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)\[\dot \epsilon_{cr} = A\ \bar \sigma^n\]

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

(2)\[\Delta \sigma_{ij}^{d} = 2G (\dot \epsilon_{ij}^{d} - \dot \epsilon_{ij}^{c}) \Delta t\]

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

(3)\[\dot\epsilon_{ij}^{c} = \left({3 \over 2}\right) \dot \epsilon_{cr} \left({\sigma_{ij}^{d} \over \bar \sigma} \right)\]

with \(\dot \epsilon_{cr}\) and \(\bar \sigma\) defined as above.

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

(4)\[\Delta \sigma_{kk} = 3 K \Delta \epsilon_{v}\]

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

(5)\[\dot \epsilon_{cr} = \dot \epsilon_1 + \dot \epsilon_2\]

where:

\[\begin{split}\dot\epsilon_1 = \begin{cases} A_1 \bar{\sigma}^{n_1} & \bar{\sigma} \ge \sigma^{ref}_1 \\ & \\ 0 & \bar{\sigma} < \sigma^{ref}_1 \\ \end{cases}\end{split}\]
\[\begin{split}\dot\epsilon_2 = \begin{cases} A_2 \bar{\sigma}^{n_2} & \bar{\sigma} \le \sigma^{ref}_2 \\ & \\ 0 & \bar{\sigma} > \sigma^{ref}_2 \\ \end{cases}\end{split}\]

With these two terms, several options are possible:

  1. The Default Option
\[\sigma_1^{ref} = \sigma_2^{ref}\]

\(\bar \sigma\) is always positive, so this is the one-component law with

\[\dot \epsilon_{cr} = A_1\ \bar \sigma^{n_1} \qquad \bar \sigma \ge \sigma_1^{ref}\]
  1. Both Components Active
\[\sigma_1^{ref} = 0\]
\[\sigma_2^{ref} = {\hbox{"large"}}\]
\[\dot \epsilon_{cr} = A_1\ \bar \sigma^{n_1} + A_2\ \bar \sigma^{n_2} \qquad \sigma_1^{ref} < \bar \sigma < \sigma_2^{ref}\]
  1. Different Law for Different Stress Regimes
  1. \(\sigma_1^{ref} = \sigma_2^{ref} = \sigma^{ref} \ >\ 0\)
\[\begin{split}\dot\epsilon_{cr} = \begin{cases} A_2 \bar{\sigma}^{n_2} & \bar{\sigma} < \sigma^{ref} \\ & \\ A_1 \bar{\sigma}^{n_1} & \bar{\sigma} > \sigma^{ref} \\ \end{cases}\end{split}\]
  1. \(\sigma_1^{ref} < \sigma_2^{ref}\)
\[\]
\[\begin{split}\dot\epsilon_{cr} = \begin{cases} A_2 \bar{\sigma}^{n_2} & \bar{\sigma} \le \sigma^{ref}_1 \\ A_1 \bar{\sigma}^{n_1} + A_2 \bar{\sigma}^{n_2} & \sigma^{ref}_1 < \bar{\sigma} < \sigma^{ref}_2 \\ A_1 \bar{\sigma}^{n_1} & \bar{\sigma} \ge \sigma^{ref}_2 \\ \end{cases}\end{split}\]
  1. \(\sigma_1^{ref} > \sigma_2^{ref}\)
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:

(6)\[\sigma_{kk}^{(t + \Delta t)} = \sigma_{kk}^{(t)} + 3K \dot \epsilon_{kk} \ \Delta t\]

Deviatoric Component:

(7)\[\sigma_{ij}^{d (t + \Delta t)} = \sigma_{ij}^{d (t)} + 2G(\dot \epsilon_{ij} - \dot \epsilon_{ij}^{c}) \ \Delta t\]

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.