FLAC3D Theory and Background • Constitutive Models

Anisotropic (Transversely) Elastic Model

The transversely isotropic model takes a plane of isotropy into consideration. Let the axis of rotational symmetry, normal to the plane of isotropy, correspond to the local 3 axis. This axis is a principal direction of elasticity. Also, any two perpendicular directions 1,2, which are principal directions of elasticity, can be selected in the isotropic plane. With this convention, the transversely isotropic model may be considered as a particular case of the orthotropic model for which

(1)E1=E2
(2)G13=G23
(3)ν13=ν23
(4)G12=E12(1+ν12)

If, for clarity, we write

E=E1=E2 Young’s moduli in the plane of isotropy
E=E3 Young’s moduli in the direction normal to the plane of isotropy
ν=ν12 Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied in this plane
ν=ν13=ν23 Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied in the direction normal to it
G=G12 shear modulus for the plane of isotropy
G=G13=G23 shear modulus for any plane normal to the plane of isotropy

The strain-stress relations in the local axes take the form

(5){Δϵ11Δϵ22Δϵ332Δϵ122Δϵ132Δϵ23}=[1EνEνEνE1EνEνEνE1E1G1G1G]{Δσ11Δσ22Δσ33Δσ12Δσ13Δσ23}

The model involves the five independent elastic constants E,E,ν,ν, and G. The shear modulus, G, is calculated by the code from the relation G=E/2(1+ν) (See Equation (4)). In addition to those five properties, the user prescribes the orientation of the isotropic plane by giving its dip and dip direction. Default values for all properties are zero.

The cross shear modulus, G13, for anisotropic elasticity must be determined. Lekhnitskii (1981) suggests the following equation, based on laboratory testing of rock:

(6)Gxy=ExEyEx(1+2νxy)+Ey

assuming the xz-plane is the plane of isotropy.

The FLAC3D implementation proceeds as described in the context of the orthotropic model.



anisotropic Model Properties

Use the following keywords with the zone property (FLAC3D) or block zone property (3DEC) command to set these properties of the anisotropic (transversely-elastic) model.

anisotropic
dip f

dip angle [degrees] of the plane of isotropy

dip-direction f

dip direction [degrees] of the plane of isotropy

normal v

normal direction of the plane of isotropy, (nx,ny,nz)

normal-x f

x-component of unit normal to the plane of isotropy, nx

normal-y f

y-component of unit normal to the plane of isotropy, ny

normal-z f

z-component of unit normal to the plane of isotropy, nz

poisson-normal f

Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied normal to the plane, ν = ν13 = ν23

poisson-plane f

Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied in the plane, ν = ν12

shear-normal f

shear modulus for any plane normal to the plane of isotropy, G = G13 = G23

young-plane f

Young’s modulus in the plane of isotropy, E = E1 = E2

young-normal f

Young’s modulus normal to the plane of isotropy, E = E3