Tensile Strength of Continuous Fiber-Reinforced

```Tensile Strength of Continuous
Fiber-Reinforced Lamina
Strength of a Continuous Fiber
Reinforced Lamina
For the orthotropic lamina under simple
uniaxial or shear stress, there are 5
strengths:
()
S L = Longitudinal tensile strength
()
S L = Longitudinal compressive strength
()
S T = Transverse tensile strength
()
S T = Transverse compressive strength
S LT = Shear strength
(See Fig. 4.1)
showing lamina strengths and ultimate strains.
Longitudinal Uniaxial
1
1
Tension
sL
eL
()
()
eL
1
sL
Compression
()
()
1
showing lamina strengths and ultimate strains.
Transverse Uniaxial

ST
eT

Compression
2

2
2
Tension
()
()
ST
()
eT
( )
2
showing lamina strengths and ultimate strains.
 12
 12
s LT
e LT
 12
Assuming linear elastic behavior up to failure:
SL
SL
ST
ST
()
()
()
()
 E1e L
()
 E 1e L
()
 E 2 eT
( )
 E 2 eT
()
(4.1)
S LT  G 12 e LT
( )
()
()
()
where e L , e L , eT , eT , eLTare the
corresponding ultimate strains.
Transverse tensile strength ST(+) is low because of
stress concentration in matrix at fiber/matrix
interfaces.

2

2
Fibers are, in effect, “holes” in matrix under
Typical values of lamina strengths for several composites
SL(+)
ksi(MPa)
SL(-)
ksi(Mpa)
ST(+)
ksi(Mpa)
ST(-)
ksi(Mpa)
SLT
ksi(Mpa)
Boron/5505
boron/epoxy
vf = 0.5 (*)
230 (1586)
360 (2482)
9.1 (62.7)
35.0 (241)
12.0 (82.7)
AS/3501
graphite/epoxy
vf = 0.6 (*)
210 (1448)
170 (1172)
7.0 (48.3)
36.0 (248)
9.0 (62.1)
T300/5208
graphite/epoxy
vf = 0.6 (*)
210 (1448)
210 (1448)
6.5 (44.8)
36.0 (248)
9.0 (62.1)
Kevlar 49/epoxy
aramid/epoxy
vf = 0.6 (*)
200 (1379)
40 (276)
4.0 (27.6)
9.4 (64.8)
8.7 (60.0)
Scotchply 1002
E-glass/epoxy
vf = 0.45 (*)
160 (1103)
90 (621)
4.0 (27.6)
20.0 (138)
12.0 (82.7)
E-glass/470-36
E-glass/vinylester
vf = 0.30 (*)
85 (584)
116 (803)
6.2 (43)
27.1 (187)
9.3 (64.0)
Material
Micromechanics Models for
Strength
• Strength more sensitive to material and
geometric nonhomogeneity than stiffness,
so statistical variability of strength is
usually greater than that of stiffness.
• Different failure modes for tension and
compression require different micro mechanical models.
Statistical distribution of tensile strength for boron filaments. (From Weeton, J.W., Peters,
D.M., and Thomas, K.L., eds. 1987. Engineers’ Guide to Composite Materials.
ASM International, Materials Park, OH. Reprinted by permission of ASM International.)
Tensile Failure of Lamina Under Longitudinal Stress
Representative stress-strain curves for typical fiber,
matrix and composite materials
(matrix failure strain greater than fiber failure strain)
Stress
S f1
S m1
SL
()
SL
()
()
(a) Fiber Failure Mode
Fiber
Composite( v f  v crit )
()
S mf 1
Typical of
polymer matrix
composites
Composite ( v f  v crit ) Matrix
()
ef1
()
em1
( )
Strain
Tensile Failure of Lamina Under Longitudinal Stress
Representative stress-strain curves for typical fiber,
matrix and composite materials
(fiber failure strain greater than matrix failure strain)
Stress
S f1
Fiber
()
S fm 1
SL
S m1
(a) Matrix Failure Mode
()
()
()
Composite
Typical of
ceramic matrix
composites
Matrix
em1
( )
ef1
()
Strain
Longitudinal Tensile Strength
a) Fiber failure mode (ef1(+)<em1(+)); polymer matrices
Rule of mixtures for longitudinal stress:
(3.22)
 c1   f 1v f   m 1v m
when

f1
 S f1
 m 1  S mf 1
 c1  S L
 SL
()
()
()
 Eme f 1
()
()
 S f1
()
v f  Eme f 1
 S f1
()
()
v f  S mf 1
vm
()
(only valid if vf is large enough)
(4.22)
vm
Longitudinal Tensile Strength
Critical fiber volume fraction, vfcrit
when
SL
v f crit 
()
S m1
S f1
()
()
 S m1
( )
 S mf 1
()
 S mf 1
()
(4.23)
Once fibers fail, when vf <vfcrit
SL
()
 S m1
()
vm
(4.24)
Longitudinal Tensile Strength
This defines
v f min 
S m1
S f1
()
()
 S mf 1
 S mf 1
()
()
 S m1
()
(4.25)
In most of the cases, vfcrit is very small,
so
SL
()
 S f1
()
v f  S mf 1
()
1  v 
f
(4.22)
Variation of composite longitudinal tensile strength with
fiber volume fraction for composites having
matrix failure strain greater than fiber failure strain
Strength
S m1
Equation (4.22)
S f1
()
S mf 1
()
0
Equation (4.24)
v f min
v fcrit
Fiber Volume Fraction
1.0
()
Variation of composite longitudinal tensile strength with
fiber volume fraction for composites having
fiber failure strain greater than matrix failure strain
Strength
S f1
()
Equation (4.27)
Equation (4.26)
S m1
()
v f min
Fiber Volume Fraction
S mf 1
()
Longitudinal Tensile Strength
(b) Matrix Failure Mode; ceramic matrices
SL
()
 S fm 1
()
v f  S m1
()
1  v 
f
(4.26)
Fibers can withstand ef1(+)>em1(+) and remaining
area of fibers is such that
SL
()
 S f1
()
vf
which applies for practical vf
(see Fig. 4.13 – previous two slides)
(4.27)
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