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Viscoplasticity 1

Viscoplasticity

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

materials.

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

1.[2]

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

types.[3]

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

History

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

Saint Venant (1870) and Levy

(1871)[6]

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,

Odqvist[11]

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.

Phenomenology

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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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]

i.e.,

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

[3]the

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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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

tests.

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

model.

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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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

solid

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

types:[3]

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

al.[24]

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]

[26]

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]

[35]

[36]

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]

[38]

[39]

) 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

pressure.

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

law

where

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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

with

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

References

[1] Perzyna, P. (1966). "Fundamental problems in viscoplasticity".Advances in applied mechanics 9 (2): 244368.

[2] J. Lemaitre and J. L. Chaboche (2002) "Mechanics of solid materials" Cambridge University Press.

[3] Simo, J.C.; Hughes, T.J.R. (1998), Computational inelasticity

[4] Batra, R. C.; Kim, C. H. (1990). "Effect of viscoplastic flow rules on the initiation and growth of shear bands at high strain rates".Journal of

the Mechanics and Physics of Solids38 (6): 859874.

[5] Tresca, H. (1864). "Sur l'coulement des Corps solides soumis des fortes pressions". Comptes Rendus de l'Acadmie Sciences de Paris59:754756.

[6] Levy, M. (1871). "Extrait du mmoire sur les equations gnrales des mouvements intrieures des corps solides ductiles au dela des limites ou

l'lasticit pourrait les ramener leur premier tat".J Math Pures Appl16: 369372.

[7] Kojic, M. and Bathe, K-J., (2006), Inelastic Analysis of Solids and Structures, Elsevier.

[8] von Mises, R. (1913) "Mechanik der festen Korper im plastisch deformablen Zustand." Gottinger Nachr, math-phys Kl 1913:582592.

[9] Betten, J., 2005, Creep Mechanics: 2nd Ed., Springer.

[10] Norton, F. H. (1929). Creep of steel at high temperatures. McGraw-Hill Book Co., New York.

[11] Odqvist, F. K. G. (1934) "Creep stresses in a rotating disc."Proc. IV Int. Congress for Applied. Mechanics, Cambridge, p. 228.

[12] Prandtl, L. (1924) Proceedings of the 1st International Congress on Applied Mechanics, Delft.

[13] Reuss, A. (1930). "Berucksichtigung der elastichen, Formanderung in der Plastizitatstheorie".ZaMM10: 266.

[14] Hohenemser, K. and Prager, W., (1932), "Fundamental equations and definitions concerning the mechanics of isotropic continua,",J.

Rheology, vol. 3.

[15] Bingham, E. C. (1922) Fluidity and plasticity. McGraw-Hill, New York.

[16] Hoff, ed., 1962, IUTAM Colloquium Creep in Structures; 1st, Stanford, Springer.

[17] Lubliner, J. (1990) Plasticity Theory, Macmillan Publishing Company, NY.

[18] Young, Mindness, Gray, ad Bentur (1998): "The Science and Technology of Civil Engineering Materials," Prentice Hall, NJ.

[19] Franois, D., Pineau, A., Zaoui, A., (1993), Mechanical Behaviour of Materials Volume II: Viscoplasticity, Damage, Fracture and

Contact Mechanics, Kluwer Academic Publishers.

[20] Cristescu, N. and Gioda, G., (1994), Viscoplastic Behaviour of Geomaterials, International Centre for Mechanical Sciences.

[21] Rappaz, M., Bellet, M. and Deville, M., (1998), Numerical Modeling in Materials Science and Engineering, Springer.

[22] Irgens, F., (2008), Continuum Mechanics, Springer.

[23] Johnson, G.R.; Cook, W.H. (1983), "A constitutive model and data for metals subjected to large strains, high strain rates and high" (http://

www.lajss.org/HistoricalArticles/A constitutive model and data for metals.pdf),Proceedings of the 7th International Symposium on

Ballistics: 541547, , retrieved 2009-05-13

[24] Steinberg, D.J.; Cochran, S.G.; Guinan, M.W. (1980), "A constitutive model for metals applicable at high-strain rate",Journal of Applied

Physics51: 1498, doi:10.1063/1.327799

[25] Steinberg, D.J.; Lund, C.M. (1988), "A constitutive model for strain rates from 104

to 106

s1

" (http://cat.inist.fr/?aModele=afficheN),

Journal de Physique. Colloques 49 (3): 33, , retrieved 2009-05-13

[26] Hoge, K.G.; Mukherjee, A.K. (1977), "The temperature and strain rate dependence of the flow stress of tantalum" (http://www.

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