Modeling of Cement Based Composite Laminates Barzin Mobasher Department of Civil & Environmental Engineering Arizona State University Tempe, Arizona, 85287-5306

HPFRCC-4, Ann Arbor, Michigan, June 16-18, 2003.

Scope of Presentation   

Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation  



Non-linear matrix response  



Cracking Softening

Case Studies      



Elastic response Traditional Composite laminate approach

Glass Fiber/ carbon fiber epoxy composites Cement based Glass fiber composite laminate Polypropylene fiber cement composites Fabric composites Comparison with Experimental results Repair and retrofit approach

Conclusions

Areas of Application for high performance fiber reinforced Materials      

high tensile-toughness characteristics superior impact, earthquake, and fatigue characteristics. prefabricated Structural elements, thin sheets, panels, cladding members. structural repair and retrofit. I-beams, structural members. Sound abatement walls.

Scope of Presentation  

Introduction Pultrusion Based Cement Composite Systems

Filament Winding

Fiber Spool

Fabric Reinforcing Methodology

Cement composites with 50 MPa Tensile Strength and more than 1% strain Capacity 50 Unidirectional

Stress, MPa

40 30

0/90/0 20 10

GFRC Mortar

0 0.000

0.004 0.008 0.012 Strain, mm/mm

0.016

Lamina stacking optimization for strength and toughness 60 50

Unidirectional

Stress, MPa

40

[0/-45/45/90]s

30 20 10

[45/-45]s

0 0.000

0.005

0.010 0.015 0.020 Strain, mm/mm

0.025

Flexural response of unidirectional and angle ply composite laminates

Equivalent Flexural Stress, MPa

40

30

Unidirectional

20

0-90-0

10 Paste 0

0

4

4.8% Continuous AR Glass Fibers

8 12 16 Deflection, mm

20

Polypropylene Fiber based Unidirectional Composites 20

Polypropylene Based cementitious laminates with 7% volume fraction of continuous fibers.

Stress (MPa)

16

PP #5

12 8 4 0 0.00 0.02 0.04 0.06 0.08 0.10 Strain (mm/mm)

Microcrack Toughening Mechanisms

Cracked Laminate

Distributed microcracking in unidirectional composite

Weak Interfaces Result in Strong & Tough Composites

Interfacial Microcracking in between plies is a toughening Mechanism

Various stages of cracking

Fabric Reinforced Cement composites 25

80 BT-GNSP21

Stress, MPa

60 15 40 10 20

5

0

0

0.01

0.02

Strain, mm/mm

0.03

0 0.04

Crack Spacing, mm

20

Crack Spacing for AR-Glass and Polyethylene Fabrics 100

Crack Spacing, mm

80

AR Glass Fabric Polyethylene

60 AR-Glass fabric

40 20 0

0

0.01 0.02 0.03 Strain, mm/mm

0.04 Polyethylene Fabric

Stiffness degradation and Crack Spacing Relationship 1000

Tangent Stiffness, MPa

Glass Fabrics 100

10

1 80

Polyethylene Fabric

60 40 20 Crack Spacing, mm

0

Scope of Presentation   

Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation  

Elastic response Traditional Composite laminate approach

Composite Elastic Properties 

Rule of Mixtures for longitudinal stiffness:

E1( ω ) = E f V f + Em ( ω )( 1 − V f ) 

Halpin-Tsai equations for transverse modulus E2 , G12, and ν12

E2 ( ω ) =

Em ( ω )( 1 + ξ ηV f ) 1 − ηV f

η=

E f − Em ( ω ) E f + ξ Em ( ω )

Incremental Approach 

Unidirectional approach

∆ε ij = S jki ∆σ k

( )

σ = S i k



i −1 jk

or in matrix form:

 σ1   S11  σ k =  σ 2  =  S 21 τ12  i  0 S11 =

∆ε ij + σik−1

1 E1( ω )

S12 = −

S12 S 22 0 υ12 E1( ω )

−1

0   ∆ε1   σ1   0   ∆ε 2  +  σ 2  S 66   ∆γ12  i τ12  i −1 S 22 =

1 E2 ( ω )

S 66 =

1 G12 ( ω )

Q11 Q12 0  Sij−1= Q21 Q22 0   0 0 Q66 

Orientation Effects and Stress Transformation    σ  Cos 2θ σ    1  x     2θ σ  =  σ = T Sin    ij  y    2  τ  − Sinθ Cosθ τ  12   xy  

1  R = 0  0 

0 0 1 0  0 2

Sin2θ Cos 2θ Sinθ Cosθ

   − 2Sinθ Cosθ   Cos 2θ −Sin2θ  

2Sinθ Cosθ

  σ   x σ   y    τ  xy 

m=n θn

Qij = Tij R Qij R Tij−1 hm-1

n ) A = ∑ Qijm( h − h ij m m−1 m=1 1 n 2 − h2 ) Bij = ∑ Qijm( hm m−1 2 m=1 1 n D = ∑ Qijm( h3 − h3 ) ij 3 m m−1 m=1

hm

m=1 θ1

Incremental Approach to Laminate Formulation Piecewise linear approach:  ΔN    ΔM

   

=

A   B 

B   D 

 0  Δε     Δκ   

n A = ∑ Qijm( h − h ) m m−1 ij m=1 1 n 2 − h2 ) B = ∑ Qijm( hm m−1 ij 2 m=1 1 n D = ∑ Qijm( h3 − h3 ) m m−1 ij 3 m=1

For uniaxial loads in a symmetric lamina (B=0) : For bending only in a symmetric lamina (B=0) :

N or M

ε or δ

   ΔN  = A   Δε 0             

 ΔM  = D   Δκ      

Modes of failure σ1t

σ1c

τ12

σ 2t

σ 2c

τ 23

Initial Failure Criteria   

F ( σ1 , σ 2 , τ12 ) = 1

General Yield surface Strength criteria σ1 ≥ σ1fu Tsai-Wu Criteria

σ 2 ≥ σ 2fu

τ12 ≥ τ12fu

F ( σ1 , σ 2 , τ12 ) = F11σ1 + 2 F11σ1σ 2 + F22 σ 2 + F66 σ12 + F1σ1 + F2 σ 2 = 1 2

F1 =

R := −

1 1 − σ1t σ1c

F2 =

σt1 := 50

σc2 := 15

σt2 := 10

σ12 := 5

σc1 := 15

σ13 := 5

1 1 − σ 2t σ 2 c

2

F11 =

1 σ 1t σ 1c

σ23 := 8

1 1 2 2 σ − σ σ σ + σ + σ + 150 2 112500 1 2 150 1 30 2 750 1 7

1

1

2

F22 =

1 σ 2t σ 2c

F66 =

1 2 σ 6u

Scope of Presentation   

Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation  



Elastic response Traditional Composite laminate approach

Non-linear matrix response  

Cracking Softening

Three zones of stress-strain response for the matrix 

Elastic matrix 



Cracking matrix  



failure by means of the yield surface criterion, ultimate strength, Tsai-Wu reduced stiffness, model the stiffness degradation use a scalar damage variable stress-strain response obtained using Nemat-Nasser and Hori’s approach.

σ 1

2 3

Softening matrix, distributed cracking 

strain-softening response, Karihaloo, and Hori’s softening Model

ε

Cracking Matrix- Zone 2 Stress

Damage B

1

ω A

ω O

0

0 Matrix parallel cracking

Strain

Damage Evolution Law-Zone 2 ω0  ω = β ω α ε ε + ( − ) um 1  0 k i

∀ f (σ 1 , σ 2 ,τ 12 ) < 1 , 0 ≤ ω i ≤ 1 ∀ f (σ 1 , σ 2 ,τ 12 ) = 1 1.00

Em =

Em 1+

16 ω i (1 - υ m2 ) 3

β = 0.8

0.80 E( ω) / E0

Karihaloo and Fu, 1990 α= 0.16, β= 2.3 εum= ultimate strain at failure for uniaxial conditions

β = 0.6

0.60

0.40

0.000 Nemat Nasser and Hori, Micromechanics: overall properties of Heterogeneous Materials, 1993

β = 0.4

0.002 0.004 Axial Strain, mm/mm

0.006

Computations Within a load step Compute overall stiffness Calculate the strains, stresses, curvature, and elongation Check for the failure of the entire lamina using ultimate strength update the structure

Check for the failure in the components of the lamina using the failure criteria (Tsai-Wu or ultimate strength Theory) update the lamina properties

Initialize all the geometrical and historical parameters for each lamina. Impose the strain distribution, update the stresses in this increment

 ∆N   A( Q( ω )) ∆M  =     B ( Q( ω ))

B ( Q( ω ))   D ( Q( ω ))

∆ε 0     ∆κ 

[ ]

N xj ,i = N xj ,i −1+∆N xj ,i = N xj ,i −1+ [A( Q( ω ))] i ∆ε 0

M xj ,i =M xj ,i −1+∆M xj ,i =M xj ,i −1+ [D (Q(ω ))] i [∆κ ]

Ultimate Failure & Strain Softening Response 1.0

σfu is ultimate strength of fiber •Softening Response Post peak as a fraction of ultimate matrix stress, Model by Karihaloo and Fu, 1990

σ/ f t

•Ultimate FailureFailure Criteria for each lamina: σun = max(VfσfuCos2θm, σt2)

H = 0.25 gage length (in meters) E0= 30000 MPa εB= 200e-6 εA= 0.3* ft/ E0 ω0= εB * H Cd0=.16*(1-εA/ εB ) 2.3 w0=16*Cd0 / 9

0.8

0.6

0.4

0.2 0.0000

0.0001

0.0002 0.0003 Crack Opening,

0.0004

Scope of Presentation   

Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation  



Non-linear matrix response  



Elastic response Traditional Composite laminate approach Cracking Softening

Case Studies      

Glass Fiber/ carbon fiber epoxy composites Cement based Glass fiber composite laminate Polypropylene fiber cement composites Fabric composites Comparison with Experimental results Repair and retrofit approach

Glass-epoxy Composites

Unidirectional Glass-Epoxy and Woven Carbon-Epoxy Composites Vf (%)

Strength (MPa)

Elastic Modulus (GPa)

Glass Fiber epoxy, 0º, 90º

45

σfu = 1654 σmu = 70

Ef =72, υf = 0.2 Em= 4.0, υm=.18

Woven Carbon Textile

50

σt1=35-70

Ef = 100, υm=.25 Em= 6.0, υm=.18

Glass-Epoxy composites

Woven Carbon

Experiment

Simulation

60 Theory

40

Load, KN

Load, KN

0 degree

40

Sample 1

Vf =.5 ξ =0.2 Em= 6000. MPa Ef= 100000. MPa υm=.18 υf =0.25

20

20 90 degree Theory Experiment

0 0.00

2.00 4.00 6.00 Elongation, mm

8.00

0 0.00

Sample 2

0.20

0.40

0.60

0.80

Clip Gage Elongation, mm

1.00

Unidirectional and 0/90/0 fiber compositeseffect of fiber volume fraction

Vf = 6%

400 Vf = 0%

σt1= 10 MPa σt2= 5 MPa σc1= 40 MPa σc2= 40 MPa τ12= 5 MPa τ23= 5 MPa E m= 30000 ν m= 0.18

200

500

400

0/90/0 Glass Fiber Composites Vf = 6%

Nominal Load, N/mm

Nominal Force, N/mm

600

Unidirectional Glass Fiber Composites

300 Vf = 4%

σt1= 10 MPa σt2= 5 MPa σc1= 40 MPa σc2= 40 MPa τ12= 5 MPa τ23= 5 MPa E m= 30000 ν m= 0.18

200 Vf = 2%

Vf = 2%

100 Vf =0%

Vf = 0%

0 0.000

0.001 0.002 0.003 0.004 Axial Strain, mm/mm

0

0.005

0.000

0.001 0.002 0.003 Axial Strain, mm/mm

0.004

Response of a 6 stack 0/90/0 lamina Strain Distribution

150 Nominal Stress, MPa

10 8 6

z,mm

4 2 0 -2

100

50

-4 -6 -8 -10 0

0.5

1

1.5

2

2.5

3

0

1.5

2

2.5

3

3.5 x 10

10

Transverse Stress

6

8 6 4

z,mm

4 2

z,mm

1

-3

x 10

8

0

2 0

-2

-2 -4

-4

-6

-6

-8

-8

-10 0

0.5

-3

mm/mm x

Stress 10 Distribution

0

3.5

-10 2

4

6

MPa

8

10

12

0

0.5

1

1.5

2

MPa

2.5

3

3.5

4

Response of a 6 stack 0/90/0 lamina

Response of an 8 stack [0/45/-45/90/90]s lamina Strain Distribution 10 8 6

140

2 0

120

-2 -4 -6 -8 -10 0

0.5

1

1.5

2

mm/mm x Stress Distribution 10

2.5

3

3.5 x 10

-3

8 6

Nominal stress, MPa

z,mm

4

100

80

60

40

z,mm

4

20

2 0

0

-2

0

0.5

1

1.5

2

2.5

-4

Nominal strain, mm/mm

-6 -8 -10 0

5

10

15

MPa

20

25

3

3.5 -3

x 10

Response of an 8 stack [0/45/-45/90/90]s lamina

[0/45/-45/90/90]s Composites - Effect of Vf 200 [0/45/-45/90/90]s Vf =6%

Nominal Load, N/mm

160 120

Vf = 4% Vf =2%

80

σt1= 10 MPa σt2= 5 MPa σc1= 40 MPa σc2= 40 MPa τ12= 5 MPa τ23= 5 MPa E m= 30000 ν m= 0.18

Vf = 0%

40 0 0.000

0.001 0.002 0.003 Axial Strain, mm/mm

0.004

Polypropylene Fiber Composites Effect of Vf

2000

σt1= 5 MPa σt2= 5 MPa σc1= 40 MPa σc2= 40 MPa τ12= 5 MPa τ23= 5 MPa

40 Unidrectional Polypropylene Fiber Composites

Vf = 20%

Vf = 15%

1000

Vf = 10%

0 0.00

ω 0= 3.5e-4 Em = 30000 MPa Ef = 12000 MPa νm = 0.18 νf = 0.25

0.02 0.04 Axial Strain, mm/mm

0.06

Nominal Stress, MPa

Nominal Load, N/mm

3000

Effect of Lamina Orientation

30

ω 0= 3.5e-4 Em = 30000 MPa Ef = 8000 MPa νm = 0.18 νf = 0.25 σt1 = 5 MPa

Vf = 6%

unidirectional

20 0/90/0

10 90/0/90

0 0.00

0.02 0.04 Axial Strain, mm/mm

0.06

Comparison With Experimental Results of unidirectional and 0/90/0 composites 60 50 40 Stress, 30 MPa 20

Unidirectional Experiment Theory

Theory

[0/90]s

Experiment Em = 30000 Ef = 70000 Vf = 5% νm = 0.18

10 0 0.000

σt1= 10 MPa σt2= 5 MPa σc1= 40 MPa σc2= 40 MPa τ12= 5 MPa τ23= 5 MPa

0.005 Strain, mm/mm

0.010

Comparison of PPFRC with Experiments

Model Simulation

Stress, MPa

15

Damage Evolution Law

ω ik = ω 0 + α (ε1 − ε um ) β ω 0 = 0.,α = 5., β = 0.8

10

Experiments, Pivacek, Haupt, and Mobasher, 1998

5

Vf = 6% Em = 30000 MPa Polypropylene Ef = 8000 MPa Fiber Composites νm = 0.18 νf = 0.25 σt1 = 5 MPa w0= 3.5e-4 Softening Coefficient

0.000

0.005

0.010

Strain, mm/mm

0.015

Flexural Response of Glass Fiber Composite Laminates Nominal Moment, N-mm/mm

1600

1200

σt1= 5 MPa σt2= 5 MPa σc1= 10 MPa σc2= 10 MPa τ12= 5 MPa τ23= 5 MPa

Vf = 6%

Vf = 4%

800

Vf = 2%

400

0 0.0000

Vf =0%

Em = 30000 Ef = 70000 νm = 0.18

0.0001 0.0002 0.0003 Curvature, 1/mm

0.0004

Response of an 8 stack unidirectional lamina to Flexure Strain Distribution

a b

10 8

c

-2

c

-4 -6 -8 -10-3

b

4

a

b

c

a

6 z,mm

4 2 0

2 0

c

σxx

-2 -4 -6

a

b

-8

-2

-1

0 1 mm/mm

-10 -10

2 3 x 10-3

-5

0

10 5 MPa

15

20

25

10000 8

b

a

6

Moment/unit width, N mm /mm

10

c

4 z,mm

z,mm

10 8 6

2 0 -2 -4

c

σyy

b

-6

a

-8 -10

-4

-3

-2

-1

0

MPa

1

2

3

4

5

9000 8000

c

7000

b

6000 5000 4000 3000 2000 1000 0

a 0

0.5 1.5 1 Curvature, 1/mm

2

2.5

3

3.5 x 10-4

Flexural Load-Deflection Strain Distribution

sigmax

0.25

0.25

0.2

0.15

0.1

0.1

0.05

0.05

z,mm

0.15

z,mm

Stress Distribution 8 layers 0 degree

0.2

0

0 -0.05

-0.05

Strain Distribution

-0.1 -0.15

-0.1 -0.15 -0.2

-0.2 -0.25 -1

-0.5

0

0.5 1 mm/mm

1.5

2

-0.25 -4000

2.5

-3000

-2000

-1000

0

1000

2000

3000

psi

-3

x 10

sigmax 0.25

250

0.2 0.15

200

0 degree

0.1 0.05

force, lbs

z,mm

150

0

Stress Distribution 8 layers [02/902]s

-0.05

100

[02/902]s

50

-0.1 -0.15 -0.2 -0.25 -4000

0

0

1

2

3

4 5 deflection, in

6

7

8

9 -3

x 10

-3000

-2000

-1000

0 psi

1000

2000

3000

Composite Retrofit - Reinforced Concrete beamGlass Epoxy Bonded Laminate Concrete Steel-concrete layer Concrete Glass-epoxy material

Young’s Modulus psi

Posisson’s ratio

Compressive Strength, psi

Tensile strength, Psi

Coefficient of Thermal expansion

epoxy

6.e5

0.28

20000

6000

2.6e-6

glass fibers

11.5e6

0.25

200000

200000

1.e-6

paste

3.e6

0.17

8000.

500.

1.e-6

steel

29.e6

0.3

36000

36000

6.e-6

Stress Distribution in Flexural Sample

Vs= 0.12 tgl-ep

Moment Curvature computations of a Retrofitted RC beam 60000

Composite Retrofit

Vg= 0.4

Moment, Lb-in

Vs= 0.12

tgl-ep= 0.2"

12”

40000

tgl-ep= 0.1"

10” tgl-ep= 0.0"

20000

tgl-ep 0

0

0.0001

0.0002

Curvature, 1/in

0.0003

0.0004

Modeling of Fabrics

Crack Spacing

Debonding Length

Pullout Slip response

Conclusions 





Theoretical models based on composite laminate Theory can be used to predict the mechanical response of various cement based composite systems. Combination of damage mechanics based methods and composite laminate theory can result in a useful method to evaluate the response of various fiber matrix, geometry, and loading combinations. The formulation is applicable to a wide range of materials studied including unidirectional, angle ply, sandwich, fabric reinforced, and also retrofit composite systems.