Benchmark 1

Viscoelastic relaxation of stresses resulting from an imposed uniaxial strain. No body forces are imposed. Material should have infinite bulk viscosity (i.e., should not relax in volume, only in shear).

Benchmark #1a: Solve using a Maxwell linear viscoelastic material rheology conceptually, a spring (G_M) and dashpot (η_M) in series.

Benchmark #1b: Solve using a Burgers body rheological description - conceptually, a Maxwell element in series with a Kelvin-Voigt element (a spring (G_KV) and dashpot (η_KV) in parallel).

Benchmark #1c: Solve using a Maxwell power-law material description conceptually, a spring (G_M) and dashpot (η_eff) in series. The dashpot has a nonlinear viscosity that depends on the second invariant of the stress tensor, σ. For these benchmarks, we assume a power law rheology with n=3, that is:
η_eff = η_ref (σ_ref /σ)²

Goals:

Test relevant constitutive relations
Verify shear relaxation only no bulk relaxation
Verify that mesh geometry introduces no errors

Detailed Description:

Model size: 24 km by 24 km by 24 km (0 km ≤ x; y ≤ 24 km; -24 km ≤ z ≤ 0 km)
Maxwell elastic material properties: Poisson solid, G_M = 30 GPa
Maxwell viscoelastic material properties: η_M = 10¹⁸ Pa-s
Burgers body material properties: Maxwell element as above, Kelvin-Voigt element has G_KV = 10 GPa, M = 10¹⁷ Pa-s
Power-law material properties: η_ref = 10¹⁸ Pa-s and ref = 10⁵ Pa. (Note: This value is chosen because the maximum initial elastic stress is of order 10⁶ Pa, only a fraction of that is deviatoric, and the deviatoric stress decreases with time.)
Density and Gravity: None
Boundary conditions:
- Bottom pinned
- Sides pinned in x and y; free in z
- Top pinned in x and y; +1 m of displacement imposed in z
Coarse mesh node spacing: dx = dy = dz = 2 km

Requested Output and Results:

Mesh Variations: As memory, time, and patience allow, run models at 1/2, 1/4, and 1/8, etc. the original coarse mesh spacing, investigate variable mesh spacing, and/or employ a variety of element types. For All Benchmark Variations:

Stresses along a path through (0,0,-24) and (24,24,0) at t = 0, 1, 5, and 10 years.
Displacements along a path through (0,0,-24) and (24,24,0) at t = 0, 1, 5, and 10 years.
CPU time, wallclock time, memory usage info, compiler info, and platform info

Truth:

Analytical solutions for each material rheology will be posted at geoweb.mit.edu/fe.

Additional Notes:

Benchmark #1 should be completed for each material rheology used in any of the benchmarking exercise.