A Simple Sand Model
The continuum mechanics based sign convention (tension is positive) is used for consistency.
The SimpleSand
model is a simple sand hardening model that adopts a bounding surface concept.
Readers can also refer to the corresponding section in for details on the theory.
Syntax
material SimpleSand (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) [14]
# (1) int, unique material tag
# (2) double, elastic modulus
# (3) double, poissons ratio
# (4) double, m, size of yield surface
# (5) double, A, dilatancy related parameter, often negative
# (6) double, h, dilatancy related hardening parameter
# (7) double, alpha_c, critical alpha
# (8) double, n_b, bounding surface evolution parameter
# (9) double, n_d, dilatancy surface evolution parameter
# (10) double, v_c, critical specific volume
# (11) double, p_c, critical hydrostatic stress, should be negative
# (12) double, lambda_c, the slope of critical state line
# (13) double, v_0, initial specific volume
# [14] double, density, default: 0.0
Theory
Critical State
The state parameter is defined as
ψ=v−vc+λcln(pcp) The specific volume can be expressed in terms of strain,
v=v0(1+tr ε). Thus, the bounding surface and dilatancy surface can be defined to evolve with ψ,
αb=αcexp(−nbψ),αd=αcexp(ndψ), where αc is the initial size of surfaces.
Yield Surface
The following wedge-like function is chosen to be the yield surface,
F=∣s+pα∣+mp, where s is the deviatoric stress, p is the hydrostatic stress, α is the back stress ratio and m is a constant that controls the size of the wedge.
Flow Rule
A non-associated flow rule is defined.
Δεp=Δγ(n+31DI), where n=∣s+pα∣s+pα is a unit tensor, I is the second order unit tensor and D=A(αd−α:n) is the dilatancy parameter.
Note due to the change of sign convention, a negative D leads to contractive response, thus A often needs to be negative.
Hardening Rule
The evolution of α is similar to the Armstrong-Frederick hardening law.
Δα=Δγh(αbn−α), where h is a constant that controls the speed of hardening.
Example