘8’8” _m llllTrl TT’W--p-I7-mm mm’ 0.00 0 13 0.25 0.38 0.50 0.63 0.75 0.0% 1 .oo

FIGURE 1 a

Fig. l.(a) Caption on p.274.

THE GRAPH OF THE SNEAK FUNCTION FOR P-YAuJE =‘I 25

1 1

0.9 ii

0.6 1

0.7

O.Sj

05 I

i)J’

Fig. l.(b) Caption on p.274.

274 B. BERNSI-MN et al.

THE GRAPH OF THE SNEAK FUNCTION FOR P “AWE =o 125

1

07

06

05

? 1

03

02

01

0 rrrTmrTnnmn-T.~.-rrmn~ - ,1_- ,.,~_ ., ,,,_ , .r . . ,. . , ,,_ . 0 00 ‘0.13 0.25 0.38 0.50 0.63 0.75 0.00 1 .OO

FIGURE 15

Fig. 1. Graph of the sneak function cp(_u: p) for (a) p = 0.0625; (h) ~1 = (1.25; and (c) ,I = 0.135

constant unless and until s is brought back to its yield value. Formally, equation (13) is reminiscent of the small viscosity limit of Burger’s equation: In this case also the dissipation of energy at discontinuities is represented by a combination of linear and singular functions

[141. What is remarkable about this formulation is the nature of the energy exchange during

plastic deformation. The stored energy as a function of E is illustrated in Figs 2(a)-(c) for several values of p. In these examples the stored energy becomes positive, varies in an apparently irregular fashion, and ultimately returns to zero at E = 1. This is an extreme illustration of what can be done in disconnecting macroscopic stresses, such as those in (8) and (14), from the microscopic energy transfer (4).

1 --

i 0.91

0.8 -I 0.7

0.6 1

05

04,

0.3 :

0.2

0.1 -

“0,;

THE STORED ENERGY FOR P.w.LM=0125

0.13 0.25 0.30 0.50 FIGURE 2*

Fig. 2.(a) Caption on p.275

Sneaky plasticity 275

THE STORED ENERGY FOR P-“ULE=0.25

l-

0.9-

0.8-

0.7-

0.6.

0 5-

0.4 I

-1 _.---

-/_\,, /-- ,> _--,

0.3 j /f-/

,. _ ,~

0.2 A--

1’ 1

_/J 0.1

0 1.. 0.00 0.13 0.25 0.30 0.50 0.63 0.75 0.88

FWRE a

THESTOREDENERGY FOR P-VALVE=0 c&?s

1

0.9

0.8 :

0.74 ,’ 1, -$-?,,

, ‘?.

0.6. _I

0.5 L

i)J/ _I’

0.3 _’

,’

0.2 I

/’

A/’ /

0.1 1

OLmmrr 0.00 0.13 0.25 0.38 0.50 0.63 0.75 0.68 1

FIGURE h

Fig. 2. Graph of the stored energy for (a) p = 0.125; (b) p = 0.25: and (c) p = 0.0625.

In these examples, we have implicitly restricted E to the unit interval, which is really not necessary. The simplest extension is obtained by replacing (13) with

(15)

where E” is a normalizing factor. If we also assume periodic behavior for U, i.e.

U(E + Eg) = U(E) (16) then (13) can formally be extended to arbitrary values of strain The scale of + can of course also be adjusted by multiplicative factors of either sign in order to describe energy exchanges of various magnitudes.

216 B. BERNSTEIN et al

4. THE THREE-DIMENSIONAL CASE

The energy expression (13) can be generalized so that the sneak function is combined with the three-dimensional description of plasticity. Let o,, be the stress tensor, and let e,, be the strain tensor. In addition, let s, denote the stress deviator tensor and E,, the corresponding strain deviator tensor. Then

and

s,/ = O;, - &/,b,, (17a)

E,, = e ‘, - iekk6,,. (17b)

Furthermore, let

0, = V(QQ) (l&i)

and

P = %,s,,). (1Xb)

In analogy with the one-dimensional case we assume that whenever /j i k. there is no change in E,,; but when p 2 k, the energy associated with plastic yield is given by

Instead of (14), we then obtain

In particular, when {j a k

But. in fact

C’ = k[a - @(a; p)].

au a,, = s,, = ---,

26 (20)

8,

SlI = k !!!, (21)

cl

t,, et, /i’ = s,,s,, = k -- =: k cl

(22)

when /J 3 k; so /3 actually never exceeds k. and the von Mises yield condition is satisfied. Again. a reduction of the stress below /j = k implies that E,, does not change: but if the

stress increases and reaches /3 = k, it cannot increase any further as in classical von Mises plasticity. However, unlike the situation in classical plasticity, the energy associated with ((, may be exchanged with internal degrees of freedom. Just as in the one-dimensional expression (13). equation (19) corresponds to an extreme case where energy is cyclically stored and-as shown in Figs 2(a)-(c)-returns to zero when /j = 1.

5. CONCLLWONS

Both the one- and three-dimensional energy expressions in (13) and (1Y) show how the general scheme of dividing plastic energy into a smooth macroscopic part (C.i,“,). and a not absolutely continuous microscopic part ( Url) can bc implemented. Although the identifica- tion of [I,, with the fractal sneak 4 has analogies with the description of Brownian motion by singular functions. the choice in these examples is only heuristic. However, more physics can be incorporated by adjusting the magnitude and frequency distribution of the singular

Sneaky plasticity 277

fluctuations [cf. (5)] to match the corresponding microscopic energy ‘jumps’. Clearly, these adjustments can be made in U, without affecting the macroscopic continuum mechanics governing the evolution of U,,. In this sense, introducing singular functions that are strictly monotonic but whose derivatives are macroscopically ‘invisible’ provides new flexibility in describing plastic behavior.

Acknowledgements-We thank W. F. Darsow, M. J. Frank, S. A. Guralnick, and A. Sklar for helpful conversations. T. E. also thanks the Department of Physics at UCLA for its hospitality. This work was supported in part by AFOSR, DOE and the Research Corporation.

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