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 \(\nu\).