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
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
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
with ˙ϵcr and ˉσ defined as above.
The volumetric behavior is assumed elastic. The isotropic stress increment is given by
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
where:
With these two terms, several options are possible:
- The Default Option
ˉσ is always positive, so this is the one-component law with
- Both Components Active
- Different Law for Different Stress Regimes
- σref1=σref2=σref > 0
- σref1<σref2
- 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:
Deviatoric Component:
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.
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