Phase-field modeling of cyclic behavior in quasi-brittle materials: a micromechanics-based approach

Technical Summary: Phase-field modeling of cyclic behavior in quasi-brittle materials: a micromechanics-based approach

Problem Statement
Quasi-brittle materials, such as concrete, masonry, and rock, exhibit complex failure mechanisms under cyclic loading. While traditional fatigue models address damage accumulation, they often fail to capture the progressive accumulation of inelastic strains known as ratcheting, which is critical in low-cycle fatigue scenarios. Furthermore, standard phase-field models often rely on phenomenological energy splits (e.g., spectral or volumetric-deviatoric) to distinguish between tension and compression, which may not fully reflect the underlying micromechanisms of microcrack opening and closure. The challenge lies in developing a thermodynamically consistent framework that integrates cyclic plasticity with ratcheting effects into a phase-field model for quasi-brittle materials, explicitly accounting for microstructural features like frictional sliding and microcrack evolution.

Methodology
The authors propose an extension of a micromechanics-based phase-field framework. The core of the methodology involves:

1. Micromechanics-Based Free Energy: The model distinguishes between open and closed microcrack states within a Representative Volume Element (RVE).

   • Open State: Microcracks do not transmit stress; the material behaves in a brittle manner with no plastic contribution to strain energy.
   • Closed State: Microcracks undergo frictional sliding. The free energy includes contributions from elastic strain, plastic strain, and a newly introduced ratcheting strain ($\varepsilon_r$). The ratcheting strain represents an irreversible shift in the elastic reference state due to microstructural rearrangements.
   • The formulation naturally captures asymmetric behavior (tension vs. compression) without explicit energy splitting, relying instead on the state of microcracks.

2. Non-Associative Plasticity with Ratcheting:

   • The plasticity model employs a non-associative Drucker–Prager yield criterion to capture frictional-dilational behavior, avoiding the unrealistic excessive dilation predicted by associative models.
   • A ratcheting strain tensor is introduced as an internal variable that evolves via a non-associative flow rule. The evolution is driven by the plastic strain rate but accumulates progressively over cycles.
   • The plastic potential is formulated to allow independent control of deviatoric and volumetric ratcheting components through parameters $\beta_\mu$ and $\beta_K$.

3. Fatigue Degradation:

   • Fatigue is modeled via a fatigue history variable ($F$) and a degradation function ($h(F)$).
   • Two degradation functions are implemented: an asymptotic function and a logarithmic function. The degradation reduces the fracture energy ($G_c$) once a threshold ($F_0$) is exceeded.
   • The model distinguishes between Mode I (tensile) and Mode II (shear/compressive) fracture energies, using a smoothstep function to regularize the transition between them.

4. Numerical Implementation:

   • A three-field staggered solution scheme is employed to decouple the system of partial differential equations.
   • The algorithm iteratively solves for: (1) the displacement field (mechanical equilibrium), (2) the crack phase-field (damage evolution using a history-field approach), and (3) the plastic and ratcheting strains (using an implicit return-mapping algorithm).

Key Contributions

• Integration of Ratcheting: The primary contribution is the thermodynamically consistent incorporation of ratcheting strain into a micromechanics-based phase-field model. This allows the simulation of progressive inelastic strain accumulation under asymmetric cyclic loading.
• Micromechanical Basis: The model retains a variational structure rooted in microcrack behavior (opening vs. sliding), providing a physics-based alternative to purely phenomenological energy splits.
• Independent Control of Ratcheting Modes: The formulation introduces specific parameters to independently control volumetric and deviatoric ratcheting, offering flexibility in modeling complex cyclic responses.
• Unified Framework: The model simultaneously captures fatigue damage evolution, cyclic plasticity, and ratcheting within a single continuum framework.

Results
Numerical simulations were conducted to validate the model under various conditions:

• Monotonic and Cyclic Three-Point Bending: The model successfully reproduced experimental force-displacement curves for concrete beams. Under cyclic loading, it captured crack initiation, stable propagation, and sudden failure after a specific number of cycles (755 cycles in the example), aligning with observed fatigue behavior.
• High-Cycle Fatigue (Single Element): S-N curves were generated using both asymptotic and logarithmic degradation functions. The study highlighted the sensitivity of the predicted fatigue life to the mathematical form of the degradation function, particularly noting unexpected behaviors in the logarithmic formulation at high strain levels.
• Material Point Ratcheting: Simulations under stress-controlled and strain-controlled cyclic loading demonstrated the model's ability to capture:
  • Stress Relaxation: Under strain control, where stress amplitude decreases over cycles.
  • Ratcheting: Under stress control, where strain progressively accumulates with each cycle.
  • The model showed sensitivity to the ratcheting parameters ($\beta_K, \beta_\mu$), with higher values leading to greater strain accumulation.
• Perforated Specimen (Low-Cycle Fatigue): A square specimen with a central hole subjected to force-controlled tension-compression cycles demonstrated the model's capability to simulate localized damage and inelastic strain accumulation around stress concentrations. The results showed stiffness degradation and significant ratcheting strain accumulation over 1000 cycles.

Significance and Claims
The paper claims that the proposed framework offers a robust tool for investigating the cyclic behavior of quasi-brittle materials, particularly in low-cycle fatigue regimes where inelastic strain accumulation is a dominant failure mechanism. By grounding the model in micromechanics, it provides a more physically motivated description of failure modes (tensile vs. shear) compared to conventional phenomenological approaches. The authors emphasize that the model effectively captures the interplay between fatigue damage and cyclic plasticity, including the critical phenomenon of ratcheting, without relying on heuristic energy splits. The work suggests that such models can complement experimental studies, potentially reducing the need for extensive testing by providing accurate predictions of material degradation and failure under complex loading paths. The study concludes that the independent control of ratcheting components and the thermodynamic consistency of the formulation are essential for accurately modeling the long-term deterioration of quasi-brittle structures.