Chapter 3 Micromechanical Analysis of a Lamina Ultimate Strengths of a Unidirectional Lamina

Transcription

1 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 Courtesy o the Textbook Mechanics o Coposite Materials by Kaw

2 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.

3 and, E = σ ε E = σ ε V 1- E + V = T 1 ε σ σ

4 ] [ ] [ 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

5 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.

6 Exaple 3.13 E = 85 GPa, and σ = 1550 MPa ε = =

7 Exaple 3.13 E = 3.4 GPa, and σ =72 MPa ε = =

8 Exaple 3.13 T 6-1 σ = = 1104 MP 1-0.7

9 Exaple 3.13 = % 10 = = V iniu

10 Exaple 3.13 V critical = = = %

11 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.

12 Tab Length = o Tab Thickness = 3.2 T t Gage Length = 153 Length = 229 w FIGURE 3.26 Geoetry o a longitudinal tensile strength specien.

13 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.

14 Exaple 3.13 E1= GPa, T = 2896 MPa, σ 1 T ε 1 and = 1.560%

15 a b c FIGURE 3.28 Modes o ailure o unidirectional laina under a longitudinal tensile load.

16 ε 1 = σ 1 E 1 ε 2 = ν 12 σ 1 E 1

17 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.

18 c σ 1 = E 1 T ε 2 ν 12 T ε 2 = ε T 1-V 1/3 T d E ε T = ε s E +1

19 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

20 τ 12 = τ V + τ V c σ 1 2[ τ V + τ V = ]

21 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 Type I Carbon/Epoxy Kevlar 49/Epoxy

22 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.

23 Exaple 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

24 Exaple E1= GPa ν 12= 0.23

25 Exaple d 1 2 s 40.7 = π =

26 Exaple ε = =

27 Exaple T ε 2 = =

28 Exaple T ε 2 = / =

29 Exaple σ C 1 = = MPa

30 Exaple S C 1 = = MPa

31 Exaple G= GPa S = C 2 9 = 4360 MPa

32 Exaple C σ 1 = in 21349,4360 = 4360 MPa

33 Exaple C 6 σ 1 2 [ = = 69.4 MPa 0.3]

34 Exaple 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.

35 Exaple c σ 1 E c 1= 199 GPa, = 1908 MPa, and c ε 1 = %

36 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 ± ± ± 0.1 or 6.4 ± 0.1

37 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.

38 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.

39 δ = δ c δ δ δ c = s = d + δ where : ε c ε,, and = s - d ε

40 ε ε ε c s d s d = ε ε = E E ε ε c s d E E s d = T T 2 s d E E s d = ε ε T 2 2 T 2 E = ε σ

41 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.

42 Exaple 3.15 T -2 ε 2 = E2= GPa 9 σ T -2 = 2 = MPa

43 Exaple 3.15 E2= GPa, T σ 2 = MPa, and T ε 2 = %

44 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.

45 C σ C 2 E 2 ε 2, where = C d E d C ε 2 = + 1 ε s E s

46 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.

47 Exaple 3.16 E = 85 GPa, E= 3.4 GPa, = 102 MPa, σ C E2= GPa, and d = s

48 Exaple 3.16 ε C = 9 =

49 Exaple 3.16 C ε 2 = =

50 Exaple 3.16 C 9 σ = = MPa

51 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.

52 Exaple 3.16 σ c 2 E c 2 = 93 GPa, = 198 MPa, and c ε 2 = 2.7%

53 c = + where : = sγ 12 c c, d γ 12, and = = s - d γ 12

54 d d γ 12 c = γ γ 12 s s γ G= γ G d G d γ 12 c = + 1- γ 12 s G s τ 12 = G = G d s γ G G d s γ 12 d G d γ 12 = + 1- γ 12 s G s

55 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.

56 Exaple 3.17 G = GPa G= GPa G12= GPa d s = τ 12 = 34 MPa

57 Exaple 3.17 γ 12 = =

58 Exaple 3.17 σ x 45 o 45 o σ x 45 o o Strain-Gage Rosette FIGURE 3.36 Scheatic o a [±45]2S lainate shear test.

59 Exaple τ = = MPa

60 Exaple 3.17 σ τ = x 12, and 2 γ 12 = ε x + ε y

61 Exaple 3.17 G12= GPa, = MPa, and τ 12 γ 12 = 2.619%

62 Exaple 3.17 FIGURE 3.37 Shear stress shear strain curve obtained ro a [±45]2S graphite/epoxy lainate under a tensile load.

63