The low strain behavior of most materials is approximately spring-like (Hookean). In textiles, the governing equation is a simple one:
s = E e;
where s is called the 'tenacity', E is the “Young's modulus”, and e (generally epsilon) is the degree of deformation. For simple fabrics (linen, most likely) this Hookean response is short-lived and the weave of the material will generally fail before the fibers themselves. For lycra, the material response is likely elastic for large strains. Fabrics is not my 'area' so to speak, but the basic story for most materials is told by their Young's modulus. This site:
click here has a nice description of the properties of a few standard fabrics and details how the materials are characterized as well as how the weave impacts mechanical properties as well.