Model reduction strategy applied to a problem of crack propagation in elastically cubic materials
Résumé
Fatigue affects components subjected to cyclic loadings. High pressure turbine blades are one such component that works under very aggressive environments such as: high temperature, oxidation and centrifugal forces. In order to predict its life, a damage tolerance approach must be set. This approach requires a study of crack propagation and encounters two main problems. The first is linked to the non-linear behaviour of the material especially under complex loadings where the definition of a cycle is no longer obvious. The second problem is the high cost of running elastic-plastic finite element computations on complex 3D structures over millions of cycles. In order to overcome the first issue, an incremental model was proposed. This model is related to crack tip reference and mainly based on the blunting at this region since many studies had shown that cracks in ductile materials propagate due to plasticity under fatigue loading conditions. To address the second problem, a model reduction strategy using the Proper Orthogonal Decomposition (POD) was used to reduce the cost of finite element analysis. Besides, a wide range of turbine blades are manufactured as singles crystals. This structure, in regards to its direction-dependent behaviour, creates a challenge in studying crack propagation in anisotropic materials. An elastic analysis is applied over the whole structure while a confined elastic-plastic one, fed by the previous, is processed on the K-dominance zone around the crack tip. Kinematic displacement fields from these analyses are projected on a reduced basis in order to obtain a non-local condensed model that serves as an input for the incremental model. This condensed model gives a variation of blunting intensity factors of different modes as a function of the loading given through stress intensity factors. To build this basis, many hypotheses should be taken into account. For instance; elasticity and plasticity should be kinematically independent, plasticity is confined in a small region around the crack, the geometry is self-similar at the crack tip and so for the displacement fields, etc. From the aforementioned hypotheses, kinematic displacement field is represented as the superposition of different modes, each mode represents a degree of freedom. It was shown that the first two modes of the POD are sufficient to describe the physics around the crack tip where the first mode corresponds to the elastic kinematic field and the second mode refers to the plastic field. Each one of these components is the product of an intensity factor and a spatial distribution. These spatial distributions are the reference fields that form the needed basis to reduce the model. Since analysis is made under multiaxial loadings and with the presence of anisotropy, it faces a problem of a mixed mode crack, therefore, kinematic fields are initially decomposed into three main fields that describe different crack modes. Each field is extracted from a computation model that enhances the appearance of the corresponding crack mode. Thus, mode I is obtained from an equi-biaxial test, mode II from a pure shear test and mode III from an anti-plane shear test. Based on the hypothesis of the self-similar structure at the crack tip, reference fields are considered as the products of a radial function that presents the scale and an angular function that describes the shape of the displacement field around the crack. These functions can be easily saved then used as a basis to project displacement fields extracted from more complicated and long analyses. This projection will give a non-local model that relates loading, described by stress intensity factors, to the crack tip opening and sliding, described by blunting intensity factors. Once this condensed model is obtained and validated for different aspects of anisotropy, it can be used to extract further information such as the yield surface criterion and the plastic flow direction as function of the anisotropy of the material.