Mesoscale Modelling of Confined Split-Hopkinson Pressure Bar Tests on Concrete: Effects of Internal Damage and Strain Rates

This study employs mesoscale finite element modelling to demonstrate that while increasing loading ramp rates, internal friction, and confining pressure all enhance the dynamic strength of concrete, only higher loading ramp rates significantly amplify the strain-rate effect by inducing pronounced damage in both mortar and aggregate phases, whereas the other factors weaken this effect primarily through the mortar phase.

The Gist

Imagine concrete not as a solid, boring block of gray rock, but as a giant, microscopic city.

In this city:

• The Aggregates are the sturdy skyscrapers (the rocks and stones).
• The Mortar is the concrete paste filling the streets and alleys between the buildings.
• The ITZ (Interfacial Transition Zone) is the sticky, slightly weaker glue holding the buildings to the streets.

This paper is about what happens when you hit this microscopic city with a giant hammer, but you do it so fast that the city doesn't have time to "think" about breaking. This is called dynamic loading, and it's crucial for understanding how bridges or dams survive earthquakes, explosions, or ship crashes.

Here is a simple breakdown of what the researchers did and what they found, using some fun analogies.

The Experiment: The "Hammer on a Stick" Test

The researchers used a machine called a Split Hopkinson Pressure Bar (SHPB).

• The Analogy: Imagine two long steel rods (like giant drumsticks) with a tiny concrete cylinder sandwiched between them. You hit one end of the first rod with a heavy hammer. A shockwave travels down the rod, hits the concrete, and tries to crush it.
• The Goal: They wanted to see how much stronger concrete gets when you hit it really fast compared to when you push it slowly. This strength boost is called the Dynamic Increase Factor (DIF).

The Innovation: A "Realistic" City Model

Usually, computer models treat concrete aggregates (the rocks) as perfect spheres, like marbles. But real rocks are jagged and weird-shaped.

• What they did: The team built a 3D computer model where the "skyscrapers" (aggregates) have realistic, jagged shapes, just like real rocks. They simulated the shockwave hitting this realistic city and watched exactly where the cracks started and how fast the "streets" (mortar) and "buildings" (aggregates) deformed.

The Three Big Discoveries

The researchers tested three different "knobs" they could turn to see how they changed the city's reaction to the hammer.

1. The "Ramp Rate" (How fast the hammer speeds up)

• The Setup: Instead of the hammer hitting instantly, they tested how the speed of the impact wave's rise mattered.
• The Finding: If the hammer speeds up its strike very quickly (a steep ramp), the concrete gets much stronger and the "strain rate effect" (the boost in strength) becomes huge.
• The Analogy: Think of pushing a heavy door. If you push it slowly, it's hard. If you shove it instantly, it feels like it's fighting back harder. If you accelerate that shove incredibly fast, the door (concrete) reacts violently, causing more internal chaos (damage) but also making the material act tougher overall. The faster the "shove" builds up, the more the microscopic city fights back.

2. The "Internal Friction" (How much the parts rub against each other)

• The Setup: Concrete isn't smooth inside; the rocks rub against the paste. The researchers simulated different levels of "grippiness" (friction) between these parts.
• The Finding: Higher friction actually makes the concrete stronger overall, but it makes the "speed boost" (strain rate effect) weaker.
• The Analogy: Imagine a crowd of people trying to run through a hallway.
  • Low Friction: They slide easily. If you push them fast, they all scramble wildly (high strain rate effect).
  • High Friction: They are sticky and hold onto each other. They can't move as fast, so they feel stronger and harder to crush. However, because they are already so "stuck" and resisting movement, hitting them faster doesn't change their behavior as much as it does for the slippery crowd. The friction acts like a brake, dampening the extra boost you'd usually get from speed.

3. The "Confining Pressure" (Squeezing from the sides)

• The Setup: They simulated squeezing the concrete cylinder from the sides (like putting it in a vice) while hitting it.
• The Finding: Squeezing from the sides makes the concrete much stronger, but again, it reduces the extra boost you get from hitting it fast.
• The Analogy: Imagine a bag of marshmallows.
  • No Squeeze: If you hit a loose bag, it explodes.
  • Squeezed: If you squeeze the bag tight with your hands, the marshmallows can't expand. They become incredibly hard to crush.
  • The Twist: Because the bag is already so tight and strong from the squeeze, hitting it faster doesn't make it significantly stronger than it already is. The "squeeze" did most of the work, so the "speed" doesn't add as much extra value.

The Secret Sauce: Why the "Streets" Matter Most

The most interesting part of the paper is why these things happen. The researchers looked inside the microscopic city.

They found that the Mortar (the streets) is the real hero (or villain) of the story.

• When you hit the concrete fast, the "streets" (mortar) get damaged much more than the "skyscrapers" (aggregates).
• The Ramp Rate: When the hit is super fast, the streets get smashed up wildly, which paradoxically makes the whole structure act stronger.
• Friction & Squeezing: When you add friction or squeeze the sides, the streets stop getting smashed up as much when you hit them fast. They are "protected" or "dampened." Because the streets aren't reacting as wildly to the speed, the overall "speed boost" (DIF) is smaller.

The Bottom Line

This paper teaches us that concrete isn't just a solid block; it's a complex, messy city.

1. Faster impacts generally make concrete stronger, but only if the impact wave rises sharply.
2. Squeezing it or making the inside stickier makes it stronger overall, but it stops the "speed boost" from being as dramatic.
3. The mortar (paste) is the part that reacts most to speed. If you can control how the mortar behaves, you can better predict how a bridge or dam will survive a crash or earthquake.

This research helps engineers design safer buildings that can withstand the unpredictable, high-speed forces of nature and accidents.

Technical Summary

1. Problem Statement

Concrete is a heterogeneous material composed of mortar, aggregates, and an interfacial transition zone (ITZ). Under dynamic loading conditions (e.g., earthquakes, impacts, blasts), concrete exhibits a significant increase in strength, quantified by the Dynamic Increase Factor (DIF). While Split Hopkinson Pressure Bar (SHPB) tests are the standard for characterizing this behavior, several critical gaps exist in current understanding:

• Microscopic Mechanisms: Macroscopic experimental data cannot intuitively reveal the evolution of internal damage or local strain rates within the specific material phases (mortar, aggregate, ITZ).
• Complex Loading Factors: The influence of specific loading parameters, such as the loading ramp rate (wave shape), internal friction between phases, and confining pressure, on the DIF-strain rate relationship is not fully understood or systematically quantified.
• Limitations of Experiments: It is experimentally difficult to maintain constant confining pressure during dynamic deformation or to isolate the effects of internal friction and specific wave shapes.

2. Methodology

The authors employed a mesoscale Finite Element Method (FEM) approach to simulate SHPB tests on concrete, bridging microscopic heterogeneity with macroscopic responses.

• Mesoscale Model Generation:

  • Concrete was modeled as a three-phase composite: mortar, aggregates, and ITZ.
  • Realistic Aggregate Shapes: Instead of simplified spheres, aggregates were generated using Voronoi tessellation combined with Fourier-descriptor-based methods and fractal dimensions ($F_d = 2.3$) to mimic realistic, irregular morphologies.
  • Geometry: A cylindrical specimen ($D=50$ mm, $L=50$ mm) containing ~120 aggregates (30% solid fraction) was cut from a larger cubic packing space.
  • Meshing: The model utilized ~418,000 tetrahedral solid elements for mortar/aggregates and ~34,000 zero-thickness cohesive interface elements (CIE) for the ITZ.

• Constitutive Models:

  • Mortar and Aggregates: Modeled using the Concrete Damage Plasticity (CDP) model. Material properties (strength, modulus) were made strain-rate dependent based on the FIB Model Code 2010 formulas.
  • ITZ: Modeled using Cohesive Interface Elements (CIE) with a bilinear traction-separation law and the Benzeggagh-Kenane (BK) criterion for mixed-mode failure.
  • Strain Rate Effects: Only compressive/tensile strength and fracture energy were adjusted for strain rate; the ITZ was treated as rate-independent due to a lack of experimental data.

• Loading and Simulation:

  • SHPB Setup: Simulated using ABAQUS/Explicit with steel incident and transmission bars.
  • Variables Investigated:
    1. Loading Ramp Rate ($V_L$): Controlled via trapezoidal stress wave amplitudes.
    2. Internal Friction ($f_P$): Friction coefficients between phases (0.1, 0.3, 0.5).
    3. Confining Pressure ($\sigma_c$): Lateral pressures (0, 5, 10, 15 MPa).
  • Validation: The model was validated against experimental stress-strain curves and DIF data from literature, showing excellent agreement ($R^2 = 0.99$).

3. Key Contributions

• Microscopic Evidence: The study provides direct numerical evidence linking macroscopic DIF variations to the statistical distributions of internal strain rates and local damage within specific material phases.
• Quantification of Ramp Rate: It isolates the effect of the loading ramp rate (wave shape), demonstrating that higher ramp rates significantly amplify the strain-rate sensitivity of concrete strength.
• Role of Confinement and Friction: It systematically decouples the effects of internal friction and confining pressure, revealing that while both increase absolute strength, they weaken the strain-rate sensitivity (slope of the DIF-strain rate curve).
• Statistical Correlation: The authors introduce a statistical metric, the mean internal equivalent strain rate ($M_{\dot{\varepsilon}}$), which shows a strong positive correlation ($R=0.94$) with the normalized DIF, serving as a robust microscopic indicator for macroscopic dynamic strength.

4. Key Results

A. Loading Ramp Rate ($V_L$)

• Effect: Increasing the loading ramp rate leads to higher DIF values, particularly at high strain rates ($>100/s$).
• Mechanism: A higher ramp rate induces a more rapid increase in internal strain rates and local damage in both mortar and aggregate phases. This "amplifies" the strain-rate effect, resulting in a steeper DIF-strain rate curve (higher $\beta$ value).

B. Internal Friction ($f_P$)

• Effect: Higher internal friction increases the absolute DIF (due to stress concentration and confinement effects) but reduces the slope of the DIF-strain rate curve (lower $\beta$).
• Mechanism: Higher friction suppresses local deformation and restricts the increase in internal strain rates. While it promotes fracture propagation from the ITZ into aggregates, the mortar phase shows a less pronounced increase in damage with increasing strain rate under high friction, thereby weakening the overall strain-rate sensitivity.

C. Confining Pressure ($\sigma_c$)

• Effect: Confining pressure significantly increases the dynamic strength (higher DIF) but, similar to friction, reduces the strain-rate sensitivity (lower $\beta$).
• Mechanism: Confinement suppresses the internal strain rate distribution. While it causes extensive damage in aggregates (due to lateral restraint forcing cracks into the aggregate), the mortar phase exhibits a weakened strain-rate effect on damage. The mortar's reduced sensitivity dominates the macroscopic response, flattening the DIF-strain rate curve.

D. Microscopic Indicators

• The study confirms that the mortar phase is the primary driver for the variation in strain-rate sensitivity (the slope of the DIF curve).
• The mean internal strain rate ($M_{\dot{\varepsilon}}$) serves as a reliable predictor for the DIF, validating the necessity of mesoscale modeling to understand dynamic fracture.

5. Significance

This research advances the understanding of concrete dynamics by moving beyond empirical macroscopic models to a physics-based mesoscale explanation.

• Design Implications: It highlights that the "strain-rate effect" is not a fixed material property but is highly dependent on loading conditions (ramp rate) and boundary conditions (confinement/friction).
• Theoretical Basis: The findings provide a theoretical basis for refining constitutive models for concrete under complex dynamic loading, particularly for protective structures (e.g., blast walls, offshore platforms) where confinement and impact wave shapes vary.
• Methodological Advance: It demonstrates the power of mesoscale FEM with realistic aggregate shapes and cohesive interfaces to uncover mechanisms (like internal friction and local strain rate heterogeneity) that are inaccessible to traditional experimental methods.

In conclusion, the study establishes that while confinement and friction increase concrete's static and dynamic strength, they fundamentally alter the material's sensitivity to strain rate, a phenomenon driven by the damage evolution behavior of the mortar phase.