Finite Element Investigation on the Stress State at Crack Tip by Using EPFM Parameters

Transcription

1 Finite Element Investigation on the Stress State at Crack Tip by Using EPFM Parameters FRANCESCO CAPUTO, ALESSANDRO DE LUCA, GIUSEPPE LAMANNA 1, ALESSANDRO SOPRANO Department of Industrial and Information Engineering Second University of Naples via Roma, Aversa ITALY 1 giuseppe.lamanna@unina2.it Abstract: Nowadays, the real plastic zone shape and size at the crack tip cannot be described using the linear elastic fracture mechanics theory (LEFM). In fact, one of the basic principles of the LEFM theory is to consider the Plastic Zone Size (PZS) at the crack tip as negligible with respect to the crack length. Moreover, since the Plastic Zone Size (PZS) strictly depends on many variables, an exact analytical solution, such as to take into account all of these parameters is not available. Therefore, numerical simulations analyses are mandatory. Within this work, an extensive numerical analysis, based on elastic-plastic fracture mechanics theory (EPFM), has been developed in order to study the plastic zone size both at the tip of a trough crack and at the tip of a pre-crack at the notch edge under MODE I loading condition. In particular, in this work, a parametric 3D finite element model has been carried out in order to show the influence of the crack size and of the component thickness on PZS. Key-Words: EPFM, Large Scale Yielding, Plastic radius, Crack tip, Pre-cracked notch. 1 Introduction The behaviour of damaged structures is usually studied through the Linear Elastic Fracture Mechanics (LEFM), which considers only plane stress-strain states at the crack front. The main advantage of two-dimensional theories is their analytical simplicity compared to the threedimensional ones, but for Large Scale Yielding (LSY) phenomena they aren t able to overcome some limits in describing the actual behaviour of the Figure 1: Test case geometry: trough crack (left); pre-cracked notch (right). ISBN:

2 material around the damage [1]. Therefore, one of the basic principles of the LEFM theory is to consider the Plastic Zone Size (PZS) at the crack tip as negligible with respect to the crack length, i.e. take into account all these parameters is not yet available [20]. For this reason, finite element analysis is mandatory. The main scope of the study reported in the present Small Scale Yielding (SSY) condition [2]. For this reason, the LEFM theory can t describe the behaviour of short cracks, where the state of stress at the tip is generally characterized by a Large Scale Yielding (LSY) and then by high ratios of PZS to the crack length [1, 3, 4]. Several numerical and experimental investigations [5-9] have shown that such ratio is larger for short cracks than for long ones, for a given nominal Stress paper is to describe the plastic zone size, which takes place around the crack tip under Mode I loading condition, by using the EPFM theory and a parametric 3D finite element model [21-24]. In particular, within this paper the structural behaviour of two different plates have been presented. The first one, characterized by a pre-cracked circular notch; the second one, characterized by a trough crack (without hole). Figure 2: Material properties. Intensity Factor (SIF) [10-13]. The difficulties encountered to describe the stressstrain state at the crack tip through the parameter of LEFM theory is leading to consider the Elastic- Plastic Fracture Mechanics (EPFM) theory s parameters [14, 15], as the CTOD (Crack Tip Opening Displacement), the CTOA (Crack Tip Opening Angle), the COD (Crack Opening Displacement) and the J-integral [16, 17]. However, since PZS strictly depends on many variables (the material yield stress σ y, the applied remote load σ, the crack size a and the component thickness t) [18, 19], an analytical formulation for PZS such as to 2 Problem Formulation Please, A plate with a trough crack in the middle transverse section and a plate with a pre-cracked circular notch, both subjected to a remote longitudinal stress (Mode I), whose value spans the range N/mm 2, have been numerically modeled (Fig. 1). A parametric analysis has been performed, whose allowed ranges of the considered geometrical and physical parameters are illustrated in Tab. 1. ISBN:

3 The material properties have been assumed nonlinear (Fig. 2); elastic-plastic analyses of the model have been performed by using Abaqus ver code. been kept accurately small to match with those necessary at the crack tip to reach the required resolution of the stress field (minimum average element length is about 1E-04 mm). Figure 3: FE model detail at crack tip: trough crack (left); pre-cracked notch (right). The FE models of both plate configurations, shown in Fig. 3, have been built with a number of nodes between and and a number of elements between and 42090, depending on the values assumed by the geometrical parameters. Symmetry conditions have been used for an efficient computation and therefore a quarter A number of elements between 20 and 30 have been considered along the thickness, depending on the values assumed by the geometrical parameters, to resolve consistently the out of plane stress gradient. Figure 4: rp (a=5 mm t=0,5 5). symmetric model has been adopted. The reduced integration 20-nodes brick elements (element type C3D20R by the Abaqus elements library) have been used. In all models, the element sizes have 2 Analysis of results and conclusions As matter of the fact, the plastic radius (r p ), i.e. the plastic zone size on the crack plane in the middle plane of the plate, has been evaluated through the ISBN:

4 von Mises yield criterion, by considering the distance from the crack front at which the von Mises For fixed thickness values (t = 0.5, 1, 2.5 and 5 mm) and crack size (a) equals to 5 mm (circular notch Figure 6: rp (a=2,6 mm t=0,5 5). stress, σ vm, reaches the value of the material yielding stress, σ y = MPa. radius R=2.5 mm), the evolution of plastic radius obtained by both numerical models have been shown as function of applied load (σ) and correlated ISBN:

5 Figure 7: J (a=2,6 mm t=0,5 5). between themselves (Fig. 4). In addition, the evolutions of J-Integrals have been correlated (Fig. 5). For similar crack dimension and hole radius values, the curves for both cracked plate configurations are very close. Specifically, in accord to the graphs above, the plastic radius dimension and the J- Integrals values, related to the plate with precracked notch, are slightly higher than those related to the plate without hole. It is more evident for higher values of stress and depends on the notch effect caused by the hole. However, for fixed hole radius size (R=2.5 mm), with shorter crack length values, for example a=2.6 mm, the aforementioned curves, obtained by both plates, are not in agreement (Fig. 6 for plastic radius evolution and from 7 for J- integrals evolution). It is likely to be caused by the difficulty to use the EPFM theory for describing the real plastic zone shape and size at the tip of a shortcrack. References: [1] Caputo F., Lamanna G., Lanzillo L., Soprano A., Numerical investigation on LEFM limits under LSY conditions, Key Engineering Materials, Vols , 2014, pp [2] Park H. B., Kim K. M., Lee B. W., Rheem K. S., Effects of crack tip plasticity on fatigue crack propagation, Journal of Nuclear Materials, Vol. 230, No. 1, 1996, pp [3] Hussain K., Short fatigue crack behavior and analytical models: a review, Engineering Fracture Mechanics, Vol. 58, No. 4, 1997, pp [4] McDowell D. L., An engineering model for propagation of small cracks in fatigue, Engineering Fracture Mechanics, Vol. 56, No. 3, 1997, pp [5] Zhang J. Z., Du S. Y., Elastic-plastic finite element analysis and experimental study of short and long fatigue crack growth, Engineering Fracture Mechanics, Vol. 68, No. 14, 2001, pp [6] Caputo F., Lamanna G., Soprano A., On the evaluation of the plastic zone size at the crack tip, Engineering Fracture Mechanics, Vol. 103, 2013, pp [7] Caputo F., Lamanna G., Soprano A., Geometrical parameters influencing a hybrid mechanical coupling, Key Engineering Materials, Vols , 2012, pp [8] Caputo F., Lamanna G., Soprano A., Effects of Tolerances on the Structural Behavior of a Bolted Hybrid Joint, Key Engineering Materials, Vols , 2012, pp [9] Caputo F., Lamanna G., Soprano A., Residual Strength Improvement of an Aluminium Alloy Cracked Panel, The Open Mechanical Engineering Journal, Vol. 7, 2013, pp ISBN:

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