Michael Vormwald

The paper addresses the numerical calculation of multi‐axial fatigue crack propagation in ductile materials under proportional and non‐proportional cyclic loading conditions. The analysis is based on a phase field fracture theory proposed in earlier work, which describes crack propagation on the basis of the evolution of plastic strain. A plane stress formulation of the theory is developed and used to analyze both length scale effects and comparison with experimental results. The considered loading histories comprise proportional and non‐proportional cyclic tension/compression and shear loadings. Generally, the predicted responses are in good agreement with the experimental results for proportional loading histories. Qualitative differences seem to exist, however, in what concerns the number of predicted cracks for non‐proportional cyclic loading conditions.

The paper demonstrates the analysis of the stress and strain states of an inhomogeneous structure with an axial symmetrical spheroidal inclusion in an infinite solid under a remote uniform tension load derived from the conditions of the theory of elasticity. With respect to the inhomogeneous structure, the deformations produce constraints that require a complete 3-dimensional analysis in the z, r-coordinate system. The solution generates the stress state of the inclusion and at the interface of the matrix. Spheroidal inclusions in an uniform outer tension stress field deform self-similarly to a rather elongated spheroid. With respect to the compatibility condition and depending on the different elastic moduli and Poisson’s ratios in the inclusion and matrix, the solution procedure leads to a system of two linear equations with the magnitudes of the two tractions as unknowns. In this way, the solution is shortened to a twofold statically indeterminate system. The analysis performed leads to an exact solution of the general spheroidal problem in a formulation of stress concentration factors considering the different stiffness parameters of inclusion and matrix.

The approaches used to calculate the fatigue life of components must inevitably consider multiaxial stresses. Compared to proportional loading, the calculation of nonproportional loading is particularly challenging, especially since different materials exhibit the effects of nonproportional hardening and shifts in fatigue life. In this paper, the critical plane approach of scaled normal stresses, first proposed by Gaier and Dannbauer and later published in a modified version by Riess et al., is investigated in detail. It is shown that, on the one hand, compatibilities exist or can be established with known proportional strength criteria that can account for the varying ductility of different materials. Furthermore, it is demonstrated that the scaled normal stress approach can be formulated in such a way that different strength criteria can be used therein. As an example, the generally formulated approach for scaled normal stresses is applied to test results from ductile cast iron material EN-GJS-500-14. Different correction factors accounting for nonproportional loading are investigated. Through appropriate parameterization of one of the studied corrections, proportional and nonproportional test results were observed to fall within one common scatter band.

The fundamental idea in phase field theories is to assume the presence of an additional state variable, the so-called phase field, and its gradient in the general functional used for the description of the behaviour of materials. In linear elastic fracture mechanics the phase field is employed to capture the surface energy of the crack, while in damage mechanics it represents the variable of isotropic damage. The present paper is concerned, in the context of plasticity and ductile fracture, with a commonly used phase field model in fracture mechanics. On the one hand, an appropriate framework for thermodynamical consistency is outlined. On the other hand, an analysis of the model responses for cyclic loading conditions and pure kinematic or pure isotropic hardening are shown.