The Happy Couple's Secret Weapon - According to John Gottman

In his extensive research, John Gottman has found that there are a number of key differences between happy and unhappy couples.

One of these differences relates to the presence of 'repair attempts' during arguments. When happy couples argue, if the conflict starts to escalate, one of them will break the tension. They might make a joke, apologise; express empathy or even suggest taking a break from the discussion. In happy couples the other partner responds positively to a repair attempt - treating it as a genuine effort to make things better.

The key here is that this is a two step process:

1. one of the pair makes a peace offering 
2. the other person notices it and responds positively.

John Gottman on Relationships

John Gottman does great research on relationships.  He has found that, on average, men get much more physiologically stressed by conflict than women.  They get flooded by adrenaline, their heart rates skyrocket and then they find it very hard to pay attention and process information.  They can tend to respond to this by stonewalling, which drives women crazy!
If you are in conflict with someone who has become flooded in this way.  Pause, slow things right down.  Notice your tone of voice - see if you can take a few deep breaths and get the cadence of your voice back to a gentler rhythm. Let go of trying to reach an outcome for a while. Perhaps take a break.  Write down some notes about the key issues so the topic becomes more manageable.