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I have proposed a new equation and will be implementing it into a publicly available code in the thesis. I have modified the core equation of the program (the main constitutive equation, "Johnson Cook Model"), while the numerical method (newton-raphson solver) is as implemented by the original author.

I would cite the original authors in my PhD thesis where I will explicitly talk about the functioning of the code. However, my journal paper would be only about the equation (as it is the highlight of the work) and the implementation is not the main focus. I will not talk about the implementation and just mention that the "equation was implemented using a VUMAT."

In that case, do I have to cite the original source in the paper when I would eventually submit to a journal?

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    What do you mean "have to"? Why wouldn't you? It costs nothing Commented Jan 12, 2021 at 18:41

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If you borrowed something from another source and modified it, you should definitely cite the original source in your paper.

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You should put a footnote about how you modified this from such and such. While the reality is that people tend not to cite or acknowledge even when credit to someone is amply due, that is not the practice that should be followed. In my publications I not only give credit to obvious papers I use but I try to find the genesis of the idea and give credit of inspiration to my own work. Giving credit is never going to hurt your work. Not giving credit will, at least in the long run, hurt your reputation one way or the other.

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