Scientific Part
Mechanics

On the question of the physical interpretation of material constants of hyperelastic models

S. A. Muslov^a, P. Yu. Sukhochev<sup>b

a</sup> Russian University of Medicine, 4 Dolgorukovskaya St., Moscow 127006, Russia
^b Lomonosov Moscow State University, GSP-1 Leninskie Gory, Moscow 119991, Russia

Abstract: There is a known demand for hyperelastic deformation models in the design of technical products using elastomeric materials (rubber and rubber-like polyurethanes, silicones, and TPE thermoplastic elastomers), which realize high (up to $500\%$) reversible deformations and damping capacity under cyclic and impact loading. Such products include car tires, shock absorbers, flexible gears, “compliance mechanisms” in robotics, etc. No less relevant and at the same time socially significant is the application of the theory of hyperelasticity for the purpose of developing implantable materials and devices for general, cardiac, and plastic surgery, including the replacement of soft biological tissues (skin, muscles, ligaments, etc.) with their functional analogues in the form of biocompatible synthetic materials. One of the unsolved problems in the mechanics of hyperelastic models remains the physical interpretation of their material constants. In this report, the material constants of the models are compared with the elastic moduli of the materials ($E_{0}$ and $G_{0}$) in the unstrained state. It is verified that for the neo-Hookean model, the relation $\mu=E_{0}/6$ holds, for the $2$-parameter Mooney – Rivlin model $C_{01}+C_{10}=E_{0}/6$. It has been established that the same formula is valid for the $3$-, $5$-, and $9$-parameter Mooney – Rivlin models and the nth order polynomial model. For the Ogden model $3\mu\alpha=2E_{0}$, Yeoh $C_{1}=E_{0}/6$, Veronda – Westmann $6(C_{1}C_{2}+C_{3})=E_{0}$. Material constants are indicators of the mechanical stability of hyperelastic models due to the Hill – Drucker condition. Using the example of a biomaterial, the results obtained using the derived formulas are compared with each other and with the indicators of elastic models: linear, bilinear, and exponential. A number of models characterize cases of small deformations unsatisfactorily.

Key words: hyperelastic models, material constants, elastic moduli.

UDC: 517.98

Received: 20.01.2024
Revised: 20.03.2024

DOI: 10.18500/1816-9791-2025-25-3-380-390