Constitutive Equations - Plasticity

Transcription

1 MCEN 5023/ASEN 5012 Chapter 9 Constitutive Equations - Plasticity Fall,

2 Mechanical Properties of Materials: Modulus of Elasticity Tensile strength Yield Strength Compressive strength Hardness Impact strength Creep 2

3 Mechanical Properties of Materials: Force Extension 3

4 Mechanical Properties of Materials: Time dependent, rate dependent Force Extension 4

5 Mechanical Properties of Materials: Stress Stress time time Strain Strain Elasticity time time Viscoelasticity 5

6 Mechanical Properties of Materials: Viscoelasticity Stress Creep Stress Relaxation time time Strain Strain time time 6

7 Plastic Deformation in Materials Load Elongation 7

8 Physics of plasticity of metals Some typical crystal structure of metals Body Centered Cubic (b.c.c) Face Centered Cubic (f.c.c) Hexagonal close-packed (h.c.p) 8

9 Physics of plasticity of metals 9

10 Physics of plasticity of metals- Defects Line defects 10

11 Physics of plasticity of metals- Dislocations Edge dislocation Screw dislocation 11

12 Physics of plasticity of metals - Dislocations From edge dislocation to screw dislocations 12

13 Physics of plasticity of metals- Motion of dislocations Edge dislocations Screw dislocations 13

14 Physics of plasticity of metals- Motion of dislocations Professor Hideharu Nakashima Kyushu University, Japan 14

15 Physics of plasticity of metals- Hardening Hardening is due to obstacles to the motion of dislocations; obstacles can be particles, precipitations, grain boundaries. 15

16 Physics of plasticity of metals- Hardening Three types of hardening mechanism Solid solution hardening Precipitation hardening Strain hardening 16

17 Physics of plasticity of metals- Yield σ σy is yield strength σy Question: how can relate the σy from 1D test to yield in a general 3D stress state? 0.2% e A simple tension test 17

18 Physics of plasticity of metals- Yield Criteria It has been found experimentally 1.Tresca yield condition (1864) τ max 1 σ 1 σ 2 = 3 τ y τ y Shear yield strength 2.Mises yield condition (1913) σ < σ y σ Mises stress Equivalent tensile stress 18

19 Physics of plasticity of metals- Yield Criteria Mises yield condition ( σ σ ) + ( σ σ ) + ( σ σ ) σ y 19

20 Physics of plasticity of metals- Yield Criteria Advantage of Mises criterion: Continuous function Advantage of Tresca criterion: Simple 20

21 Physics of plasticity of metals- Yield Criteria Mises criterion is more conservative 21

22 Physics of plasticity of metals- Tensile yield and shear yield Tensile yield σ y Shear yield τ y 22

23 Physics of plasticity of metals- Single crystal and polycrystal Shear yield strength of polycrystal τ y, poly Shear yield strength of single crystal τ y, single 23

24 Plastic Deformation: Elastic strain in a single crystal is the strain related to the stretching of the crystal lattice under the action of applied stress. Therefore, elastic strain is recoverable. Since the production of plastic strain requires the breakage of interatomic bonds, plastic deformation is dissipative. Crystals contain dislocations; When dislocations move, the crystal deform plastically. Plastic strain is incompressible because dislocation motion produces a shearing type of deformation; Hydrostatic pressure has a negligibly small effect on the plastic flow of metals. Plastic strain remains upon removal of applied stress, that is, plastic strain produces a permanent set. The current resolved shear stress governs the plastic strain increment, and not the total strain as in linear elasticity. 24

25 Rate-Dependent Plasticity and Rate-Independent Plasticity: Plastic deformation in metals is thermally-activated and inherently rate-dependent - Energy barrier and thermal vibration of atoms - Increasing temperature promote thermal vibration - Applying stress lowers down the energy barrier For most single and polycrystalline materials, the plasticity is only slightly ratesensitive if the temperature T < 0.35 Tm 25

26 General Requirements for a Constitutive Model: Requirements from thermodynamics First Law: The rate of change to total energy of a thermodynamic system equals the rate at which external mechanical work is done on that system by surface tractions and body forces plus the rate at which thermal work is done by heat flux and hear sources. Total energy change = Mechanical work + Heat transfer Second Law: (Entropy inequality principle). Entropy cannot decrease unless some work in done; Heat always flows from the warmer to the colder region of a body, not vice versa; Mechanical energy can be transformed into heat by friction, and this can never be converted back into mechanical energy. Energy dissipation >=0 26

27 One Dimensional Theory of Isothermal Rate-Independent Plasticity: σ e p e e e e 27

28 One Dimensional Theory of Isothermal Rate-Independent Plasticity: 1. Constitutive Equation For Stress Stress is due to elastic deformation. Or 28

29 One Dimensional Theory of Isothermal Rate-Independent Plasticity: 2. Yield Condition: Yield Function: f ( σ s) = σ s Yield Condition: f ( σ s) = σ s = 0 s: Deformation resistance. Internal variable. Non-zero, positive valued scalar with dimension of stress Internal variables: Internal variables cannot be directly observed. They describe the internal structure of materials associated with irreversible effects. 29

30 One Dimensional Theory of Isothermal Rate-Independent Plasticity: 3. Evolution Equation For e p, Flow Rule: p p e e& p & = sign e& 0 ( σ ) 30

31 One Dimensional Theory of Isothermal Rate-Independent Plasticity: 4. Evolution Equation For s, Hardening Rule: s & = he& h: Hardening function p 31

32 One Dimensional Theory of Isothermal Rate-Independent Plasticity: 5. Consistency Condition: 32

33 One Dimensional Theory of Isothermal Rate-Independent Plasticity: 5. Consistency Condition: 33

34 One Dimensional Theory of Isothermal Rate-Independent Plasticity: Summary of 1-D theory 1. Decomposition of strain p e = e + p 2. Constitutive Equation & σ = E ( e& e& ) e e 3. Yield Condition: f = σ s 0 4. Flow Rule : p e e& p sign( σ ) & = p e& 0 e& p = 0 0 g sign if f < 0 if f = 0, and, sign( σ ) trial ( σ )& σ f f = 0, and, sign( σ ) & σ & σ trial 1 trial < 0 > 0 5. Hardening Rule s: s & = he& p trial & = σ Ee& g = E + h 34

35 One Dimensional Theory of Isothermal Rate-Independent Plasticity: What are the material parameters and how do we determine them? σ 1. Uniaxial tension or compression e 35

36 One Dimensional Theory of Isothermal Rate-Independent Plasticity: s h = ds de p e p 36

37 37 Constitutive Equations One Dimensional Theory of Isothermal Rate-Independent Plasticity: Possible functions for h for metals α = * 1 0 s s h h ( ) ( ) [ ] α α α + = 1 1 * * * 1 p e s h s s s s

38 One Dimensional Theory of Isothermal Rate-Independent Plasticity: Another Hardening function s = K ( p e ) n 38

39 Three Dimensional Theory of Isothermal Rate-Independent Plasticity: Governing Variables 39

40 Three Dimensional Theory of Isothermal Rate-Independent Plasticity: 1. Decomposition of strain rate: e + p e = e e (1D) 2. Constitutive Equations & σ = E p ( e& e& ) (1D) 40

41 Three Dimensional Theory of Isothermal Rate-Independent Plasticity: 3. Yield Condition: f = σ s 0 (1D) 41

42 Three Dimensional Theory of Isothermal Rate-Independent Plasticity: 4. Flow Rule : ( σ ) p p e & = e& sign (1D) 42

43 Three Dimensional Theory of Isothermal Rate-Independent Plasticity: 4. Flow Rule (continued): 43

44 Three Dimensional Theory of Isothermal Rate-Independent Plasticity: 5. Hardening Rule s: s & = he& p 44

45 Three Dimensional Theory of Isothermal Rate-Independent Plasticity: Summary of 3-D theory 1. Decomposition of strain e = e + e ij p ij e ij 2. Constitutive Equation ij p ( e& ij e& ij ) λe& kkδ ij & σ = 2G + 3. Yield Condition: f = σ s 0 4. Flow Rule : 5. Hardening Rule s: e& p = 0 0 g σ p p 3 = ij p e& ij e& sign e& 0 2 σ if f < 0 3 σ ij & σ ij 2 σ s & = he& p if f f f = 0, and, σ & σ = 0, and, σ & σ trial ij 1 trial trial ij ij trial ij ij kk ij ij < 0 > 0 & σ = 2Ge& + λe& δ g = 3G + h 45