🔬 applied physics

High performance construction materials fracture and high cycle fatigue assessment based on accelerated PF-CZM

This paper proposes an accelerated Phase-Field Cohesive Zone Model (PF-CZM) framework incorporating a stress-based failure criterion, a fatigue degradation function, and an envelope load-based acceleration algorithm to efficiently simulate and validate the mixed-mode fracture and high-cycle fatigue behavior of Ultra-High Performance Concrete (UHPC) and High Strength Steel (HSS).

Original authors: Jiaqi Li (College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling, China), Zhihua Xiong (College of Water Resources and Architectural Engineering, Northwest A&F Un

Published 2026-06-23

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CC BY 4.0

Original authors: Jiaqi Li (College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling, China), Zhihua Xiong (College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling, China), Xuyao Liu (College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, China), Hongyu Liu (College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling, China)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are building a skyscraper or a massive bridge using the strongest materials available: Ultra-High Performance Concrete (UHPC) and High-Strength Steel (HSS). These materials are like the "superheroes" of construction—they are incredibly tough and can handle heavy loads. However, just like a superhero, they have a weakness: if you keep shaking them or bending them back and forth millions of times (like wind hitting a bridge or traffic hitting a road), they can eventually crack and fail. This is called fatigue.

The problem is that figuring out exactly when and how these materials will break is a nightmare for engineers. Traditionally, they have to build physical models and shake them until they break. This is expensive, takes forever, and you can't test every single possible scenario.

This paper introduces a new digital "crystal ball" (a computer model) that predicts these cracks and failures much faster and more accurately than before. Here is how it works, broken down into simple concepts:

1. The "Smart Map" for Cracks (Phase-Field Cohesive Zone Model)

Imagine trying to draw a crack on a piece of paper. In old computer models, you had to draw the line exactly where you thought the crack would go. If you guessed wrong, the whole simulation failed.

This paper uses a new method called PF-CZM. Think of this not as drawing a sharp line, but as painting a fuzzy, glowing zone where the material is getting weak.

Instead of a sharp crack, the computer sees a "cloud" of damage that gets darker and darker until it becomes a full break.
This "cloud" can grow, split, or change direction automatically. It doesn't need a human to tell it where to go; it follows the physics of the material.
The authors tuned this "cloud" specifically for UHPC and HSS so it mimics how these real materials actually soften and break.

2. The "Speed Booster" (Accelerated Algorithm)

Here is the biggest hurdle: High-cycle fatigue means the material is shaken millions of times. If a computer tries to simulate every single shake (every cycle), it would take years to finish the calculation. It's like trying to count every grain of sand on a beach one by one.

The authors created a "Speed Booster" (an acceleration algorithm) to solve this:

The Envelope Trick: Instead of simulating every single shake, the computer looks at the "envelope" (the outer shape) of the shaking. It calculates the average effect of thousands of shakes at once.
The Three-Stage Race: The simulation runs in three different speeds:
1. The Warm-up: When the material is fresh and nothing is happening, the computer skips ahead in giant leaps (simulating millions of cycles in a second).
2. The Warning Zone: As the material starts to get tired, the computer slows down to watch closely, taking smaller steps to catch the first signs of trouble.
3. The Finish Line: Once the crack starts to really grow, the computer slows down even more to watch the final break in high definition.
The Result: This method cuts the computing time by over 97%. It's like watching a movie at 100x speed until the action starts, then switching to normal speed for the climax.

3. Testing the "Crystal Ball"

To prove their new model works, the authors tested it against real-world experiments:

For Concrete (UHPC): They simulated beams with notches (weak spots) being bent until they broke. The computer predicted the exact path the crack took and how much weight the beam could hold, matching real lab tests perfectly. They also tested it under mixed forces (bending and twisting), which is much harder to predict.
For Steel (HSS): They tested how steel behaves in freezing cold temperatures (where it becomes brittle and snaps like glass). The model successfully predicted how the steel would crack under tension and shear forces, matching real-world data.

The Bottom Line

This paper doesn't just say "we made a faster computer." It says: "We built a specific, high-speed digital simulator that understands the unique 'personality' of super-strong concrete and steel."

By combining a smart way to visualize cracks (the fuzzy cloud) with a speed-boosting algorithm (the three-stage race), engineers can now predict when these critical materials will fail due to fatigue without spending years and millions of dollars on physical testing. It allows them to design safer bridges and buildings by knowing exactly how the materials will behave under millions of cycles of stress.

Technical Summary: Fracture and High-Cycle Fatigue Assessment of High-Performance Materials via Accelerated PF-CZM

Problem Statement
High-performance materials, specifically Ultra-High Performance Concrete (UHPC) and High Strength Steel (HSS), are critical for demanding civil and mechanical engineering applications such as wind turbines and bridges. These structures are frequently subjected to cyclic loading, where fracture and fatigue phenomena can precipitate catastrophic failures. While experimental methods are currently the primary means of assessing these behaviors, they are often costly, time-consuming, and limited in generalizability. Furthermore, numerical simulation of High Cycle Fatigue (HCF), which involves millions of load cycles, presents a significant computational bottleneck. Existing Phase-Field Models (PFM) and Phase-Field Cohesive Zone Models (PF-CZM) struggle to efficiently simulate the complete three-stage crack propagation process for these specific materials, particularly under HCF conditions.

Methodology
This study proposes a comprehensive computational framework based on the Phase-Field Cohesive Zone Model (PF-CZM) to address the fracture and HCF behavior of UHPC and HSS. The methodology integrates three core components:

Constitutive Modeling and Failure Criteria:
- For UHPC: A generic stress-based failure criterion is derived from biaxial tensile test results to calculate the crack driving force. This criterion utilizes a hyperbolic biaxial strength envelope to capture mixed-mode fracture behavior. Additionally, a constitutive model is calibrated using uniaxial tensile test data to define the energy degradation function and softening curves.
- For HSS: The framework is adapted to simulate low-temperature brittle fracture. A linear softening energy degradation function is employed, and the von Mises failure criterion is utilized for mixed-mode fracture calculations.
Fatigue Degradation Mechanism:
The model incorporates a fatigue degradation function into the Griffith energy balance equation. This approach adopts the concept of state-dependent fracture toughness, where a cumulative historical variable ( $\bar{\alpha}(t)$ ) drives the reduction in fracture toughness over time. The fatigue damage accumulation parameter ( $k_f$ ) is introduced to regulate the rate of damage accumulation.
Accelerated Algorithm:
To overcome the computational prohibitions of simulating millions of cycles, an accelerated algorithm is integrated. This method employs an envelope load to approximate real cyclic loads, avoiding the simulation of every individual cycle. The algorithm utilizes a three-stage adaptive cycle-increment adjustment strategy:
- Pre-threshold stage: Large cycle increments are used when damage is below the threshold.
- Threshold stage: Increments are reduced as damage approaches the threshold to ensure precision.
- Post-threshold stage: Adaptive increments control the maximum damage increment to maintain a moderate fatigue damage rate.
  The numerical implementation utilizes ABAQUS subroutines (UMAT and HETVAL), leveraging an analogy between heat transfer laws and the PF balance equation to solve the coupled field equations.

Key Contributions

Unified Framework: The paper extends the PF-CZM framework to simultaneously model fracture and HCF for both UHPC and HSS, bridging the gap between experimental limitations and computational efficiency.
UHPC Specifics: A new constitutive model and a generic stress-based failure criterion (hyperbolic biaxial strength envelope) are developed specifically for UHPC, calibrated against uniaxial and biaxial test data.
Efficient HCF Simulation: The integration of a fatigue degradation function with a three-stage adaptive acceleration algorithm enables the efficient simulation of high-cycle fatigue without sacrificing the accuracy of crack path prediction.
Low-Temperature HSS Modeling: The framework successfully reproduces the low-temperature brittle fracture behavior of HSS, a phenomenon often absent in robust numerical models.

Results
The proposed model was validated against relevant experimental data:

UHPC Fracture: The model accurately captured Mode-I and mixed-mode fracture behaviors of UHPC notched beams. Simulation results for load-CMOD curves and crack propagation paths showed strong agreement with experimental data across different length scales, demonstrating mesh objectivity.
UHPC Fatigue: The accelerated algorithm successfully simulated Mode-I and mixed-mode high-cycle fatigue. The fatigue life and creep curves (CMOD evolution) matched experimental results when the fatigue damage accumulation parameter ( $k_f$ ) was calibrated (specifically $k_f=1$ ). The model correctly reproduced the three-stage fatigue process and crack propagation paths.
HSS Fracture: The model effectively simulated the Mode-I and mixed-mode fracture of HSS at low temperatures. Stress-strain curves and crack propagation paths from Center Notch Tension (CNT) and Compact Tension Specimen (CTS) tests were consistent with experimental observations, including the transition from tensile to shear-dominated crack propagation under varying loading angles.

Significance and Claims
The authors claim that this study provides a computationally efficient and accurate alternative to traditional experimental methods for assessing the fracture and fatigue of high-performance materials. By integrating a fatigue degradation function with an acceleration algorithm, the framework overcomes the computational bottlenecks associated with HCF simulations. The derivation of a generic stress-based failure criterion for UHPC allows for the precise calculation of crack driving forces in mixed-mode scenarios. Furthermore, the successful extension of the model to HSS demonstrates its versatility in predicting low-temperature brittle fracture. The study concludes that the proposed PF-CZM framework is a robust tool for predicting the fracture and high-cycle fatigue behavior of both UHPC and HSS, offering a pathway to more reliable design and assessment of cyclic-loaded infrastructure.

1. The "Smart Map" for Cracks (Phase-Field Cohesive Zone Model)

2. The "Speed Booster" (Accelerated Algorithm)

3. Testing the "Crystal Ball"

The Bottom Line

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