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Why is the stress-strain relationship of rubber nonlinear at large stretches?

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Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).
mechanics of solids Figure 1: The position vector x and the velocity vector v of a material point, the body force fdV acting on an element dV of volume, and the surface force TdS acting on an element dS of surface in a Cartesian coordinate system 1, 2, 3 (see text).

The nonlinear stress-strain relationship of rubber at large stretches arises from its unique microscopic structure, which differs fundamentally from the linear elastic behavior observed in materials like steel.1

At the microscopic level, rubber consists of long-chain molecules that are highly flexible and randomly coiled in their unstressed state. When a tensile force is applied, these chains uncoil and align in the direction of the stretch.21

The nonlinearity occurs because the material's resistance to deformation changes as the chains reach their limit of extensibility. At small deformations, the response is approximately linear and can be described by Hooke’s law. However, as the material is stretched to large extensions, the molecules become increasingly oriented and straightened, significantly increasing the stiffness of the network. Because the elasticity of rubber is governed by the configuration of these polymer chains rather than the stretching of interatomic bonds, the relationship between stress and strain is inherently nonlinear, requiring a stored-energy function to accurately model its mechanical response.1