Mechanical APDL Structural Analysis Guide

Here, e 1 is the tangential component of the local coordinate system with respect to the crack surface,

while e 2 represents the normal component of the local coordinate system. The tangential component

of the material force vector Fmat, which is in the direction of e 1 , represents the scalar crack-driven force.

In the numerical evaluation of material force, the material force is calculated based on the resultant of
all the material force vectors in a user-defined domain β surrounding the crack tip.

If the plastic deformations exist in the structure, the material body forces acting on the domain are
expressed as:

mat

iw ie

e

numel

= ∏ ∫ ( ⋅ )

=

βΣΣ ∇∇ΝΝ −− ΝΝΒΒ

1

where B is the material body forces:

= p+ 

Here, Bp is the material body forces via plasticity, whereas Bt is the material body forces via thermal
stress. Then, the material body forces via plasticity are expressed as:

= ⋅ ∇ − ∇

=

=

=

If thermal strains exist in the structure, the nodal material body force vectors are:

= ⋅ ∇

=

∇ =

For a 3-D problem, integral representation of the nodal material forces becomes a volume integration,
which again is evaluated over a group of elements. The principal is similar to the 2-D problem. After
nodal material forces are evaluated, however, they are divided by a distance quantity through the
thickness.

11.3.5.1.1. Virtual Crack-Extension Nodes and Material Force Contours

Virtual crack-extension nodes are critical input data elements required for material force evaluation.
The program uses virtual crack-extension node input to evaluate tangential (crack-driven force) and
non-tangential components to the crack surface of the material force vectors. The crack-extension nodes
are typically grouped together as crack-tip node components.

For a 2-D crack problem, the crack-tip node component typically contains one node, which is also the
crack-tip node. The first contour for the area integration of the material force is evaluated over the
elements associated with the crack-tip node component. The first contour gives nodal material force.
The second contour for the area integration of the material force approach is evaluated over the elements

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Numerical Evaluation of Fracture Mechanics Parameters