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In contrast to elastic theories, plasticity is the behavior, where the material would undergo unrecoverable deformations due to the response of applied forces. There are several models for plasticity. Viscoplasticity, for instance, is one of the famous models of plasticity. It is defined as a rate-dependent plasticity model. Rate dependent plasticity is important for (high-speed) transient plasticity calculations. It should be used, however, in combination with a plasticity law. In that aspect, viscoplastic solids exhibit 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 (equilibrium is impossible). Similar to viscoelasticity, such materials are represented by a combination of non-linear dashpot elements, Hookean spring elements and sliding frictional elements. As shown in Fig1, E is the modulus of elasticity, λ is the viscosity parameter and N is another parameter that represents non-linear dashpot [σ(dε/dt)= σ = λ(dε/dt)^(1/N)]. It is good to mention that the sliding element could have a yield stress that is strain rate dependent, or even constant, as shown in Fig 1c. In that aspect, dashpot contributes to the viscosity of the material, while the spring contributes to the elastic behavior. Some rheological models will be presented throughout this text.^[1] In this text, a brief overview is presented about the viscoplasticity theory, starting with the background history, and ending with some proposed models with their response to creep, stress relaxation and strain hardening tests.

Contents 1 Background 2 Domain of validity and use 3 Phenomenological aspects 3.1 Strain hardening test 3.2 Creep test 3.3 Relaxation test 4 Rheological models 4.1 Perfectly viscoplastic solid 4.2 Elastic perfectly viscoplastic solid 4.3 Elastoviscoplastic hardening solid 5 Strain-rate dependent plasticity models 5.1 Perzyna formulation 5.2 Duvaut–Lions formulation 5.3 Flow stress models 5.3.1 Johnson–Cook flow stress model 5.3.2 Steinberg–Cochran–Guinan–Lund flow stress model 5.3.3 Zerilli–Armstrong flow stress model 5.3.4 Mechanical threshold stress flow stress model 5.3.5 Preston–Tonks–Wallace flow stress model 6 See also 7 References

[edit] Background

The research work about plasticity started in 1864, while studying the maximum shear criterion. Another criterion is the von Mises theory. In 1930, Prager and Hohenemser proposed the first model for the maximum shear criterion. This was a generalization of Bingham model. However, the application of these established theories did not begin before 1950, where limit theorems were discovered. In viscoplasticity, the development of a mathematical model heads back to 1910 with the representation of primary creep by Andrew’s law. In 1929, Norton developed a model which links the rate of secondary creep to the stress. Basically, that simple model was represented by a dashpot. In 1934, Odqvist’s generalization, of Norton’s law to the multi-axial case, was established.^[1]

In 1960, the first IUTAM Symposium “Creep in Structures” organized by Hoff provided a great 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. For instance, Perzyna, in 1963, introduced a viscosity coefficient that is temperature and time dependent.^[2] The formulated models were supported by the thermodynamics of irreversible processes and the phenomenological standpoint.^[1] Furthermore, several constitutive laws have been presented, taking into account the material characteristic such as: perfectly viscoplastic, viscoplasticity with isoropic or kinematic hardening etc.

In general, viscoplasticity theories are considered 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

[edit] Domain of validity and use

In contrast to plasticity, the viscoplasticity theory explains the flow by creep, which is time dependent. For metals and alloys, it corresponds to a mechanism linked to the movement of dislocations in grains deviation, polygonization-with superposed effects of inter-crystalline gliding. The mechanism begins to arise as soon as the temperature is greater than approximately one third of the absolute melting temperature. However, certain alloys exhibit viscoplasticity at room temperature (300K). Time effect must be taken into consideration as well. For polymers, wood, and bitumen, the theory of viscoplasticity must be used as soon as the load has passed the limit of elasticity or viscoelasticity.^[1] Some materials, where viscoplasticity is highly required, at high strain rates, are:

- Polymers

- Wood and bitumen

- Metals exposed to extremely high temperatures

[edit] Phenomenological aspects

For a qualitative analysis, several characteristic tests are performed to describe the phenomenology of viscoplastic materials. Some examples of these tests are (i) hardening tests at constant stress or strain rate, (ii) creep tests at constant force, and (iii) stress relaxation at constant elongation.

[edit] Strain hardening test

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.^[3]

For a viscoplastic material the hardening curves are not significantly different from those of plastic material. Nevertheless, three essential differences are apparent:

- At the same strain, the higher the rate of strain is, the higher the stress will be.

- A change in the rate of strain during the test results in an immediate change in the stress–strain curve. (See Fig 2b)

- The concept of a plastic yield limit is no longer strictly applicable.

The hypothesis of partitioning the strains by decoupling is still applicable in most cases (where the strains are small):

ε = ε_e + ε_p

where:

ε_e = the linear elastic strain.

ε_p = the viscoplastic strain.

[edit] Creep test

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 Fig3. 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. See Fig 3a. In general, this curve usually shows three phases or periods of behavior: ^[3]

- A primary creep stage, also known as transient creep, is the starting stage during which hardening of the material leads to a decrease in the rate of flow which is initially very high. (0 ≤ ε ≤ ε₁)

- The secondary creep stage, also known as the steady state, is where the strain rate is constant. (ε₁ ≤ ε ≤ ε₂)

- A tertiary creep phase in which the usual increase in the strain rate up to the fracture strain. (ε₂ ≤ ε ≤ ε_R)

[edit] Relaxation test

As shown in Fig 4, the relaxation test^[1] 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 viscoplasticity strain. Hence, the decompositon of strain rate is shown as follows:

dε/dt = dε_e/dt + dε_p/dt

Where the linear elasticity dε_e/dt = −(dσ/dt)/E. Thus, each point on the relaxation curve σ(t) gives the stress and rate of viscoplastic strain.

For a total of zero strain rate, subsituting both equations we get

dε_p/dt = −(dσ/dt)/E.

[edit] Rheological models

Several constitutive models for viscoplasticity have been developed. The most famous models are the following:^[1]

- Perfectly viscoplastic solid,

- Elastic perfectly viscoplastic solid

- Elastoviscoplastic Hardening Solid

Before illustrating the earlier defined models, it is good to note that:

- In series connection, the strain is additive while the stress is equal in each element.

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

[edit] Perfectly viscoplastic solid

Considering the simple Norton Model, the perfectly viscoelastic solid is represented in Fig5. The rate of stress (as for viscous fluids) is a function of the rate of permanent strain :

σ(dε/dt)= σ = λ(dε/dt)^(1/N)

N = A fitting parameter, when N = 1.0, the solid is viscoelastic. These models could be applied in metals and alloys at temperatures higher than one third of their absolute melting point (in Kelvin) and polymers/asphalt at elevated temperature. λ = The kinematic viscosity of the material. The responses for strain hardnenning, creep, and relaxation tests of such material are shown in Fig 7.^[1]

[edit] Elastic perfectly viscoplastic solid

Using Bingham–Norton model (Bingham plastic), the elasticity is no longer considered negligible but the rate of plastic strain is only a function of the stress . There is no influence of hardening. See Fig 6 for such model. The sliding material shown in the following figure represents a constant yielding stress when the elastic limit is exceeded irrespective of the strain.

|σ| < σ_y → ε = ε_e =σ⁄E

|σ| ≥ σ_y → ε = ε_e + ε_p

dε/dt =(dσ/dt) ⁄ E + f(σ)

The responses for strain hardnenning, creep, and relaxation tests of such material are shown in Fig 8.^[1]

[edit] Elastoviscoplastic hardening solid

This is the most complex schematic representation because the stress depends on the plastic strain rate and on the plastic strain itself. The elastoviscoplastic material is similar to the elastic perfectly viscoplastic solid, but after exceeding the yield stress, the stress starts to increase beyond the yielding point (Strain Hardening). Basically, the yield stress in the sliding element increases with strain, as shown below: |σ| < σ_s → ε = ε_e = σ ⁄ E |σ| ≥ σ_s → ε = ε_e + ε_p σ = Eε_e = f(ε_p, dε_p/dt) This model is adopted when metals and alloys are at medium and higher temperatures and wood under high loads.^[1] The responses for strain hardnenning, creep, and relaxation tests of such material are shown in Fig 9.

[edit] Strain-rate dependent plasticity models

Classical, phenomenological, viscoplasticity models for small strains are usually categorized into two types:^[4]

• the Perzyna formulation
• the Duvaut–Lions formulation

[edit] Perzyna formulation

In the Perzyna formulation the plastic strain rate is assumed to be given by a constitutive relation of the form

where f(.,.) is a yield function, is the Cauchy stress, is a set of internal variables, τ is a relaxation time.

[edit] Duvaut–Lions formulation

The Duvaut–Lions formulation is equivalent to the Perzyna formulation and may be expressed as

where is the closest point projection of the stress state on to the boundary of the region that bounds all possible elastic stress states.

[edit] Flow stress models

The quantity represents the evolution of the yield surface. The yield function f is often expressed as an equation consisting of some invariant of stress and a model for the yield stress (or plastic flow stress). An examaple is von Mises or J₂ plasticity. In those situations the plastic strain rate is calculated in the same manner as in rate-independent plasticity. In other situations, the yield stress model provides a direct means of computing the plastic strain rate.

Numerous empirical and semi-empirical flow stress models are used the computational plasticity. The following temperature and strain-rate dependent models provide a sampling of the models in current use:

1. the Johnson–Cook model
2. the Steinberg–Cochran–Guinan–Lund model.
3. the Zerilli–Armstrong model.
4. the Mechanical Threshold Stress model.
5. the Preston–Tonks–Wallace model.

The Johnson–Cook (JC) model ^[5] is purely empirical and is the most widely used of the five. However, this model exhibits an unrealistically small strain-rate dependence at high temperatures. The Steinberg–Cochran–Guinan–Lund (SCGL) model ^{[6] [7]}is semi-empirical. The model is purely empirical and strain-rate independent at high strain-rates. A dislocation-based extension based on ^[8] is used at low strain-rates. The SCGL model is used extensively by the shock physics community. The Zerilli–Armstrong (ZA) model ^[9] is a simple physically-based model that has been used extensively. A more complex model that is based on ideas from dislocation dynamics is the Mechanical Threshold Stress (MTS) model ^[10]. This model has been used to model the plastic deformation of copper, tantalum ^[11], alloys of steel ^{[12] [13]}, and aluminum alloys ^[14]. However, the MTS model is limited to strain-rates less than around 10⁷/s. The Preston–Tonks–Wallace (PTW) model ^[15] is also physically based and has a form similar to the MTS model. However, the PTW model has components that can model plastic deformation in the overdriven shock regime (strain-rates greater that 10⁷/s). Hence this model is valid for the largest range of strain-rates among the five flow stress models.

[edit] Johnson–Cook flow stress model

The Johnson–Cook (JC) model ^[5] is purely empirical and gives the following relation for the flow stress (σ_y)

where is the equivalent plastic strain, is the plastic strain-rate, and A,B,C,n,m are material constants.

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

where is a user defined plastic strain-rate, T₀ is a reference temperature, and T_m is a reference melt temperature. For conditions where T ^* < 0, we assume that m = 1.

[edit] Steinberg–Cochran–Guinan–Lund flow stress model

The Steinberg–Cochran–Guinan–Lund (SCGL) model is a semi-empirical model that was developed by Steinberg et al.^[6] for high strain-rate situations and extended to low strain-rates and bcc materials by Steinberg and Lund^[7]. The flow stress in this model is given by

where σ_a is the athermal component of the flow stress, is a function that represents strain hardening, σ_t is the thermally activated component of the flow stress, μ(p,T) is the pressure- and temperature-dependent shear modulus, and μ₀ is the shear modulus at standard temperature and pressure. The saturation value of the athermal stress is σ_max. The saturation of the thermally activated stress is the Peierls stress (σ_p). The shear modulus for this model is usually computed with the Steinberg–Cochran–Guinan shear modulus model.

The strain hardening function (f) has the form

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

The thermal component (σ_t) is computed using a bisection algorithm from the following equation ^{[8] [7]}.

where 2U_k is the energy to form a kink-pair in a dislocation segment of length L_d, k_b is the Boltzmann constant, σ_p is the Peierls stress. The constants C₁,C₂ are given by the relations

where ρ_d is the dislocation density, L_d is the length of a dislocation segment, a is the distance between Peierls valleys, b is the magnitude of the Burgers vector, ν is the Debye frequency, w is the width of a kink loop, and D is the drag coefficient.

[edit] Zerilli–Armstrong flow stress model

The Zerilli–Armstrong (ZA) model ^{[9] [16] [17]} is based on simplified dislocation mechanics. The general form of the equation for the flow stress is

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

where σ_g is the contribution due to solutes and initial dislocation density, k_h is the microstructural stress intensity, l is the average grain diameter, K is zero for fcc materials, B,B₀ are material constants.

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

where α₀,α₁,β₀,β₁ are material parameters that depend on the type of material (fcc, bcc, hcp, alloys). The Zerilli–Armstrong model has been modified by ^[18] for better performance at high temperatures.

[edit] Mechanical threshold stress flow stress model

The Mechanical Threshold Stress (MTS) model ^{[10] [19] [20]}) has the form

where σ_a is the athermal component of mechanical threshold stress, σ_i is the component of the flow stress due to intrinsic barriers to thermally activated dislocation motion and dislocation-dislocation interactions, σ_e is the component of the flow stress due to microstructural evolution with increasing deformation (strain hardening), (S_i,S_e) are temperature and strain-rate dependent scaling factors, and μ₀ is the shear modulus at 0 K and ambient pressure.

The scaling factors take the Arrhenius form

where k_b is the Boltzmann constant, b is the magnitude of the Burgers' vector, (g_0i,g_0e) are normalized activation energies, () are constant reference strain-rates, and (q_i,p_i,q_e,p_e) are constants.

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

where

and θ₀ is the hardening due to dislocation accumulation, θ_IV is the contribution due to stage-IV hardening, (a₀,a₁,a₂,a₃,α) are constants, σ_es is the stress at zero strain hardening rate, σ_0es is the saturation threshold stress for deformation at 0 K, g_0es is a constant, and is the maximum strain-rate. Note that the maximum strain-rate is usually limited to about 10⁷/s.

[edit] Preston–Tonks–Wallace flow stress model

The Preston–Tonks–Wallace (PTW) model ^[15] 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 τ_s is a normalized work-hardening saturation stress, s₀ is the value of τ_s at 0K, τ_y is a normalized yield stress, θ is the hardening constant in the Voce hardening law, and d 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 τ_s close to the melt temperature, () are the values of τ_y at 0K and close to melt, respectively, (κ,γ) are material constants, , (s₁,y₁,y₂) are material parameters for the high strain-rate regime, and

where ρ is the density, and M is the atomic mass.

[edit] See also

• Viscoelasticity
• Bingham plastic
• Dashpot
• Creep
• Plasticity (physics)
• Continuum mechanics

[edit] References

1. ^ J.Lemaitre and J.L.Chaboche (2002) "Mechanics of solid materials" Cambridge University Press.
2. ^ Lubliner, Jacob (1990) "PLASTICITY THEORY," Macmillan Publishing Company, NY.
3. ^ Young, Mindness, Gray, ad Bentur (1998): "The Science and Technology of Civil Engineering Materials," Prentice Hall, NJ.
4. ^ Simo, J.C.; Hughes, T.J.R. (1998), Computational inelasticity 
5. ^ ^{a b} Johnson, G.R.; Cook, W.H. (1983), "A constitutive model and data for metals subjected to large strains, high strain rates and high", Proceedings of the 7th International Symposium on Ballistics: 541–547, http://www.lajss.org/HistoricalArticles/A%20constitutive%20model%20and%20data%20for%20metals.pdf, retrieved 2009-05-13 
6. ^ ^{a b} Steinberg, D.J.; Cochran, S.G.; Guinan, M.W. (1980), "A constitutive model for metals applicable at high-strain rate", Journal of Applied Physics 51: 1498, http://link.aip.org/link/?JAPIAU/51/1498/1, retrieved 2009-05-13 
7. ^ ^{a b c} Steinberg, D.J.; Lund, C.M. (1988), "A constitutive model for strain rates from 10⁻⁴ to 10⁶ s⁻¹", Journal de physique. Colloques 49 (3): 3–3, http://cat.inist.fr/?aModele=afficheN, retrieved 2009-05-13 
8. ^ ^{a b} Hoge, K.G.; Mukherjee, A.K. (1977), "The temperature and strain rate dependence of the flow stress of tantalum", Journal of Materials Science 12 (8): 1666–1672, http://www.springerlink.com/index/Q406613278W585P7.pdf, retrieved 2009-05-13 
9. ^ ^{a b} Zerilli, F.J.; Armstrong, R.W. (1987), "Dislocation-mechanics-based constitutive relations for material dynamics calculations", Journal of Applied Physics 61: 1816, http://link.aip.org/link/?JAPIAU/61/1816/1, retrieved 2009-05-13 
10. ^ ^{a b} Follansbee, P.S.; Kocks, U.F. (1988), "A constitutive description of the deformation of copper based on the use of the mechanical threshold", Acta Metall. 36 (1): 81–93, http://www.csa.com/partners/viewrecord.php?requester=gs, retrieved 2009-05-13 
11. ^ Chen, S.R.; Gray, G.T. (1996), "Constitutive behavior of tantalum and tantalum-tungsten alloys", Metallurgical and Materials Transactions A 27 (10): 2994–3006, http://www.springerlink.com/index/D65G5112T720L7N5.pdf, retrieved 2009-05-13 
12. ^ Goto, D.M.; Garrett, R.K.; Bingert, J.F.; Chen, S.R.; Gray, G.T. (2000), "The mechanical threshold stress constitutive-strength model description of HY-100 steel", Metallurgical and Materials Transactions A 31 (8): 1985–1996, http://www.springerlink.com/index/N4820676MU264G8H.pdf, retrieved 2009-05-13 
13. ^ Banerjee, B. (2007), "The Mechanical Threshold Stress model for various tempers of AISI 4340 steel", International Journal of Solids and Structures 44 (3–4): 834–859, http://arxiv.org/pdf/cond-mat/0510330, retrieved 2009-05-13 
14. ^ Puchi-cabrera, E.S.; Villalobos-gutierrez, C.; Castro-farinas, G. (2001), "On the mechanical threshold stress of aluminum: Effect of the alloying content", Journal of Engineering Materials and Technology 123: 155, http://link.aip.org/link/?JEMTA8/123/155/1, retrieved 2009-05-13 
15. ^ ^{a b} Preston, D.L.; Tonks, D.L.; Wallace, D.C. (2003), "Model of plastic deformation for extreme loading conditions", Journal of Applied Physics 93: 211, http://link.aip.org/link/?JAPIAU/93/211/1, retrieved 2009-05-13 
16. ^ Zerilli, F.J.; Armstrong, R.W. (1994), "Constitutive relations for the plastic deformation of metals", AIP Conference Proceedings 309: 989, http://link.aip.org/link/?APCPCS/309/989/1, retrieved 2009-05-13 
17. ^ Zerilli, F.J. (2004), "Dislocation mechanics-based constitutive equations", Metallurgical and Materials Transactions A 35 (9): 2547–2555, http://www.springerlink.com/index/0406259680452456.pdf, retrieved 2009-05-13 
18. ^ Abed, F.H.; Voyiadjis, G.Z. (2005), "A consistent modified Zerilli–Armstrong flow stress model for BCC and FCC metals for elevated", Acta Mechanica 175 (1): 1–18, http://www.springerlink.com/index/X7WFH0P2PJXK17FH.pdf, retrieved 2009-05-13 
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