Danilovskaya problem: dynamic thermoelastic stress wave in a half-space
Coupled transient thermoelasticityPASS · 1.72%Sudden boundary heating of an elastic half-space in uniaxial strain generates a longitudinal thermoelastic stress wave propagating at the dilatational wave speed c_p = √(E(1−ν)/(ρ(1+ν)(1−2ν))) with a compressive precursor ahead of the wavefront x = c_p·t and a stress jump Δσ_xx = βT₀ across the wavefront. Verified against Danilovskaya's (1950) exact closed-form solution.Sudden boundary heating of an elastic half-space in uniaxial strain generates a longitudinal thermoelastic stress wave propagating at the dilatational wave speed c_p = √(E(1−ν)/(ρ(1+ν)(1−2ν))) with a compressive precursor ahead of the wavefront x = c_p·t and a… show full text
- Tolerance:
- 5%
Problem statement
- Geometry
- Prismatic bar 24 × 1 × 1 mm in uniaxial strain; quadratic tetrahedra (tet10), element size 0.3 mm (3919 elements).
- Material
- Steel: E = 200 GPa, ν = 0.3, ρ = 7800 kg/m³, α = 1.2 × 10⁻⁵ K⁻¹, c_p = 460 J/(kg·K), characteristic thermal-acoustic length ℓ = a/c_p = 2 mm (c_p = 5875.1 m/s, β = Eα/(1−2ν) = 6.0 MPa/K).
- Boundary conditions
- Initial T₀ = 0 °C; surface number 1 (x = 0) heated to 100 °C over one time step; lateral faces restrained in normal direction (u_y = 0 on surfaces 3, 4; u_z = 0 on surfaces 5, 6). Coupled dynamic HHT-α (α = −0.2), Δt = 0.5·h/c_p = 25.27 ns, t_end = 2.3 μs.
- Measured quantity
- Normalized longitudinal stress σ_xx(x, t)/(βT₀) at probes x = 6, 10, 12 mm before and after wavefront arrival, wavefront speed c_p, and temperature profile T₀·erfc(x/(2√(at))).
WebCAE Terminal
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units mm
material E=200000000000 nu=0.3 density=7800 kappa=42159717.5 specific_heat=460 alpha=0.000012
primitive box dx=24 dy=1 dz=1
mesh tet size=0.3 order=2
thermal surface number 1 temperature=100
fix surface number 3 component=y
fix surface number 4 component=y
fix surface number 5 component=z
fix surface number 6 component=z
solve thermoelastic transient mode=dynamic dt=2.52747e-8 tend=2.3e-6 t0=0 scheme=hht alpha=-0.2 amp_temperature="0:0,2.52747e-8:1" probes="6,0.5,0.5;10,0.5,0.5;12,0.5,0.5" frames=2 outputs=2000
probe transient_stress dir=x mode=3 time=1.743e-6 id="s_dan"Results comparison
| Reference value | c_p = 5875.1 m/sσ/(βT₀)(x=12 mm, t=1.74 µs) = −0.3889 |
| Our result | c_p = 5758.2 m/sσ/(βT₀)(x=12 mm, t=1.74 µs) = −0.3822 |
| Deviation | 1.72% |
| Tolerance | 5% |
| Status | PASS · 1.72% |
| Mesh details | tet10, size 0.3 mm (3919 tets); 91 steps of Δt = 25.27 ns, heat balance residual 3.76e-14 |
Updated: 2026-10-27 · version not recorded