2. Discretized formulation implemented within POST_JMOD

The domain integral formulations presented in Section 1 are conveniently implemented in discretized form. The general equations of modified*J* -integral without and with the presence of initial strain Equation 9 and Equation 10 are re-written here by the finite element method.

The first, the global coordinates*xi* may be written as:

… (14)

where*ne* is the number of nodes per element, Nk are the shape functions and*Xik* are the nodal coordinates.

The nodal fields describing the displacement*ui*, the*qi* fields and the initial strain*lè0* are written in a similar way:

where*Uik* are the nodal displacements, Qik are the nodal values of virtual crack extension, and*lèijk0* are the nodal values of initial strain.

It is noted that the strain*è* and the energy*W* are evaluated at Gauss points. For calculating their gradients, these fields are first converted to global nodal fields (XXXX_ELGA to XXXX_NOEU in*code_aster*). This conversion introduces an approximation error due to the fact that a simple arithmetic mean (with no consideration of mesh size) of the connected nodes is used to evaluate the global nodal values. However, this approximation allows us to avoid the issue that locally the strain gradients are null on the element, when considering classical finite element methods.

The gradients of these fields are given by:

math:

frac {{partialepsilon} _ {mathit {ij}}} {{partial x} _ {n}}approxsum _ {k=1} ^ {mathit {ng}} ^ {ng}}sum _ {ng}}}sum _ {l=1}} ^ {3}}frac {{partial N}} _ {l}}frac {partial {eta} _ {l}} {{partial N}} {{partial N}} _ {partial {epsilon}} _ {mathit {ijk}} |} _ {mathit {NOEU}}} (21)

math:

frac {partial W} {{partial W} {{partial x}} {{partial x} _ {n}}approxsum _ {mathit {ng}}sum _ {l=1} ^ {3}}frac {partial x}} {{partial x}} _ {3}}frac {partial {eta} {3}}frac {partial {eta} {3}}frac {partial {eta} {3}}frac {partial {eta} {3}}frac {partial {eta} {3}}frac {partial {eta} {3}}frac {partial {eta} {3}}frac {partial {eta} {{partial N} _ {k}} {{W}} {{W} _ {k} |} _ {mathit {NOEU}} (22)

wher*ηl* represents the local coordinates (η1, η2, η3) and 218*sig2*xm*is the inverse Jacobian matrix of the transformation in Equation 14, and the terms with |NOEU Indicate that the global nodal values have been used in the computation, which were obtained from Gaussian values as described earlier.

The domain integral expression of modified*J* without initial stress in Equation Error: Reference source not found is:

wher*nV* represents the number of elements in the volume*V*, nV1 represents the number of elements in the volume*V1* and*ng* is the number of gauss points per element. The quantities within {.} p are evaluated at all Gauss points in an element and*wp* are the respective weights.

The domain integral expression of modified*J* with initial stress in Equation 10 is:

In the development of POST_JMOD, the Equations 23-24 are calculated using POST_ELEM operator.