Viscoplasticity and Static Strain Ageing
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Inelastic deformation of materials is broadly classified into rate independent plasticity and rate dependent plasticity. The theory of Viscoplasticity describes inelastic deformation of materials depending on time i.e. the rate at which the load is applied. In metals and alloys, the mechanism of viscoplasticity is usually shown by the movement of dislocations in grain . From experiments, it has been established that most metals have tendency to exhibit viscoplastic behaviour at high temperatures. Some alloys are found to exhibit this behaviour even at room temperature. Formulating the constitutive laws for viscoplasticity can be classified into the physical approach and the phenomenological approach . The physical approach relies on the movement of dislocations in crystal lattice to model the plasticity. In the phenomenological approach, the material is considered as a continuum. And thus the microscopic behaviour can be represented by the evolution of certain internal variables instead. Most models employ the kinematic hardening and isotropic hardening variables in this respect. Such a phenomenological approach is used in this work too.
According to the classical theory of plasticity, the deviatoric stresses is the main contribu- tor to the yielding of materials and the volumetric or hydrostatic stress does not influence the inelastic behaviour. It also introduces a yield surface to differentiate the elastic and plastic domains. The size and position of such a yield surface can be changed by the strain history, to model the exact stress state. The theory of viscoplasticity differs from the plasticity theory, by employing a series of equipotential surfaces. This helps define an over-stress beyond the yield surface. The plastic strain rate is given by the viscoplastic flow rule. To model the hardening behaviour, introduction of several internal variables is necessary. Unlike strain or temperature which can be measured to asses the stress state, internal variable or state variables are used to capture the material memory by means of evolution equations. This must include a tensor variable to define the kinematic hardening and a scalar variable to define the isotropic variable. The evolution of these internal variables allows us to define the complete hardening behaviour of materials. In this work we consider only the small strain framework.
The basic principles of viscoplasticity are similar to those from Plasticity theory. The main difference is the introduction of time effects. Thus the concepts from plasticity and the introduction of time effects to describe viscoplasticity, as summarised by Chabocheand Lemaitre are discussed in this chapter.
- Basic principles
Considering small strains framework, the strain tensor can be split into its elastic and inelastic parts
ε = εe+ εin(2.1)
where ε is total strain, εe is the elastic strain and εin is the inelastic strain. In this work, we neglect creep and thus consider only the plastic strain to be the inelastic strain. Hence we can proceed to rewrite the above equation as :
ε = εe+ εp(2.2)
where εp is the plastic strain. Let us consider a field with stress σ = σi j(x) and external volume forces fi. Thus the equilibrium condition is given as:
∂σi j + f
= 0;i, jε
From the balance of moment of momentum equation, we know that the Cauchy stress ten- sor is symmetric in nature. The strain tensor is calculated from the gradient of displacement, uas:
εi j = 2
The Hooke's law for the relation between stress and strain tensors is given using the elastic part of the strain:
σ = E· εe(2.5)
where εe and the stress σ are second order tensors. E is the fourth order elasticity tensor.
- Equipotential surfaces
In the traditional plasticity theory which is time independent, the stress state is governed by a yield surface and loading-unloading conditions. In Viscoplasticity the time or rate dependent plasticity is described by a series of concentric equipotential surfaces. The location on the centre and its size determine the stress state of a given material.
Fig. 2.1 Illustration of equipotential surfaces from 
It can be understood that the inner most surface or the surface closest to the centre represents a null flow rate(â„¦ = 0). As shown in Figure (2.1), the outer most and the farthest surface from the centre represents infinite flow rate (â„¦ = ∞). These two surfaces represent the extremes governed by the time independent plasticity laws. The region in between is governed by Viscoplasticity. The size of the equipotential surface is proportional to the flow rate. Greater the flow, greater is the surface size. The region between the centre and the inner most surface is the elastic domain. Flow begins at this inner most surface( f=0).
In Viscoplasticity, there are two types of hardening rules to be considered: (i) Kinematic hardening and (ii) isotropic hardening. The Kinematic hardening describes the movement of the equipotential surfaces in the stress plane. From material science, this behaviour is known to be the result of dislocations accumulating at the barriers. Thus it helps in describing the Bauschinger effect  which states that when a material is subjected to yielding by a compressive load, the elastic domain is increased for the consecutive tensile load. This behaviour is represented by α which does not evolve continuously during cyclic loads and thus fails to describe cyclic hardening or softening behaviours. A schematic representation is shown in Fig.(2.2).
Fig. 2.2 Linear Kinematic hardening and Stress-strain response from 
The isotropic hardening on the other hand describes the change in size of the surface and assumes that the centre and shape remains unchanged. This behaviour is due to the number of dislocations in a material and the energy stored in it. It is represented by variable r, which evolves continuously during cyclic loadings. This can be controlled by the recovery phase. As a result, isotropic behaviour is helpful is modelling the cyclic hardening and softening phenomena. A schematic representation is shown in Fig.(2.3).
Fig. 2.3 Linear Isotropic hardening and Stress-strain response from 
From Thermodynamics, we know the free energy potential(ψ ) to be a scalar function . With respect to temperature T, it is concave. But convex with respect to other internal variables. Thus, it can be defined as :
ψ= ψ. ,T,εe,εp,Vk.(2.6)
where ε,Tare the only measured quantities that can help model plasticity. Vkrepresents the set of internal variable, also known as state variables which help define the memory of the previous stress states.
In Viscoplasticity, it is assumed that ψ depends only on εe,T,Vk. Thus we have:
ψ= ψ. e,T,Vk.(2.7)
According to thermodynamic rules, stress is associated with strain and the entropy with temperature. This helps us define the following relations:
σ = ρ
,s = −
where ρ is density and s is entropy. It is possible to decouple the free energy function and split it into the elastic and plastic parts.
ψ= ψe. e,T.+ ψp. ,r,T.(2.9) Similar to σ, the thermodynamic forces corresponding to α and r is given by:
X = ρ
,R = ρ
Here we have X the back stress tensor, used to measure Kinematic hardening. It is noted as a Kinematic hardening variable which defines the position tensor of the centre of equipotential surface. Similarly Ris the Isotropic hardening variable which governs the size of the equipotential surface.
- Dissipation potential
The equipotential surfaces that describe Viscoplasticity have some properties.
- Points on each surface have a magnitude equal to the strain rate.
- Points on each surface have the same dissipation potential.
- If potential is zero, there is no plasticity and it refers to the elastic domain.
The dissipation potential is represented by â„¦ which is a convex function. It can be defined in a dual form as:
â„¦ = â„¦. ,X,R; T,α,r.(2.11)
It is a positive function and if the variables σ,X,Rare zero, then the potential is also zero. The normalityrule, defined in  suggests that the outward normal vector is proportional to the gradient of the yield function. Applying the normality rule, we may obtain the following relations:
∂â„¦ εË™ p = ∂σ,
αË™ = ∂â„¦ ,
Considering the recovery effects in Viscoplasticity, the dissipation potential can be split into two parts:
â„¦ = â„¦p+ â„¦r(2.13)
where â„¦p is the Viscoplastic potential and â„¦r the recovery potential which are defined as :
â„¦p=â„¦p..− X. − R− k,X,R; T,α,r. ,(2.14)
â„¦r=â„¦r. ,R; T,α,r.(2.15)
σ− X=2 σ− X: σ − X
where J2 .− X. refers to the norm on the stress plane and kis the initial yield or the initial
size of equipotential surface.
Going back to the relation in (2.12) , we have:
σ′ − X ′
Here, p is the accumulated viscoplastic strain, given by :
εË™ p : εË™p(2.18)
Also applying the normality rule on eq. (2.15) we may define r as :
rË™ = pË™ −
Thus when recovery is ignored (i.e â„¦r = 0), r is equal to p.
- Perfect viscoplasticity
Let us consider pure viscoplasticity where hardening is ignored. Thus the internal variables may also be removed.
â„¦ = â„¦. ,T.(2.20)
Since plasticity is independent of volumetric stress, we may consider just the deviatoric stress σ ′ = σ − 1 tr(σ)I. Using isotropic property, we may just use the second invariant of
σ ′. Thus:
â„¦ = â„¦. (σ ),T.(2.21)
Applying the normality rule here, we may obtain the flow rule for Viscoplasticity.
2 ∂J2 .σ.
From the Odqvist's law , the dissipation potential for perfect viscoplasticity can be obtained. Here the elastic part is ignored. Thus we have:
n + 1
where λ and n are material parameters.
Using this relation in the flow rule from eq.(2.22), we get:
J2 . .
Further the elasticity domain can be included through the parameter kwhich is a measure of the initial yield:
.J2(σ) − k.nσ′
J2 . .
The <> are the Macauley brackets defined by :
âŸ¨FâŸ© = F· H(F),H(F) =
0 ifF <0
- Hardening rules
As suggested in previous sections, the Viscoplasticity can be enhanced by including internal variables, which help describe the material memory dependent behaviours. These internal variables or state variables include Kinematic and isotropic hardening. Kinematic hardening can help describe the stress hysteresis loops accurately. It governs the movement of the centre of equipotential surfaces and thus the surface by itself. Isotropic hardening helps describe the cyclic hardening and softening effects. It governs the changes in size of the surface.
We have already discussed the normality rule in (2.10), where a relation is established.
Ignoring the recovery, we have:
X = ρ
, R = ρ
Thus the dissipation potential can be written as:
â„¦ = â„¦.σ,X,R; T,α,p.(2.28)
2.2 Static strain ageing11
As suggested in , by including αand pin the free energy function and the dissipation potential, the following forms are achieved:
ρψ = ρψe + 3 c α : α+ h(p)(2.29)
â„¦ = â„¦
− X. − R− k+ 1
J2(X) − 2
c γ J2(α) ; T, p
σ2 c 292
X=c α, R = ρ
εË™ p = ∂â„¦= 3∂â„¦
∂σ2 ∂J2 .− X.
J2 σ − X
pË™ = − ∂R=
: εË™ p
αË™ = − ∂R= εË™
Substituting the above in (2.31), we obtain the final relations for Kinematic hardening variable and the Isotropic hardening variable as:
XË™ = 2 cεË™pγXpË™ 3
RË™ =b (Q − R) pË™
where b,Qare material parameters. Superposing the kinematic and isotropic hardening rules, a unified Visoplastic model was formulated by Chaboche. An extension of such a model will be used in this work.
- Static strain ageing
Strain ageing is defined as the rise in material strength and decrease of ductility in a material when heat treated at a low temperature post the cold working. It can be broadly classified into:
- Static strain ageing - If the change in material properties occurs post the ageing period.
- Dynamic strain ageing - If the change in material properties during the ageing period.
Experimental results of low carbon steel specimen subjected to tensile strain and then rested for period of time reveal interesting behaviour. A steel specimen is loaded beyond the yield point and after the hardening has evolved, it is unloaded and rested at 200c for a period of time. On applying the load again, a considerable rise in the yield point is noticed. This phenomenon is termed as Strain ageing. Higher temperatures have been observed to accelerate the ageing. Composition of a material is also known to be a factor.
- Theory of deformation
A detailed account found in  has been summarised here. Metals at a crystal structural level consist of atoms arranged together in patterns which are governed by various factors such as atomic size, temperature and the alloyed elements. Atoms are arranged in layers at different orientations. When enough force is applied on the crystals in certain favourable orientations, the layers of atoms may slip above one another. This phenomena is termed as slip. This slip happens due to the movement of defects or dislocations in the crystal lattice. Depending on the material's resistance to the slip phenomena, its material properties may be defined. If the slip occurs with less amount of force, then it results in a low yield point during a tensile load. Subsequently the material doesn't show much hardening behaviour and fractures may occur with a marginal increase in force. This kind of fracture is characterised by good ductility. If a large force is needed to cause this slip, then the material has a high yield point. Material hardening behaviour is also greater. Also greater amount becomes necessary to continue the deformation and cause fracture. This fracture is characterised by poor ductility.
- Metallurgical causes
Alloys may consist of varying alloying elements in addition to iron. These elements upon cooling, move through the dislocations due to the distortion they create in a crystal lattice. They concentrate around these defects and thereby stabilising the structure. This stability ensures that a greater amount of force is necessary to enforce a slip. Thus correspondingly the yield point rises along side material strength. Temperature is also an influence in this phenomena. Ageing of a steel does not occur at room temperature. However a temperature in the range 150-250c as a minimum, is the required to be maintained over a period of hours to initiate the ageing effect.
Constitutive viscoplastic material model
One of the initial constitutive models was proposed by Bingham  . This model describes deformations to occur only when force applied exceeds the yield limit. An im- provement was suggested by Drucker Prager in his elastic-plastic model  . Here the elastic region (i.e before the yield limit) is in accordance with the Hooke's law. But post the yield limit, the stress is defined by the plastic constitutive model. Much progress has been made in devising constitutive models to better describe the viscoplastic material behaviour. Based on the material state, these models are broadly divided into two approaches .
In the first approach, the material behaviour is defined entirely by the current state of the material. Thus it depends only on the current physical quantities and internal material parameters. These methods were used for rate-independent plasticity by applying the con- cepts of single or multiple yield surfaces [16, 14, 31]. Time dependent effects were included only through the separation of plastic and creep strains , or a combined viscoplastic model. The second approach brings into consideration the physical parameters, from the current state and from the previous time steps. Numerous applications of this method have been made, as mentioned in [9, 8, 7]. The "strain memory effect" is an example of this approach. It considers the current material state to be affected by the maximum plastic strain in the previous time steps.
The Armstrong Frederick Chaboche (AFC) model is an existing constitutive model. It has the ability to effectively describe time dependent plasticity seen in cyclic loading conditions. This is due to the presence of evolution equations describing the kinematic and isotropic hardening parameters. An extension of this AFC model for the Viscoplastic model was formulated in , with the introduction of the viscoplastic flow rule. Such a model is termed the "Chaboche Viscoplastic model", which combines the plastic strain and viscous strain to form a viscoplastic strain. This viscoplastic strain is able to describe both strain hardening and time hardening. As a result, the strain rate does not pose any problem of discontinuity during loading. Hence the model can be termed the "Unified-Chaboche viscoplastic model".
A constitutive model based on the unified model has been formulated by Wang , through the superposition of kinematic hardening, isotropic hardening evolution equations and strain memory effect. This constitutive model was implemented in this work due to the said advantages and is thoroughly discussed in following sections.
- Flow condition
From Chapter 2, it is understood that the viscous effect in a viscoplastic model is implemented through the introduction of equipotential surfaces. Differing from rate independent plasticity, viscoplasticity considers a stress state even beyond the yield surface which is viscous in nature. The von Mises yield criterion can be extended using the normality rule, wherein the kinematic hardening and isotropic hardening variables are introduced:
f = J2(σ− X) − R− k>0,with J2(σ− X) =(σ′ − X′) : (σ′ − X′)(3.1)
wherek is the initial size of the elastic domain, its value is given by the yield surface,
X is the Kinematic hardening variable, also known as the back-stress tensor,
R is the Isotropic hardening variable, which is a scalar.
The terms X and R are the two internal variables which are responsible for determining the position and size of the equipotential surfaces.
- Flow rule
The normality rule defines that post the yield point, viscoplastic strain increments are perpendicular to the equipotential surfaces. This fulfils the criterion for viscoplastic flow rule f>0. From the concept of equipotential surfaces it is known that beyond the elastic region, the stress state is given by a dissipation potential. This remains zero in the elastic domain.
The dissipation potential from Ohno is given by:
â„¦p = n + 1
.J2(σ− X) − R− k.n+1
whereZ is an internal variable, known as the drag effect; Z= K+ D, n is an internal variable, 3 ≤ n ≤ 30 
From the normality rule , the viscoplastic flow rule is given as:
∂â„¦ εË™ p = ∂σ
2 ∂J2(σ − X)
σ′ − X′ J2(σ− X)
σ′ − X′ J2(σ− X)
where pË™ is the accumulated plastic strain rate. Which can also be calculated from the
viscoplastic strain rate εË™ .
σ′ − X′
.J2(σ− X) − R− k.n+1
∂J2(σ− X) J2(σ− X)Z
εË™ : εË™
The viscous stress with respect to the dissipation potential is given as :
σvis= J2(σ− X) − R− k(3.6)
σvis= Z· pË™ 2(3.7)
- Kinematic hardening
Kinematic hardening variable is a part of the viscoplastic flow rule which describes the translation of the yield surface. Thus it is vital in defining the location of the yield surface.
- Melan-Prager's evolution law
One of the earliest models of Kinematic hardening was given by Prager . The evolution equation of kinematic hardening helps to define the hardening behaviour which is direction dependent. Prager's proposal is considered to be a linear approach in describing this phenom- ena as it considers only the current plastic strain. Thus it is also referred to as the Linear Kinematic hardening(LK) model. Prager established a relation between the kinematic hardening variable X and the viscoplastic strain rate as :
XË™ = 2 cεË™
where c is a material parameter.
- Armstrong-Frederick model
The Linear Kinematic model works based on a linear relationship between the back stress and the viscoplastic strain. Although this model is used in some constitutive models, it has it's limits.
- It fails to describe the Bauschinger effect  since it considers the material behaviour to be a function of the current plastic strain.
- It is also incapable of describing the ratcheting behaviour. This is due to the fact that the back stress and the viscoplastic strain are synchronous with each other.
To over come this, an enhancement is suggested by Armstrong and Frederick .This model assumes that the material behaviour depends on the strain history in addition to current state. The model introduces a non linearity through the introduction of a certain dynamic recallterm, which is proportional to the product of back stress and the norm of the plastic strain rate. Where both quantities are measured in the current state. Thus now, the formulation establishes a dependency on the current displacement of the yield surface in addition to the viscoplastic strain increment.
XË™ = 2 cεË™
- γ XpË™
where c and γ are material properties derived from experimental data.
With this extension, a differential equation is formulated to better describe the evolution of the back stress, which is a measure of the kinematic hardening. During loading the back
stress is activated once the yield point is reached and the first term continues to increase as long as the loading continues in the same direction. But the recall term helps to slow down this growth. When the loading is reversed, so does the direction of the back stress and recall term slows down the reduction in value.
Fig. 3.1 NLK approach in principal stress space 
An illustration of the NLK approach is given in (3.1). The increments of the kinematic hardening variable can be decomposed into two parts. The first one is normal to the equipo- tential surface and the second involves the recall term. It is to be noted that the recall term is parallel to the back stress tensor Xand the accumulated viscoplastic strain pË™. Thus with this formulation, the model has the greater capability to accurately define the radical changes in stress state during cyclic loading. As a result it is able to describe the hardening effects during a hysteresis. To summarise, the Armstrong and Frederick model provides the following advantages :
- Possibility for explicit integration.
- Explains the non linearity in the stress-strain relationship during cyclic loads. Thus is able to describe the Bauschinger effect with considerable accuracy.
- Describes the Ratcheting effect.
- Chaboche model
The Armstrong and Frederick (A-F) model is widely used in constitutive models by re- searchers. But the major disadvantage is the limited strain range âˆ†ε where the kinematic hardening can be calculated. With increase in strain, the back stress value reaches saturation under the A-F model. The rate of saturation is influenced by the material constants employed in (3.9). It is reported that the constants determined for calculations of small strains do not work well with calculations involving large strains.
Ideas to improve the A-F model and overcome the saturation effects was introduced by Chaboche  . The concept was to decompose the back stress or kinematic hardening variable X into several components. Such that when one component with a certain set of constants reaches saturation, the next component can be used to describe the hardening effect. Applying this concept in the A-F model to decompose the back stress and parameters gives:
X = ∑ X
XË™ =cεË™ − γXpË™
The formulation (3.10) provides good results and thus will be used in this work. The number of Kinematic hardening components to be used is four. And thus, the total back stress can be calculated using (3.12).
X = X
+ X + X + X
- Isotropic hardening
During cyclic loading, a metal exhibits hardening and softening behaviour which can be de- fined using the isotropic hardening variable,Rin the material model. The isotropic hardening basically helps describe the stretching and shrinking of the size of elastic domain within the equi-potential surfaces.
- Cyclic hardening behaviour
Chaboche initially introduced the evolution equation for isotropic hardening as:
RË™ = b (Q − R) pË™
where b and Q are material parameters. p is the accumulated plastic strain. This is a differential equation of first order.
The parameter Q is a constant here, which is a measure of the maximum hardening or simply the saturation value of R. However for simulating the isotropic hardening, this saturation value needs to differ based on the loading conditions. For example, consider a strain controlled loading condition with the strain amplitude being progressively increased. Thus when the isotropic hardening reaches the saturation value for a strain amplitude after certain number of cycles, the amplitude is increased in the succeeding loadstep. With the increased strain amplitude, the isotropic hardening resumes to evolve from the last saturation value towards a new saturation state. Thus it can be established that the parameter Q needs to be a function of the last attained extrema of plastic strain, rather than be a constant. The function of Q can be phrased as Strain memory effect and is discussed in later sections in a detailed manner.
- Cyclic non-saturated behaviour
Experimental results from the uni-axial loading of austenitic steel under large strain am- plitudes show that cyclic hardening develops continuously instead of being limited by a saturation state. The Chaboche formulation in (3.13) is insufficient in describing such a behaviour. Thus a modification as suggested by Krempl  is used.
RË™ = [b(Q− R) H(qs− q1)+ aH(qs− q2) p] pË™,(3.14)
.1 ifF≥ 0
0 ifF <0
where qs refers to the radius of the strain memory surface. q1, q2(q2 > q1) are material parameters defining the extremes of qs
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