Chapter 3 Micromechanical Analysis of a LaminaElastic Moduli

Dr. Autar KawDepartment of Mechanical Engineering

University of South Florida, Tampa, FL 33620

Courtesy of the TextbookMechanics of Composite Materials by Kaw

h, t = A ff

and h, t = A mm

ht = A cc

tt =

AA = V

c

f

c

ff

V - 1 =tt =

AA = V

f

c

m

c

mm

1

2 3

h

tc

tm/2 tf

tm/2

h

tc

Lc

FIGURE 3.3Representative volume element of a unidirectional lamina.

h

tc tm/2

tf tm/2

σc σc

FIGURE 3.4A longitudinal stress applied to the representative volume element to calculate the longitudinal Young’s modulus for a unidirectional lamina.

,A = F ccc σand ,A = F fff σ

A = F mmm σ

, E = c1c εσand , E = fff εσ

εσ mmm E =

F + F = F mfc

A E + A E = A E mmmfffcc1 εεε

: then),( If εεε mfc = =

AA E +

A

A E = E

c

mm

c

ff1 V E + V E = E mmff1

V E + V E = E mmff1

V EE =

FF

f1

f

c

f

,A = F ccc σand ,A = F fff σ

A = F mmm σ

, E = c1c εσand , E = fff εσ

εσ mmm E =

FIGURE 3.5Fraction of load of composite carried by fibers as a function of fiber volume fraction for constant fiber to matrix moduli ratio.

Example 3.3

Find the longitudinal elastic modulus of a unidirectional Glass/Epoxy lamina with a 70% fiber volume fraction. Use the properties of glass and epoxy from Tables 3.1 and 3.2, respectively. Also, find the ratio of the load taken by the fibers to that of the composite.

Example 3.3

Ef = 85 Gpa

Em = 3.4 GPa

GPa 60.52 =0.3 3.4 + 0.7 85 = E1 )()()()(

Example 3.3

FIGURE 3.6Longitudinal Young’s modulus as function of fiber volume fraction and comparison with experimental data points for a typical glass/polyester lamina.

Example 3.3

0.9831 = 0.7 60.52

85 = FF

c

f )(

h

tc

tm/2 tf

tm/2 σc

σc

FIGURE 3.7A transverse stress applied to a representative volume element used to calculate transverse Young’s modulus of a unidirectional lamina.

σσσ mfc = =

∆∆∆∴ mfc + =

, t = ccc ε∆

and , t = fff ε∆

mmm t = ε∆

,

E = c

c2

σε

and ,E

= f

ff

σε

E =

m

mm

σε

andtt

E1 +

tt

E1 =

E1

c

m

mc

f

f,

2

EV +

EV =

E1

m

m

f

f

2

Example 3.4

Find the transverse Young's modulus of a Glass/Epoxy lamina with a fiber volume fraction of 70%. Use the properties of glass and epoxy from Tables 3.1 and 3.2, respectively.

Example 3.4

= 85 GPaE f

Em = 3.4 GPa

GPa 10.37 = E

3.40.3 +

850.7 =

E1

2

2

FIGURE 3.8Transverse Young’s modulus as a function of fiber volume fraction for constant fiber to matrix moduli ratio.

πV4

= sd f 2

1

πV32

= sd f

21

s

d

(b)

(a)

s

d

FIGURE 3.9Fiber to fiber spacing in (a) square packing geometry and (b) hexagonal packing geometry.

FIGURE 3.10 Theoretical values of transverse Young’s modulus as a function of fiber volume fraction for a boron/epoxy unidirectional lamina (Ef = 414 GPa, vf = 0.2, Em = 4.14 GPa, vm = 0.35) and comparison with experimental values. Figure (b) zooms figure (a) for fiber volume fraction between 0.45 and 0.75. (Experimental data from Hashin, Z., NASA tech. rep. contract no. NAS1-8818, November 1970.)

h

tc

tm/2 tf

tm/2

σ1 σ1

(b)

(a)

tc

tm/2

tf

tm/2

tc + δcT

Lc

tf + δfT

h

tc

tm/2 tf

tm/2

σ1 σ1

(b)

(a)

tc

tm/2

tf

tm/2

tc + δcT

Lc

tf + δfT

FIGURE 3.11A longitudinal stress applied to a representative volume element to calculate Poisson’s ratio of unidirectional lamina.

δδδ Tm

Tf

Tc + =

,t

= f

TfT

fδ

ε

and ,t

= m

TmT

mδε

t =

c

TcT

cδε

εεε Tmm

Tff

Tcc t + t = t

,- = L

f

Tf

fεε

ν

and ,- = Lm

Tm

mεεν

εεν L

c

Tc

12 - =

ενενεν Lmmm

Lfff

Lc12c t - t- = t-

ενενεν Lmmm

Lfff

Lc12c t - t- = t-

: then, If εεε Lm

Lf

Lc = =

ννν mmff12c t + t = t

tt +

tt =

c

mm

c

ff12 ννν V + V = mmff12 ννν

Example 3.5

Find the Major and Minor Poisson's ratio of a Glass/Epoxy lamina with a 70% fiber volume fraction. Use the properties of glass and epoxy from Tables 3.1 and 3.2, respectively.

Example 3.5

0.2 = fν

0.3 = mν

0.230 = 0.3 0.3 + 0.7 0.2 = 12 )()()()(ν

Example 3.5

E1 = 60.52 Gpa

E2 = 10.37 GPa

Example 3.5

0.03941 = 60.5210.37 0.230 =

EE =

1

21221 νν

h

tc

tm/2 tf

tm/2 τc

τc

FIGURE 3.12An in-plane shear stress applied to a representative volume element for finding in-plane shear modulus of a unidirectional lamina.

δδδ mfc + =

,t = ccc γδ

and ,t = fff γδ

mmm t = γδ

,

G =

12

cc

τγ

and ,G

= f

ff

τγ

G =

m

mm

τγ

,t = ccc γδ

and ,t = fff γδ

mmm t = γδ

δδδ mfc + =

tG

+ t G

= t G

mm

mf

f

fc

12

c τττ

tG

+ t G

= t G

mm

mf

f

fc

12

c τττ

: then, If τττ mfc = =

tt

G1 +

tt

G1 =

G1

c

m

mc

f

f12

GV +

GV =

G1

m

m

f

f

12

Example 3.6

Find the in-plane shear modulus of a Glass/Epoxy lamina with a 70% fiber volume fraction. Use properties of glass and epoxy from Tables 3.1 and 3.2, respectively.

Example 3.6

GPa 85 = E f 0.2 = fν

GPa 35.42 =

0.2 + 1 285 =

+ 1 2E = G

f

ff

)(

)( ν

Example 3.6

GPa 3.4 = Em 0.3 = mν

GPa 1.308 =

0.3 + 1 23.40 =

+ 1 2

E = Gm

mm

)(

)( ν

Example 3.6

GPa 4.014 = G

1.3080.30 +

35.420.70 =

G1

12

12

FIGURE 3.13 Theoretical values of in-plane shear modulus as a function of fiber volume fraction and comparison with experimental values for a unidirectional glass/epoxy lamina (Gf = 30.19 GPa, Gm = 1.83 GPa). Figure (b) zooms figure (a) for fiber volume fraction between 0.45 and 0.75. (Experimental data from Hashin, Z., NASA tech. rep. contract no. NAS1-8818, November 1970.)

V E + V E = E mmff1

V - 1V + 1

= EE

f

f

m

2

ηηξ

ξη

+ E / E1- E / E =

mf

mf

)()(

Example 3.7

Find the transverse Young's modulus for a Glass/Epoxy lamina with a 70% fiber volume fraction. Use the properties for glass and epoxy from Tables 3.1 and 3.2, respectively. Use Halphin-Tsai equations for a circular fiber in a square array packing geometry.

Example 3.7

a

b

σ2

σ2

FIGURE 3.14Concept of direction of loading for calculation of transverse Young’s modulus by Halphin–Tsai equations.

Example 3.7

2 = ξ Ef = 85 GPa Em = 3.4 GPa

0.8889 =

2 + 85/3.41 - 85/3.4 =

)()(η

Example 3.7

GPa 20.20 = E

0.70.8889 10.70.88892+ 1 =

3.4E

2

2

))(())((

−

FIGURE 3.15 Theoretical values of transverse Young’s modulus as a function of fiber volume fraction and comparison with experimental values for boron/epoxy unidirectional lamina (Ef = 414 GPa, νf = 0.2, Em = 4.14 GPa, νm = 0.35). Figure (b) zooms figure (a) for fiber volume fraction between 0.45 and 0.75. (Experimental data from Hashin, Z., NASA tech. rep. contract no. NAS1-8818, November 1970.)

Ef/Em = 1 implies = 0, (homogeneous medium)

Ef/Em implies = 1, (rigid inclusions)

Ef/Em implies (voids)

∞→

η

η

0→

ξη 1 - =

a

b

τ12 τ12

FIGURE 3.16Concept of direction of loading to calculate in-plane shear modulus by Halphin–Tsai equations.

V + V = mmff12 ννν

V - 1V + 1

= GG

f

f

m

12

ηηξ

ξη

+ G / G1 - G / G =

mf

mf

)()(

V40 + 1 = 10fξ

Example 3.8

Using Halphin-Tsai equations, find the shear modulus of a Glass/Epoxy composite with a 70% fiber volume fraction. Use the properties of glass and epoxy from Tables 3.1 and 3.2, respectively. Assume the fibers are circular and are packed in a square array. Also get the value of the shear modulus by using Hewitt and Malherbe’s8 formula for the reinforcing factor.

Example 3.8

1 = ξ GPa 35.42 = G f GPa 1.308 = Gm

0.9288 =

1 + 835.42/1.301 - 835.42/1.30 =

)()(η

Example 3.8

GPa 6.169 = G

0.7 0.9288 10.7 0.9288 1 + 1 =

1.308G

12

12

)()()()()(

−

Example 3.8

2.130 =0.7 40 + 1 =

V 40 + 1 = 10

10f

)(

ξ

Example 3.8

0.8928 =2.130 + 1.308 / 35.42

1 - 1.308 / 35.42 = )(

)(η

Example 3.8

GPa 8.130 = G

0.7 0.8928 - 10.7 0.8928 2.130 + 1 =

1.308G

12

12

)()()()()(