Question

(32 POINTS) IV. A n ann ealed copper strip 12 inches wide and 2 inches thick, is rolled to its maximum in one pass. The following properties of annealed copper are given: strength coefficient is 80,000 psi; true strain at the onset of non-uniform deformation is 0.5; an engineering strain at yield is 0.12. The coefficient of friction between strip and rollis 0.25. The roll radius is 16 inches and the rolls rotate at 150 rpm Calculate the roll-strip contact length. Calculate the absolute value of the true strain that the strip undergoes in this operation. Determine the average true stress of the strip in the roll gap. Calculate the roll force. Calculate the horsepower required.

Answer 1

Given to us;

Width of strip, w = 12 inches

Thickness of strip, h_o = 2 inches

Radius of roller, R = 16 inches

Strength coefficient = 80,000 psi

True strain = 0.5

Engg. Strain = 0.12

Coefficient of friction, μ = 0.25

N = 150 rpm

Roll strip contact length, L = [RΔH]^{1/2 ________________________________}(1)

R = Roll radius

ΔH = Draft = ( h_o - h_f )

In this case, given that the strip is rolled to its maximum possible draft;

ΔH_max = μ²R = 0.25² x 16 = 1 inch

Theerfore, L = [16 x 1]^1/2 = 4 inches

L = 4 inches

Absolute value of true strain, ε = ln ( h_o / h_f )

ΔH = ( h_o - h_f ) = 1

or, h_f = h_o - 1

given, h_o = 2 inches

So, h_f = 2 - 1 = 1 inches

Now,  ε = ln ( h_o / h_f )

ε = ln ( 2/ 1) = 0.693