J. Cent. South Univ. Technol. (2011) 18: 1-5

DOI: 10.1007/s11771-011-0650-5

A unified criterion for yielding behavior of metallic glasses

SONG Min(宋旼), HE Yue-hui(贺跃辉)

State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: The yield behavior of metallic glasses was studied. Three yield criteria, including von Mises yield criterion, Mohr-Coulomb yield criterion and the unified yield criterion were used to describe the yield phenomena of the metallic glasses. Two classes of the experimental data were chosen to draw the yield loci using the unified yield criterion. It is shown that the unified yield criterion can be used to describe the yield behavior of the metallic glasses no matter whether the metallic glasses show strength- different effect or non-strength-different effect. Almost all the widely accepted yield criteria are the subsets of the unified yield criterion if the intermediate principle stress and/or the intermediate principle shear stress are not considered at all.

Key words: metallic glasses; yield criterion; unified criterion

1 Introduction

Metallic glasses, also known as amorphous alloys or metals, show potential as structural materials because of their unique mechanical properties, such as extremely high strength, no working hardening and high fracture toughness. Metallic glasses differ from traditional crystalline materials in that the atoms do not assemble on a crystalline lattice. Prior to the development of bulk metallic glasses, studies on the mechanical properties of metallic glasses were confined to tension and bending testing because only ribbon and wire specimens were available since high cooling rate (>10⁵ K/s) was required to prevent crystallization [1]. The development of bulk metallic glasses [2-3] enabled a wild range of mechanical tests including compression, torsion, fracture toughness and fatigue testing.

In principle, understanding yield mechanism can gain important insight into the micromechanisms of plastic deformation of metallic glasses. Up to now, von Mises and Mohr-Coulomb criteria were suggested as the yield mechanisms of metallic glasses. Both criteria were supported by many experimental results. KIMURA and MASUMOTO [4] suggested that metallic glasses obey a von Mises yield criterion based on the plastic deformation of Pd_77.5Cu₆Si_16.5 metallic glass. This was further supported by BRUCK et al [5] since the measured values of compressive and tensile yield stress are approximately equivalent and the ratio of these yield stresses to the shear yield stress is 1.84, very close to 1.73 predicted by von Mises criterion. Later, LOWHAPHAUDU et al [6] demonstrated a negligible effect of confining pressure on the flow/fracture stress, which has been predicted by von Mises yield criterion. On the other hand, DONOVAN [7] demonstrated that yielding in Pd₄₀Ni₄₀P₂₀ follows a Mohr-Coulomb criterion based on the experimental results of uniaxial compression, plane-strain compression, pure shear and tension. The sign of the normal stress acting across the slip-plane has a significant effect on the yield strength so that the yield strength in compression is substantially higher than that in tension, but the hydrostatic pressure has only a small effect on yielding. MUKAI et al [8] further showed that the deformation of metallic glasses obeys the Mohr-Coulomb criterion since the fracture plane inclines approximately 42? with respect to the compression axis during compression test, which is in contrast to 56? observed in samples tested in tension.

In this work, a unified yield criterion [9-10], first developed to describe the yield behavior of crystalline materials, was introduced to describe the yield behavior of the metallic glasses. Compared with most other studies [4-8, 11-16] concerning the mechanical behavior and yield phenomenon of the metallic glasses, the purpose of this study is trying to unify the different yield criteria to a common one, which can describe the different yield phenomena of all metallic glasses.

2 Yield criteria for yield behavior of metallic glasses

2.1 von Mises yield criterion

Metallic glasses are elastically isotropic and do not work-harden during plastic flow so that they are probably close to ideal elastic/plastic materials. For isotropic and pressure insensitive materials, the von Mises criterion can be expressed as

        (1)

where σ₁, σ₂ and σ₃ are the principle stresses and k is a constant equal to the yield stress in shear. This criterion predicts that plastic flow will commence when a critical shear strain energy density is reached, and the compressive and tensile yield stresses are equivalent with the ratio of these stresses to the shear yield stress being 1.73. BRUCK et al [5] showed that the measured values of compressive and tensile yield stresses of Zr_41.25Ti_13.75Ni₁₀Cu_12.5Be_22.5 are approximately equivalent, while the ratio of these yield stresses to the shear yield stress is 1.84, very close to 1.73. It has also been shown that the ratio of the yield stresses in plane strain compression to those in uniaxial compression is 1.12, which approaches the value of the von Mises prediction, 1.15 [5]. The work by LOWHAPHAUDU et al [6] also showed that the flow/fracture stress and flow/fracture strain of a ZrTiNiCuBe bulk metallic glass are negligibly affected by changes in stress triaxiality over the range of  from -0.33 to 0.33 and are essentially independent of superimposed hydrostatic pressure over the range of 50-575 MPa (where σ_m is the mean stress and  is the effective stress). Using uniaxial compression tests at quasistatic and high strain rates in the range of 10^-3-10³ s^-1, SUBHASH et al [17] also showed that the deformation of a Zr-Ti-Ni-Cu-Be bulk metallic glass is probably governed by pressure- insensitive von Mises criterion.

2.2 Mohr-Coulomb yield criterion

The Mohr-Coulomb criterion is based on an empirical approach that accounts for increased and decreased shear resistance caused by compressive normal stress and tension normal stress, respectively. The Mohr-Coulomb criterion can be expressed as

                               (2)

where τ_c is the shear stress on the slip plane at yielding, σ_n is the stress component in the direction normal to the slip plane, and k₀ and α are constants. Experimental observation can be readily used to test the appropriateness of the Mohr-Coulomb criterion. First, if the yield stress depends on the normal stress on the slip plane, σ_n, the slip plane will deviate from the plane of the maximum resolved shear stress (45? from uniaxial compression and tension directions based on von Mises criterion). Second, the ratio of the yield stresses in plane strain compression to those in uniaxial compression might deviate from 1.15, as predicted by von Mises criterion. The experimental results of Pd₄₀Ni₄₀P₂₀ and Zr₄₀Ti₁₄Ni₁₀Cu₁₂Be₂₄ by DONOVAN [7], WRIGHT et al [18] and MUKAI et al [8] showed that the shear bands formed at an angle of about 56? to the tension loading axis and about 42? to the compression loading, which agrees with the Mohr-Coulomb yield criterion. DONOVAN [7] further indicated that the ratio of the measured yield stresses in plane strain to those in uniaxial compression is about 1.02, which clearly demonstrates that metallic glasses do not obey the von Mises criterion (should be 1.15 predicted by von Mises criterion).

2.3 Unified yield criterion

von Mises criterion predicts that plastic flow will commence when a critical shear strain energy density is reached. Another explanation of von Mises criterion is that octahedral shear stress (τ_o) reaches a critical value. Mohr-Coulomb criterion considers the effect of the maximum shear stress (τ_c) on the slip plane as well as the normal stress (σ_n) on the same plane. Both criteria do not take into account of the effect of the intermediate principle stress (σ₂) and/or the effect of the intermediate principle shear stress (τ₁₂ or τ₂₃). Since there are only two independent principle shear stresses, the shear stress state can be converted into twin-shear stress state (τ₁₃ and τ₁₂, or τ₁₂ and τ₂₃) with

                               (3)

The unified yield criterion assumes that the yielding of materials begins when the sum of the two larger principle shear stresses and the corresponding normal stress function reaches a critical value. The mathematic expression can be expressed as [9-10]

when                (4)

when                (5)

where b is a parameter that reflects the influence of the intermediate principal shear stress τ₁₂ or τ₂₃; β is the coefficient that represents the effect of the normal stress; C is a strength parameter of material; τ₁₃, τ₁₂ and τ₂₃ are principal shear stresses and σ₁₃, σ₁₂ and σ₂₃ are the corresponding normal stresses acting on the sections where τ₁₃, τ₁₂ and τ₂₃ act. In principle, the principal shear stresses and the corresponding normal stresses satisfy the following equations [9-10]:

                           (6)

                          (7)

where σ₁, σ₂ and σ₃ are principal stresses with σ₁> σ₂ > σ₃. The parameters β, C and b can be expressed as [9-10]

                         (8)

                          (9)

                       (10)

where σ_t, σ_p and τ_s are uniaxial tension stress, compression stress and pure shear stress, respectively, and α is the ratio of the tension stress to compression stress. The unified yield criterion can also be expressed in terms of principal stresses by substituting Eqs.(5) and (6) into Eq.(4):

 when    (11)

 when  (12)

3 Relationship between unified yield criterion and other yield criteria

In the unified yield criterion, the parameter b reflects the influence of the intermediate principal shear stress and/or intermediate principle stress, while parameter β reflects whether the material is a strength- different (SD, σ_t≠σ_p or β≠0) material or non-strength- different (NSD, σ_t=σ_p or β=0) material. Thus, both parameters are very important to build a bridge among different yield criteria.

Fig.1 and Fig.2 show the yield loci on π-plane and in the plane stress state predicted by the unified yield criterion, respectively. Both SD materials (β≠0) and NSD materials (β=0) are considered. The yield loci predicted by Tresca (for NSD materials), von Mises (for NSD materials) and Mohr-Coulomb (for SD materials) yield criteria are included. It can be seen that for both SD and NSD materials, the unified yield criterion generates a series of different yield loci with varying b value. The most important feature of the unified yield criterion is that all the traditional yield criteria (including Tresca, von Mises and Mohr-Coulomb criteria) are the subsets of the unified yield criterion. For NSD materials, the unified yield criterion changes to Tresca or von Mises yield criteria when b=0 or b≈0.5. For SD materials, the unified yield criterion changes to Mohr-Coulomb yield criterion when b=0.

Fig.1 Yield loci on π-plane predicted by unified yield criterion for NSD materials (β=0) (a) and SD materials (β≠0) (b)

Fig.2 Yield loci in plane stress state predicted by unified yield criterion for NSD materials (β=0) (a) and SD materials    (β≠0) (b)

In general, yield loci of the unified yield criterion both on π-plane and in the plane stress state are the intersection lines of the yield surface in principal stress space. They will be transformed into hexagon when b=0 or b=1, and into dodecagon when 01 or b<0, which indicates that the unified yield criterion theory can also be used for non-convex yield materials as well as for Drucker convex yield materials. Thus, the unified yield criterion theory is not a single yield criterion suitable only for one specific material, but a complete system, which embraces many well-established criteria as its special cases. In principle, the SD effect, the hydrostatic stress effect, the normal stress effect, the intermediate stress effect and the intermediate principal shear stress effect have all been considered in the unified yield criterion. It should be noted that θ_b in Fig.1(b) is the stress angle with the expression of

                        (13)

4 Application of unified yield criterion in metallic glasses

Applying the unified yield criterion to metallic glasses needs experimental data to determine the corresponding parameters. Available limited experimental data in literature do not agree with each other: some experimental data indicate that metallic glasses are NSD materials [5], while others indicate that they are SD materials [7]. In this work, two classes of the experimental data, as listed in Table 1, have been chosen to draw the yield loci using the unified yield criterion. The calculated parameters are listed in Table 2.

Figs.3 and 4 show the yield loci on π-plane and in

the plane stress state for both Zr_41.25Ti_13.75Ni₁₀Cu_12.5Be_22.5 and Pd₄₀Ni₄₀P₂₀ metallic glasses predicted by the unified yield criterion. The yield loci predicted by Tresca (for NSD materials), von Mises (for NSD materials) and Mohr-Coulomb (for SD materials) yield criteria are also included. The parameters and experimental data from Tables 1 and 2 are used to draw the yield loci. It can be seen from Fig.3 that Zr_41.25Ti_13.75Ni₁₀Cu_12.5Be_22.5 metallic glass is a NSD material with the uniaxial compression yield strength equal to the uniaxial tension yield strength. The yield locus predicted by the unified yield criterion is a dodecagon on π-plane and in the plane stress (with the same length of each line on π-plane), which is located between the yield locus predicted by Tresca and that by von Mises yield criteria. Under the condition of uniaxial tension or compression, all three criteria have the same yield strength. It can also be seen from Fig.4 that Pd₄₀Ni₄₀P₂₀ metallic glass is a SD material with the uniaxial compression yield strength being lager than the uniaxial tension yield strength. The yield locus predicted by the unified yield criterion is also a dodecagon on π-plane and in the plane stress. The lines of the yield locus are located inside the yield locus predicted by Mohr-Coulomb yield criterion. Under the condition of uniaxial tension or compression, both unified criterion and Mohr-Coulomb yield criterion have the same yield strength, with the compression yield strength being larger than the tension yield strength. It should also be noted that Pd₄₀Ni₄₀P₂₀ metallic glass exhibits non-convex yield feature instead of traditional convex yield feature.

Figs.3 and 4 indicate that the unified yield criterion

Table 1 Mechanical properties and parameters of two metallic glasses

Table 2 Calculated parameters using unified yield criterion

Fig.3 Yield loci on π-plane (a) and in plane stress state (b) predicted by unified yield criterion for Zr_41.25Ti_13.75Ni₁₀Cu_12.5Be_22.5 metallic glass

Fig.4 Yield loci on π-plane (a) and in plane stress state (b) predicted by unified yield criterion for Pd₄₀Ni₄₀P₂₀ metallic glass

can be used to describe the yield behavior of metallic glasses. The main difference between the unified yield criterion and other yield criteria is that the unified yield criterion includes the effects of the intermediate principal stress and/or the intermediate principal shear stress on the yield behavior. In principle, von Mises criterion and Mohr-Coulomb criterion are the subsets of the unified yield criterion if the effect of the intermediate principal stress and the intermediate principal shear stress are not considered at all. Thus, the yield behavior of different metallic glasses can be described by the unified yield criterion, no matter whether they show difference in tension yield strength and compression yield strength, or the slip plane deviates from the plane of the maximum resolved shear stress. However, on the other hand, it should be noted that further experiments on metallic glass yielding under the other loading conditions need to be done since most of the available experimental data in the literatures are from uniaxial compression, uniaxial tension and torsion.

5 Conclusions

1) The unified yield criterion, first developed to describe the yield behavior of crystalline materials, was introduced to describe the yield behavior of metallic glasses and eliminate the contradiction in the yield behavior of metallic glasses described by von Mises and Mohr-Coulomb criteria.

2) The yield behavior of metallic glasses can be described by the unified yield criterion, no matter whether they show difference in tension yield strength and compression yield strength, or the slip plane deviates from the plane of the maximum resolved shear stress.

3) The unified yield criterion, by considering the effects of the intermediate principal stress and/or the intermediate principal shear stress on the yield behavior, generates a series of different yield loci. Almost all the widely accepted criteria such as Tresca, von Mises and Mohr-Coulomb criteria are the subsets of the unified yield criterion.

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(Edited by YANG Bing)

Foundation item: Projects(51011120053, 50823006, 50825102) supported by the National Natural Science Foundation of China

Received date: 2010-01-27; Accepted date: 2010-07-13

Corresponding author: SONG Min, Professor, PhD; Tel: +86-731-88877880; E-mail: msong@mail.csu.edu.cn