**K_field

Description

This boundary condition is used to impose on a NSET the displacement field given by the linear solution at a crack tip. The crack is loaded under mode I or II. This boundary condition is limited to 2D plane problems. It will work on axisymmetric problems , however its physical meaning will be uncertain.

The displacement field is given by:

Mode I:

(43)\[\begin{split}\begin{aligned} u_1 = \frac{K_I}{2\mu}\sqrt{\frac{r}{2\pi}}\cos{\frac{\theta}{2}} \left(\kappa-1+2\sin^2\frac{\theta}{2}\right) \\ u_2 = \frac{K_I}{2\mu}\sqrt{\frac{r}{2\pi}}\sin{\frac{\theta}{2}} \left(\kappa+1-2\cos^2\frac{\theta}{2}\right) \\\end{aligned}\end{split}\]

Mode II:

(44)\[\begin{split}\begin{aligned} u_1 = \frac{K_{II}}{2\mu}\sqrt{\frac{r}{2\pi}}\sin{\frac{\theta}{2}} \left(\kappa+1+2\cos^2\frac{\theta}{2}\right) \\ u_2 = -\frac{K_{II}}{2\mu}\sqrt{\frac{r}{2\pi}}\cos{\frac{\theta}{2}} \left(\kappa-1-2\sin^2\frac{\theta}{2}\right) \\\end{aligned}\end{split}\]

with

(45)\[\begin{split}\kappa = \begin{matrix} 3-4\nu & \hbox{ plane strain} \\ (3-\nu)/(1-\nu) & \hbox{ plane stress} \\ \end{matrix}\end{split}\]

where \(\nu\) is the Poisson’s ratio. Let \(M\) be the point where the displacement field is computed and \(T\) the crack tip. \(r=||\vect{MT}||\), \(\theta=(\vect{Ox_1},\vect{MT})\).

Syntax

The syntax is as follows:

**K_field nset center young poisson plane_state mode value [table]

nset

name of the NSET

center

crack tip position (2D vector)

young

Young modulus (double)

poisson

Poisson coefficient (double)

plane_state

plane_stress or plane_strain

mode

specifies loading mode: I or II

value

basic boundary condition value (double). This represents the basic value of \(K_{I,II}/2\mu\).

table

table name

Example

**K_field
   border  % nset name
   (0.,0.) % crack tip position
   0.3     % Poisson's ratio
   plane_stress
   II      % mode II
   10.     % basic value
   Ktab    % table name