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

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136

135

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133

132

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130

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127

126

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124

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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

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79

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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