May 2021 Volume 3

FORGING RESEARCH

Both the Tresca and von Mises yield criteria state that the metal will yield when the combination of the principal stresses equals the shear yield stress, no matter whether it is under a uniaxial loading or a complex 3D stress state. For example, yielding occurs in a uniaxial tension (1D) when σ 1 = σ 0, σ 2 = σ 3 = 0. The Tresca and von Mises criteria are, respectively: (4) where σ 0 is the yield stress in a simple tension. Substituting σ ̄ into Eq. (4) for σ 0 and then substituting Eq. (4) into Eqs. (2) and (3) for k, a general form of the Tresca yield criterion (Eq. (5)) and vonMises yield criterion (Eq. (6)) are obtained, respectively: Both th Tresca nd von Mises yield c iteria state th t the metal will yiel the principal stre es equals the shear yield stress, no matter whether it is a complex 3D stress state. For example, yielding occurs in a uniaxial tens σ 3 = 0. The Tresca and von Mises criteria are, respectively: = 0 2 for Tresca and = 0 √3 for von Mises w σ i h yield stress in a simpl tension. Substituting ̄ into Eq. (4) . ( into Eqs. (2) and (3) for k , a general form of the Tresca yield criter yield criteri n (Eq. (6)) are obtained, respectively: ̄ = 1 − 3 where 1 > 2 > 3 ̄ = √ 1 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 where ̄ = 2 or √3 , which states that if the magnitude of ̄ reaches a cr appli d stress state will cause yielding. Since ̄ can represent any state of t effec ve stress . In metal forming, the von Mises yield criterion is more of accurate when the middle principal str ss σ 2 is involved. Similarly, for the von Mises criterion, the effective strain or plastic strain ̅ = √ 3 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 Flow rules After yielding occurs, the material enters a plastic flow state. Flow rul define the direction and magnitude of the plastic straining or flow unde relationship of the stresses and plastic strains, also known as the Levy-M Eq. (8): = where dλ is a non-zero positive scalar loading parameter for the magnitu is a yield criterion or yield surface. This flow rule is now called an “asso is associated with a specific yield criterion. The second term is the d under such the plastic stresses. The dλ can be expressed as in Eq. (9) afte stress (Eq. (6)) and plastic strain increment (Eq. (7)). = 3 2 � ̅ Both the Tresca and von Mises yield criteria state that the metal will yie the principal tresses equals the shear yield stress, no matter whether it i a complex 3D stress state. For example, yi lding o curs in a uniaxial ten σ 3 = 0. The Tresca and von Mises criteria are, respectively: = 0 2 for Tr sca and = 0 √3 for von Mises where σ 0 is the yield stress in a simple tension. Substituting ̄ into Eq. (4 Eq. (4) into Eqs. (2) and (3) for k , a general form of the Tresca yield crite yield criterion (Eq. (6)) are obtained, respectively: ̄ = 1 − 3 where 1 > 2 > 3 ̄ = √ 1 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 where ̄ = 2 or √3 , which states that if the magnitude of ̄ reach s a applied stre s state will cause yielding. Sinc ̄ can rep esent any state f effective stress . In metal formi g, he von Mises yield criterion is more o ccurat when the middle principal stress σ 2 s involved. Sim larly, for the von Mises criterion, the effective strain or plastic strai ̅ = √ 3 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 Flow rules After yi l ing occurs, the material enters a plastic flow state. Flow ru define the direction and magnitude of the pl stic strai ing or flow und relationship of the stresses and plastic strains, also known as the Levy-M Eq. (8): = where dλ is a non-zero po itive scalar loading parameter for the m gnit is a yield criterion or yield surface. This flow rule is now called an “ass is associated with a specific yield criter on. The s cond term is the under such the plastic stresses. The dλ can be expressed as in Eq. (9) af stress (Eq. (6)) and plastic strain increment (Eq. (7)). = 3 2 � ̅ Both the Tresca and von Mises yield criteria state that the metal will yield whe the principal stresses equals the shear yield stress, no matter whether it is under a complex 3D stress state. For example, yielding occurs in a uniaxial tension (1 σ 3 = 0. The Tresca and von Mises criteria are, respectively: = 0 2 for Tresca and = 0 √3 for von Mises where σ 0 is the yi ld stress in a simple tension. Substituting ̄ into Eq. (4) for σ 0 Eq. (4) into Eqs. (2) and (3) for k , a general form of the Tresca yield criterion (E yield criterion (Eq. (6)) are obtained, respectively: ̄ = 1 − 3 where 1 > 2 > 3 ̄ = √ 1 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 where ̄ = 2 or √3 , which states that if the magnitude of ̄ reaches a critical applied stress state will cause yielding. Since ̄ can represent any state of the stre effective stress . In metal forming, the von Mises yield criterion is more often u accurate when the middle principal stress σ 2 is involved. Similarly, for the von Mises criterion, the effective strain or plastic strain incre ̅ = √ 3 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 Flow rules ft yielding occurs, the material enters a plastic flow state. Flow rules hav define th direction and magnitude of the plastic strainin or flow under the relationship of the stresses and plastic strains, also known as the Levy-Mises e Eq. (8): = where dλ is a non-zero positive sc lar loading parameter for the magnitude of is a yield criterion or yield surface. This flow rule is now called an “associated is associated with a specific yield criterion. The second term is the directi under such the plastic stresses. The dλ can be expressed as in Eq. (9) after con stress (Eq. (6)) and plastic strain increment (Eq. (7)). = 3 2 � ̅ Both the Tre ca and von Mises yi ld criteria state that the metal will yield wh the principal stresses equals th shear yield stress, no matter whether it is u de a complex 3D stress state. For example, yielding occurs in a uniaxial tension (1 σ 3 = 0. The Tresca and von Mises criteria are, respectively: = 0 2 for Tresca and = 0 √3 for von Mises where σ 0 is the yield stress in a simpl t nsion. Substituting ̄ into Eq. (4) for σ 0 Eq. (4) into Eqs. (2) and (3) for k , a general form of the Tresca yield criterion (E yield criterion (Eq. (6)) are obtained, respectively: ̄ = 1 − 3 where 1 > 2 > 3 ̄ = √ 1 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 wher ̄ = 2 or √3 , which states that if the magnitud of ̄ reaches a critical applied stres state will cause yielding. Since ̄ can represent a y state of th str effective stress . In etal forming, the von M es yield criterion is more often u accurate when the middle principal stress σ 2 is involved. Similarly, for the von Mises criterion, the effective strain or plastic strain incre ̅ = √ 3 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 Flow rules After yielding occurs, the material enters a plastic flow state. Flow rules a define the direction and m gnitude of he plastic straining or flow under th relationship f the stresses a d plastic strains, also known as the Levy-Mises e Eq. (8): = where dλ is a n -zero positive scalar loading parameter for the agnitude of is yield criterion or yield surface. This flow rule is now called an “associated is associated with a specific yield criterion. The second term is the directi under such the plastic stresses. The dλ can be expressed as in Eq. (9) after con stress (Eq. (6)) and plastic strain increment (Eq. (7)). = 3 2 � ̅ Both the Tresca and von Mises yield criteria state that the metal wil yield whe the p incipal str sses equals the shear yield stres , no matter whether i is under a complex 3D stress state. For example, yielding occurs in a uniaxial tension (1 σ 3 = 0. The Tresca and von Mises criteria are, respectively: = 0 2 for Tresca and = 0 √3 for von Mises where σ 0 is the yield stress in a s mple tension. Substituting ̄ into Eq. (4) for σ 0 Eq. (4) to Eqs. (2) and (3) for k , a general form of the Tresca yield criterion (E yield criterion (Eq. (6)) are obtained, respectively: ̄ = 1 − 3 where 1 > 2 > 3 ̄ = √ 1 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 wh re ̄ = 2 or √3 , which states that if the magnitude of ̄ reaches a critical pplied stress stat w ll cause yielding. Since ̄ can represent any state of the stre effective stress . In metal forming, the von Mises yield criterion is more often us accur te when the middle principal stress σ 2 is in olved. Similarly, for the von Mises criterion, the effective strain or plastic strain incre ̅ = √ 3 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 1 2 F ow rules After yielding occurs, th materi l enters a pla tic flow state. Flow rule hav define the direction and magnitude of the plastic straining or flow under the relationship of the stresses and plastic strains, also known as the Levy-Mises e Eq. (8): = where d is a non-zero positive scalar loading parameter for the magnitude of is a yield criterion or yield surface. This flow rule is now called an “associated is associated with a specific yield criterion. The second term is the directi under such the pl tic stresses. The dλ can be expr ssed as in Eq. (9) after con stress (Eq. (6)) nd plastic strain i crement (Eq. (7)). = 3 2 � ̅ Both the Tresca and von Mises yield criteria state that the metal will yield when the co the principal stresses equals the shear yield stress, no matter whether it is under a uniax compl x 3D stress state. For example, yielding occurs i a uniaxial tension (1D) when σ 3 = 0. The Tresca and von Mises criteria are, respectively: 0 2 for Tresca and = 0 √3 for von Mises where σ 0 is the yield stress in a simple tension. Substituting ̄ i to Eq. (4) for σ 0 and the Eq. (4) i to Eqs. (2) and (3) for k , a general form of th Tresca yield criteri n (Eq. (5)) a yield criterion (Eq. (6)) are obtained, respectively: ̄ = 1 − 3 where 1 > 2 > 3 ̄ = √ 1 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 12 where ̄ = 2 or √3 , which states that if the magnitude of ̄ reaches a critical value 2 applied stress state will cause yielding. Since ̄ can represent any state of the stresses, it i effective stres . In metal forming, the von Mise yield criterion is more often used beca accurate when the middle principal stress σ 2 is involved. Similarly, for the von Mises criterion, the effective str in r plastic strain increment is d ̅ = √ 3 2 [( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 ] 12 Flow rules After yielding occurs, the material ente s a plastic flow state. Flow rules have b en define the direction and magnitude of the plastic straining or flow under the plastic relationship of the stresses and plastic strains, also known as the Levy-Mises equations Eq. (8): = where dλ is a non-zero positive scalar loading parameter for the magnitude of the plas is a yield criterion or yield surface. This flow rul i now called an “associated flow ru is associated with a specific yield criterion. The second term is the directi n of th under such the plastic stresses. The dλ can be expressed as in Eq. (9) after considering stress (Eq. (6)) and plastic strain increment (Eq. (7)). = 3 2 � ̅ (5) (6) where σ ̄ =2k or √3 k, which states that if the magnitude of σ ̄ reaches a critical value 2k or √3 k, the applied stress state will cause yielding. Since σ ̄ can represent any state of the stresses, it is often called effective stress . In metal forming, the von Mises yield criterion is more often used because it is more accurate when the middle principal stress σ2 is involved. Similarly, for the von Mises criterion, the effective strain or plastic strain increment is defined as: (7) 4.3 Flow rules After yielding occurs, the material enters a plastic flow state. Flow rules have been developed to define the direction and magnitude of the plastic straining or flow under the plastic stresses – the relationship of the stresses and plastic strains, also known as the Levy-Mises equations, as shown in Eq. (8): (8) where d λ is a non-zero positive scalar loading parameter for the magnitude of the plastic flow and f is a yield criterion or yield surface. This flow rule is now called an “associated flow rule” because it is associated with a specific yield criterion. The second term is the direction of the plastic flow under such the plastic stresses. The d λ can be expressed as in Eq. (9) after considering the effective stress (Eq. (6)) and plastic strain increment (Eq. (7)). 9 9 9 (9)

4.2 Yield condition Determining the yield condition of a material is the first fundamental step in plastic analysis. “Yielding” is the moment when themetal enters the plastic deformation stage; it can be quantitatively calculated by one of two major mathematical expressions: the Tresca yield criterion (the hexagon) in Eq. (2) (1814–1885) or the vonMises yield criterion (the ellipse) in Eq. (3) (1913) (Figure 9): (2) (3) where f is a constant, σ 1, σ 2, and σ 3 are the principal stresses, and k is the current yield stress in pure shear that is usually determined by material testing. The word “current” means the metal can exhibit different yield stresses along its loading history, which manifests as the Bauschinger effect (Figure 10) and indicates that yield stress may change when the metal is reloaded after one previous loading cycle. Yield condition Determining the yield condition of a material is the first fundamental step in plastic analysis. “Yielding” is th moment when the met l enters the plastic d formation stage; it can be quantitatively ca cula d by on of two major mathematical expressions: the Tresca yield criterion (the hexagon) in Eq. (2) (1814–1885) or the von Mises yield criterion (the ellipse) in Eq. (3) (1913) (Figure 9): = max � 1 2 | 1 − 2 |, 1 2 | 2 − 3 |, 1 2 | 3 − 1 |� − (2) = ( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 − 6 2 (3) where f is a constant, σ 1 , σ 2 , and σ 3 are the principal stresses, and k is the current yield stress in pure s ar that is usually determined by m t rial test g. The word “current” means the metal can exhibit different yield stresses along its loading history, which manifests as the Bauschinger effect (Figure 10) and indicates that yield stress may change when the metal is reloaded after one previous loading cycle. Tresca Hexagon σ 2

Yield condition ermining the yield condition of a material is the first fundamental step in plastic analysis. elding” is the moment when the metal enters the plastic deformation stage; it can be quantitatively ulated by one of two major mathematical expressions: the Tresca yield criterion (the hexagon) in (2) (1814–1885) or the von Mises yield criterion (the ellipse) in Eq. (3) (1913) (Figure 9): = max � 1 2 | 1 − 2 |, 1 2 | 2 − 3 |, 1 2 | 3 − 1 |� − (2) = ( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 − 6 2 (3) re f is a constant, σ 1 , σ 2 , and σ 3 are the p incipal stresses, and k is the current yield stress in pure ar that is usually determ ned by material tes ing. The word “current” me s the metal can exhibit erent yield stresses along its loading history, which manifests as the Bauschinger effect (Figure and indicates that yield stress may change when the metal is reloaded after one previous loading e. Yield condition rmining the yield condition of a material is the first fundamental step in plastic analysis. lding” is the moment when the metal enters the plastic deformation stage; it can be quantitatively ulated by one of two major mathematical expressions: the Tresca yield criterion (the hexagon) in 2) (1814–1885) or the von Mises yield criterion (the ellipse) in Eq. (3) (1913) (Figure 9): = max � 1 2 | 1 − 2 |, 1 2 | 2 − 3 |, 1 2 | 3 1 |� − (2) = ( 1 − 2 ) 2 + ( 2 − 3 ) 2 + ( 3 − 1 ) 2 − 6 2 (3) e f is a constant, σ 1 , σ 2 , and σ 3 are the principal stresses, and k is the current yield stress in pure r that is usually determined by material testing. The word “current” means the metal can exhibit rent yield stresses along its loading history, which manifests as the Bauschinger effect (Figure and indicates that yield stress may chan e when the etal is reloaded after one previous loading e.

( 0, σ 0 )

Tresca Hexagon Tresca Hexagon

σ 2

( 0, σ 0 )

σ 2

( 0, σ 0 )

(σ 0 , 0)

σ 1

(σ 0 , 0)

(σ 0 , 0)

σ 1

σ 1

Von Mises Ellipse

Figure 9. Yield criteria

Von Mises Ellipse Von Mises Ellipse

Figure 9. Yield criteria Figure 9. Yield criteria Figure 9. Yiel i

σ

σ

σ

σ y

σ y ’

σ y

σ y ’

ε

σ y

σ y ”

σ y ’

ε

σ y ”

ε

σ y ”

Figure 10. Bauschinger effect (1885)

Figure 10. Bauschinger effect (1885) Figure 10. Bauschinger effect (1 85) Figure 10. Bauschinger effect (1885)

FIA MAGAZINE | MAY 2021 71