Chapter 3 Microechanical Analysis o a Laina Ultiate Strengths o a Unidirectional Laina Dr. Autar Kaw Departent o Mechanical Engineering University o South Florida, Tapa, FL 33620 Courtesy o the Textbook Mechanics o Coposite Materials by Kaw
Fiber σ Stress, σ Coposite σ Matrix ε Strain, ε ε FIGURE 3.24 Stress strain curve or a unidirectional coposite under uniaxial tensile load along ibers.
and, E = σ ε E = σ ε V 1- E + V = T 1 ε σ σ
] [ ] [ iniu iniu iniu V 1- E + V > V - 1 ε σ σ iniu + E - E - < V σ ε σ ε σ ] critical critical V [1- E + V > ε σ σ critical E - E - < V ε σ ε σ
Exaple 3.13 Find the iate tensile strength or a Glass/Epoxy laina with a 70% iber volue raction. Use the properties or glass and epoxy ro Tables 3.1 and 3.2, respectively. Also, ind the iniu and critical iber volue ractions.
Exaple 3.13 E = 85 GPa, and σ = 1550 MPa ε = = 6 1550 10 9 85 10 01823. 10-1
Exaple 3.13 E = 3.4 GPa, and σ =72 MPa ε = = 6 72 10 9 3. 4 10 0. 2117 10-1
Exaple 3.13 T 6-1 σ 1 1550 10 0.7 + 0.1823 10 3.4 10 9 = = 1104 MP 1-0.7
Exaple 3.13 = 0.6422% 10 = 0.6422 10 +72 10 0.1823 10 3.4-10 1550 10 0.1823 10 3.4-10 72 = V -2 6-1 9 6-1 9 6 iniu
Exaple 3.13 V critical = 6 9-1 72 10-3. 4 10 01823. 10 6 9-1 1550 10-3. 4 10 01823. 10 = 0. 6732 10-2 = 0. 6732%
FIGURE 3.25 Tensile coupon ounted in the test rae or inding the tensile strengths o a unidirectional laina. Photo courtesy o Dr. R.Y. Ki, University o Dayton Research Institute, Dayton, OH.
Tab Length = 38 60 o Tab Thickness = 3.2 T t Gage Length = 153 Length = 229 w FIGURE 3.26 Geoetry o a longitudinal tensile strength specien.
FIGURE 3.27 Stress strain curve or a [0]8 lainate under a longitudinal tensile load. Data courtesy o Dr. R.Y. Ki, University o Dayton Research Institute, Dayton, OH.
Exaple 3.13 E1= 187.5 GPa, T = 2896 MPa, σ 1 T ε 1 and = 1.560%
a b c FIGURE 3.28 Modes o ailure o unidirectional laina under a longitudinal tensile load.
ε 1 = σ 1 E 1 ε 2 = ν 12 σ 1 E 1
a Fiber Microbuckling Extensional Mode b Fiber Microbuckling Shear Mode c Transverse Tensile Failure o Matrix d Shear Failure FIGURE 3.29 Modes o ailure o a unidirectional laina under a longitudinal copressive load.
c σ 1 = E 1 T ε 2 ν 12 T ε 2 = ε T 1-V 1/3 T d E ε T = - 1 2 ε s E +1
S c 1 c c c = in[ S1,S 2 ], σ 1 where [ E V E E = 2 V + 1-V ], E 3 1-V and S c 2 = G 1-V
τ 12 = τ V + τ V c σ 1 2[ τ V + τ V = ]
Table 3.6. Coparison o experiental and predicted values o longitudinal copressive strength o unidirectional lainae, V =0.50. Source: Table 7.2 in Introduction to Coposite Materials by D. Hull, 1981, 8 Cabridge University Press, 1981, Reprinted with the perission o Cabridge University Press Material Experiental Strength Equation 3.78a Equation 3.78b MPa MPa MPa Glass/Polyester 600-1000 8700 2200 Type I Carbon/Epoxy 700-900 22800 2900 Kevlar 49/Epoxy 240-290 13200 2900
Exaple 3.14 Find the longitudinal copressive strength o a Glass/Epoxy laina with a 70% iber volue raction. Use the properties o glass and epoxy ro Tables 3.1 and 3.2, respectively. Assue ibers are circular and are in a square array.
Exaple 3.14. Fro Table3.1 E ν = 85 GPa, = 0.20, = 1550 MPa, σ τ and = 35 MPa Fro Table3.2 E= 3.4 GPa, ν = 0.30, =72 MPa, σ and τ = 34 MPa
Exaple 3.14. E1= 60.52 GPa ν 12= 0.23
Exaple 3.14. d 1 2 s 40.7 = π = 0.9441
Exaple 3.14. ε 6 72 10 = 9 3.40 10 = 0.2117 10-1
Exaple 3.14. T ε 2 = 0.2117 10-1 0.9441 3.4 10 85 10 9 9-1 +1 = 0.1983 10-2
Exaple 3.14. T ε 2 = 0.2117-1 1/3 10 1-0.7 = 0.2373 10-2
Exaple 3.14. σ C 1 = 9 60.52 10 0.1983 10 0.23-2 = 521.8 MPa
Exaple 3.14. S C 1 = 2 0.7 + 1-0.7 3.4 10 85 10 9 9 9 0.7 3.4 10 85 10 3 1-0.7 9 = 21349 MPa
Exaple 3.14. G= 1.308 GPa S = 1.308 10 1-0.7 C 2 9 = 4360 MPa
Exaple 3.14. C σ 1 = in 21349,4360 = 4360 MPa
Exaple 3.14. C 6 σ 1 2 [ 35 10 0.7 + 34 10 6 = = 69.4 MPa 0.3]
Exaple 3.14. FIGURE 3.30 IITRI ixture ounted in a test rae or inding the copressive strengths o a laina. Data reprinted with perission ro Experiental Characterization o Advanced Coposites, Carlsson, L.A. and Pipes, R.B., Technoic Publishing Co., Inc., 1987, p. 76. Copyright CRC Press, Boca Raton, FL.
Exaple 3.14. c σ 1 E c 1= 199 GPa, = 1908 MPa, and c ε 1 = 0.9550%
w L 2 L 1 Strain Gage Specien Diensions L 1, L 2, w *, FIGURE 3.31 Geoetry o a longitudinal copressive strength specien. Data reprinted with perission ro Experiental Characterization o Advanced Coposites, Carlsson, L.A. and Pipes, R.B., Technoic Publishing Co., Inc., 1987, p. 76. Copyright CRC Press, Boca Raton, FL. 12.7 ± 1 127 ± 1.5 12.7 ± 0.1 or 6.4 ± 0.1
FIGURE 3.32 Stress strain curve or a [0]24 graphite/epoxy lainate under a longitudinal copressive load. Data courtesy o Dr. R.Y. Ki, University o Dayton Research Institute, Dayton, OH.
σ 2 Matrix Fiber d Matrix s Fiber d Matrix σ 2 FIGURE 3.33 Representative volue eleent to calculate transverse tensile strength o a unidirectional laina.
δ = δ c δ δ δ c = s = d + δ where : ε c ε,, and = s - d ε
ε ε ε c s d 1 - + s d = ε ε = E E ε ε c s d 1 - + E E s d = T T 2 s d 1 - + E E s d = ε ε T 2 2 T 2 E = ε σ
Exaple 3.15 Find the iate transverse tensile strength or a unidirectional Glass/Epoxy laina with a 70% iber volue raction. Use properties o glass and epoxy ro Table 3.1 and Table 3.2, respectively. Assue the ibers are circular and are arranged in a square array.
Exaple 3.15 T -2 ε 2 = 0.1983 10 E2= 10.37 GPa 9 σ 10.37 10 0.1983 10 T -2 = 2 = 20.56 MPa
Exaple 3.15 E2= 9.963 GPa, T σ 2 = 53.28 MPa, and T ε 2 = 0.5355%
Exaple 3.15 FIGURE 3.34 Stress strain curve or a [90]16 graphite/epoxy lainate under a transverse tensile load. Data courtesy o Dr. R.Y. Ki, University o Dayton Research Institute, Dayton, OH.
C σ C 2 E 2 ε 2, where = C d E d C ε 2 = + 1 ε s E s
Exaple 3.16 Find the iate transverse copressive strength o a Glass/Epoxy laina with 70% iber volue raction. Use the properties o glass and epoxy ro Table 3.1 and Table 3.2, respectively. Assue the ibers are circular and are packed in a square array.
Exaple 3.16 E = 85 GPa, E= 3.4 GPa, = 102 MPa, σ C E2= 10.37 GPa, and d = 0.9441 s
Exaple 3.16 ε 102 10 3.4 10 6 C = 9 = 0.0300
Exaple 3.16 C ε 2 = 3.4 10 0.9441 85 10 9 9 + 1-0.9441 0.03 = 0.2810 10-2
Exaple 3.16 C 9 σ 10.37 10 0.2810 10-2 2 = = 29.14 MPa
Exaple 3.16 FIGURE 3.35 Stress strain curve or a [90]40 graphite/epoxy lainate under a transverse copressive load perpendicular to the ibers.
Exaple 3.16 σ c 2 E c 2 = 93 GPa, = 198 MPa, and c ε 2 = 2.7%
c = + where : = sγ 12 c c, d γ 12, and = = s - d γ 12
d d γ 12 c = γ 12 + 1 - γ 12 s s γ G= γ 12 12 G d G d γ 12 c = + 1- γ 12 s G s τ 12 = G = G 12 12 d s γ G G 12 + 1 - d s γ 12 d G d γ 12 = + 1- γ 12 s G s
Exaple 3.17 Find the iate shear strength or a Glass/Epoxy laina with 70% iber volue raction. Use properties or glass and epoxy ro Tables 3.1 and 3.2, respectively. Assue the ibers are circular and are arranged in a square array.
Exaple 3.17 G = 35.42 GPa G= 1.308 GPa G12= 4.014 GPa d s = 0.9441 τ 12 = 34 MPa
Exaple 3.17 γ 12 = 6 34 10 1.308 10 9 = 0.2599 10-1
Exaple 3.17 σ x 45 o 45 o σ x 45 o 1 3 2 45 o Strain-Gage Rosette FIGURE 3.36 Scheatic o a [±45]2S lainate shear test.
Exaple 3.17 9 9 1.308 10 τ 12 4.014 10 0.9441 + 1-0.9441 0.2599 9 10 35.42 10-1 = = 9.469 MPa
Exaple 3.17 σ τ = x 12, and 2 γ 12 = ε x + ε y
Exaple 3.17 G12= 5.566 GPa, = 87.57 MPa, and τ 12 γ 12 = 2.619%
Exaple 3.17 FIGURE 3.37 Shear stress shear strain curve obtained ro a [±45]2S graphite/epoxy lainate under a tensile load.