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 Fibre Composites, FS12 Masoud Motavalli
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.
Ma ter ia ls Sci ence & Technolog y
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
Ma ter ia ls Sci ence & Technolog y
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
Ma ter ia ls Sci ence & Technolog y
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Modell of a Laminate
Laminate
A Unidirectionally Reinforced Lamina(UD)
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
A Laminate
UD-Laminas
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
A Laminate
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
UD-Lamina (or UD-Ply or UD-Layer)
Ma ter ia ls Sci ence & Technolog y
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Micromechanics
Fiber
1
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Macromechanics
Nx
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Unidirectional Lamina(UD-Lamina)
Stiffness of a UD-Lamina
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
C
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Symmetrical planes of a transverse isotrop UD-Lamina
1
2 E2 and 2
E1 and 1
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Why ??
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Strain-Stress relation of a UD-Lamina, Plane stress
22
211
11
1 EE
22
11
122
1 EE
12 1212
1 G
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Engineering Constants of a UD-Lamina
1 2 12
1 1
1E
2
21
E
2 1
12
E
2
1E
12
12
1G
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Compliances Sij of a UD-Lamina :
12
662
221
111;1;1
GS
ES
ES
2
2112
1
1221 ;
ES
ES
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Compliance Matrix S of a UD-Lamina
1 2 12
1 11S 12S
2 21S 22S
12 66S
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Stiffness Matrix Qof a UD-Lamina
1 2 12
1 11Q 12Q
2 21Q 22Q
12 66Q
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Stiffnesses Qij of a UD-Lamina
12661221
222
1221
111 ;
1;
1GQEQEQ
1221
21212
1221
12121 1
;1
EQEQ
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Symmetry ?
Stiffness matrix Q:Das Bild kann zurzeit nicht angezeigt werden.
2mmN
Q =
Q11
Q21 Q22
Q66
Q12 0
0
0 0
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Symmetry ?Compliance matrix S:
N
mm2
S =
S11
S21 S22
S66
S12 0
0
0 0
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Symmetry ??
2112 QQ 2112 SS and
Coupling Terms
???
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Elastic Energy
The stored elastic energy in the UD-Lamina is:
12122221 11W
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Partial differentiation provides the following strain-stress relation:
12211221
SSSW
1111
21211221
2222
SSSW
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
or following equation is obtained for engineering constants:
2EE 12121
112
21
EE2
or:
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Independent Elasticity Constants
With the existing symmetry, the number of independent elasticity constants of a UD-Lamina is reduced from 5 to 4.
Ma ter ia ls Sci ence & Technolog y
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Stress State of a Unidirectional (UD) Lamina: Transverse Isotropy, Homogeneous
Plane Stress:
2
3
1
3
1
2
1 2 2
2
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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 a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of materialsSemi empirical equations
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Semi Empirical Equations Based on Experiments
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
According to ‚Puck‘ for isotropic fibers:
25.1
5.0
12 1/60.01
FFMF
FM GG
GG
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
According to ‚Förster‘ and ‚Schneider‘ for isotropic fibers:
21 M
MoM
EE
45.12 1/1
FFo
MF
oM EE
EE
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
According to ‚Förster‘ and ‚Schneider‘ for isotropic fibers:
45.1
5.0
12 1/4.01
FFMF
FM GG
GG
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Following ‚Tsai‘ for isotropic fibers:
whereEEF
FM
11
2
2/
1/
andEEEE
MF
MF
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Following ‚Tsai‘ for isotropic fibers:
whereGGF
FM
11
12
1/
1/
andGGGG
MF
MF
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Anisotropic fibers
Fiber axis
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
and
FGMGFF
FMGG
/25.125.11
5.025.01
12
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
and
MFFF 112
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Following equations are still valid:
2112 SS
2112 QQ
1EE 21212
221
12
EE1
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Glass-Mat Lamina
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
…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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Thermal properties of a UD-Lamina:Expansion cofficients
11
MF
FF
FMF
EE 1
111
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
and
22 FMM
MM
2M
M2
M3
M
ν11νν2Φ1.1Φ1.11ννν2
F
F
FFMF
MFM
EE
EE
1.1/1.11/
/
2
2
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Failure Theories for a UD-Lamina
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Fiber failure due to tensile stress
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Fiber failuredue tocompression-stresses
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Fiber failuredue tocompression-stresses
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Matrix failure
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Experimental determination of the UD-Lamina properties:
12
1
1
1
E
12
12
2
2
2
E
12
12
G
12
1 2
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Torsion and Tensile Samples
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
A torsion sample after the test
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Transverse Tension
2
GFRP
CFRP (P55S)
2Strain
Tran
sver
se T
ensi
on
z2
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Shear
GFRP
CFRP (P55S)
12
12
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
Combined shear and transverse stresses
12E-Glas
T 300 Aramid
2max12 mmN
2max2 mmN
T
2max2 mmN
C
Mechanics of a Lamina Fibre Composites, FS12 Masoud Motavalli
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