Physically, elastic materials are materials which return to their original state upon complete removal of applied mechanical loads under isothermal conditions.
In the book "Mechanics of Laminated Composite Plates and Shells 2nd Edition" by JN Reddy", the author defines
Materials for which the constitutive behavior is only a function of the current state of deformation are known as elastic. In the special case in which the work done by stresses during a deformation is dependent only the initial state and the current configuration, the material is called hyperelastic.
I am unable to relate the mathematical definition of elastic materials with the physical one. How is the dependence of constitutive relations on current state of deformation related to the material returning to its original state upon complete removal of loads?