Correlation between fractal dimension and strength of brittle bulk metallic glasses W. Yang1,2, Y. Zhao*1, L. Dou2, C. Dun3, J. Zhang1, M. Li1, G. Zhao1, L. Xue2, H. Bian4 and H. Liu*2 In this paper, the strength of brittle bulk metallic glasses (BMGs) was investigated, and a corresponding theoretical model was proposed by modification of Griffith theory using fractal geometry. It was shown that the fracture energy and strength of brittle BMGs not only are related to the plastic zone size, the length scales of atomic order range and the surface energy but also has significant relationship with the fractal dimension in fracture surface. The strength of typical BMGs was also calculated by our improved model, which is in excellent agreement with the experimental data. This study provides a compelling approach for investigating and understanding the high strength of BMGs. Keywords: Bulk metallic glasses, Fracture energy, Strength, Fractal dimension
Introduction Since their first synthesis in the early 1990s,1,2 bulk metallic glasses (BMGs) have attracted substantial interest for their high strength, excellent soft magnetic and chemical properties.3–5 As a significant physical property, strength plays an important role in the application of BMGs; thus, exploration of the intrinsic fracture behaviour of BMGs with high strength is especially essential.6,7 So far, extensive researches on this issue have been carried out,8–11 and it was revealed that the strength of a certain crystalline material is closely related to the intrinsic frictional stress for dislocation motion. In addition, due to the lack of defects, the strength of BMGs is believed to be directly associated with the atomic bond,12 which can determine the physical parameters that related to atomic cohesive energy.13 However, the underlying physics for the high strength of BMGs still remains unclear.14 For solids, their brittle deformation is a complex inhomogeneous process, and the dislocation motions are characterized by scale free and intermittent avalanches, which are reminiscent of the concept of self-organized criticality.15,16 Although it is not easy to represent this characteristic, the crack propagation can always leave cell or river-like vein patterns on the final fracture surface,17–20 which is statistically self-similar and has the ability to demonstrate the deformation of solids by fractal dimension.21 Many
1
School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China School of Sciences, China University of Mining and Technology, Xuzhou 221116, China 3 Department of Physics, Wake Forest University, Winston Salem, NC 27109, USA 4 Construction Company of China National Petroleum Corporation, Xuzhou 221008, China 2
*Corresponding author, email
[email protected];
[email protected]
ß 2014 Institute of Materials, Minerals and Mining Published by Maney on behalf of the Institute Received 19 May 2013; accepted 6 August 2013 DOI 10.1179/1743284713Y.0000000374
researchers attempted to study the facture properties of materials by using fractal geometry,22–25 and one of the problems remains to be solved is how to eliminate the effects of anisotropy.24 Due to the absence of defects such as dislocations or grain boundaries, brittle BMGs can be taken as model materials to investigate the fracture mechanism.26,27 However, nobody has derived a universal scaling law about the strength of BMGs, and the physical principle behind it remains unclear. In this paper, we describe the fracture behaviour of brittle BMGs by taking advantage of fractal theory. The relationship between fractal dimension in fracture surface and the strength with Griffith theory was studied and established.
Theoretical model The first successful study of microcrack propagation was conducted by Griffith for an ideal glass.28 It is found that many of these microcracks would intersect and propagate when the stress concentrations are equal to the strength of the material, leading to the failure of the material. The fracture energy GC was employed to describe the energy that is required to create a unit area of fracture surface. For ideally brittle materials, it is28 GC ~2cS
(1)
where cS is the surface energy. It should be noted that the above conclusion was drawn based on the assumption that the cracks propagate along a straight line.28 For brittle BMGs, however, the crack propagation path was irregular rather than straight,13,29–31 which needs a much higher fracture energy compared with that derived from Griffith theory.7,9 Figure 1 presents the experiments and theoretical data of the fracture energy GC and surface energy cS for a variety of BMGs. A great difference was indicated
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Fractal dimension and strength of brittle bulk metallic glasses
of e with respect to L0. As a natural fractal curve, the crack propagation length has a minimum value, the smallest scale at which the fractal description of roughness is appropriate for describing the fracture surfaces. For BMGs, the atomic cohesive energy in clusters should be larger than that between the clusters; thus, the crack initiation and propagation normally occur firstly between the clusters. Previous researchers18–20,38,39 revealed that the dimple like size w on the crack surfaces can reflect the length scale of local softening for typical BMGs. In other words, w can be regarded as a scale of fractography,32,33 with the fracture energy rewritten as GC &2cS (d=w)1{D
1 Comparisons of experiments and theoreticals between fracture energy GC and surface energy cS for variety of BMGs
such that the Griffith theory is obviously not applicable for brittle BMGs. As is known, the cell or river-like vein patterns appear on the final fracture surface in the process of crack propagation.17,32–35 The patterns are statistically selfsimilar, which demonstrates the feasibility to describe a fracture by a fractal dimension.21 Therefore, we can study the facture properties of brittle BMGs using fractal geometry.26,27 For fractal geometry, if selfsimilarity of the crack trace exists in an interval, the true length of the crack can be estimated by36 1{D L(d)~LD 0d
(6)
where d is the length scales of atomic order range in the BMGs. From the extension of Griffith theory,6 the fracture strength of brittle BMG is EGC 1=2 (7) s~ pC where C refers to the plastic zone size instead of precrack size in Orowan–Irwin’s equation,40 and E is the Young’s modulus. On the basis of equations (6) and (7), we can get the correlation between fracture strength and physical parameters. Therefore, the strength of BMGs is expressed as " #1=2 2E(d=w)1{D cs (8) s~ pC
(2)
where D is the fractal dimension, L0 is the straight path length of crack in an interval and d is a measurement scale with dimensions of length. Rearranging these terms in equation (2) yields L(e) ~L0 e1{D
(3)
where e5d/L0 is a dimensionless ratio of the measurement scale to the interval length. For a fractal trace, the real fracture area is known to be larger than the apparent area, assumed in classical fracture mechanics, and it is defined by Areal ~(L(e) =L0 )Aapp
(4)
From the above analysis, the fracture energy GC can be modified as37 GC ~2 L(e) =L0 cS (5) where L(e)/L0 is the lineal profile roughness parameter, a dimensionless quantity depending upon the relative scale
Results and discussion From the above analysis, we found that the strength of BMGs is related not only to the Young’s modulus, the plastic zone size and the surface energy but also to the length scales of atomic order range and the fractal dimension in fracture surface. To verify our model, five typical BMGs including Zr41?25Ti13?75Cu12?5Ni10Be22?5, Pd77?5Si16?5Cu6, Cu60Zr20 Hf10Ti10, Ce70Al10Ni10Cu10 and Mg65Cu25Tb10 were taken as examples. The theoretical GC and s is estimated by using equations (6) and (8) from the following material parameters: the average outer diameter of the solute centred clusters d
