Section 18.104.22.168.0. The J-Integral
In 1968, Rice [1968b] published a paper describing a path
independent integral (J) which was
noted to be equal to the negative of the change in potential energy of
deformation occurring during the infinitesimal growth of a crack in a nonlinear
elastic material, i.e. he showed that
Rice’s path independent integral J was defined by [Rice, 1968a; 1968b]
is any contour surrounding the crack tip, traversing in a counter clockwise
direction (see Figure 2.2.13), W is the strain energy density, is the traction on G, and
is the displacement
on an element along arc s.
Figure 2.2.13. J-Integral
Before elaborating on a detailed description of the parameters
involved in the calculation of the J-Integral,
it is useful to note that Equation 2.2.33 is the nonlinear elastic equivalent
of Equation 2.2.12. Thus, for linear
elastic materials, J reduces to the
value of the strain energy release rate, G,
and the J-integral is
related to the stress-intensity factor through the expression
where E¢ is given by Equation 2.2.18.
Equations 2.2.35 and 2.2.36 are noted to be valid only when the
material is behaving in a linear elastic fashion. When values of the J-Integral
are determined via Equation 2.2.34 using finite element methods applied to
linear elastic cracked structures, Equation 2.2.36 provides the engineer with a
simple energy-based method for obtaining stress-intensity factors as a function
of crack length.
In the first subsection below, the calculations associated with
developing the J-Integral for an
elastic-plastic material are detailed.
In the second subsection, some engineering approximation methods for
calculating the J-Integral are outlined.