Bibliography style with elsarticle

You should specify authoryear in the class options.

\begin{filecontents*}{\jobname.bib}
@article{Diamond1965,
 author={J. A. L. Diamond},
 title={Some Like It Hot},
 journal={J. High Level Cinema},
 year={1965},
}
\end{filecontents*}

\documentclass[authoryear]{elsarticle}

\usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref}

\begin{document}

\section{The Model}

The main environment modifies that of \cite{Diamond1965}, into two main dimensions

\bibliographystyle{plainnat}
\bibliography{\jobname}

\end{document}

The filecontents* environment is just to make the example selfcontained, you don't need it; change the \bibliography line to have the name of your .bib file.

Yes, I know that “Some Like It Hot” wasn't shot in 1965. ;-)

Adding the authoryear to natbib is not a solution, because the elsarticle class already loads it, so having

\usepackage[authoryear]{natbib}

in the preamble will only cause an error about conflicting options.

Since you're using the elsarticle document class, you may want to use the elsarticle-harv instead of the plainnat bibliography style. As its name suggests, elsarticle-harv.bst gives you authoryear-style citations -- aka harvard-style citations -- and references that are formatted according to the guidelines of most Elsevier-run journals that require authoryear-style citations.

Moreover, in order to activate full authoryear style (with years rather than numbers), authoryear should be specified as a documentclass option:

\documentclass[authoryear]{elsarticle}

(This information added after the OP posted an MWE.)

if you are using the elsevier reference style (which contains all the packages itself) you can pose bibliography options like this:

\bibliographystyle{elsarticle-harv}\biboptions{authoryear}