types of non-linearities in structural mechanics ...fast10.vsb.cz/brozovsky/data/pp2/p7en.pdf ·...
TRANSCRIPT
-
Non-linear mechanics
• types of non-linearities in structural mechanics,
• introduction to the theory of plasticity,
• solution methods for non-linear problems.
1
-
Types of non-linearities
• construction (model) non-linearities – supports and struc-
tural elements that are working only in certain conditions
(compression-only etc.),
• physical (constitutive) non-linearities – material behaviour
is not linear (differs from Hooke’s law) (non-linear elasticity,
plasticity, fracture mechanics,. . . ),
• geometric non-linearity – large deformations (displacements,
rotations,. . . )
2
-
Model non-linearity
F
F
• supports that are working only under
certain type of load,
• often used for contact problems,
• usually requires iterative solution.
3
-
Support working only for certainload (1)
4
-
Support working only for certainload (2)
Model:
podlozka
uFEM 0.2.46
Time: 1CS: CART
23. 09. 2008
x
y
z
5
-
Support working only for certainload (3)
Reactions (normal supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Results
23. 09. 2008
-7.135765e+02
-6.243795e+02
-5.351824e+02
-4.459853e+02
-3.567883e+02
-2.675912e+02
-1.783941e+02
-8.919706e+01
0.000000e+00
5.488281e+01
1.097656e+02
1.646484e+02
2.195312e+02
2.744141e+02
3.292969e+02
3.841797e+02
4.390625e+02
x
y
z
6
-
Support working only for certainload (4)
Deformed shape (normal supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Result: s_1
23. 09. 2008
-7.135765e+02
-6.243795e+02
-5.351824e+02
-4.459853e+02
-3.567883e+02
-2.675912e+02
-1.783941e+02
-8.919706e+01
0.000000e+00
5.488281e+01
1.097656e+02
1.646484e+02
2.195312e+02
2.744141e+02
3.292969e+02
3.841797e+02
4.390625e+02
x
y
z
7
-
Support working only for certainload (5)
Normal stress σx (normal supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Result: s_x
23. 09. 2008
-7.164786e+02
-6.269188e+02
-5.373590e+02
-4.477991e+02
-3.582393e+02
-2.686795e+02
-1.791196e+02
-8.955983e+01
0.000000e+00
5.410934e+01
1.082187e+02
1.623280e+02
2.164374e+02
2.705467e+02
3.246560e+02
3.787654e+02
4.328747e+02
x
y
z
8
-
Support working only for certainload (6)
Normal stress σy (normal supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Result: s_y
23. 09. 2008
-1.534330e+03
-1.342539e+03
-1.150747e+03
-9.589562e+02
-7.671650e+02
-5.753737e+02
-3.835825e+02
-1.917912e+02
0.000000e+00
3.712790e+00
7.425580e+00
1.113837e+01
1.485116e+01
1.856395e+01
2.227674e+01
2.598953e+01
2.970232e+01
x
y
z
9
-
Support working only for certainload (7)
Normal stress σ1 (normal supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Result: s_1
23. 09. 2008
-7.135765e+02
-6.243795e+02
-5.351824e+02
-4.459853e+02
-3.567883e+02
-2.675912e+02
-1.783941e+02
-8.919706e+01
0.000000e+00
5.488281e+01
1.097656e+02
1.646484e+02
2.195312e+02
2.744141e+02
3.292969e+02
3.841797e+02
4.390625e+02
x
y
z
10
-
Support working only for certainload (8)
Deformed shape (compression-only supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000CS: CART
23. 09. 2008
-8.029127e+02
-7.025486e+02
-6.021845e+02
-5.018204e+02
-4.014564e+02
-3.010923e+02
-2.007282e+02
-1.003641e+02
0.000000e+00
4.536868e+01
9.073735e+01
1.361060e+02
1.814747e+02
2.268434e+02
2.722121e+02
3.175807e+02
3.629494e+02
x
y
z
11
-
Support working only for certainload (9)
Normal stress σx (compression-only supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Result: s_x
23. 09. 2008
-8.084983e+02
-7.074360e+02
-6.063737e+02
-5.053114e+02
-4.042491e+02
-3.031869e+02
-2.021246e+02
-1.010623e+02
0.000000e+00
3.922044e+01
7.844088e+01
1.176613e+02
1.568818e+02
1.961022e+02
2.353226e+02
2.745431e+02
3.137635e+02
x
y
z
12
-
Support working only for certainload (10)
Normal stress σy (compression-only supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Result: s_y
23. 09. 2008
-1.535040e+03
-1.343160e+03
-1.151280e+03
-9.594000e+02
-7.675200e+02
-5.756400e+02
-3.837600e+02
-1.918800e+02
0.000000e+00
6.736788e+00
1.347358e+01
2.021036e+01
2.694715e+01
3.368394e+01
4.042073e+01
4.715751e+01
5.389430e+01
x
y
z
13
-
Support working only for certainload (11)
Normal stress σ1 (compression-only supports):
podlozka
uFEM 0.2.46
Set: 1: 1.000Result: s_1
23. 09. 2008
-8.029127e+02
-7.025486e+02
-6.021845e+02
-5.018204e+02
-4.014564e+02
-3.010923e+02
-2.007282e+02
-1.003641e+02
0.000000e+00
4.536868e+01
9.073735e+01
1.361060e+02
1.814747e+02
2.268434e+02
2.722121e+02
3.175807e+02
3.629494e+02
x
y
z
14
-
Physical (constitutive) non-linearity
• Non-linear elasticity
– Hooke’s law is not respected
– No non-reversible deformations
• Elasto-plastic behaviour
– No-reversible (plastic) deformations
• Viskoelasticity, viskoplasticity,. . .
• Fragile materials
– Fracture mechanics
F
u
F
u
F
u
15
-
Elastoplastic behaviour (1)
Load-displacement diagram for axially loaded member:
F
f
elasticplastic
u
16
-
Elastoplastic behaviour (2)
1. Ideally elasto–plastic
2. Elasto–plastic with harden-
ing
3. Stiff–plastic
F
F
u
F
u
u
F
17
-
Plasticity condition
F
u
f()f()
σ2
σ1
18
-
Plasticity conditions (1)
Maximal normal stresses theory (Rankine):
−σmd ≤ σ1 ≤ σmt
σ1 − σmt = 0
σ2 − σmd = 0
σ
σ2
1
σσ
σ
σ
mt
mt
md
md
19
-
Plasticity conditions (2)
Maximal shear stresses theory (Tresca):
τmax =σ1 − σ3
2− τm = 0
σ1 − σ3 − σmt = 0
(τm =σmt
2)
σmd = σmt
σ
σ1
2
σσ
σ
mtmt
mt
σmt
20
-
Plasticity conditions (3)
Shape change energy condition (von Mises)
(von Mises, Huber, Hencky):
(σ1−σ2)2+(σ3−σ2)
2+(σ1−σ3)2 = 2σ2mt
σmd = σmt
Note: ,,von Mises stress“:√
√
√
√
(σ1 − σ2)2 + (σ3 − σ2)
2 + (σ1 − σ3)2
2
σ
σ1
2
σmtσ
σ
von Mises
Tresca
mt
mt
mtσ
21
-
Plasticity conditions (4)
Mohr – Coulomb condition (for geotechnics):
σ1 −σmt
σmdσ3 − σmt = 0
σmd 6= σmt
σ
σ2
1
σσ
σ
σ
mt
mt
md
md
22
-
Plasticity conditions (4)
Chen – Chen condition (concrete):
For compression–compression
area (σ1 < 0 a σ2 < 0, σ3 < 0):
J2 +Ayc
3I1 − τ
2yc = 0
For other areas:
J2 −1
6I21 +
Ayt
3I1 − τ
2yt = 0
σ
σ1
2
fyc
fyt
fyt
fybc
fybc ycf
23
-
Hardening (1)
F
u
σ1
σ2
24
-
Hardening (2)
• Kinematic
– subsequent plasticity conditions are
moving
– no change of shape and size
• Isotropic
– subsequent conditions are changing
size proportionally
– no moves
• Combined
– kinematic + isotropic
σ
σ1
2
σ
σ1
2
25
-
Hardening (3)
Combined hardening
σ
σ1
2
26
-
Plasticity and failure conditions
1. Initial plasticity condition
2. Subseqent plasticity condition
3. Failure condition
F
u
σ1
σ2
21
3
32
1
27
-
Variants of theory of plasticity (1)
Theory of plastic deformations:
• uses relations between total deforma-
tions and stresses:
σ = DEP ε
• solution does not depend on loading
path
• (in most cases) hard derivation of rela-
tions
σ
ε
28
-
Variants of theory of plasticity (2)
Plastic flow theory:
• relation between changes (”speeds”) of
deformations and stresses:
σ̇ = Dep ε̇
• solution depends on loading path
• solution can bed divided to a set of lin-
earised steps
σ
29
-
Plastic flow theory
Unknowns – changes (”speeds”):
• stresses: σ̇ = {σ̇x, σ̇y, σ̇z, τ̇yz, τ̇yz, τ̇xy}T
• relative deformations: ε̇ = {ε̇x, ε̇y, ε̇z, γ̇yz, γ̇yz, γ̇xy}T
• displacements (and rotations): u̇ = {u̇, v̇, ẇ}
Assumptions:
• initial stress σ and strain ε state must be known
• solution have to respect boundary conditions
30
-
Elasto–plastic material matrix (1)
• Constitutive equations:
σ̇ = Dep ε̇
Dep . . . elasto–plastic material matrix (have to be found)
• Division of change of deformations to elastic and plastic part:
ε̇ = ε̇e + ε̇p
• Plastic condition (used for description of change from elastic
to plastic state):
f(σ, k) = 0
31
-
Elasto–plastic material matrix (2)
• Consistence condition of plastic material:
df =
{
∂f
∂σ
}T
{dσ}+
{
∂f
∂k
}T
{dk} = 0
32
-
Elasto–plastic material matrix (3)
• Speed of plastic deformation (plastic deformation law):
ε̇p = dλ
{
∂f
∂σ
}
• Stress changes:
σ̇ = dσ = De (ε̇− ε̇p) = De
(
ε̇− dλ
{
∂f
∂σ
})
• Equivalent plastic deformation:
dεp =
√
ε̇pT ε̇p = dλ
√
√
√
√
√
{
∂f
∂σ
}T {∂f
∂σ
}
33
-
Elasto–plastic material matrix (4)
• From consistence condition:{
∂f
∂σ
}T
Dedε−dλ
{
∂f
∂σ
}T
De
{
∂f
∂σ
}
+dλ∂f
∂εp
√
√
√
√
√
{
∂f
∂σ
}T {∂f
∂σ
}
= 0
34
-
Elasto–plastic material matrix (3)
Computation of dλ parameter:
dλ =
{
∂f∂σ
}TDeε̇
{
∂f∂σ
}TDe
{
∂f∂σ
}
+ ∂f∂εp
√
{
∂f∂σ
}T {∂f∂σ
}
If put to σ̇ = De(
ε̇− dλ{
∂f∂σ
})
:
σ̇ = De
ε̇−
{
∂f∂σ
}TDeε̇
{
∂f∂σ
}TDe
{
∂f∂σ
}
+ ∂f∂εp
√
{
∂f∂σ
}T {∂f∂σ
}
{
df
dσ
}
35
-
Elasto–plastic material matrix (4)
The equation for σ̇ can be simplified:
σ̇ = Dep ε̇ep,
where elasto–plastic material matrix Dep is:
Dep = De −De
{
∂f∂σ
} {
∂f∂σ
}TDe
{
∂f∂σ
}TDe
{
∂f∂σ
}
− ∂f∂εp
√
{
∂f∂σ
}T {∂f∂σ
}
36
-
Material models for concrete
Mechanical properties of concrete
• Near no linear elastic behaviour
• Non-linear behaviour (non-reversible deformations, crack-
ing,. . . )
• Different behaviour for different types of loading
F
u
37
-
Constitutive models for concrete
• Discrete models:
individual cracks are modelled (usually by finite element mesh
changes)
• Continuum models:
model is assumed to remain continuous but with different ma-
terial properties
– Models based on non-linear fracture mechanics (smaered
cracks, non-local continuum, microplane models)
– Elasto–plastic models
38
-
Chen plasticity condition (1)
• Usual plasticity conditions don’t satisfy concrete
behaviour (von Mises, Tresca)
• Experiment research by plane stress concrete
samples (Kupfer)
• Several approximations of experimental data
(Kupfer, Chen a Chen, Willam a Warnke,. . . )
Chen and Chen:
• Aproximation of Kupfer data by polynomic func-
tions
• The condition can be used both for plasticity and
for failure
σ2
σσ
σ
mt
mt
σmt
σ2
39
-
Chen plasticity condition (2)
For compression–compression zone
(σ1 < 0 a σ2 < 0, σ3 < 0):
J2 +Ayc
3I1 − τ
2yc = 0
For all ther zones:
J2 −1
6I21 +
Ayt
3I1 − τ
2yt = 0
where:
I1 = σ1 + σ2 + σ3
J2 =1
2
(
σ21 + σ22 + σ
23
)
σ
σ
2
fyc
fyt
fyt
fybc
fybc ycf
40
-
Chen plasticity condition (3)
Constants Ayx, τyx explanation:
Ayc =f2ybc − f
2yc
2fybc − fyc
τ2yc =fybcfyc(2fyc − fybc)
3(2fybc − fyc)
Ayt =fyc − fyt
2
τ2ut =fycfyt
6
σ
σ1
2
fyc
fyt
fyt
fybc
fybc ycf
41
-
Chen failure condition (4)
Failure condition can be defined in the same manner:
J2 +Auc
3I1 − τ
2uc = 0
J2 −1
6I21 +
Aut
3I1 − τ
2ut = 0
Auc =f2ubc − f
2uc
2fubc − fuc
τ2yc =fubcfyc(2fuc − fubc)
3(2fubc − fuc)
Aut =fuc − fut
2
τ2ut =fucfut
6
42
-
Chen failure condition (5)
Intermediate (subsequent) conditions:
Ac = αcτ2c + βc
At = αtτ2t + βt
u
y
αc =Auc −Ayc
τ2uc − τ2yc
βc =Aycτ2uc −Aycτ
2yc
τ2uc − τ2yc
αt =Aut −Ayt
τ2ut − τ2yt
βt =Aytτ
2ut −Aytτ
2yt
τ2ut − τ2yt
43
-
Related conditions
Kupfer failure condition:
• defined for 2D stress state
• uses data from standardized tests (cyllindric strength of con-
crete)
Willam–Warnke condition:
• defined for 3D
• very similar to Chen one in term of input data and shape:
f =1
3z
I1σc
+
√
√
√
√
2
5+
1
r(θ)
J2σc
− 1 = 0
44
-
Example – finite element model ofconcrete arch
45
-
Example – plastic areas on arch
46
-
Example – stress–strain curve
0
20
40
60
80
100
120
140
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Rel
ativ
e lo
ad
Relative displacement
"arc18chen.rtrack" using 3:5"arc18smc.rtrack" using 3:5
47
-
Smeared crack model (1)
• Modelling of damaged area (cracks,. . . ) by reduction of ma-
terial properties (E, ν)
• Continuous model
• Not very good for large discrete cracks
R red.
48
-
Smeared crack model (2)
Orthotropic material:
D =R2
R2 − µ2 R1
R1 µR1 0µR1 R2 0
0 0 β GR2/(R2−µ2 R1)
,
1
2
49
-
Smeared crack model (3)
One-dimensional equivalent stress–strain relation:
• 2D failute condition is used for determination
of equivalen diagram parameters:
– Chen–Chen
– Kupfer
• Crack band model can be used for limiting
of result dependence of finite element mesh
size (fracture energy is used)
σ
ε
σ1
σ2
50
-
Smeared crack model (4)
Bažant’s crack band model:
Fracture energy:
GF = AG L = const.,
GF =∫ ∞
0σn(w) dw,
Total cracks widht: w = ε L
Descending modulus form:
Ez =Eo
1− 2GFEoL σ2max
.
AG w
σ
L
σ
σ
Eo
Ez
max
ε
51
-
Smeared crack model (5)
Example – finite element model:
1
2
162
161
160
159
158
157
156
155
154
153
152
151
150
149
148
147
146
145
144
143
142
141
140
139
138
137
136
135
134
133
132
131
130
129
128
127
126
125
124
123
122
121
120
119
118
117
116
115
114
113
112
111
110
109
108
107
106
105
104
103
102
101
100
99
98
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
20. 03. 2008
CS: CARTTime: 1
uFEM 0.2.30
xz
y
52
-
Smeared crack model (6)
Example – residual stiffness Rt:
uFEM 0.2.30
Time: 39.0000
y
x
2.000000e+10
1.750000e+10
1.500000e+10
1.250000e+10
1.000000e+10
7.500000e+09
5.000000e+09
2.500000e+09
0.000000e+00
0.000000e+00
0.000000e+00
0.000000e+00
0.000000e+00
0.000000e+00
0.000000e+00
0.000000e+00
0.000000e+00
20. 03. 2008
Result: 28
z
53
-
Smeared crack model (7)
Example – load–displacement curve:
0
0.2
0.4
0.6
0.8
1
1.2
0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003
rela
tive
load
F [-
]
vertical displacement w [m]
8x416x818x9
54
-
Recomended reading
• BAŽANT, Z. P., PLANAS J. Fracture and Size Effect in Con-
crete and Other Quasibrittle Materials, CRC Press, Boca Ra-
ton, 1998
• HAN, D. J., CHEN, W. H. Constitutive Modeling in Analysis of
Concrete Structures, Journal of Engineering Mechanics. Vol.
113, No. 4, April, ASCE, 1987
• CHEN, A. C. T., CHEN, W. F. Constitutive Relations for Con-
crete, Journal of the Engineering Mechanics Division ASCE,
1975
• OHTANI, Y., CHEN, W. F. Multiple Hardening Plasticity for
Concrete Materials, Journal of the EDM ASCE, 198855