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4 Fracture mechanics

Date post: 09-Apr-2018
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    Fracture mechanics

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    Loading configuration Obreimoff: stable

    equilibrium No failure

    Griffith: unstableequilibrium Failure only for

    uniform tension

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    We need to characterize the drivingforce for fracture:

    Stress Intensity Factor, K (units: Pa m 0.5 ) Crack extension force, G (units: J m -2)

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    Crack displacement modes:

    Mode I Opening mode fractureMode II In plane shear fractureMode III Antiplane shear fracture

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    Irwins crack tip solutions Defines the

    shape of thestress fieldsurrounding the

    crack tip

    Polar or cartesiancoordinates

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    Stress intensity factor, K The stress surrounding a crack is proportional

    to one over the square root of the distance, r from the crack, hence

    The constant of proportionality is the stressintensity factor, K

    2 / 1

    r

    2 / 1= Kr

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    Stress intensity factor, K Depends on fracture displacement mode

    (I, II or III) and crack geometry

    cK cK

    cK

    yz III

    yx II

    yy I

    =

    =

    =

    y

    yy

    yx

    yz

    x

    z

    2c

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    Geometry term,

    2c 2c

    Straight crack = 1

    Penny-shaped crack = 2/

    cK

    cK

    cK

    yz III

    yx II

    yy I

    =

    =

    =

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    Irwins crack tip solutions give the shape of the stress field

    Stress intensity factor gives the magnitude

    of the stress field

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    Critical stress intensity factor (or fracture toughness), K c

    Where the stress intensity factor reaches the energy equilibrium- unstable propagation of the crack

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    Critical stress intensity factor, K c There is a K c for each displacement

    mode: K Ic K IIc K IIIc

    Units of K c are stress x crack length,MPa m 0.5

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    Typical values for K Ic ~0.7 MPa m 0.5 for glass

    ~1.0 MPa m 0.5 for marble ~1.5 MPa m 0.5 for granite ~2.5 MPa m 0.5 for basic rocks ~3.5 MPa m 0.5 for eclogite ~140 MPa m 0.5 for mild steel

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    Crack extension force, G Energy per unit area at the crack tip

    G is related to the stress intensity factor, K by:

    E

    K G I

    I

    2

    =

    (for plane stress and mode I fractures only)

    dC dU

    G m= G can be related to specific surface energy

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    Problems with the fracturemechanics approach

    Crack tip processes lower the crackextension force: distributed cracking plastic flow

    The crack behind the tip is assumed to be

    cohesionless ok for mode I fractures problematic for mode II and III

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    Measuring K Ic

    Easy to prepare Crack growth initially stable Critical crack length is constant no crack

    length measurements needed

    Chevron notch method

    -recommended by ISRM

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