The augmentation of existing finite element codes to use complex and hypercomplex variables and algebras provides an accurate and straightforward method to compute the energy release rate for linear and nonlinear solids. The basic concept is to introduce complex nodes defined by real and imaginary nodes. The real nodes define the geometry and the imaginary nodes define the perturbation to the real mesh. The crack is extended using imaginary coordinates surrounding the crack tip. The solution of the complex system of equations then yields a complex displacement with the imaginary displacement equal to the derivative of the displacement with respect to the crack length. Subsequently, the energy release rate (the derivative of the strain energy with respect to the crack length) can be determined using from the complex strains and stresses. The results indicate that the ERR results are as accurate as the J integral but the method has several advantages: there are no contours to interrogate – only one result is provided, the method works for both linear and nonlinear materials with loading and unloading, unlike the J integral, and no integral formulation must be developed and implemented. Numerical examples demonstrate the accuracy of the method.
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