The closest answer is zoomorphism, which is giving animal traits to non-animal subjects or objects, as in the sidewalk that "purrs."
There is an interesting issue bubbling under the surface of this question, though: how do we draw a line between animals and humans? Since the advent of Darwinism, humans have been understood to have evolved from animals—in short, to be animals themselves.
Nevertheless, we do make commonsense distinctions between what is usually animal and what is usually human. Thus, we are back to what might be called a problem of origins: Did the word "purr" begin life as descriptor for the sound a cat makes and then migrate over to describe certain human noises and, from there, get assigned to inanimate objects? Or did it start as a descriptor of human noises and migrate to cats? Depending on which is true, purring would become either personification or zoomorphism, and the same would be true of any other word. If we want to get technical, the best way to determine the literary category in which to place a word would be to find a good dictionary that traces word etymology and follow its wisdom.
Otherwise, common sense is a good guide. For example, to my mind, a purr is more associated with a cat than a human, so I would call a purring sidewalk an example of zoomorphism. In other cases, absent a word root, I would advise you to make an educated and context-based determination.
This is an interesting point. There really is no specific literary term for this. Although personification is closest, it gives the sense of person rather than animal; technically, though, some critics would accept personification as an accurate term for this. However, if you refer, for example, to a car engine as purring, you are comparing the car to a contented cat, not to a person.
The phrase "the engine purred" would be a metaphor, as it is an implicit comparison of engine to cat, not using explicit comparative words. The term zoomorphism, as discussed above, would work well for portraying humans in animal terms.
The Egyptian gods who have some animal characteristics, are described as theriomorphic, from Greek roots meaning having the form of an animal. This is not, however, typically used as a literary term.
Personification and anthropomorphism are typically associated with giving human characteristics to things that are not human: inanimate objects, animals, etc. However, personification is sometimes given a more broad definition. Therefore, a sidewalk purring could be an example of personification or zoomorphism. For instance, "the wind sighed" could be personification or zoomorphism as animals and humans do indeed sigh. Although it is odd to consider a human purring, it is not out of the question. But to be precise, a sidewalk purring is more specifically an example of zoomorphism.
The pathetic fallacy is a poetic convention in which natural phenomena are described as feeling emotions. Again, this could apply to instances of anthropomorphism/personification as well as zoomorphism. The...
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logic here is that both humans and animals feel emotion; therefore, the pathetic fallacy can apply to both personification and zoomorphism. But the pathetic fallacy is most often described in terms of human emotions. Thus, the "rain clouds weeping" is most often a pathetic fallacy applied to human emotions (personification) but certain animals weep as well, so the attribution of weeping to an inanimate object like a cloud could also be an example of zoomorphism.
Interesting question. The term you might be looking for is zoomorphism, which, according to one of its definitions, is "the tendency of viewing human behaviour in terms of the behaviour of animals, analogous to anthropomorphism, which views animal behaviour in human terms." Most of the other definitions of zoomorphism, however, refer more to art, sculpture, or religious deities (such as totems) which are shaped in animal form.
In practice, your example would probably be considered personification or anthropomorphism. Your reasoning, that purring is an animal rather than a human characteristic, is correct; however, the concept of a sidewalk doing something that only animate objects can do is close to personification.
Take another verb which animals do, perhaps pouncing or hissing, and personification works because humans can also pounce and hiss. Your example is extraordinary in that purring is unique to felines and is generally not an action associated with humans.
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