Mechanical APDL Structural Analysis Guide

ij

j

i

i

iw ie

ie

ne

=

∂

∂

−













∂

= ∂

∑ σ δ

1

1

1

where ne is the number of elements to be integrated, wiw is the weight function, and Aie is the area of

the element represented by ie.

For higher-order elements (such as PLANE183 and SOLID186), the q vector at midside nodes takes the
averaged values from the corresponding corner nodes.

If the thermal strains exist in the structure and the surface tractions act on crack faces, the J-Integral is
expressed as:

A 

 


C

= 

∂

∂

−

















∫

∂

∂

+∫

∂

∂

σ δ ασ −∫

θ







 ,

where α is the thermal expansion coefficient, tj is the crack face traction, and C is crack face upon which

the tractions act.

For a 3-D problem, domain integral representation of the J-Integral becomes a volume integration,
which again is evaluated over a group of elements. The implementation becomes more complicated;
however, the principal is similar to the 2-D problem.

11.3.1.1.1. Virtual Crack-Extension Nodes and J-Integral Contours

Virtual crack-extension nodes are one of the most important input data elements required for J-Integral
evaluation. It is also referred to as the crack-tip node component.

For a 2-D crack problem, the crack-tip node component usually contains one node which is also the
crack-tip node. The first contour for the area integration of the J-Integral is evaluated over the elements
associated with the crack-tip node component. The second contour for the area integration of the J-
Integral is evaluated over the elements adjacent to the first contour of elements. This procedure is re-
peated for all contours. To ensure correct results, the elements for the contour integration should not
reach the outer boundary of the model (with the exception of the crack surface).

For a 3-D crack problem, the crack-tip node component is comprised of the nodes along the crack front.
The crack-tip node component is not required to be sorted. The 3-D J-Integral contour follows a procedure
similar to that of the 2-D contour.

11.3.1.1.2. Element Selection and Material Behavior

J-Integral or stress-intensity evaluation (accessed via the CINT command) supports the following elements:

• PLANE182


• PLANE183


• SOLID185


• SOLID186


• SOLID187

Release 15.0 - © SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information

Fracture Mechanics