CH.8. PLASTICITY Continuum Mechanics Course (MMC) - ETSECCPB - UPC - - PowerPoint PPT Presentation
CH.8. PLASTICITY Continuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Introduction Previous Notions Principal Stress Space Normal and Shear Octahedral Stresses Stress Invariants Effective Stress Principal Stress
Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Introduction
Previous Notions
Principal Stress Space
Normal and Shear Octahedral Stresses Stress Invariants Effective Stress
Principal Stress Space
Normal and Shear Octahedral Stress Stress Invariants Projection on the Octahedral Plane
Rheological Friction Models
Elastic Element Frictional Element Elastic-Frictional Model
2
Rheological Friction Models (cont’d)
Frictional Model with Hardening Elastic-Frictional Model with Hardening
Phenomenological Behaviour
Notion of Plastic Strain Notion of Hardening Bauschinger Effect Elastoplastic Behaviour
1D Incremental Theory of Plasticity
Additive Decomposition of Strain Hardening Variable Yield Stress, Yield Function and Space of Admissible Stresses Constitutive Equation Elastoplastic Tangent Modulus Uniaxial Stress-Strain Curve
3
3D Incremental Theory of Plasticity
Additive Decomposition of Strain Hardening Variable Yield Function Loading - Unloading Conditions and Consistency Conditions Constitutive Equation Elastoplastic Constitutive Tensor
Yield Surfaces
Von Mises Criterion Tresca Criterion Mohr-Coulomb Criterion Drucker-Prager Criterion
4
5
A material with plastic behavior is characterized by: A nonlinear stress-strain relationship. The existence of permanent (or plastic) strain during a
loading/unloading cycle.
Lack of unicity in the stress-strain relationship. Plasticity is seen in most materials, after an initial elastic state.
6
1
2
3
1
x
2
x
3
x
1
x
2
x
3
x
1
x
2
x
3
x
11
13
12
22
23
21
33
32
31
PRINCIPAL STRESSES
Regardless of the state of stress, it is always possible to choose a special
set of axes (principal axes of stress or principal stress directions) so that the shear stress components vanish when the stress components are referred to this system.
The three planes perpendicular to the principle axes are the principal
planes.
The normal stress components in the principal planes are the principal
stresses.
1 2 3
1 2 3
7
1
2
3
1
x
2
x
3
x
1
x
2
x
3
x
1
x
2
x
3
x
11
13
12
22
23
21
33
32
31
PRINCIPAL STRESSES
The Cauchy stress tensor is a symmetric 2nd order tensor so it will diagonalize
in an orthonormal basis and its eigenvalues are real numbers.
Computing the eigenvalues and the corresponding eigenvectors :
1 2 3
v v v 1
11 12 13 12 22 23 13 23 33
det
not
= 1 1
1 1 2 2 3 3
3 2 1 2 3
I I I characteristic equation INVARIANTS
8
STRESS INVARIANTS
Principal stresses are invariants of the stress state.
They are invariant w.r.t. rotation of the coordinate axes to which the
stresses are referred.
The principal stresses are combined to form the stress invariants I : These invariants are combined, in turn, to obtain the invariants J:
1 1 2 3 ii
I Tr
2 2 1 1 2 1 3 2 3
1 : 2 I I
3
det I
1 1 ii
J I
2 2 1 2
1 1 1 2 : 2 2 2
ij ji
J I I
3 3 1 1 2 3
1 1 1 3 3 3 3 3
ij jk ki
J I I I I Tr
REMARK
The J invariants can be expressed the unified form:
1 1,2,3
i i
J Tr i i
REMARK
The I invariants are obtained from the characteristic equation
9
SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR
Given the Cauchy stress tensor and its principal stresses, the following is defined:
Mean stress Mean pressure A spherical or hydrostatic
state of stress:
1 2 3
1 1 1 3 3 3
m ii
Tr
1 2 3
1 3
m
p
1 2 3
1
REMARK
In a hydrostatic state of stress, the stress tensor is isotropic and, thus, its components are the same in any Cartesian coordinate system. As a consequence, any direction is a principal direction and the stress state (traction vector) is the same in any plane.
10
SPHERICAL AND DEVIATORIC PARTS OF THE STRESS TENSOR
The Cauchy stress tensor can be split into:
The spherical stress tensor:
Also named mean hydrostatic stress tensor or volumetric stress tensor or
mean normal stress tensor.
Is an isotropic tensor and defines a hydrostatic state of stress. Tends to change the volume of the stressed body
The stress deviatoric tensor:
Is an indicator of how far from a hydrostatic state of stress the state is. Tends to distort the volume of the stressed body
sph
1 1 : 3 3
sph m ii
Tr 1 1 1 dev
m
1
11
STRESS INVARIANTS OF THE STRESS DEVIATORIC TENSOR
The stress invariants of the stress deviatoric tensor: These correspond exactly with the invariants J of the same stress
deviator tensor:
1 1
J I
2 2 1
1 2 J I
2 2
1 2 : 2 I I
3 3 1
1 3 J I
1 2
3I I
3 3
1 1 3 3 3
ij jk ki
I I Tr
1
I Tr
2 2 1
1 : 2 I I
2 2 2 3 11 22 33 12 23 13 12 33 23 11 13 22
1 det 2 3
ij jk ki
I
12
EFFECTIVE STRESS
The effective stress or equivalent uniaxial stress is the scalar: It is an invariant value which measures the “intensity” of a 3D stress state
in a terms of an (equivalent) 1D tensile stress state.
It should be “consistent”: when applied to a real 1D tensile stress, should
return the intensity of this stress.
' 2
3 3 3 2 2
ij ij
J ´: ´
13
Calculate the value of the equivalent uniaxial stress for an uniaxial state of stress defined by:
x
x
x y z
u
u
E, G
u
14
Mean stress: Spherical and deviatoric parts
1 ( 3 3
u m
Tr
3 3 3
u m u sph m m u
2 3 1 3 1 3
u u m sph m u m u
2
3 3 4 1 1 3 2 ( ) 2 2 9 9 9 2 3
ij ij u u
u
u
15
16
The principal stress space or Haigh–Westergaard stress space is
1 2 3
17
Any of the planes perpendicular to the hydrostatic stress axis is a
Its unit normal is .
1 2 3
1 1 1 3 1 n
18
Consider the principal stress space:
The normal octahedral stress is defined as:
1 2 3 1 2 3
m
1
m
19
Consider the principal stress space:
The shear or tangential octahedral
Where the is calculated from:
2 2 2 2 2 2 2 1 2 3 2 ' 1 2 3 2
AP
1 2 2
1/2 2 2 2 2 1 2 3 1 2 3 1/2 2 2 2 1 2 2 3 1 3
1 1 3 3 1 3 3
Alternative forms of :
20
In a pure spherical stress state: In a pure deviator stress state:
sph m
esf
2
m
A pure spherical stress state is located on the hydrostatic stress axis. A pure deviator stress state is located on the octahedral plane containing the origin of the principal stress space
21
Any point in space is unambiguously defined by the three
The first stress invariant
characterizes the distance from the origin to the octahedral plane containing the point.
The second deviator stress invariant characterizes the radius of the
cylinder containing the point and with the hydrostatic stress axis as axis.
1
The third deviator stress invariant
characterizes the position of the point on the circle obtained from the intersection
2
3
3
J
22
The projection of the principal stress space on the octahedral
These are characterized by the different principal stress orders.
FEASIBLE WORK SPACE Election of a criterion, e.g.:
1 2 3
23
2 1 2
24
1
2
1 2 2 2
m
2 2
25
2 2
26
38
e p
PLASTIC STRAIN
e
elastic limit: LINEAR ELASTIC BEHAVIOUR
e
E
39
Also known as kinematic hardening.
f
e
e
41
Considering the phenomenological behaviour observed,
Lack of unicity in the stress-strain relationship.
The stress value depends on the actual strain and the previous loading
history.
A nonlinear stress-strain relationship.
There may be certain phases in the deformation process with
incremental linearity.
The existence of permanent (or plastic) strain during a loading /
unloading cycle.
42
43
The incremental plasticity theory is a mathematical model used
Developed for 1D but it can be generalized for 3D problems.
REMARK
This theory is developed under the hypothesis of infinitesimal strains.
44
Total strain can be split into an elastic (recoverable) part, ,
Also,
e
e p
e
e p
where where elastic modulus or Young modulus
e
p
45
The hardening variable, , is defined as:
Note that is always positive and:
p
p
REMARK
The function is:
sign
1
p
d d sign d
p
d d
p
d
p p
p p p
d d
46
Stress value, , threshold for the material exhibiting plastic
It is considered a material property. For
f
p f e
f
d H d
( )
f
H
HARDENING LAW
47
f e
H
( )
f
The yield function, , characterizes the state of the material:
,
f
F
, F
, F
ELASTO-PLASTIC STATE ELASTIC STATE
: , F
E R
0 :
,0
e
F
E R
: , F
E R
, F
ELASTIC DOMAIN YIELD SURFACE INITIAL ELASTIC DOMAIN: Space of admissible stresses
48
Any admissible stress state must belong to the space of
Space of admissible stresses
E
, F
E E E R
,
f
F ( ), ( )
f f
E
REMARK
49
The following situations are defined:
ELASTIC REGIME ELASTOPLASTIC REGIME – UNLOADING ELASTOPLASTIC REGIME – PLASTIC LOADING
E
, dF
E
, dF
E d E d d E d
ep
d E d
Elastoplastic tangent modulus
REMARK
The situation is not possible because, by definition, on the yield surface .
( , ) dF
E
, F
50
Consider the elastoplastic regime in plastic loading, Since the hardening variable is defined as:
p
d sign d
HARDENING LAW
f
d H d 1
p
d d H
E for
E
, ,
f
F dF
( )
,
f
sign
H
dF d d
1 d sign d H
51
e p e p
ep
E
ep
ELASTOPLASTIC TANGENT MODULUS
ep
52
Following the constitutive equation defined
ELASTIC REGIME ELASTOPLASTIC REGIME
ep
REMARK
Plastic strain is generated only during the plastic loading process.
53
The value of the hardening modulus, , determines the following
ep
Linear elasticity Perfect plasticity Plasticity with strain hardening H Plasticity with strain softening H
55
In real materials, the stress-strain curve shows a combination of
56
57
The 1D incremental plasticity theory can be generalized to a
Additive decomposition of strain Hardening variable Yield function
Loading - unloading conditions Consistency conditions
58
Total strain can be split into an elastic (recoverable) part, ,
Also,
1 : e
e p
1 : e
e p
where where
e
p
constitutive elastic (constant) tensor
59
e p e
The hardening variable, , is a scalar:
The flow rule is defined as:
,
p
f
,
p
G d
0,
, G
with
60
p
The yield function, , is a scalar defined as:
,
f
F
, F
, F
ELASTOPLASTIC STATE ELASTIC STATE
: , F
E
0 :
,0 F
E
: , F
E
, F
ELASTIC DOMAIN YIELD SURFACE INITIAL ELASTIC DOMAIN:
E E E
Space of admissible stresses Equivalent uniaxial stress Yield stress
61
1 ,
f
D F
Loading/unloading conditions (also known as Karush-Kuhn-
Consistency conditions:
; , ; , F F
For , , F dF
, 0; 0; , 0; 0; , 0; 0;
p p p
G F dF d G d F dF G d F dF
ELASTIC LOADING/UNLOADING ELASTOPLASTIC NEUTRAL LOADING IMPOSSIBLE
62
ELASTOPLASTIC LOADING
The following situations are defined:
ELASTIC REGIME ELASTOPLASTIC REGIME – ELASTIC UNLOADING ELASTOPLASTIC REGIME – PLASTIC LOADING
E
, dF
E
, dF
E : d d C : d d C :
ep
d d C
ELASTOPLASTIC CONSTITUTIVE TENSOR
( 0 and , 0) F dF ( 0) F
( 0 and , 0) F dF
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The elastoplastic constitutive tensor is written as:
: : : :
ep
G F F G H C C C C C C C C C
, , , , , , , 1,2,3
ijpq rskl pq rs ep ijkl ijkl pqrs pq rs
G F i j k l p q r s F G H C C C C C
REMARK
When the plastic potential function and the yield function coincide, it is said that there is associated flow:
, , G F
64
65
The initial yield surface, , is the external boundary of the initial
The state of stress inside the yield surface is elastic for the virgin material. When in a deformation process, the stress state reaches the yield surface, the
virgin material looses elasticity for the first time: this is considered as a failure criterion for design. Subsequent stages in the deformation process are not considered.
E
E
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The yield surface is usually expressed in terms of the following
Where: The elastoplastic behavior will be isotropic.
1 2 3
, ,
e
F I J J
F
1 1 2 3 ii
I Tr
2 2 1
1 2 J I
2 2
1 2 : 2 I I
3 3 1
1 3 J I
1 2
3I I
3 3
1 1 3 3 3
ij jk ki
I I Tr
with
1 2 3
REMARK
Due to the adopted principal stress criteria, the definition of yield surface only affects the first sector
67
The yield surface is defined as:
Where is the effective stress.
(often termed the Von-Mises stress)
The shear octahedral stress is, by definition, .
And the yield surface is given by:
( )
e
F
2
1 2 2
2 3
J
1 2 2
3 2
J
e
REMARK
The Von Mises criterion depends solely on the second deviator stress invariant.
68
e
F
The octahedral stresses characterizes the radius of the cylinder
e
e
REMARK
The Von Mises Criterion is adequate for metals, where hydrostatic stress states have an elastic behavior and failure is typically due to deviatoric stress components.
2 e
F J F
69
e
Consider a beam under a composed flexure state such that for a beam section the stress state takes the form, Obtain the expression for Von Mises criterion.
x
x
x xy xy
70
The mean stress is: The deviator part of the stress tensor is: The second deviator stress invariant is given by,
1 3 3
x m
Tr
2 3 1 esf 3 1 3 x m xy x xy xy m xy x m x
2 2 2 2 2 2 2 2
x x x xy xy x xy
71
The uniaxial effective stress is: Finally, the Von Mises yield surface is given by the expression: (Criterion in design codes for metal beams)
2 2 2
x xy
2
e
2 2
x xy e
co
(comparison stress)
72
Also known as the maximum shear stress criterion, it establishes
It can be written univocally in terms of invariants and :
1 3 max
2 2
e
1 3 e
1 3 2 3
( ) ,
e e
F J J F
2
J
3
J
Plane parallel to axis
2
73
e
REMARK
The Tresca yield surface is appropriate for metals, which have an elastic behavior under hydrostatic stress states and basically have the same traction/compression behavior.
2 3
,
e
F J J F
74
1 3 e
75
Obtain the expression of the Tresca criterion for an uniaxial state of stress defined by:
x
x
x y z
u
u
E, G
u
76
Consider: The Tresca criterion is expressed as:
u
1 3
e u e u e
u
1 3 u
u
1 3
e u e u e u
1 3 u
e
u e
Note that it coincides with the Von Mises criterion for an uniaxial state of stress.
77
u
Consider a beam under a composed flexure state such that for a beam section the stress state takes the form, Obtain the expression for Tresca yield surface.
x
x
x xy xy
78
The principal stresses are: Taking the definition of the Tresca yield surface,
2 2 2 2 1 3
1 1 1 1 , 2 4 2 4
x x xy x x xy
1 3 e
2 2 2 2 1 3
e x x xy x x xy
2 2
x xy e
co
s (comparison stress)
79
It is a generalization of the Tresca criterion, by including the
In the Mohr circle’s plane, the Mohr-Coulomb yield function takes
internal friction angle cohesion
REMARK
The yield line cuts the normal stress axis at a positive value, limiting the materials tensile strength.
80
Consider the stress state for which the yield point is reached:
1 3
cos sin 2
A A
R R
1 3 1 3 1 3 1 3 1 3
tg cos sin tg 2 2 2 sin 2 cos
A A
c c c
1 3 1 3 sin
2 cos F c
REMARK
For and , the Tresca criterion is recovered.
/2
e
c
81
REMARK
The Mohr-Coulomb yield surface is appropriate for frictional cohesive materials, such as concrete, soils or rocks which have considerably different tensile and compressive values for the uniaxial elastic limit.
1 2 3
, , F I J J F
82
It is a generalization of the Von Mises criterion, by including the
The yield surface is given by the expression:
Where:
It can be rewritten as:
1/2 2
3
m
F J
1 2 3 1
2sin 6 cos ; ; 3 3 3 3 sin 3 3 sin
m
I c
1/2 1 2 1 2
3 3 , 2
F I J I J
REMARK
For and , the Von Mises criterion is recovered.
/2
e
c
83
REMARK
The Drucker-Prager yield surface, like the Mohr-Coulomb one, is appropriate for frictional cohesive materials, such as concrete, soils or rocks which have considerably different tensile and compressive values for the uniaxial elastic limit.
1 2
, F I J F
84
85
86
1
2
3
1
x
2
x
3
x
1
x
2
x
3
x
1
x
2
x
3
x
11
13
12
22
23
21
33
32
31
The principal stress directions correspond to the set of axes that make
the shear stress components vanish when the stress components are referred to this system.
The normal stress components in the
principal planes are the principal stresses.
Mean stress: Mean pressure: A spherical or hydrostatic state of stress:
1 2 3
1 2 3
1 2 3
1 1 1 3 3 3
m ii
Tr
m
p
1 2 3
1
87
The Cauchy stress tensor can be split into: Stress invariants:
sph
:
sph m
1 dev
m
1
spherical stress tensor: stress deviator tensor:
1 1 2 3 ii
I Tr
2 2 1 1 2 1 3 2 3
1 : 2 I I
3
det I
1 1 ii
J I
2 2 1 2
1 1 1 2 : 2 2 2
ij ji
J I I
3 3 1 1 2 3
1 1 1 3 3 3 3 3
ij jk ki
J I I I I Tr
1
I Tr
2 2 1
1 : 2 I I
3
1 det 3
ij jk ki
I
1 1
J I
2 2 1
1 2 J I
2 2
1 2 : 2 I I
3 3 1
1 3 J I
1 2
3I I
3 3
1 1 3 3 3
ij jk ki
I I Tr
88
Effective stress: Principal stress space or
Haigh–Westergaard stress space:
Normal octahedral stress: Shear octahedral stress:
' 2
3 3 3 2 2
ij ij
J ´: ´
1 2 3
1 1, 1, ,1 3
T
n
1
3
m
I
1 2 2
2 3
J
89
Any point in space is unambiguously defined by the three invariants:
The projection of the principal stress space on the octahedral plane results in the division of the plane into six spaces: FEASIBLE WORK SPACE
90
Phenomenological behaviour:
Plastic strain Hardening Bauschinger Effect
Elastoplastic materials are
characterized by:
Lack of unicity in the stress-strain relationship. A nonlinear stress-strain relationship. The existence of permanent (or plastic) strain during a loading / unloading
cycle.
Perfectly Plastic Material
94
1D Incremental Theory:
Additive decomposition of strain: Hardening variable: Yield stress, : Yield function:
e p
e
E
where
p
d sign d
such that and d
p
f
f
d H d
f e
H HARDENING LAW hardening parameter
,
f
F
, F ELASTOPLASTIC STATE ELASTIC STATE
, F
95
1D Incremental Theory (cont’d):
Space of admissible stresses Constitutive Equation: ELASTIC REGIME: ELASTOPLASTIC REGIME: – UNLOADING ELASTOPLASTIC REGIME: – PLASTIC LOADING
Space of admissible stresses
, F
E E E R
E d E d
; , dF
E d E d
, F
, F
, F
; , dF
E
ep
d E d Elastoplastic tangent modulus
ep
EH E E H
96
1D Incremental Theory (cont’d):
Uniaxial Stress-Strain Curve
ELASTIC REGIME ELASTOPLASTIC REGIME
ep
97
1D Incremental Theory (cont’d):
Role of the hardening parameter
Plasticity with strain hardening Lineal elasticity Perfect plasticity Perfect plasticity Plasticity with strain softening
98
3D Incremental Theory: generalization of the 1D incremental theory.
The same concepts are used:
Additive decomposition of strain: Hardening variable: Yield function:
1 : e
e p
where
d
0, with
,
p
G d
FLOW RULE: plastic potential function plastic multiplier
,
f
F
, F ELASTOPLASTIC STATE ELASTIC STATE
, F
equivalent uniaxial stress
99
3D Incremental Theory (cont’d):
Plus, additional ones are added:
Loading - unloading conditions (Kuhn-Tucker conditions): Consistency conditions:
If , then
, , F F
F F
, F
, F
0; 0; 0; 0;
p
F dF F dF F dF
LOADING PLASTIC LOADING impossible
100
3D Incremental Theory (cont’d):
Constitutive Equation: ELASTIC REGIME: ELASTOPLASTIC REGIME: – UNLOADING ELASTOPLASTIC REGIME: – PLASTIC LOADING
E
; , dF
E
, F
; , dF
E Elastoplastic constitutive tangent tensor
, F
, F
: d d C : d d C :
ep
d d C
: : : :
ep
G F F G H C C C C C C C C C
101
The yield surface is the external boundary of the elastic domain.
1 2 3
, , F I J J F
1 1 2 3 ii
I Tr
2 2 1
1 2 J I
2 2
1 2 : 2 I I
3 3 1
1 3 J I
1 2
3I I
3 3
1 1 3 3 3
ij jk ki
I I Tr
with
1 2 3
102
Yield criteria:
Von Mises Criterion Tresca Criterion
2
3
e
F J
3 2
e
1 3 e
F
2 3
( ) , F J J F
2
F J F
103
Yield criteria:
Mohr-Coulomb Criterion Drucker-Prager Criterion
1 3 1 3 sin
2 cos F c
internal friction angle cohesion
1 2 3
, , F I J J F
1/2 2
3
m
F J
1 2 3 1
2sin 6 cos ; ; 3 3 3 3 sin 3 3 sin
m
I c
1 2
, F I J F
3 cotan c
104