Would a 1km edge length solid tungsten cube sink through the Earth crust? This question was inspired by this question on worldbuilding SE. Suppose you have a giant cube with $1$km edge length made out of pure tungsten and you put it somewhere on the ground. This is the size of a medium mountain and tungsten has much higher density than rock so it would definitely sink in quite a bit. But how far would it sink? Is this massive enough to just sink through the earth crust and eventually end up in the center of the earth? Or would it just think half way in or something like that? Does this depend on whether the ground is a solid rock surface or some loose sand? I don't even know what kind of model one would use to answer a question like that.
 A: To determine if the cube is going to sink or not, we care about the pressure it exerts on the ground and whether it's enough to make it break and sink. Let's do a quick back of the envelop calculation, because I don't know enough about earth's crust to do actual computations !
Tungsten has a density of $19.3g/cm^3$, or in more convenient units $19.3\times 10^{3}kg/m^3$. According to a quick google search, the average rock density seems to be of the order of $2.65\times 10^{3}kg/m^3$. Now, the pressure exerted on a point where the cube touches the ground is simply $height*density*g$, in this case with $g\approx 10 m/s^2$ we get $$P \approx 20\times 10^3kg/m^3*1000m*10m/s^2= 2\times 10^8 Pa \approx  2000 atm$$
Now let's take the other example of a big rock : the mount everest. The peak pressure it would exert on earth's crust would be right below its top which towers at around $9 km$. So we have by the same formula as before : $$P_{everest}\approx 2.5\times 10^{3}kg/m^3*9000m*10m/s^2 = 2.25\times 10^8 Pa \approx 2250 atm$$
Of course the real answer will be lower since the mount everest is obviously not a liquid so the internal tension forces of the rock will redistribute the pressure more evenly, but it's just to have an order of magnitude.
So your block exerts a pressure on earth's crust which is roughly of the same order of magnitude than Mount Everest. So it is quite a lot for such a small object, but definitely not enough to sink through earth crust and plunge to the core of the earth. To determine exactly if it would sink a little bit, a lot, or not at all is very difficult to answer and it will of course depend on the type of ground you are resting on.
But with this comparison we can more or less confidently say that it will not plunge to the center of the earth.
