Robert preferentially or favorably oriented. This explains

Robert Le

Continuum Mechanics

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12 December 2017

Written
Report: “Constitutive Models of Rubber Elasticity: A Review”

A paper
authored by Mary C. Boyce and Ellen M. Arruda, “Constitutive Models of Rubber
Elasticity: A Review” is essentially devoted to collecting popularly known
constitutive models for the finite deformation response of rubber elastics into
one source and addressing their unique characteristics and similarities to each
other. It is important to first explain rubber and it’s ability of elasticity. The
molecular structure of rubber is what allows for it’s unique characteristic,
the molecules are long chains that are basically tangled up in randomness while
weakly joined together and when stretched they become preferentially or favorably
oriented. This explains why when rubber is stretched, entropy decreases. Rubber
products are everywhere with a variety of different properties and purposes
that continue to grow today. Constitutive models of rubber elasticity provide
us with essential models in order to perform 3D simulations and analysis of the
Stress-Strain Behavior for rubber materials. Being able to conduct these simulations
and analyses is a necessary part of the design process for elastic products
that undergo complex forms of deformation. As the industry grows along with the
advancement of elastic materials, this will spark further interest and the need
for more development of such modeling.
            Various constitutive models
are addressed by Boyce and Arruda, all of which are compared to Treloar’s classical
data in order to measure their predictability of stress-strain responses. Leslie
Treloar was one of the leading figures in the science of rubber and elasticity,
writing the book, “The Physics of Rubber Elasticity”. Her work is models the
basic features of stress-strain behavior very well and provides a foundation in
this specific field. The three main approaches to constitutive modeling are the
statistical mechanics treatment, the invariant-based continuum mechanics
treatment, and the stretched-based continuum mechanics treatment. Firstly, the
statistical mechanical approach assumes rubber as a formation of randomly
oriented long molecular chains. There are three main models in this approach such
as the Gaussian Model which is only able to accurately predict at small
deformations, the Non-Gaussian Model which includes the 3, 4, and 8 Chain
Models that utilize a unit cell that deforms in a principle stretch space, and
lastly the Flory and Erman Model which even more accurately predicts at small
deformations that the Gaussian Model except that it becomes inaccurate as deformation
increases. It was found that a combination of the 8 Chain Model which predicts
at medium to large deformation and the Flory and Erman Model will best fit
Treloar’s data. Secondly, the invariant-based continuum mechanics approach takes
into account rubber as an isotropic, hyper elastic material and uses the fundamental
basis of continuum mechanics, meaning that the strain density is based on the
stretch of one or more invariants of the stretch tensor. The two models
discussed in this approach are the Mooney Rivlin Model and the Gent Model. It
is important to explain that Gent Model uses a higher order of  and is essentially the
same as the 8 Chain Model, showing that the statistical mechanical approach and
the invariant-based continuum mechanics have some similarities. Thirdly, the
stretch-based continuum mechanics approach uses the same fundamental basis as
the previously mentioned approach except that instead of the strain density depends
on the principal stretches. There are two models discussed by Boyce and Arruda
which are the Valanis and Landel Model and the Orgen Model, both models are
strain energy functions of principal stretch with Orgen creating a more specific
function may be adjusted in order to fit certain specified data. Lastly, Boyce
and Arruda talk about the effects of compressibility, where in real life
situations, rubber is only nearly incompressible where in contrast it is
commonly considered to be incompressible. By using a compressible model for
rubber, the numerical errors that are essentially built in incompressible formulations
can be avoided. The problem with compressible models is that they implement compressible
forms of strain energy via volumetric deformation where there is not enough
experimental data in volume changes on deformation.

            

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