Lecture #11: Introduction to Fiber-reinforced Composite ... · PDF file2/15/2016 Lecture #11...

2/15/2016 1 1Lecture #11 – Fall 2015 1D. Mohr

151-0735: Dynamic behavior of materials and structures

by Dirk Mohr

ETH Zurich, Department of Mechanical and Process Engineering,

Chair of Computational Modeling of Materials in Manufacturing

Lecture #11:

• Introduction to Fiber-reinforced Composite Materials

© 2015



Composite Materials

Composite materials are materials that feature “microstructures” that arecomposed of two or more materials. Wood is an example of a natural compositematerial: it features cellulose fibers that are embedded in a lignin matrix. Straw-reinforced clay may be considered as one of the first manmade compositematerials.

Source: http://venice.umwblogs.org/exhibit/the-conservation-of-venetian-building-materials/wood/

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwi2hofrr7XJAhWBPQ8KHY_DBcIQjRwIBw&url=http://venice.umwblogs.org/exhibit/the-conservation-of-venetian-building-materials/wood/&psig=AFQjCNGCV9_rrN6oe-9kLCuWWktILEkMEg&ust=1448877596200965

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwi2hofrr7XJAhWBPQ8KHY_DBcIQjRwIBw&url=http://venice.umwblogs.org/exhibit/the-conservation-of-venetian-building-materials/wood/&psig=AFQjCNGCV9_rrN6oe-9kLCuWWktILEkMEg&ust=1448877596200965



Fiber-reinforced Composites

Concrete is a an example for a particle-reinforced composite (aggregates such ascoarse gravel embedded in a cement matrix). In this class, we focus on fiber-reinforced and layered composites with continuous long fibers:

Unidirectional fiber Bi-directional tri-directional

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjVnaDYs7XJAhWFfw8KHSqoBRYQjRwIBw&url=http://www.123rf.com/photo_6579951_woven-carbon-fiber-texture-background-ideal-desktop-as-it-seamless.html&psig=AFQjCNGc2MQ6XmiGkxf-UriYrCoXiUOYyQ&ust=1448878636356015

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjVnaDYs7XJAhWFfw8KHSqoBRYQjRwIBw&url=http://www.123rf.com/photo_6579951_woven-carbon-fiber-texture-background-ideal-desktop-as-it-seamless.html&psig=AFQjCNGc2MQ6XmiGkxf-UriYrCoXiUOYyQ&ust=1448878636356015

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj037GNtLXJAhVDfw8KHe6aCswQjRwIBw&url=http://www.progressionboats.com/home/construction 3.html&psig=AFQjCNHkyAchdzkWn0NeZVQyN-hmuuNPiQ&ust=1448878701551400

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj037GNtLXJAhVDfw8KHe6aCswQjRwIBw&url=http://www.progressionboats.com/home/construction 3.html&psig=AFQjCNHkyAchdzkWn0NeZVQyN-hmuuNPiQ&ust=1448878701551400



0

200

400

600

800

1000

0 2 4 6 8 10

Mo

du

lus

[GP

a]

Density [g/cm3]

Fiber vs. bulk materials

Fibers often feature a high mass specific strength and stiffness as compared tobulk materials. This is due to a lower volume fraction of defects (→strength) andthe optimal alignment of the material microstructure with respect to themechanical loading direction (i.e. the fiber direction).

Carbon fibers

Kevlar

Glass fibers

Alum.

Steel

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10

Ten

sile

Str

en

gth

[G

Pa]

Density [g/cm3]

Titanium

Alum.

Steel

Titanium

Kevlar

Glass fibers

Carbon fibers



Common fibersReinforcements fibers typically are of 10-100mm diameter. E-glass fibers are thelow-cost reinforcement choice for composite materials (~1.5 CHF/kg, which is50% higher than the price for steel). They are made from amorphous glass (blendof sand, limestone and other oxidic compounds). The particular feature of E-glassfibers is a low area density of defects on the fiber surface which results in atensile strength of up to 3 GPa. Other type of glass fibers include C-glass(corrosion resistant), S-glass (high strength), D-glass (dielectric) and A-glass(alkaline resistant).

Carbon fibers are widely used in aerospace engineering. Their stiffness can be upto 10 times higher than that of glass fibers, and their price per kg can be 20 to500 times higher. High strength carbon fibers reach stress levels of up to 7 GPa.Depending on the raw material and manufacturing process, their stiffness canvary from 200 to 1000 GPa.

Carbon nanotubes also belong to the class of carbon fiber materials. They feature a cylindrical atomicstructure (1nm diameter). The modulus and strength of defect-free nanotubes may reach values ashigh as 1500 GPa and 200 GPa, respectively. At this stage, the estimated costs of about 500’000CHF/kg prohibits their wide-spread use in industrial applications.



Textile design

A 1D-textile can be a single fiber, a strand (untwistedbundle of fibers), a tow (untwisted bundle of carbonfibers), a yarn or thread (twisted bundle of fibers), or aroving (collection of fiber bundles).

Source: E.J. Barbero, Introduction to Composite Materials Design

A fabric is 2D-textile. A uniaxial woven fabric consists of parallel yarns alongthe warp direction that are held together by very thin yarns along the filldirection. In biaxial woven fabrics the same yarn is usually used along the warpand fill directions.

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjwytaPz7XJAhUDXRQKHcH3BN0QjRwIBw&url=http://www.isplc2006.org/b2b/energy_agents/2/&bvm=bv.108194040,d.ZWU&psig=AFQjCNGxezIZtm_h2d5tMn0VabIUdLDE0Q&ust=1448885983385626

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjwytaPz7XJAhUDXRQKHcH3BN0QjRwIBw&url=http://www.isplc2006.org/b2b/energy_agents/2/&bvm=bv.108194040,d.ZWU&psig=AFQjCNGxezIZtm_h2d5tMn0VabIUdLDE0Q&ust=1448885983385626



Matrix materials

The matrix material holds together the fibers and contributes to the stiffnessand strength of the composite material. It also provides many non-mechanicalfunctions (heat and electrical conductivity, appearance, corrosion resistance,fire resistance, etc.).

Source: E.J. Barbero, Introduction to Composite Materials Design

Matrix materials can be subdivided into two categories: thermoplastics andthermoset matrices, i.e. polymer matrices that solidify through irreversiblecross-linking at the molecular level, such as:

• Polyester resins (high performance/cost ratio, cross-linking at room temperature through use of

accelerators, fumes can be toxic)

• Vinyl ester resins (high chemical resistance)

• Epoxy resins (often toughed through additives, excellent electrical insulation, tensile strength of

up to 100 MPa, Young’s modulus of 1 to 4 GPa, density about 1.2g/cm3)

• Phenolic resins (low flammability and smoke production)



Laminae and laminate

In most applications, multiple laminae (matrix impregnated 2D textile layer) arestacked up to built a laminate. The characteristic orientation of the laminae isvaried from layer-to-layer to tailor the laminate strength for different loadingdirections.

Lamina Laminae Laminate



Wet lay-up process

Low cost composite components are often made using a wet lay-up process usingambient curing resins (e.g. polyester). A mold surface needs to be smooth andcovered by a release agent to avoid the bonding to the mold surface. The fiberreinforcement layers (e.g. woven fabrics) are laid onto the mold layer-by-layer,while the resin is applied using a hand roller.

26 x 10-6

Source: Principles of the Manufacturing of Composite Materials , Suong V. Hoa



Autoclave process

In aerospace engineering. composite parts are usually manufactured usingautoclave processing.




Prepregging

Prepregs are fiber sheets with partially cured resins (usually epoxy). The prepregis stored in a freezer to slow down the curing (i.e. cross-linking) of the resins.


Spreading

• Separation of fibers (spreading)

Impregnation

• Fibers are wetted by liquid resin

Heater

• Partial curing of resin (to a highly viscous state)

Roll up

• Backing paper is placed on both sides of the sticky prepreg sheets

Tensioning



Prepregging

Source: Giant bicycles



Prepregging

Source: https://www.youtube.com/watch?v=P-zI9xbwxZY&feature=player_detailpage



Prepregs

Pepregs are available with uniaxial fiber orientation or woven fibers. The width ofcommercially available prepregs typically varies from 1 to 12 inches (25.4 to 305mm), while these are about 150mm thick before full cure, and 125mm after fullcure under pressure (due to “bleeding out of resin”).




Mold material

Composites structures are typically cured at 180°C and pressures of up to 600kPa. Due to the high temperature of the curing process, the molds used inautoclave processing are often made from Invar, which is a Ni-Fe alloy of aparticularly low Coefficient of Thermal Expansion (CTE):

Material CTE (1/K)

Steel 13 x 10-6

Aluminun 26 x 10-6

Invar 0.3 x 10-6



Autoclave

An autoclave is a heated pressure vessel with a vacuum systems for the curing ofbagged prepreg lay ups which are positioned onto a mold.

Source: www.ycp.co.jp

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwim56eInLPJAhWJ1hoKHeIGBoAQjRwIBw&url=http://www.ycp.co.jp/gijyutu_kaihatu_eng.htm&psig=AFQjCNHSYQLtRb4xOEuqIr_TTQdBuSiVow&ust=1448803479044112

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwim56eInLPJAhWJ1hoKHeIGBoAQjRwIBw&url=http://www.ycp.co.jp/gijyutu_kaihatu_eng.htm&psig=AFQjCNHSYQLtRb4xOEuqIr_TTQdBuSiVow&ust=1448803479044112



Making of the Boeing 787 fuselage



Making of the 787 fuselage




Source: https://www.youtube.com/watch?feature=player_detailpage&v=_GDqxnahwbk

http://www.google.ch/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjnrM6L9LLJAhXDDg8KHe59DB4QjRwIBw&url=http://www.jyi.org/issue/the-boeing-787-dreamliner-designing-an-aircraft-for-the-future/&psig=AFQjCNF7UQzAQ8g6CwmYdHUxy8I1ygyyjA&ust=1448792348096555




Resin-Transfer Molding (RTM)

Carbon-fiber composite “Life-module” of BMW i3

• Resin matrix: Araldite LY 3585 epoxy withHardener XB 3458 injected at more than40bars

• Cure in less than 10 min at 100°C

Source: http://www.compositesworld.com/articles/bmw-leipzig-the-epicenter-of-i3-production-

Source: http://www.jjmechanic.com



Elasticity of Fiber-reinforced Composites



Anisotropic Elasticity

In 3D, the linear elastic stress-strain relationship may be written as

Cεσ or

23

13

12

33

22

11

2323

13231313

122312131212

3323331333123333

22232213221222332222

112311131112113311221111

23

13

12

33

22

11

C

CC

CCC

CCCC

CCCCC

CCCCCC

sym

For a fully-anisotropic material, the above coefficients are all independent, i.e.21 elasticity constants must be determined to describe its elastic response. Fullyanisotropic materials are rare and seldom used in engineering practice.



Isotropic Elasticity

Isotropic materials are more common, in which case two elasticity constants(e.g. the Young’s modulus E and the elastic Poisson’s ratio n) are sufficient tocharacterize the material’s elastic response.

23

13

12

33

22

11

23

13

12

33

22

11

21

021

0021

0001

0001

0001

)21)(1(

E

sym



Orthotropic Elasticity (3D)

Fiber-reinforced composites are often made from orthotropic laminae. Fororthotropic materials, it can be shown that nine elasticity constants aresufficient to characterize the material’s elastic response.

sym

23

13

12

33

22

11

2323

1313

1212

3333

22332222

113311221111

23

13

12

33

22

11

0

00

000

000

000

C

C

C

C

CC

CCC



Compliance tensor for an orthotropic material (3D)The inverse of the stress-strain relationship, i.e. the strain-stress relationship isdescribed through the compliance tensor S. For an orthotropic material, we have

SσσCε 1 or

23

13

12

33

22

11

23

13

12

3

3

32

2

3

31

2

21

1

23

13

12

33

22

11

2

1

02

1

002

1

0001

0001

0001

n

nn

G

G

G

E

EE

EEE

sym

An orthotropic material is fully characterized through three Young’s moduli, threePoisson’s ratios and three shear moduli, i.e. a total of nine elastic constants.



DefinitionsDenote the three orthotropy direction by i, j, k. Then consider a uniaxial tensionexperiment along the orthotropy direction xi.

ii1

jj1

iiii

Plus thickness strain measurement

ii

iiiE

ii

jj

ij

n

kkt

t 1

0

We can define (& determine):

• Young’s modulus

• Poisson’s ratios

ii

kkik

n



Poisson’s ratios for orthotropic materialsThree Young’s moduli and six Poisson’s ratios can be determined from uniaxialtension experiments along the orthotropy directions

11

11

111

E

11

2212

n

11 11

3313

n

22

22

222

E

22

1121

n

22

3323

n

33

33

333

E

33

2232

n

3333

1131

n

22



Poisson’s ratios for orthotropic materialsAmong the nine measured elasticity constants from uniaxial tension experimentsalong the orthotropy axes, only six are independent. Due to the symmetry of theelastic compliance tensor, we have three reciprocal relationships:

23

13

12

33

22

11

23

13

12

32

23

1

13

3

32

21

12

3

31

2

21

1

23

13

12

33

22

11

2

100000

02

10000

002

1000

0001

0001

0001

nn

nn

nn

G

G

G

EEE

EEE

EEE

2

21

1

12

EE

nn

3

31

1

13

EE

nn

3

32

2

23

EE

nn



Shear moduli for orthotropic materialsThe shear moduli are defined from pure shear experiments where the shearingaxes are aligned with the orthotropy axes.

ij 2ij

ij

ij

ijij

ij

jiij GG

2



Plane stress law for an orthotropic material (2D)

12

22

11

12

2

23

1

13

2

2

21

1

23

13

12

33

22

11

000

000

2

100

0

01

sym

01

G

EE

E

EE

Under plane stress conditions, it is assumed that 0132333

The elastic stress-strain relationship for an orthotropic material then reduces to

In other words, the relationship among the in-plane stress and strain componentsis defined by four constants only: Young’s moduli E1 and E2, the Poisson’s ration21, and the shear modulus G12.

In case the thickness strain must be determined as well, two additional Poisson’s ratios, n13 and n23,must be known.

12

22

11

12

1

2

212

2

2

1

2

212

2121

1

2

212

21

12

22

11

2

0

0

n

n

n

n

G

EE

E

EE

EE

EE

EE

sym



Plane stress law for an orthotropic material (2D)

For notational convenience, the elastic stress-strain relationship for anorthotropic material is also written as

1221

2

21

2

21

2

1

2

212

2

222

1/1 nnnn

E

EE

E

EE

EQ

with

12

22

11

66

2212

1211

12

22

11

00

0

0

Q

QQ

QQ

1221

1

21

2

21

1

1

2

212

2111

1/1 nnnn

E

EE

E

EE

EEQ

1221

121

1

2

212

212112

1 nn

n

n

n

E

EE

EEQ

1266 2GQ



Plane stress law for an isotropic material (2D)

For an isotropic material, we have

2

1221

222

11 nnn

EEQ

with

12

22

11

2

12

22

11

100

01

01

1

n

n

n

n

E

2

1221

111

11 nnn

EEQ

2

1221

12112

11 n

n

nn

n

EEQ

212661

)1(1

2n

nn

EE

GQ

EEE 21

nnn 2112

)1(212

n

EG



Isotropy check

2221 n

E

Q

2111 n

E

Q

2121 n

n

EQ

2661

)1(n

n

E

Q

1122 QQ

11

12

Q

Qn

121111

11

1266 1 QQQ

Q

QQ

66

2212

1211

00

0

0

Q

QQ

QQ

Q1122 QQ

121166 QQQ defines an isotropic material if



Material rotation

1e2e

xe

ye

The material orthotropy axes in the lamina plane are aligned with the vectors e1and e2. Suppose that the components of the Cauchy stress tensor are known inthe coordinate frame defined by the vectors ex and ey. The correspondingcomponents in the (e1, e2)-frame are then given by the linear transformation T,

xy

yy

xx

xy

yy

xx

sccscs

cscs

cssc

T22

22

22

12

22

11

2

2

with ]cos[c

]sin[s



Material rotation

1e2e

xe

ye

The inverse relationship reads

xy

yy

xx

xy

yy

xx

sccscs

cscs

cssc

T22

22

22

12

22

11

2

2

12

22

11

1

12

22

11

22

22

22

2

2

T

sccscs

cscs

cssc

xy

yy

xx



Constitutive equation for lamina As the same transformations are valid for the strain vector, the relationshipamong the stress and strain components in the (ex, ey)-frame is then given by

xy

yy

xx

xy

yy

xx

xy

yy

xx

Q

QQ

QQQ

Q

QQ

QQ

66

2622

161211

66

2212

1211

1

ˆ

ˆˆ

ˆˆˆ

00

0

0

TT

22

4

6612

22

11

4

11 )(2ˆ QsQQscQcQ

Upon evaluation, we find the components

12

44

662211

22

12 )()2(ˆ QcsQQQscQ

)(2)(2ˆ662212

3

661211

3

16 QQQcsQQQscQ

22

4

6612

22

11

4

22 )(2ˆ QcQQscQsQ

)(2)(2ˆ662212

3

661211

3

26 QQQscQQQcsQ

66

44

66122211

22

66 )()2(2ˆ QscQQQQscQ

Note that the mathematical definition of the shear strain is applied. Many textbooks and FE programs adopt theengineering definition. In that case, the third column of the stiffness matrix Q needs to be multiplied with 0.5.

sym



Membrane response of laminates

Let denote the stiffness matrix of the i-th lamina in the (ex, ey)-coordinateframe, the stress-strain relationship for the laminate is then given by the rule ofmixtures:

xy

yy

xx

tot

xy

yy

xx

Q̂

iQ̂

with

N

i

i

tot

itot

t

t

1

ˆˆ QQ

ttot①②③④⑤⑥

Thickness fraction



Special Layup Configurations

• Balanced laminate: for each + ply, there exist a – ply of the same thickness.As a result, we have

0ˆˆ2616 tottot QQ

• Symmetric laminate: ply sequence is symmetric about the laminate mid-plane

• Cross-ply laminate: the laminate contains only 0 ° and 90° laminae

• Quasi-isotropic laminate: the laminae orientations are distributed such thatthe in-plane response of the laminate becomes loading direction independent



Reading Materials for Lecture #11• E.J. Barbero (2011), Introduction to Composite Materials Design

• R. Talreja, C. Veer Singh (2012), Damage and Failure of Composite Materials

Lecture #11: Introduction to Fiber-reinforced Composite ... · PDF file2/15/2016 Lecture #11...

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