If you are using the term "elastic" to means something flexible and not necessarily stretchy, consider that leather is a good choice. Leather can be bonded to wood using mechanical means (bolts, furring strips, etc) as well as with various adhesives. Using mechanical fasteners allows for easy replacement as required.
I've constructed leather hinges which have lasted for years, although under occasional use rather than constant use. Periodic treatment with an oil or leather conditioner also improves lifespan.
I'd like to see video of the dog gluing the material of choice to the wood.