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).

Examples

Pressurized Cylindrical Cavern
Power Model: Cylindrical Cavity
Power Model: Spherical Cavity

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.