The Johnson–Holmquist-II (JH-2) model is a widely used constitutive model for predicting the behavior of brittle materials, such as annealed glass, under high-strain-rate conditions, including dynamic compression tests, impacts, and explosions. It was developed by G. R. Johnson and T. J. Holmquist to address the limitations of traditional material models in capturing the complex response of ceramics and glasses under such extreme loading.

🔹 Introduction to the Johnson–Holmquist-II (JH-2) Model

The JH-2 model is a continuum damage mechanics model specifically designed to simulate the dynamic response of brittle materials subjected to large strains, high strain rates, and high pressures. It builds upon the original JH-1 model by incorporating damage evolution, pressure-dependent strength, and strain-rate effects.

The model is particularly suitable for:

• Ceramics and glasses (e.g., borosilicate glass, fused silica, annealed glass)
• High-velocity impacts and shock loading
• Ballistic simulations and protective armor design

🔹 Key Features of the JH-2 Model

1. Three Strength Regimes:
   • Intact material: before damage initiates.
   • Damaged material: as microcracks evolve.
   • Fully fractured material: residual strength dominates.
2. Pressure-Dependent Strength:
   • The yield strength increases with increasing pressure, reflecting how brittle materials can carry higher loads under confinement.
3. Damage Evolution:
   • Damage accumulates progressively with plastic strain and is a function of the maximum tensile hydrostatic pressure experienced.
4. Strain-Rate Sensitivity:
   • The model includes a logarithmic term to account for the rate of deformation, which influences strength and damage thresholds.
5. Equation of State (EOS):
   • The model works in conjunction with an EOS (often a Mie-Grüneisen or tabular EOS) to describe the material’s volumetric response under pressure.

🔹 Mathematical Form of the Model

The normalized equivalent stress in the JH-2 model is expressed as:

The intact and fractured strengths are typically modeled as:

The damage evolution is given by:

Where D1, D2 are material constants defining the plastic strain to fracture as a function of pressure.

🔹 Application to Annealed Glass in Dynamic Compression Tests

In dynamic compression tests on annealed glass, the JH-2 model captures:

• The initial high strength due to intact microstructure
• The progressive degradation as damage (e.g., microcracking) accumulates
• The transition to a low-strength fractured state
• Rate effects relevant at high loading speeds

Annealed glass, being brittle and amorphous, shows very limited plasticity. The JH-2 model’s damage-based formulation is thus well-suited to simulate its behavior in shock and high-pressure environments, where standard plasticity models fail.

This example is close to the paper here: Johnson–Holmquist-II model of annealed glass and its verification in dynamic compression test