Accepted Manuscript

Constitutive Description of Bulk Metallic Glass Composites at High Homolo-gous Temperatures

Kianoosh Marandi, P. Thamburaja, V.P.W. Shim

PII: S0167-6636(14)00073-8DOI: http://dx.doi.org/10.1016/j.mechmat.2014.04.008Reference: MECMAT 2267

To appear in: Mechanics of Materials

Received Date: 10 June 2013Revised Date: 8 April 2014

Please cite this article as: Marandi, K., Thamburaja, P., Shim, V.P.W., Constitutive Description of Bulk MetallicGlass Composites at High Homologous Temperatures, Mechanics of Materials (2014), doi: http://dx.doi.org/10.1016/j.mechmat.2014.04.008

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1

Constitutive Description of Bulk Metallic Glass Composites at

High Homologous Temperatures

Kianoosh Marandia,∗

, P. Thamburajab, V.P.W. Shim

a

a Department of Mechanical Engineering, National University of Singapore, Singapore 117576, Singapore

b Department of Mechanical and Materials Engineering, National University of Malaysia (UKM), Bangi

43600, Malaysia

-------------------------------------------------------------------------------------------------

Summary

Experimental stress-strain responses of La-based in-situ Bulk Metallic Glass (BMG)

composites within the supercooled liquid region reveal initial post-yield hardening,

followed by softening and subsequent strain-hardening. This behavior contrasts with that

of monolithic La-based BMGs, which reach a steady stress level after an initial

overshoot. XRD analysis of BMG composites shows the formation of intermetallic

compounds during compressive deformation. These intermetallic compound

formation/interactions are associated with storage of energy in the material and affect the

stress-strain response. In this study, an elastic-viscoplastic, three-dimensional, finite

deformation constitutive model is also established to describe the behavior of a recently

developed La-based in-situ BMG (La-Al-Cu-Ni) composite, within the supercooled

liquid region, at ambient pressure and a range of strain rates. The constitutive model is

incorporated into a finite element program (ABAQUS/Explicit) via a user-defined

material subroutine. Numerical predictions are compared with compression test results on

BMG composites cast in-house. The comparison shows that the model is able to describe

the material behavior observed.

Keywords: finite deformation, constitutive equation, bulk metallic glass composites, finite

element, viscoplasticity

1 Introduction

Under high cooling rates, some metallic alloys solidify to yield a disordered

microstructure referred to as an amorphous metallic alloy or Bulk Metallic Glass (BMG).

The inherent brittleness of BMG at low temperatures has limited its structural

applications. BMG composites have been introduced to improve ductility and to prevent

catastrophic failure, and they fall into two groups: intrinsic (or in-situ) and extrinsic (or

ex-situ). Ex-situ composites involve mechanically combining glass-forming alloys with

other materials, such as ceramic fibers, particles, or metal wires, such as W, Ta, Nb. In-

∗ Corresponding author

E-mail address: k.marandi@yahoo.com (K. Marandi); currently at Department ArGEnCo, Division MS2F, University of Liege (ULG), Chemin des Chevreuils 1, 4000 Liege, Belgium

2

situ composites are made by nucleation of a crystalline reinforcement phase from the

solution melt during cooling and solidification. In both cases, the amorphous BMG phase

acts as a matrix that provides high strength for the ductile-phase component, which is

expected to suppress catastrophic failure. Moreover, several researchers have shown that

the ductility of BMG composites is dependent on the overall sample size, as well as the

ductile phase size/spacing (Fan and Inoue, 2000; Hays et al., 2000; Sarac and Schroers,

2013; Telford, 2004).

At ambient temperatures, BMG composites are essentially rate-independent, have higher

ductility than monolithic BMGs, and are characterized by strain softening and the

formation of shear bands in the material. On the other hand, BMG composites at high

temperatures (near and above the glass transition temperatures) are highly rate-dependent

and show fluid-like flow, which endow them with great potential for superplastic forming

and fabrication of complex near-net shapes via injection molding, die casting, etc. A

systematic study of various mechanical properties of Zr-based in-situ BMG composites at

high temperatures has shown that they are dominated by deformation of the amorphous

matrix phase. This study also demonstrates that the compressive stress-strain response

reaches a steady state after an initial stress overshoot (Fu et al., 2007b). An examination

of the homogeneous deformation response of La-based BMG and in-situ BMG

composites near the glass transition temperature of ~0.9�� (�� is the glass transition

temperature), for different strain rates, shows that although the volume fraction of the

lanthanum dendrite phase in the composite varies from 37% to 52%, the compressive

stress-strain response of the material remains similar, both qualitatively and

quantitatively (Fu et al., 2007a).

In several experimental studies on amorphous alloys and BMG composites asymmetric

responses for compression and tension have been observed. These differences constitute

the basis of the hypothesis that in BMGs the component of stress normal to the slip plane

and/or hydrostatic pressure affects material behavior and the onset of plasticity (Anand

and Su, 2005; Donovan, 1989; Zhang et al., 2003). Nanocrystal formation and

aggregation have been reported for some amorphous alloys. These phenomena have an

important effect on the mechanical properties at room temperatures and in the

supercooled liquid region (between the glass transition and crystallization temperatures).

At room temperatures, it can lead to significant plasticity and strain hardening, and in the

supercooled region, it is associated with non-Newtonian behavior and an increase in the

flow stress (Bae et al., 2002; Chen et al., 2006; Mondal et al., 2008; Nieh et al., 2001;

Wang et al., 1999).

Spaepen (1977) asserted that in BMGs, the shear flow rate depends on dynamic

equilibrium between stress driven creation and diffusional annihilation of free volume;

other researchers have also contributed to extension of the original theoretical

formulation (de Hey et al., 1998; Fornell et al., 2009). Anand and Su (2005), Yang et al.

(2005) and Thamburaja and Ekambaram (2007) formulated three-dimensional finite

deformation models for metallic glass, of which the first two are for isothermal

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conditions, and the third considers temperature changes for application to BMGs

undergoing large deformation within the supercooled liquid region. There are a few

constitutive models describing the deformation of BMG composites; they are one-

dimensional and only predict the onset of plasticity using the Coulomb-Mohr or Drucker-

Prager yield criterion by assuming simple elastic-perfectly plastic behavior (Lu, 2002;

Trexler and Thadhani, 2010). Recently, Marandi and Shim (2013) developed a three-

dimensional constitutive equation for in-situ BMG composites, based on finite-

deformation macroscopic theory and experimental data, for application at ambient

temperature and pressure, as well as different strain rates.

In the current study, the compressive stress-strain response of a La-based in-situ BMG

composite (La-Al-Cu-Ni), cast in-house, with a 50% volume fraction of the crystalline

phase, is examined by subjecting samples to uniaxial compression in the supercooled

liquid region at various strain rates (0.001/s-0.01/s) and ambient pressure. This reveals

post-yield hardening-softening (stress overshoot) followed by secondary strain-

hardening. This secondary strain-hardening is not observed in monolithic La-based

BMGs subjected to the same loading conditions, where the stress settles to a plateau after

the initial stress overshoot. Analysis of XRD spectra for as-cast and deformed BMG

composite samples display an increase in the volume fraction of crystalline lanthanum

and the formation of some binary intermetallic crystalline compounds during

deformation. This strain-induced intermetallic/nanocrystal formation is believed to be the

cause of the secondary strain-hardening observed, and is associated with storage of some

energy in the material.

A major objective of the present work is to develop a three-dimensional constitutive

equation for in-situ BMG composites based on finite-deformation macroscopic theory

and experimental data, consistent with the Second Law of Thermodynamics, applicable

to supercooled temperatures and ambient pressure, as well as various low strain rates (� < 0.1/ ). This area of study appears to be hitherto unexplored.

2 Kinematics and balance laws

In order to develop a constitutive equation for an in-situ BMG composite, the material is

considered homogenous and isotropic. Considerations are limited to isothermal situations

at a fixed temperature within the supercooled liquid region and at ambient pressure, in the

absence of temperature gradients. Macroscopic theories for rate-dependent elastic-

viscoplastic deformation are developed to obtain the kinematics of the motion in the

material. It should be noted that although homogenization by assuming affine

deformation of the constituent phases in the composite can be used to predict the overall

behavior of in-situ BMG composite at room temperatures (Marandi and Shim, 2013),

experimental results show that this homogenization assumption is not valid for high

4

homologous temperatures (this will be expanded on in the Section on Experimental

Procedures and Finite-Element Simulations).

Some essential variables1 for development of the constitutive model are introduced here:

(a) the total deformation gradient �; (b) the multiplicative Kröner-Lee decomposition � = ����, where �� is the inelastic deformation distortion, and �� is the elastic

deformation distortion; (c) The elastic deformation distortion is decomposed using the

right polar decomposition, i.e. �� = ����, where ��=���� and �� = ��� are the

orthogonal elastic rotation, and positive-definite symmetric elastic stretch tensors,

respectively; (d) the velocity gradient is � = ���� = �� + ��������, where �� =� ����� and �� = ������ represent elastic and inelastic velocity gradients, respectively;

(e) �� can be further decomposed into �� = �� +��, where �� = sym(��) is the

plastic stretch rate and �� = skw(��) the plastic spin rate tensor. (f) In the development

of this analysis, it is assumed that plastic flow is irrotational, so �� = # ⇒ �� = ��,

(Gurtin and Anand, 2005). The change in volume of the deformed configuration with

respect to the reference configuration is defined by the determinant of the deformation

gradient % ≝ det �.

The right Cauchy-Green deformation tensor and elastic right Cauchy-Green deformation

tensors are defined by

* = ��� and *� = ����� = (��)+ (1)

The spectral representation of the elastic stretch ��can be expressed as

�� = ,Λ.� /.� ⊗/.�1.23

(2)

where /.� (4 =1,2,3) are eigenvectors of �� and Λ.� are the positive eigenvalues of ��. A frame-invariant measure of elastic strain is now introduced:

5� = (1 2⁄ )ln(*�) = ln(��) ⇒ 5� = , ln(Λ.� )/.�⊗ /.�1.23

(3)

where 5: is the symmetric logarithmic elastic strain tensor. The spectral representation of *:can be expressed as

1 Notation: Consider a second order tensor ;, then;��, ;� and tr(;), are the inverse, transpose, and trace

of the tensor, and (;��)� = ;��. The inner product of tensors ; and > is denoted by ; ∶ >; and the

magnitude of ; by |;| = √; ∶ ;. In addition, sym(;) =(1/2)(; + ;�) denotes the symmetric portion of

tensor;, whereas skw(;) =(1/2)(; − ;�) denotes the skew portion of tensor ;. ;C = ;− (1/3)tr(;)�

is the deviatoric portion of tensor;, and symC(;) is the symmetric-deviatoric part of tensor ;.

5

*� = , ω.�/.� ⊗/.�E.23 whereω.� ≝ Λ.�G (4)

2.1 Kinematics of deformation

The in-situ BMG composite is considered isotropic, and inelastic flow is taken to be

similar to that in amorphous materials (Thamburaja and Ekambaram, 2007). It is

associated with the evolution of free volume, and is not isochoric,

�� = HIJ + (1 3⁄ )ξ� (5)

where J denotes the inelastic flow direction, tr(J) = 0 and J = JL, with the restriction

that |J| = 1. I represents the plastic shear rate and H > 0 is a constant of

proportionality. ξ (with unit of 1/time) represents the rate of evolution of the free volume

concentration. The inelastic dilatation-rate in the material is given by tr(��) = tr(��) =ξ. Note that the first and second terms on the right-hand side of Eq. (5) are respectively

purely isochoric and purely volumetric2.

The Second Law of Thermodynamics within a purely mechanical framework in the

reference configuration is written as N: � − P ≥ 0; where P is the Helmholtz free energy

per unit volume in the reference configuration and N = JS��� is the first Piola-Kirchoff

stress. Substitution of N = JS��� and � = ���3 into the preceding relationship for the

Second Law of Thermodynamics, and considering symmetry of S yields

JS: � − P ≥ 0 (6)

Equation (6) is the basis for development of the constitutive model.

3 Free energy

The expression for the free energy of the BMG composite is modified with respect to

existing theories of viscoplasticity for high homologous temperatures and isotropic

deformation at the macroscopic level (Anand and Su, 2007; Marandi and Shim, 2013;

Thamburaja and Ekambaram, 2007; Yang et al., 2005 ). The macroscopic free energy

density for the in-situ BMG composite is assumed to be

P = PT(*�, ξ, U) (7)

2 It is assumed that crystal formation during inelastic deformation does not induce strain. Otherwise,

another term should be introduced to the right side of Eq. (5) to capture the effect of crystallization strains.

6

where *�is the elastic Cauchy-Green deformation tensor, ξthe free volume concentration,

and ϰ is a measure of the crystallization fraction3. ξ and ϰ are scalar internal-state

variables which evolve during plastic deformation and define the current microstructure

of the material. Eq. (7) asserts that storage of energy in the material is assumed to be a

function of elastic deformation, free volume and crystallization fraction. Taking the time

derivative of Eq. (7) results in

P = WPTW*� : * � + WPTWξ ξ + WPTWU U (8)

By substitution of Eq. (8) into inequality Eq. (6), the Second Law of Thermodynamics

(cf. Marandi and Shim, 2013) can be written as

X%S − 2�� XWPTW*�Y���Y :� + Z2*� XWPTW*�Y[ :�� − WPTWξ ξ − WPTWU U ≥ 0 (9)

Furthermore, noting the symmetry of \%S − 2��\WPT W*�⁄ ]���] in the preceding

relationship, and combining � = �� + �������3 and �� = ���� with the time derivative

of Eq. (1)2 (i.e. * � = 2��� � = 2�������) and substitution into the inequality Eq. (9),

followed by some rearrangements yields

^: \� ���_`] + ��^��_`: �� + Z2*� XWPTW*�Y[ : �� − WPTWξ ξ − WPTWU U ≥ 0 (10)

where

^ ≝ Xa− 2�� XWPTW*�Y��Y (11)

where the Mandel stress a (with unit of stress) is defined as a ≝ %���S��. From

rational thermodynamic argument, it is possible to obtain a = 2��\WPT W*�⁄ ]��. The

preceding relationship can be re-arranged to obtain the constitutive equation for the

Cauchy stress

S = %�3�� X2 WPTW*�Y��� (12)

Moreover, from the symmetry of \WPT W*�⁄ ] and ��, and assuming that the free energy is

an isotropic function of the elastic stretches, �� and \WPT W*�⁄ ] have the same

3 In this work, the term crystallization fraction is a measure of crystal (regular

crystals/nanocrystals/intermetallics) nucleation during plastic deformation. It is not to be confused with the

overall volume fraction of the crystalline phase in the BMG composite.

7

eigenvectors and they are commutative i.e.\WPT W*�⁄ ]�� = ��\WPT W*�⁄ ] (e.g. Neto et

al., 2008), therefore the Mandel stress can also be expressed as

a = 2��+ XWPTW*�Y = 2*� XWPTW*�Y (13)

Note that the theoretical formulation is invariant to a change of reference frame (cf. Marandi

and Shim, 2013).

Furthermore, if all plastic work during plastic deformation is dissipated, the remaining

terms in Eq. (10) define the total mechanical power dissipation per unit volume in the

reference configuration. Thus, substitution of the Mandel stress (Eq. (13)) into the

inequality Eq. (10) yields

a:�� − WPTWξ ξ − WPTWU U ≥ 0 (14)

Equation (14) will subsequently be used to determine the kinetic relationship for the

plastic shear rate in the material.

4. Specific form of constitutive equations

4.1 Specific form of free energy

The storage of free energy in a BMG composite is assumed to be attributed to elastic

deformation, evolution of free volume concentration and crystallization. The free energy

per unit reference volume is also assumed to be separable according to

PT(*�, ξ, U) = PT�(*�) + PTb(ξ) + PTc(U) (15)

where PT�is the elastic portion of the free energy density; PTb and PTdcan be considered as

micro-defect energies due to evolution of the internal state variables in the material. PTb is

the portion related to the evolution of free volume, and PTd is due to crystallization. A

specific functional form of the free energy within a purely mechanical framework is

chosen:

PT�(*�) = e|dev(5�)|+ + (1 2⁄ )ghtr(5�)i+ (16)

PTc(U) = (1/2)ℋcU+ +ℋk (17)

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PTb(ξ) = (1/2)sblξ+ − sblξξ� (18)

where e = eT(ξ, U) and g = gm(ξ, U) are respectively, the elastic shear and bulk moduli; ℋc = ℋmc(U) is the energetic interaction coefficient (with unit of energy per unit

volume), and ℋk is the thermal transformation energy of crystallization with unit of

energy per unit volume4. sbl ≥ 0 is a material parameter (with unit of energy per unit

volume) that relates the change in flow-defect energy due to the change in free volume

concentration, andξ� ≥ 0 (dimensionless) represents the thermal equilibrium (fully

annealed) free volume concentration, and it varies with temperature5:

ξ� = ξn + ko\θ − θn] (19)

where � > 0 is the absolute temperature, ξn is the thermal equilibrium free volume

concentration at the glass transition temperature θn, and kθ > 0 is a constant of

proportionality with unit of 1/temperature (Masuhr et al., 1999)

The elastic portion of the free energy for an isotropic material may alternatively be

expressed in terms of the principal stretches PT�(*�) = Pq�(Λ3�,Λ+� ,ΛE�), where Λ3� , Λ+�

and ΛE� are eigenvalues of *�. By using the chain-rule and combining Eqs. (2), (3), and

Eq. (4), the Mandel stress a (Eq. (13)) can be expressed as

a = 2��+ r, WPq�WΛ.�E.23

WΛ.�W*�s = ��+ r, 1Λ.�WPq�WΛ.�

E.23

Wω.�W*�s= ,tXΛ.� WPq�WΛ.�Y /.�⨂/.�v

E.23

= ,tX WPq�W(lnΛ.� )Y /.�⨂/.�vE.23 = XWPq�(E3�, E+�, EE�)W5� Y

(20)

where E.� ≝ lnΛ.� , and the relationship Wω.� W*�⁄ = /.�⨂/.� has been used. Substituting

Eq. (16) into Eq. (20) (Anand, 1979) results in

a = 2edev(5�) + gtr(5�)� (21)

4 The functional form of PTc(U) has been chosen with a view toward application, and for simplicity in the

present study, ℋk is assumed to be negligible. 5 It is assumed that the size of the BMG composite specimens (several millimeters) is several orders of

magnitude larger than the length scale for free-volume diffusion (typically several nanometers) and

therefore free energy is assumed to be independent of free-volume gradient. By introducing this gradient

energy (e.g. Thamburaja, 2011), it is possible to model the deformation behavior of sub-micron sized BMG

composite samples.

9

This equation will be used later to calculate the equivalent shear stress and driving force.

A combination of Eq. (21) and a = %���S�� yields

S = %�3��x2edev(5�) + gtr(5�)�y��� (22)

This is the specific form of the constitutive equation for the Cauchy stress in an in-situ

BMG composite at high temperatures.

4.2 Specific forms of kinetic relations

It is known that crystallization in BMGs is temperature and strain/strain rate dependent,

and also hydrostatic pressure assisted (Jiang et al., 2003; Nieh et al., 2001; Wang et al.,

1999). Compressive experiments on La-based BMG composites at different strain rates

reveal crystallization of intermetallic compounds in the material. This crystallization is

believed to be the reason for the secondary rise in stress observed in the compressive

stress-strain responses, and it becomes noticeable after a certain degree of deformation.

Therefore, it is assumed that the evolution of crystallization is strain-induced and is a

function of the plastic shear rate defined by

U = zI (23)

where z = z(I,a, P}) is the coefficient governing crystal formation, and P} ≝−(1 3⁄ )tr(a) is the hydrostatic pressure.

Investigations suggest that the deformation mechanism associated with homogenous flow

in the BMG and BMG composite is similar, and free volume evolves similarly in both

(Chen et al., 2009). The evolution of free volume concentration can be described as

(Thamburaja and Ekambaram, 2007):

ξ = ζI + ξ� (24)

Here, ζI is the creation of free volume concentration due to plastic shear deformation (de

Hey et al., 1998), ζ = ζ�(a, ξ) is the coefficient of free volume creation (dilatancy

function) and ξ� is the change in free volume concentration due to other mechanisms like

relaxation, diffusion, hydrostatic pressure, etc., which will be determined later (see Eq.

(39)).

By substitutingtr(��) = ξ, combined with Eqs. (24), (23), (5), into the inequality Eq.

(14) results in

x� + ζ(−P} − ��) − z�cyI + x−P} − ��yξ� ≥ 0 (25)

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where �� ≝ \WPT Wξ⁄ ] is the viscous stress, � ≝ H(symC(a). J) is the equivalent shear

stress in the BMG composite and �c ≝ WPT WU⁄ is the crystallization stress. The

preceding relationships for �� and �c coupled with Eq. (15) result in

�� = sbl(ξ − ξ�) and �c = ℋcU (26)

Taking each dissipation mechanism related to plastic shearing and free volume evolution

to be strictly dissipative, the inequality Eq. (25) becomes

x� + ζ(−P} − ��) − z�cyI ≥ 0andx−P} − ��yξ� ≥ 0 (27)

If the inequalities defined by Eq. (27)1 and (27)2 are satisfied at all times, the inequality

Eq. (25) will also be satisfied.

By defining H ≝ �1/2 and the scalar function π� ≝ x� + ζ(−P} − ��) − z�cy, the

inequality Eq. (27)1 can be expressed as π�I ≥ 0, where π� is the driving force for plastic

shear deformation in a BMG composite. The preceding relationship for π�, combined

with the definition of equivalent shear stress (� = H(symC(a). J)), can be rearranged as

H(symC(a).J) = π� + ζ(P} + ��) + z�c (28)

The following relationship can be used to satisfy Eq. (28):

H\symC(a)] = xπ� + ζ(P} + ��) + z�cyJ (29)

Noting that |J| = 1, and taking the magnitude of both sides of Eq. (29) yields

J = symC(a)|symC(a)| (30)

Eq. (30) defines the direction of plastic flow in the BMG composite.

The inequality π�I ≥ 0 is satisfied ifsign(π�) = sign(I). To fulfill this and to determine

the kinetic relation for the amorphous phase, a possible kinetic relation for the plastic

strain rate, noting the relationship for π�, is proposed in the form of:

I = ITC(I , ξ, U) X⟨� − ζ(P} + ��) − z�c⟩ (I , ξ, U) Y3 �⁄ (31)

where (I , ξ, U) > 0 (with unit of stress) is the resistance to plastic flow; ITC(I , ξ, U) > 0

(with unit of 1/time) is a reference flow-rate, m = m�(I , ξ, U) is a macroscopic rate

sensitivity parameter (0 < m ≤ 1), �� and �c are stresses due to free volume and

crystallization evolution. The relationship for π� and Eq. (31) indicates that the driving

11

force for plastic shear deformation is dependent on the hydrostatic-pressureP} , viscous

stresses�� and crystallization stress�c . This pressure-dependent and viscous stress

dependent behavior of the BMG composite can be captured by either the driving force or

the resistance to plastic flow. The latter approach is chosen for this study, based on

previous work on BMGs (Huang et al., 2002; Marandi and Shim, 2013; Thamburaja and

Ekambaram, 2007). Therefore, Eq. (31) can be written in the form of I = ITC(I , ξ, U)(⟨� − z�c⟩ (I , P}, ξ, U, ��)⁄ )3 �⁄ .To determine the kinetic relationship for

the BMG composite at high homologous temperatures, it is proposed that:

I = IC X⟨� − z�c⟩ + μP} Y3 �⁄ (32)

where in Eq. (32), the variable > 0 is the resistance to plastic flow, and the µ > 0 is a

dimensionless pressure sensitivity parameter. IC > 0 is a reference flow-rate with unit of

1/time, and m is a macroscopic rate-sensitivity parameter (0 < m ≤ 1); The term ( + µP}) in Eq. (32) constitutes the resistance to plastic flow in the BMG composite.

Here, m → 0 makes Eq. (32) rate independent, and can be regarded as a Drucker-Prager

criterion, since (� − z�c) = + µP}, while m = 1 renders Eq. (32) linearly viscous. The

values of the material parameters must ensure that the inequality π�I ≥ 0 is always

satisfied. Furthermore, Eq. (32) implies that for I ≥ 0 when � ≥ z�c , the values of the

material parameters must satisfy the condition + µP} > 0.

The evolution of resistance to plastic flow , can be expressed as

= ℎI (33)

where ℎ = ℎT( ) is a strain hardening/softening function for the BMG composite, which

is proposed to evolve according to:

ℎ = ℎC( �∗ − ) (34)

where ℎC > 0is a dimensionless material parameter; �∗ is a critical resistance (with unit

of stress). The resistance to plastic flow has an initial value of C, which needs to be

determined.The following phenomenological relationship is introduced to describe the

critical resistance �∗:

�∗ ≝ � r1 − ��exp �− 1��� + ω�s (35)

where � is defined as a key resistance, a positive material parameter with unit of stress. �� (0 ≤ �� < 1) is a dimensionless material parameter which controls the ratio of

softening with the evolution of free volume.�� is taken as the decrease in the resistance

from the key resistance to the steady-state level, normalized with respect to the key

12

resistance; � > 0 is a material parameter with units of unit volume per energy. ω =ω�(ξ, U) is a dimensionless parameter which can relate the effect of static resistance (in the

absence of stress) to the plastic flow. In this work, ω is assumed to be a constant with a

very small value (~10��). In Eq. (35), if ξ ≫ ξ(�2C), then �∗ → �(1 − ��), which

ensures that the critical resistance never reaches near-zero values6.

Moreover, it is assumed that the evolution of crystallization (Eq. (23)) can be described

by introducing the following simple phenomenological relationship for the crystallization

fraction U:

U =  ¡¢ 0 I < I�I − I�I£ − I� I� ≤ I < I£1 I£ ≤ I

¤ (36)

where I� is a critical strain at which crystallization initiates, and I£ is the final strain at

which all possible potential crystal sites have been activated in the BMG composite. It

should be noted that for simplicity in developing Eq. (36), the effects of stress,

temperature, and hydrostatic pressure on crystallization have been neglected, and

crystallization is considered to be primarily strain induced.

Incorporation of Eq. (36) into Eq. (23) results in7

z = 1I£ − I� (37)

In this study, it is assumed that the thermal transformation energy of crystallization ℋk in

Eq. (17) can be neglected, and the energetic interaction coefficient ℋc is taken to be

6 The critical resistance in the monolithic BMG alloys has been described phenomenologically by a linear

relationship with free volume evolution (e.g. Anand and Su, 2007). However, current simulation results for

the BMG composite show that the expression for the critical resistance can be described by introducing an exponential expression (Eq.(35)). It can be argued that interactive forces between the amorphous and

crystalline phases may not permit the resistance in the composite alloy to vary linearly with respect to free

volume creation. Moreover, Equation (35) has been modified with respect to its counterpart for BMG

composites at room temperature (Eq. (54) in Marandi and Shim, 2013), where the critical resistance is

related to ~exp(−1 ξ⁄ ) rather than ~exp(−1 τ¦§⁄ ). Simulation results indicates that at high temperatures,

viscous stress plays a dominant role in plastic flow, rather than free volume concentration ξ. Further

investigation needs to be undertaken to obtain experimental data on the evolution of free volume in such

composites. 7 For simplicity and because of absence of experimental data on the rate of crystallization, the current

definition of the coefficient governing the crystallization χ (Eq. (37)) is assumed to be constant; this results in almost rate-insensitive prediction crystallization. However, by assuming that nanocrystal/crystal

formation, for example, to be stress-driven (Nieh et al., 2001) and introducing a χ ≝ (1 sd⁄ )(τ¦/G) type

formulation, where sd is a dimensionless material parameter, crystallization becomes rate-sensitive. Further

investigations need to be undertaken to obtain experimental data on crystallization.

13

constant. Furthermore, the inequality Eq. (27)2 issatisfied if sign(−P} − ��) = sign(ξ�); the following relationship can be used to satisfy the inequality Eq. (27)2

−P} − �� = sbll Xξ�ICY (38)

where sbll = sªbll(ξ) > 0 represents a material parameter with units of energy per unit

volume.

Substituting Eq. (38), and Eq. (26)1 into Eq. (24) yields

ξ = ζI − X ICsbllYP} − XICsblsbll Y (ξ − ξ�) (39)

The first term on the right-hand side of Eq. (39) represents the creation of free volume

arising from plastic deformation, and the second term is associated with generation of

free volume linked to hydrostatic pressure. The last term corresponds to annihilation of

free volume because of relaxation. It should be mentioned that in deriving Eq. (39), it is

assumed that diffusion of free volume can be neglected. The coefficient of free volume

creationζ, is an increasing function of the Cauchy stress (Heggen et al., 2004); as an

approximation, the following functional form for the coefficient of free volume creation

is used (Ekambaram et al., 2010)

ζ = ζC exp � ��∗� (40)

where ζC (dimensionless) and �∗(with unit of stress) are material parameters derived from

fitting with experimental data in the softening region of the stress-strain response. This

completes the derivation of the kinetic relations for the BMG composite at high

homologous temperatures.

5 Experimental procedures and finite-element simulations

To evaluate the model, three variants of La-based materials were produced: (a) in-situ

BMG composite (La74Al14Cu6Ni6) with a 50% volume fraction of crystalline lanthanum8

(Lee et al., 2004); (b) a monolithic La-based amorphous alloy with a composition of

La61.4Al15.9Cu11.35Ni11.35 (Tan et al., 2003) and; (c) pure polycrystalline lanthanum (La100).

The crystalline dendrite phase in the BMG composite is identified as lanthanum

precipitated out from the amorphous phase (Lee et al., 2004).

8 Based on image analysis using the commercial ImageJ Software, the volume percentage of the dendrite

crystalline phase of as-cast samples (Fig. 6) is between 47%-52%; a value of 50% is taken as an overall

value.

14

All three materials were fabricated by arc-melting in an argon atmosphere. To produce

each type according to its weight composition, the desired mixture of La (99.9999%), Al

(99.98%), Cu (99.9999%) and Ni (99.98%) was placed in a quartz crucible and melted in

an induction furnace. They were then chill-cast by pouring the molten alloy into the

cavity of a copper mold. A mold with a cylindrical cavity of ϕ5 × 60mm was used to

cast lanthanum and BMG composite samples, and one with a cavity of 4 × 6 × 45mm

was used to produce monolithic BMG samples. Detailed procedures for producing these

BMG and BMG composite alloys can be found in the work of Tan et al. (2003) and Lee

et al (2004).

The ϕ5 × 60°° cast rods were machined to produce BMG composite and lanthanum

specimens measuring of ϕ4 × 4 mm. Cutting fluid was used for cooling during

machining. Reducing the diameter from ϕ5°° to ϕ4°° removes the thin ~0.5°°

layer of fully amorphous alloy on the outer surface of the BMG composite rods. This

eventually yields a cross-section of homogenous material. The 4 × 6 × 45mm BMG

plates were machined to cubic samples of 4 × 4 × 4mm.9

The rod cross-sections of the monolithic amorphous La-based alloy and its composite

were examined using X-ray diffraction (XRD), and the XRD spectrogram is depicted in

Fig. 1(a). The diffraction spectrum for the composite shows distinct peaks corresponding

to crystalline hcp lanthanum phases in the form of dendrites. There is no sharp peak in

the spectrum for monolithic BMG alloy, and this is characteristic of an amorphous

structure.

Fig. 1 – (a) XRD pattern for La-based in-situ BMG composite and monolithic BMG alloy; (b) Differential

Scanning Calorimetry (DSC) at a heating rate of 20 K/min for La61.4Al15.9Cu11.35Ni11.35 BMG alloy,

La74Al14Cu6Ni6 BMG composite and pure Lanthanum La100

9 Initially, samples with length to width (diameter) aspect ratios of 2:1 (i.e. ϕ4 × 8 mm and 4 × 4 × 8mm) were used for compression tests at high temperatures. However, these samples, especially the BMG

composite, exhibited significant buckling when the strain exceeded ~10%. Hence, samples with an aspect

ratio of 1:1 were used to preclude buckling during deformation. Specimens with a 1:1 aspect ratio deformed

uniformly.

0

20

40

60

80

20 70 120

Inte

nsi

ty [

a.u

.]

2 Theta [°]

in-situ BMG composite

monolithic BMG alloy

a)

0

5

10

15

20

330 430 530 630

Exo

the

rmic

[m

W/g

]

Temperature [K]

in-situ BMG composite

monolithic BMG alloy

pure lanthanum

θgθx θm

b)

15

In order to minimize the influence of thermal history (experienced by specimens during

production) on the mechanical properties, all machined specimens were annealed using

an Instron 3119 series environmental chamber. They were heated from a room

temperature of 24 ºC at a rate of 20 ºC/min to 165 ºC (438.15 K), then kept at this

temperature for exactly 8 mins before experiments were performed.10

It is assumed that 8

min of annealing allows the free volume in both the monolithic BMG and in-situ BMG

composite to reach to an equilibrium state ξ�2C ≈ ξ�, (Lu et al., 2003).

Figure 1(b) and Table 1 show results from Differential Scanning Calorimetry (DSC) tests

on La-based BMG and BMG composite, which confirm that a working temperature of

165 ºC (438.15 K) is within the supercooled liquid region of these alloys.

Table 1 - Results of DSC analysis for a heating rate of 20 K/min, for La74Al14Cu6Ni6 and

La61.4Al15.9Cu11.35Ni11.35, where Vf is the volume fraction of crystal phase in the alloy, θg the glass transition

temperature, θx the crystallization temperature and θm the melting temperature

Alloys Vf θg (°K) θx (°K) θm (°K)

La61.4Al15.9Cu11.35Ni11.35 0% 419 477 659

La74Al14Cu6Ni6 50% 430 478 661

It should be noted that the glass transition temperature is not a fixed physical quantity and

may change slightly as a function of heating rate (Lu et al., 2003).

5.1 Compression tests

The specimen surfaces which come into contact with the testing machine were ground

parallel and polished using superfine 1200 grit silicon carbide paper. A thin film of

molybdenum disulphide was also applied to ensure near-frictionless condition. Samples

were subjected to quasi-static compression within the supercooled liquid region at

ambient pressure and different strain rates (� < 0.1/ ). Tests were conducted using an

Instron 3119 series environmental chamber in conjunction with an Instron 8874

axial/torsional servo hydraulic machine. The temperature was maintained at 165 ºC

during the compression tests and the force-displacement signals were used to obtain the

compressive stress-strain curves. Strain data was calibrated by measuring the final length

of the compressed samples using Vernier calipers immediately after testing and

comparison with machine readings.

10

It is well-recognized that for BMGs, the stress over-shoot followed by strain softening phenomenon is a

function of the annealing applied to the specimen; this controls the amount of initial free volume which can

give rise to strain hardening (de Hey et al., 1998). In La-Based BMGs, both the peak stress and steady-state

flow stress values are affected by annealing time; varying the annealing time from 1 minute to 60 minutes

causes the peak-stress and steady-state stress values to decrease continuously (Ekambaram, 2009).

16

Figure 2 shows experimental data in terms of compressive stress-strain responses for the

BMG composite at different strain rates – 0.001/s, 0.003/s, 0.006/s and 0.009/s11

.

Fig. 2 - Stress-strain response of in-situ BMG composite with 50% volume fraction of crystalline phase, at

165 ºC, for different strain rates.

It can be seen that the in-situ BMG composite at supercooled temperatures generally

exhibits a short post-yield hardening phase followed by softening and secondary

hardening; it is rate sensitive within the strain rate range of 0.001/s to 0.009/s. A

comparison of the compressive stress-strain responses of the three types of samples, i.e.

(a) BMG composite, (b) monolithic BMG and (c) pure crystalline lanthanum, at 165 ºC

and different strain rates is shown in Fig. 3.

11

In the present study, compression tests at each strain rate were conducted on several samples; each stress-

strain curves shown is representative of three stress-strain curves that do not deviate from each other by

more than 5% of the average value.

0

100

200

300

400

500

0 0.2 0.4 0.6 0.8

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.009/s BMG Composite, Expm0.006/s BMG Composite, Expm0.003/s BMG Composite, Expm0.001/s BMG Composite, Expm

17

Fig. 3 - Compressive stress-strain response of BMG composite, monolithic BMG and lanthanum at 165 ºC

and different strain rates. X indicates the point of failure.

For BMG composites, the stress rises again after the initial stress overshoot. It is believed

that the intermetallic compounds formed during deformation are the main contributors to

the secondary strain hardening observed (this will be expanded on in the Section on

Microstructural Features). This behavior contrasts with that of the monolithic BMG alloy,

where the stress reaches a steady state instead of exhibiting secondary hardening. It is

widely accepted that in the stress plateau region, there is a balance between the creation

and annihilation of free volume, and free volume concentration attains a steady value.

Investigations suggest that the deformation mechanism in the homogenous flow of the

BMG and BMG composite are similar, and free volume evolves similarly in both alloys

(Chen et al., 2009). Therefore, the free volume in the composite is also presumed to reach

equilibrium in the post-overshoot region.

The initial linear portions of the stress-strain curves for the BMG composite and

monolithic BMG coincide, and they have a higher slope than the corresponding response

for pure lanthanum; it is also observed that lanthanum has the lowest strength. These

experimental results indicate that the stress-strain responses of crystalline lanthanum and

monolithic BMG cannot be combined to yield the behavior of the in-situ BMG

composite. This is because the slopes of linear portion of the stress-strain curves for the

BMG and BMG composite are identical, also the strength of the BMG composite cannot

be obtained by combining the strengths of the BMG and polycrystalline lanthanum. This

contrasts with the stress-strain response of BMG composites at room temperature,

whereby a homogenization approach by assuming affine deformation of the amorphous

and crystalline phases can be used to obtain the constitutive equation for the stress, which

obeys a rule of mixtures relationship (Marandi and Shim, 2013). This can be explained by

noting that in the supercooled liquid region a BMG composite is in a transition state and

the material changes from solid to liquid as the stress changes. When the stress increases,

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.003/s BMG, Expm0.003/s BMG Composite, Expm

0.001/s BMG Composite, Expm0.001/s BMG, Expm

0.01/s Lanthanum, Expm0.001/s Lanthanum, Expm

XX

18

the material behaves and flows like a liquid, and when the stress decreases, it becomes

solid. This behavior is unlike that at room temperatures, where the material is in a solid

state and it is possible to assume that both phases in the composite experience affine

deformation. In the supercooled liquid region, the assumption of affine deformation may

be applicable only to the initial stage of deformation, when the stress is very small.

Figure 3 also shows that monolithic BMG has a higher degree of rate-sensitivity than in-

situ BMG composite, while pure lanthanum displays lowest rate sensitivity. The

crystalline phases (dendrites) in the composite increase the viscosity of the material, and

atoms in the amorphous phase are more constrained than those in the monolithic BMG;

consequently, the rate-sensitivity is decreased. It is also observed that the BMG

composite displays a smaller stress overshoot compared to monolithic BMG. This is

because the secondary crystalline phase interrupts the flow of the amorphous matrix and

the atoms are not as mobile as those in the monolithic BMG, with regard to jumps into

and out of interatomic spaces to create/annihilate significant free volume; this

consequently results in a smaller stress overshoot, which is a strong function of free

volume evolution.

Shear localization and sudden failure occur in BMG composite samples at strain rates

greater than ~0.01/s, (Fig. 4), and the stress-strain responses exhibit only a short phase of

initial post-yield softening, with the stress reaching a maximum of ~495 MPa before

failure occurs. Monolithic BMG samples undergo deformation localization and sudden

failure at lower strain rates of ~0.003/s, where the maximum stress reaches ~575 MPa,

and a linear stress-strain response before failure prevails.

Fig. 4 - Stress-strain responses of in-situ BMG composite and monolithic BMG at 165 °C and strain rates at

which failure (inhomogeneous deformation) initiates – 0.006/s for BMG, and 0.01/ for BMG composite

It can be concluded that in-situ BMG composite samples are able to accommodate larger

strains at higher strain rates before failure compared to monolithic BMG samples, and the

crystalline phase of the composite delays the formation of shear bands and sudden failure.

0

100

200

300

400

500

600

0 0.2 0.4

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.006/s BMG, Expm

0.01/s BMG Composite, Expm

X

X

19

5.2 Microstructural Features

Figure 5 shows XRD spectra for as-cast and deformed in-situ BMG composite samples.

They reveal that new intermetallic crystalline phases are formed during deformation.

Fig. 5 - XRD spectra for La-based BMG composite – as-cast and deformed

The diffraction patterns show that peaks corresponding to crystalline hcp lanthanum are

much more prominent in the deformed sample; this can be interpreted as an increase in

the volume fraction of the lanthanum phase precipitated out from the composite during

deformation. In addition, two intense peaks corresponding to LaNi2.28 and Cu13La are

evident in the XRD pattern. The orthorhombic LaNi2.28 intermetallic compound has been

reported to have a formation temperature of 660 °C-730 °C (Paulboncour et al., 1987),

while the Cu13La intermetallic compound has a cubic NaZn13 structure (Bloch et al.,

1981), and it has been proposed that this phase forms peritectically at 873 °C (Okamoto,

1991). In this study, the experimental test temperature of 165 °C is much lower than the

temperatures required to form these intermetallic compounds; hence, it is concluded that

mechanical work assisted in providing energy for this, and some of the energy is stored in

the material via crystallization. In addition, XRD examination was performed on

undeformed BMG composite samples annealed and held at a temperature of 165 °C for

10 min – the approximate time required to complete a compression test at the lowest

strain rate of 0.001/s. This XRD examination on undeformed samples did not reveal any

crystallization, and supports the assumption that crystallization is mainly strain-induced

for the experimental conditions in this study. XRD spectrographs of the deformed

monolithic BMG samples after compression to 0.8 [mm/mm] of strain also did not

exhibit any crystallization (these XRD results are not presented in the paper for the sake

of brevity).

It is known that nanocrystals can form in BMGs during deformation in the supercooled

liquid region. These nanocrystals are much harder than the amorphous matrix phase and

can enhance the mechanical properties of BMGs. When they are formed, they do not

0

50

100

150

200

250

10 30 50 70 90

inte

nsi

ty [

a.u

.]

2 theta [°]

La

LaNi2.28 Cu13La

La

La deformed in-situ BMG composite

as-cast in-situ BMG composite

20

carry any load initially and just move with the viscous flow of the matrix phase. Some

crystals then make contact with one another and agglomerate, forming nanocrystal

aggregate bands with stronger interfaces; these not only interrupt the viscous flow of the

amorphous matrix but also interact with each other. Since these bands are surrounded by

the amorphous phase, they will be aligned in the direction of flow of the amorphous

phase. The bands can strengthen the material by growing and increasing in size and/or

continuously contacting and interacting with neighboring bands (Bae et al., 2002).

Similarly, Cu-La, and Ni-La intermetallic compounds, and possibly nanocrystals that

may not be detected by XRD because of their low volume fractions, form during

deformation and are believed to be stronger phases that generate the secondary work

hardening observed in the stress-strain response of La-based in-situ BMG composites.

Cyclic loading-unloading compression tests were also performed on BMG composite

samples at various stages of significant plastic deformation. No noticeable change in the

Young’s modulus was observed in the loading-unloading cycles. Therefore, it is

concluded that the influence of nanocrystal/intermetallic compound formation on the

elastic response is negligible, and the secondary work hardening is not due to changes in

the elastic stiffness of the composite.

Figure 6 shows optical microscopy images of cross-sections of a polished as-cast La-

based in-situ BMG composite sample.

Fig. 6 - Optical microscopy images of cross-section of a polished as-cast La-based in-situ BMG composite

sample. The brighter phase is the amorphous matrix phase and the crystalline phase is darker

The oriented lanthanum dendrites (darker phases) are distributed homogeneously and

randomly within the amorphous matrix (brighter phases) in all directions across the cross-

section of the specimen. Optical microscopy images of cross-sections of a deformed

sample after compression are shown in Fig. 7.

21

Fig. 7 - Optical microscopy images of polished La-based BMG composite after compression to 0.8

[mm/mm] strain

In the deformed sample, the darker dendrites have been broken into smaller pieces, lost

their connectivity and are no longer homogeneously distributed. Furthermore, the work

by Fu et al. (2007a) on compression of La-based BMG composites with a 37%-52%

lanthanum crystalline phase near the glass transition temperature of ~0.9��, showed that

there is no intermetallic formation, based on XRD results; they also reported no work

hardening after the initial stress overshoot. Therefore, it can be concluded that although

the lanthanum dendrites contribute to delaying sudden failure via the formation of shear

bands, they are unable to form a structural framework to bear the loads imposed on the

material, and their contribution to the secondary work hardening is not significant.

5.3 FEM Simulation

The VUMAT user-subroutine facility in the commercial finite element program

ABAQUS/Explicit was employed to incorporate the constitutive model proposed to

describe the mechanical response of the in-situ BMG composite. Three-dimensional brick

elements (Abaqus C3D8R) were used to simulate simple compression. The initial

undeformed finite-element mesh of a BMG composite specimen is shown in Fig. 8.

Compression is applied along the top of the specimen12

, while the nodes at the bottom

surface are fixed in the loading (height) direction, and a velocity profile is imposed on the

top surface to simulate the desired strain rate. Figure 8 also shows a comparison between

the experimental data and simulation results in terms of the compressive stress-strain

response at 165ºC and different strain rates – 0.001/s, 0.003/s, 0.006/s and 0.009/s.

12

The simulation results for uniaxial compression tests are independent of the geometry of the 3D FEM

model; for example cylindrical and cubic models yield identical stress-strain responses for the same loading

conditions. Here, a cubic model is shown, and is representative of the sample.

22

Fig. 8- Initial undeformed mesh of La-based in-situ BMG composite specimen, and comparison of

simulation and experimental compressive stress-strain responses at different strain rates of 0.001/s, 0.003/s,

0.006/s and 0.009/s, at 165 °C.

The material parameters for the La-based in-situ BMG composite at high homologous

temperature are presented in Table 2

Table 2 - Material parameters for a La-based in-situ BMG composite in the supercooled liquid region.

e = 3.24GPa � = 230MPa ´ = 0.12 ℋc = 33.33MPa g = 10.33GPa m = 0.09 I£ = 1.5 ξ� = 0.0057 ℎC = 50 IC = 0.001 ⁄ I� = 0.3 sbl = 383 GJ mE⁄ C = 150MPa ζC = 0.0009 �� = 0.37 sbl¶ = 3220GJ mE⁄

�∗ = 500MPa ω = 1 × 10�� � = 1.201 × 10�� mE J⁄

The material parameters that have been determined are: e,g,IC,m, C, �,ℎC,ζC,ξ�,´,sbl , sbl¶ , �� , �, I� , I£ , ℋc ,ω,�∗.One parameter that can be extracted from the linear

compressive stress-strain response phase of the BMG composite is the Young's modulus,

which was found to be 8.80 GPa at 165 ºC. An approximate value of 0.358 for the

Poisson's ratio of La-based metallic glass is assumed (Chen et al., 2008); hence the value

for the shear modulus e is 3.24GPa, and the bulk modulus g is 10.33 GPa. The thermal

equilibrium free volume concentration ξ� is calculated from ξ� = ξn + ko\θ − θn],where the value of the glass transition temperature is θn = 430K (Table 1); the thermal

equilibrium free volume concentration at the glass transition temperature ξn = 5.58 ×10�E and the constant of proportionality kθ = 1.54 × 10�¹ K⁄ for La-based BMG

amorphous alloys were adopted from the work of Ekambaram et al. (2010).Hence at an

ambient temperature of 438 K, ξ� is 0.0057. The pressure sensitivity parameter is taken to

be ´ = 0.12, which is the value determined for a Vitreloy 1 metallic glass (Lu, 2002).

The reference flow-rate IC is taken to be 0.001s�3, which is close to the experimental

strain rates. Other undetermined material parameters in the constitutive model for the

0

50

100

150

200

250

300

350

400

450

500

0 0.2 0.4 0.6 0.8

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.009/s BMG Composite, FEM, Predicted0.009/s BMG Composite, Expm

0.006/s BMG Composite, FEM, Predicted0.006/s BMG Composite, Expm0.003/s BMG Composite, FEM, Fitted0.003/s BMG Composite, Expm0.001/s BMG Composite, FEM, Fitted0.001/s BMG Composite, Expm

23

BMG composite (i.e. m,ºC, sbl,sbl¶ , C, �, ℎC, ��, �, I� , I£ , ℋc,ω, �∗) were obtained

by fitting the stress-strain curves from simulations with experimental data. The rate

sensitivity parameter m for the supercooled liquid region was initially estimated from m ≈ log\σ½¾¿G σ½¾¿`⁄ ] log(�+ �3)⁄⁄ where σ½¾¿ À is the stress in the post-overshoot

region for the experimental strain rate of �Á; the value was then calibrated by fitting and

found to be m = 0.09. The value for ºC in the function for free volume creation º =ºCexp(� �∗)⁄ is taken to be constant at 0.0009, which is close to the value that

Thamburaja and Ekambaram (2007) used in their simulations for Vitreloy 1, and the

value for �∗ = 500MPa was obtained by fitting with experimental data in the softening

region of the stress-strain response. To determine the values of sbl and sbl¶, an approach

similar to that by Thamburaja and Ekambaram (2007) was used. It is assumed that

hydrostatic pressure has a negligible effect on free volume generation compared to

creation of free volume from plastic shear and relaxation (see Eq. (39)). Furthermore,

crystallization is temporarily retarded by assigning the energetic interaction coefficient

the value of ℋc = 0; this precludes secondary work hardening in the stress-strain

response of the BMG composite (see Eq. (26)2 and Eq. (32)). It is also assumed that the

free volume concentration in BMG composites is able to attain a steady state, such that

free volume creation is equal to its annihilation. These assumptions and the steady state

condition for free volume evolution reduces Eq. (39) to

ζI = XICsblsbll Y (ξ − ξ�) (41)

Since experimental measurement of free volume evolution during deformation is not

feasible, it is assumed that the variation of free volume is of the same order of magnitude

as the free volume in Pd-Ni-P BMGs, measured in the work of de Hey et al. (1998); the

ratio of sbl sbll⁄ is determined from Eq. (41). Consequently, the term responsible for

generation of free volume due to hydrostatic pressure (i.e. −\IC sbll⁄ ]P} in Eq. (39)) is re-

activated and the minimum value of sbll determined such that the steady-state free

volume in simple tension and simple compression at a given applied strain rate are

approximately equal (i.e. [ξÂÂ,¾�ÃÂ. − ξÂÂ,ÄÅ�Æ.i < 1 × 10�¹ (Ekambaram et al., 2008)).

With sblldetermiend and sbl sbll⁄ known, sbl can then be determined. The value of sbl was

found to be 383GJ/mE, and sbll = 3220 GJ °E⁄ . The critical strain I� is assumed to be

0.3, which is around the strain value at which secondary hardening in the stress-strain

response of in-situ BMG initiates; the final strain I£ is assumed to be 1.5, which implies

that at this strain, all possible potential crystal sites have been activated in the material.

The value of ℎC, which governs the slope of the initial post-yield hardening in the stress-

strain curves, is taken to be 50, which is obtained by fitting the stress-strain curve with

experimental data. �� controls the ratio of the peak stress attained to the steady-state

plateau stress, and is found to be 0.37 from the experimental stress-strain curves. The

dimensionless parameter ω is taken to be 1 × 10��; and ensure that if the viscous stress

approaches a zero value (in the absence of stress), error resulting from division by a near-

zero value will not occur (see Eq. (35)). Re-activation of crystallization by setting the

24

energetic interaction coefficient ℋc to 33.33MPa, enables the secondary hardening

response to fit the experimental stress-strain data. C, � and � were also derived from

fitting the simulated stress-strain curve with experimental results.

Figure 8 shows that the simulated stress-strain curves correlate well with the

experimental responses. The experimental results for strain rates of 0.001/s and 0.003/s

were used to obtain the material parameters and based on these, FEM simulation was

employed to predict the stress-strain curves for strain rates of 0.006/s and 0.009/s. A

comparison of the simulation results with experimental data shows that beyond a certain

strain (~0.55), the rate of secondary work hardening observed experimentally is smaller

than that predicted by the model. This can be accounted for by considering the integration

and growth of micro-cracks, which reduce the strength of the material; it is also possible

that the resistance to plastic flow also approaches a saturation value at large plastic

strains. Figure 10 shows micro-cracks found in severely deformed in-situ BMG

composite samples.

Fig. 9 - Optical micrographs of typical cracks observed in severely compressed in-situ BMG composite

samples

It is noted that the cracks have propagated through both the amorphous matrix and the

dendritic phases. Further investigation would need to be undertaken to identify

appropriate fracture criteria for BMG composites.

Figure 10(a) shows the corresponding variation of the normalized free volume

concentration ≡ (ξ ξn⁄ ) with strain for the simulation results in Fig. 8. Free volume

creation is responsible for the strain-softening observed in the BMG composite. This

trend of free volume evolution is similar to the experimental findings reported for

amorphous alloys (e.g de Hey et al. ,1998).

25

Fig. 10 – (a) Predicted variation of normalized free volume with strain at a temperature of 438 K for

various strain rates; (b) Predicted crystallization fraction κ as a function of strain for a temperature of 438 K

The variation of the crystalline fraction U with strain for a temperature of 438 K is shown

in Fig. 10(b). This evolution of crystallization is responsible for the secondary work

hardening observed in the stress-strain response of the BMG composite. (Further

investigation is required to obtain experimental data on the evolution of free volume and

crystallization in such composites.)

6 Conclusions

A three-dimensional constitutive model for in-situ BMG composites based on finite-

deformation macroscopic theory and experimental data to describe the stress-strain

response at high homologous temperature and various strain rates has been established.

The constitutive model was implemented in ABAQUS/Explicit finite element software

by writing a user-defined material subroutine using the VUMAT facility. To the best of

knowledge, this is the first time a constitutive model has been formulated to model the

behavior of semi-crystalline BMG composite in the supercooled liquid region. The model

is able to describe the stress-strain responses observed and there is good correlation with

experimental data.

The experimental stress-strain behavior of a La-based in-situ BMG composite within the

supercooled liquid region reveals post-yield hardening-softening followed by secondary

strain-hardening. This contrasts with monolithic La-based BMG samples, for which the

stress drops to a plateau after an initial stress overshoot. XRD results for the BMG

composite reveal the formation of intermetallic compounds, and an increase in the

volume fraction of crystalline lanthanum precipitating out during deformation. It is

believed that the formation of intermetallic compounds is associated with the storage of

energy in the material, and affects the stress-strain response through the generation of

secondary strain-hardening.

1.02

1.025

1.03

1.035

1.04

1.045

1.05

0 0.2 0.4 0.6 0.8

True strain [mm/mm]

0.009/s, FEM simulation

0.006/s, FEM simulation

0.003/s, FEM simulation

0.001/s, FEM simulation

No

rma

lize

d fr

ee

vo

lum

e (

ξ/ξ g

) [-

]

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8

True strain [mm/mm]

0.009/s, FEM simulation0.006/s, FEM simulation0.003/s, FEM simulation0.001/s, FEM simulation

Cry

sta

lliz

ati

on

fra

ctio

n (κ

) [-

]

(b)

26

Experimental data also indicates that the stress-strain response of the BMG composite in

the supercooled liquid region cannot be obtained using a homogenization approach which

combines the behavior of monolithic BMG and crystalline lanthanum – i.e. the

amorphous and crystalline phases of the composite. This contrasts with the employment

of homogenization to describe the overall stress-strain behavior of BMG composites at

room temperatures, which is a notable difference that this study has identified.

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29

NOT FOR PRINTING>>>

NOT FOR PRINTING! JUST FOR ENDNOTE software

(Neto et al., 2008)

(Spaepen, 1977)

(Ekambaram, 2009)

(Thamburaja, 2011)

Fig. 1 – (a) XRD pattern for La-based in-situ BMG composite and monolithic BMG alloy; (b) Differential Scanning Calorimetry (DSC) at a heating rate of 20 K/min for La61.4Al15.9Cu11.35Ni11.35 BMG alloy, La74Al14Cu6Ni6 BMG

composite and pure Lanthanum La100

0

20

40

60

80

20 70 120

Inte

nsi

ty [

a.u

.]

2 Theta [°]

in-situ BMG composite

monolithic BMG alloy

a)

0

5

10

15

20

330 430 530 630

Exo

the

rmic

[m

W/g

]

Temperature [K]

in-situ BMG composite

monolithic BMG alloy

pure lanthanum

θgθx θm

b)

Fig. 2 - Stress-strain response of in-situ BMG composite with 50% volume fraction of crystalline phase, at 165 ºC,

for different strain rates.

0

100

200

300

400

500

0 0.2 0.4 0.6 0.8

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.009/s BMG Composite, Expm0.006/s BMG Composite, Expm0.003/s BMG Composite, Expm0.001/s BMG Composite, Expm

Fig. 3 - Compressive stress-strain response of BMG composite, monolithic BMG and lanthanum at 165 ºC and

different strain rates. X indicates the point of failure.

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.003/s BMG, Expm0.003/s BMG Composite, Expm

0.001/s BMG Composite, Expm0.001/s BMG, Expm

0.01/s Lanthanum, Expm0.001/s Lanthanum, Expm

XX

Fig. 4 - Stress-strain responses of in-situ BMG composite and monolithic BMG at 165 °C and strain rates at which

failure (inhomogeneous deformation) initiates – 0.006/s for BMG, and 0.01/ for BMG composite

0

100

200

300

400

500

600

0 0.2 0.4

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.006/s BMG, Expm

0.01/s BMG Composite, Expm

X

X

Fig. 5 - XRD spectra for La-based BMG composite – as-cast and deformed

0

50

100

150

200

250

10 30 50 70 90

inte

nsi

ty [

a.u

.]

2 theta [°]

La

LaNi2.28 Cu13La

La

La deformed in-situ BMG composite

as-cast in-situ BMG composite

Fig. 8- Initial undeformed mesh of La-based in-situ BMG composite specimen, and comparison of simulation and

experimental compressive stress-strain responses at different strain rates of 0.001/s, 0.003/s, 0.006/s and 0.009/s, at

165 °C.

0

50

100

150

200

250

300

350

400

450

500

0 0.2 0.4 0.6 0.8

Tru

e S

tre

ss [

MP

a]

True Strain [mm/mm]

0.009/s BMG Composite, FEM, Predicted0.009/s BMG Composite, Expm

0.006/s BMG Composite, FEM, Predicted0.006/s BMG Composite, Expm0.003/s BMG Composite, FEM, Fitted0.003/s BMG Composite, Expm0.001/s BMG Composite, FEM, Fitted0.001/s BMG Composite, Expm

Fig. 10 – (a) Predicted variation of normalized free volume with strain at a temperature of 438 K for various strain

rates; (b) Predicted crystallization fraction κ as a function of strain for a temperature of 438 K

1.02

1.025

1.03

1.035

1.04

1.045

1.05

0 0.2 0.4 0.6 0.8

True strain [mm/mm]

0.009/s, FEM simulation

0.006/s, FEM simulation

0.003/s, FEM simulation

0.001/s, FEM simulation

No

rma

lize

d fr

ee

vo

lum

e (

ξ/ξ g

) [-

]

(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8

True strain [mm/mm]

0.009/s, FEM simulation0.006/s, FEM simulation0.003/s, FEM simulation0.001/s, FEM simulation

Cry

sta

lliz

ati

on

fra

ctio

n (κ

) [-

]

(b)

Table 1 - Results of DSC analysis for a heating rate of 20 K/min, for La74Al14Cu6Ni6 and La61.4Al15.9Cu11.35Ni11.35,

where Vf is the volume fraction of crystal phase in the alloy, θg the glass transition temperature, θx the crystallization

temperature and θm the melting temperature

Alloys Vf θg (°K) θx (°K) θm (°K)

La61.4Al15.9Cu11.35Ni11.35 0% 419 477 659

La74Al14Cu6Ni6 50% 430 478 661

Table 2 - Material parameters for a La-based in-situ BMG composite in the supercooled liquid region.

� = 3.24GPa �� = 230MPa � = 0.12 ℋ� = 33.33MPa

� = 10.33GPa m = 0.09 �� = 1.5 ξ� = 0.0057

ℎ� = 50 ��� = 0.001 �⁄ � = 0.3 s"# = 383 GJ m&⁄

�� = 150MPa ζ� = 0.0009 () = 0.37 s"#* = 3220GJ m&⁄

+∗ = 500MPa ω = 1 × 10/0 1 = 1.201 × 10/0 m& J⁄

Highlights

• A three-dimensional constitutive equation to describe BMG composites at high

homologous temperatures has been developed.

• Response of BMG composites under compression at various strain rates in the supercooled

liquid region was investigated.

• XRD analysis of BMG composites shows the formation of intermetallic compounds during

compression.

• The constitutive equation established was incorporated into a finite-element program to

simulate experiments carried out.

• The simulation results display good agreement with experiments in terms of stress-strain

behaviour.