7.masat 2012-2013 smearedcrackmodel
TRANSCRIPT
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Multi-fixed smeared crack model for brittle materials
U n i v e r s i t y o f M i n h o
Department of Civil EngineeringGuimares, Portugal
www.civil.uminho.pt
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Type of models
1. Discrete crack models to simulates the behaviour of structures
with already existing cracks; When the localization of cracks are
known a priori.
2. Smeared crack models When diffuse crack partners are
expected to occur.
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
3. Hybrid model Combine discrete and smeared concepts.
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:: Example
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
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Strain decomposition concept
cr co = +
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
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Stress and strain in the crack
x2
t n
cr
n
cr
t
s
= llllT
cr cr cr n t
= llll
cr cr T
[ ]1 2 12T
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
x1
rac
cr
tcr
n w
= llllTcr cr cr
n t 1 2 12 =
Tcr cr cr cr
= llllT
cr cr cr T
2 2
2 2
cos sin 2sin cos
sin cos sin cos cos sin
crT
=
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Concrete model for concrete in between cracks
co coD =
( )2
1 0
1 01
0 0 1 2
c
co cc
c
c
ED
=
Constitutive matrix for theconcrete between cracks
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
Model for the cracks
= l ll ll ll l
cr cr cr D
0
0
cr
cr I
cr
II
DD
D
=
Constitutive matrix for the crack
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Model (I) fracture parameter
- Tangent stiffness corresponding to the fracture mode I
0
0
cr
cr I
cr
II
D
D D
=
cr
ID
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
- Tangent stiffness corresponding to fracture mode II
The opening-sliding mutual effect is not simulated.
cr
IID
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Mode (II) fracture parameters
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
Softening laws: (a) tri-linear diagram; (b) exponencial diagram
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crnncr
fct fct
Discrete approach
h
Smeared approach
w
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
Softeningdiagrams for discrete crack models (a) and distributed cracks (b).
ncr
G /hffGw
(a) (b)
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To assure mesh objectivity (independence of the results from the
mesh refinement), the crack band width should be dependent of
the finite element geometry. Several researchers have
recommended the following proposal:
Area of the finite element=h
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
Contudo, para evitar a instabilidade porsnap-back: 2f c
ct
G Ehb f
Area of the integration point=h
Constant value=h
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cr co = +
1 1 1cr
c II cG D G
= +
1
cr
II cD G
=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
1
,
1
=
p
crn
cr
n ult
Shear retention factor
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crcoD =
( )1
T Tcrco co co cr cr cr co cr cr co
D D D T D T D T T D
= +
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
crcoD Evaluation of the stiffness matrix
Evaluation of the nodal forces
T
V
K B D B dV = T
V
F B dV =
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Failure surface
II
fct
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
Rankine failure criteria
I
fct
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Extention to several cracks
( ) ( ) ( )1 1 2 2 = cr cr Tcr cr cr cr
n nT T T T
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
1
2
0 00 0
0 0
=
cr
cr
cr
cr
cr
n
DD
D
D
tne crack n. ncr
Crack constitutive
law for the ncr
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Criteria for the crack opening
1. The maximum principal stress,I, should exceedfct
2. The angle between existing cracks and the direction of I,should exceed thethreshold angle
3. Both 1. and 2. conditions (recommended criteria)
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
Teh values recommended for the threshold angle verie in the
interval 30 to 60.
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Crack history
ncr,1
ctf
fct
cr,2
n
(1)2
p
next
, ,
prev
f a f f cG G G=
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
f,cGcr,1
cr,1n
ncr,2
cr,2
Gf
/h
/h f,acr,1
G /h
, ,
2
f f f a f a
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Stress updated
, =llllcr cr
m m mT
( ), 1 , 1 + = + l ll ll ll lcr cr cr
m m m m mT
Tcr cr cr co cr cr
Joaquim Barros
Modeling and Advanced Structural Analysis Techniques
University of Minho Department of Civil Engineering
, 1 , 1 , l l ll l ll l ll l lm m m m m m m
( ), 1 , , , 1 0 + + = l l l l l l l l l l l l l l l l T
cr cr cr cr co cr cr cr cr co
m m m m m m m m m mT D T T T D
( )
( )
,
, 1 , , , 1
cr
m
Tcr cr cr cr co cr cr cr cr co
m m m m m m m m m m
f
T D T T T D
=
+ +
llll
l l l l l l l l l l l l l l l l
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Stress updated (cont.) Zero the iteration counter:
Calculate the initial solution: ( )0
, llllcr
m
0q
?
( )( ),