Viscoplasticity and Static Strain Ageing

Visco waxyity and Static broth AgeingViscoplasticityInelastic deformation of naturals is broadly classified into roll free lance plasticity and rate dependent plasticity. The theory of Viscoplasticity describes inflexible 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 ordinarily shown by the movement of dislocations in grain 21. From experiments, it has been established that or so metals accommodate tendency to exhibit viscoplastic doings at high temperatures. Some alloys be found to exhibit this deportment even at room temperature. Formulating the essential laws for viscoplasticity bear be classified into the corporeal approach and the phenomenological approach 23. The physical approach relies on the movement of dislocations in crystal lattice to sit around the plasticity. In the phenomenological approach, the material is considered as a continuum. And thus the microscopi c behaviour tummy be represented by the evolution of certain essential unsettleds instead. Most pretenses employ the kinematic bent and isotropic hardening variables in this respect. such(prenominal) a phenomenological approach is used in this work too.According to the neoclassic theory of plasticity, the deviatoric testes is the main contribu- tor to the yielding of materials and the volumetric or hydrostatic tenseness does non influence the inelastic behaviour. It also introduces a yield come in to differentiate the elastic and plastic universes. The size and position of such a yield surface can be changed by the strain history, to model the exact judge fix. The theory of viscoplasticity differs from the plasticity theory, by employing a serial of equi potentiality surfaces. This answers define an over-stress beyond the yield surface. The plastic strain rate is assumption by the viscoplastic flow bump. To model the hardening behaviour, introduction of several int imate variables is necessary. Unlike strain or temperature which can be bank billd to asses the stress accede, internal variable or state variables argon used to inhibit the material memory by means of evolution equations. This must take 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 release hardening behaviour of materials. In this work we consider only the dinky strain framework.The basic principles of viscoplasticity are similar to those from Plasticity theory. The main dispute is the introduction of time make. thereof the concepts from plasticity and the introduction of time dos to describe viscoplasticity, as summarised by Chabocheand Lemaitre21 are discussed in this chapter.Basic principlesConsidering depressed strains framework, the strain tensor can be split into its elastic and inelastic split = 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 revision 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 outdoor(a) volume forces fi. Thus the equilibrium condition is minded(p) asi j + fxii= 0i, j1,2,3(2.3)From the balance of heartbeat of momentum equation, we know that the Cauchy stress ten- sor is symmetric in nature. The strain tensor is reason 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 utilise 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 rate dependent plasticity is described by a series of coaxal 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 familiar near surface or the surface closest to the centre represents a null flow rate( = 0). As shown in Figure (2.1), the outer most and the farthest surface from the centre represents infinite flow rate ( = ). These 2 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 empyrean. Flow begins at this inner most surface( f=0).In Viscoplasticity, there are two types of hardening rules to be considered (i) Kinematic hardening an d (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 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 receipt from 11The isotropic hardening on the another(prenominal) hand describes the change in size of the surface and assumes that the centre and fix remains unchanged. This behaviour is due to the number of dislocations in a material and the force stored in it. It is represented by variab le 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 11From Thermodynamics, we know the free energy potential( ) to be a scalar economic consumption 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 expect that depends only on e,T,Vk. Thus we defecate= . 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 and 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 comparable to and r is given byX = ..,R = ..r(2.10)Here we nominate 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 each surface have a order allude to the strain rate.Points on each surface have the very(prenominal) profusion 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 i n a dual form as = . ,X,R T,,r.(2.11)It is a positive function and if the variables ,X,Rare zero, then the potential is also zero. The northwardrule, defined in 22 suggests that the outward averageal vector is proportional to the gradient of the yield function. Applying the due north 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.Going 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 ignored (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