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As I understand it, both of these approximately mean healthy or robust. I originally thought that 丈夫 was more appropriate for inanimate objects and 元気 for living things, but now I get the impression it's not so clear cut.
I'd like to know if there's a rule of thumb for when one or the other should be used?
I'm quite a new Japanese learner (as may be obvious) so please bear with me if the answer should be evident.
I really like example sentences for sussing out differences like this. A lot of things become clearer with context - for example:
As you can see, 丈夫 is indeed used more for inanimate objects, but it's not because there is a hard and fast rule about what each adjective can be used to describe, it's because the two words mean subtly different things.
元気 can mean healthy, but it most often means things like energetic, cheerful, vigorous, lively, etc. It primarily describes the way that someone or something acts or presents, so it mostly gets used with living things.
丈夫 in general has more physical nuances - it tends to mean things like sturdy, durable or rugged. Consequently, you see it mostly describing things.
Both words can be applied to people, and might both be translated as healthy in some cases, but they mean different things. 元気 evokes images of someone who is cheerful, where as 丈夫 evokes an image primarily of physical health. You can be frequently hospitalized for a health condition and still 元気 if your attitude is good, or depressed and still 丈夫 if you work out a lot, but probably not vice versa.
丈夫 is more associated with physical aspects. It is used for physical objects and their stability also. It is a minimum for well-being, it’s not necessarily positive but there are no problems to complain about. It’s often used to dismiss concern:
大丈夫ですか？
Are you okay?
大丈夫です。
I’m alright.
On the other hand, 元気 is a state of being energetic. 気 is a life energy or spirit. It’s closely tied to Japanese Shinto religion and culture and uses exclusively for living things (plants and animals as well as people). It’s meaning is difficult to describe in English but has the feeling of vitality or excitement.
子犬はとても元気ですね。
The puppy is very energetic.
If you have been injured but have a positive optimistic outlook, you can still be 元気. You can be physically okay but exhausted and your 気 would be drained. If you are feeling sick, such as an infection, you would be 病気 and not 元気.
久しぶりです。お元気ですか？
I haven’t seen you in ages. How are you doing? (Is your spirit well).
はい、元気です。
Yes, I’m doing great. (I have good vitality)
