Mesh independence is done in relation to an output. It is most often done in relation to stress, but there are system models that are too coarsely meshed to be adequate for accurate stress output, yet are adequate to capture the stiffness of the structure for accurate deflection or modal frequency output.
If the quantity of interest is deflection and the maximum deflection is plotted versus element length, you will see the graph converge at relatively large elements lengths. If you now switch to maximum equivalent stress, you may see that the stress has not converged and smaller elements are required. Many models contain a stress singularity, which means that the stress will never converge, it will only grow larger and larger as the element size gets smaller and smaller. In that case, the geometry or boundary condtions must be changed for the stress in that location to converge. A typical example is a sharp interior corner. Note that the deflection can converge in a model that contains a stress singularity.
It’s often helpful to solve a model using linear elastic material properties first before adding nonlinear properties. If the stress goes above the yield point, that may be an indication that the design needs to change if the goal is to have a structure that does not exceed the yield strength. In that case, you don’t need to add a nonlinear property such as plasticity to the material model.