53856

Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli

Table of content of printouts:

Introduction Materials and Properties of Polymer Matrix Composites Mechanics of a Lamina Laminate Theory Ply by Ply Failure Analysis Externally Bonded FRP Reinforcement for RC Structures: Post Strengthening Flexural Strengthening Strengthening in Shear Column Confinement FRP Strengthening of Masonry CFRP Strengthening of Aluminum Profiles FRP Strengthening of Wooden Structures Design of Flexural Post-Strengthening of RC: Swiss Code 166 and Other Codes/Guidelines Design of FRP Profiles and all FRP Structures An Introduction to FRP Reinforced Concrete Monitoring and Testing of Civil Engineering Structures Composite Manufacturing Testing Methods


Mechanics of a Lamina

Book Geoff Eckold, Chapter 3, pp 49-65

Ma ter ia ls Sci ence & Technolog y

A Laminate is consisting of several Laminas or Plies or Layers.

A Lamina is consisting of Fibers and Matrix.

Micromechanics (in m-mm range) is dealing for example with the determination of Lamina constitutive properties from those of Fiber and Matrix, Fiber-Matrix interface stresses, etc.


Assumptions:

- Linear Elasticity: Matrix and Fiber behave as linear elastic material (viscoelasticity of Matrix: see previous chapter)

- Perfect bond, no strain discontinuity across interface

- Fibers are arranged in a regular or repeating array


Functional requirements for Fibers:

- High E-Modulus

- High ultimate strength

- Low variation between individual fibers

- Retain the strength during handling and fabrication

- Uniform diameter and surface


Functional requirements for Matrix:

- Bind together the fibers and protect their surfaces

- Transfer stresses to the fibers efficiently

- Chemically compatible with fibers over a long period

- Thermally compatible with fibers


Modell of a Laminate

Laminate

A Unidirectionally Reinforced Lamina(UD)



A Laminate

UD-Laminas


A Laminate


UD-Lamina (or UD-Ply or UD-Layer)


Definitions:

- Homogeneous: Properties are not function of the position of the material points

- Isotropy: Properties are not function of the orientation. 2 independent material constants: E and

- Anisotropy: Properties are function of the orientation with no planes of symmetry.

21 independent material constants


Micromechanics

Fiber

1


Macromechanics

Nx



Unidirectional Lamina(UD-Lamina)

Stiffness of a UD-Lamina






C


Symmetrical planes of a transverse isotrop UD-Lamina

1

2 E2 and 2

E1 and 1



Why ??


Strain-Stress relation of a UD-Lamina, Plane stress

22

211

11

1 EE

22

11

122

1 EE

12 1212

1 G


Engineering Constants of a UD-Lamina

1 2 12

1 1

1E

2

21

E

2 1

12

E

2

1E

12

12

1G


Compliances Sij of a UD-Lamina :

12

662

221

111;1;1

GS

ES

ES

2

2112

1

1221 ;

ES

ES


Compliance Matrix S of a UD-Lamina

1 2 12

1 11S 12S

2 21S 22S

12 66S


21221

2121

1221

11 11

EE

21221

21

1221

1212 11

EE

12 1212G

Inversion of the Compliance Matrix

Stress-Strain relation of a UD-Lamina, Plane stress


Stiffness Matrix Qof a UD-Lamina

1 2 12

1 11Q 12Q

2 21Q 22Q

12 66Q


Stiffnesses Qij of a UD-Lamina

12661221

222

1221

111 ;

1;

1GQEQEQ

1221

21212

1221

12121 1

;1

EQEQ


Symmetry ?

Stiffness matrix Q:Das Bild kann zurzeit nicht angezeigt werden.

2mmN

Q =

Q11

Q21 Q22

Q66

Q12 0

0

0 0


Symmetry ?Compliance matrix S:

N

mm2

S =

S11

S21 S22

S66

S12 0

0

0 0


Symmetry ??

2112 QQ 2112 SS and

Coupling Terms

???


Elastic Energy

The stored elastic energy in the UD-Lamina is:

12122221 11W


Replace the strains with stresses using the compliance matrix as follow:

We obtain :

21212

222222112

2

21 SSSSSW 1111

1 2 12

1 11S 12S

2 21S 22S

12 66S


Partial differentiation provides the following strain-stress relation:

12211221

SSSW

1111

21211221

2222

SSSW


Compare the equations with the strain-stress relation through compliance matrix

we obtain:

2112 SS

1 2 12

1 11S 12S

2 21S 22S

12 66S


2112 QQ

If we use the stiffness matrix

and replace stresses with strains, we obtain in a similar way the following equation:

1 2 12

1 11Q 12Q

2 21Q 22Q

12 66Q


or following equation is obtained for engineering constants:

2EE 12121

112

21

EE2

or:


Independent Elasticity Constants

With the existing symmetry, the number of independent elasticity constants of a UD-Lamina is reduced from 5 to 4.


Definitions:

- Orthotropy: Anisotropy with 3 orthogonal planes of symmetry. 9 independent constants

- Transverse Isotropy: Orthotropy with a plane at which there is Isotropy 5 independent constants

- Transverse Isotropy and Plane Stress: 4 independent constants


Stress State of a Unidirectional (UD) Lamina: Transverse Isotropy, Homogeneous

Plane Stress:

2

3

1

3

1

2

1 2 2

2


Independent Elasticity Constants

If there are more symmetry conditions, there will be further reductions in the number of constants. For a cross-ply laminate with E1=E2, there are 3and for an isotropic material (for example mat-laminate with randomly distributed fibers) 2independent constants.


Mechanics of materialsSemi empirical equations


Elasticity constants of a UD-Lamina are dependent on the following parameters:

EF = Fiber E-Modulus

F = Fiber Poisson‘s Ratio

EM = Matrix E-Modulus

M = Matrix Poisson‘s Ratio

= Fiber Volume Fraction





The longitudinal E-Modulus (parallel to the fiber direction) can be derived from the following so called rule of mixture:

where F = fiber volume content of the UD-Lamina

MFFF EEE 11


The major poisson‘s ratio 12 caused by longitudinal stresses following the rule of mixture is:

MFFF 112

And the minor poisson‘s ratio is:

1EE2

1221


Semi Empirical Equations Based on Experiments


The transverse E-modulus for isotropic fibers according to ‚Puck‘ can be obtained:

25.1

2

2 1/85.01

FFo

MF

FoM EE

EE

21 M

MoM

EE


According to ‚Puck‘ for isotropic fibers:

25.1

5.0

12 1/60.01

FFMF

FM GG

GG


According to ‚Förster‘ and ‚Schneider‘ for isotropic fibers:

21 M

MoM

EE

45.12 1/1

FFo

MF

oM EE

EE


According to ‚Förster‘ and ‚Schneider‘ for isotropic fibers:

45.1

5.0

12 1/4.01

FFMF

FM GG

GG


Following ‚Tsai‘ for isotropic fibers:

whereEEF

FM

11

2

2/

1/

andEEEE

MF

MF


Following ‚Tsai‘ for isotropic fibers:

whereGGF

FM

11

12

1/

1/

andGGGG

MF

MF


Comparisons for E2

25.1

2

1/85.01

FFo

MF

FoM EE

EE

45.11/1

FFo

MF

oM EE

EE

F

FMEE

11

E2

E2

E2Fiber Volume Fraction

E 2 k

N/m

m2


Comparisons for G12

25.1

5.0

# 1/60.01

FFMF

FM GG

GG

F

FMGG

11

#

45.1

5.0

# 1/4.01

FFMF

FM GG

GG

G12

G12

G12

G12

kN

/mm

2

Fiber Volume Fraction


Anisotropic fibers

Fiber axis


For anisotropic fibers like C-fibers following equations can be applied:

F

oMFF

Fo

M

EEEE

/611

75.0

3

2

21 M

MoM

EE


and

FGMGFF

FMGG

/25.125.11

5.025.01

12


and

MFFF 112


Following equations are still valid:

2112 SS

2112 QQ

1EE 21212

221

12

EE1


Glass-Mat Lamina


Following equations can be applied to E-Glass Mat Lamina according to Puck :

920'3710'4630'29 2 FFE

F 075.034.0

370'1970'10 FG


Elasticity constants of some UD-Laminas

Lamina type T 300/5208

B (4)/5505 AS/3501 Scotchply 1002

Kevlar 49 / Epoxy

Fiber C-Fibers from Toray

Boron C –Fibers from Hercules

E-Glass Aramid from E.I. Dupont de Nemours

Matrix EP from Narmco

EP-Prepreg from Avco

EP-Prepreg from Hercules

EP-Prepreg from 3M EP

Fiber volume fraction (%) 70 50 66 45 60

Density (g/cm3) 1.6 2.0 1.6 1.8 1.46

E1 (N/mm2) E (N/mm2)

12 G12 (N/mm2)

181'000 10'300 0.28 7'170

204'000 18'500 0.23 5'590

138'000 8'960 0.30 7'100

38'600 8'270 0.26 4'140

76'000 5'500 0.34 2'300


…Elasticity constants of some UD-Laminas

Lamina type T 300/5208 B (4)/5505 AS/3501 Scotchply 1002 Kevlar 49 / Epoxy

E1 (N/mm2)

E2 (N/mm2)

12

G12 (N/mm2)

181'000

10'300

0.28

7'170

204'000

18'500

0.23

5'590

138'000

8'960

0.30

7'100

38'600

8'270

0.26

4'140

76'000

5'500

0.34

2'300

S11 (mm2/N)

S22 (mm2/N)

S12 (mm2/N)

S33 (mm2/N)

5.52510-6

97.0910-6

-1.54710-6

139.510-6

4.90210-6

54.0510-6

-1.12810-6

172.710-6

7.24610-6

111.610-6

-2.17410-6

140.810-6

25.9110-6

120.910-6

-6.74410-6

241.510-6

13.1610-6

181.810-6

-4.47410-6

434.810-6

Q11 (N/mm2)

Q22 (N/mm2)

Q12 (N/mm2)

Q33 (N/mm2)

181'800

10'340

2'897

7'170

205'000

18'580

4'275

5'790

138'000

9'013

2'704

7'100

39'160

8'392

2'182

4'140

76'640

5'546

1'886

2'300


Thermal properties of a UD-Lamina:Expansion cofficients

11

MF

FF

FMF

EE 1

111


and

22 FMM

MM

2M

M2

M3

M

ν11νν2Φ1.1Φ1.11ννν2

F

F

FFMF

MFM

EE

EE

1.1/1.11/

/

2

2


where: 1F = Thermal expansion coefficient of fibers in fiber longitudinal direction

2F = Thermal expansion coefficient of fibers perpendicular to fiber

longitudinal direction M = Thermal expansion coefficient of matrix M = Poisson’s ratio of matrix 1FE = Elasticity modulus of fibers in fiber longitudinal direction

2FE = Elasticity modulus of fibers perpendicular to fiber longitudinal direction

EM = Elasticity modulus of matrix F = Fiber volume content


Failure Theories for a UD-Lamina


Failure Theories for a UD-Lamina

Following simple criteria can be applied to examine the fiber failure:

1max = Failure stress of a UD-Lamina in fiber direction

1max1

1

maxmaxmax1 )1( MF


Matrix failure:

max12

max2

max2

max1max1

2

max122

max2max2

max2max2

max2max2

22

2

max1

1

T

C

MFM

TC

TC

TCM

Ewhere

121

Compression strength perpendicular to fiber direction

Tensile strength perpendicular to fiber direction

Shear strength


Fiber failure due to tensile stress


Fiber failuredue tocompression-stresses


Fiber failuredue tocompression-stresses


Matrix failure


Experimental determination of the UD-Lamina properties:

12

1

1

1

E

12

12

2

2

2

E

12

12

G

12

1 2


Torsion and Tensile Samples


A torsion sample after the test


Transverse Tension

2

GFRP

CFRP (P55S)

2Strain

Tran

sver

se T

ensi

on

z2


Shear

GFRP

CFRP (P55S)

12

12


Combined shear and transverse stresses

12E-Glas

T 300 Aramid

2max12 mmN

2max2 mmN

T

2max2 mmN

C


Strength of some UD-Laminas

Lamina type T 300/5208 B (4)/5505 AS/3501 Scotchply 1002 Kevlar 49 / Epoxy

1Tmax(N/mm2)

1Cmax(N/mm2)

1500

1500

1260

2500

1447

1447

1062

610

1400

235

2Tmax(N/mm2)

2Cmax(N/mm2)

40

246

61

202

51.7

206

31

118

12

53

12max(N/mm2)

68 67 93 72 34

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