Solution Manual. for. Fracture Mechanics. C.T. Sun and Z.-H. Jin

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Millicent Gregory
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1 Solution Mnul for Frcture Mechnics by C.T. Sun nd Z.-H. Jin

2 Chpter rob.: ) 4 No lod is crried by rt nd rt 4. There is no strin energy stored in them. Constnt Force Boundry Condition The totl strin energy per unit width stored in rt nd rt is L ( L ) U dx + dx A E + 0 A E A E A E A h, A h / Thus the totl strin energy is ( L ) U + Eh Eh Finlly, the energy relese rte is U G Eh Fixed End Boundry Condition dw e 0 since dδ 0, then energy relesed due to the crck extension is du ds d ds he d Eh Finlly, the energy relese rte is G d he

3 b) / / /, 4 4 / No lod is crried by rt. There is no strin energy stored in it. Constnt Force Boundry Condition The totl strin energy per unit width stored in rt nd rt is ( ) U + dx + 0 AE AE AE 8AE A h / Thus the totl strin energy is 9 U + Eh 8Eh 4Eh Finlly, the energy relese rte is U 9 G 8Eh Fixed End Boundry Condition Bsed on the fixed end boundry condition, we cn hve dw e 0 dw s -du Thus the energy relesed due to the crck extension is du ( ) ds + ds d d ds, ds he he d 4EA Eventully, the energy relese rte is 9 G d 8Eh

4 rob.: ) Symmetric cse The strin energies stored in ech prt re M ( x) U U dx dx, 0 EI 0 EI 6EI So the totl strin energy is 4 U U + U + U EI h The energy relese rte is given by du G I t d Eh t b) Anti-symmetric cse th I The strin energies stored in ech prt re ( x) U U dx 0 EI 6EI th, I L ( x) U dx ( L ), I EI EI So the totl strin energy is U U + U + U ( L + ) EI The energy relese rte is given by du 9 G II t d h th 4

5 rob.: Since the structure is symmetric (both loding nd geometry), we cn represent the given crcked bem s two crcked bems s shown in Figure. M is the moment needed to ensure zero slope t the loction where we mde the free body cut. Assuming tht the bem mrked "" is cntilever bem with tip lod /, the slope t the tip is given by ( ) θ EI The slope t the tip of cntilever subjected to tip moment is given by M θ M EI To ensure tht the system shown in Figure ccurtely represents the ctul center crcked structure with mid-section lod, we need to stisfy θ + θ 0 ( ) M 0 M EI EI 4 The moment distribution long the length of bem is M ( x) x (x0 being the tip of the cntilever bem). 4 The strin energies cn be written s p M M ( x) dx x U U dx EI EI 4 96EI 0 0 nd U 0. The totl strin energy is therefore, U U + U + U 48EI 5

6 Strin energy relese rte cn be clculted s du G t d 6 EIt 0 4 The moment of inerti for the specified dimensions is, I m The crck will propgte if G Gc. Hence, we cn clculte the minimum lod min required to propgte the crck s given below. min G Gc 6EIt 6EItGc min 6 N 6

7 rob.4: Let us ssume tht the crck extends by d s shown bove Totl distnce d moved by the loding point is given by dδ Elstic extension + peeled off length ( ) ( ε ) ε L + d ε L + d d + d + ; ssuming unit width At Work done by the externl force is therefore, dwe dδ + d Chnge in strin energy is given by Hence, du U U L+ d ( L + d) A ( L) A A EA A EA d L dw dw du s e + d 7

8 Strin energy relese rte is given by G + d Note tht in this prticulr problem, du 8

9 rob.5: dδ Let the crck extend by d s shown bove. The incrementl displcement cn be written s d dδ, tking unit width Work done by externl force, d dwe dδ Chnge in strin energy is given by du U U Hence, Strin energy relese rte is In this cse, du. L+ d ( L + d) A ( L) A A EA A EA d s L dw dw du e G. d d 9

10 Compring rob.4 nd.5 As seen in the two problems bove, the strin energy relese rte nd the strin energy gined by the film re not equl for prob.4 nd re equl for prob.5. This is due to nture of the -δ curves shown in Figure 4. In prob.4, the nture of the loding mkes the system non-conservtive. It should lso be noted tht for the sme lod, prob.4 hs higher G nd hence it is esier to propgte the crck in prob.4 thn in prob.5. d d d L δ ( L + d) + d L ( L + d) δ rob.4 rob.5 0

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