AUTOMATIC FATIGUE CRACK PROPAGATION USING A SELF-ADAPTATIVE STRATEGY Carlos V. Carvalho * , Tereza D. de Araújo * , Joaquim B. Cavalcante * , Luiz F. Martha * and Túlio N. Bittencourt † *
Department of Civil Engineering and Computer Graphics Technology Group (TeCGraf) Pontifical Catholic University of Rio de Janeiro (PUC-Rio) 22453-900 - Rio de Janeiro - RJ - Brazil e-mail: alencar, denyse, joaquim,
[email protected], web page: http://www.civ.puc-rio.br † Department of Structure and Foundation Engineering (PEF) and Computational Mechanics Laboratory (LMC) Polytechnic School of University of São Paulo (EPUSP) 05508-030 - São Paulo - SP - Brazil e-mail:
[email protected], web page: http://www.lmc.ep.usp.br
ABSTRACT This work presents a computational tool, based on the finite element method, for the analysis of two-dimensional structural components subjected to fatigue. This tool, called QUEBRA2D, estimates, through well-known empiric models for fatigue life prediction, the number of cycles necessary for a crack to reach a specific size when subjected to constant amplitude cyclic loading. Automatic crack propagation is based on the numerical computation of mixedmode stress intensity factors and crack extension direction. QUEBRA2D is an interactive graphics computational system that integrates modeling, analysis, and visualization capabilities. It is based on a self-adaptive mesh generation strategy that uses recursive spatial enumeration techniques: a binary tree partition for the boundary and the crack-line curves refinement and a quadtree partition for domain mesh generation. The self-adaptive procedure takes into account arbitrarily generated crack geometry and finite element error estimation. In each step of crack propagation, the system regenerates the entire mesh automatically according to the evolving geometry. This paper shows a simulation example that matches closely a laboratory experiment.
INTRODUCTION This work proposes a self-adaptive methodology for simulating automatic fatigue crack propagation in twodimensional structural components. The adaptive procedure provides a regular mesh refinement for the free-boundary curves (including cracks) and is based on a posteriori error estimation. An h-refinement strategy is utilized in this process [2,4]. One of the main objectives in fatigue design is the prediction of crack growth rate with respect to the number of cycles of a cyclic solicitation. Traditionally, empiric models (Paris, Walker, Forman and Priddle) [1,3] are used for fatigue crack growth prediction. These models need the stress intensity factor history along the crack path, which is only available for typical (and simple) component geometry and loading [14]. The main objective of this work is the development of an interactive graphics computational tool, called QUEBRA2D, that computes the stress intensity factor history for arbitrary component and crack geometry (2D) and predicts the crack growth rate for arbitrary cyclic loading under constant amplitude. The system was implemented using the IUP user interface system [9] and the CD graphic system [15]. It has a flexible user interface that allows the visualization of the model and its results and responses at any time during the simulation. In automatic fatigue crack propagation [5], the model geometry changes in each step of propagation. Therefore, it is necessary to modify the finite element mesh and to perform a new analysis at each step. Each propagation simulation step consists of: • finite element analysis of the initial mesh with the initial crack defined for by the user; • computation of stress intensity factors; • determination of direction of crack extension and new tip location according to a specified crack increment; • updating of crack geometry;
• automatic remeshing; The numerical computation of stress intensity factors is implemented in QUEBRA2D through the following techniques: (a) the displacement correlation technique [13], (b) computation of potential energy release rate by the modified crack closure integral [10] and (c) J-integral computation by the equivalent domain integral method (EDI)[11]. The crack direction in each step of propagation is determined through the following interaction theories: (a) the maximum circumferential stress (σθmáx ) [6], (b) the maximum potential energy release rate [7], and (c) the minimum strain energy density [12]. The following figures illustrate an automatic crack propagation simulation using QUEBRA2D. Figure 1 shows the model with attributes and geometric information. Figure 2 shows the initial mesh with an initial crack. Figure 3 shows the experimental results of this model observed in the laboratory [8]. Quarter-point singular elements are used at the crack tip.
E = 300 Ksi
1
ν = 0.30
1.25 0.5 2.0
t = 0.5 inch 0.5
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Figure 1 - Model with attributes and geometric information (dimensions in inches).
Figure 2 - Model with initial crack and mesh.
Figure 3 – Experimental trajectory observed.
Figure 4 shows simulation trajectories obtained with different crack propagation increments (0.5, 0.8 and 1.0 inches). The trajectories, predicted with different propagation increment sizes show convergence to experimental observation.
6.00
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Figure 4 - Trajectory obtained with increments of 1.0, 0.8 and 0.5 inches (EDI and σθmax). Figure 5 shows trajectories of the propagation using the three mixed-mode interaction theories for computing crack direction. The trajectories according to these theories are very similar. Figures 6 and 7 show the mesh for the last step of propagation. 6.00
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Maximum circumferential stress Maximum potential energy release rate Minimum strain density
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Figure 5 - Trajectories obtained using the three mixed-mode interaction theories with increments of 0.5 inches (EDI).
Figure 6 Figure 7 Figure 6 and Figure 7 - Final mesh model after the crack propagation. For fatigue analysis, Figures 8, 9 and 10 show the number of cycles obtained with the Paris, Walker, Forman and Priddle models. In this analysis, the following values have been adopted as their empiric parameters C, m e p : 7⋅ 10-9, 3 and 1, respectively. It has been adopted a null minimum stress for the cyclic loading.
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Figure 8 - Fatigue analysis using Paris and Walker models.
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Figure 9 - Fatigue analysis using using Forman model.
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Figure 10 - Fatigue analysis using using Priddle model.
CONCLUSION The methodology developed for simulating automatic fatigue crack propagation in two-dimensional models has been proven very practical and robust. The proposed system computes stress intensity factor histories for arbitrary model and crack geometries. The number of loading cycles in a fatigue analysis are estimated using well-known empiric fatigue models. It has been observed that, for small ratios of mode II to mode I stress intensity factors (KII/KI), crack trajectories are very similar for the three implemented interaction theories. This ratio is always small when the simulation allows change of crack orientation. It has been also observed that the predictions of crack trajectories improve as the crack increment size decreases.
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