FLAC3D Theory and Background • Constitutive Models

Power Model

The two-component Norton power law (Norton 1929) is commonly used to model the creep behavior of salt. The standard form of this law is

(1)˙ϵcr=A ˉσn

where ˙ϵcr is the creep rate, A and n are material properties, and ˉσ is the von Mises stress. By definition, ˉσ=3J2, and J2 is the second invariant of the effective deviatoric-stress tensor, σdij (i.e., J2=12σdijσdij).

The deviatoric stress increments are given by

(2)Δσdij=2G(˙ϵdij˙ϵcij)Δt

where G is the shear modulus and ˙ϵdij is the deviatoric part of the strain-rate tensor.

The creep strain-rate tensor is calculated as

(3)˙ϵcij=(32)˙ϵcr(σdijˉσ)

with ˙ϵcr and ˉσ defined as above.

The volumetric behavior is assumed elastic. The isotropic stress increment is given by

(4)Δσkk=3KΔϵv

where K is the bulk modulus and Δϵv=Δϵ11+Δϵ22+Δϵ33.

Usually, the amount of data available does not justify adding any more parameters to the creep law. There are cases, however, in which it is justifiable to use a law based on multiple creep mechanisms. FLAC3D, therefore, includes an option to use a two-component law of the form

(5)˙ϵcr=˙ϵ1+˙ϵ2

where:

˙ϵ1={A1ˉσn1ˉσσref10ˉσ<σref1
˙ϵ2={A2ˉσn2ˉσσref20ˉσ>σref2

With these two terms, several options are possible:

  1. The Default Option
σref1=σref2

ˉσ is always positive, so this is the one-component law with

˙ϵcr=A1 ˉσn1ˉσσref1
  1. Both Components Active
σref1=0
σref2="large"
˙ϵcr=A1 ˉσn1+A2 ˉσn2σref1<ˉσ<σref2
  1. Different Law for Different Stress Regimes
  1. σref1=σref2=σref > 0
˙ϵcr={A2ˉσn2ˉσ<σrefA1ˉσn1ˉσ>σref
  1. σref1<σref2
˙ϵcr={A2ˉσn2ˉσσref1A1ˉσn1+A2ˉσn2σref1<ˉσ<σref2A1ˉσn1ˉσσref2
  1. σref1>σref2
NOTE: Do not use option (c). It implies that creep occurs for ˉσ<σref2 and for ˉσ>ˉσref1, but not for σref2<ˉσ<σref1.

The two-component power law is implemented in FLAC3D by the following procedure.

Let σ(t)ij be the stress tensor at time t, and let ˙ϵij=˙ϵeij+˙ϵcij be the strain-rate tensor, which consists of an elastic component (˙ϵeij) and a creep component (˙ϵcij).

The stress σ(t+Δt)ij at time t+Δt, is calculated:

Volumetric Component:

(6)σ(t+Δt)kk=σ(t)kk+3K˙ϵkk Δt

Deviatoric Component:

(7)σd(t+Δt)ij=σd(t)ij+2G(˙ϵij˙ϵcij) Δt

where ˙ϵcij is given by Equation (3), and K and G are the elastic bulk and shear moduli.

References

Norton, F.H. Creep of Steel at High Temperatures. New York: McGraw-Hill Book Company (1929).



power Model Properties

Use the following keywords with the zone property (FLAC3D) or zone property (3DEC) command to set these properties of the power model.

power
bulk f

bulk modulus, K

constant-1 f

power-law constant, A1

constant-2 f

power-law constant, A2

exponent-1 f

power-law exponent, n1

exponent-2 f

power-law exponent, n2

poisson f

Poisson’s ratio, v

shear f

shear modulus, G

stress-reference-1 f

reference stress, σref1

stress-reference-2 f

reference stress, σref2

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 v.
  • The creep behavior is triggered by deviatoric stress, while the volumetric behavior does not consider creep.