8/13/2019 BMG orig

1/22

8/13/2019 BMG orig

2/22

1. Introduction and Motivation

Amorphous metals (metallic glasses) are metallic alloys which, unlike the crystalline

materials, do not have ordered atomic structure. They are formed by extremely rapid cooling

of the order of millions of degrees per second. Therefore, crystals cannot be formed and

glassy state of material is produced. In 1960, the first amorphous metal was reported by

Klement et al at Caltech by rapid quenching of Au-Si alloy. Metallic glass formation requires

extremely rapid quenching, furthermore their thermal conductivity is lower than crystalline

materials. Therefore, they can be produced only in certain forms (e.g. foils, wires) with

thickness limited to less than 100 micrometres. However, more recently, some alloycompositions have been explored with ever lower critical cooling rates to produce thick

layers (over 1 millimetre) and these alloys are called bulk metallic glasses (BMG).

They have numerous unique properties by virtue of their amorphous structure. Few of

them are, a very high yield strength ranging between 1840 2100 MPa, high yield strain() , a high elastic modulus (E) in the range of 47 102 GPa and non-negligible toughnesswhich depends on alloy composition. Vitreloy (Zirconium based alloy), a new metallic alloy,

has tensile strength twice as that of high grade titanium, which was developed as a part of

Department of Energy andNASA research of new aerospace materials in 1992.

Depending upon the temperature, stress and glass condition the plastic deformation is

classified as homogeneous and inhomogeneous deformation. At low stresses and the

temperatures exceeding half the glass transition temperature (), the metallic glassesundergo viscous flow under which plastic strain is distributed uniformly in different volume

elements within the material. This type of deformation is termed as homogeneous

deformation. Additionally, at high stresses and temperatures below , resulting plasticstrain, which most likely occurs at the site of free volume concentrated in narrow shear

bands, is distributed non-uniformly. This localised deformation is termed as inhomogeneous

deformation. This phenomenon has become an interesting area of research. Due to its'

superior properties, BMG have promising future with potential applications ranging from

sports equipment to MEMS. Therefore, unique properties of BMG was motivation for this

study wherein the aim is to understand fracture behaviour and development of the plastic

zone ahead of crack tip and stress field variation in near crack tip in BMG specimen.

http://en.wikipedia.org/wiki/Vitreloyhttp://en.wikipedia.org/wiki/Vitreloyhttp://en.wikipedia.org/wiki/United_States_Department_of_Energyhttp://en.wikipedia.org/wiki/NASAhttp://en.wikipedia.org/wiki/NASAhttp://en.wikipedia.org/wiki/United_States_Department_of_Energyhttp://en.wikipedia.org/wiki/Vitreloy

8/13/2019 BMG orig

3/22

2. Literature review

Schuhet al.

(2007) gave detailed insight into microscopic behaviour which resultsinto formation of shear bands during plastic deformation. The atomic bonds in the metallic

glasses are of metallic character, which can easily accommodate strain. The relative energy

required for local rearrangement of atoms is higher than that of crystalline materials. In 1979

Argon proposed mechanism of local arrangement of atoms called 'Shear transformation zone'

(STZ). STZ are local clusters of atoms which accommodate shear strain. Atomistic

simulation on number of composition suggest that STZ deformation is common in all

metallic alloys regardless of structure, size and energy scale of STZ. Unlike crystalline

materials, STZ deformation is not a structural deformation. Rather it is transient in nature

which is characterised in local volume. Site of higher free volume leads to activation of STZ

which in turn accommodates local shear, accompanied by dilatation. The free volume

adequately describes the structure of deformation in metallic glasses. Hence the plastic

deformation is described by accumulation of local strain in shear bands which accumulate

through the operation of STZ and redistribution of the free volume. Since the dilatation is

accompanied with plastic deformation in metallic glasses, the effect of hydrostatic stress is

needed to be taken into account.

Subramanya et al. (2005) carried out the 3D finite element study of mixed mode

(I&II) crack tip field in elastic plastic solid using boundary layer formulation under small

scale yielding condition. They have investigated the structure of stress field and established

the validity of 2D plane stress and plane strain approximations. The evolution of plastic zone

in term of its size and shape with increasing load has been discussed as well. They conclude

that in general, the plane stress conditions prevail at a distance from crack front exceedinghalf the plate thickness, although it could be slightly smaller for predominantly mode II

loading.

Subramanya et al. (2006) performed the finite element study of 3D crack tip fields in

pressure sensitive plastic solids (such as polymers or metallic glasses) under mode I, small

scale yielding condition. They have used the Extended Drucker-Prager yield model, assuming

material to obey a small strain condition. The effect of pressure sensitivity and plastic

dilatancy on evolution of plastic zone and stresses is systematically studied. They have found

8/13/2019 BMG orig

4/22

that pressure sensitivity enhances the plastic strain and crack opening displacements, which

leads to significant drop in hydrostatic stress all along the 3D crack front.

Anand & Su (2004) have developed a constitutive model for elastic viscoplastic

material to capture the response of to pressure sensitivity and plastic dilatancy. They have

developed a finite deformation, Mohr-Coulomb type constitutive theory which captures the

response of metallic glasses accurately. The deformation of amorphous metallic glasses is

studied by implementing constitutive model in a finite element program. They concluded that

the numerical simulations of an amorphous metallic glass in tension, compression, strip-

bending and indentation qualitatively capture the major features of corresponding results

from physical experiment available in literature.

Tandaiya et al. (2007) studied the stationary crack tip fields in metallic glasses to

understand the effect of factors, which controls the crack tip plasticity and imparts the

toughness to material. The finite element analysis is performed under plane strain small scale

yielding condition. The constitutive model developed by Anand & Su, which accounts for

pressure sensitivity and plastic strain localised into discrete shear bands is employed to

capture the behaviour of this material. They have observed the influence of internal friction

factor on plastic zone, stress and deformation field and concluded that higher friction

parameter enhances the plastic strain, which substantially decreases the opening stress which

in turn leads to larger plastic zone and enhances the fracture toughness of material. For the

material the plastic deformation is localised in form of shear bands. In this study simulation

of shear bands are performed which qualitatively matches with those in the experiments.

Tandaiya et al. (2010) developed an efficient algorithm to examine the pressure

sensitivity and plastic dilatational response of bulk metallic glasses. This algorithm is based

on Mohr Coulomb type material which accounts the finite deformation framework. The

accuracy and performance is verified by several benchmark problems, like in single element

test, where a single hexahedral finite element is subjected to uniaxial tension and

compression. The plot of true stress versus logarithmic strain represents the behaviour of bulk

metallic glasses.

1. As expected from pressure sensitivity model, the yield stress in compression is higher than

that in tension.

8/13/2019 BMG orig

5/22

2. The rapid strain softening followed by perfectly plastic behaviour, which is a typical

feature of BMGs.

Fig. 1 Plot of True stress vs. Log. Strain

3. Issues of investigation and objectives

The study regarding the fracture behaviour, stress fields and plastic deformation near

crack tip in bulk metallic glasses is confined to plane strain condition. Which shows that

plastic deformation is localised in shear band are assumed to occur on six potential slipplanes. But the actual stress field near the crack tip are essentially three dimensional in

nature, which are the conditions between plane strain and plane stress. The real condition of

plastic deformation which are assumed to occur in slip planes is unknown. The variation of

stress fields from plane strain (middle plane) to plane stress (free surface) condition is not

established. The evolution of plastic zone in term of size and shape on different planes

through the thickness is required to be recognised to know the actual fracture behaviour of

bulk metallic glasses. As the practical BMG structures are 3D in nature and to exploit the

8/13/2019 BMG orig

6/22

unique properties of this material, the analysis in practical conditions (i.e. 3D) is required to

be done.

Based on above argument following are the objectives of the study-

1. Mode I and mixed mode (I&II) simulation of crack tip fields in 3D under small scaleyielding using boundary layer formulation of fracture mechanics.

2. Simulation of 3D structure of shear band patterns near the crack tip in 3D SSYboundary layer formulation under mode I and mixed mode (I&II) loading and its'

comparison with experimental observations.

3. Comparative study of crack tip fields in SSY and in actual fracture specimens ofdifferent geometries: Dominance of elastic-plastic crack tip fields in actual specimens

(of different geometries) in 2D and 3D.

4. Simulation of 3D structure of shear band patterns near the crack tip in actual fracturespecimens of different geometries and its comparison with experimental results.

4. Report on present investigation

4.1 Material model

Mohr coulomb model, considering finite deformation, recently proposed by AS[], is

the basis of constitutive equations used in this study. The deformation gradient is given by

multiplicative decomposition of elastic and plastic part as

(1)

The elastic part of deformation is assumed negligible. And the plastic flow is assumed to be

occurred by shearing accompanied by dilatation relative to some slip system. These slip

systems with conjugate pair are defined relative to principal direction of stress (denoted by

unit vectors ), lying in planes formed by ( ), ( ) and ( ). Thedirection of slip system is represented by , where denotes slip system number, whichmake an angle of

( is the angle of internal friction), with maximum principal

stress direction in that plane. On the potential slip system, the resolved shear stress andcompressive normal traction is given by

8/13/2019 BMG orig

7/22

and

respectively, where is normal to slip plane for th slip system and is Krichhoffstress. The slip rates developed by conjugate pair are equal, since they have equal resolved

shear stresses and normal compressive tractions.

The flow rule for plastic part of spatial velocity gradient is given by

[( ) ( ) ], (2)where, is plastic shearing rate and is dilatancy function. The viscoplastic law is given

by

(3)

where, is reference plastic shearing rate and is strain rate sensitivity parameter. Inthe limit as And is the friction parameter, called cohesion is stress likeinternal variable which represent the yield strength in pure shear. When , it representsthe associated flow rule. The dilatancy parameter is given as function of plasticvolumetric strain as

{ } (4)

So as changes from to , it varies smoothly from to .

The plastic volumetric strain , which defined as ,where is associatedby change in local free volume of amorphous material, which governs their plastic

deformation. The evolution of is governed by

The cohesion is assumed to vary with as

(6)

8/13/2019 BMG orig

8/22

It is evident from equation (4)-(6), that would increase monotonically from virgin phase

initially and eventually saturate to equilibrium value of . On other hand, would smoothly

decrease from and become stable at value of the above model is

incorporated in general purpose non-linear code ABAQUS/Standard through written user

material subroutine UMAT. Fully implicit backward Eular approach is used for integration of

constitutive equation.

The objective of this study is to examine the stress field and shape to plastic zone in mode I

near crack tip under small scale yielding condition by assuming a uniform value of initial

cohesion for entire domain.

4.2 Modelling and analysis aspects

To investigate the behaviour of 3D crack tip field under small strain yield (SSY)

condition boundary layer formulation is used.

4.2.1 Model details

In this study 3D circular disc of radius and thickness containing crack

along one of its radii is examined (refer Fig.). The crack has initial notch of diameter at the

crack root. The plate is located in X-Y plane and the crack front is along the Z axis. The ratio

of radius to thickness of plate is chosen to be 40, ensuring the plastic zone remains

well inside the plate boundary even if the plastic zone size increased five times the thickness

of plate. The plate radius is 4000 times to initial crack root notch diameter. The disc is

modelled using finite element mesh containing 8-noded isoparametric hexahedral element.

To alleviate the effects of mesh locking due to near plastic incompressibility, the hybrid

element formulation feature, available in ABAQUS, is employed.

8/13/2019 BMG orig

9/22

Fig.1 Finite element mesh used in simulation showing full domain

Fig.2 Finite element mesh near notch tip in enlarged view

Due to symmetry arising because of Mode I condition, only one fourth of the plate

(i.e. one half of the thickness of upper half plane) is modelled by applying appropriate

boundary conditions. The notch surface is assumed to be traction free. The mesh contains 10

layers of elements through half of the thickness of plate. The layers become thinner towards

the free surface in order to capture large stress variations of stress on free surface. Each layercomposed of 80 rings of element, which coarsened very fast while moving along the radius

from crack tip to periphery. The mesh is well refined at crack tip with smallest size of

element in radial direction is about . The element size in direction is

between to . The mesh comprised of total 48640 elements and 54923 nodes.

The in-plane displacement components based on leading term of mode I is prescribed on the

outer boundary, which are given by

8/13/2019 BMG orig

10/22

For 3D, in the aforementioned equations.

The load is applied gradually with increasing value of stress intensity factor .

4.2.2 Choice of material parameter

The material parameters, which are employed in constitutive equation to study the behaviour

of crack tip field in amorphous solid are as follows:

Young modulus

Initial cohesion

Poissons ratio = 0.36

Friction parameter

Pressure sensitivity parameter

Strain rate sensitivity exponent

Since the value of strain rate sensitivity exponent is small, the response of material is rate

independent. The variations of stress are shown in term of Cauchy stress components

normalised by initial value of cohesion , which is stress like quantity.

5. Results and discussion

With increasing load, the evolution of plastic zone in terms of shape and size is

studied. The maximum extend of plastic zone) with corresponding normalizedstress intensity factor is recorded. As the load was applied gradually with increasing value of

8/13/2019 BMG orig

11/22

stress intensity factor , this shows that the load level is well characterised by the ratio ofmaximum extend of plastic zone to the thickness of plate. Accordingly, three load levels with

corresponding value of ) is defined -

Load level )High 1.36

Moderate 1

Low .1

5.1 Comparison of stress field in BMG and Elastic model

(b)

(a)

(c)

8/13/2019 BMG orig

12/22

Fig. 4 Comparison of stress field in BMG and Elastic model at high load level at at ) = 1.36 (a)Angular variation of Hydrostatic and tangential stress (b) Radial variation of Hydrostatic and tangential stress

at ) = 1.36 and (c) Variation of stress through thickness at ) = 1.36 and

Unlike the crystalline materials, BMGs are devoid of grain boundaries. These grain

boundaries are treated as barrier in shear band propagation. So in absence of such barriers,

the shear band propagates readily and cause failure without any discernible plastic

deformation. It is observed that at room temperature, under tensile loading most of the BMG

deforms elastically. Thus these materials are assumed to be elastic. But depending upon the

loading condition, stress and temperature level, large scale of plastic deformation is also

observed. Thus appropriate investigation must be carried out to justify this material before

accounting this assumption. Hence here the analyses are conducted on both BMG and elastic

material considering the same model, loading and boundary conditions. The variations of

stresses are plotted around the crack tip in all dimensions to show, how both materials varied

in terms of stresses. Fig.4(a) show the angular variation of hydrostatic and tangential stresses,

both stresses in both materials show the decreasing stress from ahead of crack tip but the

magnitude and trend both are different. In Fig.4(b), for elastic material, the hydrostatic and

tangential stress are limiting to higher magnitude at whereas in BMG these valuesstart from zero. Fig.4(c) shows variation of stress through thickness at . So theabove trend of stresses in all dimensions invalidate the assumption to consider the BMG as

elastic material.

5.2 Comparisons of stress field and plastic strain value in 2D plane strain and 3D

The stress field and plastic strain value are compared between 3D and 2D plane

strain condition. And then the 3D and 2D plane strain variation are itself compared at two

8/13/2019 BMG orig

13/22

different load levels, first at low load and second at high load corresponding to the value of

and respectively. As the loadincreases from low to high,stresses and plastic strain value undergo the transition from plane strain to plane stress.

Because with increase in load, the plastic zone size increases, pertaining to plane straincondition throughout the thickness. At low load level the 2D plane strain condition should

well match with that at middle plane. But here the meshing of the model through the

thickness is done with 10 layers of elements, which are coarsening from free surface to mid-

plane. So while calculating the stresses and plastic strain on mid-plane, the integration points

are far away from the that plane and interpolated values are calculated, which are different

from the actual. Hence the difference in magnitude is observed in 2D plane strain and 3D

mid-plane stress and plastic strain values at low load level, however the trends are same.

5.2.2 Angular stress variation

Fig.5 Angular variation of plastic strain in 3D

for different planes and 2D plane strain (a) (b)

At low load the trend of plastic strain value for plane strain is in good agreement with

that of mid-plane and intermediate plane. These show the flat peak at which graduallydecrease from to . For free surface steep peak appear at , which at the highload shifted at from back to ahead of crack tip and mid-plane value steeply decreasesahead of crack tip.

8/13/2019 BMG orig

14/22

Fig.6 Angular variation of hydrostatic stress in 3D for different planes and 2D plane strain (a)

(b)

Fig.7 Angular variation of tangential stress in 3D for different planes and 2D plane strain (a)

(b)

In fig.6 At low load, the plane strain and mid-plane show the flat peak which

decreases while moving toward the back of crack tip. At high load the width of flat peakreduces, which shows the transition from plane strain to stress and at free surface the stress

become uniform around the tip. In Fig.7 it show similar trend for mid plane as that in

hydrostatic, flat peak reduce and gradually decrease from front to back of crack tip, which

show transition. The plane strain condition matches well with mid-plane value at low load.

5.2.3 Radial variation in theta=0

8/13/2019 BMG orig

15/22

Fig.7 Radial variation of plastic strain in 3D for different planes and 2D plane strain (a)

(b)

Fig.8 Radial variation of hydrostatic in 3D for different planes and 2D plane strain (a)

(b)

Fig.9 Radial variation of tangential stress in 3D for different planes and 2D plane strain (a)

(b)

In Fig.7 at low load, the variation of plastic strain on different plane on 3D can easily be seen

by different curves. Whereas at high load the transition from plane strain to plane stress caneasily be noticed. And the plastic strain value decrease very sharply comparatively at low

8/13/2019 BMG orig

16/22

load. In Fig.8 at low load the plane strain show good agreement with 3D mid-plane value and

show the peak near to crack tip. The stress value decrease very steeply while moving along

the radial direction. At high load the peak is shifted in radial direction. The difference

between curves on plane strain and 3D planes are evident of transition at high load. In Fig.9

the trend followed by tangential stress is same as that of hydrostatic at low load as well as arhigh load.

5.2.5 Thickness variation

Fig. 10 Variation through the thickness at and (a)plastic strain (b) hydrostatic stress (c) tangential stressat

The variation of stresses and plastic strain through thickness at the different normalised

radial distance from crack front are plotted. The values of are choosen such that theycome well inside the plastic zone at . In fig. 10(a) for , which isfar from crack front, the plastic strain is almost constant through the thickness. While for

other two values of , it show some gradient from mid-plane to free surface.in Fig10(b)the

8/13/2019 BMG orig

17/22

5.2.6 Contour of plastic zone

At lowload

8/13/2019 BMG orig

18/22

At moderate

8/13/2019 BMG orig

19/22

8/13/2019 BMG orig

20/22

At high load

Fig. 3 Plastic zone contour in plane strain condition

Fig.4 Plastic zone contour at middle plane in 3D

8/13/2019 BMG orig

21/22

Fig.5 Plastic zone contour at free surface in 3D

Fig.6 Plastic zone contour through the thickness (upper plane is free surface)

8/13/2019 BMG orig

22/22