Viscoplasticity
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Viscoplasticity is a theory in
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is
plastic deformation In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain ...
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 In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
spring Spring(s) may refer to: Common uses * Spring (season) Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
elements. Rate-dependence can be represented by nonlinear
dashpot A dashpot, also known as a damper, is a mechanical device that resists motion via viscous friction. The resulting force is proportional to the velocity, but acts in the opposite direction, slowing the motion and absorbing energy. It is commonly us ...
elements in a manner similar to
viscoelasticity In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
. Plasticity can be accounted for by adding sliding
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
al elements as shown in Figure 1. In the figure E is the
modulus of elasticity An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
, λ is the
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
parameter and N is a power-law type parameter that represents non-linear dashpot (dε/dt)= σ = λ(dε/dt)(1/N) The sliding element can have a
yield stress In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
y) that is
strain rate In materials science, strain rate is the change in strain ( deformation) of a material with respect to time. The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change ...
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. In these models, the stress is allowed to increase beyond the rate-independent
yield surface A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of ''inside'' the yield surface is elastic. When the stress state lies on the surface the materi ...
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 In materials science, strain rate is the change in strain ( deformation) of a material with respect to time. The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change ...
dependence to the yield stress and use the techniques of rate independent plasticity to calculate the response of a material For
metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s and
alloy An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductility, ...
s, viscoplasticity is the
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenomena a ...
behavior caused by a mechanism linked to the movement of
dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to sl ...
s in
grain A grain is a small, hard, dry fruit (caryopsis) – with or without an attached hull layer – harvested for human or animal consumption. A grain crop is a grain-producing plant. The two main types of commercial grain crops are cereals and legum ...
s, 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 alloys exhibit viscoplasticity at room temperature (300K). For
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
s,
wood Wood is a porous and fibrous structural tissue found in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulose fibers that are strong in tension and embedded in a matrix of lignin th ...
, and
bitumen Asphalt, also known as bitumen (, ), is a sticky, black, highly viscous liquid or semi-solid form of petroleum. It may be found in natural deposits or may be a refined product, and is classed as a pitch. Before the 20th century, the term a ...
, the theory of viscoplasticity is required to describe behavior beyond the limit of elasticity or
viscoelasticity In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
. 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.


History

Research on plasticity theories started in 1864 with the work of
Henri Tresca Henri Édouard Tresca (12 October 1814 – 21 June 1885) was a French mechanical engineer, and a professor at the Conservatoire National des Arts et Métiers in Paris. Work on plasticity He is the father of the field of plasticity, or non-recov ...
,
Saint Venant Saint-Venant ( vls, Papingem) is a commune in the Pas-de-Calais department (administrative division) in the Hauts-de-France region of France. Geography Saint-Venant is situated some northwest of Béthune and west of Lille, at the junction of ...
(1870) and
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(1871) on the maximum shear criterion.Kojic, M. and Bathe, K-J., (2006), Inelastic Analysis of Solids and Structures, Elsevier. An improved plasticity model was presented in 1913 by
Von Mises Mises or von Mises may refer to: * Ludwig von Mises, an Austrian-American economist of the Austrian School, older brother of Richard von Mises ** Mises Institute, or the Ludwig von Mises Institute for Austrian Economics, named after Ludwig von ...
von Mises, R. (1913) "Mechanik der festen Korper im plastisch deformablen Zustand." ''Gottinger Nachr, math-phys Kl'' 1913:582–592. which is now referred to as the
von Mises yield criterion The maximum distortion criterion (also von Mises yield criterion) states that yielding of a ductile material begins when the second invariant of deviatoric stress J_2 reaches a critical value. It is a part of plasticity theory that mostly applie ...
. In viscoplasticity, the development of a mathematical model heads back to 1910 with the representation of primary creep by Andrade's law.Betten, J., 2005, Creep Mechanics: 2nd Ed., Springer. In 1929, NortonNorton, F. H. (1929). Creep of steel at high temperatures. McGraw-Hill Book Co., New York. developed a one-dimensional dashpot model which linked the rate of secondary creep to the stress. In 1934, OdqvistOdqvist, F. K. G. (1934) "Creep stresses in a rotating disc." ''Proc. IV Int. Congress for Applied. Mechanics'', Cambridge, p. 228. 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 Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of ...
(1924)Prandtl, L. (1924) Proceedings of the 1st International Congress on Applied Mechanics, Delft. and Reuss (1930). In 1932, Hohenemser and
Prager Prager (variants: Praeger, Preger) is a surname, which may refer to: Prager * David Prager (born 1977), American TV producer and blogger * Dennis Prager (born 1948), U.S. conservative radio talk show host, columnist and public speaker ** PragerU, ...
proposed the first model for slow viscoplastic flow. This model provided a relation between the
deviatoric stress In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
and the
strain rate In materials science, strain rate is the change in strain ( deformation) of a material with respect to time. The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change ...
for an incompressible Bingham solidBingham, E. C. (1922) Fluidity and plasticity. McGraw-Hill, New York. 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 HoffHoff, ed., 1962, IUTAM Colloquium Creep in Structures; 1st, Stanford, Springer. 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. The formulated models were supported by the
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
of
irreversible process In science, a thermodynamic processes, process that is not Reversible process (thermodynamics), reversible is called irreversible. This concept arises frequently in thermodynamics. All complex natural processes are irreversible, although a phase ...
es 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 #hardening tests at constant stress or strain rate, #creep tests at constant force, and #stress relaxation at constant elongation.


Strain hardening test

One consequence of yielding is that as plastic deformation proceeds, an increase in
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
is required to produce additional
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
. This phenomenon is known as Strain/Work hardening. 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. # At the same strain, the higher the rate of strain the higher the stress # A change in the rate of strain during the test results in an immediate change in the stress–strain curve. # 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, i.e., : \boldsymbol = \boldsymbol_ + \boldsymbol_ where \boldsymbol_ is the elastic strain and \boldsymbol_ 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

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 # 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 \le \boldsymbol \le \boldsymbol_1 ). # The secondary creep stage, also known as the steady state, is where the strain rate is constant. (\boldsymbol_1 \le \boldsymbol \le \boldsymbol_2). # A tertiary creep phase in which there is an increase in the strain rate up to the fracture strain. (\boldsymbol_2 \le \boldsymbol \le \boldsymbol_R).


Relaxation test

As shown in Figure 4, the relaxation testFrançois, D., Pineau, A., Zaoui, A., (1993), Mechanical Behaviour of Materials Volume II: Viscoplasticity, Damage, Fracture and Contact Mechanics, Kluwer Academic Publishers. 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 decomposition of strain rate is : \cfrac = \cfrac + \cfrac ~. The elastic part of the strain rate is given by : \cfrac = \mathsf^~\cfrac For the flat region of the strain-time curve, the total strain rate is zero. Hence we have, : \cfrac = -\mathsf^~\cfrac 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 has plateaued 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 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 \cfrac = 0 in a test requires considerable delicacy.Cristescu, N. and Gioda, G., (1994), Viscoplastic Behaviour of Geomaterials, International Centre for Mechanical Sciences.


Rheological models of viscoplasticity

One-dimensional constitutive models for viscoplasticity based on spring-dashpot-slider elements include the perfectly viscoplastic solid, the elastic perfectly viscoplastic solid, and the elastoviscoplastic hardening solid. The elements may be connected in
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
or in
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...
. 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 \dot and \dot, respectively.


Perfectly viscoplastic solid (Norton-Hoff model)

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., \boldsymbol_e = 0 and hence there is no initial yield stress, i.e., \sigma_y = 0. The viscous dashpot has a response given by : \boldsymbol = \eta~\dot_ \implies \dot_ = \cfrac where \eta is the viscosity of the dashpot. In the Norton-Hoff model the viscosity \eta is a nonlinear function of the applied stress and is given by : \eta = \lambda\left cfrac\right where N is a fitting parameter, λ is the kinematic viscosity of the material and , , \boldsymbol, , = \sqrt = \sqrt. Then the viscoplastic strain rate is given by the relation : \dot_ = \cfrac\left cfrac\right In one-dimensional form, the Norton-Hoff model can be expressed as : \sigma = \lambda~\left(\dot_\right)^ When N = 1.0 the solid is
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
. If we assume that plastic flow is
isochoric Isochoric may refer to: *cell-transitive, in geometry *isochoric process In thermodynamics, an isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which ...
(volume preserving), then the above relation can be expressed in the more familiar formRappaz, M., Bellet, M. and Deville, M., (1998), Numerical Modeling in Materials Science and Engineering, Springer. : \boldsymbol = 2 K~\left(\sqrt\dot_\right)^~\dot_ where \boldsymbol is the
deviatoric stress In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
tensor, \dot_ is the von Mises equivalent strain rate, and K, m are material parameters. The equivalent strain rate is defined as :\dot = \sqrt These models can be applied in metals and alloys at temperatures higher than two thirds 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.


Elastic perfectly viscoplastic solid (Bingham–Norton model)

Two types of elementary approaches can be used to build up an elastic-perfectly viscoplastic mode. In the first situation, the sliding 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.Irgens, F., (2008), Continuum Mechanics, Springer. 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 : \begin & \boldsymbol = \mathsf~\boldsymbol & & \mathrm~\, \boldsymbol\, < \sigma_y \\ & \dot = \dot_ + \dot_ = \mathsf^~\dot + \cfrac\left - \cfrac\right & & \mathrm~\, \boldsymbol\, \ge \sigma_y \end where \eta is the viscosity of the dashpot element. If the dashpot element has a response that is of the Norton form : \cfrac = \cfrac\left cfrac\right we get the Bingham–Norton model : \dot = \mathsf^~\dot + \cfrac\left cfrac\right\left - \cfrac\right \quad \mathrm~\, \boldsymbol\, \ge \sigma_y Other expressions for the strain rate can also be observed in the literature with the general form : \dot = \mathsf^~\dot + f(\boldsymbol, \sigma_y)~\boldsymbol \quad \mathrm~\, \boldsymbol\, \ge \sigma_y The responses for strain hardening, creep, and relaxation tests of such material are shown in Figure 8.


Elastoviscoplastic hardening solid

An elastic-viscoplastic material with
strain hardening In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context. This strength ...
is described by equations similar to those for an 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 : \begin & \boldsymbol =\boldsymbol_ = \mathsf^~\boldsymbol = ~\boldsymbol & & \mathrm~, , \boldsymbol, , < \sigma_y \\ & \dot = \dot_ + \dot_ = \mathsf^~\dot + f(\boldsymbol,\sigma_y,\boldsymbol_)~\boldsymbol & & \mathrm~, , \boldsymbol, , \ge \sigma_y \end . 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.


Strain-rate dependent plasticity models

Classical phenomenological viscoplasticity models for small strains are usually categorized into two types: * the Perzyna formulation * the Duvaut–Lions formulation


Perzyna formulation

In the Perzyna formulation the plastic strain rate is assumed to be given by a constitutive relation of the form : \dot_ = \cfrac \cfrac = \begin \cfrac \cfrac& ~f(\boldsymbol, \boldsymbol) > 0 \\ 0 & \rm \\ \end where f(.,.) is a yield function, \boldsymbol is the Cauchy stress, \boldsymbol is a set of internal variables (such as the plastic strain \boldsymbol_), \tau is a relaxation time. The notation \langle \dots \rangle denotes the
Macaulay brackets Macaulay brackets are a notation used to describe the ramp function :\ = \begin 0, & x < 0 \\ x, & x \ge 0. \end A popular alternative transcription uses angle brackets, ''viz.'' \langle x \rangle. and has the form : \dot_ = \left\langle \frac \right\rangle^n sign(\boldsymbol-\boldsymbol) where f_0 is the quasistatic value of f and \boldsymbol is a ''backstress''. Several models for the backstress also go by the name ''Chaboche model''.


Duvaut–Lions formulation

The Duvaut–Lions formulation is equivalent to the Perzyna formulation and may be expressed as : \dot_ = \begin \mathsf^:\cfrac & \rm~f(\boldsymbol, \boldsymbol) > 0 \\ 0 & \rm \end where \mathsf is the elastic stiffness tensor, \mathcal\boldsymbol is the closest point projection of the stress state on to the boundary of the region that bounds all possible elastic stress states. The quantity \mathcal\boldsymbol is typically found from the rate-independent solution to a plasticity problem.


Flow stress models

The quantity f(\boldsymbol, \boldsymbol) represents the evolution of the
yield surface A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of ''inside'' the yield surface is elastic. When the stress state lies on the surface the materi ...
. 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 example is
von Mises Mises or von Mises may refer to: * Ludwig von Mises, an Austrian-American economist of the Austrian School, older brother of Richard von Mises ** Mises Institute, or the Ludwig von Mises Institute for Austrian Economics, named after Ludwig von ...
or J_2 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: #the Johnson–Cook model #the Steinberg–Cochran–Guinan–Lund model. #the Zerilli–Armstrong model. #the Mechanical threshold stress model. #the Preston–Tonks–Wallace model. The Johnson–Cook (JC) model 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 is semi-empirical. The model is purely empirical and strain-rate independent at high strain-rates. A dislocation-based extension based on is used at low strain-rates. The SCGL model is used extensively by the shock physics community. The Zerilli–Armstrong (ZA) model 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. This model has been used to model the plastic deformation of copper, tantalum, alloys of steel, and aluminum alloys. However, the MTS model is limited to strain-rates less than around 107/s. The Preston–Tonks–Wallace (PTW) model 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.


Johnson–Cook flow stress model

The Johnson–Cook (JC) model is purely empirical and gives the following relation for the flow stress (\sigma_y) :\text \qquad \sigma_y(\varepsilon_,\dot,T) = \left + B (\varepsilon_)^n\rightleft + C \ln(\dot^)\right \left - (T^*)^m\right where \varepsilon_ is the equivalent plastic strain, \dot 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 : \dot^ := \cfrac \qquad\text\qquad T^* := \cfrac where \dot 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 \dot^ non-dimensional.Schwer http://www.dynalook.com/european-conf-2007/optional-strain-rate-forms-for-the-johnson-cook.pdf T_0 is a reference temperature, and T_m is a reference melt temperature. For conditions where T^* < 0, we assume that m = 1.


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. for high strain-rate situations and extended to low strain-rates and bcc materials by Steinberg and Lund. The flow stress in this model is given by :\text \qquad \sigma_y(\varepsilon_,\dot,T) = \left sigma_a f(\varepsilon_) + \sigma_t (\dot, T)\right \frac; \quad \sigma_a f \le \sigma_ ~~\text~~ \sigma_t \le \sigma_p where \sigma_a is the athermal component of the flow stress, f(\varepsilon_) is a function that represents strain hardening, \sigma_t is the thermally activated component of the flow stress, \mu(p,T) is the pressure- and temperature-dependent shear modulus, and \mu_0 is the shear modulus at standard temperature and pressure. The saturation value of the athermal stress is \sigma_. The saturation of the thermally activated stress is the
Peierls stress Peierls stress (also known as the lattice friction stress) is the force (first described by Rudolf Peierls and modified by Frank Nabarro) needed to move a dislocation within a plane of atoms in the unit cell. The magnitude varies periodically as the ...
(\sigma_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 : f(\varepsilon_) = + \beta(\varepsilon_ + \varepsilon_i)n where \beta, n are work hardening parameters, and \varepsilon_i is the initial equivalent plastic strain. The thermal component (\sigma_t) is computed using a bisection algorithm from the following equation. : \dot = \left frac\exp\left[\frac ____\left(1_-_\frac\right)^2\right+_ ____\frac\right.html" ;"title="frac \left(1 - \frac\right)^2\right">frac\exp\left[\frac \left(1 - \frac\right)^2\right+ \frac\right">frac \left(1 - \frac\right)^2\right">frac\exp\left[\frac \left(1 - \frac\right)^2\right+ \frac\right; \quad \sigma_t \le \sigma_p where 2 U_k is the energy to form a kink-pair in a dislocation segment of length L_d, k_b is the Boltzmann constant, \sigma_p is the
Peierls stress Peierls stress (also known as the lattice friction stress) is the force (first described by Rudolf Peierls and modified by Frank Nabarro) needed to move a dislocation within a plane of atoms in the unit cell. The magnitude varies periodically as the ...
. The constants C_1, C_2 are given by the relations : C_1 := \frac; \quad C_2 := \frac where \rho_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 In materials science, the Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted as , that represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice. The ve ...
, \nu is the
Debye frequency In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (Heat capacity) in a solid. It treats the vibrations of the atomic lattice (hea ...
, w is the width of a kink loop, and D is the
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equ ...
.


Zerilli–Armstrong flow stress model

The Zerilli–Armstrong (ZA) model is based on simplified dislocation mechanics. The general form of the equation for the flow stress is :\text \qquad \sigma_y(\varepsilon_,\dot,T) = \sigma_a + B\exp(-\beta T) + B_0\sqrt\exp(-\alpha T) ~. In this model, \sigma_a is the athermal component of the flow stress given by : \sigma_a := \sigma_g + \frac + K\varepsilon_^n, where \sigma_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, B_0 are material constants. In the thermally activated terms, the functional forms of the exponents \alpha and \beta are : \alpha = \alpha_0 - \alpha_1 \ln(\dot); \quad \beta = \beta_0 - \beta_1 \ln(\dot); where \alpha_0, \alpha_1, \beta_0, \beta_1 are material parameters that depend on the type of material (fcc, bcc, hcp, alloys). The Zerilli–Armstrong model has been modified by for better performance at high temperatures.


Mechanical threshold stress flow stress model

The Mechanical Threshold Stress (MTS) model ) has the form :\text \qquad \sigma_y(\varepsilon_,\dot,T) = \sigma_a + (S_i \sigma_i + S_e \sigma_e)\frac where \sigma_a is the athermal component of mechanical threshold stress, \sigma_i is the component of the flow stress due to intrinsic barriers to thermally activated dislocation motion and dislocation-dislocation interactions, \sigma_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 \mu_0 is the shear modulus at 0 K and ambient pressure. The scaling factors take the Arrhenius form :\begin S_i & = \left - \left(\frac \ln\frac\right)^ \right \\ S_e & = \left - \left(\frac \ln\frac\right)^ \right \end where k_b is the Boltzmann constant, b is the magnitude of the Burgers' vector, (g_, g_) are normalized activation energies, (\dot, \dot) are the strain-rate and reference strain-rate, and (q_i, p_i, q_e, p_e) are constants. The strain hardening component of the mechanical threshold stress (\sigma_e) is given by an empirical modified Voce law :\text \qquad \frac = \theta(\sigma_e) where :\begin \theta(\sigma_e) & = \theta_0 1 - F(\sigma_e)+ \theta_ F(\sigma_e) \\ \theta_0 & = a_0 + a_1 \ln \dot + a_2 \sqrt - a_3 T \\ F(\sigma_e) & = \cfrac \\ \ln(\cfrac) & = \left(\frac\right) \ln\left(\cfrac\right) \end and \theta_0 is the hardening due to dislocation accumulation, \theta_ is the contribution due to stage-IV hardening, (a_0, a_1, a_2, a_3, \alpha) are constants, \sigma_ is the stress at zero strain hardening rate, \sigma_ is the saturation threshold stress for deformation at 0 K, g_ is a constant, and \dot is the maximum strain-rate. Note that the maximum strain-rate is usually limited to about 10^7/s.


Preston–Tonks–Wallace flow stress model

The Preston–Tonks–Wallace (PTW) model 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 :\text \qquad \sigma_y(\varepsilon_,\dot,T) = \begin 2\left tau_s_+_\alpha\ln\left[1_-_\varphi ________\exp\left(-\beta-\cfrac\right)\rightright.html" ;"title=" - \varphi \exp\left(-\beta-\cfrac\right)\right">tau_s + \alpha\ln\left[1 - \varphi \exp\left(-\beta-\cfrac\right)\rightright"> - \varphi \exp\left(-\beta-\cfrac\right)\right">tau_s + \alpha\ln\left[1 - \varphi \exp\left(-\beta-\cfrac\right)\rightright \mu(p,T) & \text \\ 2\tau_s\mu(p,T) & \text \end with : \alpha := \frac; \quad \beta := \frac; \quad \varphi := \exp(\beta) - 1 where \tau_s is a normalized work-hardening saturation stress, s_0 is the value of \tau_s at 0K, \tau_y is a normalized yield stress, \theta 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 :\begin \tau_s & = \max\left\ \\ \tau_y & = \max\left\ \end where s_ is the value of \tau_s close to the melt temperature, (y_0, y_) are the values of \tau_y at 0 K and close to melt, respectively, (\kappa, \gamma) are material constants, \hat = T/T_m, (s_1, y_1, y_2) are material parameters for the high strain-rate regime, and : \dot = \frac\left(\cfrac\right)^ \left(\cfrac\right)^ where \rho is the density, and M is the atomic mass.


See also

* Viscoelasticity * Bingham plastic * Dashpot * Creep (deformation) * Plasticity (physics) * Continuum mechanics * Quasi-solid


References

{{Reflist, 2 Continuum mechanics Plasticity (physics)