Final answer:
Hooke's law ceases to apply in a force-extension graph when larger forces cause a curve in the graph, indicating non-proportional elastic deformation, and when permanent deformation occurs beyond the elastic limit.
Step-by-step explanation:
Hooke's law is obeyed in the linear region of a force-extension graph, where the relationship between the applied force (F) and the resulting deformation (∆L) is directly proportional, and the material returns to its original shape when the force is removed. This linear relationship is represented by the equation F = -kx, where k is the stiffness of the system, and the graph appears as a straight line.
However, the law is no longer obeyed when larger forces are applied that cause the graph to curve, indicating that the material is still deforming elastically but in a non-proportional manner. Permanent deformation occurs when even greater forces are applied, and at this point, Hooke's law no longer applies as the deformation will not return to zero when the force is removed. Ultimately, if force continues to be applied, the material will fracture. The slope of the graph just before fracture increases sharply, indicating that a small increase in force results in a large deformation, clearly deviating from Hooke's law.