Lec27 Maxwell and Voight-Kelvin models

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Maxwell and Voight-Kelvin Models

Transcript of Lec27 Maxwell and Voight-Kelvin models

  • Maxwellian models oversimplied viscoelas3c responses and thus give approximate predic3ons.

    FALL 2014: EMCH 315 1

    Creep

    o

    t

    Real

    Maxwell Model

    To overcome limitations of Maxwell model alternative arrangements of elements have been proposed: e.g. ______________________ Model with_____________________ arrangement of spring and dashpot.

    Most materials exhibit nonlinear creep rates

    models steady-state creep (i.e. constant creep rate )

    Voight-Kelvin parallel

  • Voight-Kelvin Material Model: beLer represents ___________________ _______________________ creep.

    Ini3ally the dashpot must carry the en3re force because the spring can carry a force only when extended. The force in the V-K model will be equal to the force in the dashpot________the force in the spring: hence ____________ {_______________________} Strains are no longer______________ as the dashpot will __________ the spring to have the same deforma3on thus deforma3on compa3bility: _____________

    FALL 2014: EMCH 315 2

    plus = s + d statically indeterminant

    independent constrain

    = s = d

    non-steady state transient

  • Equilibrium and compa3bility arguments can be rewriLen to convey that the response is ____________________.

    force equilibrium: (t) = s(t) + d(t)

    compa3bility: (t) = s(t) = d(t)

    stress-strain rela3onship for the spring:

    stress strain rela3onship for the dashpot:

    FALL 2014: EMCH 315 3

    _______________

    _______________

    _______________

    _______________

    s(t) = Es(t)

    dd/dt = (1/)d(t)

    s(t) = E(t)

    d/dt = (1/)d(t)

    time-dependent

  • To determine the governing cons3tu3ve equa3on, subs3tute the equa3ons for the spring and dashpot into the equilibrium equa3on. The governing stress-strain dieren3al equa3on: Stress depends not only on the strain, but also the strain rate Solu3on to the rst-order linear dieren3al equa3on (see next slide)

    FALL 2014: EMCH 315 4

    = d(t )dt + E(t )

    (t ) = E 1 e t / where = E

  • FALL 2014: EMCH 315 5

  • An instantaneous strain cannot be imposed as the dashpot must deform prior to the spring in the V-K model.

    FALL 2014: EMCH 315 6

    t

    Imposed condition

    o

    t

    Non-linear creep response

  • at t = 0 at time t as t

    An instantaneous strain ___________ be imposed because the model can only deform with 3me as the dashpot must deform prior to the spring in the V-K model.

    FALL 2014: EMCH 315 7

    t

    (t ) = E 1 e t /

    cannot

    = /E

    = 0

  • FALL 2014: EMCH 315 8

    t

    Now we remove the stress at some arbitrary time

    1

    models __________________ creep: ____________ down as time passes

    strain ___________ toward ____________ due to spring in parallel arrangement __________ on the dashpot o

    o

    p

    p

    t1

    Maxwell Model

    Voight-Kelvin

    transient slows

    decreases zero

    pulling

  • FALL 2014: EMCH 315 9

    t

    = E + ddt = 01

    ddt = E

    The time-dependent strain can be determined after we remove the stress at a new t=0, while = 1.

    cant account for __________________ for a constant strain, d/dt = 0 and E =

    = 1e t /

    stress relaxation

  • FALL 2014: EMCH 315 10

    Viscoelas3c Materials Their uses Asphalt Roads (moves with t and T) Silly puLy Toys Polymeric foams Seat Cushions, maLress topper Glass Dinnerware, labware Rubber Bands Hold things together Metals (T >Tm/2) W lament in light bulbs, turbine blades

    Plas3c/polymer items Sun glasses Rubber bumpers in cars Absorbs energy from impact Wood Sagging with 3me as viscoelas3c response Nylon Guitar String Rockin Out! Steel plates coated with viscoelas3c polymer

    Damping of vibra3onal energy

    Discs in human spines Slip disc, rupture of discs Skin Holds us together New born babys skull Protects the brain and other important parts

  • Examples and Applica3ons of Viscoelas3c Materials

    Creep and Recovery

    Materials which behave elas3cally at room temperature oien aLain signicant viscoelas3c proper3es when heated. Such is the case with metal turbine blades in jet engines, which reach very high temperatures and need to withstand very high tensile stresses. Conven3onal metals can creep signicantly at high temperatures and this has led to the development of creep-resistant alloys; turbine blades are now oien made of so called superalloys which contain some or all of nickel, cobalt, chromium, aluminum, 3tanium, tungsten and molybdenum.

    FALL 2014: EMCH 315 11

  • Examples and Applica3ons of Viscoelas3c Materials

    Creep and Recovery

    Disks in the human spine are viscoelas3c. Creep under normal body weight and get shorter with 3me. Lying down allows the spinal disks to recover, and thus most people are taller in the morning than in the evening.

    Skin 3ssue is viscoelas3c. Ex. Pinching the skin at the back of the hand; it takes 3me to recover back to its original at posi3on. The longer the skin is held in the pinched posi3on, the longer it takes to recover. The more rapidly it is pinched, the less 3me it takes to recover it behaves more elas3cally.

    FALL 2014: EMCH 315 12

  • Examples and Applica3ons of Viscoelas3c Materials

    Creep and Recovery

    Wood is viscoelas3c. Creep under the weight of the roof and garavity can take many decades or centuries to be no3ceable.

    Polymer foams used in seat cushions. Seat cushions creep to allow progressive conforma3on of the cushion to the body shape.

    FALL 2014: EMCH 315 13

  • Examples and Applica3ons of Viscoelas3c Materials

    Energy Absorp3on

    Viscoelas3c materials have the property of absorbing vibra3onal energy, damping the vibra3ons.

    Used in tall buildings which can vibrate when dynamically loaded by wind or earthquakes.

    Viscoelas3c materials are excellent impact absorbers. A peak impact force can be reduced by a factor of two if an impact buer is made of viscoelas3c, rather than elas3c materials. Elastomers (any of various substances resembling rubber) are highly viscoelas3c and make good impact absorbers.

    FALL 2014: EMCH 315 14

  • Examples and Applica3ons of Viscoelas3c Materials

    Energy Absorp3on Viscoelas3c materials are used in automobile bumpers, on computer drives to protect from mechanical shock, in helmets (the foam padding inside), in wrestling mats, etc. Viscoelas3c materials are also used in shoe insoles to reduce impact transmiLed to a person's skeleton. The car3lage at the ends of the femur and 3bia, in the knee joint, is a natural shock absorber. In an osteoarthri3c knee, the car3lage has degraded - some3mes the bones grind against each other causing great pain. Synthe3c viscoelas3c materials can be injected directly into an osteoarthri3c knee, enveloping car3lage-decient joints and ac3ng as a lubricant and shock absorber

    FALL 2014: EMCH 315 15

  • FALL 2014: EMCH 315 16

    The behavior of many viscoelastic materials lies between the behavior of a spring or a dashpot and is described by a combination of the Hookean (spring) and Newtonian (dashpot) elements. A couple of possibilities:

    time __________ response

    time __________ response

  • The models described so far provide a qualita3ve illustra3on of the viscoelas3c behavior of polymers.

    The Maxwell element is the most suited to represent uid polymers: the permanent ow predominates on the longer term, while the short-term response is elas3c.

    The Voight-Kelvin element, with an added spring and, if necessary, a dashpot, is beLer suited to describe the nature of a solid polymer.

    FALL 2014: EMCH 315 17

  • Both models, the Maxwell element and the Voight-Kelvin element, are limited in their representa3on of the actual viscoelas3c behavior.

    Maxwell Capable of modeling both

    stress relaxa3on and creep. Predicts that the stress relaxes

    to zero: In reality, the stress levels o at some nite value.

    Predicts that creep strain accumulates linearly with 3me; in reality, strain can accumulate non-linearly.

    Voight-Kelvin Predicts the more realis3c

    case of transient (non-linear) creep.

    Predicts recovery. Cannot impose instantaneous

    strain, thus, cannot model stress relaxa3on.

    FALL 2014: EMCH 315 18

  • elas3c (ability to stretch and return to its original length)

    plas3c (permanent deforma3on) viscoelas3c (elas3c and 3me-dependent response) creep (length changes over 3me under constant load)

    FALL 2014: EMCH 315 19

    General descrip3ons of mechanical responses are

  • E MCH 315 Mechanical Response of Engineering Materials

    Lecture 27 Creep I Chap. 10

    FALL 2014: EMCH 315 Lecture 27: Slide 20

  • FALL 2014: EMCH 315 21

  • FALL 2014: EMCH 315 22

    Creep Experiment

  • Deni3on: _______________ that increases as a func3on of _____________ and _____________ under ________________. For _____________ solids, service _____________________ exceed ___________ of Tm.

    FALL 2014: EMCH 315 23

    Permanent strain time temperature constant stress

    temperatures (absolute) 30 - 40%

    Response

    tr

    0

    t

    t

    Imposed condition

    TIME

    crystalline

  • Three stages of creep behavior: Stage I

    FALL 2014: EMCH 315 24