A physical construction of the Ogden-Hill equation for soft elastomeric networks

The theory of rubber elasticity stands as a cornerstone in the study of soft matter. Despite nearly a century of development, many models remain largely phenomenological in nature, lacking a firm physical grounding. Addressing this fundamental challenge, this study advances a strain energy function theory of molecular basis, drawing from the worm-like chain model and the tube-like entanglement model. Diverging from classical approaches like the freely-jointed chain (FJC) model, the proposed framework offers a nuanced analysis of semi-flexible chains based on their end-to-end distance distributions, thus providing a more comprehensive understanding of polymer mechanics. Moreover, the entanglement of chains is characterized through the assessment of tube potential energy. Through a defined set of parameters, the model adeptly predicts the large deformation behaviors of vulcanized rubber across various experimental conditions including uniaxial tension, uniaxial compression, pure shear, and equi-biaxial tension. Additionally, it offers analytical insights into phenomena such as uniaxial tension and the inflation of an ideal balloon based on a set of parameters. During its application, the model’s resemblance to the Ogden-Hill form was noted, prompting a comparative analysis with well-known equations in this form (e.g., Varga equation, Neo-Hookean equation, Mooney-Rivlin equation, Mullins-Tobin equation, classical Ogden-Hill equation), thereby elucidating the physical underpinnings of the strain energy function. The proposed model posits the strain energy function as comprising three distinct components, i.e., affine motion, semi-flexibility, and entanglement—each manifesting distinct mechanical characteristics denoted by invariants. Furthermore, comparative assessments against the Carroll model, Pucci-Saccomnadi model, full-chain FJC model and the semi-flexible worm-like chain (WLC) model underscore the advantages of the proposed framework. Notably, the model exhibits a capacity to accurately predict rubber stress responses under large deformations without encountering singularities, thus rendering it amenable to finite element analysis (FEA). Finally, the efficacy of the proposed constitutive models is corroborated through rigorous comparisons with experimental data drawn from a spectrum of literature sources encompassing vulcanized rubber, natural rubber, elastomeric hydrogel, supramolecular elastomeric networks, and highly entangled elastomeric networks.