Model reduction strategy applied to a problem of crack propagation in anisotropic materials
Résumé
Fatigue is a phenomenon that affects components subjected to cyclic loadings. High pressure turbine blades are one such component that works in very aggressive environments such as: high temperature, oxydation and centrifugal forces. In order to predict their life, a damage tolerance approach must be set. This approach requires a study of crack propagation and encounters two main problems. The first one is linked to the non linear behavior of the material (Nickel base super-alloy), especially under complex loadings where the definition of a cycle is no longer obvious. The second problem is the obligation to run long and expensive elastic-plastic finite element computations on complex 3D structures over millions of cycles. In order to overcome the first problem an incremental model was proposed. It's mainly based on the blunting of the crack tip and hence, the crack propagates due to plasticity. 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. However, metallurgists found that the fabrication of turbine blades using coarse grains of Nickel can cause failure as cracks can initiate and propagate from grain boundaries. They proposed to ban these boundaries by modeling a single crystal turbine blade using a directionally solidified alloy. This structure, in regards to its elastic and plastic anisotropy, creates a challenge in studying crack propagation in anisotropic materials. Instead of running expensive elastic-plastic computations, 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 in account. For instance; elasticity and plasticity should be kinematically independent, 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 contains a first degree of freedom for elasticity and a second for plasticity. 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. For each mode, the reference field is extracted from a computation model that enhances the appearance of this mode. Thus, mode I is obtained from an equi-biaxial test by extracting the symmetric part of the displacement field and mode II from the skew-symmetric part of a pure shear test. Taking into account 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 analysis. 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 usefull to extract more information such as the yield surface criterion and the plastic flow direction as function of the anisotropy of the material.