Finite element modelling of fatigue crack growth of surface cracked plates: Part I: The numerical technique
Introduction
Surface cracks, which are most likely to be found in many structures in service, such as pressure vessels, pipeline systems, off-shore structures and aircraft components, have been recognised as a major origin of potential failure for such components. The study of fatigue crack propagation from such defects has been an important subject during recent decades. The surface crack in a finite thickness plate subjected to remote tension, bending or combined loading, as shown in Fig. 1, is a well-studied configuration, to which most effort for surface crack problems has so far been devoted, and is commonly considered to be representative of the surface crack family.
Based on the experimental fact that the crack shape of propagating surface cracks in a plate under cyclic tension, bending or combined loading is approximately semi-elliptical, a theoretical model with two degree of freedom, as shown in Fig. 2(a), has been proposed [1] for predicting the fatigue crack growth of surface cracks. The model assumes the growing crack to be of semi-elliptical shape, but allows for change in crack aspect ratio. Fatigue crack growth is then calculated along both the depth and surface directions through two coupled Paris fatigue crack growth equations. This method has been widely used in practical assessments of fatigue crack propagation for surface cracks.
Obviously, the theoretical model contains a crack shape assumption, which may lead to an uncertain error in the prediction of fatigue crack growth, especially when a complex stress distribution across the plate thickness is involved. In this case the crack may significantly deviate from the assumed semi-elliptical shape. On the basis of the initial work by Smith and Cooper [2], a sophisticated predictive method has recently been developed by the authors. The method integrates an appropriate fatigue crack growth law at a set of points along the surface crack front instead of its two extremes, which enables the crack front of growing cracks to be directly traced so that the crack shape assumption mentioned above can be avoided. The stress intensity factors (SIFs) along the crack front are calculated by the finite element (FE) method, and an automatic technique has been developed in order to automatically recreate finite element models as the crack propagates.
This paper describes some theoretical background and technical details associated with the numerical simulation method developed by the authors. Three important problems in the estimate of stress intensity factors along the front of planar cracks by finite element analyses, i.e. the SIF sensitivity to crack front shape, the influence of a non-orthogonal mesh on the SIF accuracy, and the J-integral path independence, are also discussed. The fatigue crack growth characteristics of surface cracks in plates, such as crack shape development and deviation from semi-elliptical shape, aspect ratio change and stress intensity factor variation, will be examined, and their differences from those obtained through the method with a semi-elliptical shape assumption will be revealed in the following papers [18], [19].
Section snippets
Fatigue crack growth law
Paris and Erdogan [3] have constructed a quantitative framework of fatigue fracture mechanics, which correlates the fatigue crack growth rate to the range of stress intensity factor as follows:where the SIF range, ΔK, is related to the crack geometry and the applied load, whilst constants C and m account for stress ratio R (minimum to maximum applied stress), the material and environmental effects. The experimental relation described above can be obtained for a particular material from
Numerical simulation technique
The modelling is principally based on finite element analyses together with the step-by-step Paris law described previously. The variation of stress intensity factors along the crack front is estimated using the 1/4-point crack opening displacement method or the J-integral method. A Paris law is subsequently applied to evaluate the local normal outward increments of crack growth in terms of , , in which the technique of specifying a maximum increment of crack growth, Δamax, along the crack
Sensitivity to crack front shape
The above section has described the two methods of approximating a crack front shape within a FE model, i.e. the polygonal line and cubic spline methods. It has been found by the authors that a large error may occur in the estimate of stress intensity factors when using the polygonal line crack front approximation. In order to clarify this point and the difference between the two definitions, an internal penny-shaped defect in an infinite body under remote uniform tension and a semi-elliptical
Accuracy and efficiency of the technique
The accuracy of the present simulation technique depends not only on how accurate stress intensity factor estimates can be achieved by FE analyses, but also on how small crack growth increments can be chosen during the fatigue crack growth process. A wide comparison of stress intensity factors with published numerical or theoretical solutions has been made in the work by Lin [17]. The comparison showed that the stress intensity factor results obtained by the present numerical technique are
Conclusions
A 3D finite element based numerical technique which can automatically simulate surface cracks in plates has been established. The theoretical background related to the technique has been described, covering the basic linear elastic fracture mechanics principles, the Paris fatigue crack growth law and its extension to a planar crack problem, and the methods of evaluating stress intensity factors by the finite element method. Most technical details have also been presented, which demonstrate the
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