Solution approaches for large strain mechanics

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I’m studying large strain and deformation solid mechanics and I have a (seemingly) basic question on solution approaches. Is my interpretation below correct?

The governing equation of momentum for solid mechanics can be solved using a total or updated Lagrangian approach. The updated Lagrangian form is sometimes expressed in rate form. The rate form (of the updated Lagrangian approach) permits use of the small strain approximation since only the increment of displacement is considered from the current time t to time t+dt. However, when using a rate form, the stress rate appears in the equation of virtual work, and therefore an objective rate must be used.
 
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No, you can only use small strain "linear elasticity" if the material is undergoing small elastic strains. If it's a polymer or something that is undergoing large elastic strains, then you have to use a hyperelastic relation. In rate form, this would be called "hypo"-elasticity.