7.MASAT 2012-2013 SmearedCrackModel

8/11/2019 7.MASAT 2012-2013 SmearedCrackModel

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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

[email protected]

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




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Strain decomposition concept

cr co = +

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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



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



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



- 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

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Softening laws: (a) tri-linear diagram; (b) exponencial diagram


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crnncr

fct fct

Discrete approach

h

Smeared approach

w

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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



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

=

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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

= +

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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

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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

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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



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=

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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



, 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

?

( )( ),

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7.MASAT 2012-2013 SmearedCrackModel