Ductile versus brittle behavior of amorphous metals

OO22-5096/83$3.00 +O.oO 0 1983 PergamonPressLtd.

.I. Mech. Phys. Solids Vol. 31, No. 5, pp. 359-388, 1983 Printedin Greal Brilain.

DUCTILE

VERSUS

BRITTLE BEHAVIOR METALS PAUL

Division of Applied

Sciences, Harvard

OF AMORPHOUS

s. STEIFt

University,

(Received 9 November

Cambridge,

MA 02138, U.S.A

1982)

ABSTRACT

DUCTILE behavior of amorphous metals, their ability to sustain localized flow at high nominal stresses, is attributed to a mechanism which alleviates the severe stress conditions prevailing near potential cleavage flaws. The model problem of the plane strain deformation of an infinite block of non-linear visco-elastic material containing an elliptical hole is studied numerically and analytically. A strong dependence of the viscosity on the hydrostatic tension, a result of the increase in the number of viscous flow defects with dilatation, is the principal source ofnon-linearity. The analysis reveals that, under a constant remote strainrate, the initial elastic stress distribution ahead of the hole gives way, with time, to a more uniform stress distribution. Altering the stress distribution permits the remote (nominal) stress to achieve higher values before critical stress conditions are reached locally at the concentrator. At ordinary temperatures, amorphous metals are not in thermodynamic equilibrium; this motivates a modification of the constitutive law that reflects the kinetic difficulty of maintaining thermodynamic equilibrium under conditions of varying hydrostatic tension. Re-solving the elliptical hole problem with the modified constitutive law reveals a delay in the stress redistribution in front of the concentrator which may explain brittle fracture.

1.

INTRODUCTION

AMORPHOUS metals, also referred to as metallic glasses when produced by quenching from a melt, differ from ordinary structural metals in that their constituent atoms are not assembled on a crystalline lattice. The flexibility in the nearest-neighbor arrangement of atoms results in properties that are quite distinct from those of crystalline metals. The mechanical properties of amorphous metals have been investigated extensively, and the phenomenology of the various flow regimes is now well established. For a thin rectangular strip pulled in tension, essentially two modes of deformation are observed : homogeneous ilow and inhomogeneous flow. Homogeneous flow occurs at relatively low stresses and high temperatures, and the deformation is uniform throughout the sample. The strain-rate is proportional to sinh (stress) (TAUB, 1980); in the limit of low stresses the material is Newtonian with the strain-rate proportional to the stress itself. Fracture occurs when the sample eventually necks down to a narrow cross-section. Inhomogeneous flow takes place at higher stresses and lower temperatures. The deformation is localized in a thin band oriented at a 45” angle with respect to the tensile

t Present address : Carnegie-Mellon Pittsburgh, PA 15213, U.S.A.

University,

Department

359

of Mechanical

Engineering,

Schenley

Park,

360

P. S.

STEIF

axis. The flow stress is insensitive to the temperature. Intense shear strains in the band decrease the load bearing cross-section, and fracture occurs along the plane of the shear band. The mechanism of fracture is a Taylor fluid instability as evident from the river pattern on the fracture surface (SPAEPEN, 1975; ARGON and SALAMA, 1976). The localization of deformation into a shear band, as well as the occurrence of fracture by the fluid instability along the plane of the shear band, suggests that a severe softening of the material constituting the band has taken place. Differential etching of the fracture surface indicates chemical changes have occurred; this supports the conjecture of a local softening introduced by a disordering of the atoms. SPAEPEN and TURNRULL (1974) have suggested that the softening needed to initiate a shear band may develop at a microscopic flaw where the hydrostatic component of the concentrated stress locally dilates the material. Mechanically dilating the material would presumably have the same effect as raising the temperature, which is known to decrease the viscosity via the thermal expansion : the augmented volume per atom increases the likelihood of atomic transport events that lead to deformation. Not all amorphous metals, however, undergo inhomogeneous flow. Some fail by brittle fracture with little visible plastic deformation, and at stresses lower than those at which inhomogeneous flow typically occurs. Materials that flow readily in an inhomogeneous mode exhibit great ductility, which is measured experimentally by a bending test. A ribbon that can be bent back onto itself is considered ductile. A measure of the degree of brittleness is taken to be the minimum radius of curvature attained during a bend test (DAVIS, 1978). Amorphous metals of certain compositions are brittle as made (CHEN and POLK, 1974), and others are brittle at very low temperatures. Of significance here is the embrittlement caused by annealing below the glass transition temperature. Iron based alloys embrittle in this manner (LUBORSKY and WALTER, 1976), but other alloys, such as Pd,Si and CuZr experience no embrittlement with annealing at comparable temperatures. The reverse process has also been observed ; irradiation of brittle (Mo,,,Ru,~,),,B,, with neutrons makes them ductile (KRAMER, JOHNSON and CLINE, 1979). Some workers have suggested that ductile amorphous metals can sustain high stresses because the localized flow blunts potential cleavage cracks. Besides this approach, however, little work has been directed towards examining carefully the question of ductile vs brittle behavior. A theory that explains this variety of mechanical behavior is presented here that is based on a view of the critical fracture processes, which differs from the idea of localized flow blunting potential cleavage cracks. Our point of departure is the observation that localized flow in ductile samples commences rather suddenly at nominal stresses in excess of the nominal stress at which cleavage occurs typically in embrittled samples. Hence, the localized flow itself could not blunt the crack since the issue of whether or not cleavage will occur presumably arises at a stage of the deformation preceding localized flow. It is proposed that at a stress concentrating flaw, a potential initiation site for brittle fracture, the hydrostatic tension causes a reduction in viscosity, as suggested by SPAEPEN and TURNBULL (1974), that serves in turn to reduce the stress concentration at the flaw. This effect enables the macroscopic (nominal) stress to reach higher values without brittle fracture occurring, and eventually to initiate a shear band at perhaps such a softened site. The deformation history is separated conceptually into two stages: (i) achieving a more mild stress

Ductile YS brittle behavior of ~~0~~~~s

metals

361

concentration at a potential cleavage flaw through the viscosity reduction, and (ii) initiating and propagating a shear band. The process of lessening the stress concentration to avoid brittle fracture is explored in sections 3 and 4 by considering the following model problem: the plane strain deformation of an infinite body of non-linear visco-elastic material, capable of softening under hydrostatic tension, containing an elliptical hole. Under conditions of constant remote strain-rate, the elastic stress distribution in front of the hole, which would lead to cleavage at the hole, will be seen to give way to a milder stress distribution. The reduced stress concentration at the hole permits the remote stress to achieve considerably higher values before cleavage can occur at the concentrator. SPAEPEN (&2) has remarked that the viscosity’s dependence on the hydrostatic tension is so strong it would seem no material coufd be brittle. Indeed, the resufts presented in section 4.1, which indicate a considerable stress r~ist~bution, will be seen to imply that all amorphous metals would be ductile. The effect of the hydrostatic tension on the viscosity was tacitly assumed by SPAEPENand TURNBULL(1974), and in the results presented in section 4.1, to be a complete and instantaneous one, but it is argued in section 2.3 that the viscosity probably does not respond instantaneously to stress changes. Re-solving the elliptical hole problem reveals that a delay in the hydrostatic tension-induced softening delays the transition from the elastic stress distribution to a more uniform distribution. If the transition is delayed too long, brittle fracture can intervene. We submit that differences in the rate or the extent of the viscosity’s response to changes in the hydrostatic tension may distinguish ductile and brittle materials. 2.

CONSTITUTIVEEpu~rro~s

In this section the free volume model of COHENand TURNRUCC(1959, 1961, 1970}, which underlies the approach to viscous flow adopted here, is briefly presented. We then review the flow equation given by SPAEPEN(1977), which relates the viscous strainrate to the stress. Contained in the flow equation is an internal parameter, the average free volume per atom, which depends on the quenching temperature, as well as on the current temperature and pressure. In fact, the theory for ductile vs brittle behavior offered here hinges on the relation between free volume and pressure. We propose equations modeling the ekct of pressure on the free volume which are consistent with the relevant experimental observations.

2.1 Free volume model The average free volume per atom in a sample, or in some region of a sample, is defined as the average atomic volume minus the average atomic volume in the ideally ordered structure. Qualitatively, the free volume can be thought of as extra volume in which atomic movement can occur freely. This extra volume is distributed, without energy change, throughout the system, with some atoms having more local free volume than others. When the system is in thermodynamic equilibrium the free volume is distributed such that the entropy is a maximum. The equilibrium distribution of free

362

P. s. !hEIF

volume is given by (2.1)

where p(v) is the probability of an atom having a free volume between v and t’+ du, z+is the average free volume per atom, and CIis a geometric factor between 0.5 and 1.O.

The flow equation employed throughout this work is the same as that proposed by SPAEPEN(1977), and the notation here follows his closely. Flow is modeled as occurring as a result of many atomic scale rearrangements each contributing a small focal shear strain. In general, the macroscopic shear strain-rate, 9, is given by : $ = (strain produced at each jump site)

x (fraction of potential jump sites) x (net number of forward jumps at each site per second).

(2.2)

Following SPAEPEN(1977), we take the shear strain at each potential jump site to be I. A potential site is a region in which the free volume is greater than some critical value (the effective hard-sphere volume of an atom, for example). Such an amount of free volume would allow the atoms participating in the rearrangement leading to the local shear strain to “get by” one another. [SPAEPEN (I982) gives a more general treatment in which potential sites are considered flow defects.] The atom fraction of potential sites can be calculated from equation (2.1) with the result that

: -cm* i 1

the fraction of potential jump sites = exp -

3

f2.3)

Of

where u* is the critical (hard-sphere) volume, The net forward jump rate may be calculated from simple rate theory (CLASSTONE, LAIDLER and EYRING, 1941). in the absence of an applied shear stress, the numbers of forward and backward jumps are identical (no net shear accumulates), and the number of jumps in one direction per second is given by v exp where s is the attempt frequency (z Debye frequency), and AC” is the activation energy, In the presence of an applied shear stress the system is biased. The stress working through the local shear strain allows the system to lower its potential energy by ziz (where z is the applied shear stress, I;zis the atomic volume, and the shear strain is unity). An increase in the potential energy occurs when a shear strain is accomplished against an opposing shear stress. This biasing results in an extra term in the exponential of expression (2.4), hence giving the net rate of forward jumps in a biased system as

(2.5)

Ductile vs brittle behavior

of amorphous

363

metals

Therefore, the irreversible (viscous) part of the strain-rate, jl’, can be represented by the following general flow equation

(2.6)

it” = 2v exp[G]exp[+]sinh[&]. A stress dependent viscosity q may be defined as follows :

P+ 2v exp [$I

expi$]

sinh [~I’

(2’7)

Note that if the free volume uf is a small fraction of au* ( z a), then changes in of can have a large effect on the viscosity. The flow law equation (2.4) is generalized to multiaxial stress states through the *J, invariant of the stress deviator according to

where g$ are the Cartesian components of the creep strain-rate tensor, Sij is the stress deviator, and Z, is the effective shear stress defined by (2.9) For multiaxial stress states, the viscosity is given by equation (2.7) with r replaced by Z, such that (2.10) The total strain-rate is the sum of the elastic strain-rate and the creep strain-rate (2.11) where C is the elastic shear modulus, and, as discussed below, the dilatational strainrate has been neglected.

2.3 Free volume dependence on pressure Since the average free volume per atom is defined as the average atomic volume minus a reference atomic volume, the average free volume per atom ur at a hydrostatic tension (negative pressure) of a,,/3 is (2.12) where vOis the free volume at zero pressure (dependent on the quenching temperature), and K is the bulk modulus.

364

P. s.

STEIF

The flow equation (2.8) will be used in conjunction with the free volume equation (2.12) in the model stress concentration problem. The redistribution of the tensile stress, a consequence of the softening introduced by the dilatation, will be sufficient to explain the ability of amorphous metals to sustain high overall stresses. In fact, with (2.12) governing the free volume, all reasonable choices of parameters lead to a stress redistribution that is substantial enough to preclude brittle fracture from occurring at the typically observed stress levels. The explanation of brittle fracture seems to require some modification of the free volume equation (2.12) that has the effect of inhibiting the softening and stress redistribution process, at least for some choices of relevant parameters. It is tacitly assumed in (2.12) that the material is in thermodynamic equilibrium, that is, the internal state is uniquely related, at least statistically, to the imposed macroscopic quantities such as pressure. At most temperatures of interest, however, amorphous metals are not in thermodynamic equilibrium. The departure from equilibrium is discussed now, and we argue that the experimental observations suggest a plausible modification of the free volume equation. To first order the average free volume per atom uf is related to the temperature T by Uf = %9(T-

T,),

(2.13)

where ati, is the coefficient of thermal expansion and T, is the temperature at which a system in equilibrium becomes ideally ordered (the free volume vanishes). The equilibrium temperature dependence of the viscosity in the Newtonian regime, calculated by substituting (2.13) for vf into (2.7) and letting zR/ZkT + 0, is plotted as curve A in Fig. 1. For convenience, the inverse temperature is normalized with respect to the glass transition temperature T,, which for a given metal is generally chosen to be the temperature at which the viscosity has the value lOi* Ns m-*. Consider a metallic glass cooled from a melt : for temperatures greater than Tgthe viscosity is found to traverse the theoretical equilibrium curve A. When the melt has been cooled to a particular temperature (E T,), which depends on the substance and the cooling rate, its viscosity is observed to deviate from the equilibrium curve A and follow a curve such as B. If one were to stop cooling at some temperature, say TR,the viscosity would increase with time as the sample relaxes towards the equilibrium state corresponding to r,. This phenomenon is referred to as structural relaxation. In explaining the observed deviations from the equilibrium viscosity, it is useful to separate the thermal expansion conceptually into two parts: (i) expansion due to anharmonicity of the interatomic potentials, that is, increases in the atomic bond lengths without major changes in the relative arrangement of atoms, and (ii) expansion that involves major atomic rearrangements. The two contributions differ in that the latter requires atomic mobility. Experimental measurements of the viscosity agree with the equilibrium model only as long as the atomic mobility permits both contributions to cl,,, in (2.13). In fact, when T reaches Tg the mobility (which is inversely proportional to the viscosity) becomes too low: the atomic rearrangements which would further compact the system and increase the viscosity cannot take place fast enough. Ideally, for an infinitely slow rate of cooling, equilibrium can be maintained, but, for any finite rate of cooling, deviations from equilibrium occur once the required atomic jumps take place on a time scale long compared with the time scale of cooling. Note that the viscosity continues to increase along curve B since the anharmonic contribution to the

Ductile vs brittle behavior

of amorphous

365

metals

21

I!

P, .L” 8 F B I(

I

I

I

.5

10

FIG. 1. Viscosity

vs temperature

I

T9’TB

I

1.5

curve for a typical amorphous

Tg/T

metal.

thermal expansion is still operative. Curve B is referred to as an “isoconfigurational” curve because the structure or configuration remains, in some sense, constant. In addition to the low mobility preventing the system from attaining its equilibrium average free volume, there is a further source of dis-equilibrium. Even the nonequilibrium average free volume may not be distributed according to the equilibrium distribution equation (2.1). As has been pointed out by TSAO and SPAEPEN (1982), deviations from the equilibrium free volume distribution imply significant deviations from the equilibrium viscosity. We extend the notion of an incomplete (non-equilibrium) response to temperature changes to variations in the pressure. In section 4, the ellipse problem is re-solved replacing (2.12), which represents the equilibrium response of the free volume to changes in the hydrostatic tension, by an equation modeling a delayed, or nonequilibrium, response. The new free volume equation is

where 2, is the time constant

for free volume

rearrangements.

Equation

(2.14) was

P. S.

366

STEIF

chosen because it seemed to be the simplest first order differential equation with one time constant that reduced to equation (2.12) as the time constant approached zero. It should be emphasized that equation (2.14) is also approximate when used in conjunction with the flow equation (2.6), in that all types of dis-equilibrium are represented by a constrained equilibrium distribution of local free volume with a nonequilibrium value of the average free volume. This is because the flow equation (2.6) assumed an equilibrium distribution of free volume. From the results of the solution of the elliptical hole problem based on equation (2.14), a plausible range of values for z, will emerge. With this range of values, we attempt in section 5.2 to ascribe the time constant r’, to a kinetic process which is experimentally observed.

3.

ELLIPTICAL HOLE PROBLEM

To study the effect of a hydrostatic tension-induced softening at a potential cleavage flaw, we consider the plane strain deformation of an infinite block of non-linear viscoelastic material containing an elliptical hole. At infinity a constant strain-rate is imposed, and the remote stress state is one of plane strain tension. The material is taken to obey equation (2.11) in which the total strain-rate is the sum of the elastic strain-rate and the creep strain-rate. The creep strain-rate is given by (2.8), a multiaxial version of the flow equation based on the free volume model. Results will be presented for both the instantaneous free volume response, equation (2.12), and the delayed response, equation (2.14). The proposed problem is solved numerically with a procedure similar to that employed by BUDIANSKY, HUTCHINSON and SLUTSKY (1982). First, a velocity-based variational principle for incremental equilibrium, valid for unbounded regions, is formulated. To facilitate the implementation of the variational principle, we take the material to be incompressible. This permits the components of the velocity vector to be expressed as spatial derivatives of a stream function. The stream function is written as a truncated series of functions with undetermined coefficients, and this representation is substituted into the functional in the variational principle. Minimization of the functional leads to a set of linear equations for the coefficients. One can then calculate the velocities, the strain-rates, and the deviatoric stress-rates, and integrate numerically with time the quantities of interest. The equations of mechanical equilibrium allow the pressure (negative of the hydrostatic tension) to be calculated from the deviatoric stresses. Ajinite bulk modulus is then taken to relate the hydrostatic tension to the free volume, as in (2.12). The assumption of incompressibility, which enables the numerical solution to be carried through, is at odds with the dilatation-induced decrease in viscosity. For the elastic case, with stress prescribed at infinity and a traction-free hole, the in-plane stresses are independent of Poisson’s ratio. In the case of a non-linear visco-elastic solid, however, the extent to which compressibility would alter the solution is not known. Nevertheless, it seems unlikely that a more realistic analysis including compressibility would alter the conclusions reached here that the stress redistribution is possible and that varying rates of free volume response may explain ductile vs brittle behavior.

Ductile vs brittle behavior

of amorphous

metals

367

3.1 Vari~tiu~u~principle

Consider a finite body occupying a volume V and bounded by a surface S. In the absence of body forces, the equations of incremental equilibrium are cfijnj = R on S,,

dij,j = 0 in V,

(3.1)

where the summation convention is assumed, and i and j range over the values 1-3. oij are the components of the Cauchy stress tensor with respect to Cartesian axes, ( ),j denotes differentiation with respect to the jth spatial coordinate, and (‘) denotes time differentiation. S, is that part of S on which traction-rates z are prescribed, and nj is the normal to the surface S. The velocities take on prescribed values on the remaining portion of the surface S,. In what follows, a small strain formulation is used, i.e. no distinction is made between deformed and undeformed configurations. The strain-rates sij are related to the velocities ui by Eij

=

~(Vi,j

+

(3.2)

~j,i).

The notation of this paper is simplified by omitting the usual dot from the &ij.~~~ from this point on denotes strain-rate (regardless of what other superscripts it might have). When the variables are non-dimensionalized, however, the dimensionless strain-rate will be written as off. A form of the statement of virtual work is

s

ci, hij d V =

V

ss

$ Sz+dS.

(3.3)

Satisfying equation (3.3) for all variations in strain-rate and velocity related by equation (3.2) implies the equilibrium equations (3.1). The material is taken to be incompressible, hence only velocities that satisfy UiTi

=

(3.4)

0

are admissible. The stress-rates may be separated into deviatoric and hydrostatic parts according to ~ij

=

(3.5)

Sij-P6ij*

Since the material is in~ompressibie, only the deviatoric stress-rates S, contribute to the left-hand side of (3.3). The material law may be expressed in the general form Sij

Eij =

2~

+

Et,

(3.6)

where G is the elastic shear modulus and E~jis the creep strain-rate. afj may, in general, depend on the entire stress history to the present time, but not on the current stress increment. For the constitutive equation (3.6), the velocities minimize the functional F(q) =

G(EijEij- 2~~j~ij)d V/ijvi dS, sY s ST

(3.7)

where the zfisatisfy (3.4) and take on the prescribed values on Su. Because the creep

368

P. S. STEIF

strain-rates are independent of the current stress-rates, the ~~~are fixed functions of the spatial coordinates at any increment. The problem we desire to solve is that of an infinite body containing an elliptical hole. The above functional must be modified so as to be valid for an unbounded region The modified functional will be formulated for a three-dimensional body containing a cavity, and later the functional is specialized to the plane strain problem. Consider a traction-free cavity inside a spherical body V, of radius R bounded by a surface &. Subsequently, the body wiI1 be permitted to become unbounded. Traction-rates, * r Z.Z$?n. 1.l I’ are prescribed Let

on S,, consistent

(J.8)

with the stress state at inftnity.

z+ = r?i”+ i&

Fii = EZ$+ ZJj, CTj= EFJ7+gj>

(3.9)

where the superscripted quantities ( )” are the uniform fields due to c?T in the absence of the elliptical hole, The quantities with a tilde represent the perturbation of the solution due to the hole. By subtracting off quantities that do not change during a variation, the functional to be minimized can be written as &vi) = G t’ [(cijcij- 2~&~)i x By an application

of virtual

(c; E; - ~c;~E$)] d L’-

ct$nj(v;-v;+)

dS, (3.10)

work one obtains (3.11)

where nj in the last integral is the unit normal at the void surface pointing towards V,, and S,, denotes the surface of the hole. With equations (3.9) and(3.1 l), F may be written as a functional of uII,

(3.12) where the boundary of the body has been permitted to recede to infinity, and V denotes the unbounded region exterior to the hole. For the plane problem considered here, the first integral remains bounded provided Esjand ETjdecay at infinity faster than i/r. (Note that for the elastic problem the additional strains go as l/r’ as r + a.) 3.2 Sol&m

at infirmity

The quantities at infinity must be solved for at each increment. can be written as sij

sij Fij

=

7

26

+-

--,

2y

The constitutive

law

(3.13)

where the stress history dependence of the creep strain-rate is incorporated into the viscosity q (see section 2.2). Under conditions ofplane strain tension, the remote stress

Ductile vs brittle behavior

of amorphous

369

metals

state is

If a remote strain-rate

s2z = E is applied, the stress oZ2 satisfies (3.14)

Other required

quantities

are the pressure p

E

-&T&,,

p, =

-

022 2’

(3.15)

and the effective shear stress z,,

Equation (3.14) is solved for 022 at each increment so that the 6; may be calculated for the variational principle. (For typical values of parameters the material at infinity remains essentially elastic, so that a constant applied strain-rate is tantamount to constant ti$.)

3.3 Solution procedure For the given geometry, the procedure described above is facilitated by a change of variables such that the original region exterior to the ellipse occupies the interior of the unit circle in the plane of the mapped variables (see Fig. 2). This allows for a simple choice of series functions for the stream function, and it permits the easy integration of the terms in the functional (3.12). For convenience, the change of variables is written as a function of the complex variable z. The mapping is given by ml+-

z=x+iy=o([)=R (

Physical FIG. 2. Mapping

-R 2-c

plane of physical

mpeid+pe-i@, l

Mappmg plane onto interior

(3.17) >

plane

of circle in the c-plane.

P. s. STEIF

370

where x and y are Cartesian coordinates in the original z-plane with the hole, p and 4 are, respectively, the radial and azimuthal coordinates in the i-plane, and R and m are given by R=--

a+b

m=-’

2

u-b a+b’

where a and b are the semi-major and semi-minor axes, respectively. (Note : the first quadrant in the z-plane maps onto the fourth quadrant of the circle in the c-plane.) The additional velocities are written as derivatives of a stream function $ according to

61= *,29 I& = -$,I. A complete given by

representation

of the stream function

$=.44+

5 j=l

t

(3.19)

in terms of the mapped

Ajkpj-’

variables

sin 2k$.

is

(3.20)

k=l

This expression for the stream function satisfies the appropriate symmetries about the coordinate axes in the z-plane. For the case of a crack, m = 1, which is not considered here, a bounded horizontal velocity at the crack-tip requires that A be not independent of Aj, (see HE and HUTCHINSON, 1981). For convenience (and anticipating the non-dimensionalization) equation (3.20) is written in the form

$ = R’v, 1 A.~‘%, 41,

(3.21)

n

where R is given by (3.18), and vc = v exp [ - AG”‘/kT]. Similarly, strain-rates are written respectively as Ci =

the velocities and the

c A,Vi”’

(3.22)

n

and Eij =

c A&;‘.

(3.23)

n The V$“’and E$’ are related to the f’@) by Vl”’ = R2v,

f !"z' > Vy)= -j+J!;',

El”! = R2vG f !;jl, E:": = +R’v,[f With an application

(The transformation

,22 -f!;'J. @I)

of the chain rule, the above derivatives fl;) = f !;)p,l + f derivatives

(3.24)

may be written

I;) d~,~,etc.

p,r, etc. are found in Appendix

(3.25) as (3.26)

I.)

Ductile vs brittle behavior

For the two-dimensional

of amorphous

region considered G[eijgij-

P(C,) =

371

metals

here, the functional

F is written

6; njCi ds,

2E”;jE”ij]d/l -

as (3.27)

sr

sA

where A denotes the two-dimensional region exterior to the hole, and I denotes the curve bounding the hole. Equations (3.22) and (3.23) are substituted into (3.27) yielding

If the following

identifications

are made

s s

E!‘!)E!‘?) dA r, ,,

Km = Km = 2G

(3.29)

2

A

6%

tn =

n lJJ1

I/!“) ds

)

(3.30)

I-

equation

(3.28) can be put in the form p = $1

P is minimized

A,A,K,,-1

(3.31)

A,,&,.

by requiring dF -?z aAk

implying

A,b,-x

0,

(3.32)

the equations c K,,A, m

= b, + t,.

(3.33)

Because the K,, depend only on the functions f(“) and the transformation derivatives, they may be determined once and for all given the ellipse aspect ratio. During a program of deformation, the b,, which depend on the creep strain-rates, and the t,, which depend on the stress increment at infinity, will change, in general, at each increment. Equations (3.33) are solved for the A, at each increment, all quantities are updated, the b, and t, are calculated anew, and the cycle repeats. (For details of the time integration see Appendix II.) 3.4 Calculation of the pressure The constitutive law used here involves taken to be incompressible, the variational the stress. The pressure must be determined traction is zero, hence

the pressure but, because the material was principle yields only the deviatoric part of by mechanical equilibrium. At the hole the

p = n.s..n.. 1 V J

Furthermore,

at any interior

point, the equations P,i = sij 3j .

(3.34)

of mechanical

equilibrium

imply that (3.35)

372

P. S. STEIF

The equations (3.35) must be integrated with (3.34) as the boundary condition at the hole. The integration was carried out as follows. Although the deviatoric stress components can be calculated in principle at every point interior to the unit circle in the i-plane, in practice they are known at a grid of points in the region 0 < p d 1, -7(/2 < 4 ,< 0. A bicubic spline interpolation of sij in the p-4 plane allows for derivatives such as s1 l,p, etc. to be calculated. Since the transformation derivatives at each point are known, the derivatives appearing in (3.35) can be calculated, e.g. Sll,l

=

~11,pP,1+~11.~4,1.

(3.36)

The pressure is known at p = 0, which corresponds to the point at infinity, from the solution in the last section. The pressure at p = 1, corresponding to the hole, is known for grid values of 4 from (3.34). The pressure in the interval 0 < p < 1 is determined separately for each grid value of 4. The derivative P,~ can be written as P,P = P.XX,P+ P.YY.P’

(3.37)

in which x,~ and y,, are known transformation derivatives, and the p., and p.y are calculated from (3.35) and (3.36). The derivative P,~ is evaluated for discrete stations in 0 < p < 1. The pressure in the interval 0 < p d 1 is approximated by the polynomial that is a least squares fit for P,~at the grid stations, and which is constrained to take on the appropriate values at p = 0 and p = 1. (See Appendix II for more details on the numerical technique, and a discussion of its accuracy.) To simplify the analysis the variables are redefined as dimensionless quantities. All stress quantities and moduli are redefined as CT

dij

and the time according

32

A 2kT

+

3

etc.

to ttv,t.

The dimensionless strain-rates &ij are defined time derivatives are denoted by ( )‘. The free volume is redefined as Vf +-.

by &ij = vG&ij, and other dimensionless

Vf R

The spatial variables

are redefined x+g

as and

yti,

so that the undetermined coefficients and the functions in (3.21) are dimensionless. In subsequent sections, the variables will denote dimensionless quantities unless otherwise noted.

Ductile vs brittle behavior 4.

of amorphous

373

metals

RESULTS

Numerical results are presented in this section for both the instantaneous free volume response to stress and the delayed response. In all cases, one finds that the material initially responds elastically, and, under conditions of constant remote strainrate, the material remains essentially elastic everywhere (by comparison with the known elastic solution) until noticeable deviations from the elastic solution first appear at the notch. 4.1 Numerical

results for instantaneous

free volume response

The tensile stress at the notch is plotted against the remote tensile stress in Figs 3-5, the different graphs demonstrating the influence of different relevant parameters. Consider the curve labeled m = .7 in Fig. 3. For values of @JG less than approx. 0.003, the material at the notch is essentially elastic, since the notch stress and remote

___

Numerical

- - ----

Approximate

Solution Solutlon

notch C22 G .O? >-

G =200 &’ = 10-14 vg = ,012

I 002

I

I

004

006

% G FIG. 3. Notch

stress vs stress at infinity for various ellipse aspect ratios (instantaneous

free volume response).

374

P. S. Srw notch =22 G

Numerical

- --

Solution

-Approwmote

Solution

.05

I

002

I

I

,004

006

! ____L__L 008 010

012

a% G FIG. 4. Notch

slress vs stress at inlinity

for various

dimensionless

strain-rates

(instantaneous

free volume

response).

stress are proportional. At o,“,/G z 0.003, softening at the notch becomes substantial, and the stress concentration factor (the ratio of notch stress to remote stress) begins to diminish. The softening is due primarily to the increase in free volume; this is indicated by the relative contributions to the reduced value of the viscosity. In a representative case, the value ofexp (c(v*/z+) decreased by 7 orders ofmagnitude from its initial value to softening, while z,/sinh z, decreased by only two orders of magnitude during the same period. Since the stress at the notch increases at a slower rate once substantial softening sets in, but the far-field stress continues to increase at its original rate, the increased load must be borne by material elements ahead of the notch. This redistribution of the tensile stress is illustrated in Fig. 6, in which the notch stress, normalized by the remote stress, is plotted as a function of distance ahead of the notch for several values of the parameter 3ois the remote stress at which deviations from the elastic solution are first 0~&?. CY observed at the notch. (0: is the value of oT2 at which the curves in Figs. 3-5 “turn over”.) An approximate expression for a>? in terms of relevant parameters is derived in

Ductile vs brittle behavior ~

metals

375

-..Solution -1

Numerical

------

of amorphous

Approximate

Solution

I

notch

522 G .OC

G = 200 &‘z 10-‘4 m =,5 I

I

FIG. 5. Notch

I .002

I -004

stress vs stress at infinity for various volume

I

,006

I ,008

dimensionless

I

010

I ,012

initial free volumes

I

(instantaneous

free

response).

section 4.2. The quantity o,mZ/tr;,referred to here as the reduced stress, is a measure of the extent to which the far-field stress exceeds the value needed to produce deviations from elasticity at the notch. Initially, i.e. at G&/G,” = O+, the elastic stress distribution prevails, and until @Z/~T = 1 the numerical solution still gives essentially the elastic distribution. Note that the remote stress and the notch stress both continue to increase, but the normalization used in Fig. 6 focuses attention on the distribution of stress. At rJe/up = 1.85, the stress concentration is 4, and the load shed by the material at the notch is taken up by adjacent material elements, as indicated by the crossing of the two curves o?~/(T: = O+ and c$~/u; = 1.85. Further redistribution of the load has taken place by the time e$$/a,” = 2.53, when the stress concentration factor is 3. 4.2 Approximate

solutions

forinstantaneous free volume response

Approximate expressions are derived now for the notch stress as a function of the

P. S. STEIF

316 7

R=+(a+b)

+Y

m =o-b E

-

5

G

= O+ (elastic

solution)

4 7” I

522 UC0 22 4 .\

3

2

/

i

1

C

0

I

I

I

I

.5

10

1.5

20

i 2.5

x-a R FIG. 6. Normalized

stress distribution

for successive

values of reduced

stress.

stress at infinity. The following remarks explain the softening and stress redistribution, and they motivate the derivation of the approximate solutions. When straining first begins at infinity in the non-linear visco-elastic solid considered here, the stress distribution is identical with that prevailing in a linear elastic solid. Immediately, the material starts to creep, and strains in addition to the elastic strains accumulate. The elastic stress distribution would now give rise to incompatible strains, but because the creep strains are small initially, the required adjustments in the stress distribution are minor. Subsequently, due to the stress concentration, and to the exponential dependence of the viscosity on the hydrostatic tension, the creep strainrates become very large at the notch. However, as long as the effectively elastic material surrounding the notch can deform fast enough to accommodate the elastic strain-rate at the notch, in addition to the creep strain-rate given rise to by the stress of the elastic solution, then the stress distribution remains the elastic one. When the outer material can no longer accommodate further increases in the deformation rate, the elastic part of the strain-rate at the notch decreases. This causes the stress-rate to decrease, and, with the remote stress-rate remaining essentially fixed, causes the curves in Figs 3-5 to “turn over”. (Note that diminishing the creep strain-rate would require a stress reduction.) Though the accommodation process described above occurs continuously from the start of deformation, we will say that creep effects are sufficient to disturb significantly the elastic stress distribution when the creep strain-rate at the notch becomes equal to the initial elastic strain-rate at the notch (equal to the remote strain-rate times the elastic stress concentration factor). An approximate formula for the stress at which these strain-rates balance is obtained by calculating the creep strain-rate based on the

elastic stress distribution. is expressed by

Ductile vs brittle behavior

of amorphous

In the dimensionless

variables

metals

371

defined earlier, this condition

(4.1) factor, and ay” again is the where C, - ~“2”;~~/& is the elastic stress concentration dimensionless stress at infinity at which significant deviations from the elastic solution first appear. The relevant component of the deviatoric stress, and the mean stress, are both equal to C,(oFJ2) at the notch. For large values of the argument of sinh, the approximation, sinh x z $ eX, may be used. This permits (4.1) to be inverted,

yielding

PY = & (In (2C&i) - rcuO+ J{ [ln (2C&J]2 + 4~)). e The points at which the various dotted lines in Figs 3-5 intersect the elastic solution curve represent the respective values of 0; given by (4.2). The creep strain-rate depends very sensitively on the stress, in that the creep strain-rate can change by orders of magnitude in a narrow stress range. Thus, although the accommodation process described above happens gradually, beginning when the creep strain-rate is within, say, an order of magnitude of the initial elastic strain-rate, this still corresponds to a stress very near to ~7. This accounts for the high accuracy of the approximate solution. The trends in Figs 4 and 5 for the stress ay” are consistent with the accommodation process described above. Increasing the applied strain-rate increases the creep strainrate (and stress) that can be accommodated before the elastic solution is significantly disturbed. Similarly, a lower initial free volume means that the hydrostatic tension must reach higher values to produce the creep strain-rate required to perturb the elastic solution. After softening begins, the stress concentration at the notch diminishes, but the stress continues to increase. An approximate formula for the stress after softening is now presented. This formula is primarily motivated by a remarkable similarity between the numerical results for many different choices of parameters, provided the quantities of interest are normalized properly. In Fig. 7, the stress concentration at the notch, normalized with respect to the elastic stress concentration, is plotted against the reduced stress. To construct this plot, the quantity ay” must be calculated from (4.2) for each set of parameters. Since the solution is essentially elastic for aT2 < (T; (by definition), the ordinate is equal to 1 for 0T22/~F from 0 to 1. The remarkable coincidence of the results for CJ~“~/CJ,” > 1 suggests that a formula for the notch stress after softening may be derived by choosing an appropriate relation between the normalized variables C/C, and ~?~/a,“. From Figs 3-5 it is seen that the stress after softening increases almost linearly with the remote stress, but with a greatly diminished slope. The bilinear relation between notch stress and remote stress,

(4.3)

378

P. S. STEIF

.8

-

Approximate

Solution

.7

.6 q

m =.5

.5

FIG.

7. Fractional

reduction

in stress concentration dimensionless

as a function parameters.

of reduced

stress for various

sets of

with a definite value for p, corresponds to a single relation between C/C, and cr?Jo;. With /3 = 0.07 (which seemed to fit the numerical results best), (4.3) is plotted as the solid line in Fig. 7, and as the dotted lines in Figs 3-5. It will be seen in section 5.1 that the stress subsequent to softening does not play a role in determining ductile vs brittle behavior, assuming (4.3) is even approximately obeyed. The results to be presented in section 4.3 based on a modified constitutive law appear to bear more directly on the fracture question. Thus, for the sake of brevity, additional comments concerning the post-softening regime are omitted here. A qualitative picture of the post-softening regime, as well as some justification for the coincidence of the numerical results as plotted in Fig. 7, is given by STEIF (1982). 4.3 Numerical results for delayed free volume response In this section, results are presented for the elliptical hole problem assuming the free volume does not respond instantaneously to the stress, but is delayed according to (2.14). As mentioned in the Introduction, the delay in the free volume response forces the stress to reach higher values before softening and stress redistribution occur. This delay is exhibited by the curves in Fig. 8, in which the notch stress is plotted as a function of the stress at infinity. The curve labeled z, = 0 corresponds to the case of instantaneous response, and increasing values of z, represent decreasing rates of response to stress. In Fig. 9, the response rate is held fixed (z, = 10i3), and the notch stress vs the remote stress is plotted for various values of initial free volume. As is

Ductile vs brittle behavior of amorphous metals

379

l

r, = 10’3

CLEAVAGE ------

G-200 E’: 10-14 “^Z n17 -0 ‘-‘-

/

’

m =.5

FIG. 8. Notch stress vs stress at infinity for various free volume response rates.

evident, a lower value of initial free volume means that the material is initially stiffer, so the stress must attain higher values before softening occurs. 4.4 Approximate solution for delayed free volume response

The qualitative picture of softening and redistribution offered in section 4.2 also describes the ease of delayed free volume response. When the creep strain-rate becomes of the order of the elastic strain-rate, the surrounding material forces the stress-rate at the notch to decrease, and the stress must be redistributed. The earlier calculation of the creep strain-rate, equation (4.1), is not valid, however, because it assumes the free volume responds instantaneously to the stress. An approximate expression for the creep strain-rate is now derived which takes account of the free volume delay. As earlier, the stress is assumed to be given by the elastic solution up to the instant of softening. Within this approximation, the free volume equation (2.14) can be integrated exactly. In the dimensionless variables, the free volume equation for the material element at the notch can be written as (4.4)

380

P. S. STEIF .18

.l 6 -

-

notch u22 G

-

v(y =.012

CLEAVAGE _

-

t-

.l4G=200 p. 10-14

.12 -

T” =10’3 m = .5

.lO -

.08

INHOMOGENEOU.

-

FLOW

J .06

-

.04

-

I

I I

I

.Ol

I

I

I

.02

.03

@

22 G

FIG.

9. Notch

stress vs stress at infinity

for various dimensionless response rate).

initial free volumes (fixed free volume

where 0’ is the dimensionless stress-rate & at the notch. The initial condition is that u,(O) = uO. The solution of the differential equation subject to this initial condition is Vf

If (4.5) is substituted softening is C,EF;

=

=

vg +

ozv 2K ’ [e-“‘“+4_r”

into the equation

exp

1

1

for the creep strain-rate,

(4.5) then the condition

for

--cI (4.6)

cc’ vo+~e(T22Z, 2K

,-WV i

where t, is the time at which softening occurs. For large values of the argument of the sinh, (4.6) becomes (4.7)

Ductile vs brittle behavior of amorphous metals

381

For given values of parameters (with c& = AGE,“), the above equation may be solved numerically for t,. From t, one can calculate the stress at softening; this approximate solution is represented by the points on the elastic curve in Figs 8 and 9. 5.

DISCUSSION

As mentioned in the Introduction, a wide variety of mechanical behavior is observed : some materials are always ductile, some are always brittle, and others are susceptible to embrittlement. In order to interpret the results of the elliptical hole problem, contact is made with the thin strip used in a tensile test. The tensile axis of the elliptical hole problem is intended to coincide with the tensile axis of the strip, and the x,-axis of the model problem is coincident with the direction of the smallest (thin) dimension of the strip. The strip problem might be more accurately modeled as being one of plane stress (or generalized plane stress). However, if the small flaw, that is modeled by the elliptical hole, is of linear dimensions much smaller than the thin dimension of the strip, and wholly contained within the strip, then a state of approximate plane strain would prevail near it. Localized flow in the tensile strip occurs on a plane inclined with respect to the tensile axis and to the direction of the smallest dimension. For the model problem, this would correspond to displacements along the x,-axis and the x,-axis, but the latter are ruled out here by the plane strain assumption. Ductile vs brittle behavior might be explored by solving a threedimensional problem that somehow permitted localized flow on an inclined plane, along with the in-plane deformations considered here. A brittle fracture criterion would be imposed, and the competition between localized flow and brittle fracture could be studied. Instead, we use the plane strain elliptical hole problem alone to get insight into ductile vs brittle behavior. Two criteria are imposed, one for brittle fracture and one for localized flow, both based on in-plane quantities. For particular values of strain-rate and material parameters, a hypothetical sample subjected to monotonically increasing strain is brittle if the brittle fracture criterion is met before the strain localization criterion, and conversely. Thus, we sidestep the need for analyzing the out-of-plane deformations associated with a developing shear band, and instead we adopt an approximate criterion for incipient localized flow. This approximate approach is consistent with the hypothesis put forth here for ductile vs brittle behavior. We propose that differing rates of free volume response explain the d@erent types of mechanical behavior. The failure mode “chosen” by a particular sample is not dependent on whether localized flow can be achieved, but, rather, on whether brittle fracture can be avoided. As will be seen from the strain localization criterion, we are assuming that all materials are theoretically capable of strain localization, but only those that can redistribute the stresses in front of potential cleavage flaws quickly enough to avoid brittle fracture can go on to localized flow. 5.1 Brittle and ductile criteria

We take brittle fracture to occur when the tensile stress at the notch reaches a critical cleavage stress. This seems an appropriate criterion for cleavage at a blunt notch. For a

382

P. s.

STEIF

very narrow ellipse this criterion breaks down, and achieving a critical stress intensity factor may be a more suitable fracture criterion. However, the concept of singular stress fields surrounding a crack-tip is useful only if the linear dimensions of the flaw being analyzed are large compared with the radius of curvature at the stress concentrating portion, i.e. that the flaw be crack-like. Fracture events studied here probably occur at such irregularly-shaped flaws (perhaps a microscopic roughening of the sample edge, or an interior microvoid) that the use of stress intensity factors may be of questionable value. Hence, we adopt the critical stress criterion as suitable for the situation considered here. In addition, the approximate formulas derived in sections 4.2 and 4.4, which compare well with the numerical results, may be applied with confidence to blunt notches. SPAEPEN and TURNBULL (1974) originally proposed the mechanism of a pressureinduced viscosity drop as an initiator of localized flow. This effect has been adopted here as the mechanism whereby cleavage is forestalled. This by no means excludes the possibility that such a softened region is the precursor of localized flow, The results for the case of instantaneous free volume response, for which the numerical solution may be carried far enough, show a softened zone directly ahead of the elliptical hole which spreads out with time. The relative viscosity distribution in front of the notch is plotted in Fig. 10 as a function of position for several values of reduced stress. For C&/O;;” = 2.63, a zone of effectively constant viscosity extends into the region ahead of the notch. Such a zone seems to be a potential site for localization to begin. Strain localization in amorphous metals has been studied by STEIF, SPAEPEN and HUTCHINSON (1982), and they showed that the shear strain can concentrate in a band of material that has been weakened by slightly reducing the initial viscosity. The HOW law

-A04 /-------a3

G = 200 &‘= 10‘‘4

1

v,

= ,012

m= .5 0 I

0

.5

I

I

I

1.0

15

2.0

x-0

R FIG. IO. Normalized viscosity distribution for successive values of reduced stress.

J

2.5

Ductile vs brittle behavior of amorphous

metals

383

they employed is the same as (2.6) except that free volume is irreversibly created by shearing deformation, as suggested by SPAEPEN (1977) and others. STEIF, SPAEPEN and HUTCHINSON (1982) also analyzed a homogeneous body modeled by the same flow law and free volume creation equation, subjected to a constant shear strain-rate ; they found the linear elastic response was followed by a catastrophic softening after which the shear stress dropped precipitously. Their principal result is that, to a high degree of accuracy, strain localization in a body with a band of initially softened material occurs at the same stress at which catastrophic softening occurs in the identical body without a softened band. Strain localization was found to depend on the overall mechanical state of the material, and not on, say, the initial angle of inclination of the shear band or other details of the kinematics of localized deformation. Though localization is probably initiated at a softened zone such as depicted in Fig. 10, no quantitative analysis of this process is yet available. In any event, one might argue that in order for a softened zone to propagate, the material which the propagating zone is entering must be receptive to the severe change in deformation pattern. So, besides the need for an initiating zone, we suggest, by analogy with the conclusion reached in the studies of strain localization, that the overall mechanical condition also plays a role. These considerations motivate the criterion for localized flow adopted here: localization occurs when the overall (macroscopic) stress ayZ attains a critical value. This value is chosen to be the tensile stress at which inhomogeneous flow is typically observed to occur. These criteria for brittle and ductile behavior are applied to the numerical results in Figs 8 and 9. If the stress at infinity exceeds the localization stress, here taken to be 0.025 G, before the notch stress reaches the cleavage stress, taken to be 0.15 G, then the hypothetical sample crosses the inhomogeneous flow boundary. Such a sample is ductile. A sample crossing the cleavage boundary would be brittle according to this criterion. Consider the curve labeled r, = 0 in Fig. 9: it corresponds to the case of instantaneous free volume response to stress. If the curve labeled r, = 0 is extended (with its slope not increasing), one sees that the material it represents is ductile. As z, increases (the values in the plot are dimensionless), the softening occurs at increasing values of notch stress. Although the numerical solutions were only carried as far as shown, curves for much smaller values of z, peaked and returned to follow the curve r, = 0. In any event, the peak shown is the maximum value the notch stress reaches. [Occasionally, a material element adjacent to the notch attains a slightly higher stress (by 1 or 2%) than does the notch.] Thus, as long as the peak is below the cleavage stress, the material is ductile. If the peak exceeds the cleavage stress, as does the curve labeled 7, = 10’3, the material is brittle. Similarly, one may interpret decreasing the initial free volume in Fig. 9 as increasing the likelihood of cleavage. The results displayed in Figs 8 and 9 are compiled in a useful form in Fig. 11. Consider a hypothetical sample with initial viscosity qinitia, and response time r,, containing a flaw with elastic stress concentration factor C,. A sample deformed at a strain-rate E?~, such that the curve labeled C,E~~ lies above the point (log z,, log ninitial), is predicted to be ductile. If the curve labeled C,E~~ lies below this point, the sample is brittle. (All quantities in Fig. 11 are dimensional, with the following choices of parameters: AC” = 0.2 ev, 0 = 5 x 10mz9 m3, v = lOi s-l, au*/0 = 0.6, IC= 9.87 x 10” N m-‘, T = 300°K.)

P. S.

g 18

STEIF

Hypothetical annealing

;-" ," 17 4

0

2.0

Log IO FIG. 1 I. Regions ofductile

and brittle fracture

6.0

4.0

in parameter time.

8.0

T” (5) space of initial viscosity and free volume response

The curves in Fig. 11 correspond to setting the peak stress a?* = 4GE,“;t,, where t, is given implicitly by (4.7), equal to the cleavage stress crcl (taken to be 0.15 G). With this value for the peak stress, (4.7) may be rearranged in the more convenient form, gel

% = (a&)-lan(L?C&)

A

2 + lc c1-e-

a‘Ai2A

1,

(5.1)

where A = 2GC&z,. A is readily interpreted as the hydrostatic tension achieved in time z, at the notch of an elastic material. The value of A relative to the cleavage stress (or some elastic modulus) determines the rate at which softening occurs. A --f 0 corresponds to the instantaneous free volume response. The initial free volume above which the localization criterion is met before the cleavage criterion is calculated from (5.1). Since the stress is zero before deformation begins, the corresponding value of initial viscosity is calculated from (2.7), assuming the stress is vanishingly small. Also, note that C,& appears only as a product. For A -+ 0, and sensible values of other parameters, u. calculated from (5.1) is less than zero. This should be interpreted as follows: if the free volume responds instantaneously, it cannot be initially reduced to a low enough value to embrittle. Actually, for the case of z, = 0, the stress continues to increase after softening (see section 4.1). Hence, the initial free volume need not be reduced quite as low as predicted by (5.1) to embrittle. From the approximate solution for the post-softening notch stress, equation (4.3, one can calculate 0,” needed for cleavage, and, hence u0 from (4.2). (Now C, and E?~ appear separately.) The result remains, however, that, for the case of TV= 0,

Ductile vs brittle behavior of amorphous metals

the initial viscosities needed exclusively for calculating the as a product. Schematically depicted in material is initially at the start would be ductile if deformed volume, as well as the response anneal, that sample would be

5.2 Interpretation

385

to embrittle are inaccessible. This justifies using (5.1) curves in Fig. 11, and the appearance of C, and ~7~ only Fig. 11 is the embrittlement with annealing. of the hypothetical annealing path shown : the at a rate corresponding to C& = 10m4 s-l. time, are expected to increase during annealing. brittle if deformed at the same rate.

Say the material The free After the

of 2,

As discussed in section 2.3, sluggish atomic movement at ordinary temperatures prohibits amorphous metals from remaining in equilibrium when cooled from a melt. This observation motivated the idea that the system cannot respond immediately to an applied hydrostatic tension. To model this delay, a first order equation governing the free volume response to stress was introduced. The equation has a single time constant, on the softening at a notch was 7 and the effect of varying that time constant mbestigated. It is our conjecture that differing values of r, (free volume response rate) underlie differing mechanical behaviors. For this to be true, it is seen from Fig. 11 that z, ought to lie typically in the range of 10 s < r, < lo5 s. Values of z, much less than 10 s are excluded if we insist that embrittlement occurs with sufficient annealing. (Some materials, however, seem to remain ductile after very long anneals ; values of r, < 10 s might be attributed to these materials.) If a typical range for z, were much greater than lo5 s, then alterations in 7” would have no effect. Since it is our contention that different values of z, explain different mechanical behaviors, the range of T, >> 10’ s is excluded. With this range of values for ry, a few tentative remarks are made here concerning the kinetic process underlying the free volume response to stress changes. As mentioned in section 2.3, when an amorphous metal is held at a constant temperature below the glass transition temperature the sample relaxes slowly towards the equilibrium structure. Changes in the structure manifest themselves in changes in many physical properties, particularly in those involving atomic transport. Since measurements of the rate of change of viscosity during structural relaxation have been made (TAUB and SPAEPEN, 1980; TSAO and SPAEPEN, 1982), it would be convenient if the kinetics of structural relaxation were the same as those underlying the free volume response to stress changes. However, since one must wait days or weeks for the viscosity to change by one or two orders of magnitude, one is inclined to view structural relaxation as occurring on a time scale long compared with events in the tensile test, for which processes involving changes of viscosity of several orders of magnitude occur in a few minutes. An approximate comparison of the kinetics of the two processes given by STEIF (1982) also suggests strongly that the kinetics of structural relaxation are unrelated to the process of free volume response to stress. More likely, it is the kinetics associated with altering the free volume distribution that are relevant to the process we attempt to describe here, and not changes in the average free volume (such as occur in structural relaxation). Again, with reference to remarks made in section 2, a non-equilibrium distribution of local free volume is a possible source of disequilibrium. In fact, by comparing model calculations with

386

P. S. STEIF

experimentally determined iso-configurational curves, TSAO and SPAEPEN (1982) concluded that the free volume is only partially redistributed during isoconfigurational cooling. In practice, cooling is carried out by lowering the temperature abruptly from one value to another, and the iso-configurational viscosity at the new temperature is taken to be the steady value of the viscosity after initial transients have died out. Though one can attribute the transient period to the experimental apparatus, this period may be interpreted as the time required for the partial redistribution of free volume to occur. The transient period is generally several minutes in duration (SPAEPEN,private communication) and thus within the acceptable range of values for z,. Perhaps a more careful examination of the observed transients, or of isoconfigurational measurements in general, may shed light on the atomic movements responsible for the free volume response to stress.

ACKNOWLEDGEMENTS The author wishes to thank Professors J. W. Hutchinson and F. Spaepen for valuable discussions during the course of this work. Several comments by Professor B. Budiansky have also proved helpful. This work was carried out while the author was a National Science Foundation Pre-Doctoral Fellow, and was supported in part by the National Science Foundation under Grants CME-78-10756 and DMR-80-20247, and by the Division of Applied Sciences, Harvard University.

REFERENCES 1976 ARGON, A. S. and SALAMA,M. BUDIANSKY,B., HUTCHINSON,J. W. 1982 and SLUTSKY,S.

Mat. Sci. Eng. 23, 219. Mechanics of Solids, The Rodney Hill 60th Annioersary Volume (edited by H. G. Hopkins and M. J. Sewell) p. 13. Pergamon Oxford.

CHEN, H. S. and POLK, D. E. COHEN, M. H. and TURNBULL,D. DAVIS, L. A.

1974 1959 1978

GLASSTONE,S., LAIDLER,K. J. and EYRING, H. HE, M. Y. and HUTCHINSON,J. W.

1941

Press,

J. non-tryst. Solids 15, 174. J. them. Phys. 31, 1164. Metallic Glasses (edited by J. J. Gilman

and H. J. Leamy) p. 190. ASM, Metals Park, Ohio. The Theory of Rate Processes, p. 480. McGrawHill, New York.

1981 1979

J. appl. Mech. 48, 830. Appl. Phys. Lett. 35, 815.

LUBORSKY,F. E. and WALTER, J. L. SPAEPEN,F.

1976

J. appl. Phys. 47, 3648.

1975 1977 1982

Acta. metall. 23, 615. Acta. metall. 25, 407. Physics ojDej&cts (edited by R. Balian et al.) p. 136.

SPAEPEN,F. and TURNBULL,D. STEIF,P. S. STEIF,P. S., SPAEPEN,F. and HUTCHINSON,J. W. TAUB, A. I. TAUB, A. I. and SPAEPEN,F.

1974 1982 1982

Ser. metall. 8, 563.

1980 1980

Acta metall. 28,633. Acta metall. 28, I78 1.

KRAMER, F. A., JOHNSON,W. L. and CLINE, C.

North-Holland,

Amsterdam.

Ph.D. Thesis, Harvard Acta metall. 30,447.

University.

Ductile vs brittle behavior of amorphous metals TSAO, S. S. and SPAEPEN, F.

1982

TURNBULL, D. and COHEN, M. H.

1961 1970

387

4th Jnt. Conf. Rapidly Quenched Metals (edited by T. Masumoto and K. Suzuki) p. 463. Jap. Inst. Metals, Sendai. J. &em. Phys. 34,120. J. them. Phys. 52,3038.

Proc.

Spatial derivatives of mapped variables The mapping

and an additional

function

is

function

is defined where ( )’ denotes differentiation with respect to the complex variable <. With z = x -t- iy and [ = p e’@,the required derivatives are given by

where Re( ) denotes

the real part of the indicated

complex

quantity.

APPENDIX II Accuracy of numerical technique A few brief remarks concerning the numerical technique are made here; more details can be found in STEIF (1982). The integrals in (3.29) and (3.30) were evaluated numerically using tenpoint Gaussian quadrature (in each of the radial and azimuthal directions). The stress-rates were integrated with time using a first-order Euler method, for which the time step was chosen so that the creep strain increment nowhere exceeded a chosen value. Since pointwise quantities such as stress were required here, and not just a global quantity [such as the value of the J-integral found by HE and HUTCHINSON (1981) with a similar technique], comparison with an exact solution was almost essential in assessing the accuracy of the numerical technique. For the elastic case, the exact and numerical solutions along the x-axis agreed to within 1 or 2% when we took J = 5 and K = 4, that is, 5 functions in the radial direction x 4 functions in the azimuthal direction. Comparison with the exact solution for the

3X8

P. S. STEIF

linear visco-elastic case (which tests the time integration and the calculation of the b,) revealed similar accuracy. It was found that a least squares fit of the pressure (with fixed ends) led to higher accuracy, at least for the elastic case, than actually fixing the values of the interior stations. Perhaps this is because a least squares fit averages out the errors in the stress deviator and its spatial derivatives, and fixing the values would retain these errors. We took the curve

P(P)= P(O)+rP(l)-P(O)lP+cIP(P--1)+c2P2(~--1)+~,~2(~--1)2+ .” to approximate the pressure p on the interval 0 < p < 1, with the pressure taking on fixed values d0) and p(l) at the end points. The coefficients ci (i = 1,2,. . .) were determined by the last squares tit. Polynomials of order 4 and 5 gave pressure distributions that agreed with the exact elastic solution to within a few percent.