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Day of Research 2010 – February 10 – Labo Soete, Ghent University, Belgium 1 1 root type ( & ). In such a case, the quarter point singularity elements [10], in which the midside nodes at the crack tip are shifted to the quarter position in order to produce square-root stress r r singularity, as show in Figure 7(b). In Elastic-Plastic Fracture Mechanics (EPFM), the stress singularity is not of the square root type and has an order of 1 , where m is the power law constant in the m 1 1 Ramberg–Osgood equation [11] ( ) and the strain singularity has an order of m 1 m 1 m1 r 1 ( ). Also, in composite materials when cracks terminate at an interface, the order of the singularity m m1 r is no more of the square root type and has a general order ( u r & r 1 ), which can be a real number 0 1, or a complex number, depending on the geometry of the problem and material properties. To approximate the general power type singularity in FEA, special crack tip elements as the one shown in Figure 7(c) have been developed [12, 13]. After obtaining the FE results, fracture mechanics parameters such as stress intensity factor, strain energy release rate and J-integral can be calculated. There are many computational techniques to extract fracture mechanics parameters from FE results. We present herein two simple techniques to calculate the stress intensity factor and the strain energy release rate using the nodal displacements behind the crack tip and the nodal forces ahead of it. Consider the arrangement of nodes in a quarter point element as shown in Figure 7(b), a two-point formula for calculating the stress intensity factor for a symmetric cark is given by [14]: E 2 K (4 ) 1 2 4 I uy i u y (12) i L Where E is the Elastic modulus and u and u are the crack opening displacements of nodes i-1 and i-2, y i1 y i2 respectively. For the same arrangement of nodes shown in Figure 7(b), the strain energy release rates for mode I and mode II, derived from a virtual crack closure approach, are given by [15]: 2u yi1 GI ( Fy 1.5 4 F ) i1 y (13) i L 2ux i1 G ( F 1.5 4 F (14) II xi 1 ) L Where F x and F y are the crack closure nodal forces in the x and y directions, respectively, and u x and u y are the crack sliding and opening displacements, respectively. Deformation near an interface crack in a lap strap joint [16] is shown in Figure 7(d). xi (a) (b) (c) (d) Figure 7. Fracture mechanics analysis: (a) FE mesh at a crack tip, (b) Quarter point, (c) General power singularity element and (d) Deformation near a crack tip. 7 FATIGUE ANALYSIS Fatigue analysis combines stress analysis with either continuum damage mechanics for fatigue crack initiation analysis or fracture mechanics for fatigue crack propagation analysis. In continuum damage mechanics, the fatigue damage accumulation is represented by a damage variable D, which is equal to zero for a virgin material and 1 for a fully damaged material. The damage variable can be calculated from FE stress analysis results, providing that some damage material parameters are given. For low cycle fatigue, the damage variable (D) is expressed as [17]: 1 / 2 1 m D 1 A m 1 m1 eq R V N (15) 35 Copyright © 2010 by Labo Soete
Day of Research 2010 – February 10 – Labo Soete, Ghent University, Belgium Where N is the number of cycles, eq is the range of von-Mises stress, R V is the triaxiality function, m is the power constant in Ramberg-Osgood equation, and A and are damage parameters, which can be determined experimentally. For given N, m, A and , the range of von-Mises stress eq and triaxiality function R V can be determined from stress analysis results, and therefore D can be calculated using Equation (15). An example of the evolution of D in an adhesive layer with fillet used for a scarf joint [18] is shown in Figure 8. N = 1 D max = 0.002 N = 15 D max = 0.038 Figure 8. Fatigue damage evolution in adhesive layer N = 28 D max = 0.204 In fatigue crack propagation analysis, a crack growth equation (e.g. Paris law) is numerically integrated within a FE stress analysis. A modified Paris law that includes the threshold and accelerating crack growth regions is given by [19]: n 1 G th da 1 n Gmax DG (16) max n2 dN G max 1 Gc Where da/dN is the crack growth rate, G max is the maximum strain energy release rate at a specific stress ratio, G th is the fatigue threshold, G c is the fracture toughness, and n, n 1 and n 2 are constants. A procedure that is used to numerically integrate the crack growth law in FE analysis is shown in Figure 9. This procedure has been implemented in the FE package ANSYS. The application of the crack propagation procedure to a single lap joint is shown in Figure 10 [19]. Only half of the joint was modelled with rotational symmetric boundary conditions in the mid-plane. The number of cycles to failure were calculated twice for each load value; a) using the total strain energy release rate (G max or G T ) and b) using the mode I strain energy release rate (G I ). Initial condition a=a o and applied load F=F i FE stress analysis, Crack length=a i Determine G max or G I Small crack increment a Calculate (da/dN) from equation (16) Draw F-N f (load - number of cycles to failure) diagram No Yes Another load be considered? (F i =F i-1 +F) No Yes Has ‘a’ reached the maximum allowable crack length (a f )? Figure 9. Fatigue crack propagation procedure Calculate new ‘N’ and ‘a’ as: N i+1 =N i +N a i+1 =a i +a Calculate N i as: N i =a i /(da/dN) Figure 10. Fatigue crack propagation in a single lap joint 36 Copyright © 2010 by Labo Soete
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