. we address herein various hardening models and their suitability in computational modelling of deep drawing process wherein magnesium alloy AZ31 as blank material. insight on all basic and advanced hardening models. The basic models as well as advanced models were illustrated in usefulness of them in metal forming. It is purely depends on the researcher to select the appropriate hardening model for extracting the real computational behavior resembling the true hardening property. MS received xxxx; revised xxxx; accepted xxxx Keywords. Strain hardening, Isotropic hardening, kinematic hardening, mixed hardening, deep drawing. 1 Introduction In practice, di erent metal forming processes such as deep drawing, stamping, or bending operations are required in man- ufacture of automotive parts, beverage or food cans, pan- els of aerospace etc. In the last few decades the design of sheet metal manufacturing processes such as deep drawing had been signiﬁcantly inﬂuenced by modern tools, e.g. the large ﬁeld of numerical simulations that were based on FEM. The yield behaviour of material is essentially known and transformation of it into material model is signiﬁcant during the numerical simulation. Hardening behavior is one such an important material parameter to be considered in numer- ical modeling of the forming process for simulation. The car body components for example becomes more complex in geometry point of view and leading to a mixture of dif- ferent stress conditions during forming like deep drawing, stretching and shearing. As a consequence of these reasons the material models pertaining to hardening had become im- portant. The inferior formability and /or larger spring-back are also technical obstacles to overcome. Spring-back is a critical factor in the quality of ﬁnal products, making the de- signing of forming tools more di cult and expensive. Since the sheet spring-back is the elastic unloading response af- ter complex, large-strain deformation paths such as those en- countered in sheet metal forming operations [1], its accurate simulation requires a proper constitutive description incorpo- rating complex behavior such as Bauchinger e ect, transient behavior, permanent softening etc. The Computational mod- els considering all these e ects can strive to replace the ex- pensive and time consuming trial and error methods in study of forming behavior that were used so far in conventional de- sign. Through understanding of di erent models to be used for inducing the hardening behavior in constitutive equations *For correspondence: acsreddy64@gmail.comis of complex task in computational simulation of deep draw- ing and other processes had successfully implemented in the last couple of decades. The ﬁnite Element Methods (FEM) is quite successful to simulate metal forming processes, but ac- curacy of the drawn parts depends upon the constitutive laws and their parameters identiﬁcation. The development of ma- terial models and their use in computational plasticity is of much importance in reducing the cost and time.[2,3] Metals loaded beyond the yield point deform plastically and it is often essential to predict the behavior of plastic de- formation that necessitates formulations to be used in compu- tational models for evaluation and assessment of sheet metal deformation behavior. The di erent hardening models that were used for computer simulation are described in the next section for proper understanding the deformation behavior. 2 DIFFERENT TYPES OF HARDENING MODELS There are various models that were developed for evaluation of the inﬂuence of strain hardening and other factors inﬂu- encing of deformation. The di erent types of models studied in this paper are as follows. 1. Perfectly elastic model 2. Ideal rigid plastic material model 3. Elastic- perfectly plastic model 4. Hardening material models under monotonic loading (a) Rigid linear hardening model (b) Elastic linear hardening model (c) Ludwig power law 5. Hardening models under cyclic loading (a) Isotropic hardening model (b) Kinematic hardening model and (c) Mixed hardening model. 2333 2334 author name for running head Figure 1 . Perfectly elastic model Figure 2 . Rigidly Plastic material model 2.1Perfectly elastic model A basic engineering notion is that material behavior in the ﬁrst stage of deformation is approximately elastic. i.e., the material returns to its initial stage after the external cause (force) is removed. The perfectly elastic material which sig- niﬁcantly obeys Hooks law is as shown in ﬁg. and follows the law =E: (1) Brittle materials like glass ,ceramics and some other cast irons can be modeled with this model. For that have short rupture elongation( upto 2%) and goes to ruptrue immedi- ately after yield point perfectly elastic model is used. This model is is only suitable where the deoframation is within the elastic deforamation and is not suitable for sheet metal forming in which plastic deformation is the main essentiality in bringing permanent deofrmation. 2.2Rigid Plastic Model The deformation should at least partially permanent. For metals, this pattern of permanent deformation is known as plasticity. A rigidly plastic material does not have any elas- tic nature of deformation and it is rigid up to yield point. Once the applied stress reaches the yield limit, the material subjected to 100% plastic deformation and this deformation continuous further without any increment of the stress ap- plied. The rigid plastic material model is as shown in the Fig. 2. In this rigidly plastic material model the specimen exhibits no deformation until the applied stress reaches the critical value i.e., the yield point. As soon as the applied stress in the specimen reaches the yield limit the deforma- tion takes place uninterruptedly as long as the applied stress Figure 3 . Elastic – perfectly plastic material model is maintained at yield stress(Y). if the stress lessened due to unloading the deformation seizes. 2.3Elastic-Perfectly Plastic Material model An elastic – perfectly plastic material model is one which takes into account the elastic nature of the material and does not account for strain hardening of the material. While us- ing this model, the stress increment is linear until the yield point, and beyond the yield point the material o ers no re- sistance to deformation as well as no strain hardening of the material and hence uninterrupted deformation takes place as long as the stress applied is maintained at yield point. Per- fectly elasto–plastic material can yield a constant stress state that exhibits no hardening and essentially the yield surface remains constant. 2.4Hardening models under cyclic loading As a material yields, hardening deﬁnes the change of yield surface with plastic straining. Experiments show that if you plastically deform a solid, then unload it, and then try to re- load it so as to induce further plastic ﬂow, its resistance to plastic ﬂow will have increased. This is known as strain hard- ening. The clear understanding of stress-strain behavior of metals under a cyclic loading is very essential for computa- tional modeling of materials. There are many di erent mod- els that have been developed for the case of cyclic plasticity. Material models developed for correct description of cyclic plasticity is complicated due to cyclic hardening or softening e ect. Some materials show very strong cyclic softening (ex. Copper ) or cyclic hardening (ex. Stainless steels) and others less pronounced steels(medium carbon steels). A ductile ma- terial subjected to stresses beyond the elastic limit will essen- tially yield for larger plastic deformations. The phenomenon of hardening models is very essential to describe the com- putational deformation behavior of the material. Indeed, the appropriate hardening model is necessarily being selected for accurate description of the ﬂow stress as well as the yield- ing behavior of the material. In essence, the hardening phe- nomenon occurs due to micro level behavioral changes dur- ing dislocation slip between crystallographic planes. Flow stress increases with increasing plastic deformation and it is manifested due to hardening tendency of the material. In short title for running head 2335 Figure 4 . Rigid linear work hardening model Figure 5 . Elastic linear work hardening model general, the hardening behavior can be classiﬁed into two ba- sic types that deal with translation and expansion of the yield surface. Distortional hardening is yet another type of hard- ening behavior based on which the shape of the yield sur- face evolves during plastic deformation. The very important hardening models that were used are 1) isotropic hardening 2) kinematic hardening and 3) mixed hardening models. 2.5RIGID LINEAR WORK HARDENING MODELS In the below ﬁgure true stress-true strain diagram of a rigid linear work hardening material is given. In such a material deformation is not observed until tensile stress reaches to yield point. When tensile stress reaches to yield point plastic deformation starts and in order to increase deformation stress should be increased also. In this model stress varies linearly with plastic strain (linear work hardening). As in rigid per- fectly plastic model elastic deformation is neglected in this model. This model is applied to plastic bending analysis of beams. 2.6Elastic linear work hardening model This model shows elastic linear hardening behavior. 2.7 Ludwig Power Law Some empirical equations that ﬁt to the experimentally ob- tained true stress-true strain curves have been developed. One of them is developed by Ludwig and valid in con- stant temperature and strain rate situations; =Y+Hn(2) where Y: Yield strength H: Material dependent strength coe cient Figure 6 . Ludwig power law Figure 7 . Di erent forms of Ludwig law (for Y =0) n: Work hardening power. For di erent work hardening power values di erent stress- strain curves can be generated as shown in Fig. 2.8Swift power law The work hardening law recommended by Swift dipicted by the following relation =A(B+)n(3) Where B: Prestrain coe cient n: Work hardening power i.e., a measure of work hard- ening A: is a function of direction of stress. In the operations where the large deformations take place the Swift law yields results closer to the real deformation pro- cess. However, the Swift law is more complex than other models. Isotropic hardening It is evident that the easiest way to model the strain hard- ening behavior is to consider the yield surface to increase in size but its shape remains constant while plastic deformation. This method is easy to implement in numerical simulation code and has been widely used in describing the work hard- ening behavior of sheet metal in stamping process like deep drawing. However, the drawback with this method is that it over predicts the yield stress in reverse loading. At any point 2336 author name for running head Figure 8 . : Swift curve Figure 9 . (a) isotropic hardening (b) Schematic equivalent stress-strain curve considered on the surface of the material loaded for deforma- tion can be subjected to any one among the three conditions may occur i.e., unloading, neutral loading, and loading. The stress state under unloading can move back to original sur- face to the elastic domain. In this condition, plasticity will not occur. If neutral loading occurs, the state of stress will move on the yield surface, causing no plasticity to occur. A material plastically deformed under applied load can be re- versed by reversed load. Upon reloading after reversed load manifests for higher (increased) yield stress in comparison to initial loading and it’s elastic deformation limit will increase in the subsequent loading. This increase is of same nature for both tension and compression and can be termed as isotropic hardening. In other wards isotropic hardening is a type of hardening when the initial yield surface expands uniformly in all direc- tions in stress space during plastic deformation as shown in the Fig below. Basically, isotropic hardening means that a material loaded in tension past yield, when it is unloaded and then subjected to compression, it will not yield in compres- sion until it reaches the level past yield that reached when loading it in tension. In other words if the yield stress in ten- sion increases due to hardening the compression yield stress grows the same amount even though you might not have been loading the specimen in compression. Isotropic hardening states that the yield surface expands uniformly during plas- tic ﬂow. Isotropic hardening is often used for large strain or proportional loading simulations. In essence the isotropic Figure 10 . Evolution of yield surface under Isotropic hardening in biaxial stress space hardening is where the yield surface remains the same shape but expands with increasing stress as shown in above Fig 99. In isotropic hardening, the yield function can be deﬁned as: fi j=Y2(4) where Y denotes the yield stress of the material. Y also rep- resents the radius of the yield locus. As the magnitude of Y increases, the yield locus expands in all directions. Y can be expressed as a function of strain hardening quantities such as the plastic work per unit volume or the equivalent incremen- tal plastic strain. The yield function can also be expressed by taking hard- ening factor K into account as f(i j;Ki)=fo(i j)K=0 (5) According to Swift, the general law can be as follows. f=C(+o)n(6) wherefis ﬂow stress, is the plastic strain, 0is the elas- tic strain and n is strength coe cient The yield surface ac- cording to V on Mises yield criteria, the initial yield the yield surface can be deﬁned as f0(i j)=1p 2p (12)2+(23)2+(31)2Y (7) where Y is the yield stress in uni-axial tension. In biaxial testing any combination of stress inside the ini- tial yield surface, inner ellipse in the Fig 10 is in the elastic region. Once the part is taken beyond the initial yield sur- face, the part experiences plastic deformation. With isotropic hardening, the center of the yield surface remains ﬁxed but the size of the surface increases. Any stress state inside the new yield surface (outer ellipse in Fig 10) will experience elastic deformation and new plastic deformation occurs when the stress state reaches outer ellipse. Isotropic hardening model is an appropriate model for the e ective simulation of monotonous processes in which short title for running head 2337 the load direction does not change. The di erent types of isotropic hardening laws suitable for elasto plastic material are as depicted in the table 1. Among the above isotropic Table 1 . Di erent Isotropic hardening models S.No Hardening Equation Name of the Law 1iso=o+KpnLudwig (1909) 2 iso=KpnHollomon (1944) 3iso=o+Q(1ebp) Ludwig (1909) hardening functions the V oce and Swift laws have been widely used as hardening deﬁnitions for sheet metals. The V oce hardening law is widely used for steels which generally ex- hibit a saturating hardening behavior in a sense that the rate of increase of stress decreases with additional plastic defor- mation and turns into zero at very high plastic strains. On the other hand, the Swift hardening law is preferred for most aluminum alloys that exhibit a non-saturating hardening be- havior. A combined Swift-V oce (CSV) law incorporates both saturating and non-saturating hardening functions through the use of a weighting factor. This results for a superior ﬁt for both steels as well as aluminum alloys. It is to be noted that appropriate selection of isotropic hardening function gives appropriate ﬁt for monotonic loading conditions such as un- axial tensile test conditions for all orientations of the test sample with respect to the rolling direction. But many metal forming processes involve reversed loading (e.g. imposed by draw beads in deep drawing) or other non-proportional loading conditions (e.g. imposed by a sequence of stamp- ing processes). In case of using an isotropic hardening func- tion for such load reversal conditions, the hardening in re- versed loading is overestimated resulting in an exaggeration of the predicted springback and residual stresses. Using a kinematic hardening function might be an alternative. This is discussed in the following section. But the Bauschinger e ect can be observed in cyclic loading is the reason for not using of isotropic hardening models for cyclic loading. Nomenclature yYield stress B1Alphabet b Y Yield strength n Strength hardening exponent 3 Conclusion Various hardening models were described in this article and it is essential to select suitable hardening model during the simulation of sheet metal operations Acknowledgement I thank the Principal and the management of Sreyas Institute of Engineering and Technology for the research support in bringing this paper.