Answer to Question #110177 in Mechanical Engineering for Sumit

Question

15 mm dia SS (Young’s modulus = 200 Gpa) rod is coated with HA (Young’s modulus = 300 Gpa).
A) What should be the coating thickness so that the Young’s modulus of the composite is 210 Gpa (use mixture rule)?
B) If the densities are 7.9 and 4.0 gm/cc, what is the average density?

Answer 1

We know that, for the composite material

$Y_c= Y_1V_1+Y_2V_2$

Here , $Y_1, Y_2$ are the young modulus for composite material and V1 and V2 are the

volume here length will be same only cross sectional area is changing

$210(15+x)^2=15^2\times 200 +(x)^2\times300$

$47250 +210 x^2+6300=45000+300x^2$

$90x^2-6300x-2250=0$

$x^2-70x-25=0$

x=0.35 mm

(ii) For average density we have formula as

$\delta_{av}=\frac{\delta_1\times\delta_2}{\delta_1x_1 +\delta_2x_2}$

$\delta_{av}=\frac{7.9\times4.0}{7.9\times15+4\times.35}$

$\delta_{av}=0.238 \frac{g}{cm^3}$