Difference between the meanings of the equations Stress=Young's modulus x Strain and F=-kx According to Hooke's law: $$F=-kx .$$ 
Is this equation the same as the equation: $$\textrm{stress}~= ~\textrm{young's modulus}~\times~\textrm{strain}$$ (i.e. are the two equations different forms of Hooke's law) or are they different? 
 A: The Young's modulus equation is a more detailed version of Hooke's law.  In Hooke's law, you just subsume the details of the spring into its spring constant k, but in Young's approach, you generalize to allow springs of different lengths and cross sectional areas, but otherwise similar.  You also treat springy material like sponges.  The key point of Young's is, it's not the stretch that produces the force, it is the stretch relative to the unstretched length-- a fact glossed over by Hooke.  Also, the force should be proportional to the cross sectional area because two springs side by side will produce twice the force.  So Young's lets you generalize the spring as long as you keep its basic properties the same and vary length and cross section.  For example, Young's shows you right away why two identical springs, attached end to end, produce a single spring with half the k-- it's because a stretch of such a longer spring is only half what it was for the original spring, when the stretch is considered relative to the length of the whole spring (which we call stress).  In short, a spring force comes from stress, not stretch, but Hooke glosses over that.
A: The equations are different. There are multiple versions of Hooke's Law or elastomechanic constitutive relations:


*

*$F=-kx$ for a spring (i.e., an idealized, perfectly elastic lumped component)

*$\sigma=E\epsilon$ for elastic (i.e., small) axial deformation of a elongated rod

*$\epsilon_{ij}=\frac{1+\nu}{E}\sigma_{ij}-\frac{\nu}{E}\sigma_{kk}\delta_{ij}$, generalized Hooke's Law, for elastic deformation of any isotropic material

*$\sigma=C\epsilon$, where $\sigma$ and $\epsilon$ are vectors and $C$ is the stiffness tensor, for elastic deformation of any material

