Two issues here: first, as cormullion pointed out in comments, you need the argument of the first `ContourPlot` to be `f[x,y]==1`, not just `f==1`. The reason is that the `x` and `y` in your function defition for `f` are just placeholders, that can substitute for whatever arguments are passed. *Mathematica* doesn't assume that the `x` you used in the function definition is the same as the `x` used in a plot range. So: 

    ContourPlot[f[x, y] == 1, {x, -30 Pi, 30 Pi}, {y, -45 Pi, 20 Pi}, 
     Axes -> {True, True}]

![enter image description here][1]

For your second plot, you need the `ContourStyle` option to style the contours, unsurprisingly enough. You also have several undefined variables, including `s45`, which I've set to be 50 in the following. There is also a capital `S`. Please be aware that it is not good practice to name variables with single capital letters as it might clash with a built-in function.

Having defined `s45`, the following works:

    ContourPlot[{(x/s1t)^2 - x*y/(s1t*s2t) + (y/s2t)^2 == 
       1, (x/s1t)^2 + (y/s2t)^2 + x*y*(1/(s1t^2) - 1/(s2t^2)) == 1, 
      x^2/(s1t^2) + y^2/(s2t^2) - x*y/(s1t^2) == 1, 
      x^2/(s1t*Abs[s1c]) + 
        y^2/(s2t*Abs[s2c]) + (1/s1t - 1/Abs[s1c])*
         x + (1/s2t - 1/Abs[s2c])*y + 
        2*x*y*1/(2*s45^2)*(1 - (1/s1t - 1/Abs[s1c] + 1/s2t - 1/Abs[s2c])*
            s45 - (1/(s1t*Abs[s1c]) + 1/(s2t*Abs[s2c]))*s45^2) == 1, 
      x^2/Abs[(s1t*s1c)] + 
        y^2/Abs[(s2t*s2c)] + (1/s1t - 1/Abs[s1c])*
         x + (1/s2t - 1/Abs[s2c])*y - 1/(2*Sqrt[s1t*s1c*s2t*s2c]) == 1, 
      x^2/Abs[(s1t*s1c)] + 
        y^2/Abs[(s2t*s2c)] + (1/s1t - 1/Abs[s1c])*
         x + (1/s2t - 1/Abs[s2c])*y + 1/Sqrt[s1t*s1c*s2t*s2c] - 
        1/(2*S^2) == 1, (x/s1t)^2 + (y/s2t)^2 == 
       1, (x/s1t)^2 + (y/s2t)^2 - x*y/(s1t*s2t) == 
       1, (x*y - x^2)/(s1t*s1c) - (y^2/(s2t*s2c)) + 
        x*(s1t + s1c)/(s1t*s1c) + y*(s2c + s2t)/(s2t*s2c) == 
       1, (1.95*x*y - x^2)/(s1t*s1c) - (y^2/(s2t*s2c)) + 
        x*(s1c + s1t)/(s1t*s1c) + y*(s2c + s2t)/(s2t*s2c) == 
       1}, {x, -165 Pi, 70 Pi}, {y, -75 Pi, 30 Pi}, Axes -> {True, True}, 
     FrameLabel -> {sigma1, sigma2}, PlotLabel -> "failure surface", 
     AspectRatio -> Automatic, ContourStyle -> {Red, Blue}]

![enter image description here][2]

If you additionally provide a value for `S` (I used $2$ for the purposes of the next plot), you'll get more contours.

![enter image description here][3]

As a side note, the `PlotLabel` should be a string in quote marks. You only got away with it in this case because multiplying `failure` and `surface` results in `failure surface` in canonical (alphabetical) order.


  [1]: https://i.sstatic.net/XHIur.png
  [2]: https://i.sstatic.net/7QBLO.png
  [3]: https://i.sstatic.net/WzSGU.png