Peridynamic modeling of the crack velocity dependence via an incubation time fracture criterion

This study utilizes a peridynamic approach with an incubation time fracture criterion to model Ravi-Chandar and Knauss's experiments on Homalite-100, revealing that variations in the Mode-I stress intensity factor at constant crack velocities and the onset of micro-branching at higher velocities provide new insights into the nature of crack-velocity dependence in dynamic fracture.

The Gist

Imagine you are watching a crack race across a piece of brittle plastic, like a sheet of Homalite-100. In the old days of physics, scientists thought that if you knew how fast the crack was moving, you could calculate exactly how much "stress" (or pressure) was pushing it forward. It was like thinking that if a car is going 60 mph, the engine must be producing exactly 100 horsepower. Simple, right?

But experiments in the 1980s showed this wasn't true. Sometimes, the crack was moving at the exact same speed, but the pressure pushing it was wildly different. It was as if two cars were both doing 60 mph, but one had a tiny engine and the other had a rocket booster. Scientists were puzzled: Why does the same speed have different "pushes"?

This paper is a detective story where the authors use a new kind of computer simulation to solve this mystery.

The Detective Tool: Peridynamics

Most computer models of cracks are like a chain of dominoes. If one domino falls, it pushes the next one. But if a domino is missing (a crack), the chain breaks, and the math gets stuck.

The authors used a method called Peridynamics. Think of this not as a chain, but as a swarm of bees. Every bee can talk to every other bee within a certain distance, even if there is a gap in the middle. If a bee flies away (a crack forms), the other bees just stop talking to it, but the rest of the swarm keeps moving perfectly fine. This allows the computer to handle breaking and cracking without getting confused.

The Secret Ingredient: The "Incubation Time"

The real breakthrough in this paper is how they decided when a crack should actually break.

In the old way, if the pressure got high enough, the material broke instantly. But the authors used a rule called the Incubation Time Criterion.

Imagine you are trying to snap a dry twig. You don't just pull and it snaps instantly. You pull, hold it there for a split second while the fibers stretch and weaken, and then it snaps. That split second is the "incubation time."

The authors programmed their computer swarm to remember the last few microseconds of pressure. The material only breaks if the average pressure over that short "incubation" period is high enough. This accounts for the fact that materials need a tiny bit of time to "decide" to break.

What They Found

They ran simulations of the plastic plates being pulled apart, just like the real experiments. Here is what they discovered:

1. The Speed vs. Pressure Puzzle: Just like the real experiments, their computer showed that for the same crack speed, the pressure (Stress Intensity Factor) wasn't a single number. It was a range. Sometimes it was low, sometimes high.
2. The "Micro-Branching" Effect: When the crack moved slowly, it went straight. But when it sped up (over 400 meters per second), it started to get jittery. It began to sprout tiny, microscopic side-cracks, like a tree branch splitting into twigs.
   • The Analogy: Imagine a runner sprinting. At a steady jog, they run in a straight line. But when they sprint at top speed, they start to wobble and zigzag slightly to maintain balance.
   • The Result: These tiny "wobbles" (micro-branches) caused the pressure reading to jump up and down wildly. This explained why the pressure wasn't unique for a given speed; the crack was physically changing its shape slightly as it raced.

The Conclusion

The paper concludes that the reason we see different pressure values for the same crack speed is because the crack isn't a smooth, perfect line. It's a chaotic, living thing that fluctuates.

• At lower speeds: The crack is steady, and the pressure is relatively stable.
• At higher speeds: The crack starts to "micro-branch" (sprout tiny side-cracks). This chaos causes the pressure to bounce around, creating the scatter seen in the experiments.

By using this "swarm of bees" (Peridynamics) combined with the "waiting period" (Incubation Time), the authors successfully recreated the messy, non-unique relationship between crack speed and pressure that real-world experiments had shown for decades. They proved that the "noise" in the data isn't a mistake; it's a real physical feature of how fast-moving cracks behave.

Technical Summary

Technical Summary: Peridynamic Modeling of Crack Velocity Dependence via an Incubation Time Fracture Criterion

Problem Statement
This study addresses a central unresolved issue in dynamic fracture mechanics: the dependence of the instantaneous Mode-I stress intensity factor (SIF, $K_I$) on crack propagation velocity ($\dot{a}$). Classical Linear Elastic Fracture Mechanics (LEFM) and dynamic criteria based on Freund and Rosakis often suggest a unique relationship between $K_I$ and $\dot{a}$. However, experimental data from Ravi-Chandar and Knauss on brittle Homalite-100 polymer plates demonstrate a non-unique dependence, where a single crack velocity corresponds to a range of SIF values. Furthermore, at higher velocities, micro-branching occurs, leading to significant scatter in SIF measurements. The challenge lies in numerically reproducing this non-uniqueness and the associated fluctuations without relying on empirically adjusted rate-dependent material properties.

Methodology
The authors employ a non-local peridynamic framework to simulate dynamic crack propagation in Homalite-100, replicating the experimental conditions of Ravi-Chandar and Knauss. The core methodological innovation is the integration of the Incubation Time Fracture Criterion (ITFC) into the peridynamic formulation.

• Peridynamic Framework: The simulations utilize an ordinary state-based peridynamic model implemented in the open-source software Peridigm. The material is discretized into points interacting within a spherical horizon ($\delta$), allowing for the natural handling of discontinuities like cracks without spatial differential operators.
• Incubation Time Fracture Criterion (ITFC): Unlike classical criteria that assume instantaneous failure, ITFC posits that fracture requires a finite time ($\tau$) for preparatory processes (e.g., microcracking) to occur. The criterion evaluates the time-averaged stress over the incubation period $\tau$ and a characteristic length $d$.
  • The characteristic length $d$ is derived from the quasi-static Irwin criterion to ensure consistency with the critical static SIF ($K_{Ic}$).
  • The peridynamic implementation adapts the Remote Stress (RS) fracture criterion. Bond failure is determined by integrating the normal bond stress over the incubation time $\tau$ (Eq. 11).
  • To manage computational complexity and focus on micro-branching rather than full macro-branching, the criterion is evaluated only on planes parallel ($0^\circ$) and perpendicular ($90^\circ$) to the main crack axis.
• Simulation Setup: The model simulates a 300 mm $\times$ 500 mm specimen with a 50 mm pre-notch. Dynamic loading is applied via a trapezoidal volumetric force pulse with a 25 $\mu$s rise time. The dynamic SIF is calculated using a moving rectangular contour for the dynamic J-integral, ensuring the integration path encloses the main crack tip while excluding developing micro-branches.

Key Results
The numerical simulations successfully reproduce the experimentally observed phenomena regarding the $K_I$-$\dot{a}$ relationship:

1. Non-Unique $K_I$-$\dot{a}$ Dependence: The results confirm that for a nearly constant crack propagation velocity, the instantaneous SIF exhibits significant fluctuations. Consequently, the model predicts a range of SIF values for a given velocity, mirroring the non-uniqueness observed in experiments.
2. Velocity Stabilization and Transients: While experiments show crack velocity stabilizing almost instantaneously, the simulations exhibit an initial acceleration phase lasting approximately 30–40 $\mu$s before reaching a steady state. During this transient phase, the SIF history shows pronounced variations.
3. Micro-branching and Scatter: At higher loading rates (resulting in velocities > 400 m/s), the simulations exhibit the onset of micro-branching. This phenomenon correlates with a marked increase in the amplitude of SIF oscillations. The numerical scatter in SIF values at high velocities (0.53–1.25 MPa$\sqrt{m}$) aligns reasonably well with experimental ranges (0.45–1.5 MPa$\sqrt{m}$).
4. Validation of Mechanism: The study demonstrates that the observed scatter in SIF is physically justified by the dynamics of the crack tip, specifically the acceleration phase and the onset of micro-branching, rather than being an artifact of measurement error.

Significance and Claims
The paper claims that the peridynamic implementation of the ITFC provides a valuable approach for assessing dynamic fracture toughness and modeling crack propagation across a wide range of loading rates. The primary significance lies in the ability of the model to capture the rate-sensitivity of brittle fracture and the non-uniqueness of the $K_I$-$\dot{a}$ relationship without introducing arbitrary rate-dependent material parameters.

The authors modestly note that while the model qualitatively captures the experimental trends, quantitative discrepancies remain, particularly regarding the timing of velocity stabilization and the precise magnitude of SIF fluctuations. They attribute the successful prediction of the non-unique dependence to the inclusion of the incubation time, which accounts for the temporal history of stress, and the non-local nature of peridynamics, which naturally handles the singular fields and discontinuities associated with dynamic fracture. The study concludes that the interplay between the crack acceleration phase, SIF fluctuations, and the onset of micro-branching are the governing mechanisms behind the experimentally observed $K_I$-$\dot{a}$ behavior.