print design - What printing process is better to avoid t-shirt logo damage?

I don't like tshirts whose logos melt when you iron them, or crack if you put them in the dryer for too long.

What do you call the printing process where the logo becomes part of the shirt itself, and is less susceptible to damage?

Answer

I believe what you are referring to is heat transfers from off the shelf that use an iron on graphic which is designed for personal usage.

There is also sublimation printing which still uses heat to transfer and typically the graphic is digitally printed on a special film:

Sublamination Infographic Process:

DTG (direct to garment) uses spray on ink and typically a heat press to assure dried ink.

DTG Printing Inforgraphic:

All printing techniques are subjective to damage but if you are looking for the best print quality and life screen printing is typically suggested over them all. Screen printing doesn't fade as much as DTG printing but there are pros and cons for them. If you have a full color or technically anything over 3 colors some professional shops can create accurate color results for shirts with screen printing but it is hard. Also, to add to that some shops require a min. to screen print because the setup is intuitive and not usually worth doing a run of less than 25.

Screen Print Infographic:

I believe these were originally found on Pinterest but per someones comment they were created by Row Apparel

Location of infographics.