MCEN 5023/ASEN 5012
Chapter 9
Constitutive Equations - Plasticity
Fall, 2006
1
Constitutive Equations
Mechanical Properties of Materials:
¾Modulus of Elasticity
¾Tensile strength
¾Yield Strength
¾Compressive strength
¾Hardness
¾Impact strength
¾Creep
2
Constitutive Equations
Mechanical Properties of Materials:
Force
Extension
3
Constitutive Equations
Mechanical Properties of Materials:
Time dependent, rate dependent
Force
Extension
4
Constitutive Equations
Mechanical Properties of Materials:
Stress
Stress
time
time
Strain
Strain
time
Elasticity
time
Viscoelasticity
5
Constitutive Equations
Mechanical Properties of Materials:
Viscoelasticity
Creep
Relaxation
Stress
Stress
time
Strain
time
Strain
time
time
6
Constitutive Equations
Plastic Deformation in Materials
Load
Elongation
7
Constitutive Equations
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
Constitutive Equations
Physics of plasticity of metals
9
Constitutive Equations
Physics of plasticity of metals- Defects
Line defects
10
Constitutive Equations
Physics of plasticity of metals- Dislocations
Edge dislocation
Screw dislocation
11
Constitutive Equations
Physics of plasticity of metals
- Dislocations
From edge dislocation
to screw dislocations
12
Constitutive Equations
Physics of plasticity of metals- Motion of dislocations
Edge dislocations
Screw dislocations
13
Constitutive Equations
Physics of plasticity of metals- Motion of dislocations
Professor Hideharu Nakashima Kyushu University, Japan
14
Constitutive Equations
Physics of plasticity of metals- Hardening
Hardening is due to obstacles to
the motion of dislocations;
obstacles can be particles,
precipitations, grain boundaries.
15
Constitutive Equations
Physics of plasticity of metals- Hardening
Three types of hardening mechanism
Solid solution hardening
Precipitation hardening
Strain hardening
16
Constitutive Equations
Physics of plasticity of metals- Yield
σy is yield strength
σ
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
Constitutive Equations
Physics of plasticity of metals- Yield Criteria
It has been found experimentally
1.Tresca yield condition (1864)
τ max =
1
σ1 −σ 3 ≤ τ y
2
τy
Shear yield strength
2.Mises yield condition (1913)
σ <σy
σ
Mises stress
Equivalent tensile stress
18
Constitutive Equations
Physics of plasticity of metals- Yield Criteria
Mises yield condition
(σ 1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 1 − σ 3 )2 ≤ 2σ y2
19
Constitutive Equations
Physics of plasticity of metals- Yield Criteria
Advantage of Mises criterion: Continuous function
Advantage of Tresca criterion: Simple
20
Constitutive Equations
Physics of plasticity of metals- Yield Criteria
Mises criterion is more conservative
21
Constitutive Equations
Physics of plasticity of metals- Tensile yield and shear yield
Tensile yield
σy
Shear yield
τy
22
Constitutive Equations
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
Constitutive Equations
Plastic Deformation:
¾
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.
¾
Since the production of plastic strain requires the breakage of interatomic bonds,
plastic deformation is dissipative.
¾
Crystals contain dislocations; When dislocations move, the crystal deform
plastically.
¾
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.
¾
Plastic strain remains upon removal of applied stress, that is, plastic strain
produces a permanent set.
¾
The current resolved shear stress governs the plastic strain increment, and not the
total strain as in linear elasticity.
24
Constitutive Equations
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
Constitutive Equations
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
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
σ
ee
ep
e
e
27
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
1. Constitutive Equation For Stress
Stress is due to elastic deformation.
Or
28
Constitutive Equations
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
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
3. Evolution Equation For ep, Flow Rule:
p
&
e& = e sign (σ )
p
&e p ≥ 0
30
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
4. Evolution Equation For s, Hardening Rule:
p
&
s& = he
h: Hardening function
31
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
5. Consistency Condition:
32
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
5. Consistency Condition:
33
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
Summary of 1-D theory
1. Decomposition of strain
e = e p + ee
2. Constitutive Equation
σ& = E (e& − e& p )
3. Yield Condition:
f =σ −s ≤ 0
4. Flow Rule :
e& p = e& p sign (σ )
⎧0
⎪
e& p = ⎨0
⎪ g −1sign(σ )σ& trial
⎩
if f < 0
if f = 0, and , sign(σ )σ& trial < 0
f f = 0, and , sign(σ )σ& trial > 0
σ& trial = Ee&
5. Hardening Rule s:
e& p ≥ 0
g = E+h
s& = he& p
34
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
What are the material parameters and how do we determine them?
σ
e
1. Uniaxial tension or compression
35
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
ds
h= p
de
s
ep
36
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
Possible functions for h for metals
α
s ⎞
⎛
h = h0 ⎜1 − ⎟
⎝ s*⎠
s = s − [(s − s )
*
1−α
*
0
*
(
)
+ α − 1 h0 s e
]
1
p 1−α
37
Constitutive Equations
One Dimensional Theory of Isothermal Rate-Independent Plasticity:
Another Hardening function
( )
s=K e
p n
38
Constitutive Equations
Three Dimensional Theory of Isothermal Rate-Independent Plasticity:
Governing Variables
39
Constitutive Equations
Three Dimensional Theory of Isothermal Rate-Independent Plasticity:
1. Decomposition of strain rate:
e = e p + ee
2. Constitutive Equations
σ& = E (e& − e& p )
(1D)
(1D)
40
Constitutive Equations
Three Dimensional Theory of Isothermal Rate-Independent Plasticity:
3. Yield Condition:
f = σ −s≤0
(1D)
41
Constitutive Equations
Three Dimensional Theory of Isothermal Rate-Independent Plasticity:
4. Flow Rule :
e& p = e& p sign (σ )
(1D)
42
Constitutive Equations
Three Dimensional Theory of Isothermal Rate-Independent Plasticity:
4. Flow Rule (continued):
43
Constitutive Equations
Three Dimensional Theory of Isothermal Rate-Independent Plasticity:
5. Hardening Rule s:
s& = he& p
44
Constitutive Equations
Three Dimensional Theory of Isothermal Rate-Independent Plasticity:
Summary of 3-D theory
1. Decomposition of strain
eij = eijp + eije
2. Constitutive Equation
σ& ij = 2G (e&ij − e&ijp ) + λe&kk δ ij
3. Yield Condition:
f = σ −s≤0
4. Flow Rule :
⎛ 3 σ ij′ ⎞
p
&
⎟⎟
e& = e sign ⎜⎜
⎝2 σ ⎠
p
ij
⎧
⎪0
⎪
e& p = ⎨0
⎪ −1 3 σ ij′ trial
σ& ij
⎪g
2σ
⎩
if f < 0
if f = 0, and , σ ij′ σ& ijtrial < 0
f f = 0, and , σ ij′ σ& ijtrial > 0
σ& ijtrial = 2Ge&ij + λe&kk δ ij
5. Hardening Rule s:
s& = he& p
e& p ≥ 0
g = 3G + h
45