Mechanical Behaviour of Materials Chapter 5 Plasticity Theory

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1 Mechanical Behaviour of Materials Chapter 5 Plasticity Theory Dr.-Ing. 郭瑞昭

2 Yield criteria Question: For what combinations of loads will the cylinder begin to yield plastically? The criteria for deciding which combination of multiaxial stresses will cause yielding are called yield criteria. f (,,,,, ) C x y z xy yz zx 0 y f (,, ) C y 0

3 Rankine Criteria (Maximum normal stress) Yielding occurs when the largest principal normal stress is equal to the yield strength 0 in simple tension, or - 0 in compression. 0 Yielding will occur the state of stress is on the boundary of the rectangle.

4 Tresca Criteria Maximum-shear stress criterion: plastic flow takes place when the maximum shear stress in a complex state of stress reaches the maximum shear stress at the onset of flow in uniaxial tension. Yielding occurs when the maximum shear stress reaches a critical. Yielding will occur when one of the following six conditions is reached: ± ± ± 0 0 0

5 Tresca Criteria 0) ( 0) ( 0) ( 0) ( 0) 0, ( 0) 0, ( < < < < > > > > > < < >

6 von Mises Criteria Yielding occurs when the distortion energy equals the distortion energy at yield in simple tension. y +

7 von Mises Criteria Maximum-Distortion-Energy Criterion: The maximum-distortion energy theory is founded on the concept of strain energy. It assumes that only the strain energy that produces a change of shape is responsible for the plastic flow of the material. [( ) ( ) ( ) ] the total energy a volume change energy a distortion energy

8 von Mises Criteria [ ] p p p p p p u ε ε ε + + the total energy the volume change energy the distortion energy ε v v p u [ ] ε ε + ε + ε v p p p p p p d p u

9 von Mises Criteria u the total energy u E [ ε + ε + ε ] { [ ν( + )] + [ ν( + )] + [ ν( + )]} u E { + + ν( + + )}

10 von Mises Criteria: the energy of volume change ) ( + + p ( ) ν ε E p u v v p E v ) ( ν ε

11 von Mises Criteria: The distortion energy u E { + + ν( + + )} u v pεv ν 6E ( + + ) u d 6 E [ ( ) ( ) ] ( + + ) 6ν + + ( ν + + ) u d + ν 6E [ ( ) ( ) ] ( + + ) 6ν + + ( ν + + )

12 von Mises Criteria: The distortion energy ( ) ( ) ( ) [ ] ν E u d J E G J u d ν + ( ) ( ) ( ) [ ] J

13 von Mises Criteria: The distortion energy For tensile test ( ) ( ) ( ) [ ] y y y J + + ( ) ( ) ( ) [ ] + + y 0 y

14 von Mises Criteria: The distortion energy For pure shear y 0 J 6 [( ) ( ) ( )] y y y 0 y y y 6 [( ) ( ) ( ) ] + +

15 von Mises Criteria: The distortion energy y J For biaxial stress y 0 ( ) ( ) ( ) [ ] J + + y ( ) ( ) ( ) [ ] + + y

16 Comparison of Rankine, von Mises, and Tresca Criteria

17 Comparison of Rankine, von Mises, and Tresca Criteria Al, Cu Cast iron (a) Rankine, von Mises, and Tresca criteria. (b) Comparison of failure criteria with experimental results. (Reprinted with permission from E. P. Popov, Mechanics of Materials, nd ed. (Englewood Cliffs, NJ: Prentice-Hall, 976), and G. Murphy, Advanced. Mechanics of Materials (New York: McGraw-Hill, 964), p. 8.)

18 Tresca maximum shear stress criterion Example problem 5.

19 Tresca maximum shear stress criterion Example problem 5.

20 Von Mises criterion Example problem 5.

21 Von Mises criterion Example problem 5.

22 Von Mises criterion Example problem 5.

23 Von Mises criterion Example problem 5.

24 Flow rules The general relations between plastic strains and the stress states dε ij dλ f ij Levy-Mises eqs [ ( ) ( ) ] 4 dλ [ ( ) ] [ ( ) ] dε dλ + dε dλ + [ ( ) ] dε dλ + d ε : dε : dε : 0 :

25 Flow rules Example problem 5.4

26 Flow rules Example problem 5.4

27 Flow rules Example problem 5.5

28 Flow rules Example problem 5.5

29 Flow rules Example problem 5.6

30 Flow rules Example problem 5.6

31 Flow rules Example problem 5.7

32 Flow rules Example problem 5.7

33 Principle of normality dε d ε

34 Effective stress and effective strain d d d d d ε ε ε ε + + w For loading paths other than uniaxial tension ( ) ( ) ( ) [ ] + + y eff ( ) ε ε ε ε + + eff ε dw d

35 Effective stress and effective strain Example problem 5.8

36 Effective stress and effective strain Example problem 5.8

37 Effective stress and effective strain If straining is proportional ε ( ) ( ) ε +ε +ε In terms of non-principal strains (von Mises) ε ( ) ( ε x +ε y +ε z )+ ( y ) z + z x ( ) ( ) γ yz ( +γ +γ ) zx xy ( ) + ( x ) y +6 τ yz ( +τ +τ ) zx xy Tresca dε dε i max Tresca v. von Mises dε i max dε Mises.5 dε i max Strain hardening depends only on strain rate f ( ε)

38 Effective stress and effective strain Example problem 5.9

39 Effective stress and effective strain Example problem 5.9

40 Other isotropic yield criteria a + a + a Y a a, 4 a, von Mises Tresca For a even integer a a a a ( ) + ( ) + ( ) Y

41 Other isotropic yield criteria Example problem 5.0

42 Other isotropic yield criteria Example problem 5.0

43 Anisotropic plasticity Hill criterion (generalized von Mises) F ( y ) + G z z x If principal stress on symm. axes; planar isotropy ( y ) + z z x ( x ) + Lτ y yz ( ) + H ( x ) R + y ( ) + R ( )X + Mτ zx + Nτ xy Flow rules dε x : dε y : dε z R + ( ) x R y z ( ) y R x z : R + : z y x

44 Anisotropic plasticity

45 Anisotropic plasticity (cont.) ρ ( R + )α R ( R + ) Rα α ( R + )ρ + R ( R + )+ Rρ If R High-exponent criterion dε x : dε y : dε z R { ( y ) z + ( z x ) +R( x ) y ( R+) } { x ( α ) ( R+) } a a ( x y ) + ( x z ) α ++R ε ε x ( )+ αρ ( ) a a a ( y z ) + ( z x ) + R( x y ) R + a a ( y z ) ( + R y x ) ( )X a : : { a ( a y ) a a z + ( z x ) +R( x ) y ( R+) } ε ( x ) + αρ ( ) aeven integer, a> ( ) a a ( z x ) + z y

46 Anisotropic plasticity

47 Anisotropic plasticity Example problem 5.

48 Anisotropic plasticity Example problem 5.

49 Effect of strain-hardening on the yield locus

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Dr. D. Dinev, Department of Structural Mechanics, UACEG

Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents