University Courses on Svalbard. AT-204 Thermo-Mechanical Properties of Materials, 3 vt, 9 ECTS EXAMINATION SUGGESTED SOLUTION (PROBLEM SETS 2 AND 3)

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1 Page 1 of 7 Univerity Coure on Svalbar AT-204 Thermo-Mechanical Propertie of Material, 3 vt, 9 ECTS EXAMINATION SUGGESTED SOLUTION (PROBLEM SETS 2 AND 3) May 29, 2001, hour: Reponible: Sveinung Løet, Phone: / Permitte ai: No written material. Calculator permitte. In cae of obviou mitake or omiion in the problem et, make your own aumption. You may anwer the quetion in one of the following language: Norwegian, Englih, Danih or Sweih. Rock mechanic (25%): Problem Set 1 a) Santone from the Svea Nor mine ha been tete in the laboratory with the following reult: Uniaxial compreive trength: c = 150 MPa Tenile trength: t = 15 MPa Contruct the Mohr failure envelope curve, coniering the curve to be a traight line. Fin from thi the coheion an internal friction of the rock material, the failure angle, the effective hear tre an the normal tre at the failure plane at failure in compreion. b) The antone wa ry an thawe uring teting. How woul firtly water aturation an then freezing of the rock pecimen affect the compreive trength in the two cae? c) In ome of the tunnel in the Svea Nor mine hear failure ( palling ) of the roof rock may be oberve. What i the reaon for thi? ) In the mine rock tre meaurement have been carrie out with the 2-D oortopper overcoring metho an by hyraulic fracturing. Explain briefly the principle for the two metho. Ice mechanic (25%): Problem Set 2 a) Aume continuum behaviour of polycrytalline ice an a total train given by tot e v ε = ε + ε + ε (1) where the upercript refer to e elatic elaye elatic v vicou What procee in the crytal lattice oe each of the three term correpon to? Elatic: The elatic eformation are caue primarily by change in intermolecular itance uner the applie tre. The eformation of the lattice i recoverable. Delaye elatic: Grain bounary liing, time-epenent an recoverable (reverible).

2 Page 2 of 7 Vicou: The effect of the movement of ilocation in the crytal. Irreverible. b) Aume that ice i loae at t = 0 with a contant loa P (preure ). The train veru time i hown in Fig. 1. What term in Eq. (1) refer to the behaviour hown in Fig. 1. Sketch the rheological repreentation of thi behaviour. What i thi unit commonly calle? The econ term, elaye elatic. The rheological repreentation i a pring an a ahpot in parallel. Thi unit i commonly calle a Kelvin-Vogt unit. c) Show that the train in b) can be expree by ( E t ) η where - tre E - elatic moulu (Young moulu) η- vicoity t - time ε 10 5 ε = (1 e ) (2) E t () Fig. 1. Strain veru time. The tre i contant an Further, we have: = Eε = ηε ε = ε, = + = Eε + ηε = ( E+ η ) ε t or η ε + Eε = t Thi i a linear firt-orer ifferential equation, with the following olution when we aume that for t = 0, ε = 0 ε = E ( E/ η ) t (1 e )

3 Page 3 of 7 q.e.. ) Etimate the vicoity of thi material bae on Eq. (2) an value taken from Fig. 1 when auming E = 9 GPa, = 800 kpa (a preure below the cruhing trength of ice). ε E ( E/ η) t = 1 e ( E/ η ) t ε E e = 1 E ε E tln e= ln(1 ) η Et η = ε E ln(1 ) 9000MPa 60 = MPa ln(1 ) 0.8MPa 9000MPa 60 = = MPa e) Compute the train in the ice after 1 minute an after 60 minute? What i the average train rate in thi perio? How o you explain thi train from the crytal lattice point of view? ε (60 ) = (1 e ) = ε (3600 ) = The average train rate for thi 59 minute perio i imply taken a the average between thee two value: 7 1 ε (60 ) = ε (3600 ) = ε avg = The train of the crytal i explaine from bounary gliing. Ice loa (25%): Problem Set 3 a) Explain why an how ice trength, tructure form an type of ice feature influence the loa ice may exert on a tructure? Ice trength: = ( T, poroity, alinity, grain ize, train rate) An increae in the ice temperature, poroity an grain ize ecreae the trength of the ice. For a low train rate (< ) the ice behave in a uctile manner. There i enough time to reorganie bon in the crytal lattice. For a relatively high train rate (> ) the ice behave in a brittle manner an the trength of the ice ecreae with increaing train rate.

4 Page 4 of 7 For intance, fracturing occur intea of creeping. In the tranition zone, a train rate of about , the ice trength i at it maximum. Structure form: For a loping tructure, the ice normally fail in bening. The bening trength of ice i le than the cruhing trength. For a loping tructure the force from the ice that act on the tructure can be plit into a vertical an horizontal component. Very often the key parameter in eign i the horizontal loa exerte by the ice on the tructure. Thi force component will be le in cae of a loping tructure. Ice feature: Firt-year ice i more aline an porou than multi-year ice. That i the major reaon why firt-year ice exert le loa on a tructure. (in aition the ice thickne of firt-year ice i le etc.). b) Explain the meaning of the main cenario of ice/tructure interaction an ecribe conition where they can be met? What cenario control the maximum loa an why? Scenario of interaction: Limit tre (cruhing of ice at the tructure urface) Limit momentum (the momentum of the ice feature i not ufficient to provie cruhing along the whole uptream ie of the tructure). Thu the loa i le than in the cruhing cae. Limit force (the ice i failing away from the tructure, the riving force i not ufficient to provie cruhing at the tructure urface). (Limit) plitting (the ice fail in plitting, normally when the ice i relatively warm). All thee conition can be met in the rift ice. The maximum loa i normally met in the cruhing cenario. The following figure can be ueful for a) an b), but i not irectly requete. ICE LOADS Ice feature Ice propertie Scenario of interaction Structure geometry Level Rafte Rige Stamucha Iceberg Lanfat Pack ice Concentr. Spee Dimen. Thickne Strength Compreion/tenion Temperature Salinity Poroity Morphology Strain rate Lateral compreion Limit tre Limit momentum Limit force Splitting Vertical Sloping Aheion Material Roughne Non-homogenity

5 Page 5 of 7 c) Korzhavin formula. Mention avantage an iavantage with it? What o the inentation an contact factor repreent an how o thee factor epen on ice propertie an apect ratio? The Korzhavin equation rea: F = IK1K2 c Dh where I K 1 K 2 D h - inentation factor, - contact factor, - hape factor. - tructure iameter, - ice thickne. NB: Here value are not require from the tuent. The inentation factor I involve the effect of apect ratio (D/h) an the ice grain aniotropy. [The inentation factor i in the range of 3.5 (high apect ration) to 4.5 (low apect ratio) for columnar ice an 1.2 (high apect ratio) to 3 (low apect ratio)]. The apect ratio eal with confinement of the ice affecte by the tructure (take into account the ifference between the real an the unconfine tre itribution). K 1 repreent the amount of contact between the tructure an the ice an i low for col, brittle ice an cloer to 1 for warm, uctile ice. K 2 i normally 1 for a rectangular tructure an 0.9 for a circular tructure. [Korzhavin formula i bae on the ice loa meaurement on mall pile. The iameter of thee pile wa everal time le than the offhore tructure iameter. Therefore, the cale of obervation i inufficient. When cale increae, the ice trength ecreae ignificantly a it wa hown above. Korzhavin wa one of the firt who oberve cale effect.] ) What are the key element of probabilitic metho for ice loa calculation? Why i the probabilitic metho in ice loa calculation ue? In brief: 1. Ditribution of the input parameter 2. A formula to calculate the loa after 3. Ue a ranom generator to pick the ingle input parameter 4. Calculate the loa 5. Start on 3) again an run 4). Do thi looping, ay 1000 time, an a loa itribution i achieve. The metho i ue to get a more repreentative picture of the loa. Example: Say that you ue the Korzhavin eq. to calculate the loa. Normally the ice i tronget in February an it i thicket in April. It woul then be wrong to ue the highet trength value an the thicket ice when you calculate the loa. Thi kin of wrong repreentation i avoie when a probabilitic approach i ue. Problem Set 4 Deformation & fracture of metallic material (25 %)

6 Page 6 of 7 a) Dicu the ifference between the tre-train curve hown in Fig. 2. Fig. 2. b) Aume that the true tre-train curve for a material i efine by e.g. the equation I AA n where A an n are contant. Explain how the (UTS) ultimate tenile trength can be etermine from thi equation. c) Determine the UTS for a material with A = 1000 MPa, n=0.15. ) Dicu (briefly) the ifference in yieling behaviour of a contructional teel uner loaing ituation that give uniaxial, biaxial an triaxial tree. e) With reference to quetion ) match the tet pecimen in Fig. 3 to the tre-train curve. All pecimen are mae of the ame material. f) Show in a ketch what the tre-train curve woul look like if pecimen IV ha a harp crack intea of a eep notch. Stre A B C D E F Fig. 3 Average train

7 Page 7 of 7 g) Explain how you woul calculate the fracture trength for the pecimen with the harp crack. What material propertie o you nee, what requirement nee to be atifie before the fracture trength can be calculate? h) Show in a ketch how the fracture trength of a teel varie with temperature. What other material property can be ue to how the uceptability to brittle failure?