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  • 8/6/2019 Visco Plasticity Wikipedia


    Viscoplasticity 1


    Figure 1. Elements used in one-dimensional models of viscoplastic


    Viscoplasticity is a theory in continuum mechanics

    that describes the rate-dependent inelastic behavior of

    solids. Rate-dependence in this context means that the

    deformation of the material depends on the rate at

    which loads are applied.[1]

    The inelastic behavior that is

    the subject of viscoplasticity is plastic deformation

    which means that the material undergoes unrecoverable

    deformations when a load level is reached.

    Rate-dependent plasticity is important for transient

    plasticity calculations. The main difference between

    rate-independent plastic and viscoplastic material

    models is that the latter exhibit not only permanent

    deformations after the application of loads but continue

    to undergo a creep flow as a function of time under the

    influence of the applied load.

    The elastic response of viscoplastic materials can be

    represented in one-dimension by Hookean spring

    elements. Rate-dependence can be represented by

    nonlinear dashpot elements in a manner similar to

    viscoelasticity. Plasticity can be accounted for by

    adding sliding frictional elements as shown in Figure


    In the figure E is the modulus of elasticity, is the viscosity parameter and N is a power-law type parameter that

    represents non-linear dashpot [(d/dt)= = (d/dt)(1/N)]. The sliding element can have a yield stress (y) that is

    strain rate dependent, or even constant, as shown in Figure 1c.

    Viscoplasticity is usually modeled in three-dimensions using overstress models of the Perzyna or Duvaut-Lions


    In these models, the stress is allowed to increase beyond the rate-independent yield surface upon application

    of a load and then allowed to relax back to the yield surface over time. The yield surface is usually assumed not to be

    rate-dependent in such models. An alternative approach is to add a strain rate dependence to the yield stress and use

    the techniques of rate independent plasticity to calculate the response of a material[4]

    For metals and alloys, viscoplasticity is the macroscopic behavior caused by a mechanism linked to the movement of

    dislocations in grains, with superposed effects of inter-crystalline gliding. The mechanism usually becomes dominant

    at temperatures greater than approximately one third of the absolute melting temperature. However, certain alloysexhibit viscoplasticity at room temperature (300K). For polymers, wood, and bitumen, the theory of viscoplasticity is

    required to describe behavior beyond the limit of elasticity or viscoelasticity.

    In general, viscoplasticity theories are useful in areas such as

    the calculation of permanent deformations,

    the prediction of the plastic collapse of structures,

    the investigation of stability,

    crash simulations,

    systems exposed to high temperatures such as turbines in engines, e.g. a power plant,

    dynamic problems and systems exposed to high strain rates.
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    Viscoplasticity 2


    Research on plasticity theories started in 1864 with the work of Henri Tresca,[5]

    Saint Venant (1870) and Levy


    on the maximum shear criterion.[7]

    An improved plasticity model was presented in 1913 by Von Mises[8]

    which is now referred to as the von Mises yield criterion. In viscoplasticity, the development of a mathematical

    model heads back to 1910 with the representation of primary creep by Andrade's law.[9]

    In 1929, Norton[10]

    developed a one-dimensional dashpot model which linked the rate of secondary creep to the stress. In 1934,


    generalized Norton's law to the multi-axial case.

    Concepts such as the normality of plastic flow to the yield surface and flow rules for plasticity were introduced by

    Prandtl (1924)[12]

    and Reuss (1930).[13]

    In 1932, Hohenemser and Prager[14]

    proposed the first model for slow

    viscoplastic flow. This model provided a relation between the deviatoric stress and the strain rate for an

    incompressible Bingham solid[15]

    However, the application of these theories did not begin before 1950, where limit

    theorems were discovered.

    In 1960, the first IUTAM Symposium Creep in Structures organized by Hoff[16]

    provided a major development in

    viscoplasticity with the works of Hoff, Rabotnov, Perzyna, Hult, and Lemaitre for the isotropic hardening laws, and

    those of Kratochvil, Malinini and Khadjinsky, Ponter and Leckie, and Chaboche for the kinematic hardening laws.

    Perzyna, in 1963, introduced a viscosity coefficient that is temperature and time dependent.[17]

    The formulated

    models were supported by the thermodynamics of irreversible processes and the phenomenological standpoint. The

    ideas presented in these works have been the basis for most subsequent research into rate-dependent plasticity.


    For a qualitative analysis, several characteristic tests are performed to describe the phenomenology of viscoplastic

    materials. Some examples of these tests are[9]

    1. hardening tests at constant stress or strain rate,

    2. creep tests at constant force, and

    3. stress relaxation at constant elongation.
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    Viscoplasticity 3

    Strain hardening test

    Figure 2. Stress-strain response of a viscoplastic material at different strain rates.

    The dotted lines show the response if the strain-rate is held constant. The blue line

    shows the response when the strain rate is changed suddenly.

    One consequence of yielding is that as

    plastic deformation proceeds, an increase in

    stress is required to produce additional

    strain. This phenomenon is known as

    Strain/Work hardening.[18] For a

    viscoplastic material the hardening curves

    are not significantly different from those of

    rate-independent plastic material.

    Nevertheless, three essential differences can

    be observed.

    1. At the same strain, the higher the rate of

    strain the higher the stress

    2. A change in the rate of strain during the

    test results in an immediate change in thestressstrain curve.

    3. The concept of a plastic yield limit is no

    longer strictly applicable.

    The hypothesis of partitioning the strains by

    decoupling the elastic and plastic parts is

    still applicable where the strains are small,[3]


    where is the elastic strain and is the viscoplastic strain. To obtain the stress-strain behavior shown in blue in

    the figure, the material is initially loaded at a strain rate of 0.1/s. The strain rate is then instantaneously raised to

    100/s and held constant at that value for some time. At the end of that time period the strain rate is dropped

    instantaneously back to 0.1/s and the cycle is continued for increasing values of strain. There is clearly a lag between

    the strain-rate change and the stress response. This lag is modeled quite accurately by overstress models (such as the

    Perzyna model) but not by models of rate-independent plasticity that have a rate-dependent yield stress.

    Creep test

    Figure 3a. Creep test

    Figure 3b. Strain as a function of time in a creep test.
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    Viscoplasticity 4

    Creep is the tendency of a solid material to slowly move or deform permanently under constant stresses. Creep tests

    measure the strain response due to a constant stress as shown in Figure 3. The classical creep curve represents the

    evolution of strain as a function of time in a material subjected to uniaxial stress at a constant temperature. The creep

    test, for instance, is performed by applying a constant force/stress and analyzing the strain response of the system. In

    general, as shown in Figure 3b this curve usually shows three phases or periods of behavior[9]

    1. A primary creep stage, also known as transient creep, is the starting stage during which hardening of thematerial leads to a decrease in the rate of flow which is initially very high. .

    2. The secondary creep stage, also known as the steady state, is where the strain rate is constant.


    3. A tertiary creep phase in which there is an increase in the strain rate up to the fracture strain.


    Relaxation test

    Figure 4. a) Applied strain in a relaxation test and b)

    induced stress as functions of time over a short period

    for a viscoplastic material.

    As shown in Figure 4, the relaxation test[19]

    is defined as the stress

    response due to a constant strain for a period of time. In

    viscoplastic materials, relaxation tests demonstrate the stress

    relaxation in uniaxial loading at a constant strain. In fact, these

    tests characterize the viscosity and can be used to determine the

    relation which exists between the stress and the rate of viscoplastic

    strain. The decompositon of strain rate is

    The elastic part of the strain rate is given by

    For the flat region of the strain-time curve, the total strain rate is zero. Hence we have,

    Therefore the relaxation curve can be used to determine rate of viscoplastic strain and hence the viscosity of the

    dashpot in a one-dimensional viscoplastic material model. The residual value that is reached when the stress hasplateaued at the end of a relaxation test corresponds to the upper limit of elasticity. For some materials such as rock

    salt such an upper limit of elasticity occurs at a very small value of stress and relaxation tests can be continued for
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    Viscoplasticity 5

    more than a year without any observable plateau in the stress.

    It is important to note that relaxation tests are extremely difficult to perform because maintaining the condition

    in a test requires considerable delicacy.[20]

    Rheological models of viscoplasticityOne-dimensional constitutive models for viscoplasticity based on spring-dashpot-slider elements include


    perfectly viscoplastic solid, the elastic perfectly viscoplastic solid, and the elastoviscoplastic hardening solid. The

    elements may be connected in series or in parallel. In models where the elements are connected in series the strain is

    additive while the stress is equal in each element. In parallel connections, the stress is additive while the strain is

    equal in each element. Many of these one-dimensional models can be generalized to three dimensions for the small

    strain regime. In the subsequent discussion, time rates strain and stress are written as and , respectively.

    Perfectly viscoplastic solid (Norton-Hoff model)

    Figure 5. Norton-Hoff model for perfectly viscoplastic solid

    In a perfectly viscoplastic solid, also called the Norton-Hoff model of viscoplasticity, the stress (as for viscous

    fluids) is a function of the rate of permanent strain. The effect of elasticity is neglected in the model, i.e.,

    and hence there is no initial yield stress, i.e., . The viscous dashpot has a response given by

    where is the viscosity of the dashpot. In the Norton-Hoff model the viscosity is a nonlinear function of the

    applied stress and is given by

    where is a fitting parameter, is the kinematic viscosity of the material and .

    Then the viscoplastic strain rate is given by the relation

    In one-dimensional form, the Norton-Hoff model can be expressed as

    When the solid is viscoelastic.

    If we assume that plastic flow is isochoric (volume preserving), then the above relation can be expressed in the more

    familiar form[21]
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    Viscoplasticity 6

    where is the deviatoric stress tensor, is the von Mises equivalent strain rate, and are material

    parameters. The equivalent strain rate is defined as

    These models can be applied in metals and alloys at temperatures higher than one third of their absolute melting

    point (in kelvins) and polymers/asphalt at elevated temperature. The responses for strain hardening, creep, and

    relaxation tests of such material are shown in Figure 6.

    Figure 6: The response of perfectly viscoplastic solid to hardening, creep and relaxation


    Elastic perfectly viscoplastic solid (Bingham-Norton model)

    Figure 7. The elastic perfectly viscoplastic material.

    Two types of elementary approaches can be

    used to build up an elastic-perfectly

    viscoplastic mode. In the first situation, thesliding friction element and the dashpot are

    arranged in parallel and then connected in

    series to the elastic spring as shown in

    Figure 7. This model is called the

    Bingham-Maxwell model (by analogy with

    the Maxwell model and the Bingham model) or the Bingham-Norton model.[22]

    In the second situation, all three

    elements are arranged in parallel. Such a model is called a Bingham-Kelvin model by analogy with the Kelvin


    For elastic-perfectly viscoplastic materials, the elastic strain is no longer considered negligible but the rate of plastic

    strain is only a function of the initial yield stress and there is no influence of hardening. The sliding element

    represents a constant yielding stress when the elastic limit is exceeded irrespective of the strain. The model can be

    expressed as

    where is the viscosity of the dashpot element. If the dashpot element has a response that is of the Norton form

    we get the Bingham-Norton model
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    Viscoplasticity 7

    Other expressions for the strain rate can also be observed in the literature[22]

    with the general form

    The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 8.

    Figure 8. The response of elastic perfectly viscoplastic solid to hardening, creep and

    relaxation tests.

    Elastoviscoplastic hardening


    An elastic-viscoplastic material with

    strain hardening is described by

    equations similar to those for a

    elastic-viscoplastic material with

    perfect plasticity. However, in this case

    the stress depends both on the plastic

    strain rate and on the plastic strain

    itself. For an elastoviscoplastic

    material the stress, after exceeding the

    yield stress, continues to increase beyond the initial yielding point. This implies that the yield stress in the sliding

    element increases with strain and the model may be expressed in generic terms as


    This model is adopted when metals and alloys are at medium and higher temperatures and wood under high loads.

    The responses for strain hardening, creep, and relaxation tests of such a material are shown in Figure 9.

    Figure 9. The response of elastoviscoplastic hardening solid to hardening, creep and

    relaxation tests.

    Strain-rate dependent

    plasticity models

    Classical phenomenological

    viscoplasticity models for small strains

    are usually categorized into two


    the Perzyna formulation

    the Duvaut

    Lions formulation
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    Viscoplasticity 9

    JohnsonCook flow stress model

    The JohnsonCook (JC) model[23]

    is purely empirical and gives the following relation for the flow stress ( )

    where is the equivalent plastic strain, is the plastic strain-rate, and are material constants.

    The normalized strain-rate and temperature in equation (1) are defined as

    where is the effective plastic strain-rate of the quasi-static test used to determine the yield and hardening

    parameters A,B and n. This is not as it is often thought just a parameter to make non-dimensional.[34]

    is a

    reference temperature, and is a reference melt temperature. For conditions where , we assume that


    SteinbergCochranGuinanLund flow stress model

    The SteinbergCochranGuinanLund (SCGL) model is a semi-empirical model that was developed by Steinberg et


    for high strain-rate situations and extended to low strain-rates and bcc materials by Steinberg and Lund.[25]

    The flow stress in this model is given by

    where is the athermal component of the flow stress, is a function that represents strain hardening, is

    the thermally activated component of the flow stress, is the pressure- and temperature-dependent shear

    modulus, and is the shear modulus at standard temperature and pressure. The saturation value of the athermal

    stress is . The saturation of the thermally activated stress is the Peierls stress ( ). The shear modulus for

    this model is usually computed with the SteinbergCochranGuinan shear modulus model.The strain hardening function ( ) has the form

    where are work hardening parameters, and is the initial equivalent plastic strain.

    The thermal component ( ) is computed using a bisection algorithm from the following equation.[25]


    where is the energy to form a kink-pair in a dislocation segment of length , is the Boltzmann constant,

    is the Peierls stress. The constants are given by the relations

    where is the dislocation density, is the length of a dislocation segment, is the distance between Peierls

    valleys, is the magnitude of the Burgers vector, is the Debye frequency, is the width of a kink loop, and

    is the drag coefficient.
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    Viscoplasticity 10

    ZerilliArmstrong flow stress model

    The ZerilliArmstrong (ZA) model[27]



    is based on simplified dislocation mechanics. The general form of the

    equation for the flow stress is

    In this model, is the athermal component of the flow stress given by

    where is the contribution due to solutes and initial dislocation density, is the microstructural stress intensity,

    is the average grain diameter, is zero for fcc materials, are material constants.

    In the thermally activated terms, the functional forms of the exponents and are

    where are material parameters that depend on the type of material (fcc, bcc, hcp, alloys). The

    ZerilliArmstrong model has been modified by[37]

    for better performance at high temperatures.

    Mechanical threshold stress flow stress model

    The Mechanical Threshold Stress (MTS) model[28]



    ) has the form

    where is the athermal component of mechanical threshold stress, is the component of the flow stress due to

    intrinsic barriers to thermally activated dislocation motion and dislocation-dislocation interactions, is the

    component of the flow stress due to microstructural evolution with increasing deformation (strain hardening), (

    ) are temperature and strain-rate dependent scaling factors, and is the shear modulus at 0 K and ambient


    The scaling factors take the Arrhenius form

    where is the Boltzmann constant, is the magnitude of the Burgers' vector, ( ) are normalized

    activation energies, ( ) are constant reference strain-rates, and ( ) are constants.

    The strain hardening component of the mechanical threshold stress ( ) is given by an empirical modified Voce


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    Viscoplasticity 11

    and is the hardening due to dislocation accumulation, is the contribution due to stage-IV hardening, (

    ) are constants, is the stress at zero strain hardening rate, is the saturation threshold

    stress for deformation at 0 K, is a constant, and is the maximum strain-rate. Note that the maximum

    strain-rate is usually limited to about /s.

    PrestonTonksWallace flow stress model

    The PrestonTonksWallace (PTW) model[33]

    attempts to provide a model for the flow stress for extreme

    strain-rates (up to 1011

    /s) and temperatures up to melt. A linear Voce hardening law is used in the model. The PTW

    flow stress is given by


    where is a normalized work-hardening saturation stress, is the value of at 0K, is a normalized yield

    stress, is the hardening constant in the Voce hardening law, and is a dimensionless material parameter that

    modifies the Voce hardening law.

    The saturation stress and the yield stress are given by

    where is the value of close to the melt temperature, ( ) are the values of at 0 K and close to melt,

    respectively, are material constants, , ( ) are material parameters for the high

    strain-rate regime, and

    where is the density, and is the atomic mass.

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    Viscoplasticity 12


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    Article Sources and ContributorsViscoplasticity Source: Contributors: Alan Liefting, Alansohn, Badgernet, Bbanerje, BenFrantzDale, Chris the speller, D6, Drilnoth,

    Elkman, Gene Nygaard, Geniac, Headbomb, Hebrides, Hqb, J Milburn, JForget, JaGa, JamesAM, Jekshadow, Juliancolton, LilHelpa, Mentifisto, Michael Hardy, Mild Bill Hiccup, NDSUCE,

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