Final answer:
The question relates to elasticity in Physics, concerning how materials, including biological cells and membranes, respond to stress and strain. Concepts such as Young's modulus, Poisson's ratio, and JKR-theory in contact mechanics are central to understanding the mechanical behavior of materials when subjected to forces.
Step-by-step explanation:
The student's question centers on the concepts of elasticity and mechanical deformation, specifically in the context of vibrations of a rectangular membrane, contact mechanics, and cellular and material deformations under forces such as tension and compression. The provided references discuss various aspects of elasticity including the Young's modulus E, Poisson's ratio v, and shear and bulk moduli, which are fundamental concepts in the field of solid mechanics within the broader discipline of Physics. These properties characterize how materials respond to external forces and deformations. For instance, the balance of adhesion and elastic forces can be described using JKR-theory for contact mechanics. Additionally, cellular elasticity highlighted by a modulus in the range of ~10kPa demonstrates the comparative softness of biological materials set against materials like metals or diamond.
A linear relationship between stress and strain defined by Young's modulus is often assumed for small deformations in materials that obey linear elasticity. However, the need for more complex constitutive relationships, such as the Neo-Hookean model, arises when dealing with materials that exhibit non-linear behavior under stress. Other concepts such as the bending modulus and inextensibility constraints are also touched upon, indicating various modes of mechanical response in different contexts of elasticity.