Field Expansion

Abaqus/Standard requires field expansion coefficients, ${\alpha }_f$ , that define the total field expansion from a reference value of the predefined field variable n, $f_n^0$ , as shown in Figure 1.

Figure 1. Definition of the field expansion coefficient.

The field expansion for each specified field generates field expansion strains according to the formula

${\alpha }_f\left(\theta ,f_{\beta }\right)$

is the field expansion coefficient;

$f_n$

is the current value of the predefined field variable n;

$f_n^I$

is the initial value of the predefined field variable n;

$f_{\beta }$

are the current values of the predefined field variables;

$f_{\beta }^I$

are the initial values of the predefined field variables; and

$f_n^0$

is the reference value of the predefined field variable n for the field expansion coefficient.

The second term in the above equation represents the strain due to the difference between the initial value of the predefined field variablen, $f_n^I$ , and the corresponding reference value, $f_n^0$ . This term is necessary to enforce the assumption that there is no initial field expansion strain for cases in which the reference value of the predefined field variable n does not equal the corresponding initial value.

EXPANSION, FIELD=n, ZERO=$f_n^0$

Total field expansion coefficients can be provided directly as outlined in the previous section. However, you may have field expansion data available in differential form:

that is, the tangent to the strain-field variable curve is provided (see Figure 1). To convert to the total field expansion form required by Abaqus, this relationship must be integrated from a suitably chosen reference value of the field variable, $f_n^0$ :

For example, suppose ${\alpha }_f^{\prime }$ is a series of constant values: ${\left({\alpha }_f^{\prime }\right)}_1$ between $f_n^0$ and $f_n^1$ ; ${\left({\alpha }_f^{\prime }\right)}_2$ between $f_n^1$ and $f_n^2$ ; ${\left({\alpha }^{\prime }\right)}_3$ between $f_n^2$ and $f_n^3$ ; etc. Then,

The corresponding total expansion coefficients required by Abaqus are then obtained as

If an element has been removed and subsequently reactivated (Element and Contact Pair Removal and Reactivation), ${\theta }^I$ and $f_{\beta }^I$ in the equation for the field expansion strains represent temperature and predefined field variable values as they were at the moment of reactivation.

If the field expansion coefficient is defined directly, only one value of ${\alpha }_f$ is needed at each temperature and/or predefined field variable. If user subroutine UEXPAN is used, only one isotropic field expansion strain increment ( $\mathrm{\Delta }{\epsilon }^f=\mathrm{\Delta }{\epsilon }_{11}^f=\mathrm{\Delta }{\epsilon }_{22}^f=\mathrm{\Delta }{\epsilon }_{33}^f$ ) must be defined.

EXPANSION, FIELD=n, TYPE=ISO

EXPANSION, FIELD=n, TYPE=ISO, USER

If the field expansion coefficients are defined directly, the three expansion coefficients in the principal material directions ( $\alpha _f{}_{11}$ , $\alpha _f{}_{22}$ , and ${\alpha }_{f_{33}}$ ) should be given as functions of temperature and/or predefined field variables. If user subroutine UEXPAN is used, the three components of field expansion strain increment in the principal material directions ( $\mathrm{\Delta }{\epsilon }_{11}^f$ , $\mathrm{\Delta }{\epsilon }_{22}^f$ , and $\mathrm{\Delta }{\epsilon }_{33}^f$ ) must be defined.

EXPANSION, FIELD=n, TYPE=ORTHO

EXPANSION, FIELD=n, TYPE=ORTHO, USER

If the field expansion coefficients are defined directly, all six components of ${\alpha }_f$ ( $\alpha _f{}_{11}$ , $\alpha _f{}_{22}$ , ${\alpha }_{f_{33}}$ , $\alpha _f{}_{12}$ , $\alpha _f{}_{13}$ , $\alpha _f{}_{23}$ ) must be given as functions of temperature and/or predefined field variables. If user subroutine UEXPAN is used, all six components of the field expansion strain increment ( $\mathrm{\Delta }{\epsilon }_{11}^f$ , $\mathrm{\Delta }{\epsilon }_{22}^f$ , $\mathrm{\Delta }{\epsilon }_{33}^f$ , $\mathrm{\Delta }{\epsilon }_{12}^f$ , $\mathrm{\Delta }{\epsilon }_{13}^f$ , $\mathrm{\Delta }{\epsilon }_{23}^f$ ) must be defined.

EXPANSION, FIELD=n, TYPE=ANISO

EXPANSION, FIELD=n, TYPE=ANISO, USER