Interpretation of Stiffness Matrix and Mass Matrix in Finite Element Method

I would like to have a general interpretation of the coefficients of the stiffness matrix that appears in FEM. For instance if we are solving a linear elasticity problem and we modelize the relation between a node $i$ and a node $j$ as a spring system, then $K_{i,j}$ (where $K$ is the stiffness matrix of the system) can be seen as the stiffness constant of the virtual spring between the two nodes. But does there exist a more general interpretation? Perhaps in terms of internal work?

Another similar question is: What could be an interpretation of the coefficients $M_{i,j}$ of the mass matrix $M$?

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