Final answer:
Without additional details or context, it is not possible to prove σ' = σ'' for normal stresses in a rotated state with the provided information.
Step-by-step explanation:
The question of proving that σ' = σ'', where σ' and σ'' are the normal stresses in a rotated state, involves understanding the transformation of stress under rotation. However, without specific information on the rotation, material properties, or the stress state, such a general proof is not feasible with the given information. The provided excerpts discuss different concepts such as uniaxial deformation, stress tensors, and shear stress but do not directly offer the needed equations or context for proving σ' = σ'' for a rotated stress state.
For a uniaxial deformation, normal stress in the direction of loading can be related to strain through a material's properties and for small deformations, it behaves similarly to an ideal gas law. Shear stress, on the other hand, involves deformation perpendicular to the force applied and is proportional to that force divided by the product of the shear modulus and the original length. Yet, these do not directly inform us about stress in a rotated state without additional details on stress transformation equations or conditions under which σ' = σ'' could hold true.