13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- properties of MGs is still lacking for general stress state. In this paper, we attempt to derive the critical condition of the shear band initiation and direction for MGs by constructing a bridge between the microscopic origin and the loss-of-ellipticity instability in the constitutive law in continuum mechanics. 2. Theoretical model For initial homogeneously deformed material, the loss of constitutive stability will cause strain localization into a shear band [16-19]. Therein, the homogeneous deformation develops to a bifurcation point, at which the discontinuity in deformation rate is incipient across a band. As to MGs, two internal factors are critical for the shear instability, namely, free volume and thermo [1, 2, 13-15]. The onset of shear banding in MGs can be reasonably described as a result of constitutive instability induced by dramatic change of internal state variables. 2.1 Constitutive model For the instability analysis, a proper constitutive is required for MGs. In this section, a new constitutive model is developed for MGs due to the following considerations. At macroscopic scale, MGs exhibit inherent pressure sensitivity and shear-dilatancy during plastic deformation, which usually renders a non-associated flow. Microscopically, the nucleation and coalescence of free volume decreases the flow stress of MGs and further leads to shear localization. This is actually quite similar to the void evolution mechanism in the continuum damage mechanics. One of the best known micro-mechanical models is that of Gurson [22], who studied the plastic flow of a void-containing material and established a yielding function reflecting the softening effect due to the presence of voids. Here, by introducing the free volume evolution into the framework of continuum mechanics, we can establish a new constitutive model to comprehensively and satisfactorily describe the deformation in MGs. “Free volume” as the topological disorder in MGs, can be simply considered as randomly distributed atomic voids in material. Treating those voids to be spherical, the yield function of Gurson is reasonably extended to MGs by taking into the pressure sensitivity of the matrix. The free volume evolution is assumed to obey the self-consistent dynamic free volume model proposed by Johnson et al.[23]. The thermal effect is neglected since the structural disorder induced softening precedes thermal softening at the origin of the shear banding [13], especially for the present quasi-static loading case. For a pressure-independent material containing spherical voids, Gurson model gives the following yield function [22]: 1 0 2 3 2 cosh 2 2 f y m f y e F , (1) where 2 3 2 3J s s ij ij e ( ij m ij ij s ) is the effective stress, 3 ii m is the mean stress, f is the current fraction of voids, equivalent to free volume concentration, and y is the yield stress of the matrix. Numerous studies have demonstrated clearly that pressure affects the yield behavior of MGs [24-27]. This is easily reflected by the tension-compression asymmetry of failure [28]. Since the Gurson model considers the von Mises matrix, an additional pressure-dependent term i.e. m , should be taken into account (analogous to the Drucker-Prager criterion) for a proper description of MGs. For simplicity, one can modify the above criterion as below by neglecting the minor term 2 2(3 2 )) (1 cosh f y m . We define the initial free volume in
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