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

$$\dot \epsilon_{cr} = A_1\ \bar \sigma^{n_1} \qquad \bar \sigma \ge \sigma_1^{ref}$$

2. Both Components Active

$$\sigma_1^{ref} = 0$$

$$\sigma_2^{ref} = {\hbox{"large"}}$$

$$\dot \epsilon_{cr} = A_1\ \bar \sigma^{n_1} + A_2\ \bar \sigma^{n_2} \qquad \sigma_1^{ref} < \bar \sigma < \sigma_2^{ref}$$

3. Different Law for Different Stress Regimes

1. $\sigma_1^{ref} = \sigma_2^{ref} = \sigma^{ref} \ >\ 0$

$$\begin{split}\dot\epsilon_{cr} = \begin{cases} A_2 \bar{\sigma}^{n_2} & \bar{\sigma} < \sigma^{ref} \\ & \\ A_1 \bar{\sigma}^{n_1} & \bar{\sigma} > \sigma^{ref} \\ \end{cases}\end{split}$$

2. $\sigma_1^{ref} < \sigma_2^{ref}$

$$\begin{split}\dot\epsilon_{cr} = \begin{cases} A_2 \bar{\sigma}^{n_2} & \bar{\sigma} \le \sigma^{ref}_1 \\ A_1 \bar{\sigma}^{n_1} + A_2 \bar{\sigma}^{n_2} & \sigma^{ref}_1 < \bar{\sigma} < \sigma^{ref}_2 \\ A_1 \bar{\sigma}^{n_1} & \bar{\sigma} \ge \sigma^{ref}_2 \\ \end{cases}\end{split}$$

3. $\sigma_1^{ref} > \sigma_2^{ref}$

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.

power Model Properties

Use the following keywords with the zone property 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, $\sigma^{ref}_1$

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.