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 \(\dot \epsilon_{cr}\) is the creep rate, \(A\) and \(n\) are material properties, and \(\bar \sigma\) is the von Mises stress. By definition, \(\bar \sigma = \sqrt{3J_2}\), and \(J_2\) is the second invariant of the effective deviatoric-stress tensor, \(\sigma_{ij}^d\) (i.e., \(J_2 = {1 \over 2} \sigma_{ij}^d \sigma_{ij}^d\)).
The deviatoric stress increments are given by
where \(G\) is the shear modulus and \(\dot \epsilon_{ij}^{d}\) is the deviatoric part of the strain-rate tensor.
The creep strain-rate tensor is calculated as
with \(\dot \epsilon_{cr}\) and \(\bar \sigma\) defined as above.
The volumetric behavior is assumed elastic. The isotropic stress increment is given by
where \(K\) is the bulk modulus and \(\Delta \epsilon_{v} = \Delta \epsilon_{11} + \Delta \epsilon_{22} + \Delta \epsilon_{33}\).
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
\(\bar \sigma\) is always positive, so this is the one-component law with
- Both Components Active
- Different Law for Different Stress Regimes
- \(\sigma_1^{ref} = \sigma_2^{ref} = \sigma^{ref} \ >\ 0\)
- \(\sigma_1^{ref} < \sigma_2^{ref}\)
- \(\sigma_1^{ref} > \sigma_2^{ref}\)
NOTE: Do not use option (c). It implies that creep occurs for \(\bar \sigma < \sigma_2^{ref}\) and for \(\bar \sigma > \bar \sigma_1^{ref}\), but not for \(\sigma_2^{ref} < \bar \sigma < \sigma_1^{ref}\).
The two-component power law is implemented in FLAC3D by the following procedure.
Let \(\sigma_{ij}^{(t)}\) be the stress tensor at time \(t\), and let \(\dot \epsilon_{ij} = \dot \epsilon_{ij}^{e} + \dot \epsilon_{ij}^{c}\) be the strain-rate tensor, which consists of an elastic component (\(\dot \epsilon_{ij}^{e}\)) and a creep component (\(\dot \epsilon_{ij}^{c}\)).
The stress \(\sigma_{ij}^{(t + \Delta t)}\) at time \(t + \Delta t\), is calculated:
Volumetric Component:
Deviatoric Component:
where \(\dot \epsilon_{ij}^{c}\) 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, \(A_1\)
- constant-2 f
power-law constant, \(A_2\)
- exponent-1 f
power-law exponent, \(n\)1
- exponent-2 f
power-law exponent, \(n\)2
- poisson f
Poisson’s ratio, \(v\)
- shear f
shear modulus, \(G\)
- stress-reference-1 f
reference stress, \(\sigma^{ref}_1\)
- stress-reference-2 f
reference stress, \(\sigma^{ref}_2\)
- 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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