I’ve discovered that website like ResearchGate offer the possibility to ask the paper editor to provide his paper for free. Many are collaborating.
Are there any other ways for researchers to obtain free access to papers?
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I always use the following methods to get many inaccessible papers.
Another method, I always use when I really need an inaccessible paper, is just to write an e-mail to the corresponding author to send that paper. My experience is that I have always received the paper in a return email.
It turns out that a related survey was posted last year in this PLOS blog post on how paleontologists access the (non-open access) literature:
I put together an informal, non-scientific survey. The survey asked questions about how people access the literature, the kinds of journals they can access most easily, and basic demographics. I advertised the survey via Twitter and Facebook. I wouldn’t count it as a scientific sample by any means, but I do feel that I got reasonably good coverage of various types of paleontologists at various types of institutions (as well as non-paleontologists who follow the literature). 115 individuals responded, during the course of about a week.
Note that it doesn't indicate the legality.
The website mentioned here is obviously and blatantly illegal, but gives access to almost all journals ever published. It's very easy to use and more complete than our direct subscriptions. I might almost use it for stuff I have legal but slow access to.
Use or do not use depending on your views on the "illegal" consumption of research.
Ask your public library (maybe not a local branch, but a large city's one). In some places, libraries are able to help citizens access journals. Another option might be national libraries - eg the British Library.