2. A crack which is oblique (Swedish sned ) with respect to the xy coordinate system is to be analysed. TMHL

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1 (Del I, teori; 1 p.) 1. In fracture echanics, the concept of energy release rate is iportant. Fro the fundaental energy balance of a case with possible crack growth, one usually derives the equation where is the energy release rate, is the work done by external forces, is the elastic strain energy and is the crack area. Rewrite the equation in ters of only for (a) loading by fixed loads and (b) loading by fixed grips. ( ) ( ) (Del I, teori; 1 p.) 2. A crack which is oblique (Swedish sned ) with respect to the xy coordinate syste is to be analysed. (a) Which operation ust first be done in order to analyse the crack as a ixed K I /K II ode crack? (b) If the loading of the crack is sufficiently high, it will grow. Give a short explanation of the axiu tangential stress principle for finding the direction of this growth. Note: No equations or coputations needed; only short explanations in few words. (a) Introduce a new x y coordinate syste with its x axis along the crack and the y axis perpendicular to the crack. Transfor the stresses into the x y syste. (b) Find the polar angle in the x y syste in which the shear stress is. This is obviously a principal stress direction, and the noral stress in the orthogonal direction is the axiu tangential stress. (It is a frequently used principle that the direction and agnitude of this ax. tangential stress can be used as a criterion of direction and start of crack growth.)

2 (Del I, teori; 1 p.) 3. We study the three load cases ) ( ) ) ( ) ) ( ) (in all cases expressed in MPa). Order the three load cases after the risk of fatigue failure, easured by the safety factor in aplitude and provided that the coponent is anufactured fro a aterial that has the following fatigue data: (Draw the Haigh diagra on the teplate appended below). Answer: I (ost dangerous), II:, III (least dangerous): See figure above! c b a

3 (Del I, teori; 1 p.) 4. A coponent of a achine is repeatedly loaded by the typical load sequence of Fig. 4a. The Wöhler diagra of the aterial is shown in Fig. 4b. Use the linear Palgren-Miner theory to copute how any such load sequences the coponent can be expected to stand. Fig. 4a Fig. 4b 1st cycle ( The 100 cycles following ( ) corresponds to ) correspond to The load sequence therefore gives a daage, and the coponent can therefore stand (Del II, proble; 3 p.) 5. A pressure vessel is to be designed by a leak before break criterion. Leak before break eans that before a crack gets large enough to fracture by the criterion, it will have been detected by gas leakage. A frequent assuption is that a crack with length equal to the wall thickness of the vessel will give a gas leakage that is safely detectable. Design crack, crack length = wall thickness

4 Use this assuption to copute the necessary leak before break -criterion wall thickness for a circular-cylindrical pressure vessel with geoetry and data as in the figure to the right. (No testing of the validity of LEFM is needed). Data: Fro the theory of thin-walled pressure vessels, we have an expression of the tangential stress : A crack with therefore gives Our requireent that this ust be therefore gives (Del II, proble; 3 p.) 6. A bea containing a long, central crack ( and ) is loaded by a pair of concentrated loads. See figure! The aterial is linearly elastic with elastic odulus. Copute the energy release rate!

5 Energy ethod: Horizontal and vertical syetries the whole structure can be considered to consist of 4 equal beas. Study, for instance, the upper left bea: The end section force and oent at the right-hand end of it will be { Free-body diagra + oent equilibriu ( ) ( ) ( ) ( ) ( ) The strain energy of the whole configuration will then be of the above upper left bea ( ) ( ) [ ] and the energy release rate becoes ( ) [ ]

6 (Del II, proble; 3 p.) 7. A coponent, which for siplicity can be treated as a flat specien with a rectangular cross section, has an area reduction with notch geoetry; see Fig. 1. The load is a tie-varying longitudinal force, which in the thinner right-hand half gives an evenly distributed stress ( ), which is high enough that there is a risk of LCF. The coponent is to be designed for an LCF life of 5000 cycles. Copute the stress aplitude that can be allowed! Instructions: 1. Knowing the required LCF life, use the Morrow equation to find the allowable local strain aplitude. 2. Use the Raberg-Osgood curve diagra (Fig. 2) to find the corresponding local stress aplitude. 3. Use the Neuber equation with the and known fro above to copute the allowed aplitude of the externally applied stress. Fig. 1 Data: a Surface: polished MPa Raberg-Osgood curve, see Fig. 2 Fig. 2

7 Morrow s equation ( ) ( ) ( ) ( ) Raberg-Osgood Reading in the diagra of Fig. 2 gives. (An extreely careful plotting could possibly have given instead, but the relevance of such extree detail is questionable; let us keep with 450 MPa.) Neuber We need Accurate reading in the table on top left of p. 64 gives { ( ) The Neuber equation then gives PS. Note that with arrived at the sae answer! and, again, coputing to 3 significant digits, we woluld have (Del II, proble; 3 p.) 8. A bea with a crack at its lower edge is loaded by a tie-varying concentrated load ( ). See figure and data table! Copute the nuber of cycles that the bea can be expected to stand before the crack grows to fatigue failure. Data: Hint: Assue that the crack will never grow to be larger than 20 ; you will later be able to confir this fro the coputed results. (S )

8 I Stress intensity factor range K I ( ) By case No. 6 of the Forula Table, we get Note that we have used the proposed assuption that. We then see in the diagra that 1.12 will be the axiu value of. Setting will therefore give a conservative (i.e., safe) coputation. As soon as we have coputed the critical crack length, we will be able to confir whether or not the assuption was correct. II Critical crack length a cr Fracture criterion ( ) ( ) I.e., the crack length assuption ade above is justified, and we can use the value. III Crack growth coputation ( ) ( ) ( ) [ ] ( ) ( )