A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials

TLDR: In this article, an eight-chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network is proposed.

Abstract: Aconstitutive model is proposed for the deformation of rubber materials which is shown to represent successfully the response of these materials in uniaxial extension, biaxial extension, uniaxial compression, plane strain compression and pure shear. The developed constitutive relation is based on an eight chain representation of the underlying macromolecular network structure of the rubber and the non-Gaussian behavior of the individual chains in the proposed network. The eight chain model accurately captures the cooperative nature of network deformation while requiring only two material parameters, an initial modulus and a limiting chain extensibility. Since these two parameters are mechanistically linked to the physics of molecular chain orientation involved in the deformation of rubber, the proposed model represents a simple and accurate constitutive model of rubber deformation. The chain extension in this network model reduces to a function of the root-mean-square of the principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. The results of the proposed eight chain model as well as those of several prominent models are compared with experimental data of Treloar (1944, Trans. Faraday Soc. 40, 59) illustrating the superiority, simplicity and predictive ability of the proposed model. Additionally, a new set of experiments which captures the state of deformation dependence of rubber is described and conducted on three rubber materials. The eight chain model is found to model and predict accurately the behavior of the three tested materials further confirming its superiority and effectiveness over earlier models.

Figures

FIG. 4. Eight chain rubber elasticity model for (a) undeformed, (b) uniaxial extension and (c) biaxial extension configurations.

Frequently asked questions

Q1. What have the authors contributed in "A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials" ?

The chain extension in this network model reduces to a function of the root-mean-square of tho: principal applied stretches as a result of effectively sampling eight orientations of principal stretch space. I. I '' 'TRODUCTION THE AIM OF THIS WORK has been to develop a fully three-dimensional. The eight chain model is found to model and prL-dict accurately { he behavior of the three tested materials further conlirming its superiority and ctf.. 

Q2. What is the main obstacle to characterization of rubber elasticity?

The strong state of deformation dependence of the large and limiting stretch behavior of the polymeric network materials is shown in the data of Figs 1, 8, 9 and 10 and remains a primary obstacle to true characterization of rubber elasticity. 

Q3. What was the numerical treatment used for the tetrahedron model?

The numerical treatment used for the tetrahedron model followed the early method of TRELOAR (1954) which allowed the central junction point to seek an equilibrium position for affine displacements of the tetrahedron corners. 

Q4. What is the description of the eight chain model?

The eight chain model successfully accounts for the state of deformation dependence using a rubbery modulus and a locking stretch as its only two parameters, both of which can be determined from a single experiment. 

Q5. How many parameters are required to fit the three deformation states shown in Fig. A I?

A minimum of six independently adjustable parameters are required by the Ogden model to fit these three deformation states shown in Fig. 

Q6. What is the simplest way to determine the deformation state?

A The author; more than one deformation state may have been considered in determining the six constants as only the first four constants are necessary to produce the uniaxial extension result. 

Q7. What is the reason why the stress-stretch behavior is observed?

Some debate has persisted over whether the characteristic upturn in the observed stress-stretch behavior is solely due to crystallization and therefore. 

Q8. What is the nature of the state of deformation dependence of a rubbery chain model?

The eight chain model presented here is formulated such that the nature of the state of deformation dependence is clearly seen to be the result of a network of chains reaching the individual chain extensibility limit at different imposed global stretch levels for different stretch states. 

Q9. What is the simplest way to predict the state of deformation of rubber?

5. SuMMARYA physically based constitutive model for large stretch rubber deformation has been proposed which has been specifically designed to account for the three-dimensional state of deformation dependence in networked solids. 

Q10. How was the model compared to existing models?

In addition the model was compared to existing rubber elasticity models in the ability to capture the response of data in the literature (TRELOAR, 1944) in three states of deformation. 

Q11. What is the inverse Langevin function for the chain stretch?

The three stretch invariants are given byII= ).f+A~+).~,I2 = JciJc~+Jc~Jc~+2iJcLI3 = JctJciJc~.(17)(18)(19)The expression for chain stretch is seen to reduce to a function of the first stretch invariant, I~> and (16) may be rewritten as- The author1/2 Achain - J3 The author1 • (20)The strain energy of the proposed model may be found from integration of (15) using the series expansion form for the inverse Langevin function given for example in TRELOAR (t 954).