BILINEAR ISOTROPIC AND BILINEAR KINEMATIC HARDENING

Venkata Sai Prashanth Sudulaapplication. AZ60 grade is used mostly in the automotive wheel as elongation properties inthe alloy are high and toughness in material is considerably good with tensile and yieldcombined properties. While AZ91 and AZ60 grades have limited range of application due tovery minimal extrusion property, AZ31 grade application is high in automobile and aerospaceparts in view of its high extrusion capable property due to the high yield strength.Table 2 The mechanical properties of magnesium AZ91 and AZ31 alloy are outlined in the followingtable. (Material, 2012)The mechanical properties of magnesium AZ91 and AZ31 alloys are outlined in thefollowing table.AZ91PropertiesTensile strengthYield strength (atstrain 0.200 %)Elastic modulusShear modulusPoisson's ratioElongation at break (in50mm)Hardness, BrinellAZ31Metric230 MPaImperial33400 psiMetric260 MPaImperial37700 psi150 MPa21800 psi200 MPa29000 psi44.8 GPa17 GPa0.356500 ksi2470 ksi0.3544.8 GPa17 GPa0.356498 ksi2470 ksi0.353%3%15%15%636349492.3. Yielding and Yield StrengthAccording to Callister, most of the materials are designed to ensure only elastic deformationwhen stress is applied. Every element has a transition point in between elastic and plastictransformation which is known as proportional limit denoted by P in the graph. The point atwhich plastic deformation starts after elasticity is called yield. The stress at which the stressstrain curve bends in the plastic region is defined as plastic strength y. The elastic-plasticstrain transition is well defined and occurs abruptly. The abrupt occurrence of elastic-plasticstrain transition is termed as yield point phenomenon (Callister &Rethwisch, 2007). Theupper yield point is a point at which elastic strain changes to plastic strain. And when thestrain value has been increased, the stress will fluctuate at lower yield point. The averagestress associated with lower yield point is considered as yield strength. Eventually, acomponent that has experienced a permanent change in shape may not be capable offunctioning as intended in Figure1(a), Figure 1(b).Figure 1 (a) Typical stress strain behaviour for metal showing elastic and plastic deformation, the proportionallimit P, the yield strength σy as determine using the 0.002strain offset method. (b) represented stress-strainbehaviour found for some steels demonstrating the yield point phenomenon (Callister & Rethwisch, 2007)2.4. ditor@iaeme.com

Bilinear Isotropic and Bilinear Kinematic Hardening of AZ31 Magnesium AlloyDuctility is one of the important mechanical properties under plastic deformation. It is ameasure of the degree of plastic deformation that has been sustained during fracture. Amaterial with no plastic deformation is termed as brittle. The tensile stress strain behavioursfor both brittle and ductile are explained.Ductility can be expressed either as a percentage elongation or percentage reduction inarea or percentage elongation of plastic strain at fracture. The magnitude of percentageelongation depends on specimen gauge length. The shorter the original length, the greater isthe fracture of total elongation from neck and higher the value of percentage elongation andvice versa. The plastic deformation at fracture is confined to neck region.𝑙𝑓 𝑙𝑜% 𝐸𝐿 () 100𝑙𝑜Where 𝑙𝑓 fracture length𝑙𝑓original gauge length Percentage reduction in area is defined as ratio of final area and initial area of thespecimen.𝐴𝑜 𝐴𝑓) 100% 𝑅𝐴 (𝐴𝑜To determine the cyclic deformation of the material it is essential to understand materialbehaviour with the help of cyclic plasticity model. The cyclic plasticity model is generallybuilt by three components: Yield function Flow rule Hardening ruleEmbraco detailed about the bilinear isotropic hardening (BISO) uses von mises yieldcriteria coupled with an isotropic hardening assumption. The stress strain curve uses just twolines; one is to determine elastic region and the other is to determine plastic region. Multilinear Isotropic hardening (MISO) uses multiple curves to determine elastic and plasticcurves. (Embraco, 2006). In his paper, the author discussed about characteristics of both BISOand MISO curves. He stated that use of these curves will help to get better results by changingthe mesh characteristics of the specimen. The difference between both the curves areexplained below in Figure 2.Figure 2 Uni axial behaviour of BISO & MISO curves (Embraco, r@iaeme.com

Venkata Sai Prashanth Sudula2.5. Bilinear (Isotropic, Kinematic) Hardening CurveIn the bilinear isotropic hardening process, both stress and strain change even after reachingmaximum plastic deformation. But the change in shape cannot be observed clearly in thebilinear hardening process. In this simulation, inputs are the yield strength and the tangentmodulus from the stress-strain curve of the material.The model for bilinear hardening is described by Pragers equation and the yield details forthe equation are defined by Fredrick Armstrong model. The Fredrick Armstrong model isChaboche model and hardening equation is described with strain hardening variables. 𝑓𝑦 is thelinear function of 𝐼2 which is the stress invariant for metals. Equation by Pragers is givenbelow,𝑓 𝐼2 (𝜎 𝑋) 𝑘Where 𝜎 is the vector form of stress, k is the yield point and hardening point is the X.(Viktor Budaházy / László Dunai, 2013)But the bilinear kinematic hardening is collinear with strain of plastic, so the equation is2𝑓 𝐼2 (𝜎 𝑋) 𝑘 𝑤ℎ𝑒𝑟𝑒 𝑑𝑋 . 𝐶. 𝑑𝜀𝑃𝐿3Where dX is the increment of hardening tensor, 𝜀𝑃𝐿 is the plastic tensor.2.5.1. Tangent ModulusKelly explained the tangent modulus as the slope of the stress and strain curve in the plasticregion which will change due to deformation. (Kelly,2019) Tangent modulus is equivalent toyoung’s modulus when the limit lies in the permissible value of stress strain curve. Tangentmodulus is mostly used to define the stiffness of the material commonly denoted by Et, Figure3.Figure 3 Tangent modulus𝑦2 𝑦1𝑥2 𝑥1The points (1.301,600) and (313.2,0.7264) from the stress strain curve tangent are derivedfrom curve fit option in MATLAB.600 313.2𝐸𝑡 1.301 0.7264𝐸𝑡 500𝑀𝑃𝑎𝐸𝑡 me.com

Bilinear Isotropic and Bilinear Kinematic Hardening of AZ31 Magnesium AlloyAfter calculating the values from the graph, the obtained tangent modulus is 500MPa.This value is used in ANSYS in bilinear simulation to determine plasticity of AZ31 alloy withmesh size as 0.4mm and 0.6mm with element size as linear and quadratic rials/strength chars/tangent.htm (Dated on 18/05/2020)The stress and strain curve of monotonic test result used in MATLAB using curve fit toolmatched the stress and strain curve with 8-degree polynomial equation. Without changing anyvalues of monotonic result curve these points are used to calculate true stress and plastic strain(Figure 4).Figure 4 Bilinear (isotropic, kinematic) hardening curve in MATLAB – curve-fit toolIn this paper BISO curve is used to assess cyclic behaviour instead of other hardeningproperties to observe and understand plasticity of AZ31. To obtain tangent modulus of AZ31alloy, MATLAB curve fit tool is used to draw single degree polynomial curve. The stressstrain curve is plotted by using adjusting the linear least squares. From a point around stress250MPa, a tangent is drawn using curve fit tool to determine tangent modulus for bilinearhardening for input value in ANSYS.3. MATERIAL AND MATERIAL PROPERTIESTo study plasticity of AZ31 type magnesium material, cyclic loading behaviour is applied.The chemical composition of AZ31type magnesium alloy used in this study is given in table3. The yield strength of AZ31 alloy is 250 MPa, compression stress is 97MPa, ultimate tensilestrength is 296MPa, the poison’s ratio is 0.32.Table 3 Composition of AZ31 ce: https://www.azom.com/article.aspx?ArticleID 6707Ni0.1Other0.4MgBalance4. METHODOLOGYIn isotropic hardening, yield surface expands uniformly in all directions with plastic flow.Using stress-strain curve of monotonic tensile test, the yield strength is observed as 250MPaand tangent modulus calculated as 500MPa. From the above Figure 5 we can notice bilinear(isotropic, multi-linear) hardening or@iaeme.com

Venkata Sai Prashanth SudulaTo perform this simulation has considered static structural analysis with 466 sub steps forinput. One end of the specimen is fixed and at other end of the specimen continuous cyclicdisplacements are applied. From monotonic tensile test huge data few displacement points areconsidered as input for cyclic loading condition with time as independent variable.Displacement from 1mm to 7.5mm has been given to cyclic loading condition. Strain hasbeen increased from 0 –1mm to 1mm to 0, repeat this procedure from 0-2mm to 2mm-0 andcontinue same procedure up to 7.5mm to make input as cyclic loading.Figure 5 Bilinear isotropic hardening and bilinear kinematic hardening curve data input in ANSYS5. BILINEAR ISOTROPIC HARDENING AND BILINEAR KINEMATICHARDENINGIn bilinear isotropic hardening and bilinear kinematic hardening, cyclic loading behaviouracting on AZ31 alloy using linear and quadratic mesh properties is discussed. In this group ofsimulations, when a load is applied on specimen, it stores the energy in specimen as adisplacement from cycle to cycle. It implies that at the end of the simulation, the stress actingon specimen is not that high when compared to bilinear isotropic hardening.5.1. Bilinear Isotropic Hardening - 0.4mm Linear MeshMaximum principal stress vs plastic strain is plotted with the help of chart tool in ANSYSsoftware. In this simulation, the maximum principal stress is 2670 MPa during tensile loadingand the minimum principal stress during compression is -1309.7MPa can be observed with0.4mm mesh size using linear element type. The maximum middle principle stress duringtensile cycle is 2113.8 MPa and during last compression cycle maximum-middle principlestress -1777 MPa. The maximum equivalent plastic strain of 0.356mm can also be observed inthis plot in Figure aeme.com

Bilinear Isotropic and Bilinear Kinematic Hardening of AZ31 Magnesium AlloyFigure 6 Bilinear isotropic hardening – 0.4mm linear mesh, Maximum principal stress vs Equivalentplastic strain, Middle principal stress vs Equivalent plastic strain5.2. Bilinear isotropic hardening – 0.6mm linear element meshThe plot is between maximum principal stress and equivalent plastic strain of AZ31 specimenafter cyclic loading. The maximum principal stress by the end of tensile displacement is2197.7MPa, and the maximum stress during compression cycle is -1198.6MPa by the end ofthe simulation. The tensile and compressive loading acting on specimen is observed.Maximum middle principle stress by the end of eighth tensile cycle is 1419.7MPa and duringcompressive displacement, the middle principle stress is -1550.1MPa. Plastic strain observedduring this cyclic loading is 0.3845mm. Eventually, at step 466 the permanent strain in thespecimen is around 0.06 mm (Figure 7).Figure 7 Bilinear isotropic hardening – 0.6mm linear mesh, Maximum principal stress vs Equivalentplastic strain, Middle principal stress vs Equivalent plastic strain.5.3. Bilinear Isotropic hardening – 0.4mm quadratic meshFigure 8 Bilinear isotropic hardening – 0.4mm quadratic mesh, Maximum principal stress andequivalent plastic strain, Middle principal stress and equivalent plastic strainThe Figure 8 depicts plastic strain in 0.4mm quadratic mesh specimen under bilinearisotropic hardening curve. In this simulation, when the plastic strain increases with increase intime, it is observed that the strain during 8th cycle is 0.375mm. The maximum principle or@iaeme.com

Venkata Sai Prashanth Suduladuring last tensile cycle is 3213.2MPa and during compression cycle, the maximum stress isaround -1143.1MPa. The graph is plotted between middle principal stress and equivalentplastic strain. The maximum stress observed during 8th cyclic loading under tensile conditionis1927.8MPa. The maximum stress under compression cycle by the end of the simulation is 2061MPa with plastic strain being 0.375mm.5.4. Bilinear Isotropic hardening – 0.6mm quadratic meshThe plot in Figure 9, is between maximum principal stress (MPa) and equivalent plastic strain(mm). In this simulation, the increase in stress during tensile cycle for the first four cycles isbelow 1000MPa but later, the increase in stress per cycle is around 500MPa. The final cyclestarted at around 2000MPa and closed at 2850MPa during tensile cycle. The compressioncycle ends at around -1124.5MPa by the end of simulation. The equivalent plastic strainduring 1st two cycles is 0.1mm and by the end of 8th cycle, the plastic strain increased by0.05mm. During last cycle, we can observe the equivalent plastic strain at around 0.371mm.Under cyclic loading condition the middle principal stress acting on AZ31 during tensile andcompression cycles is almost same. The maximum middle principal stress during tensile lastcycle is 1552MPa and during compression cycle the maximum stress is -1660MPa.Figure 9 Bilinear isotropic hardening – 0.4mm quadratic mesh, Maximum principal stress vsequivalent plastic strain, Middle principal stress vs equivalent plastic strain5.5. Bilinear kinematic hardening - 0.4mm linear meshIn this simulation mesh size assigned to specimen is 0.4mm with linear element type and theresults plotted from this simulation are maximum principal stress, middle principal stress andequivalent plastic strain.Figure 10 Bilinear kinematic hardening – 0.4mm linear mesh, Maximum principal stress vsEquivalent plastic strain, Middle principle stress vs Equivalent plastic strain.The above Figure 10 presents the plot between maximum principle stress and equivalentplastic strain acting on AZ31 under cyclic loading condition. The maximum tensile loadingobserved during the last cycle is around 1300MPa. In the graph, we can observe tensile or@iaeme.com

Bilinear Isotropic and Bilinear Kinematic Hardening of AZ31 Magnesium Alloyalmost following the same path for every cycle with stress variation in between 200300MPa.During compression cycle, the maximum compression stress can be observed as 162.7MPa in between 2nd to 4th cycle and by the end of experiment, the stress has beenreduced to around -100MPa. The middle principle stress versus equivalent plastic strain,observes that the maximum stress during tensile loading is around 1064MPa at the end of 8thcycle. The maximum-middle compression stress in between the cycles is observed as 210MPa and by the end of the simulation it is around 150MPa. The equivalent plastic strainacting on specimen during this simulation is 0.442mm by the end of the simulation. In thissimulation, the stress stored in specimen AZ31 after each tensile cycle can also be observed.5.6. Bilinear kinematic hardening - 0.4mm quadratic meshFigure 11 Bilinear kinematic hardening – 0.4mm quadratic mesh, Middle principle stress vsequivalent plastic strain.The above figure 11, is between middle principal stress (MPa) and equivalent plasticstrain(mm). The maximum middle principal stress during last tensile cycle is around 565MPaand maximum middle principal stress during compression cycle is around -282MPa. Whencompared to bilinear kinematic hardening 0.4mm linear mesh, stress acting on specimenduring quadratic mesh cycle is almost half. The equivalent plastic strain acting on specimen is0.4377mm. The tensile cycles are following linear plot for every cycle starting from the firstcycle, The maximum stress during tensile cycle is 850MPa. During compression cycles thestress acting on the specimen is almost 186MPa.5.7. Bilinear kinematic hardening curve - 0.6mm linear meshIn below Figure 12, the plot is between maximum principal stress (MPa) and equivalentplastic strain (mm) acting on AZ31. In this simulation, the tensile cycles are following linearpattern from first cycle to last cycle. The maximum principal stress by the end of the lasttensile cycle is observed as 566.81MPa. During compression cycles, the maximum stressobserved is around 148MPa in 2nd and 3rd compression cycles and by the end of thesimulation, the maximum compression stress is around 100MPa. The maximum stress isobserved during initial cycles during compression because in kinematic hardening curve loador displacement applied on specimen will be retained in specimen and will not come tooriginal position even after removing load or displacement. The maximum-middle principalstress can be observed during 8th tensile cycle as around 300MPa. The maximum-middlecompression is observed during 2nd cycle is -165MPa and by the end of the simulation, stressfor compression cycle is around 125MPa. In this stress strain curve, we can observe bothtensile and compression cycles with minimum stress acting on AZ31 specimen. Themaximum equivalent plastic strain observed in this simulation is ditor@iaeme.com

Venkata Sai Prashanth SudulaFigure 12 Bilinear kinematic hardening – 0.4mm linear mesh, Maximum principle stress vs equivalentplastic strain, Middle principle stress vs equivalent plastic strain.5.8. Bilinear kinematic hardening - 0.6mm quadratic meshIn this simulation, the discussion is about equivalent plastic strain acting on AZ31 usingbilinear kinematic hardening curve property. The mesh size used is 0.6mm whereas elementtype is quadratic.Figure 13 Bilinear kinematic hardening – 0.6mm quadratic mesh, Maximum principal stress andequivalent plastic strain, Middle principal stress vs equivalent plastic strain.The above graph 13 is plotted between maximum plastic strain (MPa) vs equivalentplastic strain (mm). The maximum stress acting on specimen is 786MPa during tensile cycleand -171MPa during compression cycle. The average increase in plastic strain per cycle isaround 0.6mm. The increase in stress during tensile cycle is linear whereas in compressioncycle it is almost flat. The maximum-middle principal stress acting on AZ31 during tensileloading is around 495MPa during the last cycle. The maximum-middle principal stress duringcompression cycle in the last two cycles is around -263MPa. The equivalent plastic strainacting on specimen is 0.43338mm during last cycle.6. DISCUSSION AND CONCLUSIONThe above simulation reflects the performance of cyclic loading behaviour acting on AZ31alloy to determine plastic strain. From the above graph, we can notice equivalent plastic strainacting on specimen for 2,4,6,7,8 cycles. During 2nd cycle maximum plastic strain of0.1143mm is observed while performing linear element 0.6mm mesh simulation andminimum plastic strain observed is 0.10625mm for quadratic element 0.6mm mesh. When itcomes to 4th cycle, the maximum plastic strain is observed during linear element 0.6mm meshbut minimum plastic strain is observed during linear element 0.4mm me

magnesium alloys mainly to reduce weight, increase speed and efficiency. Plasticity of magnesium alloy using multi-linear and bilinear hardening properties and the behaviour of the alloy under cycle