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Maste R's D E G Re E Thesis Stability of End Notched Flexure Specimen THESIS E RE G E D Master’s (one year) Degree Thesis in Applied Mechanics Level 30 ECTS Spring term, 2010 Arun Gojuri Supervisor: Ulf Stigh Examiner: Thomas Carlberger MASTER’S School of Technology & Society Skövde University BOX 408 SE-541 28 Skövde Sweden 1 Preface This work has been carried out during the spring semester year 2010 at the Department of Mechanical Engineering at the University of Skövde, Sweden. First and foremost, I would like say thanks to my supervisor Prof. Ulf Stigh not only for his help and support, but also for sharing the knowledge and for always being able to help. Moreover, for the opportunity to perform this thesis work and also for his way of making the topic exciting and interesting. In a deep thanks to Dr. Kent Salomonsson, Dr. Tobias Andersson and Dr. Anders Biel for their great supports during my theoretical lectures works and commitment concerning various software related issues. Also thanks to my friends Tomas Walander, Daniel Svensson, and Saidul Gundeboina for help and sharing knowledge with me during this work. Finally yet importantly, wide thanks are dedicated to the examiner, Dr. Thomas Carlberger for giving an opportunity to present my thesis work here. Skövde. December 2010 Arun Gojuri 2 Abstract This paper deals with two-dimensional Finite Element Analysis of the End Notched Flexure (ENF) specimen. The specimen is known to be unstable if the crack length is shorter than some critical crack length acr. A geometric linear two-dimensional Finite Element (FE) analysis of the ENF specimen is performed to evaluate acr for isotropic and orthotropic elastic materials, respectively. Moreover, the Mode II Energy Release Rate (ERR) JII and the compliance of the specimen are calculated. The influence of anisotropy is studied. Comparisons are made with the results from beam theory. This work is an extension of previous work. Keywords: Energy release rate, Finite element method, ENF-specimen, Delamination, Composite materials isotropic & orthotropic, Controlled displacement. 3 Table of Contents Preface ................................................................................................................. 2 Abstract ............................................................................................................... 3 1. Introduction ................................................................................................. 5 1.1 ENF specimen .......................................................................................................................... 5 1.2 Contour integral J .................................................................................................................... 7 1.3 Isotropic material .................................................................................................................. 8 1.4 Orthotropic material ............................................................................................................ 9 2. Materials and method ............................................................................. 10 2.1 Geometry of specimen ..................................................................................................... 10 2.2 Material properties ............................................................................................................ 10 2.3 Finite element method simulations ............................................................................ 11 2.4 Evaluation of J ....................................................................................................................... 12 3. Numerical studies ................................................................................... 13 4. Discussion and Conclusion .................................................................. 16 5. References .................................................................................................. 17 4 1. Introduction Much work has recently been devoted to the failure mechanism of composites. Consequently, many new fracture tests have been devised for measuring the fracture properties. Most such tests and standard test procedures are limited to studies of delamination where a crack propagates between the plies. Fracture mechanics are commonly employed to accommodate crack tip singularities, and the energy release rate (ERR) is a physically well-defined quantity that is experimentally measurable using the compliance technique (Russell & Street, 1982 and Broek, 1984). There are three types of loadings of the crack tip, called Mode I, Mode II and Mode III, respectively, cf. Fig. 1. Mode I is the opening mode. Here tensile stresses act on the plane heading the crack tip. Mode II is an in-plane sliding mode. Here the shear stresses act in the plane of the crack and perpendicularly to the crack front. Mode III is another shearing (tearing) mode. Here the shear stresses act in the plane of the crack and directed parallel to the crack front. A cracked body can be loaded in any one of these modes, or a combination of two or three modes called mixed mode. Figure 1: Fracture mechanics failure modes: opening Mode I, shearing Mode II, and tearing Mode III. In this work, Mode II loading is considered. In 1956, Irwin defined the strain ERR, G. This is a measure of the energy available for an increment of crack surface extension. The strain ERR can be calculated using the compliance, C, according to (1) Here b is the thickness, a is the crack length, and is the acting load; subscript II denotes the Mode. The compliance C is used in Eq. (1). C is defined as the ratio of the displacement , under the central loading point and the applied load P as (2) This is the inverse of the stiffness, K=1/C. The compliance is a function of the crack length a, i.e. C(a). 1.1. ENF specimen The End Notched Flexure (ENF) specimen was introduced by Russell and Street (1982) as a pure mode II delamination specimen for testing of composites, see Fig. 2. 5 Figure 2: Geometry of End Notched Flexure (ENF) specimen The ENF specimen gives an almost pure mode II shear loading at the crack tip. It requires a quite simple test and the specimen is easy to manufacture. Using the ENF specimen, the experimentalist is able to measure the fracture energy, . Experiments show that the specimen is sometimes stable and sometimes unstable under controlled displacement. It is noted that the specimen is always unstable under prescribed force. If the pre-crack is too short it will be unstable at prescribed displacement. That is, the crack length must be longer than some critical crack length to achieve stable crack growth. The critical crack length is the topic of the present thesis. In experiments performed in 2002, Alfredsson et al. (2003) experienced instability with a too short crack. Carlsson, et al. (1986), and Chai & Mall (1988) developed a stability criterion. Carlsson, et al. (1986) argues that for tough resin systems, interlaminar shear effects may be significant. In turn, this requires slender specimen, i.e. large length to thickness relations. Furthermore, they argue that interlaminar shear deformation might influence the evaluation of the interlaminar Mode II fracture toughness. In an analysis based on beam theory, they show that the crack growth is stable if . In this paper, we study the same problem using 2D elasticity using the finite element method (FEM). The compliance of ENF-specimen is derived by Alfredsson and Stigh (2009) based on the J-integral and Euler-Bernoulli beam theory. Noting the fact that G = J for linear elasticity they derive (3) Here C(0) = is the compliance without a crack where E is the elastic modules and b the width. Apply C(0) in Eq. (3), the beam theory compliance of the ENF specimen is given by, (4) BT The compliance CBT and the Mode II ERR, GII are also given by Russell and Street (1982) for the Beam theory. Based on Eq. (1) they derive 6 (5) Here h is the height of specimen, and P is the applied load. In addition, BT denotes the Beam theory Presently, two different methods are widely used for calculating the ERR using FEM. One of these is the J -integral method, which is based on a path independent surface integral (line integral in a 2D formulation), and another one is the virtual crack extension method, which models a crack extension by a shift of node points in a Finite Element model. In this work we use the J -integral method to evaluate the ERR with the Finite Element Method (FEM). 1.2. Contour Integral - J The J -integral is used to calculate the ERR GII in a finite element (FE) analysis. In Anderson (2005), the J integral is defined as (7) Here, W is the strain energy density, t is the traction vector, U the displacement vector, and x1, x2 the coordinate directions. With x1 directed in the extension of the crack. The theoretical concept of the J- integral was developed independently in 1967 by Cherepano and in 1968 by Rice. Rice showed that the value of this integral equals the ERR in a non-linear elastic body that contains a crack. Now J(a) informs on how J changes with a and with everything else constant. If the crack propagates, J equals the fracture energy Jc of the material. If J increases more than the fracture energy, Jc, the specimen is unstable. Thus, the condition for stability reads There is a formula derived for this stability condition by Chai (1988) and Carlsson et al. (1986) that gives a critical crack length based on beam theory and Linear Elastic Fracture Mechanics (LEFM). In this case, J is given by (8)
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