Acta Materialia 55 (2007) 3735–3747

www.elsevier.com/locate/actamat

A constitutive theory for metallic glasses at high

homologous temperatures
Lallit Anand *, Cheng Su
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 15 August 2006; received in revised form 27 December 2006; accepted 18 February 2007
Available online 6 April 2007

Abstract

The elastic–viscoplastic constitutive theory of Anand and Su [Anand L, Su C. J Mech Phys Solids 2005;53:1362] for metallic glasses
has been extended to the high homologous temperature regime. The constitutive equations appearing in the theory have been specialized
to model the response of metallic glasses in the temperature range 0.7#g ~ # ~ #g and strain rate range [105, 102] s1. The material
parameters appearing in the theory have been estimated for the metallic glass Pd40Ni40P20 from the experimental data of De Hey et al.
[De Hey P, Sietsma J, Van Den Beukel A. Acta Mater 1998;46:5873]. The model is shown to capture the major features of the stress–
strain response, and the evolution of an order-parameter for this metallic glass. In particular, the phenomena of stress overshoot and
strain softening in monotonic experiments at a given strain rate and temperature, as well as strain rate history effects in experiments
involving strain rate increments and decrements, are shown to be nicely reproduced by the model.
 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Amorphous metals; Bulk metallic glasses; Constitutive modeling; Finite element method

1. Introduction mechanical properties of metallic glasses, and references to

the pertinent literature, see the articles in a recent viewpoint
Under slow-to-moderate cooling rates most metallic set in Scripta Materialia [5].
materials solidify in a polycrystalline form. However, When a metallic glass is deformed at ambient tempera-
under high cooling rates certain metallic alloys solidify in tures, well below its glass transition temperature, its inelas-
a disordered form; such disordered metals are referred to tic response is almost rate-independent and characterized
as amorphous metals or metallic glasses. The first genera- by strong strain softening, which results in the formation
tion of amorphous metallic glasses were developed in thin of intense, localized shear bands. Fracture typically occurs
ribbon form using very high cooling rates (105– after very small inelastic strain in tension, but substantial
106 K s1) [2], but more recently it has been discovered that inelastic strain levels can be achieved under states of con-
metallic glasses can be processed in bulk form at relatively fined compression, such as in indentation experiments
slow cooling rates (1100 K s1) in certain multicompo- [6,7,9,10]. However, when a metallic glass is deformed at
nent alloy systems due to the sluggish crystallization kinet- an absolute temperature # in the range 0.7#g ~ # ~ #x,
ics in these alloys [3,4]. The lack of long-range order makes where #g and #x are the glass transition and crystallization
the mechanical behavior of amorphous metals considerably temperatures, respectively, then its inelastic response is
different from that of crystalline metals. For a survey of the highly rate-dependent and it deforms approximately
‘‘homogeneously’’ [6], and indeed in the supercooled liquid
range #g ~ # ~ #x many metallic glasses are known to
*
Corresponding author. Tel.: +1 617 2531635; fax: +1 617 2588742. show superplastic behavior at sufficiently slow strain rates
E-mail address: anand@mit.edu (L. Anand). [11].

1359-6454/$30.00  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.actamat.2007.02.020
3736 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747

The micro-mechanisms of inelastic deformation in bulk Some important experimentally observed transient fea-
metallic glasses are not related to dislocation-based mech- tures of the tensile or compressive true stress–strain response
anisms that characterize the plastic deformation of crystal- of metallic glasses in the strain rate regime 104 to 102 s1 at
line metals. The plastic deformation of amorphous metallic high temperatures are: (i) the phenomenon of stress over-
glasses is fundamentally different from that in crystalline shoot and strain softening (which is a function of pre-anneal-
solids because of the lack of long-range order in the atomic ing history) in monotonic experiments at a given strain rate;
structure of these materials. The underlying atomistic and (ii) strain-rate history effects in experiments involving
mechanisms of the inelastic deformation of metallic glasses strain-rate increments and decrements. In this regard, the
have been under intense investigation for the past three experimental study conducted by De Hey et al. [13] is partic-
decades, and are still not completely understood (e.g. [6– ularly noteworthy. Their study neatly demonstrates such
10,12–19]). However, atomistic simulations reported in behaviors in the amorphous metal Pd40Ni40P20 (#g  578–
the literature (e.g. [8,17–19]) show that at a micromechan- 597 K). De Hey et al. used various annealing histories prior
ical level, inelastic deformation in metallic glasses occurs by to conducting tensile experiments at various strain rates
local shearing of clusters of atoms (10–30 atoms), this and temperatures, and also conducted attendant differential
shearing is accompanied by deformation-induced scanning calorimetry (DSC) experiments to study the effects
microstructural disordering and inelastic dilatation that of disordering of the material during deformation. Some of
produces strain softening, which at low homologous tem- their major findings are as follows:
peratures leads to the formation of intense shear bands.
The importance of dilatancy in the inelastic response and  For specimens pre-annealed at 564 K for 5000 s (which is
shear localization of soils and other granular materials, long enough to bring the material into its equilibrium
which consist of randomly packed grains, goes back to metastable state at this temperature), tensile stress–strain
Reynolds in 1885 [20], who applied the term ‘‘dilatancy’’ curves from experiments conducted at 564 K and strain
to the property possessed by a mass of granular material rates of (0.083, 0.17, 0.42, 0.83) · 103 s1, Fig. 1a,
to alter its volume in accordance with a change in the exhibited a large amount of stress overshoot and strain
arrangement of its grains. While the density changes in softening, after which the flow stress leveled off to a ‘‘pla-
shear bands in soils are large enough to be experimentally teau’’ value at about 15–20% strain; both the magnitude
measurable, those in the shear bands in metallic glasses are of the strain softening and the plateau value of the flow
usually quite small, ~0.5%, and difficult to measure exper- stress increase with increasing strain rate.
imentally [21], and even difficult to discern in atomistic sim-  For specimens pre-annealed at 556 K for 120, 720 and
ulations [18]. 10,000 s (the two shorter times are not long enough to
Deformation-induced microstructural disordering in a bring the material into its equilibrium metastable state
metallic glass also occurs in the high-temperature range at this temperature), tensile stress–strain experiments
0.7#g ~ # ~ #g. A macroscopic manifestation of the struc- conducted at 556 K and a strain rate of 1.7 · 104 s1,
tural disordering in this temperature range is that in strain- Fig. 1b, show markedly different characteristics in the
controlled isothermal compression or tension experiments manner in which the flow stress approaches its plateau
at constant strain-rate, the microstructural disordering value at this strain rate and temperature. As in
leads to strain softening; however, because of the high Fig. 1a, the pre-equilibriated sample annealed for
strain-rate sensitivity of the material at elevated tempera- 10,000 s exhibited a large amount of strain softening.
tures, it does not exhibit macroscopic localized shear bands The strain-softening effect was less pronounced in the
and the deformation appears as nominally homogeneous. specimen pre-annealed for 720 s, while the sample with
Experimental examples of such macroscopic responses the shortest pre-annealing time of 120 s showed strain
may be found in [13] for the amorphous metal Pd40Ni40P20, hardening instead of strain softening in its approach to
and in [14] for the commercial Zr-based alloy Viterloy-1. the plateau level of the flow stress. The plateau value
More recently, Heggen et al. [15] have reported on com- of the flow stress itself was essentially independent of
pression creep experiments under constant stress to study the pre-annealing time.
the macroscopic manifestations of the microstructural dis-  For specimens pre-annealed at 556 K for 3600 s, tensile
ordering process in Pd41Ni10Cu29P20; they report accelerat- stress–strain curves from strain-rate increment and
ing creep strain-rates due to the attendant softening of the decrement experiments conducted at 556 K and axial
material. The constant stress creep experiments of Heggen strain rates of 8.3 · 105 ! 4.2 · 104 ! 8.3 · 105 s1,
et al. [15] result in steady-state strain rates ranging from Fig. 1c, showed a pronounced strain-rate history effect
108 to 104 s1; cf. their Figs. 1–3 and 5. We note that with strong overshoot and undershoot relative to the
the transient aspects of inelastic deformation, structural monotonic experiments conducted at 4.2 · 104 s1
disordering and strain softening are not as well revealed and 8.3 · 105 s1.
in the constant stress creep experiments of Heggen et al.
[15], as they are in the constant strain rate compression Based on these experimental observations and their esti-
experiments of De Hey et al. [13] and Lu et al. [14] in the mates of the changes in the free volume from their DSC
strain rate range 104 to 102 s1 . measurements, De Hey et al. [13] concluded that:
 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747 3737

 The material disorders during deformation, and that at  The strain-softening phenomenon is directly related to
a given temperature and strain rate the free volume (a the disordering of the material, and that the decrease
measure of the degree of disorder) reaches an equilib- in the flow stress during a tensile test is related to the cre-
rium value that is different from and independent of ation of additional free volume during the deformation.
its thermal equilibrium volume prior to deformation.
Similar conclusions from experimental observations on
a 700
other metallic glasses have also been recently reported in
the literature (e.g. [16]). A recent detailed atomistic simu-
-3 -1
600 Strain rate (10 s ) lation study of the plasticity of amorphous silicon by
0.83
0.42 Argon and Demkowicz [17] also leads to qualitatively sim-
500 0.17 ilar conclusions for this covalently bonded material. Note,
0.083
Stress, MPa

however, that these authors phrase their discussion in

400
terms of a ‘‘liquid-like mass fraction’’ rather than the free
volume. Other recent atomistic studies in the literature
300
also indicate that the notion of ‘‘free volume’’ is not a
200 direct measure of atomic rearrangements that influence
the inelastic deformation of metallic glasses [18,19]. Never-
100 theless, in this paper we shall continue to use the term
564K ‘‘free volume’’ for a microstructural ‘‘order-parameter,’’
0
0 0.05 0.1 0.15 0.2 which will be an internal variable of our continuum
Strain theory.
The atomistic studies available in the literature (e.g.
b 600
[8,12,17–19]) provide valuable insight into the micromech-
10,000s anisms of inelastic deformation in amorphous metallic
500
materials, but by themselves do not provide specific forms
for continuum-level constitutive equations which can faith-
400
Stress, MPa

720s fully reproduce the experimentally measured temperature

and strain-rate-dependent stress–strain response of these
300
materials. A continuum-level one-dimensional model
120s which has long been used to represent the inelastic response
200
of metallic glasses is the free volume model proposed by
Spaepen [6], and it was this model that was used by De
100
556K , 1. 7. 10 -4 s -1 Hey et al. [13] to analyze their experimental results. Briefly,
let j _ p j denote the magnitude of the plastic strain rate in a
0
0 0.05 0.1 0.15 0.2
one-dimensional setting. In such a setting, the flow equa-
Strain tion proposed by Spaepen has the form
 
c 550
p  0 t0 jrj0 t0
556K j_ j ¼ 2cf k f sinh : ð1Þ
500 X 2k B #
450
Here jrj is the absolute value of the stress, 0 is a local
400
transformation strain, t0 is an activation volume, X is an
Stress, MPa

350
atomic volume, kf is a temperature-dependent rate factor,
300 kB is Boltzmann’s constant, # is the absolute temperature,
250 and
   
200 1 cv
150
cf ¼ exp   exp  ð2Þ
x vf
100
8.3 . 10 -5 s -1 4.2 . 10 -4 s -1 8.3 . 10 -5 s -1 is the concentration of flow defects defined in terms of a
50
normalized free-volume parameter x = vf/(cv*), where vf is
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 the average free volume per atom, v* is a critical value of
Strain the free volume, and c is a geometrical overlap factor with
Fig. 1. True stress–strain curves for Pd40Ni40P20 from Ref. [13]: (a) pre-
a value between 0.5 and 1. In [13], the evolution of the de-
annealed at 564 K for 5000 s, then tested at 564 K at the different strain fect concentration is taken as
rates indicated in the figure; (b) pre-annealed at 556 K for 120, 720, and
10,000 s, respectively, then tested at 556 K at a strain rate of 2
c_ f ¼ ðax cf ðln cf Þ Þj_p j  k r cf ðcf  cf ;eq Þ ; ð3Þ
_ ¼ 1:7  104 s1; (c) pre-annealed at 556 K for 3600 s, and then |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
subjected to a strain-rate increment and decrement experiment at 556 K. dynamic defect creation static thermal recovery
3738 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747

with ax a temperature-dependent parameter, kr a tempera- sonable values of the material parameters, cf. [13]) is
ture-dependent rate factor, and tens of hours. While one cannot dispute the importance
  of the static thermal recovery krcf(cf  cf,eq) term, espe-
1
cf ;eq ¼ exp  ; ð4Þ cially for long-term creep tests such as those considered
xeq
by Heggen et al. [15] which last for times measured in
where days, in order to match the experimental stress–strain
#  #0 data such as that shown in Fig. 1, which are completed
xeq ¼ ð5Þ in matters of minutes, the dynamic defect creation term
B
in (3) needs to be modified to allow for dynamic recov-
is the value of the free volume in thermal equilibrium at a ery. The hardening-thermal recovery format of the evo-
temperature #; here #0 and B are two material constants lution equation (3) goes back to the classical Bailey [22]–
known as the Vogel–Fulcher–Tamann (VFT) parameters. Orowan [23] form used in modeling creep of crystalline
A study of the paper by De Hey et al. [13] reveals that metals. However, as shown by Brown et al. [24] (and
the flow equation (1) and the evolution equation (3), when others), a static recovery function alone is insufficient
suitably calibrated, are not able to produce stress–strain to describe the stress–strain/strain-rate history response
curves that match their corresponding experimental of crystalline metals at high homologous temperatures;
stress–strain data which show the phenomena of stress inclusion of a dynamic recovery function is essential.
overshoot and strain softening in monotonic experiments While the underlying micromechanisms of inelastic
at a given strain rate, and strain-rate history effects in deformation in dislocation-mediated plasticity of crys-
experiments involving strain-rate increments and talline metals and the atomic disorder-mediated plastic-
decrements.1 ity of amorphous metals are of course quite different, it
In our opinion, even though the widely used flow and is our opinion that at the macroscopic mathematical-
evolution Equations (1) and (3) are physically reasonably modeling level, the evolution equation for the defect
well motivated, they leave out some important coupling concentration needs to be modified to include the effects
effects. Specifically: of dynamic recovery; we base this belief on our substan-
tial previous experience with modeling the strain-soften-
 In the flow equation (1) the ‘‘effective activation vol- ing stress–strain response of granular materials [26],
ume’’ 0t0 is assumed to be a constant. In our opinion amorphous polymeric materials[27] and amorphous
this should also be a function of the local state of the metals [1] at low homologous temperatures.
material at any given instant, especially during the tran-
sients associated with structural disordering. Although In two recent papers [1,25], we developed a continuum-
the flow equation employs the concentration of flow level constitutive theory aimed at modeling the room-tem-
defects cf as an internal variable, there is no direct cou- perature response of metallic glasses. The purpose of this
pling between this term and the ‘‘effective activation vol- paper is to present a development which extends our theory
ume’’ 0t0. to high homologous temperatures. The special constitutive
 In the evolution equation (3) (or its variant (6)), the functions in our theory, although similar in spirit to the
defect concentration cf continues to increase as long as free-volume theory of Spaepen [6] considered by De Hey
there is plastic flow j_p j > 0; this increase is balanced et al. [13], are quite different in detail. Some specific major
only by the static thermal recovery term krcf (cf  cf,eq). differences are itemized below:
Consider the monotonic compression experiments of
De Hey et al. [13] at 564 K and strain rates in the range  In contrast to assuming a constant ‘‘effective activation
0.83–0.083 · 103 s1 shown in Fig. 1a. In such experi- volume’’ 0t0 in the flow equation, we introduce an inter-
ments one would expect that defect concentration cf nal variable s which we call the slip resistance.
does not substantially change after a strain of 0.2 when  There is another internal variable of our theory, an
the stress–strain curves achieve a fully developed state in ‘‘order-parameter,’’ denoted by g, which is loosely
the plateau region. The time required to reach a strain of equivalent to the reduced free volume parameter x in
0.2 at these strain rates is only 4-40 min, but the time [13]; cf. Eq. (2).
constant in the static thermal recovery term (with rea-  The evolution equation for the slip resistance s is cou-
pled with the evolution equation for g; cf. Eq. (28).
 In contrast to the evolution equation (3) for the defect
1
A similar comment applies to the paper by Heggen et al. [15], who in concentration, in our evolution equation for g there is
analyzing their creep data have considered (3), as well as a similar
allowance for dynamic recovery; cf. Eqs. (28)–(30).
expression for the rate of defect creation:
 Finally, ours is a fully three-dimensional, thermodynam-
c_ f jþ ¼ ðax cf ðln cf Þ2 Þjrjj_p j; ð6Þ ically consistent, finite-deformation continuum theory,
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
dynamic defect creation ready for implementation in finite element programs to
based on the rate of plastic work j r jj _ p j, rather than the plastic strain solve technologically important boundary-value
rate j _ p j, as in (3). problems.
 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747 3739

Our paper is organized as follows. The new constitutive g. However, the effects of including such a depen-
theory is outlined in Section 2; this theory is fairly general, dence on (G,K) are expected to be small, and for sim-
so in Section 3, with the aim of modeling the experimental plicity we neglect such a dependence of w on g here
observations of [13], we specialize the theory for applica- (cf. [1] for a theory including such a dependence).
tion in the temperature range 0.7#g ~ # ~ #g and strain- (2) Equation for the stress:
rate range [105, 102] s1 of interest. In Section 4 we use ow
experimental data for Pd40Ni40P20 from [13] to estimate Te ¼ ¼ 2GEe0 þ KðtrEe Þ1: ð8Þ
oEe
the material parameters appearing in our specialized con-
This symmetric stress tensor has the spectral
stitutive equations, and using these material parameters
representation
we show how well the numerically calculated stress–strain
curves compare against the corresponding experimental X
3

results. We have implemented our constitutive model in Te ¼ ri^ei  ^ei ; ð9Þ

i¼1
the finite element program ABAQUS/Explicit [28], and as
a simple but important representative example of our where {riji = 1, 2, 3} are the principal values, and
numerical simulation capability, in Section 5 we report f^ei j i ¼ 1; 2; 3g the corresponding orthonormal prin-
on a finite-element simulation of a plane-strain tension test. cipal directions. We assume that the principal stresses
We close in Section 6 with some final remarks. {riji = 1, 2, 3} are strictly ordered such that
r1 P r2 P r3 : ð10Þ
2. Constitutive model
(3) Flow rule: We assume that plastic flow occurs by
We limit our considerations to isothermal situations at a shearing accompanied by dilatation relative to some
fixed temperature in the absence of temperature gradients. ‘‘slip systems’’, and take the evolution equation for
Our constitutive equations relate the following basic fields2: Fp to be given by
x = v(X,t), motion; F = $v with J = detF > 0, deformation F_ p ¼ Lp Fp ; Fp ðX; 0Þ ¼ 1; ð11Þ
gradient; F = FeFp, multiplicative elastic–plastic decompo-
sition; Fp with Jp = detFp > 0, plastic distortion; Fe with with
X
Je = detFe > 0, elasticPdistortion; Fe = ReUe, polar decom- Lp ¼ mðaÞ sðaÞ  mðaÞ þ dðaÞ mðaÞ  mðaÞ ; and dðaÞ ¼ bmðaÞ :
e 3
position of FP ; Ue ¼ a¼1 kea ra  ra , spectral decomposition a
e e 3 e
of U ; E ¼ a¼1 ðln ka Þra  ra , logarithmic elastic strain; T, ð12Þ
Cauchy stress; Te = JeReTTRe, stress conjugate to elastic
strain Ee; w, free energy density per unit volume of interme- Each slip system is specified by a slip direction sa and
diate space; # > 0, absolute temperature; and g > 0, an a slip plane normal ma, with (sa,ma) orthonormal,
‘‘order-parameter’’.3 with ma denoting the shearing rate.4 The dilatation
The set of constitutive equations is summarized below: rate associated with shearing on each slip system is
d(a) = bm(a), with b a shear-induced plastic dilatancy
(1) Free energy under isothermal conditions at a tempera- function; positive values of b describe plastically
ture #: dilatant behavior, while b < 0 describes behavior that
is plastically compacting. For an amorphous isotro-
2 1 2 pic material there are no preferred directions other
w ¼ GjEe0 j þ KðtrEe Þ ; with
2 than the principal directions of stress; accordingly
Gð#Þ > 0 and Kð#Þ > 0; ð7Þ we consider plastic flow to be possible on the follow-
ing six potential slip systems defined relative to the
where G and K denote the temperature-dependent
principal directions of stress Te:
shear and bulk moduli, respectively. In general, we
expect that the free energy, and hence also the elastic 9
sð1Þ ¼ cos n^e1 þ sin n^e3 ; mð1Þ ¼ sin n^e1  cos n^e3 ; >
constants (G,K), will depend on the order-parameter >
>
sð2Þ ¼ cos n^e1  sin n^e3 ; mð2Þ ¼ sin n^e1 þ cos n^e3 ; >
>
>
>
>
2
s ¼ cos n^e þ sin n^e ; m ¼ sin n^e  cos n^e ; =
ð3Þ
1
ð3Þ
2 1 2
Notation: We use the standard notation of modern continuum
mechanics. The symbols $ and Div denote the gradient and divergence s ¼ cos n^e1  sin n^e2 ; m ¼ sin n^e1 þ cos n^e2 ; >
ð4Þ ð4Þ
>
>
>
with respect to the material point X in the reference configuration; grad sð5Þ ¼ cos n^e2 þ sin n^e3 ; mð5Þ ¼ sin n^e2  cos n^e3 ; >
>
>
>
and div denote these operators with respect to the point x = v(X, t) in the ð6Þ ð6Þ
;
deformed configuration; a superposed dot denotes the material time- s ¼ cos n^e2  sin n^e3 ; m ¼ sin n^e2 þ cos n^e3 ;
derivative. Throughout, we write Fe1 = (Fe)1, FpT = (Fp)T, etc. We ð13Þ
write symA, skwA, A0 and sym0A, respectively, for the symmetric, skew,
deviatoric and symmetric-deviatoric parts of a tensor A. Also, the inner with
product of tensors A and B is denoted by A:B, and the magnitude of A by
pffiffiffiffiffiffiffiffiffiffiffi
jAj ¼ A : A.
3
Caution: g denotes neither the entropy density, nor a viscosity
4
parameter. The shearing rate is often denoted as c_ ðaÞ in the literature.
3740 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747

p /
def where
n¼ þ ; ð14Þ
4 2 def
X
where m¼ mðaÞ ð24Þ
a
def
/ ¼ arctan l ð15Þ is the sum of the shearing rate on all the slip systems.
is an angle of internal friction, and l P 0 an internal
friction coefficient. We emphasize that these slip sys-
tems are not the classical slip systems of crystal plastic- 3. Specialization of the constitutive equations—application
ity, but are constructs of our mathematical model for to the metallic glass Pd40Ni40P20
isotropic amorphous materials; they are related to
the principal directions of the stress, and they change The constitutive theory outlined in the previous section
both spatially and temporally as the principal direc- is fairly general. With the aim of modeling the experimental
tions of stress change in a non-homogeneously deform- observations of [13], we specialize the scalar constitutive
ing material. With the resolved shear and compressive functions (17), (18) and (23) for application in the temper-
normal traction on each slip system defined by ature range 0.7#g ~ # ~ #g and strain-rate range
def [105,102]s1 of interest.
sðaÞ ¼ sðaÞ Te mðaÞ ; rðaÞ def mðaÞ Te mðaÞ ; ð16Þ
¼

the corresponding shearing rate is given by a flow 3.1. Scalar shearing rate m(a)
function
The shearing rate on each slip system is taken in a simple
mðaÞ ¼ ^mðaÞ ðsðaÞ ; rðaÞ ; #; s; l; gÞ P 0; ð17Þ thermally activated power-law form:
where s > 0 is a stress-dimensioned internal variable   1=m
ðaÞ Q sðaÞ
representing the slip resistance, assumed for an iso- m ¼ m0 exp  P 0; ð25Þ
k B # s þ lrðaÞ
tropic material to be the same for all slip systems. It
is convenient to write K for the list of variables where m0 a reference shear strain rate, Q is an activation en-
K ¼ ðTe ; #; s; l; gÞ: ergy, kB Boltzmann’s constant and m ¼ mð#Þ ^ > 0 is a tem-
perature-dependent strain-rate sensitivity parameter.5
Using this notation, we assume that the dilatancy For m(a) > 0, the flow Eq. (25) may be inverted to read
parameter b depends on K,  ðaÞ   m
^
b ¼ bðKÞ: ð18Þ ðaÞ ðaÞ m Q
s ¼ ðs þ lr Þ exp ; ð26Þ
m0 kB#
The dissipation inequality in the theory is
which shows that the term lr(a) accounts for the pressure
Te : Lp > 0 for Lp 6¼ 0: ð19Þ
sensitivity of plastic flow. Also, the limits m ! 1 and
With Lp given by (12) and (13), the dissipation m ! 0 correspond to the linearly viscous and rate-indepen-
inequality requires that dent limits, respectively.
X6
Te : Lp ¼ ½sðaÞ  brðaÞ
mðaÞ > 0 ð20Þ 3.2. Evolution equations for l, s and g; dilatancy function b
a¼1

whenever plastic flow occurs. We assume that the For the amorphous metallic materials under consider-
material is strongly dissipative in the sense that ation we take the internal friction l to be a constant,
½sðaÞ  brðaÞ
mðaÞ > 0 for each a: ð21Þ l ¼ l0 P 0; ð27Þ
Thus, whenever m(a) > 0, we must have
there is not enough experimental information to be more
½sðaÞ  brðaÞ
> 0; ð22Þ specific here.
For s and g we consider evolution equations in the fol-
which is a restriction that the dilatancy function lowing special coupled form:
^
b ¼ bðKÞ must satisfy. 9
(4) Evolution equations for the internal variables: The s_ ¼ hm
|{z}  rs ;>
>
|{z} >
=
internal variables of the theory are the internal fric- dynamic evolution static recovery
tion coefficient l P 0, the slip resistance s > 0 and ð28Þ
g_ ¼ bm  rg ;>
>
the order-parameter parameter g > 0. We assume that |{z} |{z} >
;
dynamic evolution static recovery
the evolution of these internal variables is given by
coupled differential equations: with
9
l_ ¼ f ðK; mÞ; >
=
s_ ¼ hðK; mÞ; ð23Þ 5
This simple power-law form is easily replaced by sinh-type, or other
>
; forms to match data over a wider range of strain rates, for which a power-
g_ ¼ gðK; mÞ;
law form may prove to be inadequate.
 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747 3741

)
h ¼ h0 ðs  sÞ; s ¼ ^s ðm; #; gÞ > 0; To summarize, we have considered specialized constitu-
 ð29Þ tive equations for the slip rates ma which involve the mate-
b ¼ g0 1  gg ; g ¼ ^g ðm; #Þ > 0;
rial parameters
also fm0 ; Q; m; l0 g;
rs ¼ ^rs ðs; g; #Þ; rg ¼ ^rg ðs; g; #Þ; ð30Þ
and the material parameters
and temperature-dependent initial values
sðX; 0Þ ¼ s0 ð#Þ; gðX; 0Þ ¼ g0 ð#Þ: ð31Þ fh0 ; ~s; n; b; g0 g:

In these evolution equations h represents the strain-harden- in the evolution equations for s and g, with the particular
ing/softening function for the slip resistance during plastic function g ¼ ^g ðm; #Þ and the attendant material parame-
flow, m > 0: the material hardens (h > 0) if s* > s, and soft- ters yet to be determined.
ens (h < 0) if s* < s. The critical value s* of s controlling
such hardening/softening transitions is assumed to depend
on the current values of the plastic strain rate, temperature 4. Estimates of material parameters for Pd40Ni40P20
and free volume. In the dilatancy function the parameter g*
represents a strain rate and temperature-dependent critical We have estimated the material parameters appearing in
value for the order-parameter: the material dilates (b > 0) our model from experimental data and results available in
when g < g*, and compacts (b < 0) when g > g*. In a mono- the literature for Pd40Ni40P20. The following specific values
tonic experiment at a given strain rate and temperature the for the material parameters were chosen:
shear-induced dilatancy vanishes (b = 0) when g = g*.
However, in an experiment in which the strain rate and  Elastic moduli: Davis et al. [29] reported values of
temperature are varying (e.g. strain-rate or temperature- E = 96 GPa and mPoisson = 0.36 for the room-tempera-
jump experiments), the material will in general dilate or ture values of Young’s modulus and Poisson’s ratio.
compact, depending on the strain-rate and temperature his- The corresponding values for the shear and bulk moduli
tory, and because of the coupling between the evolution are
equations for s and g the slip resistance will also vary.
The functions rs and rg represent static thermal recovery G ¼ 35:3 GPa and K ¼ 114:3 Gpa:
functions for the slip resistance and the free volume at a
For simplicity we assume that the change in the values
given temperature, whenever there is no macroscopic plas-
of the elastic moduli of amorphous metals for tempera-
tic flow (m = 0).
tures in the range from room temperature to #g is small,
The tension experiments by De Hey et al. [13] on
and use the values above for all temperatures below #g.
Pd40Ni40P20 that we shall consider in the next section were
 Friction coefficient: Donovan [10] quotes a value of
all performed at macroscopic strain rates greater than
l = 0.11 for Pd40Ni40P20 from his estimates of this
8 · 105s1 to strain levels of less than 25%. For this strain
parameter based on measured shear-band orientations
rate and strain regime, we assume that the timescale of the
in compression at room temperature. However, we have
static recovery processes is sufficiently slow so that effects
shown previously that shear-band orientations are con-
of static recovery on the evolution of s and g may be
trolled not only by the friction coefficient l, but also
neglected. Accordingly, we do not consider further specifi-
by the dilatancy parameter b, and estimates for the fric-
cations of the recovery functions rs and rg, and set them to
tion parameters from shear band orientations typically
zero for the application under consideration.6
yield abnormally high values [1,25]. For a Zr-based
As a particular form for the critical value s* of s in the
metallic glass we estimated a value
hardening function (29), we consider
  n l ¼ 0:04:
m Q
s ¼ ~s exp þ bðg  gÞ; ð32Þ We use this estimate also for the Pd-based glass. No
m0 kB#
high-temperature measurements for the pressure sensi-
with n ¼ ^
nð#Þ not necessarily equal to m ¼ mð#Þ,
^ but (m0,Q) tivity of plastic flow for this material appear to have
the same as in Eq. (25). been reported in the literature.
A simple analytical form for the dependency of the crit-  Viscoplasticity parameters:We have used the simple ten-
ical value g* on m and # is elusive. In the next section, sion stress–strain curves of [13], Fig. 1a, to calibrate the
guided by the experimental data of [13] for the metallic viscoplasticity parameter for Pd40Ni40P20. In simple ten-
glass Pd40Ni40P20, we will construct and curve-fit a simple sion the principal stresses are
empirical form for this function.
r1 > 0; r2 ¼ r3 ¼ 0: ð33Þ
Straightforward calculations using (13) and (16) show
6
Of course, for long-term creep experiments, such as those reported by that in this case the resolved shear stresses and compres-
Heggen et al. [15], the effects of static recovery must be included. sive normal tractions on the slip systems are given by
3742 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747

def 1
s ¼ sð1Þ ¼ sð2Þ ¼ sð3Þ ¼ sð4Þ ¼ sinð2nÞr1 ; sð5Þ ¼ sð6Þ ¼ 0; m ¼ n ¼ ð2:6375  103 Þ#  1:356:
2
ð34Þ With these estimates of Q, m and n, (40) and the data of
and [13] then allows us to estimate the following values for m0
and ~s :
rð1Þ ¼ rð2Þ ¼ rð3Þ ¼ rð4Þ ¼ sin2 nr1 ; rð5Þ ¼ rð6Þ ¼ 0: ð35Þ
Thus, (25) dictates that the shearing rate on the slip sys- m0 ¼ 2:47  1013 s1 ; ~s ¼ 500MPa:
tems must obey, The quality of the fit using (40) and these material
m ð1Þ
¼m ð2Þ
¼m ð3Þ
¼m ð4Þ
> 0; and m ð5Þ
¼m ð6Þ
¼ 0: ð36Þ parameters to the ðrss ; _ Þ data of [13] is shown in Fig. 2.
 Steady-state free volume as a function of strain rate and
For l = 0.04, the angle of internal friction is / = 0.04 rad, temperature: In order to estimate the function
and if we neglect the effects of this small value, it is easy to
verify that the non-zero resolved shear stresses and shear- g ¼ ^g ðm; #Þ
ing rate on the slip system may be approximated as
(cf. Eq. (30)), we use the data in Fig. 5 of [13]. Their data
ð1Þ ð2Þ ð3Þ ð4Þ 1 are expressed in terms of the defect concentration ratio
s¼s ¼s ¼s ¼s  r; and
2 (cf. Eq. (2))
1
mð1Þ ¼ mð2Þ ¼ mð3Þ ¼ mð4Þ  _ p ; ð37Þ cf ; expð1=g Þ
2 ¼ ; ð41Þ
cf ;eq expð1=geq Þ
where r > 0 and _ p > 0 are the axial stress and axial plastic
strain rate in a tension test. Thus, in a fully developed flow as a function of _ , where geq is the thermal equilibrium
state at an axial strain rate _ , when the axial stress reaches value of the free volume (cf. Eq. (5)) with VFT
the steady-state ‘‘plateau’’ stress rss and _  _ p , we have parameters
def 1 def
X
4
sss ¼ rss ; and m¼ mðaÞ ¼ 2_: ð38Þ #0 ¼ 355K; B ¼ 6600K
2 a¼1
for Pd40Ni40P20. The ln(cf,*/cf,eq) vs. ln _ data of [13] at
At steady state in a monotonic tension test at a given three different temperatures is shown in Fig. 3. At each
strain rate and temperature, s = s*, g = g*, the term temperature this data may be approximated as a linear
b(g*  g) vanishes, and substituting the corresponding dependence of ln(cf,*/ cf,eq) on ln _ . By fitting such a lin-
value of s* from (32) in (26), one obtains ear relationship to their data we obtain
 m
 mþn
1 m Q  
sss ¼ ~s exp : ð39Þ cf ;
4 m0 kB # ln ¼ k ln m þ l; ð42Þ
cf ;eq
By taking logarithms on both sides of (39) we obtain
ln sss ¼ ðm þ nÞ ln m where k and l are linear functions of temperature,
     m 
Q 1
þ ðm þ nÞ  ln m0 þ ln ~s ; 600
kB# 4
ð40Þ 549 K
500
556 K
which shows that at a constant temperature lnsss is lin- 564 K
ear in lnm. Tuinstra et al. [30] provide the following esti-
400
mate for the activation energy for Pd40Ni40P20
σss (MPa)

Q ¼ 2:66  1019 J:
300
A fit of (40) to the steady-state flow stress rss at various
strain rates _ and temperatures data provided by De Hey
et al. [13] (their Fig. 6), allows us to determine the slope
(m + n) as a function of temperature. For simplicity we
assume that the two rate-sensitivity parameters m and n 200
are equal to each other. Then at 564K we find that
0
10-2 10-1 10
m ¼ n ¼ 0:1316; -3
Strain rate (10 s ) -1

and that the temperature sensitivity of these parameters Fig. 2. Steady-state flow stress rss as a function of strain rate _ at three
in the range 549–564K may be approximated by the different temperatures. The symbols represent experimental results from
empirical relation Ref. [13], and the lines are from the model.
 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747 3743

10
4
s_ ¼ h0 ðs  sÞm; sð0Þ ¼ s0 ;
  n
m Q
549 K s ¼ ~s exp þ bðg  gÞ;
m0 kB#
10
3
556 K  
g
564 K g_ ¼ g0 1  ; gð0Þ ¼ g0 :
g
cf, / cf,eq

2
10 The initial value g0 depends on the pre-annealing history
*

of the material, and at thermal equilibrium is estimated

using (5) and the values of the VFT parameters #0 and B
10
1
listed above. Also, with the function g* and the material
parameters fm0 ; Q; n; ~sg as estimated above, it remains to
determine the parameters {s0, h0, b, g0}. These are deter-
10 -3
0
mined by fitting the complete stress strain curves at
-2 -1 0
10 10 10 10 564 K and four different strain rates,
-3 -1
Strain rate (10 s ) (0.83, 0.42, 0.17, 0.083) · 103 s1, Fig. 1a. The material
Fig. 3. The normalized steady-state flow defect concentration as a in these experiments was pre-annealed at 564 K for
function of strain rate at three different temperatures. The symbols 5000 s which is long enough for it to be in thermal equi-
represent experimental results from Ref. [13], and the lines are from the librium; in this case (5) gives
model.
g0 ¼ 0:0317:
Noting that s0 controls the beginning of the nonlinearity
k ¼ 10:8  0:0179#; l ¼ 253  0:435#: in the stress–strain curves, h0 controls the strain-harden-
ing slope of the curves, b controls the peak value, and g0
The quality of the curve-fit is shown in Fig. 3. Next, controls how quickly the strain softening occurs; a few
using (5)2 and (41), Eq. (42) may be expressed in terms trials using different values of these parameters yield
of the free volume as the following estimates:
" #1
1 #  #0 s0 ¼ 20 MPa; h0 ¼ 75; b ¼ 1:4  105 MPa; g0 ¼ 0:55;
g ¼  ðk ln m þ lÞ with geq ¼ : ð43Þ
geq B
which provide the acceptable fits to the complete stress
The steady-state free volume as a function of strain rate strain curves shown in Fig. 5. The model captures the
and temperature using this simple model is plotted in essential features observed in the experiments: the extent
Fig. 4 and compared with the corresponding data of of the stress overshoot, the strain softening, and the dif-
[13] (their Fig. 4). ferent plateau levels of the flow stress after 15–20%
 Material parameters in the evolution equations for the slip strain at the different strain rates are all reproduced very
resistance and the free volume: Recall that the coupled well by the model.
evolution equations for s and g are

700
0.038
–3 –1
549 K 600 Strain rate (10 s )
0.83
0.036 556 K 0.42
500 0.17
564 K 0.083
Stress, MPa
*
Free volume, η

0.034 400

300
0.032
200

0.03 Experiment
100
Simulation

0
0.028 -3 -2 -1 0
0 0.05 0.1 0.15 0.2
10 10 10 10 Strain
Strain rate (10-3 s-1 )
Fig. 5. True stress–strain curves for Pd40Ni40P20, pre-annealed at 564 K
Fig. 4. The steady-state free volume g*, as a function of strain rate at three for 5000 s, tested at 564 K at different strain rates. The solid lines represent
different temperatures. The symbols represent experimental results from experimental results from Ref. [13], and the dashed lines are from the
Ref. [13], and the lines are from the model. model.
3744 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747

Using the material parameters so determined, in the next Thus, the specimen that is annealed for 10,000 s has an
two subsections we verify the predictive capability of the initial free volume g0 = 0.031 that is substantially less than
model to reproduce the additional experimental results g* = 0.0344. Using the terminology of soil mechanics, it is
of [13] shown in Fig. 1b and c. in an ‘‘over-consolidated’’ state, and it is this over-consol-
idated initial state that leads to the large amount of stress
4.1. Effects of pre-annealing history overshoot. The specimen that is annealed for 720 s has an
initial value g0 = 0.0336, which is only slightly smaller than
The pronounced effect of the initial value of the free vol- g* = 0.0344. This specimen is therefore in a slightly over-
ume on stress–strain curves is further illustrated with consolidated condition, and correspondingly the stress
results from tension tests conducted at the same tempera- overshoot exhibited by this specimen is much less pro-
ture and strain rate, but on specimens with different pre- nounced than that in the previous case. Finally, for the
annealing histories. Fig. 1b shows stress–strain curves for specimen that is annealed for 120 s, the initial value of
specimens pre-annealed at 556 K for 120, 720 and the free volume is g0 = 0.0346, which is larger than
10,000 s, and subsequently tested at a temperature of g* = 0.0344, so the specimen is ‘‘under-consolidated.’’ In
556 K and a strain rate of 1.7 · 104 s1. this case there is no stress overshoot and strain softening;
The initial values of the free volume for the three speci- indeed, the material strain hardens with the stress–strain
mens with the different pre-annealing times are different. curve increasing monotonically to approach the steady-
The specimen that is annealed for 10,000 s will have an ini- state flow stress, which is a function only of temperature
tial free volume very close to the thermal equilibrium value and strain rate of the experiment, but independent of the
at this temperature, while the specimens that are annealed pre-annealing history.
for shorter periods will have higher initial free volumes. In
our simulations of these experiments we assigned initial 4.2. Effects of strain rate history
values
g0 ¼ 0:0346; 0:0336; and 0:0310 Fig. 1c shows the stress–strain curve from a strain-rate
increment and decrement experiment conducted at 556 K
for the specimens pre-annealed for 120, 720 and 10,000 s, and axial strain rates of 8.3 · 105 ! 4.2 · 104 ! 8.3 ·
respectively. The stress–strain curves calculated using these 105 s1, with the jumps occurring at strain levels of
initial values of g0 and with the values of the other material 0.125 and 0.27 [13]. The specimen was pre-annealed at
parameters fixed as in the previous section are compared 556 K for 3600 s, and (5) yields a value of
against the corresponding experimental measurement in
Fig. 6; the numerically simulated results are in excellent g0 ¼ 0:0334:
agreement with the experimental measurements.
The steady-state free volume for this test condition cal- Also, for the strain rates of 8.3 · 105 s1 and
culated from (43) is 4.2 · 104 s1, the steady-state values of g at 556 K using
(43) are
g ð1:7  104 s1 ; 556 KÞ ¼ 0:0344:
g ð8:3  105 s1 ; 556 KÞ ¼ 0:0337;
and g ð4:2  104 s1 ; 556 KÞ ¼ 0:0354:
At the beginning of the experiment g0 = 0.0334 is
600
slightly less than g*(8.3 · 105 s1, 556 K) = 0.0337, so
104 s the specimen is slightly ‘‘over-consolidated,’’ and the first
500
portion of the stress–strain curve shows a small amount
of overshoot. By a strain level of 0.125 the specimen has
400 720 s
almost reached the steady-state value of the free volume
Stress, MPa

for this strain rate. When the strain rate is suddenly

300 increased to 4.2 · 105 s1, the steady-state free volume
120 s also increases from g* = 0.0337 ! g* = 0.0354, and hence
200 Experiment the specimen is now significantly ‘‘over-consolidated’’ rela-
Simulation tive to the steady-state value g* at the higher strain rate.
100 This this manifests itself in a clear overshoot in the
stress–strain curve followed by strain softening until the
0
0 0.05 0.1 0.15 0.2
flow stress reaches its steady value at the higher strain rate.
Strain By a strain level of 0.27 the specimen is almost at its steady-
state value of g* = 0.0354 for the higher strain rate. Thus,
Fig. 6. True stress–strain curves for Pd40Ni40P20, pre-annealed at 556 K
for 120, 720 and 10,000 s, respectively, and tested at 556 K and
when the strain rate is suddenly decreased back to its lower
_ ¼ 1:7  104 s1. The solid lines represent experimental results from value of 8.3 · 105 s1, the value of g* at the lower strain
Ref. [13], and the dashed lines are from the model. rate is again g* = 0.0337, and the specimen is now in an
 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747 3745

in Fig. 8a. In order to simulate the spatial variations in

500 Experiment the initial free volume of actual materials, the initial free
Simulation volume g0 in the simulation was statistically assigned in
400 every element using a Gaussian distribution with a mean
value of 0.0316, and a standard deviation of 0.0001. A con-
Stress, MPa

300
tour plot of this initial free volume distribution is shown in
Fig. 8b. Fig. 9 shows the overall engineering stress–strain
200

100
a
–4 –1 –5 –1
8.3×10–5s–1 4.2×10 s 8.3×10 s

0
0 0.1 0.2 0.3 0.4
Strain

Fig. 7. True stress–strain curves for Pd40Ni40P20, pre-annealed at 556 K

for 3600 s, and then subjected to a strain-rate increment and decrement
experiment at 556 K. The solid lines represent experimental results from
Ref. [13], and the dashed lines are from the model.

‘‘under-consolidated’’ condition. Correspondingly, the

stress–strain curve now shows a stress undershoot as it
approaches the new steady-state value. The numerically
simulated stress–strain curve shown in Fig. 7 captures the
complex experimentally measured strain-rate history
response quite well.

5. Finite-element simulation of a plane-strain tension test

We have implemented our constitutive model in the

finite-element computer program ABAQUS/Explicit [28]
by writing a user material subroutine. As a simple but
important representative example of our numerical simula-
tion capability to model high-temperature deformation of b
metallic glasses, in this section we report on a finite element
simulation of a plane-strain tension test.
When a metallic glass is deformed at a low homologous
temperature, its inelastic response is almost rate indepen-
dent and characterized by strong strain softening, which
results in the formation of intense localized shear bands;
and in tension the material typically fails abruptly with
very little evidence of plastic strain at the macroscopic
level. One of the simulations that we performed in our ear-
lier paper [1] was that for plane-strain tension of a Zr-based
metallic glass at room temperature. In that simulation we
used 5000 ABAQUS-CPE4R plane-strain elements to rep-
resent the tension specimen (see Fig. 4 of [1]). As shown in
our previous simulation, once a few shear bands initiate at
heterogeneities in the microstructure, further inelastic
deformation occurs mainly inside these bands and the
material fails shortly thereafter, with the material outside
the shear bands not experiencing much plastic deformation
(see Fig. 5 of [1]).
We have repeated a similar plane strain tension simula-
tion for the Pd-based glass deformed at a high homologous Fig. 8. (a) Finite element mesh consisting of 5000 ABAQUS-CPE4R
temperature of 556 K and a strain rate of 1.7 · 104 s1. elements for the two-dimensional plane strain tension simulation. (b)
The finite element mesh used for this simulation is shown Contour plot of the distribution of the initial free volume.
3746 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747

curve and the contour plots of the equivalent plastic the maximum equivalent plastic strain varies only a small
strain in the vicinity of the stress peak, point (a), and at amount between 23.2% and 27.3%.
a strain level of 0.2, point (b). Fig. 9a shows that in the The response of a metallic glass at high temperatures is
vicinity of stress peak, multiple shear bands have formed, in stark contrast to the numerically predicted response at
with the location and ‘‘waviness’’ of the bands controlled low temperatures (e.g. Fig. 5 of [1]). Even though the mate-
by the initial heterogeneity of the free volume in the spec- rial exhibits relatively strong strain softening in its macro-
imen. Note,R t however, that the maximum equivalent plastic scopic stress–strain response, because of the substantially
strain ( 0 mðvÞdv) only varies a small amount between higher strain-rate sensitivity of the material at high temper-
3.1% and 3.59%. Fig. 9b at a nominal axial strain of atures, the tendency to form intense shear bands is substan-
20% (after substantial macroscopic strain softening) tially diminished, and the material deforms approximately
shows that a diffuse shear-band pattern persists, but again ‘‘homogeneously.’’

700
(a)
600

500
Stress, MPa

400 (b)

300

200

100

0
0 0.05 0.1 0.15 0.2
Strain

a b

Fig. 9. Engineering stress–strain curve from a two-dimensional plane-strain tension simulation. Contour plots of the equivalent plastic strain keyed to two
points on the stress–strain curve are also shown: (a) in the vicinity of the stress peak; (b) when the stress reaches the steady-state value.
 L. Anand, C. Su / Acta Materialia 55 (2007) 3735–3747 3747

6. Concluding remarks Acknowledgment

We have extended the elastic–viscoplastic constitutive Financial support was provided by grants from ONR
model of Anand and Su [1] for amorphous metals to (N00014-01-1-0808) and NSF (CMS-0555614).
the high homologous temperature regime, and specialized
the constitutive equations appearing in this theory to
model the response of metallic glasses in the temperature References
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