FLAC3D Theory and Background • Constitutive Models

# Isotropic Elastic Model

In this isotropic elastic (or just, elastic, for simplicity) 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)$E = {{9KG} \over {3K+G}} \;\;\;\;\;\;\;\; \nu = {{3K-2G} \over {2(3K+G)}}$

elastic Model Properties

Use the following keywords with the zone property (FLAC3D) or block zone property (3DEC) 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 $$\nu$$.