May 2021 Volume 3

FORGING RESEARCH

It is important to know that the plastic flow vectors are always directed along the normal to the yield surface, as shown in Figure 11, so that the plastic flow direction can be calculated once the directions of the stress vectors are known. If the von Mises yield criterion is used for f, after some mathematical operations, the relationship of the plastic stresses and strains is: It is important to know that the plastic flow vectors are always directed along the normal to the yield surface, as shown in Figure 11, so that the plastic flow direction can be calculated once the directions of the stress vectors are known. If the von Mises yield criterion is used for f , after some mathematical operations, the relationship of the plastic stresses and strains is: 1 = ̄ ̅ � 1 − 1 2 ( 2 + 3 )�, 2 = ̄ ̅ � 2 − 1 2 ( 3 + 1 )� and 3 = ̄ ̅ � 3 − 1 2 ( 1 + 2 )� (10) It can be seen that Eq. (10) has the same format as Hooke’s Law in elastic deformation. This is the second fundamental step in plastic analysis. (10) It canbe seen that Eq. (10) has the same format asHooke’s Law in elastic deformation.This is the second fundamental step in plastic analysis.

forming analysis. In some cases the mixed hardening rule, a more complex hardening rule that combines the features of both the isotropic and kinematic hardening rules, is used instead.

Figure 12. Illustration of kinematic hardening rule [5] The general form of hardening rules can be expressed as an ellipse equation, shown below (Eq. (11)): (11) where k2 is the current yield stress in pure shear and defines the size of a yield surface, α defines the location and σ controls the shape of the yield surface. For isotropic hardening, the equation becomes Eq. (12) and for kinematic hardening, it becomes Eq. (13). (12) Work hardening effects are more significant when a metal is worked at room temperature. Deformation is more difficult when a steel is work hardened because along the deformation (increasing strain), higher stresses (flow stress) are necessary to move the increasingly finer dislocation segments (dislocation cell size decreasing and density increasing) produced by the dislocation interactions. The work hardening continues to increase flow stresses with increasing strain until the ultimate tensile strength is reached. Until this point, the deformation is uniform throughout the entire piece. After this point, localized necking occurs. The work-hardening curve is usually obtained by experiment. In cold forming, the strain hardening is not relieved and the flow stress increases continuously with deformation. Therefore, the total deformation before fracture in cold forming is much less than that of hot forming. To achieve a larger deformation, sometimes intermediate annealing is applied. The general form of hardening rules can be expressed as an ellipse equation, sh � , � = � − � − 2 = 0 where k 2 is the current yield stress in pure shear and defines the size of a yield location and σ controls the shape of the yield surface. For isotropic hardening, Eq. (12) and for kinematic hardening, it becomes Eq. (13). � , � = � � − 2 = 0 � , � = � − � = 0 Work hardening effects are more significant when a metal is worked Deformation is more difficult when a steel is work hardened because a (increasing strain), higher stresses (flow stress) are necessary to move dislocation segments (dislocation cell size decreasing and density increas dislocation interactions. The work hardening continues to increase flow stresse until the ultimate tensile strength is reached. Until this point, the deformation the entire piece. After this point, localized necking occurs. The work-harde obtained by experiment. In cold forming, he strai h rdeni is not relieved and the flow stress increa deformation. Therefore, the total deformation befor fracture in cold forming of hot forming. To achieve a larger deformation, sometimes intermediate anne Influence of strain rate and temperature on flow stress Many metals and steels display a behavior in which their flow stress increases rate, as shown in Figure 13, but decreases with increasing working tempe addition, the influence of the strain rate on the flow stress increases wit Figure 12. Illustration of kinematic hardening rule [5] The general form of hardening rules can be expressed as an ellipse equation, sh � , � = � − � − 2 = 0 where k 2 is the current yi ld stress in pure shear and defines the size of a yield location a σ cont ols the shape of the yield s rface. For isotropic hardening, Eq. (12) and for kinematic hardening, it becomes Eq. (13). � , � = � � − 2 = 0 � , � = � − � = 0 Work hardening effects are more significant when a metal is worked Deformation is more difficult when a steel is work hardened because a (increasing strain), higher stresses (flow stress) are necessary to move isl ti segments (dislocation cell size decreasing and density increas d slocation interactions. The wo k hardening c tinu s to increase flow stresse until th ultimate t nsil strength is reached. Until this po t, the def rmation the entire piece. After this point, localized necking occur h work-harde obtained by experime t. In cold forming, the strain hardening is not relieved and the flow stress increa deformation. Therefore, the total deformation before fracture in col forming of hot forming. To achieve a larger deformation, sometimes intermediate anne Influence of strai rate and temperature on flow stress Many metals and st els display a behavior in hich their flow stress increases rate, as shown in Figure 13, but decreases with increasing working tempe addition, the i flue ce of th strain rate on the flow stress increases wit (13) Figure 12. Illustration of kinematic hardening rule [5]

Figure 11. Illustration of flow rules [5]

Figure 11. Illustration of flow rules [5]

4.4 Hardening Rules In this fundamental step in plastic analysis, the hardening rules describe “work hardening,” a behavior seen in many metals and steels wherein yield stress increases with further plastic strain. In plasticity’s terms, the hardening rules illustrate how the yield surface changes with changing stress levels. There are two major hypotheses for hardening rules. One is isotropic hardening, which assumes that the yield surface remains the same shape and stays at the same origin but expands with increasing stresses. The other is kinematic hardening, which assumes that the yield surface remains the same shape and size but translates to another location in the stress space. Figure 12 explains the kinematic hardening rule. As the stress point moves along its loading path fromA to B, the yield surface translates (no rotation) as a rigid body. It represents the most current yield function. When unloading, the material behaves elastically from B to C but then begins to flow again before the stresses are completely relieved, which is due to the Bauschinger effect described previously. Of the two, the isotropic hardening rule is more often used in metal Hardening rules In this fundamental step in plastic analysis, the hardening rules describe “work hardening,” a behavior seen in many metals and steels wherein yield stress increases with further plastic strain. In plasticity’s terms, the hardening rules illustrate how the yield surface changes wit cha ging stress levels. There are two major hypotheses for hardening rules. One is isotropic hardening, which assum s that the yield surface remains the same shape and stays at the same origin but expand with incr asing str sses. The other is kinematic harde ing, which as umes that the yie d surfac remains the s me shape and size but translates o another lo ation in the stress spac . Figure 12 xplains the kinematic hardening rule. As the stress po t moves along its loading path from A to B, th yield surf ce translates (no rota io ) s a rigid body. It represents the most current yield fun tio . W n unloading, the material behaves elastically from B to C but then begins to flow again before the stresses are completely relieved, which is due to the Bausching r effect described previously. Of the two, the isotropic hardening rule is more often used in etal forming analysis. In some cases the mixed hardening rule, a more complex hardening rule that combines the features of both the isotropic and kinematic hardening rules, is used instead. 11

72

FIA MAGAZINE | MAY 2021