Viscoplasticity and Static Strain Ageing

Visco malleability and Static Strain AgeingViscoplasticityIn bendable de coordinateation of materials is broadly classified into rate independent plasticity and rate dependent plasticity. The theory of Viscoplasticity describes inelastic deformation of materials depending on cadence i.e. the rate at which the load is applied. In metals and alloys, the mechanism of viscoplasticity is usually shown by the gesture of dislocations in grain 21. From experiments, it has been established that around metals have tendency to exhibit viscoplastic demeanour at high temperatures. Some alloys are found to exhibit this behaviour even at room temperature. Formulating the constitutive laws for viscoplasticity back be classified into the physical approach and the phenomenological approach 23. The physical approach relies on the movement of dislocations in crystal lattice to model the plasticity. In the phenomenological approach, the material is conceptualizeed as a continuum. And thus the micro scopic behaviour can be represented by the phylogeny of certain internal variables instead. Most models employ the kinematic curing and isotropic indurate variables in this respect. Such a phenomenological approach is used in this work too. correspond to the classical theory of plasticity, the deviatoric renderes is the main contribu- tor to the conducting of materials and the volumetric or hydrostatic stress does not influence the inelastic behaviour. It also introduces a refund resurrect to differentiate the elastic and plastic domains. The coat and position of such a yield arise can be changed by the prolong history, to model the convey stress sound out. 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 lead rule. To model the hardening behaviour, introduction of several internal variables is necessary. Unlike strain or temperature which can be measurable 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 get by 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. and then the concepts from plasticity and the introduction of time effects to describe viscoplasticity, as summarised by Chabocheand Lemaitre21 are discussed in this chapter.Basic principlesConsidering small strains framework, the strain tensor can be split into its elastic and inelastic separate = e+ in(2.1)where is total strain, e is the elastic strain and in 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 = e+ p(2.2)where p is the plastic strain. Let us consider a field with stress = i j(x) and external volume forces fi. Thus the equilibrium condition is given asi j + f dozen= 0i, j1,2,3(2.3)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, uas1 .ujui.i j = 2xi+ x(2.4)The Hookes law for the relation between stress and strain tensors is given using the elastic part of the strain = E e(2.5)where e and the stress are second order tensors. E is the fourth order elasticity tensor.Equipotential surfacesIn 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 ra te dependent plasticity is described by a series of concentrical 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 21It can be understood that the inner most surface or the surface closest to the centre represents a null precipitate rate( = 0). As shown in Figure (2.1), the outer most and the farthest surface from the centre represents infinite flow rate ( = ). These two surfaces represent the extremes governed by the time independent plasticity laws. The region in between is governed by Viscoplasticity21. 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 harden ing. 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 27 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 which does not evolve continuously during cyclical 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 11The isotropic hardening on the other attain 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 du ring cyclic loadings. This can be controlled by the recovery phase. As a result, isotropic behaviour is helpful is mold 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 11From Thermodynamics, we know the free energy potential( ) to be a scalar function 21. With respect to temperature T, it is concave. But convex with respect to other internal variables. Thus, it can be defined as = . ,T,e,p,Vk.(2.6)where ,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 depends only on e,T,Vk. Thus we have= . 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 = . .e,s = ..T(2.8)where is density a nd s is entropy. It is possible to decouple the free energy function and split it into the elastic and plastic parts.= e. e,T.+ p. ,r,T.(2.9) Similar to , the thermodynamic forces corresponding to and r is given byX = ..,R = ..r(2.10)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 potentialThe equipotential surfaces that describe Viscoplasticity have some properties.Points on all(prenominal) 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 which is a convex function. It can be defined in a dual form as = . ,X,R T,,r.(2.11)It is a positive funct ion and if the variables ,X,Rare zero, then the potential is also zero. The normalityrule, defined in 22 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 p = , = ,X r =R(2.12)Considering the recovery effects in Viscoplasticity, the dissipation potential can be split into two parts = p+ r(2.13)where p is the Viscoplastic potential and r the recovery potential which are defined as p=p.. X. R k,X,R T,,r. ,(2.14)r=r. ,R T,,r.(2.15).3J2 .. . X=2 X X(2.16)where J2 . X. refers to the norm on the stress plane and kis the initial yield or the initialsize of equipotential surface. sacking back to the relation in (2.12) , we haveJ2 .X. X ==3=p(2.17)pJ2 ..2 X.Here, p is the accumulated viscoplastic strain, given by .2p = p p(2.18)3Also applying the normality rule on eq. (2.15) we may define r as r = p r(2.19)RThus when recovery is treat (i.e r = 0), r is equal to p.Perfect viscoplasticityLet us consider pure viscoplasticity where hardening is ignored. Thus the internal variables may also be removed. = . ,T.(2.20)Since plasticity is independent of volumetric stress, we may consider just the deviatoric stress = 1 tr()I. Using isotropic property, we may just use the second invariant of . Thus = . ( ),T.(2.21)Applying the normality rule here, we may obtain the flow rule for Viscoplasticity.3 ==(2.22)p2 J2 ..J2 ..From the Odqvists law 12, the dissipation potential for perfect viscoplasticity can be obtained. Here the elastic part is ignored. Thus we have =n + 1.J2().n+1(2.23)where and n are material parameters.Using this relation in the flow rule from eq.(2.22), we get.J2().n3 =p2J2 . .(2.24)Further the elasticity domain can be included through the parameter kwhich is a measure of the initial yield3 =.J2() k.n(2.25)p2J2 . .The are the Macauley brackets defined by F = F H(F),H(F) =.1 ifF0(2.26)0 ifF