32 The cross-sections of a yield surface are the intersection curves between the yield surface and the deviatoric plane ρ, θ which is perpendicular to the hydrostatic axis ξ and with ξ =
Trang 112 In terms of the principal stresses, those invariants can be simplified into
I1 = σ1+ σ2+ σ3 (19.24)
I2 = −(σ1σ2+ σ2σ3+ σ3σ1) (19.25)
I3 = σ1σ2σ3 (19.26)
13 Similarly,
J1 = s1+ s2+ s3 (19.27)
J2 = −(s1s2+ s2s3+ s3s1) (19.28)
J3 = s1s2s3 (19.29)
19.4.1.2 Physical Interpretations of Stress Invariants
14 If we consider a plane which makes equal angles with respect to each of the principal-stress directions,
π plane, or octahedral plane, the normal to this plane is given by
n = √1
3
1 1 1
The vector of traction on this plane is
toct=√1
3
σ1
σ2
σ3
and the normal component of the stress on the octahedral plane is given by
σ oct= toct ·n = σ1+ σ2+ σ3
1
or
σ oct= 1
15 Finally, the octahedral shear stress is obtained from
τ oct2 =|t oct |2− σ2
oct= σ
2
3 +
σ2
3 +
σ2
3 − (σ1+ σ2+ σ3)2
Upon algebraic manipulation, it can be shown that
9τ oct2 = (σ1− σ2)2+ (σ2− σ3)2+ (σ1− σ3)2= 6J2 (19.35) or
τ oct=
1 2
and finally, the direction of the octahedral shear stress is given by
cos 3θ = √
2 J3
τ3
oct
(19.37)
Trang 2U = U1+ U2 (19.38) where
U1=1− 2ν
2
U2= 1 + ν
19.4.1.3 Geometric Representation of Stress States
Adapted from (Chen and Zhang 1990)
17 Using the three principal stresses σ1, σ2, and σ3, as the coordinates, a three-dimensional stress space
can be constructed This stress representation is known as the Haigh-Westergaard stress space,
Fig 19.4
3
1 Cos
J 2
−1
(s ,s ,s )
3
1 2 3
1
σ
N(p,p,p)
ρ
ξ
O
σ
2
σ
Hydrostatic axis
deviatoric plane
P( , , )
σ σ
σ1 2 3
Figure 19.4: Haigh-Westergaard Stress Space
18 The decomposition of a stress state into a hydrostatic, pδ ij and deviatoric s ij stress components
can be geometrically represented in this space Considering an arbitrary stress state OP starting from
O(0, 0, 0) and ending at P (σ1, σ2, σ3), the vector OP can be decomposed into two components ON and
NP The former is along the direction of the unit vector (1√
3, 1/ √
3, 1/ √
3), and NP⊥ON.
19 Vector ON represents the hydrostatic component of the stress state, and axis Oξ is called the
hy-drostatic axis ξ, and every point on this axis has σ1= σ2= σ3= p, or
ξ = √
Trang 3to the ξ axis Any plane perpendicular to the hydrostatic axis is called the deviatoric plane and is
expressed as
1
√
and the particular plane which passes through the origin is called the π plane and is represented by
ξ = 0 Any plane containing the hydrostatic axis is called a meridian plane The vector NP lies in a
meridian plane and has
ρ =
0
s2+ s2+ s2=
21 The projection of NP and the coordinate axes σ i on a deviatoric plane is shown in Fig 19.5 The
’
1200
1200
1200
P’
θ N’
σ ’1
Figure 19.5: Stress on a Deviatoric Plane
projection of NP of NP on this plane makes an angle θ with the axis σ1
cos 3θ = 3
√
3 2
J3
22 The three new variables ξ, ρ and θ can all be expressed in terms of the principal stresses through their invariants Hence, the general state of stress can be expressed either in terms of (σ1, σ2, σ3), or (ξ, ρ, θ).
For 0≤ θ ≤ π/3, and σ1≥ σ2≥ σ3, we have
σ1
σ2
σ3
p p p
+
2
√
3
J2
cos θ cos(θ − 2π/3) cos(θ + 2π/3)
=
ξ ξ ξ
+
1 2
3ρ
cos θ cos(θ − 2π/3) cos(θ + 2π/3)
(19.44-c)
19.4.2 Hydrostatic Pressure Independent Models
Adapted from (Chen and Zhang 1990)
23 For hydrostatic pressure independent yield surfaces (such as for steel), their meridians are straigth lines parallel to the hydrostatic axis Hence, shearing stress must be the major cause of yielding for
Trang 4direction, it follows that the elastic-plastic behavior in tension and in compression should be equivalent for hydorstatic-pressure independent materials (such as steel) Hence, the cross-sectional shapes for this
kind of materials will have six-fold symmetry, and ρ t = ρ c
19.4.2.1 Tresca
24 Tresca criterion postulates that yielding occurs when the maximum shear stress reaches a limiting
value k.
max
1
2|σ1− σ2|,1
2|σ2− σ3|,1
2|σ3− σ1|
from uniaxial tension test, we determine that k = σ y /2 and from pure shear test k = τ y Hence, in Tresca, tensile strength and shear strength are related by
25 Tresca’s criterion can also be represented as
2
J2sin
θ + π
3
− σ y = 0 for 0≤ θ ≤ π
26 Tresca is on, Fig 19.6:
Plane
π
σ
σ
3
σ
1
2
ξ
ρ
ρ
0
ξ
ρ
σ1’ ρ
0
σy
σ2
−σy
−σy
σ1
σ
τ
σ3
σ’ 2
’
Figure 19.6: Tresca Criterion
• σ1σ2σ3 space represented by an infinitly long regular hexagonal cylinder.
Trang 5ρ = 2
J2=√ σ y
2 sin%
θ + π3& for 0 ≤ θ ≤ π
a regular hexagon with six singular corners
• Meridian plane: a straight line parallel to the ξ axis.
• σ1σ2 sub-space (with σ3 = 0) an irregular hexagon Note that in the σ1 ≥ 0, σ2 ≤ 0 the yield
criterion is
• στ sub-space (with σ3= 0) is an ellipse
σ
σ y
2 +
τ
τ y
2
27 The Tresca criterion is the first one proposed, used mostly for elastic-plastic problems However, because of the singular corners, it causes numerous problems in numerical analysis
19.4.2.2 von Mises
28 There are two different physical interpretation for the von Mises criteria postulate:
1 Material will yield when the distorsional (shear) energy reaches the same critical value as for yield
as in uniaxial tension
=
1
(σ1− σ2)2+ (σ2− σ3)2+ (σ1− σ3)2
2 ρ, or the octahedral shear stress (CHECK) τ oct, the distance of the corresponding stress point
from the hydrostatic axis, ξ is constant and equal to:
ρ0= τ y √
29 Using Eq 19.52, and from the uniaxial test, k is equal to k = σ y / √
3, and from pure shear test k = τ y Hence, in von Mises, tensile strength and shear strength are related by
σ y =√
Hence, we can rewrite Eq 19.51 as
f (J2) = J2− σ2
30 von Mises is on, Fig ??:
• σ1σ2σ3 space represented by an infinitly long regular circular cylinder.
• π (Deviatoric) Plane, the yield criterion is
ρ =
1 2
a circle
Trang 6π Plane
1
σ3 σ
2
σ
ξ
ρ
ρ
0
ξ
1
ρ
0
σ
τ
σ2
σ’
3
σ1
x y
Figure 19.7: von Mises Criterion
• Meridian plane: a straight line parallel to the ξ axis.
• σ1σ2 sub-space (with σ3= 0) an ellipse
σ21+ σ22− σ1σ2= σ y2 (19.57)
• στ sub-space (with σ3= 0) is an ellipse
σ
σ y
2 +
τ
τ y
2
Note that whereas this equation is similar to the corresponding one for Tresca, Eq 19.50, the
difference is in the relationships between σ y and τ y
19.4.3 Hydrostatic Pressure Dependent Models
31 Pressure sensitive frictional materials (such as soil, rock, concrete) need to consider the effects of both the first and second stress invariants frictional materials such as concrete
32 The cross-sections of a yield surface are the intersection curves between the yield surface and the
deviatoric plane (ρ, θ) which is perpendicular to the hydrostatic axis ξ and with ξ = constant The
cross-sectional shapes of this yield surface will have threefold symmetry, Fig 19.8
33 The meridians of a yield surface are the intersection curves between the surface and a meridian plane
(ξ, ρ) which contains the hydrostatic axis The meridian plane with θ = 0 is the tensile meridian, and passes through the uniaxial tensile yield point The meridian plane with θ = π/3 is the compressive
meridian and passes through the uniaxial compression yield point.
34 The radius of a yield surface on the tensile meridian is ρ t , and on the compressive meridian is ρ c
Trang 7σ ’ 2 σ 3 ’
ρ ρ
ρ
c
t
θ
DEVIATORIC PLANE
ξ
MERIDIAN PLANE
Tensile Meridian
Compressive Meridian
θ=0
θ=π/3
Figure 19.8: Pressure Dependent Yield Surfaces
19.4.3.1 Rankine
35 The Rankine criterion postulates that yielding occur when the maximum principal stress reaches the tensile strength
σ1= σ y; σ2= σ y; σ3= σ y; (19.59)
36 Rankine is on, Fig 19.9:
• σ1σ2σ3 space represented by XXX
• π (Deviatoric) Plane, the yield criterion is
2
ρ t = √1
2(√ 3σ Y − ξ)
ρ c = √
2(√
a regular triangle
• Meridian plane: Two straight lines which intersect the ξ axis ξ y=√
3σ y
• σ1σ2 sub-space (with σ3= 0) two straight lines
σ + 1 = σ y
• στ sub-space (with σ3= 0) is a parabola
σ
σ y +
τ
σ y
2
19.4.3.2 Mohr-Coulomb
37 The Mohr-Coulomb criteria can be considered as an extension of the Tresca criterion The maximum shear stress is a constant plus a function of the normal stress acting on the same plane
Trang 8c Cot
π Plane
3 1
σ
2
σ
ξ
Φ
1
σ2
σt
σ’
3
σ1
ξ
θ=π/3
σ τ
Figure 19.9: Rankine Criterion
where c is the cohesion, and φ the angle of internal friction.
38 Both c and φ are material properties which can be calibrated from uniaxial tensile and uniaxail
σ t = 2c cos φ 1+sin φ
39 In terms of invariants, the Mohr-Coulomb criteria can be expressed as:
1
3I1sin φ +
J2sin
θ + π
3
+
1
J2
3 cos
θ + π
3
sin φ − c cos φ = 0 for 0 ≤ θ ≤ π
3 (19.65)
40 Mohr-Coulomb is on, Fig 19.10:
• σ1σ2σ3 space represented by a conical yield surface whose normal section at any point is an
irregular hexagon
• π (Deviatoric) Plane, the cross-section of the surface is an irregular hexagon.
• Meridian plane: Two straight lines which intersect the ξ axis ξ y = 2√
3c/ tan φ, and the two
characteristic lengths of the surface on the deviatoric and meridian planes are
2
ρ t = 2
√ 6c cos φ−2 √
2ξ sin φ 3+sin φ
ρ c = 2
√ 6c cos φ−2 √
2ξ sin φ 3−sin φ
(19.66)
Trang 9c Cot
π Plane
3
2
1
σ
σ
ξ
ρ
σ
Φ
1
ρ ty ρ
cy
σt
σt
σ1
σ2
−σc
σ’
3
ξ
σ
θ=0 θ=π/3
ρ cy
ρ ty
τ
Figure 19.10: Mohr-Coulomb Criterion
• σ1σ2 sub-space (with σ3= 0) the surface is an irregular hexagon In the quarter σ1≥ 0, σ2≤ 0,
of the plane, the criterios is
where
m = σ c
σ t =
1 + sin φ
• στ sub-space (with σ3= 0) is an ellipse
σ + m−1 2m σ c
m+1 2m σ c
2
+
'
τ
√ m 2m σ c
(2
(19.69-a)
19.4.3.3 Drucker-Prager
41 The Drucker-Prager postulates is a simple extension of the von Mises criterion to include the effect
of hydrostatic pressure on the yielding of the materials through I1
F (I1, J2) = αI1+ J2− k (19.70)
The strength parameters α and k can be determined from the uni axial tension and compression tests
2
σ t = √ 3k
1+√ 3α
σ c = √ 3k
1− √ 3α
(19.71)
Trang 10α = √
3(m+1)
k = √ 2σ c
3(m+1)
(19.72)
42 Drucker-Prager is on, Fig 19.11:
c Cot
π Plane
1
σ3
Φ
2
σ
σ
ξ
ρ
σ1’ ρ
0
σ2
σ1
σt
σ’ 3
σ’ 2
ξ ρ
y
x
σ
τ
−σc
Figure 19.11: Drucker-Prager Criterion
• σ1σ2σ3 space represented by a circular cone.
• π (Deviatoric) Plane, the cross-section of the surface is a circle of radius ρ.
ρ = √
• Meridian plane: The meridians of the surface are straight lines which intersect with the ξ axis
at ξ y = k/ √
3α.
• σ1σ2 sub-space (with σ3= 0) the surface is an ellipse
'
x + 1−12α6√ 2kα2
√ 6k 1−12α2
(2
+
√ y 2k
√ 1−12α2
2
(19.74)
where
x = √1
y = √1
Trang 11σ + 3kα 1−3α2
√ 3k 1−3α2
(2 +
'
τ
k
√ 1−3α2
(2
43 In order to make the Drucker-Prager circle coincide with the Mohr-Coulomb hexagon at any section
Outer
2
α = √ 2 si n φ
3(3−sin φ)
k = √ 6c cos φ
3(3−sin φ)
(19.77-a)
Inner
2
α = √ 2 si n φ
3(3+sin φ)
k = √ 6c cos φ
3(3+sin φ)
(19.77-b)
19.5 Plastic Potential
44 Once a material yields, it exhibits permanent deformations through the generation of plastic strains According to the flow theory of plasticity, the rate of generation of these plastic strains is governed by the flow rule In order to define the direction of the plastic flow (which in turn determines the magnitudes
of the plastic strain components), it must be assumed that a scalar plastic potential function Q exists such that Q = Q( σ p , ε p)
45 In the case of an associated flow rule, Q = F , and in the case of a non-associated flow rule, Q
Since the plastic potential function helps define the plastic strain rate, it is often advantageous to define
a non-associated flow rule in order to control the amount of plastic strain generated by the plasticity formulation For example, when modeling plain concrete, excess plastic strains may lead to excess dilatancy (too much volume expansion) which is undesirable The plastic potential function can be formulated to decrease the plastic strain rate, producing better results
19.6 Plastic Flow Rule
46 We have established a yield criterion When the stress is inside the yield surface, it is elastic, Hooke’s law is applicable, strains are recoverable, and there is no dissipation of energy However, when the load
on the structure pushes the stress tensor to be beyond the yield surface, the stress tensor locks up on the yield surface, and the structure deforms plastically (if the material exhibits hardening as opposed
to elastic-perfectly plastic response, then the yield surface expands or moves with the stress point still
on the yield surface) At this point, the crucial question is what will be direction of the plastic flow (that is the relative magnitude of the components of ε P This question is addressed by the flow rule,
or normality rule.
47 We will assume that the direction of the plastic flow is given by a unit vector m, thus the incremental
plastic strain is written as
˙p = ˙λ p ∂Q
∂ σ
(19.78)
where ˙λ p is the plastic multiplier which scales the unit vector mp, in the direction of the plastic flow evolution, to give the actual plastic strain in the material Note analogy with Eq 18.42
Trang 12this vector as the gradient of a scalar potential Q which itself is a function of of the stresses
m = ∂Q
∂ σ = ∂σ ∂Q11
∂Q
∂σ22
∂Q
∂σ33
∂Q
∂σ12
∂Q
∂σ23
∂Q
where Q is called the plastic flow potential and is yet to be defined Hence, the direction of plastic
flow is always perpendicular to the plastic potential function
49 We have two cases
Non-Associated Flow when F p p which corresponds to the general case
Associated Flow when when F p = Q pwhich is a special case This gives rise to the Associated flow
rule
˙p = ˙λ p ∂F p
In this context, the difficulty in determining the normal of a yield surface with a sharp corner should be noted
50 The incremental plastic work (irrecoverable) is given by
which can be rewritten as
51 Hence, the evolution of the stress stateσ is given by the stress rate relation,
˙
19.7 Post-Yielding
19.7.1 Kuhn-Tucker Conditions
52 In the elastic regime, the yield function F must remain negative, and the rate of plastic multiplier ˙λ p
is zero On the other hand, during plastic flow the yield function must be equal to zero while the rate
of plastic multiplier is positive
plastic loading : F˙p = 0; ˙λ p > 0
elastic (un)loading : F˙p < 0; ˙λ p= 0 (19.84)
53 Hence, both cases can be simultaneously covered by the loading-unloading conditions called
Kuhn-Tucker conditions
˙
F p ≤ 0, ˙λ p ≥ 0 and F˙p ˙λ p= 0 (19.85)
Trang 1354 A hardening rule describes a specific relationship between the subsequent yield stress σ y of a material and the plastic deformation accumulated during prior loadings
55 We define a hardening parameter or plastic internal variable, which is often denoted by κ.
κ = ε p=* √
dε p dε p Equivalent Plastic Strain (19.86-a)
κ = W p=*
κ = ε p=*
56 A hardening rule expresses the relationship of the subsequent yield stress σ y , tangent modulus E t and plastic modulus E p with the hardening parameter κ.
19.7.2.1 Isotropic Hardening
57 In isotropic hardening, the yield surface simply increases in size but maintains its original shape Hence, the the progressively increasing yield stresses under both tension and compression loadings are always the same
19.7.2.2 Kinematic Hardening
58 In kinematic hardening, the initial yield surface is translated to a new location in stress space without change in size or shape Hence, the difference between the yield stresses under tension loading and under
compression loading remains constant If we denote by σ t y and σ y c the yield stress under tension and compression respectively, then
σ y t (κ) − σ c
or alternatively
where c(κ) represents the track of the elastic center and satisfies c(0) = 0.
59 Kinematic hardening accounts for the Baushinger effect, Fig 16.2
19.7.3 Consistency Condition
60 Fig 19.12 illustrates the elastic dE e and plastic dE p strain increments for a given stress increment dσ Unloading always follows the initial elastic stiffness E o The material experiences plastic loading once the stress state exceeds the yield surface Further loading results in the development of plastic strains and stresses However, the total stress state cannot exceed the yield surface Thus, during plastic flow the stress must remain on the yield surface, and hence the time derivative of the yield function must vanish whenever and this limit is enforced through the consistency condition,
d
dt F p(σ, ε p , κ) = d
dt F p(σ, λ, κ) = 0 (19.90) or
∂F p
∂ σ
np
: ˙σ + ∂F p
∂λ p
−H p
: ˙λ p+∂F p
∂κ
∂κ
... p= (19.85) Trang 13< /span>54 A hardening rule describes a specific relationship