Fracture Mechanics - Part 1

00:38:15
https://www.youtube.com/watch?v=G5mcTw-PLEI

概要

TLDRLecture iyi inotaura nezve fracture mechanics, iyo yakakosha mukunzwisisa kuti sei zvinhu zvichiitika mu brittle manner uye sei kukanganisa kunowedzera kuburikidza nemaburi kunokonzeresa kukundikana nekukurumidza. Semienzaniso, kuvakomana veLiberty neBoeing 737. Inotsanangura kuti fracture mechanics inobatsira sei pakudzivirira kukundikana uye inoongorora mechanics yematanda emusasa, kutyorwa mukati memigero mwaka mudomu nevari mode off. Inosuma Linear Elastic Fracture Mechanics (LEFM), iine fracture criteria uye proportionality pakati pe K1 nechubura cheInput. Zvokushandisa zvakaita se fracture energy rate, irving criterion, nenzira yekuwedzera, nonlinear fracture models inotaurwa. Inotaura nezve brittle to ductile transition uye nezve migumisiro yekushisa, kurodwa nekukurumidza uye constraint on fracture toughness.

収穫

  • 🏛️ Historical structures like the Minakshi Sundareshwar Temple show evidence of fracture mechanics in action.
  • ✈️ Aircraft fuselage failures illustrate the risk of crack propagation under stress.
  • 🚢 Liberty ships in WWII are classic examples of brittle fracture in metals.
  • 🔍 Fracture mechanics explains why actual material strength is often lower than theoretical predictions.
  • 📈 Linear Elastic Fracture Mechanics (LEFM) provides a framework for understanding crack propagation.
  • 🌡️ Temperature variations can cause a transition from ductile to brittle behavior.
  • ⚠️ Stress concentration due to defects leads to increased crack propagation risk.
  • 📉 Fast loading rates decrease fracture toughness, increasing brittleness.
  • 🔧 Understanding mode I, II, III crack propagation is key to predicting failures.
  • 📊 Fracture energy quantifies a material's resistance to crack growth.
  • 🏗️ Materials like concrete and ceramics are more prone to brittle fracture.
  • 🔬 New models beyond LEFM, including nonlinear fracture models, explore complex failure behaviors.

タイムライン

  • 00:00:00 - 00:05:00

    Muvhuro uyu unotaura nezve fracture mechanics, kukosha kwekuziva mabridges ari mu tempuro yekare anosanganisira akaoma nezvimwe. Mujekiseni izvi zvinokonzera mabridges kubatwa nenyaya dzekukanganiswa kwepasi uye zvakare kusimba kwezvakatikomberedza. Nyaya dzakasiyana-siyana dzinoratidzwa dzine zvivako zvakadambuka uye zvinoratidza ngozi yakakosha yezvimwe zvinovaka simende zvinodonha zvakadaro.

  • 00:05:00 - 00:10:00

    Fracture mechanics inosiyana nedzimwe nzira dzekuvakira nekutarisira zvinhu zvine makweme uye kuvapo kwezvikanganiso. Zviedzo zvinoratidzawo mutsauko uripo pakati pehukuru hwemabhuku uye hwaicho chemidziyo yekusimba. Izvi zvakanyanya nevezvinhu zvakaita seceramics uye girazi asi zvinogona kubata zvakare zvimwe zvinhu pasi pemamiriro akakodzera.

  • 00:10:00 - 00:15:00

    Mazano ehukuru hwematambo akatonyanya kutarisisa kumanikidza uye displacement fields mudenderedzwa regomba zvakakosha mu fracture mechanics. Kudzidza kwevastreams kunobatsira kunzwisisa kuwanda kwezviedzo. Kuvharirwa kwe geometry kunowedzera kumanikidzika uye kunogona kukonzera kubuda kwekunyatsonamira kwemadziro. Kuderedzwa kwekudzokorora kwakapihwa kuomesa kwechinhu zviri nyore. Griffith akaratidza kuti zvinhu zvine makweme anodzokororwa anoita kuti stress ikurumidze.

  • 00:15:00 - 00:20:00

    Humbowo hwemigwagwa yakaoma kana kuwanda kwechisarudzo chinokonzeresa mukati mezvinhu zvinounza failure nekuda kwekumanikidzirwa kwakanyanya panzvimbo yemakweme kana kupindera. Stress inosangana inowedzera padhuze pagomba uye inogona kukonzera failure kunyange kana kumanikidzwa kuri pazasi penguva chero iyo theoretical fracture stress. Mienzaniso yakasiyana-siyana yekuratidza fracture inotsanangurwa pamwe nekusuma connect k1 uye zvirinani zve fracture mechanics zvakakurukurwawo.

  • 00:20:00 - 00:25:00

    Mhando nhatu dzeyekupwanya dzokudzidza dzinosanganisira: tension, shear, tearing. Mode imba yekuvhara ndiyo inonyanya kune zvinhu zvine tensile forces uye zvinowanzo sungirwa nekuputika nekuda kwetensile forces. Mode inotsanangurwa. Ndiyo inonyanya kusangana nezvekutieta-kutieta kumusoro kurwendo kunodzima.

  • 00:25:00 - 00:30:00

    Mode imba yemunhu isina kurongeka ndiyo inoziva failure, kazhinji mune tensile kuona. Zvinhu zvisinganyanyo kuputika zviri pakati pekudonhedza anokweva muLCD. Konsa yemasangano musanganiswa wemasumbu edombo edombo edzizione asiririke. Kuvapo kwetambo yeatomic bonds padyo nec rack point inobatsira kujekesa zvehunyanzvi hunoonekwa ne fracture mechanics. Sangano rakachinjika re failure materials rine gravity re stress.

  • 00:30:00 - 00:38:15

    Linear elastic fracture mechanics inoratidza kuwanda kwefracture toughness uye kupatsanurwa kwe fracture failure factor kana kachut kesheye yemukati yemu body. Zvinetso zveAlternative nekuraramisa K1 zvakanyatsorongedzana ne geometrical property inotaridza ratendo pakuregera kuchikosha kweK1 Zvinonzi load pasina damage.

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ビデオQ&A

  • What is fracture mechanics?

    Fracture mechanics is a field of mechanics concerned with studying the propagation of cracks in materials and is crucial for understanding material failure, especially in brittle materials.

  • Why is fracture mechanics important?

    Fracture mechanics is important because it helps predict potential catastrophic failures in materials that are prone to cracking, ensuring safety and reliability.

  • What are some historical examples of fracture mechanics?

    Examples include the Minakshi Sundareshwar Temple with its granite columns, a Boeing 737 in Hawaii, and the failure of Liberty ships during World War II.

  • What factors contribute to brittle fracture?

    Brittle fracture can be caused by defects like microcracks, environmental influences, and stress concentrations. It can also occur due to material properties and applied stress.

  • What are the modes of crack propagation?

    Cracks can propagate in three pure modes: Mode I (opening mode), Mode II (sliding or in-plane shear), and Mode III (tearing or out-of-plane shear).

  • What is Linear Elastic Fracture Mechanics (LEFM)?

    LEFM is a concept where fracture criteria involve a single material parameter related to stress and energy distribution around crack tips, assuming materials behave in an elastic manner.

  • How are stress intensity factor and energy release rate related?

    The stress intensity factor (K1) and the energy release rate (G) are related in that they both can predict crack propagation but focus on local and global properties, respectively.

  • How does temperature affect fracture toughness?

    Fracture toughness generally decreases with decreasing temperature, making materials more brittle at lower temperatures.

  • Why do some materials fail in a brittle manner under certain conditions?

    Materials may fail in a brittle manner due to low temperatures, high loading rates, or geometrical constraints that cause stress concentrations, even if they are typically ductile.

  • What is the significance of fracture energy?

    Fracture energy is a measure of a material's resistance to crack propagation, with higher values indicating more ductile behavior and lower values leading to brittle behavior.

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  • 00:00:01
    [Music]
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    [Music]
  • 00:00:15
    welcome to lecture number nine of modern
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    construction materials today we're going
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    to talk about fracture mechanics and uh
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    fracture mechanics is important because
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    lot of materials fail in a brittle
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    Manner and when defects occur in them
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    then things get worse these de defects
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    propagate into cracks and you have
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    sudden catastrophic
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    failure I start with this picture of the
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    minakshi sundareshwar temple in Mad
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    which is about 600 to 800 years old and
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    uh most of it is made out of granite and
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    around this Temple tank we have many
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    many many columns of
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    stone some of these stone columns have
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    started to crack and they are being
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    replaced and this cracking is due to
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    settlement of the ground around the pond
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    and many other environmental
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    factors what else can fracture cause
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    brittle facture can cause sudden failure
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    as we see in this picture we cracks
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    occurred in the fuselage of this plane
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    while it was being uh while it was in
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    flight
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    and part of the fuselage just came
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    off most of the people surprisingly
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    survived because they had their seat
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    belts
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    on what happened was in this Boeing 737
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    297 that was flying in Hawaii was
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    that a crack propagated along the
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    section that was riveted after many
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    flights due to fatigue loading defects
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    had started to propagate
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    along the
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    circumference of the
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    fuselage where it was reted a crack
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    suddenly developed and the fuselage was
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    Tor away unfortunately some people were
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    killed during this
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    accident another case that we've already
  • 00:02:20
    discussed is that of the failure of the
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    Liberty ships and some T2 tankers that
  • 00:02:27
    were built in World War II and a similar
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    Behavior also happened in the Titanic or
  • 00:02:34
    what they say happened to the Titanic
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    when it sunk here the
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    weld that was used due to the cold
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    weather and fatigue became brittle and a
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    crack propagated this is what was seen
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    in one of these ships in the Boston
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    Harbor crack developed and suddenly this
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    ship broke into two
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    brittle facture can occur in smaller
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    levels smaller scales but with
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    disastrous consequences like in this
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    case of a pipeline failing in Australia
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    which led to the spilling of about 2
  • 00:03:13
    million lers of crude oil causing uh
  • 00:03:17
    ecosystem damage in the mangrove lined
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    Canal flowing into the Brisbane River in
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    2003 huge financial loss and huge
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    ecological loss
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    another very interesting case of a
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    catastrophic failure occurred in two of
  • 00:03:35
    the three steel gers of the Hanan bridge
  • 00:03:38
    in Milwaukee in the US in the year 2000
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    and here a generally ductile material
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    such as steel due to the triaxial
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    constraint that was brought about by the
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    bracing system led to brittle crack
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    propagation instead of a ductile failure
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    it was confined so much it was
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    constrained so much that the material
  • 00:03:59
    would not yield but failed due to
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    brittle
  • 00:04:04
    cracking and these are some of the
  • 00:04:06
    cracks that you see in the flung and in
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    the web
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    plate this occurred over a period of a
  • 00:04:12
    few hours and the bridge was then made
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    unserviceable more brittle materials are
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    more prone to
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    fracture these are pictures of the wooi
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    Bridge during the Chichi earthquake in
  • 00:04:32
    Taiwan in 1999 where the there was a
  • 00:04:36
    fault very near the bridge and the
  • 00:04:40
    bridge had a lot of lateral loads
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    resulting in severe cracking of the pi
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    large movements about 2 m that the
  • 00:04:49
    bridge was not designed for and the
  • 00:04:52
    concrete though it was reinforced failed
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    by
  • 00:04:58
    cracking so why fracture mechanics what
  • 00:05:01
    is different about fracture
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    mechanics conventional design procedures
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    that are based on the maximum stress
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    Criterion are not always
  • 00:05:12
    adequate failure theories and Concepts
  • 00:05:15
    that we discussed in the previous two
  • 00:05:17
    lectures always consider a material free
  • 00:05:20
    of defects and free of
  • 00:05:24
    discontinuities fracture mechanics now
  • 00:05:26
    determines failure based not only on the
  • 00:05:29
    applied
  • 00:05:30
    stress but also on the crack or the
  • 00:05:34
    floor that is present and brings in new
  • 00:05:36
    materials parameters like fracture
  • 00:05:39
    toughness instead of just the stress or
  • 00:05:41
    the strain
  • 00:05:43
    limit instead of the magnitude of stress
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    or
  • 00:05:46
    strain which are very difficult to
  • 00:05:49
    determine have very high values in any
  • 00:05:51
    case near the crack tip fracture
  • 00:05:54
    mechanics is concerned primarily with
  • 00:05:56
    the distribution of stresses the stress
  • 00:05:58
    fields and the the displacement fields
  • 00:06:01
    in the vicinity of the crack
  • 00:06:05
    tip this is particularly applicable to
  • 00:06:08
    brittle
  • 00:06:10
    materials like concrete Rock glass
  • 00:06:13
    Ceramics and so on but under certain
  • 00:06:15
    circumstances other materials also fail
  • 00:06:18
    in a brittle manner like we saw in some
  • 00:06:19
    of the examples of the pipes and the
  • 00:06:22
    gers fracture mechanics also helps us
  • 00:06:25
    understand why there is such a large
  • 00:06:27
    difference between the theor itical
  • 00:06:30
    material fracture strength that we can
  • 00:06:33
    calculate from the bond energy remember
  • 00:06:36
    when we looked at the Condon M diagram
  • 00:06:40
    linking the energy with the interatomic
  • 00:06:42
    distance there was a bonding energy and
  • 00:06:44
    we could also see what happens to this
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    bonding energy as the distance between
  • 00:06:49
    the atoms changes so if we were to
  • 00:06:52
    calculate the strength of the material
  • 00:06:54
    from those values of bond energy we
  • 00:06:57
    would find that that is much much higher
  • 00:07:00
    than what is actually measured in the
  • 00:07:02
    lab on a material probably two to three
  • 00:07:04
    orders of magnitude higher so there is a
  • 00:07:08
    big difference between the actual
  • 00:07:11
    strength and the theoretical strength
  • 00:07:13
    the actual strength being much lower
  • 00:07:16
    than the theoretical
  • 00:07:20
    strength the reason for this was
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    categorically stated by Griffith in 1920
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    when he concluded that any material any
  • 00:07:29
    real material has flaws microcracks or
  • 00:07:34
    some other defects and these defects
  • 00:07:37
    concentrate the stress so much that the
  • 00:07:41
    theoretical fracture stress is reached
  • 00:07:44
    in small points small locations in small
  • 00:07:49
    areas localized around these
  • 00:07:52
    defects so as the theoretical fracture
  • 00:07:55
    stress is reached at those points the
  • 00:07:58
    cracks start to propagate
  • 00:07:59
    even though the applied stress is less
  • 00:08:03
    than the theoretical fracture stress and
  • 00:08:06
    finally fracture
  • 00:08:10
    occurs now let us try to
  • 00:08:13
    visualize how stress concentration is
  • 00:08:15
    induced and for this we look at this
  • 00:08:17
    simple
  • 00:08:19
    diagram on the left we have a panel
  • 00:08:23
    subjected to a tensile load so these are
  • 00:08:26
    the applied stresses Sigma outside the
  • 00:08:30
    panel now imagine a Channel with water
  • 00:08:35
    flowing in
  • 00:08:36
    it okay so these lines would then be the
  • 00:08:39
    stream
  • 00:08:41
    lines water is Flowing similarly we can
  • 00:08:44
    visualize in the case of a panel under
  • 00:08:48
    stress that these would be the stress
  • 00:08:50
    flow lines so stress has to go from one
  • 00:08:53
    end to the other through the body so
  • 00:08:55
    these are can be called the stress flow
  • 00:08:57
    lines now going back to the Chan with
  • 00:09:00
    water flowing example think of what will
  • 00:09:02
    happen if you put your hand at one of
  • 00:09:05
    the edges you put your hand into the
  • 00:09:07
    flowing water and what would
  • 00:09:10
    happen you would have now water moving
  • 00:09:13
    faster along the edge of your hand
  • 00:09:16
    because you have blocked a part of the
  • 00:09:18
    flow and you will find some currents Ed
  • 00:09:20
    currents forming around the
  • 00:09:23
    edges so there is a concentration of
  • 00:09:25
    flow there is a faster flow occurring at
  • 00:09:28
    the edge of your and because you have
  • 00:09:30
    blocked part of the
  • 00:09:32
    channel now coming back to our stress
  • 00:09:35
    example a similar thing happens suppose
  • 00:09:38
    I have this panel and I make a cut in it
  • 00:09:41
    so I decrease the section I force the
  • 00:09:44
    flow of stress to go around this cut
  • 00:09:47
    that I have made or a notch that I have
  • 00:09:49
    made so here you have a zone of stress
  • 00:09:54
    concentration so the stress
  • 00:09:57
    concentration occurs at
  • 00:10:00
    here and we find that the stress here
  • 00:10:05
    can be actually much higher than the
  • 00:10:08
    stress applied far away from this defect
  • 00:10:11
    or the far field
  • 00:10:13
    stress this stress now can reach at a
  • 00:10:17
    small area the theoretical fracture
  • 00:10:20
    stress even though the stress applied
  • 00:10:23
    far away is much less and this causes
  • 00:10:25
    the propagation of the crack makes
  • 00:10:28
    things worse higher stress concentration
  • 00:10:30
    and the failure
  • 00:10:33
    continues so stress concentration can be
  • 00:10:36
    visualized as a concentration of stress
  • 00:10:39
    flow
  • 00:10:40
    lines due to some geometrical
  • 00:10:43
    discontinuity in the
  • 00:10:47
    continum now there are lot of equations
  • 00:10:50
    which describe the stresses that form
  • 00:10:53
    round
  • 00:10:55
    defects we can look at the case of an
  • 00:10:58
    elliptical defect or void in a plate say
  • 00:11:02
    an infinite plate subjected to a tensile
  • 00:11:05
    stress Sigma subt you have now a defect
  • 00:11:09
    or a crack
  • 00:11:10
    inside and this defect is now we assume
  • 00:11:13
    in the form of a of an
  • 00:11:16
    ellipse defined by a length of 2 c and a
  • 00:11:20
    width of 2
  • 00:11:22
    B the radius at the end is row
  • 00:11:30
    so let us see what happens to the
  • 00:11:32
    stresses in the vicinity of the crack
  • 00:11:34
    tip so we go here this is now the crack
  • 00:11:39
    tip with a semi length of c a semi
  • 00:11:44
    length of B this is as I told you the
  • 00:11:46
    radius of the crack tip and if we
  • 00:11:49
    consider a case of C = 3B the solution
  • 00:11:54
    given in Yang we find that the stress in
  • 00:11:58
    this direction Direction in the
  • 00:12:00
    direction of the pull caused by the
  • 00:12:02
    applied stress we find that at the crack
  • 00:12:06
    tip this stress is almost five times the
  • 00:12:10
    stress appli on the x-axis here we have
  • 00:12:14
    Sigma which is the stress at any point
  • 00:12:16
    divided by the applied stress Sigma
  • 00:12:18
    subt this is the sigma y y and this is
  • 00:12:21
    Sigma XX the stress this
  • 00:12:24
    direction so we find that Sigma y y is
  • 00:12:27
    Amplified almost five times at the
  • 00:12:29
    vicinity of this defect
  • 00:12:32
    tip Sigma XX also is quite high
  • 00:12:38
    in in here far away Sigma
  • 00:12:42
    xx and sigma y y will become smaller but
  • 00:12:46
    near the crack tip you have very high
  • 00:12:48
    values and what we see is that if the
  • 00:12:53
    defect becomes sharper and sharper that
  • 00:12:56
    is C becomes very large compared to row
  • 00:13:00
    the radius that is for a defect that is
  • 00:13:02
    very sharp tending to be a crack this
  • 00:13:07
    stress now reaches Infinity in the
  • 00:13:10
    theoretical sense that is there is a
  • 00:13:12
    singularity of stress in the crack tip
  • 00:13:16
    the crack tip stresses become
  • 00:13:23
    singular so this can also be understood
  • 00:13:26
    by looking at the lce or the atomic
  • 00:13:29
    bonds near a crack tip we have a far
  • 00:13:32
    stress that is
  • 00:13:35
    Sigma that is applied far away from the
  • 00:13:37
    defect or crack now the stresses have to
  • 00:13:40
    go around the crack tip like we saw in
  • 00:13:43
    these stress flow diagrams and it will
  • 00:13:46
    happen that the bond which is nearest to
  • 00:13:48
    the crack tip or forms the crack tip
  • 00:13:50
    will be very highly stressed lot of
  • 00:13:52
    stress goes through here the stress here
  • 00:13:55
    now is much more than Sigma and this
  • 00:13:59
    reaches a point of failure so this Bond
  • 00:14:01
    breaks and then the crack now has
  • 00:14:03
    advanced by one latis plane and next the
  • 00:14:07
    stress will be taken by this Bond CD and
  • 00:14:11
    so on so this crack now keeps
  • 00:14:13
    propagating through the body this is the
  • 00:14:16
    reason why even though when you have a
  • 00:14:19
    stress that is applied which is smaller
  • 00:14:22
    than the theoretical fracture stress
  • 00:14:24
    failure stress you have failure because
  • 00:14:26
    the bonds that are near the defect
  • 00:14:29
    take very high stresses and start
  • 00:14:32
    failing one by
  • 00:14:34
    one there are different ways that
  • 00:14:37
    fracture can occur and the three pure
  • 00:14:41
    modes into which all the other modes can
  • 00:14:44
    be put into our mode one where we have a
  • 00:14:50
    tension or opening mode this this is
  • 00:14:53
    called the opening
  • 00:14:55
    mode where you have a crack which is
  • 00:14:58
    propagating due to tensile forces so
  • 00:15:01
    something is opening this we are pulling
  • 00:15:03
    on either side and the crack opens this
  • 00:15:05
    is the most common mode of failure of
  • 00:15:07
    most materials due to tension the crack
  • 00:15:10
    develops perpendicular to the applied
  • 00:15:12
    tension and you have
  • 00:15:14
    failure now this is
  • 00:15:17
    a shearing
  • 00:15:20
    mode where you have a in plane here
  • 00:15:30
    we have a case where we are propagating
  • 00:15:33
    a crack by shearing this is not very
  • 00:15:36
    common one example is what you would
  • 00:15:39
    have seen in a direct Shear test in your
  • 00:15:41
    soils lab when you did a test on a rock
  • 00:15:44
    joint or in over
  • 00:15:47
    soil where two sides of the material are
  • 00:15:52
    slipped with respect to each other which
  • 00:15:54
    are sheared and a crack forms here along
  • 00:15:58
    this Shear plane without much opening so
  • 00:16:00
    this is called inlan Shear this has
  • 00:16:02
    applications in earthquake engineering
  • 00:16:05
    where there are slipping of
  • 00:16:08
    joints mode three is called the tearing
  • 00:16:10
    mode this is out of plain
  • 00:16:19
    Shear where you have something tearing
  • 00:16:22
    like when you tear a piece of paper the
  • 00:16:26
    material now twists
  • 00:16:29
    and you have shearing occurring out of
  • 00:16:32
    plane so there is some torsion here
  • 00:16:36
    occurring so these are the three
  • 00:16:38
    principal modes the most common is mode
  • 00:16:41
    one most of our materials that we
  • 00:16:43
    consider
  • 00:16:44
    are failing in mode one or in the
  • 00:16:47
    tensile mode with an opening crack let
  • 00:16:50
    us look more in detail in what happens
  • 00:16:53
    in mode one or for an opening crack what
  • 00:16:56
    we can imagine is now we have a crack
  • 00:16:59
    a sharp crack and let us see what
  • 00:17:03
    happens in front of this crack so the
  • 00:17:05
    crack is advancing this
  • 00:17:07
    way and if we look at a stress state of
  • 00:17:11
    a point ahead of the crack tip defined
  • 00:17:14
    by this distance from the crack tip R
  • 00:17:17
    and an angle Theta we would find that
  • 00:17:20
    the stresses become
  • 00:17:24
    infinite as it approaches as we approach
  • 00:17:28
    the crack tip
  • 00:17:30
    and this increase in stress is defined
  • 00:17:33
    by the factor R to the^ of minus
  • 00:17:37
    1/2 so there is a singularity in the
  • 00:17:40
    stresses the stresses increase
  • 00:17:45
    as R to the^ of -2
  • 00:17:49
    increases so when R becomes zero the
  • 00:17:52
    stresses become
  • 00:17:53
    infinite that's why it's called a
  • 00:17:55
    singular stress field and the equations
  • 00:17:57
    for this are given
  • 00:18:00
    here where we have Sigma X Sigma Y and
  • 00:18:03
    to XY the stresses ahead of the crack
  • 00:18:07
    tip the point that we are discussing is
  • 00:18:11
    defined in terms of R being the distance
  • 00:18:14
    from the crack tip to that point Theta
  • 00:18:17
    being the angle from the crack
  • 00:18:20
    plane and what we find here is we have a
  • 00:18:24
    term here which is k1 K1 being the stret
  • 00:18:29
    intensity
  • 00:18:30
    factor divided by 2 pi r to the^ of 1/2
  • 00:18:34
    and here you see the term R to ^ of
  • 00:18:37
    minus
  • 00:18:38
    1/2 coming
  • 00:18:40
    in for the stress Singularity as R
  • 00:18:44
    becomes smaller as we approach the crack
  • 00:18:48
    tip these values become larger and
  • 00:18:50
    larger and when R becomes zero these
  • 00:18:53
    stresses become
  • 00:18:55
    infinite the other
  • 00:18:57
    stresses in this case would be tox to y
  • 00:19:01
    equal to 0 and sigma Z the stress
  • 00:19:04
    perpendicular to the plane would be mu *
  • 00:19:07
    Sigma x + Sigma y mu being the pound
  • 00:19:10
    ratio what is important here is that we
  • 00:19:13
    have introduced the term K1 K1 for the
  • 00:19:17
    mode one which is the stress intensity
  • 00:19:20
    factor this is one of the terms that we
  • 00:19:22
    will continue to give importance as we
  • 00:19:25
    discussed fracture mechanics
  • 00:19:29
    K1 can be considered as a single
  • 00:19:31
    parameter describing the stress in the
  • 00:19:34
    displacement fields near the crack
  • 00:19:37
    tip to calculate it we generally
  • 00:19:40
    consider the material to be linear
  • 00:19:42
    elastic and both isotropic and
  • 00:19:44
    homogeneous that is the properties are
  • 00:19:46
    uniform and the material is behaving in
  • 00:19:49
    a linear elastic manner even though it
  • 00:19:51
    is fracturing this has limitations and
  • 00:19:54
    it can be incorrect for many materials
  • 00:19:57
    but generally we assume that the
  • 00:19:59
    approximations involved in the
  • 00:20:02
    application of linear elastic fracture
  • 00:20:04
    mechanics is reasonable we get
  • 00:20:06
    reasonable values and we can use these
  • 00:20:08
    values and modify them later to bring in
  • 00:20:11
    other nonlinearities
  • 00:20:13
    K1 has the dimension of stress times the
  • 00:20:16
    square root of
  • 00:20:18
    length such as MEAP Pascal square root
  • 00:20:21
    of
  • 00:20:25
    meter how we can apply or calculate K1
  • 00:20:29
    say we have a panel here with a c a
  • 00:20:32
    crack of length a and we have a far
  • 00:20:36
    field stress applied of
  • 00:20:38
    Sigma K1 will then be equal to Sigma
  • 00:20:41
    which is the applied stress times F
  • 00:20:43
    which is a function of the geometry say
  • 00:20:46
    the ratio between the width and the
  • 00:20:47
    length and crack length times square
  • 00:20:51
    root of Pi a a again being the crack
  • 00:20:54
    length so through tests of
  • 00:20:56
    this type of element we can get K1 we
  • 00:20:59
    can find out when failure occurs and
  • 00:21:01
    that would be the critical value of
  • 00:21:03
    K1 and if we know this we can calculate
  • 00:21:06
    K1 for any applied Sigma and
  • 00:21:10
    a in another form suppose you have a
  • 00:21:13
    beam we can put K1 in form of load p is
  • 00:21:18
    the applied load again we have a beam
  • 00:21:21
    now with a certain depth and a certain
  • 00:21:26
    width B is the width of the beam the
  • 00:21:29
    thickness of the beam D is the depth of
  • 00:21:31
    the beam and K1 could be put in the form
  • 00:21:33
    of P divided by
  • 00:21:35
    BD times the square root of d f alpha
  • 00:21:39
    alpha being the a by D Ratio or the
  • 00:21:43
    relative crack depth ratio that is a
  • 00:21:47
    this is now a and a divided by D is
  • 00:21:52
    Alpha F Alpha is a function that depends
  • 00:21:54
    on the span depth ratio this is now the
  • 00:21:57
    span
  • 00:22:02
    the span depth ratio defines this
  • 00:22:04
    function f Alpha and this can be
  • 00:22:06
    calculated from numerical analysis or
  • 00:22:10
    otherwise so K1 can be determined
  • 00:22:13
    depending on the geometry of the element
  • 00:22:16
    and the defect length or the crack
  • 00:22:20
    length until now what we've discussed is
  • 00:22:22
    called linear elastic fracture mechanics
  • 00:22:24
    or lefm the main features to summarize
  • 00:22:28
    are that the fracture Criterion involves
  • 00:22:32
    only one material
  • 00:22:34
    parameter which is related to the near
  • 00:22:37
    tip stress field and the energy of the
  • 00:22:41
    structure the stresses near the crack
  • 00:22:43
    tip have an R to the ^ of
  • 00:22:48
    min-2 Singularity and become infinite at
  • 00:22:51
    the crack
  • 00:22:53
    tip during fracture we assume that the
  • 00:22:56
    entire body remains elastic
  • 00:22:59
    and whatever energy is dissipated during
  • 00:23:02
    fracture occurs only at the crack tip
  • 00:23:05
    that is fracture occurs at a point which
  • 00:23:07
    is the CCP these are the main features
  • 00:23:09
    of linear elastic fracture mechanics and
  • 00:23:12
    these have limitations they cannot be
  • 00:23:13
    applied to all materials and all
  • 00:23:18
    situations in this framework when do we
  • 00:23:22
    consider the crack to
  • 00:23:25
    propagate crack propagation according to
  • 00:23:27
    linear elastic fracture mechanics occurs
  • 00:23:29
    when K1 is greater than or equal to K1 C
  • 00:23:33
    so this is the fracture
  • 00:23:37
    Criterion K1 if you remember comes from
  • 00:23:41
    the geometry of the element the stress
  • 00:23:46
    applied or the load applied and the
  • 00:23:50
    defect K1 C is a material property it's
  • 00:23:53
    a material parameter called the critical
  • 00:23:56
    stress intensity factor of fracture t
  • 00:23:58
    toughness it's a material property so on
  • 00:24:00
    the right side you have a material
  • 00:24:02
    property and the left you have a
  • 00:24:04
    parameter which depends on the load that
  • 00:24:05
    you apply the type of structure and the
  • 00:24:08
    defect that it has when K1 increases and
  • 00:24:13
    reaches a value of k1c or surpasses the
  • 00:24:16
    value of k1c the crack starts
  • 00:24:19
    propagating so this is the failure
  • 00:24:21
    Criterion according to linear elastic
  • 00:24:23
    fracture mechanics
  • 00:24:28
    we can look at a table of
  • 00:24:31
    values of fracture
  • 00:24:34
    toughness we see that the values of
  • 00:24:38
    materials that we always think of as
  • 00:24:41
    brittle like glass cement ice rocks
  • 00:24:46
    other Ceramics are lower the fracture
  • 00:24:48
    toughness is low so that's why these
  • 00:24:50
    materials crack rather than
  • 00:24:53
    yield on the other hand we have Metals
  • 00:24:56
    ductile Metals the pure metals are most
  • 00:24:58
    ductile right at the top where the
  • 00:25:01
    fracture toughness is so high that these
  • 00:25:03
    materials also always yield they do not
  • 00:25:05
    crack or
  • 00:25:11
    rupture again at the bottom we have
  • 00:25:13
    materials like epoxies which are very
  • 00:25:17
    brittle slightly higher up would be
  • 00:25:20
    other polymers which are not so brittle
  • 00:25:22
    like polypropylene and
  • 00:25:24
    nylon in the middle we have Composites
  • 00:25:28
    and and wood wood again parallel to
  • 00:25:30
    grain cracks more easily that is the
  • 00:25:33
    grain separate we'll look at this again
  • 00:25:35
    when we talk about
  • 00:25:37
    Timber perpendicular to the grain there
  • 00:25:40
    is a higher crack resistance the
  • 00:25:42
    fracture toughness is higher Composites
  • 00:25:45
    which are polymers reinforced with
  • 00:25:47
    different types of fibers are more at
  • 00:25:49
    the
  • 00:25:51
    top Boron fiber glass fiber carbon fiber
  • 00:25:54
    reinforced polymers
  • 00:26:01
    some materials that we can deal with in
  • 00:26:03
    civil
  • 00:26:05
    engineering are listed here we have like
  • 00:26:09
    we said before higher values of fracture
  • 00:26:12
    toughness for
  • 00:26:15
    Metals pressure vessel
  • 00:26:18
    steel 210 megapascal root of meter
  • 00:26:22
    copper 110 and so on so metals have
  • 00:26:25
    relatively higher fracture toughness
  • 00:26:27
    values
  • 00:26:29
    and if you see here the more brittle
  • 00:26:31
    materials have much lower almost two
  • 00:26:35
    orders of magnitude
  • 00:26:37
    lower glass cement paste concrete have
  • 00:26:43
    very low fracture toughness values so
  • 00:26:45
    they crack more easily and undergo
  • 00:26:48
    brittle failure nylon for reference here
  • 00:26:51
    would be intermediate slightly higher
  • 00:26:54
    than these brittle materials but still
  • 00:26:56
    not comparable to those
  • 00:26:58
    of
  • 00:27:04
    metals another way of considering
  • 00:27:07
    fracture involves the energy release
  • 00:27:09
    rate instead of the K1 which is the
  • 00:27:11
    stress intensity
  • 00:27:12
    factor we can look at fracture mechanics
  • 00:27:15
    in terms of the energy release
  • 00:27:18
    rate whenever a new crack is formed a
  • 00:27:22
    surface is formed and for this we
  • 00:27:25
    require energy so all crack extend
  • 00:27:29
    ition requires energy to be available
  • 00:27:32
    for the crack growth to occur this
  • 00:27:36
    energy should be sufficient to overcome
  • 00:27:38
    whatever resistance that comes from the
  • 00:27:40
    material itself and this material
  • 00:27:42
    resistance can come from the new energy
  • 00:27:45
    that is going to be created plastic work
  • 00:27:48
    or the work that goes into the yielding
  • 00:27:51
    and any other type of energy dissipation
  • 00:27:53
    like heating noise and so on all that is
  • 00:27:56
    some for form of energy that accompanies
  • 00:27:59
    a
  • 00:28:01
    crack we Define what is called the
  • 00:28:03
    energy release rate as the rate of
  • 00:28:06
    change in potential energy with crack
  • 00:28:10
    area we find what is the rate of change
  • 00:28:13
    of potential energy for every unit crack
  • 00:28:16
    area to be created this again for a
  • 00:28:19
    linear elastic
  • 00:28:21
    material what Irvin found in 1956 was a
  • 00:28:26
    fracture Criterion could be given in
  • 00:28:28
    this form where G is the energy release
  • 00:28:33
    rate this now depends on the stresses
  • 00:28:38
    the load applied and the body the shape
  • 00:28:42
    of the structure shape of the element
  • 00:28:44
    and the
  • 00:28:45
    defects this has to be greater than or
  • 00:28:48
    equal to a material parameter the
  • 00:28:51
    critical energy release rate or the
  • 00:28:54
    fracture energy so this also could be a
  • 00:28:57
    failure criter but these two failure
  • 00:29:00
    criteria that we've seen K1 greater than
  • 00:29:02
    or equal to K1 C and G greater than or
  • 00:29:05
    equal to GC are not
  • 00:29:07
    independent it was found that K1 which
  • 00:29:12
    characterizes the stress and
  • 00:29:13
    displacement fields near the crack tip
  • 00:29:16
    which is a local parameter we looking at
  • 00:29:18
    what happens near the crack tip and G
  • 00:29:21
    which quantifies the net change over the
  • 00:29:23
    body how the potential energy changes
  • 00:29:26
    due to this crack extension
  • 00:29:29
    which can be described as a global
  • 00:29:31
    Behavior are related that K and G K1 and
  • 00:29:34
    G are
  • 00:29:36
    related in irin in 1957 showed that
  • 00:29:40
    there is a unique relation between K1
  • 00:29:42
    and
  • 00:29:43
    G G is equal to K1 s/ e Prime okay so
  • 00:29:49
    what we looked at previously are not too
  • 00:29:51
    independent failure Criterion but they
  • 00:29:52
    are
  • 00:29:53
    related and K1 is a parameter which is
  • 00:29:57
    coming from the near tip stresses and
  • 00:30:00
    displacements and G gives us the global
  • 00:30:04
    change in
  • 00:30:05
    energy e Prime here is given as e for
  • 00:30:09
    plain stress and E / 1 - new squ for
  • 00:30:13
    plain strain new being the poison
  • 00:30:17
    ratio fracture energy can also be uh
  • 00:30:22
    determined and this is a chart again
  • 00:30:24
    from ashb and Jones showing typical
  • 00:30:27
    fracture values we find again that the
  • 00:30:30
    fracture energy for brittle materials
  • 00:30:33
    like Ceramics rocks cement glass are at
  • 00:30:36
    the bottom we have here the group of
  • 00:30:40
    brittle
  • 00:30:41
    materials and at the top we have the
  • 00:30:43
    pure ductile metals and the Alloys that
  • 00:30:47
    follow so again we find that these
  • 00:30:50
    materials fail in a brittle manner these
  • 00:30:52
    materials will not fail in a brittle
  • 00:30:53
    Manner and would rather yield polymers
  • 00:30:56
    are somewhere in the middle we have
  • 00:30:59
    epoxies at the bottom and less brittle
  • 00:31:03
    polymers at the top like polypropylene
  • 00:31:05
    and Composites in the again in the
  • 00:31:09
    middle the units of fracture energy or
  • 00:31:12
    the critical energy release rate are
  • 00:31:14
    Jews per square meter or Newton per
  • 00:31:18
    meter we had discussed brittle ductile
  • 00:31:21
    transition that can also be now related
  • 00:31:24
    with the fracture toughness and Fracture
  • 00:31:26
    energy we find like you see in the graph
  • 00:31:30
    on the left that fracture energy
  • 00:31:32
    increases with an increase in
  • 00:31:34
    temperature so fracture energy increases
  • 00:31:38
    with an increase in
  • 00:31:39
    temperature so a ductile metal when it
  • 00:31:42
    becomes colder will have a fracture
  • 00:31:45
    energy that is decreasing and can become
  • 00:31:48
    brittle and fail in a brittle M fracture
  • 00:31:52
    energy also increases when there is a
  • 00:31:54
    decrease in loading rate that is as the
  • 00:31:58
    loading rate increases a material will
  • 00:32:02
    have lower fracture energy it can become
  • 00:32:04
    brittle the fracture energy increases as
  • 00:32:08
    the loading rate decreases if you load
  • 00:32:11
    slower the fracture energy is higher
  • 00:32:13
    there is a less tendency for brittle
  • 00:32:15
    failure when you load very very slowly
  • 00:32:18
    when you load load very fast it is a you
  • 00:32:22
    see a decrease in fracture energy and a
  • 00:32:24
    higher tendency for brittle failure to
  • 00:32:27
    occur
  • 00:32:30
    a third case which shows a increase in
  • 00:32:33
    fracture energy is a decrease in
  • 00:32:36
    triaxiality triaxiality is the
  • 00:32:38
    confinement and the degree of constraint
  • 00:32:41
    as you remove
  • 00:32:43
    constraints the fracture energy
  • 00:32:48
    increases opposed to when you have
  • 00:32:51
    increasing triaxiality the fracture
  • 00:32:53
    energy decreases and you have brittle
  • 00:32:55
    failure this was like in the case of the
  • 00:32:57
    bridge that we saw from
  • 00:33:00
    Milwaukee where there was higher
  • 00:33:02
    triaxiality leading to brittle type of
  • 00:33:04
    failure even though the material is
  • 00:33:07
    characteristically
  • 00:33:09
    ductile so this is something interesting
  • 00:33:12
    that you should remember what causes a
  • 00:33:14
    brittle to ductile transition and how it
  • 00:33:17
    can be related to the fracture
  • 00:33:20
    energy whenever the fracture energy is
  • 00:33:22
    lower the tendency for brittle failure
  • 00:33:25
    is higher
  • 00:33:28
    when the fracture energy is high you can
  • 00:33:30
    have more of yield type failure ductile
  • 00:33:33
    failure rather than brittle
  • 00:33:36
    failure here you see how fracture
  • 00:33:39
    toughness varies with temperature for a
  • 00:33:42
    low alloy structural steel in this curve
  • 00:33:45
    you have on the y- axis fracture
  • 00:33:47
    toughness x-axis you have temperature
  • 00:33:50
    and you find that as temperature
  • 00:33:53
    decreases the fracture toughness
  • 00:33:55
    decreases the material is becoming less
  • 00:33:57
    ductile and more brittle as the
  • 00:34:00
    temperature is
  • 00:34:01
    decreasing this is the corresponding
  • 00:34:04
    trend for the eeld stress which keeps
  • 00:34:08
    decreasing as we increase the
  • 00:34:10
    temperature heel stress increases as the
  • 00:34:13
    temperature decreases whereas the
  • 00:34:14
    fracture toughness decreases as
  • 00:34:16
    temperature decreases the material
  • 00:34:18
    becomes Little Bit Stronger but more
  • 00:34:21
    brittle as the temperature drops and
  • 00:34:24
    this is what gave rise to the failures
  • 00:34:26
    of Steel elements in the ships that we
  • 00:34:29
    saw in very low
  • 00:34:33
    temperatures in terms of loading rate
  • 00:34:35
    when the loading rate becomes faster
  • 00:34:38
    here we have on the x-axis uh the
  • 00:34:41
    loading rate and Y AIS we have the
  • 00:34:43
    fracture toughness of Steel we find that
  • 00:34:46
    as we load faster the loading rate is
  • 00:34:48
    higher there the fracture toughness
  • 00:34:50
    decreases that is there is a tendency
  • 00:34:53
    for more brittle failure when the
  • 00:34:55
    loading rate is faster
  • 00:34:58
    and the failure would be more ductile
  • 00:35:01
    when you slowly load it this is the
  • 00:35:03
    reason that we intuitively always try to
  • 00:35:06
    break things very fast when we want them
  • 00:35:08
    to crack we apply a very fast load when
  • 00:35:11
    we want something to crack into
  • 00:35:13
    two where we are intuitively decreasing
  • 00:35:17
    the fracture toughness by increasing the
  • 00:35:19
    loading rate so we'll stop here with
  • 00:35:22
    this part we've introduced linear
  • 00:35:24
    elastic fracture mechanics we've looked
  • 00:35:26
    at the concepts and very interestingly
  • 00:35:28
    we've seen how at the tip of a crack we
  • 00:35:32
    can have very high stresses much higher
  • 00:35:34
    than what is applied far away and in the
  • 00:35:37
    case of linear elastic fracture
  • 00:35:39
    mechanics the stress can become singular
  • 00:35:43
    or infinite at the crack tip we then
  • 00:35:46
    went on to Define failure criteria in
  • 00:35:49
    terms of the stress intensity factor
  • 00:35:52
    K1 and the energy release rate G and in
  • 00:35:57
    both the cases fracture occurs when
  • 00:36:00
    these
  • 00:36:01
    parameters surpass or equal the
  • 00:36:04
    corresponding material
  • 00:36:06
    property fracture occurs when K1 is
  • 00:36:09
    equal to or greater than K1 C K1 c is
  • 00:36:13
    now the fracture toughness of material
  • 00:36:15
    parameter or alternatively we can see
  • 00:36:18
    that fracture will occur or crack
  • 00:36:21
    propagation will occur when G the energy
  • 00:36:24
    release rate is greater than or equal to
  • 00:36:26
    G Sub C which is the fracture toughness
  • 00:36:29
    Or the critical strain energy release
  • 00:36:31
    rate we also saw what happens to these
  • 00:36:35
    fracture parameters under conditions of
  • 00:36:38
    increasing temperature triaxiality and
  • 00:36:41
    loading rate and we found that under
  • 00:36:44
    these changing conditions there can be a
  • 00:36:48
    ductile to brittle transition or brittle
  • 00:36:49
    to ductile transition and this could
  • 00:36:52
    change the way the material fails a
  • 00:36:55
    ductile material could end up failing in
  • 00:36:57
    a Brit Manner and vice versa in the
  • 00:37:00
    second part of this lecture we'll go on
  • 00:37:02
    to see how different materials fail in
  • 00:37:06
    fracture what happens at the crack tip
  • 00:37:08
    what controls the crack
  • 00:37:11
    resistance we look at Metals we look at
  • 00:37:14
    polymers we look at concrete and we look
  • 00:37:16
    at some of the models which go beyond
  • 00:37:19
    just linear elastic fracture mechanics
  • 00:37:21
    these could be called nonlinear fracture
  • 00:37:23
    models and we'll see some applications
  • 00:37:27
    of these models and at the end of the
  • 00:37:29
    next lecture we'll also bring in the
  • 00:37:32
    effect of probability the variations in
  • 00:37:35
    the defects that we see in different
  • 00:37:38
    materials and how the probability of the
  • 00:37:41
    defect occurring can change the strength
  • 00:37:44
    that we get when you have a brittle
  • 00:37:46
    failure thank you
  • 00:37:53
    [Music]
  • 00:38:12
    [Music]
タグ
  • Fracture Mechanics
  • Brittle Failure
  • Crack Propagation
  • LEFM
  • Fracture Energy
  • Stress Intensity Factor
  • Temperature Effects
  • Brittle-Ductile Transition
  • Material Failure
  • Mechanical Engineering