SimpleSand

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 Constitutive Modelling Cookbook 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

$$\psi=v-v_c+\lambda_c\ln\left(\dfrac{p}{p_c}\right)$$

The specific volume can be expressed in terms of strain,

$$v=v_0\left(1+\mathrm{tr}~\varepsilon\right).$$

Thus, the bounding surface and dilatancy surface can be defined to evolve with $\psi$,

$$\alpha^b=\alpha^c\exp\left(-n^b\psi\right),\qquad \alpha^d=\alpha^c\exp\left(n^d\psi\right),$$

where $\alpha^c$ is the initial size of surfaces.

Yield Surface

The following wedge-like function is chosen to be the yield surface,

$$F=|s+p\alpha|+mp,$$

where $s$ is the deviatoric stress, $p$ is the hydrostatic stress, $\alpha$ is the back stress ratio and $m$ is a constant that controls the size of the wedge.

The following is a visualization of the yield surface in the $p-q$ plane.

Flow Rule

A non-associated flow rule is defined.

$$\Delta\varepsilon^p=\Delta\gamma{}\left(n+\dfrac{1}{3}DI\right),$$

where $n=\dfrac{s+p\alpha}{|s+p\alpha|}$ is a unit tensor, $I$ is the second order unit tensor and $D=A\left( \alpha^d-\alpha:n\right)$ 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 $\alpha$ is similar to the Armstrong-Frederick hardening law.

$$\Delta\alpha=\Delta\gamma{}h\left(\alpha^bn-\alpha\right),$$

where $h$ is a constant that controls the speed of hardening.

Example

Please refer to triaxial-compression-of-sand.

Iso-error Map

The following example iso-error maps are obtained via the following script.

from plugins import ErrorMap
# note: the dependency `ErrorMap` can be found in the following link
# https://github.com/TLCFEM/suanPan-manual/blob/dev/plugins/scripts/ErrorMap.py

young_modulus = 1e5
ref_stress = 20

with ErrorMap(
    f"material SimpleSand 1 {young_modulus} .2 .01 -.7 5. 1.25 1.1 3.5 1.915 -130. .02 2. 0.",
    ref_strain=ref_stress / young_modulus,
    ref_stress=ref_stress,
    contour_samples=40,
) as error_map:
    error_map.contour(
        "simple.sand.uniaxial", center=(-4, -2.5, -2.5), size=2, type="rel"
    )

relative error uniaxial