Original ArticleRelationship between fractography, fractal analysis and crack branching
Introduction
Materials that fail in a brittle manner behave in a very characteristic manner [[1], [2], [3]]. There is generally a single crack that starts the propagation process. It proceeds to gain velocity until it reaches a critical velocity at which time the main crack branches [4]. Although the initial fracture surface appears relatively smooth (in the “mirror” region) because the undulations on the surface are in general less than the wavelength of light, we do not see them in an optical microscope. We can observe them using an atomic force microscope [5]. There is an increase in the size of the series of undulations such that they appear to form a fine mist in glass (called the “mist” region) and proceed to get even larger in a region called the hackle region [6]. All of these perturbations along the crack front occur prior to macroscopic crack branching [7]. Over the past few years we have made much progress in understanding the role of the fracture surface markings in identifying the history of the fracture event in materials that fail in a brittle manner [1]. A.A. Griffith [8] was the first to attribute the differences in strength variations to the existence of flaws or cracks. He used an energy balance approach to describe the relationship between strength, , crack size, , and fracture energy, . Many years later, George Irwin [9] introduced the concept of stress intensity and showed that the strength of a material that failed in a brittle manner was related to the crack size through a critical stress intensity factor, , also called the fracture toughness. He also showed that the fracture toughness was related to the fracture energy of Griffith through the Strain Energy Release Rate, , i.e, where . Although the fracture toughness can be a function of the crack size in some brittle materials [2], for the purposes of this paper we consider a constant of the material. This does not in any way limit the presentation of the derivations. Fracture mechanics equations can be applied to obtain the fracture toughness of materials [10]. Quantitative fractography, applying the principles of fracture mechanics, can be used to determine the strength from unexpected field failures [11,12].
Over a series of experimental observations, it has been determined that all fracture surfaces are fractal [[13], [14], [15]]. This discovery implies that the fracture process from the atomic to the macroscopic length scales is related in a self-similar (or self-affine) manner and is length-scale invariant. We have observed evidence of this through the scaling of the crack-size-to-mirror-size ratio [16]. The fractal dimension can be determined from the fracture surface. There are several excellent reviews of fracture mechanics and fractal geometry [[17], [18], [19]]. The reviews cover many of the aspects of fracture processes including fast fracture [18], ductile failure [17], microcracking [19], etc. The present paper only addresses a small part of the vast studies in fractal fracture. Most of the studies on fractal fracture treat the tortuosity of the fracture surface and measure the fractal dimension, where is where is the fractal dimensional increment showing the amount of tortuosity. In fact, the fractal dimensional increment is directly related to the crack-to-mirror size ratio [20]. In addition, the fractal dimensional increment is related to the toughness of the material [14]. The fractal dimension is independent of the stress at fracture. In other words, it is constant for a given monolithic material.
Several researchers have noted the relationship between the angles of crack branching and the far field stress applied, i.e., the method of loading [21,22]. It turns out that the crack branching pattern is a fractal process as well [[23], [24], [25]]. However, the pattern is dependent on the stress at fracture. Thus, even though we measure the fractal dimension, in order to avoid confusion, the fractal dimension parameter is called the Crack Branching Coefficient (CBC). Interestingly, the general reviews mentioned [[17], [18], [19]] do not point out the difference between the fractal dimension of fracture surfaces and the fractal dimension of crack branching.
The observation that there is a fractal dimension of the fracture surface and a fractal dimension of the crack branching pattern leads to a dilemma. How can the fractal dimension of the fracture surface be independent of strength and the crack branching coefficient (which is really a fractal dimension) be a function of strength? This paper addresses the relationship between the fractal dimension of the fracture surface and the crack branching coefficient. We also demonstrate how the knowledge of both the fractal dimension of the fracture surface and the crack branching information aids in the forensic analysis of fracture events.
Section snippets
Theoretical background
Fracture mechanics principles [26] can be used to relate the stress, , to the crack size, :where is the stress at fracture (strength) and is the critical stress intensity factor, (fracture toughness). There are several ways to use Eq. (1). If the fracture toughness has been previously determined using various techniques [7,26], then the strength of a fractured component can be determined by measuring the flaw or crack at the fracture origin. Conversely, if the strength and
Crack branching and fractal geometry
Early studies have shown that the angle of crack branching is a function of the type of loading [21,22,30]. The distance to branching is related through Eq. (2). The combination of the change in angle and the distance to branching leads to a fractal pattern in the branching of cracks as shown in Fig. 1. To investigate this branching pattern, MgF2 disks which were previously fractured in ring-on-ring biaxial flexure were examined for the crack branching coefficient. The strength was found to be
An analysis of the crack branching coefficient in terms of fractography
Combining Eqs. (6) and (2), we may eliminate to obtain:
It has been experimentally determined that there is a linear relationship between and [29] given by:
Combining Eqs. (4) and (8), we obtain:
Substituting from Eq. (9) into Eq. (7) we obtain a useful relationship between and :
Equation (10) relates the crack branching coefficient to the fractal dimensional increment through the branching distance
Experimental verification of CBC-D* relationship
We can compare the results from experiments with the predictions of Eq. (10) for MgF2 and borosilicate data previously studied [23,27]. The values for were obtained from Kirchner [31]. The values for in the MgF2 data were determined from the strength values in Rice et al. [32] which is the basis for the Mecholsky et al. [23] data for fractal analysis. The values for for the borosilicate disks were obtained from strength values (water data only) in J. Quinn [27]. The values for ,
Implications for forensic analysis
Once some basic parameters ( and ) and constants ( and ) are determined for materials of interest, the above equations can be used for forensic analysis. If an unexpected fracture occurs and only limited pieces are left, you can still determine whether or not the material failed from an overload or from processing by determining . If the value is less than the established value, then that implies a lower fracture toughness (Eq. 4) due to processing effects. If a number of pieces
Summary and conclusions
We have derived an equation for showing the relationship between the fractal dimension of the fracture surface and the fractal dimension of crack branching formation (called the Crack Branching Coefficient). Combining equations related to fracture mechanics, quantitative fractography and fractal analysis led to a relationship between the crack branching coefficient and the fractal dimensional increment. Data from previous measurements support the use of this equation. In using Eq. (6), it was
Declaration of Competing Interest
We have not received any outside funding for this research.
Acknowledgements
We thank George Quinn for discussing the research and providing access to the borosilicate data from the late Dr. Janet Quinn’s paper.
References (42)
- et al.
Fractal fracture mechanics--a review
Eng. Fract. Mech.
(1995) Applying fractography and fracture mechanics to the energy and mass of crack growth for glass in the mirror region
J. Eur. Cer. Soc.
(2014)The strain intensity criterion for crack branching in ceramics
Eng. Fract. Mech.
(1978)- et al.
Models of fragmentation and stochastic fractals
Phys. Lett. A
(1995) Fractography of Ceramics and Glasses, NIST
(2016)- et al.
Fracture of ceramics
Adv. Eng. Mater.
(2008) Fractography of Brittle Materials, Issue 15 of Measurement Good Practice Guide
(1999)- et al.
Instability in the propagation of fast cracks
Phys. Rev. B
(1992) - et al.
Estimation of fracture energy from the work of fracture and fracture surface area: I. Stable crack growth
Int. J. Fract. Mech.
(2009) Crack front stability and hackle formation in high velocity glass fracture
J. Am. Ceram. Soc.
(1995)
Fracture mechanics of fracture mirrors
J. Am. Ceram. Soc.
The phenomenon of rupture and flow in solids
Philos. Trans. Math. Phys. Eng. Sci.
Fracture
Standard Test Methods for Determination of Fracture Toughness of Advanced Ceramics at Ambient Temperature
Criteria for crack branching in cylindrical rods: I tension
J. Am. Ceram. Soc.
Fracture of Brittle Materials: Testing and Analysis
Fractal character of fracture surfaces of metals
Nature
Quantitative analysis of brittle fracture surfaces using fractal geometry
J. Am. Ceram. Soc.
Using fractal dimensions to characterize atomic force microscope images of epoxy resin fracture surfaces
Scanning
Effect of grinding on flaw geometry and fracture of glass
J. Am. Ceram. Soc.
Fractal Fracture Mechanics Applied to Materials Engineering
Cited by (13)
Small plastic fragments: A bridge between large plastic debris and micro- & nano-plastics
2023, TrAC - Trends in Analytical ChemistryExperimental study on crack irregularity of hollow shell particle under impact loading
2023, Powder TechnologyAcoustic emission technology-based multifractal and unsupervised clustering on crack damage monitoring for low-carbon steel
2023, Measurement: Journal of the International Measurement ConfederationExploiting fractal features to determine fatigue crack growth rates of metallic materials
2022, Engineering Fracture MechanicsInfluence of pulse TIG welding thermal cycling on the microstructure and mechanical properties of explosively weld titanium/steel joint
2022, VacuumCitation Excerpt :Fig. 13 shows the crack distribution characteristics of the Ti/steel explosive welding interface at Tm = 1283 °C. It was reported that, crack branching indicates the direction of crack propagation [30,31]. Consequently, it can be determined that the crack source was roughly generated between Fe2Ti and FeTi, and then the crack propagated into the Fe2Ti and FeTi layer.