1.2 Deformation mechanism

the liquid so fast that there will not be sufficient time to form the crystal. By this method, the liquid solidifies as MG, which is metal without any particular arrangement of atoms. When Jun et al. [3] performed their classic experiment of rapidquenching on Al-Si alloys MGs came to reality. MGs are opaque, shiny, smooth, gray in color and less brittle than conventional glass. Yield strength of MGs is reported to be as high as 1.9 GPa [4] and is found to be more resistant to fracture than ceramics. As there are no grain boundaries in MGs they are resistant to corrosion and wear. They have high electrical sensitivity which leads to low eddy current losses. Also, MGs formed from alloys of Fe, Co, or Ni has soft magnetic

In spite of all the attractive properties, commercial use of MGs was not possible until the 1990s. This is due to the fact that it was not practically possible to attain

[4]. Even if the cooling rates were attained, the thickness of MGs that was cast was very small because of slow heat conduction. The main focus was on how to decrease

researchers found that if a different metal alloy is added to the liquid metal, then the amorphous structure can be achieved at lower cooling rates. Presence of more number of elements increases the size and complexity of the unit cell which leads to amorphous formation. If there is a large difference between sizes of a radius of different atoms then packing density increases which again favors amorphous arrangement. Using this technique, around 1990, first commercial MG was developed and named "Vitreloy," also known as Liquidmetal [4]. Following the success of "Vitreloy," there was tremendous development in reducing the critical cooling rate (minimum rate to suppress the formation of crystalline lattice). Due to the increase in the achievable thickness (>1 cm) of MGs, they are also known as bulk metallic glasses (BMGs) [5]. A large variety of BMGs is found out using different

MGs are used in golf club heads, baseball bats and tennis rackets due to low damping and ability to transfer a large fraction of impact energy to the ball. Its high strength to weight ratio allows the mass to be distributed differently, which means different shapes can be achieved for club heads. They are also used in bicycle

As a result of being very good corrosion and wear resistant, use of MGs in luxurious items is increasing day by day. Also, they have excellent scratch resistance along with smooth and shiny surface which makes them ideal for watch cases,

MGs are biocompatible with a non-allergenic form which allows using it as a knee-replacement device or pacemaker casing. Also, they are used in surgical blades as they are less expensive than diamond blades as well as sharper and long-lasting than steel. Due to the lack of grain structure, a blade can be sharpened to an

Micro electro mechanical systems offer a very wide range of application for MGs. They are used in micromirrors used in digital light processor (DLP) technology for data projectors. MGs are lighter, stronger and easily molded due to which they are used in components of liquid crystal display (LED), ultra-personal computer screens and casing of cameras. Also, high hardness and lack of grain structure

exceptional edge. They can also be used in razor blades and knives.

the cooling rate and simultaneously increase the casting thickness. Finally,

°C/s) so that there will not be any time for crystallization

properties as well.

Metallic Glasses

required cooling rate (10<sup>6</sup>

alloys in varying proportions.

frames, hunting bows, fishing equipment and guns.

allow it to use as information storage of digital data.

spectacle frames, rings, pens and mobile cases.

1.1 Applications of MGs

64

Fracture toughness of MGs exhibits a very wide range from 1 to 140 MPa ffiffiffiffi mp [6]. Lewandowski et al. [7] found that the fracture toughness can be correlated with Poisson's ratio. As Poisson's ratio decreases fracture energy decreases resulting in brittle fracture. Brittle fracture surface shows very fine corrugations of the length scale of nanometers on the other hand ductile fracture surface shows a coarse pattern with deep impressions of the length scale of micrometers. Thus higher Poisson's ratio implies higher fracture toughness and a critical Poisson's ratio around 0.34 is found for the transition between brittle and ductile behavior [8]. In order to understand the effect of Poisson's ratio on the deformation mechanism of MGs, Murali et al. [8] conducted atomic scale simulations on two MGs, FeP and CuZr with Poisson's ratio 0.33 and 0.39, respectively. They found that FeP with Poisson's ratio 0.33 fractures in brittle fashion according to cavitation mechanism. Cavitation is the process of nucleation of voids inside a solid. In brittle fracture, the crack propagates by continuous series of nucleation of voids in front of the crack tip. This nucleation of void takes place in void-free solid when hydrostatic stress at any point reaches a critical value known as cavitation stress σc. Firstly, single void nucleates and grows and after sufficient growth of the void, next one nucleates at some distance from the new crack tip and crack propagation continuous. On the other hand, in CuZr with Poisson's ratio 0.39 instead of the propagation of crack tip, they observed blunting of the crack tip. Shear bands were formed near the crack tip as the applied strain was increased. This extensive shear banding near crack tip gives rise to ductile failure at large strains. There could be a large amount of dissipation of energy inside shear bands which can cause localized melting observed on the ductile fracture surface.

Hence two different type of fracture mechanism were observed depending on Poisson's ratio. Brittle MGs shows crack propagation while ductile shows crack blunting with extensive shear banding. In this chapter, the focus will be on the modeling of shear bands causing ductile failure.

### 1.3 Microscopic deformation models

A large amount of research has been done to understand the nature of mechanism which causes shear band, but still, it is not totally clear. Gilman [9, 10] modeled shear bands in MGs in terms of glide of dislocations. After that Chaudhari et al. [11] and Shi [12] investigated the stability of dislocations in MGs but results did not follow the results came from theoretical work. They found that both edge and screw dislocations are rather stable. Recently Takeuchi and Edagawa [13] did some atomic simulations to review so far proposed deformation models and to understand deformation induced softening, shear band formation and its development.

In the late 1970s, two classic theories were developed for the understanding of deformation in case of MGs assuming that the fundamental process responsible for deformation is a local arrangement of atoms which accommodates the local shear strain. The first theory is known as the shear transformation zone model by Argon. This model depends on the shear transformation zone (STZ) which is a local cluster of atoms undergoing plastic shear distortion. Argon [14] was first to develop a model of STZ where he treated the problem as an Eshelby inclusion. When MGs undergoes uniform stress STZ will be selected on the basis of energies which in turn must be dependent upon local atomic arrangements.

The second theory was developed by Spaepen which was based on the free volume model. The basic free volume model was developed by Turnbull [15, 16] and later on applied to the MGs by Spaepen [17]. This model is based on free volume concentration as an order parameter which is nothing but extra volume where atom can move freely. The distribution of free volume controls the deformation in MGs and regions having more free volume would localize the shear deformation. Deformation in MGs is considered a series of atomic jumps which in addition to temperature can be driven by applied shear stress.

and Greer et al. [37] gives a comprehensive review of recent advances in understanding the mechanical behavior of MGs, with particular emphasis on the defor-

Assuming no body force, governing equations for quasi-static deformation in

Under small deformation regime, strain displacement relation is given by

ui,j þ uj,i

where i, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, 3 and ðÞ<sup>j</sup> means differentiation with respect to jth spatial coordinate. σij are components of Cauchy stress tensor, ui are the components of displace-

<sup>ε</sup>ij <sup>¼</sup> <sup>1</sup> 2

ment, εij are components of strains. For MGs, strain is decomposed as [21],

ij þ ε p ij þ 1

the inelastic dilatation strain (strain due to excess free volume), where ξ is local concentration of free volume and ξ<sup>0</sup> is the free volume concentration at the reference state with no strain. Dividing the stress tensor into mean stress and deviatoric

p

<sup>σ</sup><sup>m</sup> <sup>¼</sup> <sup>1</sup> 3

τ<sup>e</sup> ¼

ij þ

in the same direction as deviatoric stress tensor Sij, with its rate depending upon concentration of free volume ξ<sup>f</sup> , effective shear stress τ<sup>e</sup> and mean stress σ<sup>m</sup> as:

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>f</sup> <sup>ξ</sup><sup>f</sup> , <sup>τ</sup>e, <sup>σ</sup><sup>m</sup>

Elastic strain and stress are relates to each other by Hooke's law,

<sup>σ</sup>ij <sup>¼</sup> <sup>2</sup>μ ε<sup>e</sup>

Following Mises theory [39] flow of deviatoric plastic strain ε

∂ε p ij

where μ is shear modulus and ν is Poisson's ratio.

ffiffiffiffiffiffiffiffiffiffiffiffi 1 2 SijSij <sup>r</sup>

> ν 1 � 2ν

� � Sij

εe

2τ<sup>e</sup>

<sup>ε</sup>ij <sup>¼</sup> <sup>ε</sup><sup>e</sup>

σij,j ¼ 0 (1)

� � (2)

<sup>2</sup> <sup>ξ</sup> � <sup>ξ</sup><sup>0</sup> ð Þδij (3)

ij is the deviatoric plastic strain and, ξ � ξ<sup>0</sup> ð Þ is

ð Þ σ<sup>11</sup> þ σ<sup>22</sup> þ σ<sup>33</sup> (4)

(6)

(8)

Sij ¼ σij � σmδij (5)

kkδij � � (7)

p

ij is assumed to be

2. General formulation for infinitesimal deformation

2.1 Governing equations and constitutive laws

Adiabatic Shear Band Formation in Metallic Glasses DOI: http://dx.doi.org/10.5772/intechopen.87437

isotropic and homogeneous materials are [38]:

ij is the elastic strain, ε

The Mises effective shear stress is

mation and fracture mechanics.

where, ε<sup>e</sup>

stress tensor,

67

This model was applied to study inhomogeneous deformation during shear load by Taub and Spaepen [18], Steif et al. [19], where they assumed a band of slightly weaker material in their analysis where strain can be localized. Vaks [20] proposed the mechanism for the origin of shear bands in amorphous alloys. Vaks supposed that the localized plastic flow is connected with the preceding formation of weakened band region in which free volume is increased. Huang et al. [21] developed a general theory for modeling of inhomogeneous deformation by considering it as the isothermal process and solved 1D simple shear problem as an example. They showed that initial inhomogeneity of free volume increases with applied strain and cause localization of strain in that region leading to the creation of the shear band. Kulkarni and Bhandakkar [22] extend this theory to finite deformation. Gao [23] extended the above idea and solved 2D plate problem for inhomogeneous deformation still treating the deformation as the isothermal process. Evolution of free volume concentration during high-temperature deformation was studied by De Hey et al. [24], Yang et al. [25, 26] and Lewandowski and Greer [27] found a temperature rise of 0.25°C in a hot band of about 0.4 mm width signifying the importance of thermo-mechanical model for modeling of shear bands. Then Gao et al. [28] solved homogeneous deformation case of simple shear by considering deformation as an adiabatic process where plastic work was done was considered a source of heat. Jiang and Dai [29] using the approach of Steif et al. [19] have shown that initial formation of shear band depends dominantly on the free volume concentration but after yield point temperature also plays an important role and initial slight distribution of free volume concentration can lead to significant strain localization resulting a shear band. Ruan et al. [30] investigated the physical origin of the shear band by assuming that instability is due to bifurcation of the constitutive model at a particular stress state. They established a new constitutive model based on the rugged free energy landscape of MGs.

In addition to continuum models, beam theory has also been used to model deformation in MGs. Conner et al. [31] modeled the experimentally observed inverse dependence of ductility to plate thickness by treating the shear band as mode II crack and calculating the conditions for their formation when subjected to plane strain bending. Ravichandran and Molinari [32] treated the beam as elasticplastic with yielding based on Tresca yield criteria and included shear band dissipation. They studied the shear banding phenomenon in thin plates of MGs subjected to plane strain banding. Dasgupta et al. [33] explains the mechanism of shear bands on the basis of considering the shear band as an Eshelby inclusion. They found out that under compressive test, Eshelby inclusions in MGs arrange themselves in a line which is at 45° to the compressive stress and in this process energy is minimized.

Apart from above, Murali et al. [8] showed that in case of brittle MGs deformation occurs through cavitation mechanism, on that basis to explain that behavior, Singh et al. [34] propose a model of a heterogeneous solid containing a distribution of weak zone. Huang et al. [35] also presented a theoretical description of void growth undergoing hydrostatic tension. They found out that cavitation instabilities prefer to occur in solid with higher pressure sensitivity coefficient. Schuh et al. [36]

The second theory was developed by Spaepen which was based on the free volume model. The basic free volume model was developed by Turnbull [15, 16] and later on applied to the MGs by Spaepen [17]. This model is based on free volume concentration as an order parameter which is nothing but extra volume where atom can move freely. The distribution of free volume controls the deformation in MGs and regions having more free volume would localize the shear deformation. Deformation in MGs is considered a series of atomic jumps which in

This model was applied to study inhomogeneous deformation during shear load by Taub and Spaepen [18], Steif et al. [19], where they assumed a band of slightly weaker material in their analysis where strain can be localized. Vaks [20] proposed the mechanism for the origin of shear bands in amorphous alloys. Vaks supposed that the localized plastic flow is connected with the preceding formation of weakened band region in which free volume is increased. Huang et al. [21] developed a general theory for modeling of inhomogeneous deformation by considering it as the isothermal process and solved 1D simple shear problem as an example. They showed that initial inhomogeneity of free volume increases with applied strain and cause localization of strain in that region leading to the creation of the shear band. Kulkarni and Bhandakkar [22] extend this theory to finite deformation. Gao [23] extended the above idea and solved 2D plate problem for inhomogeneous deformation still treating the deformation as the isothermal process. Evolution of free volume concentration during high-temperature deformation was studied by De Hey et al. [24], Yang et al. [25, 26] and Lewandowski and Greer [27] found a temperature rise of 0.25°C in a hot band of about 0.4 mm width signifying the importance of thermo-mechanical model for modeling of shear bands. Then Gao et al. [28] solved homogeneous deformation case of simple shear by considering deformation as an adiabatic process where plastic work was done was considered a source of heat. Jiang and Dai [29] using the approach of Steif et al. [19] have shown that initial formation of shear band depends dominantly on the free volume concentration but after yield point temperature also plays an important role and initial slight distribution of free volume concentration can lead to significant strain localization resulting a shear band. Ruan et al. [30] investigated the physical origin of the shear band by assuming that instability is due to bifurcation of the constitutive model at a particular stress state. They established a new constitutive model based on the rugged free

In addition to continuum models, beam theory has also been used to model deformation in MGs. Conner et al. [31] modeled the experimentally observed inverse dependence of ductility to plate thickness by treating the shear band as mode II crack and calculating the conditions for their formation when subjected to plane strain bending. Ravichandran and Molinari [32] treated the beam as elastic-

Apart from above, Murali et al. [8] showed that in case of brittle MGs deformation occurs through cavitation mechanism, on that basis to explain that behavior, Singh et al. [34] propose a model of a heterogeneous solid containing a distribution of weak zone. Huang et al. [35] also presented a theoretical description of void growth undergoing hydrostatic tension. They found out that cavitation instabilities prefer to occur in solid with higher pressure sensitivity coefficient. Schuh et al. [36]

plastic with yielding based on Tresca yield criteria and included shear band dissipation. They studied the shear banding phenomenon in thin plates of MGs subjected to plane strain banding. Dasgupta et al. [33] explains the mechanism of shear bands on the basis of considering the shear band as an Eshelby inclusion. They found out that under compressive test, Eshelby inclusions in MGs arrange themselves in a line which is at 45° to the compressive stress and in this process

addition to temperature can be driven by applied shear stress.

energy landscape of MGs.

Metallic Glasses

energy is minimized.

66

and Greer et al. [37] gives a comprehensive review of recent advances in understanding the mechanical behavior of MGs, with particular emphasis on the deformation and fracture mechanics.
