소성재료역학 (Metal Plasticity) Chapter 2: Plasticity characteristics

446.631A

소성재료역학 (Metal Plasticity)

Chapter 2: Plasticity characteristics

Myoung-Gyu Lee

Office: Eng building 33-309 Tel. 02-880-1711

myounglee@snu.ac.kr

TA: Chanyang Kim (30-522)

Real Word vs. Continuum Mechanics

• Discrete vs. Continuum

 Real world is discrete in atomistic scale

 Simplification about averaged physical value : continuum scale

 Averaged description = mathematical description = phenomenological description

 Materials, such as solids, liquids and gases, are composed of molecules.

 On a microscopic scale, materials have cracks and discontinuities.

 Assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of space it occupies.

 A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material.

Multi-scale concept: scale bridging

Fe atom Aircraft

Can

Metal texture

Scale

Numerical analysis

MD FEM FDM

Classification of continuum mechanics

Rigid body statics & dynamics

Deformable body dynamics

F₁

F₂

σ,ε,…

Continuum Mechanics - Stress

, ,

σ σ σ σ

     

     

     

   

   

   

   

         

       

       

xx yx zx xx yx zx

x xy y yy x zy xy yy zy

xz yz zz xz yz zz

Continuum Mechanics - deformation

x y

z O

X

u x

P P

Q Q

dX

dx

dX du^{( )x}

zx^yx

y

y yx

du u dX dX

X 

 



x

x xx

du u dX E dX X

 



z

z zx

du u dX dX

X 

 

 x

y

z

y

y yy

du u dY E dY Y

 



dY

du(y) x

x xy

du u dY dY

Y 

 



z

z zy

du u dY dY

Y 

 



zy

xy

dZ

du(z)

xz

yz y

y yz

du u dZ dZ

Z 

 



x

x xz

du u dZ dZ

Z 

 



z

z zz

du u dZ E dZ Z

 



Deformable body statics

• Equilibrium equations

 3 equations

• Conservation of moment of momentum

 Symmetry of stress tensor

• Kinematics

 6 equations

• Constitutive equations

 6 equations

15 equations for 15 unknowns !!

,

0

ij j

b

  

ij ji

  

1 ( )

2

i j ij

j i

u u

  ^  ^

 

ij

C

  

Simple tension test

Sheet Bulk

Stress in tension: 1D

• True stress

• Engineering strain



 F A





o

F A

0 0 0 0 0 0 0 0



 

 

  

 

 

σ

Simple tension test

Eng. S-S curve of metals

Simple tension test

Nonlinear elastic behavior of rubber

Strain in tension: 1D

1 ln

o

t

t l

l o

d dl

l dl l

l l







  

o 0 e

o e l

l o

d dl

l

l l dl

l l







   

• True strain • Engineering strain

Stress in tension: Eng. vs. True

• Tension case

 

ln ln 1 ln 1

t o e

o o

l l l

l l

   ^  ^   ^   

 

 

1 1

( : volume constant)

t o e o e e

o o o o

o o

A l l

F F F l

A A A A l l

A l Al

      ^  ^   ^    

 



Stress in tension: Eng. vs. True

 

ln ln 1 ln 1

t o e

o o

l l l

l l

   ^ ^  ^  

 

 

 

 

 

0

0

1 1 1 1

ln 1 ln1 ...

1 2! 1 2

for small deformation,

 

     

 

 

 

        

 



e

e

t e e e e e

e e

t e

• Relation for small deformation

With Taylor series,

Stress in tension: Eng. vs. True







t

1

e

e

  

 

ln 1

1

t e e

e

       

Stress in tension: Eng. vs. True





Stress in tension: Eng. vs. True

-1





e e





t e





 ¹ 

t e e

 ^  ^ 

 

1: 1   ^e    ^e : ^t

Stress in compression: Eng. vs. True

• Compressive case

 

 

0 ln 0

t

t o t

d dl

l l

l



 

  

    

 

1 0 1

e o e

o o

l l l

l l

  ^     





( : volume constant)

t o e e

o o o

o o

F F A F l

A A A A l

A l Al

     



l₀

Stress in compression: Eng. vs. True

1

ln 1 1

1

t t

e o

t o e

e

l l

l e e

l

 



 





 

     



e

t

1 ^t

e e ^

   ^ 1

Stress in compression: Eng. vs. True





t t

  

t e

  

1

t e

  

Stress in compression: Eng. vs. True

e

 t

t e

 

e e

 

e ₁^e

 e

1





t e e







 

1: 1  

  

:

Tension vs. Compression

1-D (Metal) Plasticity: characteristics

• Consists of linear and non-linear regions

• Permanent deformation; vs. elasticity

• Yield stress

• Bauschinger effect

 The yield strength of a metal decreases when the direction of strain (strain path) is changed; i.e., normally loading is

reversed.

 Mostly in polycrystalline metals.

 The basic mechanism is related to the dislocation structure in the cold worked metals.

 As deformation occurs, the dislocations will accumulate at barriers and produce dislocation pile-ups and tangles.

0 200 400 600 800 1000 1200

0 0.025 0.05 0.075 0.1

DP980-10-3

True Stress (MPa)

True Strain DP980 -1.43

See close-up

1-D (Metal) Plasticity

1-D (Metal) Plasticity

1-D (Metal) Plasticity: Simplification

• Elastic-linear plasticity

• Rigid-linear plasticity

• Elastic-perfect plasticity

• Rigid-perfect plasticity

1-D (Metal) Plasticity: Simplification

• Ludwick

• Hollomon

• Voce

• Swift

• Prager

• Ramberg and Osgood

n

Y

Hε

σ

σ   Hε

σ 

 ^{σ σ}   ^{1 - exp}   ^{- nε} 

σ

σ 





 ^ε

^ε 

σ  H 

 

 



 

Y

Y

σ

tanh Eε σ

σ

n

E H σ

E

ε σ 



 



 



Viscoelasticity

Viscous behavior of fluid

Viscoelasticity – Creep response Input - stress

Ouput - strain

Viscoelasticity – classical models

Viscoelasticity – Maxwell model

Spring Dashpot

Maxwell fluid model

s

E

   



Serial connection

s d

    

s d

E

    

    

Viscoelasticity – Kelvin model

Kelvin solid(or Voight model) Spring Dashpot

s

E

   



Parallel connection

s d

    

s d

E

        

Viscoelasticity – Dirac delta function

y

 ^1.0

t

differentiation

integration

t

y

t

_t

0

0 0

0 *( ) 0 0

0

( ) ( ) ( ) 1

0

 0 





  

  



at t t at t t

t t t t dt H t t

at t t elsewhere

Viscoelasticity – Creep



Maxwell model

Kelvin model





0

t

E

 

 ^  ^

0

(1 )

Et

E e



  

t t



t

Viscoelasticity – Relaxation



Maxwell model

Kelvin model





0

Et

E e

 ^ 

0 0

( 0)

E t

      

t t



t

Generalization of Kelvin/Maxwell model

Generalization of Kelvin model

Generalization of Maxwell model