Logarithmic Differentiation

When a car’s weight and driving forces act on a tire, they impose an external load on the rubber material. This load is resisted internally by forces distributed throughout the tire structure, which are defined as stress. The resulting deformation of the rubber due to this stress is quantified as strain. The relationship between stress and strain governs how the tire deforms under load and is central to understanding its mechanical response during operation.

Rubber exhibits a nonlinear mechanical behavior that differs significantly from that of linear elastic materials. This behavior is commonly described using idealized constitutive relationships that incorporate material parameters capturing resistance to deformation. In the present physical theory, the stress–strain relationship is expressed as

The dependence of stress on strain in both the base and the exponent reflects the nonlinear nature of the material response and allows the model to account for large deformations typical of rubber.

To evaluate how stress evolves as strain increases, the stress–strain relationship must be differentiated with respect to strain. Direct application of standard differentiation rules is challenging because the variable appears simultaneously in multiple functional roles. Logarithmic differentiation provides a systematic approach by transforming the expression through the natural logarithm, resulting in separating the components of the function into simpler terms.

Differentiating the transformed equation using the product and chain rules yields an expression for the rate of change of stress with respect to strain. Substituting the original stress function back into the differentiated result provides a compact mathematical description of how the internal resistance of the tire rubber changes with deformation, forming the basis for analyzing tire behavior under applied loads.