In nuclear fission, by bombarding the heavy nucleus with smaller nuclei will result in the production of energy, nuclei. That means the number of neutrons in the heavy nucleus increases. But addition of neutrons increases the strong force. Then how does the nucleus get torn apart by repulsion of the positive charge, when the strong force will be stronger?
Handwaving: the spherical nucleus deforms into two lobes, which become separated enough that the short distance strong force acts little between them, less than the repulsive electrostatic force, which eventually result in the separation of the lobes, and the production of two nuclei. About 10 MeV are necessary for that to happen. The binding of the nucleon by the strong force provides about half of that. Then there are basically two mechanisms for getting the other half. Fast neutrons is one of them, in which case the kinetic energy of the neutron makes the difference. The other mechanism works only for nuclei with an odd number of neutrons. In that case, there is one neutron occupying alone the highest nuclear energy level and the incoming neutron can go on the level too and form a pair, resulting in an extra binding energy that is enough to make the nucleus go "over the hill" and start the mechanism I was describing to start with. The first mechanism is what happens in nuclear weapons, whereas the second one is how nuclear reactor can work, at least those with moderators that slow neutrons.
The U-235 nucleus is unstable with respect to fission into Barium-141 and Krypton-92. You can see this by looking at the graph of binding energy per nucleon:
(this graph is all over the Internet - I got it from the question Why only light nuclei are able to undergo nuclear fusion not heavy nuclei?)
I've marked the U-235 nucleus and it's two fission products by red circles, and it's obvious from the graph that fission increases the binding energy per nucleus so it should happen spontaneously. The reasin it doesn't happen is because there is a large energy barrier to the fission process. Splitting a U-235 nucleus in two requires a wholesale rearrangement of the nucleons and that costs energy. The final state will have a lower energy but the intermediate states have a higher energy and present a barrier.
It's important to note that the neutron doesn't produce a more fissile nucleus. In fact absorption of a nucleus produces U-236 and U-236 is not fissile. What the neutron does is provide a pathway for the U-235 nucleus to get round the energy barrier and allow the fission to occur. This isn't simply a matter of adding some energy since even low energy thermal neutrons will cause fission. Luc refers to an energy of 10MeV but neutrons with just a few eV of energy will cause fission and these can't possibly be providing enough energy to get over the barrier.
Exactly what goes on I don't know, but I would guess that the nucleus rearranges when an extra neutron is added, and in that process the nucleus passes through a configuration where the energy barrier is much reduced.
Here is an one of the possible U235 fission processes:
In one of the most remarkable phenomena in nature, a slow neutron can be captured by a uranium-235 nucleus, rendering it unstable toward nuclear fission. A fast neutron will not be captured, so neutrons must be slowed down by moderation to increase their capture probability in fission reactors. A single fision event can yield over 200 million times the energy of the neutron which triggered it!
The crux is that the neutrons should be thermal order of 0.25 eV, so it is a soft neutron that destroys the stability and in total three thermal neutrons are released , leading to the chain reaction possibilities.
In nuclear fission, how does the strong force get dominated by the electrostatic repulsive force?
There are various models which describe how the electromagnetic repulsion, the strong force attraction, the Pauli exclusion principle generate stable and metastable nuclei. For example the shell model:
The evidence for a kind of shell structure and a limited number of allowed energy states suggests that a nucleon moves in some kind of effective potential well created by the forces of all the other nucleons. This leads to energy quantization in a manner similar to the square well and harmonic oscillator potentials. Since the details of the well determine the energies, much effort has gone into construction of potential wells for the modeling of the observed nuclear energy levels.
So it is not just the strong force pulling one way, and the electromagnetic the other. There are effective quantum mechanical potentials. Here is an old paper discussing fission within a shell model:
THE importance of the ‘magic’ neutron numbers 50 and 82 has been repeatedly stressed1. It has been suggested that these numbers of neutrons form shells of considerable stability, and that additional neutrons, if present, are more loosely bound. It seems possible that such a shell may persist as a definite structure at moderate excitation of the nucleus, the excitation energy being shared by the nucleons not contained in the closed shell.
And here is a book excerpt which discusses the deformations of the fissible nuclei.
Nuclear physics is a branch of theoretical physics by itself.