〜も構わず
without caring about, heedless of
Noun / Plain form + の + も構わず
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も構わず means doing something without caring about something that would normally hold you back: 人目も構わず、泣き出した, "she burst into tears, not caring who saw".
It comes from 構う, "to mind" (N4 ても構わない). The first half is something people usually worry about: others' eyes, getting dirty, getting wet, bothering people, the time of day.
After a verb, add の: 服が汚れるのも構わず, "not caring that their clothes got dirty".
The tone is often critical of other people's behaviour, or describes a strong emotion overriding normal restraint. Compare を問わず, "regardless of (a category)", which is neutral.
Examples
人目も構わず、泣き出した。
She burst into tears, not caring who saw.
服が汚れるのも構わず、子どもたちは遊んだ。
The children played, not caring that their clothes got dirty.
雨にぬれるのも構わず、走って行った。
They ran off without caring about getting soaked.
周りの迷惑も構わず、大声で話している。
They're talking loudly without caring whether they bother anyone.
夜中も構わず、電話をかけてくる。
They call me even in the middle of the night, not caring what time it is.