I really would like to know the very first appearance of the concept of a Tensor. Because I read that was on the studies of stress on materials by Bernoulli conversely other times by Gibbs. I have also another question: about the very first appearance of a rigorous and well-defined structure of Tensor products or other "modern definitions" using linear algebra. I read that was Cartan that have made this "axiomatization" (from old differential geometry to differential forms and then to tensor concept).
Feel free to use mathematics as you pleased.
The differential theory of tensors was created by Ricci-Cubastro with his "Absolute Differential Calculus" heavely criticized as useless for 30 years before Einstein used it in the general theory of relativity. Levi-Civita has extended it.
In that same period of Bernoulli there was Euler who conceived the concept of interial tensor in his Theoria Motus Corporum Solidorum seu Rigidorum (Theory of the Movement of the Solid and Rigid Bodies) published in 1760. While the term "tensor" probably has been coined very later by Hamilton in On some extensions of Quaternions. Philosophical magazine (4th series), vol. 7, pp. 492-499; vol. 8 (1854), pp. 125-137, 261-269; vol. 9 (1855), pp. 46-51, 280-290