Size effect on stability of shear-band propagation in bulk metallic glasses: an overview

Abstract

Understanding of the size effect on shear banding in bulk metallic glasses (BMGs) is currently the topic of active research but also remains under intense debates. In this article, we provide an overview of the recent research findings from experiments, theoretical modeling, and atomistic/continuum simulations which are intended to advance our knowledge related to the size effect on the stability of shear-band propagation in BMGs. Through the compilation of and comparison among the results reported in the literature, we aim at providing a comprehensive understanding of the underlying mechanisms and a unified physical picture of the size effect on shear-band propagation and its resultant ductility in BMGs.

Appendix: The α parameter and the extrinsic length scale L ext

The theoretical derivation of the α parameter has been detailed in our recent work [60]. For the information of readers, the general expression for the α parameter is introduced here. According to the dimensional analysis, the α parameter can be expressed as follows for a cylindrical BMG sample compressed between two cylindrical compression platens:

$$ \alpha = \left( {\sum\limits_{i = 1}^{2} {\Uppi_{i} \left( {\frac{D}{{D_{i} }},\frac{{H_{i} }}{{D_{i} }},\nu_{i} } \right)} \frac{E}{{E_{i} }}} \right)\frac{\pi }{4} $$

(7)

where the symbols D, H, E, and ν denote the object diameter, the object height, the Young’s modulus, and the Poisson’s ratio, respectively. For the BMG sample, those symbols carry no subscript; while for the compression platens, they are labeled with a subscript (1 = upper platen and 2 = lower platen), as shown in Fig. 11. Here Π represents a dimensionless function of which the exact functional form is not known yet.

Fig. 11

The illustrated experimental set-up for a compression test on a BMG sample

For microcompression tests, the upper platen is a diamond punch that may be assumed as a rigid body (E ₁ = ∞), and the lower one is the base material, which can be regarded as an elastic half infinite space (D ₁ = ∞ and H ₁ = ∞) and possesses the same material properties as the micropillar. In such a case, the functional form of the α parameter is reduced to $\alpha = \alpha \left( \nu \right)$, which only depends on the Poisson’s ratio of the BMG. Through the FE simulation, one may fit the simulated values of the α parameter to a linear function, which reads $\alpha = {{4\left( {5.6 - 4\nu } \right)} \mathord{\left/ {\vphantom {{4\left( {5.6 - 4\nu } \right)} \pi }} \right. \kern-\nulldelimiterspace} \pi }$ [26]. Assuming an average Poisson’s ratio of ~0.35 for BMGs, the α parameter can be then estimated as around ~5. As such, the corresponding extrinsic length scale L _ext can be expressed as $L_{\text{ext}} = H + 5D$.

For regular compression tests, the α parameter is a multivariable function. For simplicity, let us assume that both upper and lower platens are made of the same material and possess the same size and geometry (E ₁ = E ₂, D ₁ = D ₂, H ₁ = H ₂, and ν ₁ = ν ₂). As such, Eq. 7 can be simplified to:

$$ \alpha = \Uppi_{1} \left( {\frac{D}{{D_{1} }},\frac{{H_{1} }}{{D_{1} }},\nu_{1} } \right)\frac{E}{{E_{1} }}\,\frac{\pi }{2} $$

(8)

In principle, FE simulations can be performed to attain a proper functional form for Π ₁ that fits the elastic boundary conditions set in a real experiment as seen in [26]. After that, the corresponding extrinsic length scale L _ext can be derived and then utilized for studying the size effect in a quantitative manner. Here, for the sake of demonstration, we assume the following functional form for Π ₁ for a finite platen size without resorting to the FE simulations:

$$ \Uppi_{1} = \left( {\frac{D}{{D_{1} }}} \right)^{x} \left( {\frac{{H_{1} }}{{D_{1} }}} \right)^{y} \frac{{2f\left( {\nu_{1} } \right)}}{\pi } $$

(9)

where the exponent x, y and the single-variable function f are to be fitted out by FE simulations. Substituting (9) into (3a) then gives the corresponding extrinsic length scale L _ext:

$$ L_{\text{ext}} = H + \left( {\frac{D}{{D_{1} }}} \right)^{x - 1} \left( {\frac{{H_{1} }}{{D_{1} }}} \right)^{y - 1} \frac{\pi E}{{4k_{\text{M}} }}f\left( {\nu_{1} } \right)D^{2} $$

(10)

in which $k_{\text{M}} = {{\pi E_{1} D_{1}^{2} } \mathord{\left/ {\vphantom {{\pi E_{1} D_{1}^{2} } 4}} \right. \kern-\nulldelimiterspace} 4}H_{1}$ is the machine stiffness. Now, if we further assume x = y = f(ν₁) = 1, (10) is therefore simplified to:

$$ L_{\text{ext}} = H + \frac{\pi E}{{4k_{\text{M}} }}D^{2} $$

(11)

Comparing (11) to Eq. 2 in the main text, it can be easily seen that the general size-effect model described here is now reduced to the size-effect model developed by Cheng et al. [56] and their proposed shear-band instability index (S _new) is equivalent to the extrinsic length scale, L _ext, of a special form.