Report on: A Physically Based Method to Represent the
Thermo-Mechanical Behavior of Elastomers
Author: Alexander Lion
Journal: Acta Mechanica, 123.1-4 (1997): 1-25
written by: Dor Rettig Grun 208044727
Introduction
Elastomers are polymers with highly elastic properties, commonly used in engineering
applications where flexibility, resilience, and durability characteristics are crucial. This paper
particularly talks about filled rubbers (carbon black, like car tiers). Understanding their
mechanical behavior is essential for reliable design or simulation, especially under changing
temperatures and loading conditions. Traditional models have struggled to accurately
represent the full thermo-mechanical response of elastomers, more prevalent at large strains
or under thermal loading. In this article, Alexander Lion presents an improved, physicallybased, model that addresses these challenges by incorporating microstructural principles and
thermo-viscoelasticity into a unified framework.
Motivation
The motivation for the paper stems from the need for a better model that incorporates the
nonlinear, time-dependent, and temperature-sensitive behavior of elastomers. In addition, the
model needs to be grounded in physically and not purely theoretical knowledge and
assumptions. The expected goal for this model is that it could be applied to practical
engineering problems involving finite deformations and varying environmental conditions
like high temperature and pressure. In previous models were often lacking in a
thermodynamically consistent formulation and failing to adequately represent relaxation and
hysteresis effects. Lion’s model suggest fills this gap by basing the approach on molecular
theory, using internal variables to represent irreversible processes, and allowing coupling
between mechanical and thermal fields.
Methods
First, lion is describing the microstructure as a polymer network, using a series of nonlinear
elastic springs and viscous dashpots, representing the molecular chain behavior. the springs
are assumed to be nonlinear functions of the corresponding strain variables 𝜀𝑒𝑝 , 𝜀 𝑎𝑛𝑑 𝜀𝑒𝑣 .
picture 1. One-dimensional rheological model of viscoplasticity
Secondly, internal Variables is incorporated to track the non-equilibrium state of the material
and allow for history-dependent behavior, crucial for modeling phenomena like the Mullins
effect (also known as stress softening and describes the phenomenon where a filled rubber material softens upon
initial deformation and then becomes more stable under repeated loading ).
Third, using integral Constitutive Law, the stress is calculated via an integral over past
deformations, weighted by a temperature-dependent relaxation function.
Fourth, the model includes temperature as an internal state variable, influencing both the
elasticity and the rate of viscous flow (thermal Coupling).
Fifth, lion uses fundamental mechanics of Soft Materials for the calculations of the model,
such as expanding the idea for three dimension or addressing the finite strain theory. Mainly
lion use greens' strain tensor law and decompositions of the deformation gradient.
picture 2. Two different decompositions of the deformation gradient
Finally, lion's model parameters are identified utilizing experimental uniaxial tension tests
and stress-relaxation data and simulations. Doing such in order to verify the model and its
actual ability to be used in mechanical and engineering applications.
** also known as stress softening and describes the phenomenon where a filled rubber material softens upon initial
deformation and then becomes more stable under repeated loading
Main Results
The model was tested using uniaxial tensile tests at various temperatures and loading rates.
results include:
1) The model was accurate at predicting stress-strain behavior across a wide range of
strains and temperatures.
2) It was successful at capturing of the Mullins effect.
3) It gave a realistic modeling of stress-relaxation and hysteresis loops.
4) Good correlation with experimental data in both transient and steady-state regimes.
5) Its week point is in predicting the increase of the mean temperature in frequencies
higher than 10Hz, as the assumption of temperature independent material parameters
no longer applies.
*Comparison of experiments and simulations in the appendices
Applications and Significance
This model has several scientific significance and practical applications:
•
Engineering wise, firs and for most this model useful for simulating rubber
components in automotive, aerospace, and biomedical devices. Help to predict the
behavior under cyclic stress and heat increase.
•
Material studies, it can aid in the understanding of elastomeric materials and what we
can expect a black carbon rubber like material to behave in an expiriment.
•
Computational Mechanics: The model's thermodynamic consistency and finite strain
formulation make it suitable for implementation different kind pf computation
methods and thus help develop and understand them.
This work represents a significant advancement in constitutive modeling by utilizing physical
insight with mathematical insight using previous work in elastic and polymer mechanics,
paving the way for a more predictive and reliable simulations of elastomeric systems,
especially rubber like materials.
Conclusion
Alexander Lion’s article successfully gives a physical framework for modeling the complex
thermo-mechanical behavior of elastomers. By addressing the limitations of prior models and
grounding the approach in polymer physics, the proposed method achieves both accuracy and
applicability. Its adoption can enhance the predictive capabilities of simulations involving
elastomeric materials under realistic service conditions. As further research in the filled can
improve upon this model as more data and experimentation is done.
** examples for important equations of the model:
** example of a comparison between simulations and experiment