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  1. Material
  2. Material3D
  3. Sand

SimpleSand

PreviousSandNextDafalisaManzari

Last updated 3 years ago

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⁡(ppc)\psi=v-v_c+\lambda_c\ln\left(\dfrac{p}{p_c}\right)ψ=v−vc​+λc​ln(pc​p​)

The specific volume can be expressed in terms of strain,

v=v0(1+tr ε).v=v_0\left(1+\mathrm{tr}~\varepsilon\right).v=v0​(1+tr ε).

Thus, the bounding surface and dilatancy surface can be defined to evolve with ψ\psiψ,

αb=αcexp⁡(−nbψ),αd=αcexp⁡(ndψ),\alpha^b=\alpha^c\exp\left(-n^b\psi\right),\qquad \alpha^d=\alpha^c\exp\left(n^d\psi\right),αb=αcexp(−nbψ),αd=αcexp(ndψ),

where αc\alpha^cαc is the initial size of surfaces.

Yield Surface

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

F=∣s+pα∣+mp,F=|s+p\alpha|+mp,F=∣s+pα∣+mp,

where sss is the deviatoric stress, ppp is the hydrostatic stress, α\alphaα is the back stress ratio and mmm is a constant that controls the size of the wedge.

Flow Rule

A non-associated flow rule is defined.

Δεp=Δγ(n+13DI),\Delta\varepsilon^p=\Delta\gamma{}\left(n+\dfrac{1}{3}DI\right),Δεp=Δγ(n+31​DI),

where n=s+pα∣s+pα∣n=\dfrac{s+p\alpha}{|s+p\alpha|}n=∣s+pα∣s+pα​ is a unit tensor, III is the second order unit tensor and D=A(αd−α:n)D=A\left( \alpha^d-\alpha:n\right)D=A(αd−α:n) is the dilatancy parameter.

Note due to the change of sign convention, a negative DDD leads to contractive response, thus AAA often needs to be negative.

Hardening Rule

The evolution of α\alphaα is similar to the Armstrong-Frederick hardening law.

Δα=Δγh(αbn−α),\Delta\alpha=\Delta\gamma{}h\left(\alpha^bn-\alpha\right),Δα=Δγh(αbn−α),

where hhh is a constant that controls the speed of hardening.

Example

Please refer to .

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triaxial-compression-of-sand