Hooke's Law Experimental verification

Elasticity is a property of a material which allows it to return to its original shape or length after being distorted. Some materials are not at all elastic -- they are brittle and will snap before they bend or stretch. Others, like rubber, for example, will stretch a long way without significant warping or cracking. This is because the materials contain long chain molecules that are wrapped up in a bundle and can straighten out when stretched. Those are the materials we choose for things like waistbands. But in physics we like numbers. One important number that relates to elasticity is the spring constant. The spring constant is a number that represents how much force it takes to stretch a material -- materials with larger spring constants are stiffer. So these are our properties to describe the stretchiness of materials -- elasticity and the spring constant. But what happens when you're in the process of stretching out a rubber band? Do you need more force at first and less once you've stretched part way? Perhaps it's the other way around? The relationship could be almost anything -- linear, quadratic, variable... so what did scientists discover when they started investigating elastic materials?