TMHL TMHL (Del I, teori; 1 p.) SOLUTION I. II.. III. Fig. 1.1

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1 TMHL (Del I, teori; 1 p.) 1. Fig. 1.1 shows three cases of sharp cracks in a sheet of metal. In all three cases, the sheet is assumed to be very large in comparison with the crack. Note the different crack lengths and temperatures! The of the material is temperature-dependent according to Fig Order the three cases after risk of crack growth. Short mathematical explanations (but not detailed computations!) are required. I. II.. III. Fig. 1.1 Nr. 1. Highest risk of growth III I 3. Lowest risk of growth II Fig. 1.2 Therefore: Compare = for the three cases; the higher, the more dangerous. Fall I: Fall II: Fall III:

2 TMHL (Del I, teori; 1 p.) 2. A crack which is ob iqu (Sw dis s d ) wi sp o xy coordinate system is to be analysed. (a) Which operation must first be done in order to analyse the crack as a mixed K I /K II mode crack? (b) If the loading of the crack is sufficiently high, it will grow. Give a short explanation of the maximum tangential stress principle for finding the direction of this growth. Note: No equations or computations needed; only short explanations in few words. (a) Introduce a new x y coordinate system with its x axis along the crack and the y axis perpendicular to the crack. Transform the stresses into the x y system. (b) Find the polar angle in the x y system in which the shear stress is. This is obviously a principal stress direction, and the normal stress in the orthogonal direction is the maximum tangential stress. (It is a frequently used principle that the direction and magnitude of this max. tangential stress can be used as a criterion of direction and start of crack growth.) TMHL (Del I, teori; 1 p.) 3. A component is to be designed against LCF. The component is to be manufactured from a normal steel. We compare two cases (I) and (II) of required LCF life, where. Which of the corresponding ratios of allowed plastic strain amplitudes in the table to the right is probably the best guess? (Tip: think of normal values of the exponent in the Coffin-Manson law!) X Coffin-Manson:, where is of the order of Consequently:

3 TMHL (Del I, teori; 1 p.) 4. D o P is w are frequently given in non-si units. is, for instance, often given so that inserted in will give in m. Show how such a value of is converted into an SI value such that i s d i P is w i will give in m. With and according to the first case of the text: With and : TMHL (Del II, problem; 3 p.) 5. A metal sheet contains an edge crack (see figure). Data: Plane strain mm mm Pa Nm -3/2 Compute the maximum allowable tensile stress (a) without and (b) with an Irwin correction for plastic zone (No testing of validity of LEFM need be done).

4 Use crack case Nr. 5: (a): (b): Corrected crack length: o and TMHL (Del II, problem; 3 p.) 6. A beam containing a long, central crack ( and ) is loaded by an evenly distributed load. See figure! The material is linearly elastic with elastic modulus. Compute the energy release rate! Use an energy method or go via the J integral! It is (as always) a good idea to start by identifying and using all symmetries.

5 Energy method: Horizontal and vertical symmetries the whole structure can be considered to consist of 4 equal beams. Study, for instance, the upper left beam: The end section force and moment at the right-hand end of it will be { Free-body diagram + moment equilibrium We can now compute the strain energy of this beam: [ ] and the energy release rate becomes [ ]

6 J integral method 1 We therefore have [ ] 2 We thus need. We can use linear beam deflection theory:

7 We therefore have and Conclusion TMHL (Del II, problem; 3 p.) 7. A thin-walled cylindrical pressure vessel is loaded by a time-varying inner pressure { in the pressure vessel wall. The pressure vessel is to be designed against HCF, using the Sines hypothesis. Material parameters in the Sines hypothesis : and. Compute the necessary wall thickness, provided that is known.

8 [ ] [ ] ( ) TMHL (Del II, problem; 3 p.) 8. A railway line has been laid with the European standard rail profile UIC 60, whose cross section is shown in Fig. 1. The worst crack that can be imagined on a line laid with such rails would be a surface crack at the bottom of the rail, and we want to make a simple damage-tolerance design analysis for this crack. Assumptions: The rail section containing the crack can be considered to be a beam on the moment-free support of two neighbouring sleepers at a distance 0.6 m; see Fig 2. The crack is in the worst possible lengthwise position, i.e. in the middle of the beam span. The crack is a semielliptical surface crack (sectioned in Fig. 1), and for simplicity we assume that it can be treated as a semielliptial edge crack in a large sheet under tensile load (case No. 7 in the formula table). We further assume that it has an initial minor to major axis ratio and that this ratio remains constant even if the crack grows. The allowed axle load on the line is 22 tons, which gives a max. wheel load N. Since there is no practical way of inspection of the bottom of the rail on the laid line, we must assume as a worst case that the crack is there already before the laying. The line must then be able to stand maximum-load wheel passages, and we do not allow the crack to grow to more than a depth mm (since the geometry of the rail cross section then makes the ligament too small for a safe analysis). Paris data of the material: (SI units) and. Make a so b i s ssu p io di v u o for the crack and compute the maximum allowable start-crack size (i.e., the crack size that the inspection of the rail before laying must safely detect).

9 2c a Fig. 1

10 Fig. 2 Engineering assumption: ; kept constant during the crack growth process. Then and ( ) [ ] Given criterion of required life: cycles is allowed to give a maximum crack size A reasonable engineering ANSWER might be: