Are a Constitutive Model's Parameters Consistent with Its Own Material?

1. The code can be right and the result wrong

The reliability of a blast or impact simulation rests on two things: the model's code and the parameters given to it. Code is usually tested well; parameters are often copied from a table and never questioned again. In this site's hydrocode engine, the code of the Johnson–Holmquist-2 (JH2) model representing UHPC was correct; but its parameters did not agree with the material's own strength. What revealed this was not a complex analysis but a one-line check.

2. The one-line check: P = f_c/3

At failure in a uniaxial compression test the axial stress is f_c and the lateral stresses are zero. The hydrostatic pressure is then P = (σ₁ + σ₂ + σ₃)/3 = f_c/3 and the equivalent stress σ_eq = f_c. In normalised form: at P* = 1/3 the model must give σ* = 1. If a constitutive model's strength surface cannot reproduce the most basic test of its own material, there is no reason to trust it under more complex loading.

A second form of the same check is to find the model's own uniaxial strength: on the uniaxial loading path σ = 3P; wherever the strength surface crosses that line is the uniaxial strength the model "sees".

3. Finding and correction

The old parameter set (A = 0.79, N = 0.61, HEL = 3.5 GPa) overstated UHPC's compressive strength about 3.3 times. These values are close to those typical of ceramics. In protective design the direction matters: a model that overstates resistance approves a layer that may not hold, so the error is on the unsafe side.

4. Tracing the source

Searching for the origin of the parameters turned up a peer-reviewed paper applying JH2 to UHPC (Mai et al., Defence Science Journal, 2020). But the full text showed three things: the parameters were taken not from the authors' own tests but from a non-peer-reviewed conference paper; the table lacked the exponents N and M that define the strength surface; and some units were physically meaningless (K₁ = 8.5 MPa for a concrete). The site set's A, C, density and D₂ matched that table.

The rest of the literature pointed elsewhere: almost all peer-reviewed, experimentally validated calibrations for UHPC are for the HJC (Holmquist–Johnson–Cook) model. Wan et al. (2021) determined HJC strength, damage and equation-of-state parameters for UHPC with 0–3% steel fibre from published test data and validated them against contact-explosion tests on 2% fibre slabs. The strength surface was fitted to triaxial test data as σ* = 0.3 + 1.781·P*^0.810 (R² = 0.988).

5. A peer-reviewed source can be incomplete too

The table of Wan et al. needed completing in two places:

• A missing value: the third region of the equation of state needs the volumetric strain at the locking pressure (μ_plock); the table gives only μ_lock at grain density. μ_plock was derived as 0.160 from the condition that the polynomial be continuous at P_lock. The geopolymer-UHPC table of Liu et al. (2022) independently supports this: their values also make the polynomial continuous at that point within 1%.
• An internal inconsistency: the damage constant D₁ is computed from D₁ = EFMIN/(1/6 + T*). The tabulated 2% fibre values imply T* ≈ 0.096, whereas the tensile strength measured in the paper's own test gives T* = 8.74/180 ≈ 0.049. The site's engine does not model damage, so this does not affect results; but it is documented.

6. Effect of the change

In the site's civil scenarios (20 kg TNT, 4 m) results changed by less than 1%: the reflected pressure (~7 MPa) is far below UHPC's strength, so the material stays elastic in both models. The difference matters mainly for close-in and contact explosions; the old model treated UHPC as elastic up to about 3 GPa. A side result is instructive too: a low-impedance foam in front creates a tensile wave in the UHPC behind it; with the measured tensile strength the model reports yielding in that layup. The old model, with inflated strength, could not see this cracking.

7. Rules

1. Test parameters against the material's own tests. Uniaxial compression (σ*(1/3) ≈ 1) and uniaxial tension are the cheapest and strongest checks.
2. Trace the origin of a parameter set. Appearing in a peer-reviewed paper does not mean it came from a peer-reviewed calibration.
3. Fill a table's gaps openly. State how every derived value was derived and, where possible, test it against an independent source.
4. Measure the size of the effect. A correction does not change every scenario equally; show where it matters.