1. Homework # 4
Problems : 4.5 , 4.6 , 4.11
4.5
𝐸𝑓 = 230 𝐺𝑝𝑎 , 𝐸 𝑚 = 4.6 𝐺𝑝𝑎 , 𝐺𝑓 = 1.84 𝐺𝑝𝑎 , 𝐺 𝑚 = 1.77 𝐺𝑝𝑎 , 𝜗𝑓 = 0.25 ,
𝜗 𝑚 = 0.3 , 𝑉𝑓 = 0.4 , 𝑉𝑚 = 0.6
𝑹𝒆𝒒𝒖𝒊𝒓𝒆𝒅 ∶ 𝑬 𝟏 , 𝑬 𝟐 , 𝑮 𝟏𝟐, 𝝑 𝟏𝟐
𝐸1 = 𝐸 𝑚 𝑉𝑚 + 𝐸𝑓 𝑉𝑓 & 𝐸2 =
𝐸 𝑚 . 𝐸𝑓
𝑉𝑓 𝐸 𝑚 + 𝑉𝑚 𝐸𝑓
𝐺12 =
𝐺 𝑚 . 𝐺𝑓
𝐺𝑓 𝑉𝑚 + 𝐺 𝑚 𝑉𝑓
& 𝜗12 = 𝜗𝑓 𝑉𝑓 + 𝜗 𝑚 𝑉𝑚
𝐸1 = 4.6 ∗ 0.6 + 230 ∗ 0.4 = 94.76 𝐺𝑝𝑎
𝐸2 =
4.6 ∗ 230
0.4 ∗ 4.6 + 0.6 ∗ 230
= 7.5658 𝐺𝑝𝑎
𝐺12 =
1.84 ∗ 1.77
0.6 ∗ 1.84 + 1.77 ∗ 0.4
= 1.7974 𝐺𝑝𝑎
𝜗12 = 0.25 ∗ 0.4 + 0.3 ∗ 0.6 = 0.28

2. 4.6
𝑹𝒆𝒒𝒖𝒊𝒓𝒆𝒅 ∶ 𝑬 𝟏 , 𝑬 𝟐 , 𝑮 𝟏𝟐, 𝝑 𝟏𝟐 For E-glass , S-glass , Carbon T-300 :
𝐸1 = 𝐸 𝑚 𝑉𝑚 + 𝐸𝑓 𝑉𝑓 & 𝐸2 =
𝐸 𝑚 . 𝐸𝑓
𝑉𝑓 𝐸 𝑚 + 𝑉𝑚 𝐸𝑓
𝐺12 =
𝐺 𝑚 . 𝐺𝑓
𝐺𝑓 𝑉𝑚 + 𝐺 𝑚 𝑉𝑓
& 𝜗12 = 𝜗𝑓 𝑉𝑓 + 𝜗 𝑚 𝑉𝑚
𝐺23 = 𝐺 𝑚
𝑉𝑓 + Ƞ23(1 − 𝑉𝑓)
Ƞ23 1 − 𝑉𝑓 + 𝑉𝑓 𝐺 𝑚 /𝐺𝑓
& Ƞ23 =
3 − 𝜗 𝑚 + 𝐺 𝑚 /𝐺𝑓
4 1 − 𝜗 𝑚

3. 4.11
𝐸1 = 𝐸 𝑚 𝑉𝑚 + 𝐸𝑓 𝑉𝑓 & 𝐸2 =
𝐸 𝑚 . 𝐸𝑓
𝑉𝑓 𝐸 𝑚 + 𝑉𝑚 𝐸𝑓
𝑉𝑚 = 1 − 𝑉𝑓
𝐸1 > 30 𝐺𝑝𝑎 𝑎𝑛𝑑 3.5𝐸2 > 𝐸1
∴ 𝐸1 > 30𝐺𝑝𝑎 𝑎𝑛𝑑 𝐸2 < 8.5714
∴ 𝐸1 = 𝐸 𝑚 1 − 𝑉𝑓 + 𝐸𝑓 𝑉𝑓
𝐸2 =
𝐸 𝑚 𝐸 𝑓
𝑉 𝑓 𝐸 𝑚 + 1−𝑉 𝑓 𝐸 𝑓
𝒂𝒔𝒔𝒖𝒎𝒆 𝑽 𝒇 = 𝟎. 𝟒
∴ 0.6𝐸 𝑚 + 0.4𝐸𝑓 > 30
∴
−𝐸 𝑚 𝐸𝑓
0.4𝐸 𝑚 + 0.6𝐸𝑓
> −8.5714

4. 𝐸 𝑚 >
10
6
30 − 0.4𝐸𝑓 ……………………… (1
−10
6
30−0.4𝐸 𝑓 𝐸 𝑓
0.4∗
10
6
30−0.4𝐸 𝑓 +0.6𝐸 𝑓
> −8.5714 …………………….(2
From (1) and (2) we can get
𝐸𝑓 & 𝐸 𝑚 𝑎𝑛𝑑 𝑓𝑟𝑜𝑚 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐸1 𝑤𝑒 𝑐𝑎𝑛 𝑔𝑒𝑡 𝑉𝑓