Dynamic Fracture: when the going gets tough . . .
“Nothing happens, and nothing happens, and then everything happens.” - Fay Weldon
Fracture in nominally brittle materials behaves much like the above quote, in that during the build up of stresses at the crack tip, at least from the naked eye, all remains calm – until suddenly and catastrophically, the crack takes off and propagates through a material. On some level, one could argue that all fracture is dynamic; however it is generally acknowledged in the mechanics community that the dynamic nature of cracks is relevant to failure criteria when the inertia is large enough that it requires the kinetic term to be included in the total energy balance.
While not a new subject, there remains fundamental open questions in dynamic fracture, including two most relevant to this discussion:
How fast can cracks propagate?
What are the physics governing the mechanisms of dynamic crack tip instabilities?
Figure 1: Homalite 100 plate with center hole and pre-crack in slight tension (less than 4 MPa) fractured from an out-of-plane hypervelocity impact which acts like a local explosion and resulting cracks grew left to right in the field of view. Under higher speed and more energetic impacts, the cracks grew faster and had branching characteristics, as circled (top), whereas under less extreme impacts the cracks exhibited slower speeds and an oscillating crack path .
“I feel the need . . . the need for speed.” - Maverick & Goose, Top Gun (1986)
Even though the study of dynamic fracture in nominally brittle materials has been around for quite some time (often first credited to Shardin who examined dynamic crack growth around in the 1930), there are some puzzling contradictions behind some of the continuum dynamic fracture mechanics theory; particularly in regard to instabilities, or when transient effects come into play.
Pragmatically speaking, the American Society for Testing and Materials has well-established and detailed standards for static fracture toughness testing, including ASTM E399 for metals, and ASTM C1421 for ceramics [2-4]; yet no generally accepted procedure exists for these materials to examine the dynamic fracture toughness. Dynamic fracture characterization methods are generally grouped by types of high rate bending, high rate tension, and dynamic wedging, and single-point impact (although there is extensive body of work with single wire explosions for mode 2 fracture within single point) [4-9]. At the same time, there lies a key discrepancy between static and dynamic initiation fracture toughness values, some materials such as aluminum 7075-T6 have been shown to exhibit little rate dependency, while others such as Ti-6264 and 4340 steel show as much as 1.4 to 1.6 times higher toughness under dynamic loading conditions . For PMMA, the dynamic fracture toughness was shown to be three times higher than it’s static initiation value . Indeed, our research group has shown that a MAX phase (metal-ceramic nanolayered material) Ti3SiC2, exhibited no real rate dependency on its dynamic initiation toughness, whereas tungsten carbides with various levels of nickel and cobalt binders exhibited dynamic toughness increases on the order of 20% from their quasi-static values [12,13].
From these selected examples, differences appear to be a key signatures of the smaller length scale initiation mechanisms which govern the material inertia or resistance to initiation in the formation of a macrocrack, as well as a function of (given different dynamic loading conditions with various methods of evaluating dynamic fracture) wave interactions in the material and ahead of the moving crack front, where the theory used to extract relevant stress intensity values is leveraged. Regardless of these material differences, those few researching experimental dynamic fracture appear to agree that the history of loading on the crack tip, even in rate-insensitive materials, directly affects resulting crack growth .
Figure 2: MAX phase Ti3SiC2 fracturing from the pre-crack and notch on the right hand side, when impacted by a steel rod at 4 m/s on the left hand side. The DIC is used to extract deformation fields which are then optimized using a least squares overdeterministic analysis to determine the stress intensity facture in mode I (opening) and corresponding inertia absorbing microstructural features of kinking bands and delaminations in resulting SEM fractography .
Classical Dynamic Fracture Mechanics
“Aye, there’s the Rub!” - Shakespeare, from Hamlet's Soliloquy
In classical dynamic fracture mechanics, experimental evaluation leverages derived physical quantities that require adequate theory . Problems dealing with dynamically loaded stationary cracks are relatively well understood, although not straightforwardly computed in terms of linear elastodynamics. However, crack initiation and propagation under transient conditions exhibits a rapidly changing stress field at the crack tip. While the typical square root singularity exists, the field initially exists only in an asymptotically small zone around the crack tip that spreads as a function of the wave speed, influenced by the geometry of the sample . As a result, this domain has been shown to take a characteristically long time to develop, generally recognized to depend on the difference between the shear wave speed and crack propagation velocity. Thus, the faster the crack propagates (or more dynamic the case), the longer it takes for this domain to develop ahead of the moving crack front. Often, series expansion descriptions of the stress fields are employed, yet this approach implies steady-state conditions and consequently raises questions on the validity in accurately evaluating dynamic crack tip energetics by these means. At the same time, it is argued and reasonably shown in literature that at least to some degree, this condition is met and the results are still useful forms of predictive fracture criterion . In fact, you will see a lot of the experimental dynamic fracture literature employing various means such as long bars and low impedance boundaries with clay to help ensure this condition is met (and the validity of their solutions valid), but in doing are essentially mitigating the interesting and under-explored, yet more complex transient regimes where crack speeds and complex wave interactions are increased, and unstable crack fronts are instigated and branching often ensues. It should be noted that the most useful mathematical solutions considered infinite domains with asymptotic behavior of the deformation (and stress) fields near the moving crack tip, and were originally supplied by Freund and Kostrov [17,18].
Measuring Dynamic Stress Intensity Factors (SIF)
“Measure what is measurable, and make measurable what is not so.” - Galileo Galilei
Characterizing fracture behavior under dynamic conditions is a fundamentally more challenging problem than its quasi-static counterpart. Classically, the methods of photoelasticity and caustics are used to evaluate crack tip stresses, and extract stress intensity factor(s) (SIFs) by combining high-speed imaging with coherent light and optics. The former method relies on the interference of light, while the later relies on the reflection of light. Caustics is related to simple geometric measurements of the shadow spot (caustic) at the crack tip to determine a stress intensity value, whereas photoelasticity determines the SIFs non-trivially based on the isochromatic fringe pattern that is generated when the material undergoes stress. Both methods have drawbacks of requiring a transparent or reflective material, or in the case of photoelasticity, a birefringent material . Photoelasticity can be used with a greater range of materials through the application of a birefringent coating, yet issues can arise with the thickness and adherence of the coating to the true behavior of the material being investigated. In addition, while the crack tip location is known with photoelasticity, the domain at the crack tip is generally not well resolved. Caustics has the issue of being a purely local measurement, and there is still some ambiguity of the true crack tip location and sizing of the ‘z’ or out of plane focus in regards to encompassing the process zone . These drawbacks are mentioned for discussion only, and not to understate the fact that there has been a large body of impressive work using these techniques spanning over 40 years.
More recently, with the proliferation of advanced computing equipment and development of newer full-field experimental techniques (namely shearing interferometry methods such as coherent gradient sensing and coherent digital sensing, an digital image correlation) has given rise to new methods to extract fracture behavior [20-22]. Tippur’s and Yoneyama’s laboratories have more recently developed a method for extracting SIFs from DIC data taken during fracture by using the relationship between SIFs and displacement around the crack tip [23,24]. This method has been applied to SENBs of tests for several materials. Yoneyama first developed the technique based on earlier work by Sanford mapping deformation fields around a crack tip to fracture properties . Yoneyama’s experiments focused on an isotropic material loaded quasi-statically . Kirugulige and Tippur examined mixed-mode loading in syntactic foams under dynamic loading, expanding the method to work with higher loading rates . Lee, Tippur, and Bogert studied graphite/epoxy composites both prior to and during crack propagation, further broadening the applicability of the technique to anisotropic material under dynamic conditions . While the spatial resolution is dramatically increased with using a DIC based approach, the major drawback remains that the precision location of the crack tip is an unknown and must be optimized in a nonlinear formulation to determine. Our group has also pushed the metrology of DIC to extract SIFs in uncertainty quantification of influence of regions and size of regions of K-dominance datasets up, as well as error in the location of the crack tip. These newer experimental dynamic fracture investigations with increased resolution usher in a new era of hybrid numerical-experimental methods to deal with the sheer increased in magnitude of measured field data in a meaningful way.
Considerations for Dynamic Crack Behavior
"When the going gets tough, the tough get going" – English proverb
So why do these differences between quasi-static and dynamic toughness exist? How do we begin and/or continue to understand material inertia, transient effects, and their dependence on crack tip velocity and instabilities? Kalthoff in 1986 first suggested an incubation time for dynamic initiation, during which the SIF increases rapidly above the quasi-static value . Aoki and Kimura in 1993 also suggested a delay time, but further surmised that it was a pragmatic result of detecting a surface crack via a 2D optical technique, where in reality the SIF at the mid-thickness of the specimen is higher than that on the outer surface, and the time is due to the propagation of the intensity to the outer surfaces (creating a dwell time) . Others still have suggested that the higher dynamic toughness values from quasi-static can be attributed to the time needed to establish the singular crack-tip field as compared to the actual fracture time [30,31].
Another train of thought for a physical explanation of this phenomenon is that the actual crack front shape (the area being created by the moving crack) might have some roughness or lower length scale irregularities- which have since been visualized by post-mortem fractography[32,33] . Theoretically, this microstructural insight has been explored by Gao and Rice in 1989 as heterogeneity along the crack tip, in addition to a weak geometric perturbation (transient effects), illustrating a dynamic toughening effect . Broadly speaking, crack front perturbations and influence of local microstructural variation along the crack path have been addressed from a statistical physics view via crack pinning and depinning due to local obstacles . It has been pointed out that the energy dissipation increases with crack velocity, a fact that is reasoned by the increasing process zone size by Broberg .
A simpleminded way to think about this based on the above discussion is in terms of energy release rate, G, which considering a flat crack and change in potential energy with A is the crack area (where you may be used to seeing crack length a). By definition, the energy release rate is a measure of energy available for crack extension . For quasi-static regimes, this is simply stated as the change in two energy states (in equilibrium) as a function of change in crack area with the same prescribed loads:
For the dynamic case, taking into account inertial effects (where the dynamic path-independent d J –integral is equal to G ):
and F is the work done by external forces. For the sake of discussion if we further assume (for simplicity) that the forces are identical at each state, we are left with taking the limit and revealing:
This is a limited case, but the forces drop away, and only holds analytically for the advance of triangular or straight cracks as:
So using this simple means, we can demonstrate that fundamentally the kinetic term leads to a toughening effect, as was first suggested by Chandra and Krauthammer in 1995 . Further, depending on microstructural considerations which influence material resistance to fracture on a local scale (i.e. crack front shape), along with transient effects creating a gradient of forces rapidly changing across the moving crack front, one would modify the work term that was neglected here and create interplay in the overall energy balance available for crack extension. This competition is what creates interesting and useful understanding of true dynamic fracture, and perhaps sheds light on rate-sensitivity with a given type of loading, geometry and material microstructure.
Next-Generation Dynamic Fracture Mechanics
"If we knew what we were doing, it wouldn't be called research " -Albert Einstein
With the increase of spatial and temporal resolution of high-speed and ultra high-speed imaging (see a nice chart of the current specs on cameras by Dr. Phillip Reu here: as well as increasing resolution of full-field optical techniques, the future holds the possibility of accurately measuring not only displacements, but acceleration fields ahead of a moving crack under dynamic loading. This is a very powerful metric on many fronts, both in general dynamic behavior, but also in regards to dynamic fracture. Namely, that ability would lead to a direct means of assessing the kinetic term that gives rise to the aforementioned dynamic toughening effect, and allows for more quantitative tracking of the transient nature of a moving crack front under conditions with high rates of loading and complicated wave interactions. In some regard, having the capability to not only track but map evolving acceleration fields suggests a shift away from the classical dynamic fracture mechanics approach of square root singularity, or K dominant zone, and opens the door to directly extracting the energetic balance terms directly and straightforwardly. It should be noted that at present, noise in experimental resolution of accelerations due to the fact that with full-field methods like DIC or grid methods, you are taking a derivative of the measured deformation field data not once, but twice, which magnifies noise, and remains somewhat prohibitively significant (to the authors chagrin); yet again this idea shows promise in moving dynamic fracture mechanics and its fundamental understanding, with associated experimental techniques forward, leaving restrictions of singular domains and asymptotically fitted solutions (in infinite spaces), or complications of a direct J-integral approach around the keyhole contour in its wake (excuse the pun). Our group is working presently in this fringe of space (excuse the interferometry pun), demonstrating the energy based method with virtual FE modeling, examining the implication of some of the assumptions of classical dynamic fracture mechanics and correlations to the J-Integral. One can think of this approach akin to a control volume problem, but now we are celebrating transient fields instead of avoiding them  . It’s an interesting future giving life to less-new, but still unresolved field of dynamic fracture. Plus, who doesn’t like to break stuff for a living?
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