FLAC3D Theory and Background • Constitutive Models

Elastic (Isotropic) Model

In this elastic, isotropic model, strain increments generate stress increments according to the linear and reversible law of Hooke:

(1) $$\Delta {\sigma}_{ij} = 2G\ \Delta {\epsilon}_{ij} + {\alpha}_2\ \Delta {\epsilon}_{kk}\ {\delta}_{ij}$$

where the Einstein summation convention applies, ${\delta}_{ij}$ is the Kroenecker delta symbol, and ${\alpha}_2$ is a material constant related to the bulk modulus, $K$, and shear modulus, $G$, as

(2) $${\alpha}_2 = K - {{2}\over{3}}\ G$$

New stress values are then obtained from the relation

(3) $${\sigma}_{ij}^N = {\sigma}_{ij} + \Delta {\sigma}_{ij}$$

Bulk modulus, K, and shear modulus, G, are related to Young’s modulus, E, and Poisson’s ratio, ν, by the following equations:

(4) $$K = {E \over {3(1-2\nu)}} \;\;\;\;\;\;\;\; G = {E \over {2(1+\nu)}}$$

or

(5) $$E = {{9KG} \over {3K+G}} \;\;\;\;\;\;\;\; \nu = {{3K-2G} \over {2(3K+G)}}$$

isotropic Model Properties

Use the following keywords with the zone property command to set these properties of the elastic (isotropic) model.

isotropic

bulk f

bulk modulus, $K$

poisson f

Poisson’s ratio, $\nu$

shear f

shear modulus, $G$

young f

Young’s modulus, $E$

Notes:

Only one of the two options is required to define the elasticity: bulk modulus $K$ and shear modulus $G$, or, Young’s modulus $E$ and Poisson’s ratio $\nu$. When choosing the latter, Young’s modulus $E$ must be assigned in advance of Poisson’s ratio $\nu$.