Harden, solidify, indurate, petrify, cake are comparable when they mean to make or to become physically hard or solid.
Harden usually expresses an opposition to soften and therefore may be as often used of the process as of the effect. The term suggests a change in degree with an approach toward a state of firm consistency or texture, though it need not imply impenetrability or resistance to efforts to break, cut, pierce, or bend.
Solidify, although differing little from harden, usually expresses an opposition to liquefy and places more stress upon the effect produced than upon the process involved; the term, therefore, suggests a change in quality rather than in degree and is more often applied to a mass subject to compacting or consolidation.
Indurate, which means to make very hard or very compact, implies usually the making of something that is firm in texture still harder.
Petrify implies a making or becoming stone or stonelike in hardness; the word is used of organic bodies that by a process (called petrifaction ) of infiltration by water containing mineral deposits (as silica, calcium carbonate) and the replacement, particle by particle, of the organic matter by the introduced mineral become replaced by stony mineral while the original form is more or less perfectly retained.
Cake implies the formation into a firm, hard, or solid mass (as by baking, fusing, or congealing).