NorSand (Jefferies 1993) is a critical-state model for sand built around the state parameter ψ. This page sets out the NorSand equations the cavity solver uses, in the notation of the NorSand model in FLAC3D 9.0 (Itasca). The solver implements the same equations, so results can be compared directly with FLAC3D. Only the parts the solver uses are shown. Stresses are effective stresses and compression is positive.
1Stress measures
θ is the Lode angle: +30° in triaxial compression and −30° in triaxial extension. In the cavity solution the principal stresses are σ′r, σ′θ and σ′3. σ′3 is the out-of-plane stress for a cylinder and the second hoop stress for a sphere.
2Elasticity
Poisson’s ratio ν is constant. When Gref is not given, it follows from the rigidity index at the in-situ stress, Ir0 = G0/p′0: Gref = Ir0 p′0 / (p′0/pref)m.
3Critical state line and state parameter
ψ is the state parameter at the current mean stress, and ψi is the image state parameter at the image stress p′i. For the curved CSL, λ is the local slope, evaluated at p′i wherever it enters χi.
4Yield surface
The image stress p′i is the mean stress at which the yield surface crosses the image condition, Dp = 0. It is the model’s single hardening variable. N couples the image stress ratio to the state, and χi is the state–dilatancy coefficient referred to the image state.
5Lode-angle function
g = 1 in triaxial compression and g = c in triaxial extension. In spherical expansion θ stays at 30°, so g = 1 throughout. The Lode-angle switch in the app sets g = 1 for every θ. FLAC3D always applies g(θ).
6Flow rule and stress–dilatancy
Flow is associated: the plastic strain increment is normal to the yield surface F = 0 of equation 7, including the Lode-angle term ∂F/∂θ. On that surface Dp = Mi(θ) ln(p′/p′i), so the soil dilates when p′ < p′i and contracts when p′ > p′i.
7Hardening
p′b is the bounding mean stress. By equation 15, equation 13 is the familiar form dp′i/p′i = H g(θ) (p′/p′i)² (p′i,max/p′ − p′i/p′) dεqp: the image stress moves towards its limit p′i,max. It hardens while p′ > p′b (p′i < p′i,max) and softens while p′ < p′b. At the critical state ψi = 0, so p′b = p′i and p′i,max = p′.
8Cap softening (undrained TXC and DSS)
VBA (Jefferies & Shuttle)
FLAC3D
S is the softening index, 0 ≤ S ≤ 1. K is the elastic bulk modulus (equation 3). Dp = Mi(θ) ln(p′/p′i) is the plastic dilatancy (equation 11). η = q/p′. λ is the CSL slope at p′. Mi(θ), Mtc, χtc, χi, ψi and g(θ) are as in sections 3–5. ηL is the limiting stress ratio, and ω is the λ factor.
9Initial state
10Parameters
| Symbol | Meaning | FLAC3D 9.0 keyword |
|---|---|---|
| Mtc | Critical stress ratio in triaxial compression | ratio-critical |
| N | Volumetric coupling coefficient | factor-coupling |
| χtc | State–dilatancy coefficient | factor-dilatancy |
| H0, Hy | Plastic hardening modulus and its dependence on ψ | hardening-0, hardening-y |
| Γ, λ | Linear critical state line | critical-state-1, critical-state-2 (with critical-state-3 = 0) |
| Ca, Cb, Cc | Curved critical state line | critical-state-1, -2, -3 |
| Gref, m, ν, pref | Elasticity | shear-reference, exponent, poisson, pressure-reference |
| ψ0 or e0 | Initial state | state-parameter-initial, void-initial |
| OCR | Overconsolidation ratio of the image stress | over-consolidation-ratio |
| S | Cap-softening index, 0 to 1 (undrained TXC and DSS only) | index-softening |
11Parts of FLAC3D’s NorSand not used here
- The inner cap (
flag-inner) and the principal-stress-rotation term (modulus-annealing). Both are off by default in FLAC3D and absent from the solver. The cap-softening term S (index-softening) is used only in the undrained element tests (section 8). - The void-ratio-dependent elasticity options (
index-elasticity1 and 2). The solver uses equation 3 only. - FLAC3D’s limits χi ≤ 1.25 χtc (when λχtc/Mtc > 0.2) and H ≥ max(10, 0.5/(λ − κ)). The solver uses Hmin instead, and the verification cases stay clear of both limits.
References
- Jefferies, M. G. (1993). Nor-Sand: a simple critical state model for sand. Géotechnique, 43(1), 91–103. https://doi.org/10.1680/geot.1993.43.1.91
- Jefferies, M. and Been, K. (2016). Soil Liquefaction: A Critical State Approach, 2nd edition. CRC Press. https://doi.org/10.1201/b19114
- Itasca Consulting Group. FLAC3D 9.0 documentation, NorSand model. docs.itascacg.com
