Explicit dynamics: shock wave in copper, Mie-Gruneisen equation of state, Us = c0 + s Up
Explicit dynamics (shock waves)PASS · 0.08%A piston drives a velocity step Up into a confined copper bar. The volumetric response follows the Mie-Gruneisen equation of state on the shock adiabat (p = p_H (1 - Gamma0 eta/2) + Gamma0 rho0 e, p_H = rho0 c0^2 eta/(1 - s eta)^2), the deviatoric part is the elastic material kernel, artificial viscosity captures the front. The shock speed from the 0.5 Up arrival times at two geophones must equal c0 + s Up (Rankine-Hugoniot with the linear Us-Up fit); the compression behind the front must equal Up/Us (mass conservation). The script shows Up = 1500 m/s; the Up = 500 m/s run is executed by the runner.A piston drives a velocity step Up into a confined copper bar. The volumetric response follows the Mie-Gruneisen equation of state on the shock adiabat (p = p_H (1 - Gamma0 eta/2) + Gamma0 rho0 e, p_H = rho0 c0^2 eta/(1 - s eta)^2), the deviatoric part is the… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Bar 100 × 0.5 × 0.5 mm, 200 × 1 × 1 HEX8 elements (h = 0.5 mm), lateral faces confined (uniaxial strain).
- Material
- Cu: rho0 = 8900 kg/m³, c0 = 3940 m/s, s = 1.489, Gamma0 = 1.97; deviatoric stiffness nearly zero (nu = 0.4995, K = rho0 c0²), so the check is hydrodynamic.
- Boundary conditions
- Piston face x = 0: velocity step to Up = 500 and 1500 m/s over 0.2 µs; far end fixed; finite strain; quadratic/linear bulk viscosity 1.5/0.06; time-step factor 0.5.
- Measured quantity
- Level 0.5 Up of the x velocity at x = 30 and 70 mm (Us = 40 mm / Δt); compression eta from the displacement difference between x = 10 and 20 mm.
WebCAE Terminal
Paste these commands into the Terminal panel inside WebCAE. They use the same text DSL as the application; run one command per line:
units mm
material E=414480120 nu=0.4995 density=8900
box dx=100 dy=0.5 dz=0.5
mesh hex nx=200 ny=1 nz=1
explicit endTime=0.000013968527 dtScale=0.5 saveStride=1000000 bulkQ1=1.5 bulkQ2=0.06 largeStrain=on eos=cu
bc explicit_motion surface number 1 direction 1,0,0 velocity="0:0;2e-7:1500000;0.000013968527:1500000"
bc explicit_fixed surface number 2
bc explicit_symmetry surface number 3
bc explicit_symmetry surface number 4
bc explicit_symmetry surface number 5
bc explicit_symmetry surface number 6
explicit geophone at=30,0,0 name=a
explicit geophone at=70,0,0 name=b
explicit geophone at=10,0,0 name=c
explicit geophone at=20,0,0 name=d
solve explicit
probe explicit_wave_speed geophone=a geophone2=b component=vx method=threshold level=750 relative=off id="us"
probe explicit_geo_value geophone=c component=ux id="uc"
probe explicit_geo_value geophone=d component=ux id="ud"
probe explicit_steps id="steps"Results comparison
| Reference value | Us = c0 + s·Up (Hugoniot)Cu: Up = 500 → 4684.5 m/sUp = 1500 → 6173.5 m/s |
| Our result | Up=500: Us=4688.4 (c0+s·Up = 4684.5)Up=1500: Us=6176 (c0+s·Up = 6173.5) |
| Deviation | 0.08% |
| Tolerance | 5% |
| Status | PASS · 0.08% |
| Element family | Solid element |
| Mesh details | 200×1×1 HEX8 (h = 0.5 mm, confined bar), finite strain, Mie–Grüneisen EOS (Altair Radioss table: Cu), q1 = 1.5, q2 = 0.06; two explicit;: webcae_solve (native) |
Updated: 2026-10-22 · version not recorded