The indirect methods use relationships that exist between the stress-intensity factor (K) and the elastic-energy content (U) of the cracked structure.  These relationships are developed in Section 1.3.2 along with a full discussion of the strain energy release rate (G) and compliance (C), i.e. the inverse stiffness of the system.  The stress-intensity factor is related to these parameters by the following:

where B is the plate thickness and  is the elastic modulus E in plane stress and is E/ (1-n²) in plane strain.

The elastic energy content and the compliance of cracked structures are obtained for a range of crack sizes either by solving the problem for different crack sizes or by unzipping nodes.  Differentiation with respect to crack size gives K from the above equations.  The advantage of the elastic-energy content and compliance methods is that a fine mesh is not necessary, since accuracy of crack-tip stresses is not required.  A disadvantage is that differentiation procedures can introduce errors.

The strain energy release rate relationship (Equation 11.2.33) was derived based on the use of the crack tip stress field and displacement equations to calculate the work done by the forces required to close the crack tip.  The crack tip closing work can be calculated by uncoupling the next nodal point in front of the crack tip and by calculating the work done by the nodal forces to close the crack to its original size.

The concept is that if a crack were to extend by a small amount, Da, the energy absorbed in the process is equal to the work required to close the crack to its original length.  The general integral equations for strain energy release rates for Modes 1 and 2 deformations are

The significance of this approach is that it permits an evaluation of both K₁ and K₂ from the results of a single analysis.

In finite-element analysis, the displacements have a linear variation over the elements and the stiffness matrix is written in terms of forces and displacements at the element corners or nodes.  Therefore, to be consistent with finite-element representation, the approach for evaluating G₁ and G₂ is based on the nodal-point forces and displacements.  An explanation of application of this work-energy method is given with reference to Figure 11.2.12.  The crack and surrounding elements are a small segment from a much larger finite-element model of a structure.  In terms of the finite-element representation, the amount of work required to close the crack, Da, is one-half the product of the forces at nodes c and d and the distance (v_c - v_d) which are required to close these nodes.  The expressions for strain energy release rates in terms of nodal-point displacements and forces are (see Figure 11.2.12 for notations)

Figure 11.2.18.  Finite-Element nodes Near Crack Tip.