ME 649 ADVANCED MANUFACTURING PROCESSES – I

 

ASSIGNMENT 1

 

  1. The following two sets of components correspond to the identical stress tensor measured in two coordinate systems (x1, x2, x3) and (x1’, x2’, x3’) . Determine the rotation matrix to transform the components from one coordinate system to the other. Write a program to find principal directions and the rotation matrix to change component to a standard principal coordinate system.

 

 

1          1.73     1                                  0.5       1.414   0.5

           σ  =       1.73     0.75     0.433               σ ‘ =    1.414   1          1.414

                        1          0.433   0.25                             0.5       1.414   0.5

 

 

 

  1. Write a program to determine roots of a cubic equation
  2. Considering that the tensile necking occurs at the maximum load, determine the true as well as engineering strains at which necking would begin for the following material laws.
    1. σ = K(ε+εo)n                σ = 500(ε+0.05)0.25                  Swift
    2. σ = σo + k(ε+εo)n         σ = 100 + 500(ε+0.05)0.25       Ludwik            
    3. σ = σo (1-Aexp(-Bε))   σ = 500(1-0.6exp(-3ε))           Voce
    4. σ = σo                          σ = 500                                    ideal plastic
    5. σ = σo + kε                  σ = 250 + 350ε                        linear
    6. σ = ksin(Bε)                 σ = 500sin(2πε)                       trigonometric

 

  1. Find relationship between constants A, B and C in the following expressions, which represent a possible defoemaion rate in a 2D field :

ex = Ax2 (x2 + y2);   ey = By2(x2+y2); gxy = Cxy(x2 + y2)

Show that the associated velocity field is given by

U = Cx3(3/5x2 + y2) + Dy  and

V = Cy3(x2 + 3/5y2) – Dx,  D being an arbitrary constant.

 

  1. The effect of elastic deformation in the material on instability strain can be estimated from the Ramberg-Osgood equation.

 

e = s/E + 3so/7E (s/so)1/n

where so = nominal yield stress. Show that true strain at the onset of localised necking is given by

 

e = n + (7n/3)n(so/E)1/n

 

            For n = 0.05 and so/E = 0.002,  determine the percentage error with respect to the value determined using the power law s = Ken