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CONSTITUTIVE MODELING OF THE THERMAL
RESPONSE OF RUBBER-LIKE MATERIALS

By

Yuhui Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2006

ABSTRACT

CONSTITUTIVE MODELING OF THE THERMAL RESPONSE OF
RUBBER-LIKE MATERIALS

By
Yuhui Wang

The thermal conductivity and diffusivity of elastomers can change due to finite
deformation. Some studies have proposed the constitutive relationship between
thermal conduction and finite deformation for uniaxial loading of specimens. These
correlations often omit the values of the components of the diffusivity and ignore the
results of continuum mechanics in developing the constitutive relation.

The present study focuses on two means of correlating the diffusivity of
elastomers with respect to finite deformation. First, measured diffusivities fi'om other
studies are correlated with deformation, through the principal stretches or the Cauchy-
Green strains, in the current (measurement) frame and afier transforming the
diffusivity to the reference (undeformed) frame. The change in each component of the
diffusivity tensor appears to be determined by the deformation in the direction of the
component only. The correlation in the current frame suggests separate fitting
parameters for each direction of each material. In the reference frame, the two
directions seem to follow a similar trend.

In the second part, advantage is taken of the constraints introduced by the isotropy
that elastomeric samples often exhibit in their undefonned state. This material

symmetry gives rise to mathematical restrictions on the constitutive equations. By

satisfying the principle of material frame indifference and the second law of
thermodynamics, and putting the constitutive function in an undeforrned reference
configuration, a detailed form of the heat flux vector function is derived as a fimction
of the pure stretching tensor. Parameters are determined for several elastomers that

have wide use and ofien undergo large deformation.

TO my family

iv

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Neil T. Wright, for his support and help
throughout the study. It was my pleasure to conduct this interesting and challenging
research under his guidance.

I am also very grateful to my academic committee members, Dr. Andre Benard, Dr.
Dahsin Liu, and Dr. Baisheng Yan, for their valuable advice and inspiring comments.
Many of my colleagues have contributions to my dissertation. I would like to thank
Xianglan Bai, Xing Huang, and Yuwei Chi for their insightful comments and sugges-
tions.

Finally, I am especially indebted to my husband, Hongbo Zhou, who has always been
supportive. I am very lucky to have him in my life. I also would like to thank my

lovely daughter, Julina Zhou, for bringing me so much pleasure since she was born.

LIST OF TABLES
LIST OF FIGURES

1 Introduction

TABLE OF CONTENTS

1.1 Background ..................................
1.2 Helmholtz free energy ............................
1.2.1 The effects of deformation on 11) ......................
1.2.2 The effects Of temperature on 1,0 ......................
1.3 Heat flux vector q ..............................
1.3.1 Thermal diffusivity .............................
1.3.2 The effect of deformation on K or a ...................
1.3.3 The effects of temperature on K or a ..................
2 Related work

2.1 Experimental work ..............................
2.1.1 Uniaxial stretching .............................
2.1.2 Multiaxial stretching ............................
2.2 Existing models for k or a ..........................
2.2.1 Temperature dependence ..........................
2.2.2 Deformation dependence ..........................

3 Theoretical Framework

3.1 Choice of reference configuration ......................
3.2 Relationship of q between configurations ..................
3.3 Material flame Indifference .........................
3.4 Initially isotropic material ..........................
4 A Correlation of Diffusivity

4.1 Uniaxial Loading ...............................
4.2 Biaxial Loading ................................
4.3 Discussion ...................................

5 A Constitutive Relation for q

5.1 Methods ....................................
5.1.1 Material frame indifference ........................
5.1.2 Three linearly independent variables ...................
5.1.3 Application of second law of thermodynamics ..............
5.1.4 Material Symmetry .............................
5.2 Analysis ....................................
5.2.1 Heat flux vector in terms of U .......................
5.2.2 A special property of 2,111, «in, and 1123 ...................

vi

56
57
57
57
58

5.3 Invariants ................................... 61

5.4 Data Analysis ................................. 62
5.4.1 Analysis of the uniaxial stretching of elastomers ............. 63
5.4.2 Analysis of biaxially stretched elastomers ................. 66
6 Discussion 69
APPENDICES 75

vii

4.1

4.2
4.3

5.1

LIST OF TABLES

Polymers subject to uniaxial loading correlated with any,- / (134 = 0.03. . . 46
RTV polyurethane subject to biaxial loading duh/am = an- ...... 48
Correlating coefficients for equation 4.8 for neoprene rubber that is anisotropic

in the unloaded condition and then is subject to biaxial loading . . . 49
Polymers subject to biaxial loading with form a/aeq = —(z,/)1I +¢2U"1+

«mu-2) .................................. 68

viii

1.1

1.2
2.1

2.2

3.1
3.2

4.1

4.2

5.1

5.2

LIST OF FIGURES

A solid undergoing a thermomechanical process from reference configura-
tion Bo to current configuration B ...................
Typical force-extension curve for vulcanized rubber ............

The relationship between diffusivity and stretch ratio in the direction of
stretching of silicon rubber subject to uniaxial loading [2]. The error
bars denote i7.5% uncertainty in the stretch ratio ...........

The normalized components of thermal diffusivity of silicone rubber as
functions of their respective stretch ratios. Note the differing Slope
indicating that a single parameter is insufficient .............

Observer transformation diagram describing Material Frame Indifference
Two reference configurations related by a rigid body motion .......

Variation of aQfi/aeq with respect to C“- for silicone rubber subject to
uniaxial loading. .............................
Variation (aofi-i/aeq) as a function of 0;,- for polyurethane subject to bi-
axial loading. Specimens A1, A2, and A3 are shown. .........

Normalized thermal conductivity tensor as a function of the pure stretch-
ing tensor for silicone ..........................
Normalized thermal conductivity tensor as a function of pure stretching
tensor for polyurethane RTV .......................

ix

[\D

16

23

30
36

44

47

65

67

Chapter 1

Introduction

Elastomers are rubber-like materials that are widely used in the automotive, aerospace,
and medical industries in conditions of severe thermal and mechanical stress that
often involve large temperature gradients and deformation. Natural rubber, for ex-
ample, has excellent resistance to cutting, gouging, and abrasion. Another elastomer,
synthetic rubber or styrene-butadience copolymer (SBR), is used predominantly in
automobile tires because of its excellent abrasion resistance. Silicone rubber, another
elastomer, possesses a high degree of flexibility at temperatures as low as —90°C and
yet is stable to temperatures as high as 250°C, and thus can be used for high or low
temperature applications (insulation, seals, diaphragms, or tubing). Elastomers are
expected to provide trouble-free performance during their service life. Designs in—
corporating elastomers and other engineered products rely increasingly on computer
models of the thermomechanical performance, which leads to the need of constitutive

relationships for their response to varies boundary conditions.

1. 1 Background

A general theory of characterizing the thermomechanical response Of solid materials
has long been available [5]. The deformation of a solid may be described as in Figure

1.1.

Ci

33

Figure 1.1: A solid undergoing a thermomechanical process from reference configura-
tion Bo to current configuration B

Based on the conservation laws and the second law of thermodynamics, the general
theory reveals that five constitutive functions are needed to describe the response of
a solid material. These functions may include the internal energy function (5, specific
internal entropy function 7“), Cauchy stress function T, heat flux vector function 6;, and
internal state vector function ,6. These constitutive functions depend on the material
point X, deformation gradient tensor F = 8x/0X, where x is the spatial position
vector that the material point X occupies in the current configuration, temperature
T, temperature gradient VOT = BT/BX, and internal state vector ,6 = (61,62, mflN),
where m, 62"", H N are internal state variables. Since we are considering homogeneous

2

materials, the material point X can be omitted for convenience. The constitutive

functions are written as

e = é(F,T, VT, 3) (1.1)
n = 17(F, T, VT, 5) (1.2)
T = T(F, T, VT, 3) (1.3)
q = 61(15‘, T, VT, [3) (1.4)
B = 3(F,T, VT, 3) (1.5)

The superposed hat serves to distinguish these functions from their values. The
internal state vector 3 describes aspects of the internal structure of the material
associated with irreversible or dissipative effects. The evolution of internal variables
accounts for, indirectly, the effects of the history of the deformation. Because rubber-
like materials demonstrate reversible processes during deformation, 3 is unneeded
as a constitutive function for rubber-like materials with some exceptions; the other
four constitutive functions 53, 1?, T, 5; will be considered independent of B for such a
material response. Furthermore, the constitutive functions for rubber-like materials

exhibiting hyperelastic behavior can be further simplified and only two independent

constitutive functions, u and q, where 7,!) = 6 — T17 is defined as Helmholtz free energy,

are needed. As shown in the Appendix, these have the following forms

w = in: T) (1.6)

q = (1(P, T, VT) (1.7)

The other two constitutive functions can be expressed in terms of if and q as

n = filET) = —ziT(F,T> (1.8)

T = T(F, T) = §Ftl3g~‘(F, T) (1.9)

where J is the determinant Of the deformation tensor J = detF, the subscript denotes
the partial differentiation, and the superscript T denotes the transpose. Equations
1.6 and 1.7 demonstrate that the modeling of a thermomechanical response of elas-
tomers requires constitutive models that account for both the mechanical and thermal

dependence of the thermomechanical properties of these materials.

1.2 Helmholtz free energy

1.2.1 The effects Of deformation on w

Experiments have revealed that macroscopic deformations imposed on elastomeric
materials lead to molecular reorientation [30], which can induce changes in mechan-
ical and thermophysical properties [2]. Measurements have demonstrated that there
can be significant changes in the thermal conductivity or diffusivity tensors of elas-
tomers subject to finite deformation [2, 22]. Additionally, the constitutive equation
1.6 shows that deformation determines the value of 1]) through F. Many studies have
considered, experimentally and theoretically, the relationship between F and ii. A
widely accepted form of 21) is still a matter of active study, although a number of forms
have been proposed (see reviews by, for example, 'D'eloar [30] and Holzapfel [19]). The
models of ‘t/} have been developed based on either a statistical or a phenomenological
approach.

The phenomenological method describes materials as continua. This methods
fits mathematical equations to experimental data, which have been developed second
law of thermodynamics, and material frame indifference. In studying of constitutive
function if, the phenomenological method has led to, for example, the neo—Hookean,
Mooney-Rivlin, Rivlin-Saunders, and Ogden models [19]. These models describe the
mechanical response of elastomers for various boundary conditions or levels of defor-
mation. Each of these models fits well experimental data under certain conditions.
Most studies have focused on isothermal deformation, with less attention being de-

voted to the effects of temperature on 1,0. Some exceptions include Ogden [25], who

proposed a method for finding w as a function of biaxial stretches and temperature,
and Humphrey and Rajagopal [20], who showed that in-plane biaxial tests allow
measurement of thermoelastic response fimctions similar to the isothermal results of
Rivlin and Saunders [27].

Although the phenomenological method is an effective way to develop models,
it fails to capture the relationship between the mechanism of deformation and the
underlying molecular structure of the material. In contrast, the statistical method
has been recently used to offer interpretation of constitutive models in terms of mole-
cular changes. E‘ied [13], for example, gave an elementary molecular explanation for
each of the Mooney and Rivlin-Saunders theories of rubber deformation. Moreover,
Wineman developed a continuum mechanics model that considered a development
of a secondary molecular network, which results from microstructural change due to
elevated temperature and deformation [37, 38].

In contrast to the phenomenological method, the molecular-statistical method
may be used to develop constitutive models based on the molecular structural of
materials. A comprehensive review of the several studies done using this method
was given by Treloar [30]. Most studies consider the unique molecular structure of
rubber-like materials, and by analyzing the features of the conformational change of
the molecular network of the material under deformation, Gaussian or non-Gaussian
models are developed. The Gaussian model correctly describes the behavior of real
rubber to a first approximation only, within a very limited range of deformation, for
stretch ratio up to 5.5 during uniaxial stretch. The Gaussian model is equivalent to the

neO-Hookean model, which has been developed using the phenomenological approach.

Tensile force (N mm")
M
l

 

o lllllllllllll ;
0 2m 400 an

WW)

Figure 1.2: Typical force-extension curve for vulcanized rubber

By extension of the statistical approach, three-chain and four-chain network models
have been developed based on non-Gaussian theory [30]. The non-Gaussian theory
gives an improved representation of the general features of the experimental response
as compared to the Gaussian theory; the later entirely fails to represent the point of
inflection, where the concave curve changes, and subsequent rapidly increasing slope
at high strains, a typical force-deformation curve for a rubber is demonstrated as in
figure 1.2 [30]. Arruda and Boyce developed an eight-chain molecular model [1]. This
statistical-molecular method used insight Of the chemical structure of elastomers to

model the mechanical response.

1.2.2 The effects of temperature on U)

Compared to the study of F on w, few studies have examined the effects of T on 1/2,
focusing instead on the isothermal response. A notable exception is the study of mate-
rial behavior at elevated temperature by Wineman and Min [38]. Their study is based
on the experimental work of Tobolsky [29] whose experimental results demonstrate
that the molecular network changed when the temperature becomes high enough.
Wineman and Min demonstrate the importance of temperature on the mechanical re-
sponse of elastomers in studies of the development of a secondary molecular network
resulting from microstructural changes due to deformation at elevated temperature.
This secondary networks form due to consists of scission and subsequent cross-linking
of the macromolecular network bonds. At high temperature, say greater than the
temperature of onset of the ’chemorheological range’, the affected molecules will re-
coil and cross-link to form a new network. This microstructural change results in a
change in the mechanical response and leads to permanent set. Wineman and Min
present a constitutive model based on this combination of two, or possibly more,

microstructural networks.

1.3 Heat flux vector q

A phenomenological law, based on experimental Observations, that relates the heat
flux vector q to the spatial temperature gradient VT is asserted as Duhamel’s law of

heat conduction [19]

q = —K(F,T)VT (1.10)

where K(F, T) is the spatial thermal conductivity tensor written in the current con-
figuration. If the material is thermally isotropic, which means there is no preferred
direction for the heat conduction, the thermal conductivity tensor becomes K = H,

where k is a scalar, and equation 1.10 simplifies to
q = —IcVT (1.11)

which is called Fourier’s law of heat conduction, and k denotes the coefficient of
thermal conductivity.
Substituting equation 1.10 into the first law of thermodynamics for an incompress-

ible solid without internal motion or energy generation yields

mpg—31:: = V - (KVT) (1.12)
and for isotropic conduction

papal: = V - (kVT) (1.13)

where p is the material density at current configuration and 0,, the specific heat

capacity at constant temperature.

1 .3.1 Thermal diffusivity

Thermal diffusivity is another material parameter that arises for transient heat-flow
calculations. Some experiments measure the thermal diffusivity instead of the ther-
mal conductivity of elastomers because the thermal diffusivity can be measured via
transient experiments. Here, we demonstrate the relationship between the diffusivity
and conductivity so that experimental results may be compared. We denote a to be

the Spatial thermal diffusivity tensor, which is related to K by

Here we will assume that elastomers have no change of volume upon deformation,
which has been well-justified on the basis of experiments that show volume changes
on deformation to be very small, i.e. on the order 10"4 or less [30]. This assumption
is a common idealization invoked in continuum and computational mechanics, and
leads to the result that detF = 1. Therefore, p = p(), Since p = podetF, where p0 is the
material density described in referential configuration. In addition, Broerman et a1.
[2] deduce that op is independent Of stretch ratio in their uniaxial loading experiment
based on the assumption that the tensile force is a linear function of temperature as
proposed by Treloar [30]. Therefore, Similar relations hold between the deformation

and a and between deformation and K.

10

1.3.2 The effect of deformation on K or a

Heat conduction in elastomers changes substantially due to deformation. When amor-
phous polymers or rubber-like materials are stretched uniaxially, these original ther—
mally isotropic materials become anisotropic with increased conduction in the direc-
tion of stretching, which has been measured in several studies by Broerman et al.
[2], Washo [36], Tautz [28], Hennig [18], and Hands and Horsfall [16]. This property
change results from the unique molecular structure of elastomers. Elastomers consist
of long-chain molecules that are randomly oriented in their native state. When the
material is subject to a uniaxial deformation, the molecular chains are oriented pref-
erentially along the stretching direction. The thermal conductivity of the oriented
material will be greater along the back bone of the chain with its strong covalent
bonds, and weaker valences parallel to the chain axis, and smaller perpendicular
to the stretching due to the fewer van der Vaals bonds. Therefore, the orientation
causes a preference for heat conduction in direction of stretching and the material
demonstrates anisotropic thermal conduction.

In order to understand better the changes in thermophysical properties of elas-
tomers in response to deformation, some experiments have been conducted to measure
the changes in heat conduction due to deformation [36, 18, 16]. Broerman et a1. [2]
measured the thermal diffusivity of silicone rubber subject to uniaxial loading using
a thermo-optical technique. They reported that for initially isotropic silicone rub-
ber, the component of the thermal diffusivity tensor in the direction of stretching
increases by 10% for a stretch ratio of two, and the component of the diffusivity or-

thogonal to the direction of stretch decreases by 5%. This agrees qualitatively with

11

previous studies of LeGall. LeGall [23] studied multiaxial loading cases, where they
measured the three orthogonal components of diffusivity during biaxial stretching
of polyurethane, natural gum, and neoprene rubbers. The deformation was homo-
geneous in the central region of the specimen, where the diffusivity was measured.
The results showed an increased diffusivity in the direction of stretching and a de-
creased value in the orthogonal direction, in the out of plane direction. LeGall and
Wright did not provide a correlation of these data. Wang and Wright [35] studied
both the uniaxial loading and biaxial loading experimental data for initially isotropic
elastomers. By transforming those data to the respective reference configurations,
where the elastomers demonstrate isotropic thermal property, they found that the
thermal diffusivity or thermal conductivity is proportional to the inverse of Right
Cauchy deformation tensor C = FTF. Within the expectation, the coefficients are
different for different materials.

Although a few experiments have been conducted related to this area and some
observations have been made, no complete and rigorous mathematic theory has been

advanced.

1.3.3 The effects of temperature on K or a

Most properties of elastomers depend heavily level on temperature. For example, the
mechanical response changes greatly at the glass transition temperature [6]. At a tem-
peratures below the glass transition temperature, the molecular structure is random
and amorphous, but the mechanical behavior is unlike a rubber and these materials

become hard and rigid like a glass. Above the glass transition temperature, the mate-

12

rial are more compliant. The thermal conductivity also shows significant dependence
on temperature when elastomers are either above or below the glass transition tem-
perature [16, 6, 9, 10, 11, 12, 7]. These studies have assumed an isotropic thermal
response of elastomers. In addition to the experimental studies, some theoretical stud-
ies have offered quantitative relations between the thermal conductivity or thermal

diffusivity and temperature, and two different theories are proposed [16, 6, 7].

13

Chapter 2

Related work

2.1 Experimental work

2.1 . 1 Uniaxial stretching

Measurements have demonstrated that there can be significant changes in the thermal
conductivity or diffusivity tensors of elastomers subject to finite deformation. In order
to measure the thermal conductivity or diffusivity accurately, Broerman et a1. [2]
used Forced Rayleigh Scattering (PBS) to measure the thermal diffusivity of silicone
rubber subject to uniaxial stretching. In FRS, a transient optical grating is formed in
the material by a diffraction pattern that results from absorption of intersecting laser
beams, which have been split from a single source. The rate of decay of the efficiency of
diffraction is a function of one of the components of the thermal diffusivity, depending
on the relative orientation of the lasers to the specimen. Multiple experiments are

required in order to measure multiple components of the diffusivity.

14

Broerman et al. [2] measured the thermal diffusivity properties Of a crosslinked
silicone elastomers that were uniaxially stretched up to a stretch ratio of 2.1. The
stretch ratio A,- is defined as the current length 1,- divided by a undeformed length L,
the direction of stretching is 2' = 1. Motivated by the stress-optic rule [21], Broerman
et al. [2] described their results using a stress-thermal rule, which was first suggested

by van den Brule [3]. The stress-optic rule for simple elongations is

A7113 = 000’ (2.1)

where a = tn — t33, with t11 and t33 the components of the Cauchy stress, and
An13 = n11 — n33, with ml and 1133 the components of the refractive index of

refraction tensor. The stress-thermal rule is written analogously as

(011 - a33)/aeq = 00’ (2-2)

where all and 0133 are the components of thermal diffusivity tensor in the direc-
tion parallel and perpendicular to the direction of stretching, aeq is the equilibrium
thermal diffusivity, which is defined as the thermal diffusivity of the unstretched
elastomers, and c is a material parameter. For the stretched specimen, the measure-
ments of A7213 were made by using a low-power HeNe laser and measurements of the
diffusivity 0111 and 0133 were made by using a low-power HeNe laser and PBS.

By plotting the normalized thermal diffusivity 011 / aeq and 0133/0139 against A,
Broerman et al. reported two linear relationships with different coefficients. It was

shown that for initially isotropic silicone rubber, the component of the thermal dif-

15

fusivity tensor in the direction of a uniaxial stretching normalized by aeq, that is
011 /aeq increases by 10% with A1 = 2.1. The component of the diffusivity orthogo-
nal to the stretching normalized by aeq, that is 033/0164 decreases by 5% at A1 = 2.1.

Figure 2.1 shows these variations in 011 and 033 for stretching in the 1-direction.

 

 

1.15 r
l.
[ ' all/aeq
1.10 ~ ‘ “331“”
. H—l
t—I—I
l—I—l
i—I—i
“ l—I—I
J 105 t-I-t
W)
5’ 1 H2“
:3- l.
w. 100 I i=5?”
5 . . 1:64—1 l—H
H—i
l—H
H... H—l i—H
. H—i ”=1
l—H
0.95 r
0.90 1 1 A 1 1 1 1 1 1
1.0 1.5 2.0 2.5
M

Figure 2.1: The relationship between diffusivity and stretch ratio in the direction of
stretching of silicon rubber subject to uniaxial loading [2]. The error bars denote
i7.5% uncertainty in the stretch ratio

Broerman et al. cite Tautz [28] as also finding a linear relationship between con-
ductivity or diffusivity and stretch. Tautz [28] measured the thermal conductivity of

natural rubber only along the stretch direction for a stretch ratio up to 3.5. A nearly

16

linear relationship existed between the thermal conductivity and stretch ratio, but,
the data may not fit the linear relationship well with stretch ratio greater than 2.5
since all the data lie above the trend line.

Similar responses are measured in non-elastomeric polymers. Wash and Hansen
[36] also characterized the relationship between thermal conductivity of amorphous
polymers and the stretch ratio. They measured the thermal conductivity of polystyrene
along the direction parallel and orthogonal to the stretch direction with stretch ratios
of up to 3.2 and Of polymethylmethacrylate (PMMA) with stretch ratios of up to 6.0.
The thermal conductivities were normalized by the respective value of the equilib—
rium conductivity. These data showed a linear relationship with respect to stretch
ratio. Again, the thermal conductivities were greater than the equilibrium value and
those in the perpendicular direction are less than the equilibrium value. They also
examined the relationship between the thermal conductivity and molecular weights
by measuring the conductivity of PMMA along the stretch direction and perpendicu-
lar to the direction with stretch ratios of up to 3.5, B—10834 with stretch ratios of up
to 3.9, and B—10836 with stretch ratios of up to 3.0. When these data were plotted
against the stretch ratio, the similar linear results were obtained. With an increasing
molecular weight, the anisotropy of the thermal conductivity become greater. They
explain these results using the theory of Hansen-Ho [17].

Equation 2.2 correlates the difference between these orthogonal components of the
diffusivity tensor with the difference in the corresponding Cauchy stress components,
where Cauchy stress is defined as a force exerted on a unit area of an object in the

current configuration. The magnitude of each component is not uniquely described,

17

however. The change in each component of the diffusivity appearing linear with
respect to the stretch ratio in the stretching direction suggests that the stress is also
linearly correlated with stretch, which typically occurs over only limited ranges of

stretches in most elastomers.

2.1.2 Multiaxial stretching

Several studies have examined the results of uniaxial loading, which provides part of
the data needed for constitutive modeling of thermal conductivity of elastomers. In
order to develop constitutive models suitable for general deformation, measurements
for elastomers subject to more complicated loading are needed. LeGall and Wright
[23] measured the change in thermal diffusivity of several elastomers that were subject
to biaxial stretching. In order to make these multiaxial measurements of thermal
diffusivity of elastomers, a computer-controlled Optical-thermomechanical system was
designed and developed [26].

LeGall and Wright [22] measured the three orthogonal components of the dif-
fusivity tensor during biaxial stretching of polyurethane, natural gum rubber, and
neoprene rubber. Rectangular specimans, 4.5cm x 4.5cm x 3mm, were stretched.
The polyurethane was initially isotropic and subject to equibiaxial deformation, where
both in-plane directions are stretched equally. The diffusivity measurement was made
in the central region of the specimen where the deformation was homogeneous. They
used an extention of the flash method [8] to measure the orthogonal components of the
thermal diffusivity tensor 0: simultaneously. In order to get repeatable thermoelastic

results, three specimens were subject to three different mechanical preconditioning

18

protocals and measured with stretch ratios of up to 1.83, 1.98, and 2.0. These results
showed an increased diffusivity in the direction of stretch and a decreased value in
the orthogonal direction, similar to the uniaxial results. In addition to measuring the
initially isotropic elastomers, they also measured the principal components of ther-
mal diffusivity of premanufactured natural gum rubber (N GR) and neoprene sheets,
which were anisotropic in the undeformed states due to manufacture processes. Their
findings also illustrated the Mullin’s effect and the differences due to proportional

stretch [22]. LeGall and Wright did not provide a correlation of their data.

2.2 Existing models for k or a

The literature reviewed in section 1.3 demonstrate that the thermal conductivity and
thermal diffusivity are functions of deformation and temperature level. Experimental
and theoretical studies have examined the dependency on temperature [6], [9], [10],

[12] [7], and on deformation [28],[17], [32].

2.2.1 Temperature dependence

Dashora [6] examined the temperature variation of the thermal conductivity of di-
electric linear elastomers by considering their molecular structure. At temperatures
above the glass transition temperature Tg, but below the melting temperature Tm,
the temperature dependence of k is controlled by the variation of the mean free path

of phonon that depends on the structural features of the system, in which structural

19

scattering is considered to dominate. The contribution of structural scattering to the

thermal resistance can be written as

W1 = A (2.3)

where W1 is the resistance related to k as k = 1/W1, A is a constant for a given
elastomer. In the rubbery plateau between T9 and Tm, some vacant sites are occurred
with similar effect as the point defects in crystalline solids. Therefore, a rise in
temperature leads to an increase of defect density, and consequently, an increased

thermal resistance that is linearly proportional to temperature, that is

W2 = ClT (2.4)

where Cl is a constant for a given polymer. Therefore, these assuming resistances are
in series, the relationship between the total resistance W = W1 + W2 and k = 1 / W

results in

l/k=A+ClT (2.5)

The predicted decrease in k with T, above T9, is consistent with the experiment

results of Eiermann [9, 10] and Hands [16] and with the analysis of Hands et al. [15].

2.2.2 Deformation dependence

Hansen and Ho [17] proposed that the thermal conductivity of linear high polymers

increases in the direction of deformation as the molecular orientation increases in the

20

direction of deformation and decreases in the orthogonal directions. This model is
based on a molecular analysis and assumes that a single segment in an amorphous
linear polymer has two chemically bonded neighbors on the same molecule, and is
surrounded by segments of other molecules. The segment is assumed to interact with
its nearest neighboring segments at frequency V1 and with neighboring molecules at
a frequency 112, (V1 > V2) and that the energy transferred by each interaction is
proportional to the temperature difference between the interacting segments. The
energy of each segment is assumed to be a function of the local temperature gradient
and position. Based on an energy balance on each segment, and given N segments

involved into the energy transfer, Hansen and Ho propose

k = (csv1p1/2v2/3N) Z «1. (2.6)

where cs is the heat capacity per segment, p1 a proportionality factor, I) the volume
occupied by the segment, and q, the macroscopic average energy through the segment,
which is assumed to be a linear function of the position. Although the theory of
Hansen and Ho [17] fits the experimental data well, there are difficulties in applying
this theory because V1, V2, p1, and q,- are not known, nor is it apparent how to measure
these parameters.

Washo and Hansen [36] developed equations, based on the analysis of Hansen and
Ho [17], relating the components of thermal conductivity of a uniaxially stretched
elastomer to its stretch ratio. Comparison of the calculated relative thermal con-
ductivity was linearly related to the average segment position, which was assumed

to behave as an affine deformation, allowing the thermal conductivity parallel to a

21

uniaxial stretching to be calculated to vary as
kII/keq — 1 = C (1, — 1) (2.7)

where kn is the thermal conductivity along the stretching direction and keg the
conductivity in the undeformed state. Then, assuming a uniaxial stretch and an
incompressible solid, the thermal conductivity perpendicular to the stretching would
vary as

kgg/keq — 1 = C (if/2 — 1) = C(Ag — 1) (2.8)

where 11:22 is the thermal conductivity perpendicular to the stretching direction and
[€33 = 1:22 and A3 = A2, because the material is assumed to be transversely isotropic.
Assuming that the principal directions of the diffusivity correspond to those of the

stretching, then we may rewrite equations 2.7 and 2.8 as

—k——I=C(F—I) (2.9)

keg

Plotting the experimental results obtained by Broerman et al. in terms Of kg,- versus
/\,-, as illustrated in Figure 2.2, in which R2 is the R—squared error when using least
square method. We see that a single constant, as suggested in equations 2.7 and 2.8,

is insufficient to describe the variation of diffusivity with stretching.

22

  
   
 

 

 

 

- a,da,,,vs A,
1 (122/0.qu k;
1.05 ~
a“ = 0.092411] + 0.9005
5 R’=o.9a41
22"
_1.00 ~
8'
A
‘ 022:0.1193241'03791
”5 R2=0.555
a.” 11x111111m1111111141411111m1111111111144414LJ
0.5 0.7 0.9 1.1 133-1 1.5 1.7 1.9 2.1 2.3
. M

Figure 2.2: The normalized components of thermal diffusivity of silicone rubber as
functions of their respective stretch ratios. Note the differing slope indicating that a
single parameter is insufficient.

Washo and Hansen contrast equations 2.7 and 2.8 with the results of Hennig [18],

who assumed a network of parallel resistances to write

%=(E)+(E)

and Hashin and Shtrikrnan (1963) who assumed a series network of resistances to

write

23

Washo and Hansen note that these four equations 2.7, 2.8, 2.10, and 2.11 give sim-
ilar results for small deformation. Hennig also related deformation to anisotropy by

writing

 

1/k11-1/keq _ 1 2 1
1/k22 —1/keq " 5N ’\1 A1 (2'12)

where N is a material parameter, and the right hand side is based on the neo-Hookean
description of rubber elasticity.

Broerman et al. [31] suggest using a stress-thermal rule to write

keg

where they define a as the difference in stress between the parallel and perpendicular
directions. Broerman et al. leave this in terms of the stress, but for uniaxial stretching,
with traction free lateral surfaces, and assuming a neo-Hookean material response,
this may be rewritten as

k - k 1
__..11 22 = 02 (A? _ Xi) (2.14)

where C2 is a new material parameter. If the material exhibits a Mooney-Rivlin

response with uniaxial stretching, equation 2.13 may be rewritten as

k11- k22 ( 2 1 ) 1
—-—— = C )1 - —- - C — - A 2.15

24

Interestingly, if equation 2.7 is subtracted from equation 2.8, then

kn - 1922 _ 1/2
keg _ 0 (A1 — 1/,\1 ) (2.16)

The term in parentheses in equation 2.16 is a root of the parenthetical term in equation
2.14.

These descriptions have been made in the context of the stress-thermal rule. This
has a limitation that only the differences in the components for a material with a
transversely isotropic response subject to uniaxial deformation are correlated. By
considering the stress-thermal rule for neo-Hookean and Mooney-Rivlin responses,
we see differences in these variations between these models. There is a clear need to
find a new description, preferably one that describes the variation of the individual

components.

25

Chapter 3

Theoretical Framework

Several correlations relating the change in diffusivity of solids with respect to finite
deformation have been proposed. None of these have been taken advantage of the
available results of continuum mechanics and finite elasticity. Here, the utility of
choosing the unloaded, isotropic state as the reference configuration is demonstrated.
This allows the thermal conductivity and diffusitivity to be formulated using principal
stretchs, Cauchy-Green strains, or the invariants of Cauchy-Green strains. Using these
measures of deformation should allow the consitiutive description of thermal transport

to be more easily generalized to multiple dimensions and complex deformations.

3.1 Choice of reference configuration

The thermomechanical response of a material must be characterized with respect to
a configuration or coordinate frame, which describes a region in a space that the
material occupies at a given instant. Each constitutive model is developed with

respect to a specific configuration. Models of the same material response might be

26

different when described in different configurations. One consequence is that the
choice of a configuration that describes symmetries for a material in that configuration
will introduce constraints on the form of the constitutive law that may be formulated.
That is, such symmetries give rise to mathematical restrictions on the constitutive
equations.

Here, we will describe the thermomechanical response of elastomers in a reference
and a current configuration, which are convenient for the constitutive and experimen-
tal descriptions, respectively. The reference configuration is here defined by the region
of space that the material occupies at reference time to, which is arbitrary, but is here
taken as the time at which an elastomer is undeformed and in thermal equilibrium
with its surroundings. In contrast, the current configuration is defined as the region
Of space that the material occupies at some time t > to, when the elastomer may
have undergone significant deformation or changes in temperature.

For materials subject to infinitesimal deformations, constitutive models of the
thermal response may be developed with negligible error by assuming that the refer-
ence and current configurations are indistinguishable. Elastomers, however, are Often
subject to large multiaxial deformation, and thus the variables must be described
separately in reference and current configurations. Elastomers are often thermally
isotropic when undeformed. In order to take advantage of this symmetry when form-
ing the constitutive model, we will describe the heat conduction of elastomers in an
undeformed reference configuration.

In following work, the notation will follow the conventions of finite elasticity (c. f.

Gurtin [14]).

27

3.2 Relationship Of q between configurations

Conservation of energy dictates the total heat transfer across the surface of a material
boundary is unchanged whether the material boundary is described in the current or

reference configurations. Thus,

/ <1<F.T,v:r).da= / solF,T.voT)-dao (3.1)
BB 630

where do and 1‘; denote the heat flux vector in the reference and current configurations,
respectively; X and x denote position vectors in reference and current configurations,
respectively; VOT and VT are the gradients of T with respect to X and x, respec-
tively; no and a are the area vectors of the object described in reference and current
configurations, respectively; Bo and B are the body described in current and reference
configurations, respectively; and 630 and BB are the boundaries of Bo and B, respec-
tively. The relationship between da and dag is da = JF’Tdao, where J = detF and
F = Bx/BX. Because Bo and B are arbitrary, equation 3.1 relates the descriptions

of the heat flux in the reference and current configurations with the local form of

61(F.T.VT) = -}F<io(F.T,VoT) (32)

Writing the heat flux in the reference configuration, using Duhamel’s equation

(equation 1.10), yields

do(F.T.VoT) = —I‘<o(F.T)voT (3.3)

28

where K0(F, T) is the thermal conductivity tensor in the reference configuration. The

heat flux in the spatial frame can be written similarly as
61(F,T. VT) = —K(F,T)VT (3.4)

where K(F, T) is the thermal conductivity tensor in the current configuration. Sub-
stituting from equations 3.3 and 3.4 into equation 3.2, K(F, T) is related to K0(F, T)
by

~ 1

K(F, T) = JFK0(F,T)FT (3.5)

Assuming elastomers to be volume preserving, as is commonly done, .1 = 1, which

simplifies equation 3.5 to

K(F, T) = FK0(F, T)FT (3.6)

3.3 Material Frame Indifference

Material frame indifference states that two Observers in relative motion make equiv-
alent mathematical and physical deductions about the macroscopic response of a
material. As a constitutive function, the heat flux function must satisfy the principle
of material frame indifference [5]. This requires that the functional form is the same

for different observers and hence,
q*(F*,T*, V*T*) = (“1(F", T“, VT") (3.7)

29

Q(t)X+¢(t)

x*(X,t)

Figure 3.1: Observer transformation diagram describing Material Frame Indifference

where the starred variables refer to the starred configuration in Figure 3.1.

Consider two Observers describing the same motion of an object as in Figure 3.1.
One Observer describes the motion of the object as x = :l:(X, t), while the other
describes the same motion of the Object as x“ = 23* (X, t).

For the first observer, the heat flux function is written as

For the second Observer, the heat flux function is written as

q* = q*(F*, T“, V*T) (3.9)

30

Since both Observers record the same motion, x and x“ are related to each other by

an observer transformation, meaning that
x* = Q(t)x + c(t) (3.10)

where Q(t) is a proper orthogonal tensor, with the time dependence henceforth un-
derstood, such that Q‘1 = QT, detQ = 1, and c(t) is a vector representing a

translation. Therefore, F“ is related to F as

_ 8r*(X, t) __ 6:1:*(X, fig: _

F _ ax ’Tax‘QF (3'11)

The temperature, being a scalar, has the same value for different observers, that is,

T * = T. The spatial gradients of temperature for the two observers are related by

,_W_mm_
V T— 5;,— _ axed —QVT (3.12)

The heat flux is the same for both Observers, which requires that
/ q*(F*, T", V*T) -da* =f q(F,T, VT) - da (3.13)
63" BB
recall equation 3.7

[63, o(1~*,1~*,v*1~) -da* = [63 c1(F.T. VT) . da (3.14)

31

where da“ 2 JQQda. Since JQ = det Q = 1, then da“ = Qda. Because a is arbitrary,

the local form of equation 3.13 may be written

q(F*, T*, V*T) = Qq(F, T, VT)

Substituting from equation 1.10 into equation 3.15 yields

—K(F*, T*)V*T = —QK(F, T)VT

Rewriting this in terms of the unstarred variables

K(QF, T)QVT = QK(F, T)VT

Therefore,

K(QF. T) = QK(F, TmT

(3.15)

(3.16)

(3.17)

(3.18)

Equation 3.18 demonstrates that K(F, T) is an objective tensor in current con-

figuration. Because equation 3.18 holds for any proper orthogonal tensor Q, we may

take Q = RT, where R arises in the right polar decomposition of the deformation

tensor F = RU. It follows from equation 3.18 that

K(F, T) = RK(U, T)RT

(3.19)

In essence, equation 3.19 states that the intrinsic properties of the thermal conductiv-

ity are not influenced by the rotational part of the deformation. Thus, the response

32

of the thermal conductivity to an arbitrary deformation tensor F is determined by

the knowledge of its response to the positive definite symmetric stretching tensor U.

This is even clearer when the relationship is written in the reference configuration.

For the second observer, with F“ = QF, K(F"', T“) = K(QF, T) may be written

in the reference configuration by using equation 3.6 such that

main T) = (QF)K0(QF.T)(QF)T

Substituting from equation 3.18 into equation 3.20 gives

QK(F,T)QT = (QF)Ko(QF.T)(QF)T

and thus

K(F, T) = FK0(QF, T)FT

Again, taking Q = RT and using the polar decomposition F = RU yields

K(F, T) = FKO(U,T)FT

Using equation 3.6 results in

K0(11‘17‘) = K0(U, T)

33

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

Taking Q = RT in equation 3.20 and remembering that, because U is positive definite

and symmetric, UT = U, then equation 3.20 may be expressed as

K(U, T) = UK0(U, T)U (3.25)

and therefore,

K0(U, T) = U"11‘{(U,T)U“l (3.26)

Since U is unaffected by an observer transformation [19], K0(U, T) is the same for
all the observers. Thus, the principle of the material frame indifference is embodied
in equation 3.26.

Because elastomers are assumed to be incompressibile with a specific heat that
is insensitive to deformation, the thermal diffusivity 0: changes in the same manner
as K, and the relationship between a and 00 has the same form as equation 3.26,

namely

a0(U, T) = U'1d(U, T)U-1 (3.27)

Experiments have Shown, as discussed in Chapter 1, that elastomers displaying
thermally isotropic responses in an undeformed reference configuration may demon-
strate anisotropic thermal responses in a current state that results from uniaxial
or biaxial loading. By describing the thermal conductivity tensor in the reference
configuration, we can use the isotropic response in the unloaded reference configu-
ration to simplify the development of the constitutive model. Thermal conductivity

or thermal diffusivity is measured in a deformed current state and all Of the existing

34

correlations are based on these configurations. Equations 3.26 and 3.27 provide a
transformation of the thermal conductivity and thermal diffusivity from a current to
a reference configuration, by which we can use the available experimental data to

build the constitutive model in an undeformed configuration.

3.4 Initially isotropic material

The isotropy of elastomers in the undeformed reference configuration imposes restric-
tions on the allowable forms of the conductivity tensor. The response of an isotropic
material is independent of the choice of reference configuration. Consider a reference
configuration with points in space denoted by X, and 23(X) is a mapping from this
reference configuration to the current configuration denoted by x = c(X), where x is
the spatial point vector in the current configuration. Another reference configuration
is related to the first reference configuration by a rigid-body motion as X" = QX+ c,
where Q is a rotation tensor and c is a constant vector. Note that in this case, Q
and c are independent of time. The mapping from this reference configuration to
the current configuration, expressed by x = 2:(X"‘). The relationship of these two
reference configurations is shown in Figure 3.2. The deformation gradient tensor may
be written
61:(X) = 6:1:(X*)0X*

F = ‘6"):— 6x* ax = F Q (3'28)

 

35

/(X)\’

QX+c

Figure 3.2: Two reference configurations related by a rigid body motion

because x = T(X) = :r(X*). For the scalar

T" = T (3.29)
and
,, _ 0T _ 6T 6X _
VoT — DY; — 5X5)? — QVOT (3.30)

Since a material response is independent of the reference configuration

(10(F,T, V0T) = (”10(1'", T”, VET) (3-31)

It follows from equations 3.28 to 3.31 that

600‘. T. mm = solFQT. T, QVOT) (3.32)

36

Using equation 1.10

oo(F.T.voT) = —I"<o(F.T)voT (3.33)

and

60(FQ,T.QTV0T) = 60(F*,T*.V6T)
= —K0(F*,T*)V3T (3.34)

= —Ko(FQT.T)QVoT

Therefore, substituting from equations 3.33 and 3.34 into equation 3.32, we have

Rom. T) = KolFQT. T)Q (3.35)

Now consider the left polar decomposition of the deformation gradient F = VR,
where V is a positive-definite symmetric tensor and R is the same rotation as in right

polar decomposition F = RU. By taking the special case of Q = R, we can write

K0(F, T) = K0(VRRT,T)R

= K0(v, T)R (3.36)

Recall equation 3.24 and compare it with equation 3.36, the result being

37

A

K0(U, T) = K0(V,T)R

= K0(RURT,T)R (3.37)

Thus, for an initially isotropic material, the response of the thermal conductivity in
the reference configuration is determined by knowledge of its response to the stretch-
ing tensor U, and furthermore RURT. In essence, equation 3.37 states that the
thermal conductivity described in the reference configuration is not influenced by
the rotational part of the deformation. Since R is an arbitrary rotation tensor and
U is a positive definite symmetric tensor, RURT can be expressed as the principal
components of U.

Assuming that A1,/\2, and A3 are the principal values of U and noticing that U
shares its principal values with V, then the referential thermal conductivity tensor

can be written as

K0(F, T) = K501. )2, As. T)R (3.38)

Then, we can write the thermal conductivity tensor in terms of invariants of C or
B, because C = U2 and B = V2 [14]. Moreover, the thermal conductivity may be

written in terms of the invariants of C as

K0(F, T) = Ko(11.12.Is.T)R (3.39)

38

where 11 = trC, 12 = §((trC)2 — trCZ), and I3 = detC are the three invariants of
C. Note that the invariants of B are equal to those of C.

The foregoing analysis demonstrates that the referential thermal conductivity ten-
sor can depend on the deformation through the invariants Of C or B, or A1,A2, and
A3 due to the isotropy of elastomers in their reference configuration. This eases the
analysis of experimental results for development of a quantitative constitutive formu-

lation.

39

Chapter 4

A Correlation Of Diffusivity

As stated in previous sections, Broerman et al. [2], Tautz [28], and Washo and Hansen
[36] have measured the thermal conductivity or diffusivity of initially isotropic elas-
tomers undergoing uniaxial stretching. Each of them noted a linear dependence of
thermal conductivity or thermal diffusivity on the stretch ratio. We will reexamine
these data based on the continuum mechanics analysis of isotropic materials, outlined
in previous chapters.

The thermal conductivity or diffusivity must be measured in the current con-
figuration. Formulation of the constitutive equation may be easier in the reference
configuration. Therefore, we must transfer the measured values of thermal conduc-
tivity or diffusivity to its reference configuration using either equation 3.23, 3.25, or

3.27.

40

Uniaxial or biaxial stretching yields a deformation gradient tensor with nonzero

values only along the diagonal. Thus,

 

 

A1 0 0
0 A2 0
0 0 A3
.l

(4.1)

where the Ai’s are the stretch ratios in the principal directions. The right Cauchy

deformation tensor then becomes

F
A¥00

C=FTF= 0 13 0

The left Cauchy deformation tensor, B = FFT, equals C in this case.

4.1 Uniaxial Loading

 

 

00A§J

(4.2)

For a uniaxial stretching with stretch ratio A1 in the direction of stretching, the

stretch ratios in the directions perpendicular to the direction of stretching are often

assumed to be equal, that is A2 = A3. Furthermore, elastomers are usually assumed

to be volume preserving during deformation [30], and thus, J = det F = A1A2A3 = 1,

which results in

41

(4.3)

Notice that the three principal values of C are now A2, A3 = 1 / A1, and A2 = 1 / A1.

The deformation gradient tensor for uniaxial stretch is thus

F= 0 INA—1 0 (4-4)

 

 

The components of C are then

2
A1 0 0
C=FTF= 0 1/A1 0 (4-5)
0 0 l/A1

 

 

The principal values of B equal those of C.

Equation 3.38 suggests that a useful relationship between the thermal conductiv-
ity or diffusivity and the principal values of C or B may be obtained. Two other
assumptions are used in the following analysis. First, we assume that the thermal
conductivity or diffusivity tensor is diagonal during the uniaxial deformation, i.e.,
that the principal directions of the thermal conductivity tensor coincide with those of
the deformation as characterized by C. Secondly, it is assumed that each component
of thermal diffusivity depends on the corresponding component of C. For example,
the (10,22 will depend on A2 only, and will be independent of A1 or A3.

Although Broerman et al. [2] observed a linear relationship between 011/ deg and
A1, and one between 0133/08), and A1 based on their data, as shown in Figure 2.1, it

may be reasonable to consider the relationships between 0111 /aeq and A1 and between

42

a33/aeq and A3. The relationship between agg/aeq and A2 is assumed to be the
same as the one between a33/aeq and A3, because the material is assumed to be
transversely isotropic along the direction of loading. Figure 2.2 shows that a single
parameter is insufficient to describe the relationship between (15,- and A,-. Thus, neither
a correlation using A1 only, nor a correlation based on A,- is sufficient to describe the
variation of a with respect to deformation.

Using C or B to describe the deformation has advantages as compared to A5. The
advantage of choosing C is that K(C, T) and d(C, T) automatically satisfy material
frame indifference because C is unaffected by the transformation of observers. The
advantage of choosing B is that K(B, T) and (“1(B, T) are independent of the choice

of reference configuration, because

B“ = F*F*T = FQT(FQT)T = FFT = B (4.6)

erthermore, since C = U2 and B = V2, combining with equation 3.37 yields

K0(C, T) = K0(B,T)R (4.7)

Equation 4.7 demonstrates that one can choose either C or B instead of F, as the
argument in the thermal conductivity tensor. Each one of them would automatically
satisfy material frame indifference and material isotropy. In the following work, we
choose C to describe the deformation.

Broerman et al. [2] measured a/aeq of silicone rubber in the current configuration.

Figure 4.1 shows the relation between a and C for the silicone rubber data from Figure

43

2.5 r

 

 

l
1 I “an/(1.1; V3 C11
2.0 ~ ‘
. 1 (Jules/or.q vs 033
l ‘1
i A
3... _ 1
1". » 5
8 .
. A
J L ‘
\ _ A
’5. 1.0 ~
8 .
I I
t I
t I
0.5 r ' I
. I
. . - .
* I
i
0.0 A n 1 l 1 1 1 1 1 4 1 L 1 l 1111111111111111111111111111111111
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
C11! C33

Figure 4.1: Variation of 010,1,- / aeq with respect to C;,- for silicone rubber subject to
uniaxial loading.
2.1. Here, the material is assumed to be transversely isotropic, with (122 = (133,
consistent with Venerus et al. [32] reporting only a single component of a in the
orthogonal direction.

The only nonzero components of (10 are 010,11 = allA-z, (10,23 = 0122A, 01033 =
a33A. The trend in Figure 4.1 suggests a form for a constitutive relationship between
ao/aeq and C. A least-squares fit of these data shows that both the stretched and

orthogonal directions apparently follow the same trend. This trend is described well

by
00,1'1'

b
—— .. 4.
deg -aCu ( 8)

44

where a and b are dimensionless material parameters.

In addition to inspection of silicone rubber, we examined the change in diffusivity
with stretching of several few other elastomers for which data are available. The
values of components of (10 / aeq were obtained from published results. The materials
and the stretching range for each material are shown in table 4.1. These values
were transformed to their respective referential frames, the components of a/aeq
apparently follow a similar trend with respect to the components of C. Although the
parameters for each material are different, as would be expected, the functional forms
are the same. Values of a and b are listed in Table 4.1 for the different elastomers
examined.

Suppose that the material undergoes no deformation, the deformation tensor F =
Q, which leads to C = FTF = I. In the referential state, we should have do,“ = aeq.
Therefore, equation 4.8 yields

00,11

= 1 = a03- = a (4.9)
aeq

Table 4.1 shows that, for all the tested polymers, (1 is approximately 1.0, as expected.

4.2 Biaxial Loading

While data from tests with uniaxial loading are sufficient to suggest a constitutive
model, other types of loading are required to assess the predictive capability of the

model. Two convenient forms of alternative loading are biaxial loading and simple

45

 

 

 

 

 

 

 

 

 

 

Material stretch ratio a b r2 T Ref
HDPE 1.00-4.60 0.8794 -0.669 0.998 100 K [4‘
HDPE 1.00-4.53 1.10 -0.5715 0.995 300 K 4]

Plexiglas 2.53-3.53 1.007 -0.908 0.9999 313 K :36:

PS 3.20-6.27 0.992 -0.988 1.00 313 K :36:

B-10834 1.74-3.79 1.00 -0.930 0.9999 313 K :36]

PS 2.35-3.32 1.01 -0.901 0.9999 313 K :36:

 

 

 

 

 

 

 

 

 

 

Table 4.1: Polymers subject to uniaxial loading correlated with 0033/0“, = a0]:-

shear. LeGall and Wright [23] measured the thermal diffusivity of several elastomers
subject to biaxial extension experiments. They used a variation of the flash method [8]
to measure simultaneously the three principal components of a of room-temperature
vulcanizing (RTV) polyurethane that was subject to homogeneous equibiaxial load-
ing, that is, both in-plane directions are stretched equally. This variation of the flash
system measures the temperature of the specimen in several places with little contact
of the specimen, which is required for measuring the thermal diffusivity Of materials
undergoing finite stretching.

Take directions 1 and 2 as the loading directions when polyurethane sheet is
subject to biaxial loading. The corresponding stretch ratios in these two directions,
A1 and A2, are measured, while A3 in the out-Of-plane direction is calculated assuming
J = 1 by A3 = 1/(A1A2), because the polyurethane is volume preserving during

deformation. Therefore, for such a material subject to biaxial loadng

(4.10)

 

 

46

and C can be calculated as

, .
2
A1 0 0
C=FTF= 0 Ag 0 (4.11)
2
. 0 0 1/A1Agj

 

 

Since the thermal diffusivity was measured in the current configuration, we again
transform it to referential configuration by using equation 3.27. Taking the thermal
diffusivity in the undeformed state as 084 and plotting the components of ao/aeq
against the corresponding components of C, there appears to be a single relationship
for the RTV polyurethane as in Figure 4.2. A least squares fit of these data suggests

a Similar trend as in the Figure 4.1 for uniaxial loading of silicone rubber.
9.0 -
- 00.11/11” V3 C11

. 011.2210... vs 022
. cross/a... V8033

(1033/1109
9
o

9'
o

(ion/Goo.
.“
O

 

 

5 3° r ‘
3 2.0 1 ,

10 L I

.- ~
9 CI
0.0 ‘ 1 Q
0 0.5 1 1.5 2 2.5 3 3.5 4
C11. C22. C33

Figure 4.2: Variation (00,5i/aeq) as a function of Ci,- for polyurethane subject to
biaxial loading. Specimens A1, A2, and A3 are shown.

47

Table 4.2: RTV polyurethane subject to biaxial loading 0031/0121; = a03-

 

 

 

 

 

 

 

 

 

 

Specimen a b r2 T ref
A1 0.928 -0.926 0.996 [22:
A2 0.931 -0.946 0.997 293K :22:
A3 0.778 -0.512 0.896 :22]

 

 

 

 

The parameters a and b for each of the three specimens for which results are
available are listed in Table 4.2. LeGall and Wright [22] cite the three specimens
separately because each of them had undergone a different thermal-mechanical pre-
conditioning protocol. Preconditioning of elastomeric specimens is required to obtain
repeatable thermoelastic results due to the Mullins effect [8]. Again, as with Table
4.1, the coeflicients a is approximately 1.0, as expected.

LeGall and Wright measured the diffusivity of two other types of polymers: com-
mercially available sheets of neoprene and natural gum rubber (N GR). These were
thermally anisotropic in their undeformed state, because of processing during their
manufacture. Table 4.3 shows fitting parameters of the initially anisotropic neoprene
subject to biaxial loading. It follows the same trend as polyurethane during biaxial
loading, as demonstrated by the similar values of 0.

Tables 4.1, 4.2, and 4.3 illustrate that when the normalized thermal diffusivity
is transferred back to its reference configuration, and C is chosen to be the variable
describing deformation, the relationships between the components of 00/016,, and C
follow a single trend. This is true for both uniaxial and biaxial loading.

Washo and Hansen [36] showed that the thermal conductivity of polymers is sensi-

tive to the molecular weight distribution of the polymer during stretching. Wineman

48

Table 4.3: Correlating coefficients for equation 4.8 for neoprene rubber that is
anisotropic in the unloaded condition and then is subject to biaxial loading

 

Specimen a b r2 T Rcf
B3 0.927 -0.978 0.994 293K [221

B4 0.938 -0933 0.995 293K 22

 

 

 

 

 

 

 

 

 

 

 

 

and Min [38] pointed out that cross-linking also influences the thermal conductivity
property of polymers. It can be deduced that both molecular weight distribution and
cross links would influence the values of b in equation 4.8. However, the quantitative

effect of these two factors on 0 remains an open question.

4.3 Discussion

The isotropic thermal diffusivity of undeformed silicone and polyurethane rubbers
becomes anisotropic when these elastomers undergo reversible finite strain. The re-
sults for uniaxial stretching [2] and biaxial stretching [23] imply that a is transversely
isotropic after homogeneous deformation. Figure 2.2 suggests that a single correlation
in terms of A5, nearly linear in stretch, may describe the change in diffusivity during
uniaxial deformation, both in the stretching direction and orthogonal to it.

While constitutive relations for thermal diffusivity that include the influence of
finite deformation could be developed using a number of coordinate frames, relating
a to 010 and comparing it with C, i.e., in the reference configuration, can aid in the
development of the constitutive relations because of material symmetry. The ability
of Equation 4.8 to correlate the diffusivity data is expected, given the relationship

between 010 and C. That is, if a were constant with stretch, then in the reference

49

configuration, 010 or C‘l. This is revealed in Table 4.1, where b w —1, the deviation
being the result of the actual change in diffusivity. The scatter in the measurements
is revealed by the value of a, which is essentially 1 for the uniaxial data, but deviates
from 1 by as much as 10% for the biaxial data. Thus, while care must be exercised
to avoid obscuring the small variations in a with deformation, both components of
a may be correlated in the reference configuration.

Figure 2.2 and Equation 4.8 further suggest that components of diffusivity are
described by the deformation in the direction corresponding to that component. De-
formation orthogonal to the uniaxial stretch or in the out-of—plane direction for the
biaxial stretch case was computed assuming that rubber-like materials are volume
preserving on deformation. This assumption has been well-justified based on mea-
surements that have shown that volume changes are of the order of 10‘4 or less during
deformation [16]. Based on this assumption of incompressibility, the components of
F are related by J = 1 and only one component of F is needed to characterize the
deformation for the uniaxial and equibiaxial cases. This suggests that the off-axis
terms are unlikely to contribute to changes in a for initially isotropic materials.

LeGall [22] also measured the principal components of a of natural gum rubber
(N GR) and neoprene that were anisotropic in the undeformed state due to manufac-
turing. The in-plane components of a were 5 and 6% higher than the out-of-plane
coefficient for the N GR and neoprene, respectively. Materials with initial anisotropy
violate the assumptions leading to equation 4.8. Nevertheless, the response of these
materials, shown by the values of a and b for the neoprene listed in Table 4.3, is

similar to those listed in Table 4.1.

50

A simple relation between finite deformation and the change in diffusivity of elas-
tomers has been presented. Clearly, further study is required to clarify the relation-
ship between a and deformation for elastomers and other polymers. Both molecular
weight [36] and cross-linking [27] have influence on the thermomechanical response
of elastomers. Their role in the thermophysical constitutive behavior needs to be

examined.

51

Chapter 5

A Constitutive Relation for q

Duhamel’s equation (equation 1.10) and its isotropic form Fourier’s equation (equa-
tion 1.11) are phenomenological laws that have been developed based on observation
and measurements. As discussed above, several methods have been developed to
correlate changes in diffusivity due to deformation. These correlations may be un-
suitable for general deformations. In order to include deformation as an independent
parameter in Fourier’s law, Wang [33, 34] developed a generalized Fourier’s law suit-
able for Newtonian fluid flows. No such study has been done for solid materials,
specifically elastomers. The current study will examine the form of the relationship
between heat flux and deformation based on rigorous analysis, without the limitation
to small deformation. This will be applicable for elastomers, which are widely used in
industrial applications and which Often undergo large deformation. By satisfying the
second law of thermodynamics, satisfying material frame indifference, and putting
the constitutive function in an undeformed reference configuration, a detailed form

for q is derived.

52

5.1 Methods

5.1.1 Material frame indifference

As stated in section 3.3, as a constitutive function, heat flux function must satisfy
the principle of material frame indifference. Here, we only consider elastomers with
initially isotropic thermal conductivity. In order to take advantage of this isotropy,
each of the variables and the constitutive function are described in the reference
configuration. We will see some limitations that arise from the principal of material
frame indifference on the form of heat flux function in the reference configuration.

Recall that the heat flux vector in the reference configuration is written

<10 = 51007.7", VoT) (5-1)

If two observers record a motion of the same body as T(X, t) and 93*(X, t), then

9:(X, t) is related T*(X, t) by observer transformation, which is

93*(X) = C211000 + c(t) (5-2)

where Q1 is a proper orthogonal rotation and c(t) again represents a translation.

 

Therefore,
092*(X)
F 0X Q1 ( )
The scalar temperature transforms as
T" = T (5.4)

53

and the referential gradient as

6T" 8T

V0T=55=fii

= VQT (5.5)

Recalling equation 3.15, material frame indifference is satisfied when

60(F,T, VoT) = Qffio(QIF,T1V0T) (5.6)

5.1.2 Three linearly independent variables

Consider the polar decomposition of F = RU, for which U is a symmetric tensor
that has three real eigenvalues that represent the principal stretching ratios and R is
a rigid-body rotation. U is also called the pure stretching tensor. Here, the case of
three distinct eigenvalues is considered.

Lemma. VOT, U'lVoT, and U’ZVOT are linearly independent variables.

Proof. Assume Ak and fk (k = 1, 2, 3) are the eigenvalues and eigenvectors of U.

Then, U can be represented as

3
U = Z Akfk ® fk (57)
16:1
and, according to [14],
3
U‘1 = Z Aglfk ® 1“,, (5.8)
k=1
and
3
U-2 = Z A;2fk ® fk (5.9)
10:1

54

The vector VOT can be written using fk as its basis such that

VOT = T ,1), (5.10)

where the subscript ,1: indicates differentiation in the k—direction. Using equations
5.8 and 5.9,

U‘IVOT = Aglrkrk (5.11)

and

U'ZVOT = Agzrkrk (5.12)

If VOT, U-IVOT, and U'2V0T are linearly dependent variables, there exist

0; 75 0 (2' = 1,2,3) such that

ClTchk + CgAng’kfk + 63A,:2T1kfk = 0 (5.13)

Since this is true for arbitrary Ak, (3,-(2' = 1, 2, 3) must be 0. Therefore, VOT, U‘lVoT,
and U-2V0T are linearly independent vectors.

Then, the heat flux vector can be expressed using the basis of VOT, U'IVOT,

and U’2V0T as

(“10(F1T,V0T) = ¢1(F, T, VOT)VOT + ¢2(F. T. VoT)U-1V0T

+ ¢3(F, T, V0T)U_2V0T (5.14)

where ¢;(F, T, VOT) are functions whose values are the components of qo.

55

5.1.3 Application Of second law of thermodynamics

The second law of thermodynamics [33] requires

é10(1“.T.-V0T) = -<10(F.T.V0T) (5-15)

Using equations 5.14 and 5.15, the heat flux vector function may be written

€10(F1T1 -V0T) = ¢1(F1T1—V0T)(_V0T) + ¢2(F1T1 —V0T)U—1(—V0T)
+ ¢3(F. T. -V0T)U—2(-V0T)
= -¢1(F. T. V0T)(V0T) — ¢2(F.T. voT)U‘1voT

- 93(F. T. VoT)U‘2VoT (5.16)

Comparing the coefficients in equation 5.16, ¢k(F, T, —V0T) = ¢k(F, T, VOT) for k =

1,2,3. A special form, which satisfies this property, can be written as

¢k(F. T. VOT) = 7.0110“ . T. V0T ® VOT) (5-17)

Therefore, we may write

61(F. T. VoT) = 210?. T. VoT 8 voT)voT + 22(F. T. voT 8 voT)U‘1voT

+ 1123(F, T, VOT <3 VOT)U‘2V0T (5.18)

56

5.1 .4 Material Symmetry

The isotropy in the reference configuration may be used to restrict further the con-
stitutive function q0(F, T, VOT).

Consider a new reference configuration X“ = Q2X + c(t). For an isotropic ma-
terial, qo is independent of the choice of reference configuration as was demonstrated

in section 3.4. Following the similar development yields

510(F. T. VoT) = (10(F". T". VFT) (519)

where T" = T, F" = FQT,and V3*T = Q2V0T, therefore,

510(F. T. VoT) = qolFQf. T. Q2V0T) (5.20)

5.2 Analysis

5.2.1 Heat flux vector in terms of U

By considering both material frame indifference and material symmetry, and referring
both equation 5.6 and equation 5.20, the heat flux vector function q0(F, T, VOT) can

be rewritten as

616(F.T.voT) = Q?so(Q1F.T.VoT)

= ofqolczlmil‘rcsvoT) (5.21)

57

We can now use the polar decomposition F = RU such that Q1FQ; = QlRUQg.
Because Q1 and Q2 are arbitrary, we can define a rotation tensor Q = Q1R and

Q2 = Q1R such that

01m? = QUQT (5.22)

Thus, the deformation dependence of q is independent of the rotation. Then, the

heat flux function may be rewritten as

6100“. T. VoT) = RQT90(QUQT. T. QVoT) (523)

Because Q is an arbitrary tensor, it may be taken as Q = I, with the result that

510(F1T. VoT) = Rd0(U. T. VoT) (5-24)

Therefore, the heat flux vector can be expressed as

610(U. T. VoT) = QTflo(QUQT. T. QVOT) (5.25)

5.2.2 A special property of 1,01, 1L2, and 1/23

Again using the property of the second law of thermodynamics such that

510(U.T. -V0T) = -€10(U.T.V0T) (5-26)

58

we can express the heat flux vector in terms of three linearly independent vectors as

("10(U1T. -V0T) = ¢1(U.T1—V0T)(-V0T)
+ ¢2(U,T,—V0T)U‘1(—V0T)
+ ¢3(U, T, —V0T)U‘2(—V0T)
= —¢1(U.T.V0T)(V0T)
— (02(U,T,V0T)U’1V0T

— ¢3(U,T, VOT)U'2V0T (5.27)

Therefore, the coefficient functions ¢k(U, T, —V()T) and ¢k(U, T, VOT) are equal for

k = 1, 2, 3. A special form, similar to equation 5.17, that satisfies this property is

¢k(U1T1V0T) = “(11.7". VOT ‘8 VOT) (5-23)

Equation 5.27 may be thus rewritten in the form

510(U. T. VOT) = 1/11(U. T. VoT ® VOT)V0T
+ ¢2(U, T, VoT 8 VOT)U‘1V0T

+ 423m. T. voT 8 VoT)U'2VoT (5.29)

Equation 5.29 differs from equation 5.17 in that the argument characterizing the

deformation is now U, which is independent of rotation in 5.29.

59

Following the same rule in the rotated frame,

90(QUQT.T.QV0T) = 2121(QUQT,T,QV0T®QV0T)QV0T
+ 22(QUQT.T. QV0T8QV0T)QU‘1V0T

+ ¢3(QUQT.T.QV0T®QVoTiQU'2V0T (5.30)

Substituting from equations 5.29 and 5.30 into equation 5.25 yields

81 (U. T. VOT 8 VOT)V0T

+ ¢2(U, T, VOT 8 VOT)U'1V0T

+ 1143(U. T. VoT 8 VoT)U"'2V0T

= 0"" [¢1(QUQT. T. QVoT 8 QVoT)QvoT
+ ¢2(QUQT. T. (NOT 8 QVOT)QU-1VOT
+¢3(QUQT. T. QVoT 8 QVOT)QU’2V0T]
= w1(QUQT. T. QV0T 8 QVOTWOT

+ zileUQT. T. QVOT 8 QVOT)U-1V0T

+ws(QUQT.T.QvoT8QVoT)U‘2voT (5.31)

60

Because VOT, U-lVOT, and U’2V0T are linearly independent, this results in

101(U. T. VoT 8 VoT) = 141 (QUQT, T, QVOT 8 QVOT)
(low. T. voT 8 mm = ¢2(QUQT. T. QVOT 8 QVoT) (5.32)

2123(U.T.V0T®V0T) = 413(00on QvoT8QvoT)

5.3 Invariants

Next, we can define invariants to characterize further the form of the constitutive

equation. Wang [34, 33] proved a lemma that states that suppose
(MT. Q8 8 0g. 090‘") = «AT. s 8 s. G) V8 and G. (533)

then,

MT. 8 <8 a. A) = 10(T . 3 ® 8. G) (5-34)

whenever Jk(a, A) = Jk(g, G) (k = 1,2, . . . ,6), and where

J1(a,A) = trA J2(a, A) = %[(trA)2 — tr(A2)]
J3(a, A) = detA J4(a, A) = a - Aa

J5(a, A) = a - A2a J6(a, A) z] a |

and a and g are two arbitrary vectors and A and G are two arbitrary symmetric

tensors.

61

Therefore, for an initially isotropic material, by starting from the principle of
material frame indifference and the second law of thermodynamics, the heat flux
vector function can be expressed as

sow, T. VoT) = («AI + 102U"1 + 1113U—2)V0T (5.35)

where

w,=1/3,-(T,J,,(V0T,U)) (1:123 k=1,2,3,...,6) (5.36)

Equation 5.36 indicates that the heat flux vector is proportional to the gradient of
temperature in the reference configuration. The coefficients depend on the symmetric

tensor U through the invariants of U.

5.4 Data Analysis

Comparing equation 5.35 with equation 1.10, we can define K as

K = —(¢1I + ngU'1 + 1,03U‘2) (5.37)

Since U is a positive definite symmetric tensor, K is also a symmetric tensor. In the
special situation in which U is the unit tensor, which means that the Object undergoes

rigid body rotation only, equation 5.37 can be reduced to

K = ’(1P1 + 1/12 + $3” (538)

62

Writing the scalar k = 1111 + $2 + 1P3, equation 1.10 may be rewritten as

q = —kVT (5.39)

which is the classic Fourier’s law of heat conduction [19], where k denotes the coef-
ficient of thermal conductivity, and the heat flux and the temperature gradient are
with respect to the spatial fame.

As stated above, some studies have explored the relation between K and deforma-
tion [2, 22, 36]. Here, we would like to use these data to examine the new form of K
as a function of the pure stretching tensor U. In each study, the thermal conductivity
or diffusivity has been measured in its current configuration. Thus, the first step is

to transform these data back to the appropriate (i.e., undeformed) reference state.

5.4.1 Analysis of the uniaxial stretching of elastomers

Consider first the special situation of uniaxial stretching. As described in Section 4.1,
assume that Al is the stretching ratio in the loading direction and that the other two

directions have stretching ratios of A2 and A3. Then the stretching tensor U is

A10 0

U= 0 A2 0 (5.40)
A
L0 0 3;

 

 

For uniaxial stretching with a volume preserving material, the stretch ratios in the
orthogonal directions are equal, and the volume preservation yields A3 = A2 = 1/1/A1.
Therefore, U becomes

63

 

 

 

 

U= 0 INA—1 0 (5-41)

0 0 INA—1
and

PA;1 0 0 -

U‘1= 0 yr, 0 (5.42)
. O 0 NJ
PA;2 0 0 -

11-2: 0 A1 0 (5.43)

0 0 A1

. 1

 

 

Consider again the data shown in Figure 2.1 for silicone rubber that is uniaxially
stretched. Again, these data are transferred from the current configuration to the
undeformed reference configuration using equation 3.26, and using the stretching
ratio to calculate U, U“, and U"'2 via equations 5.41 and 5.43. A least squares
fitting yields coefficients oft/21 = 0.0158, $2 = -—0.132, and 1/23 = —0.879. Recall
equation 5.37 and that the negative of the coefficient corresponds to the conductivity,
Figure 5.4.1 Shows that the experimental data fit the curve well, with an error Of

132 = 0.9997.

64

(a)

ity tensor Components
N
N '01

.5
0!

.°
01

 

Normalized Thermal oonductiv

0.5 1 1.5 2 2.5
The Components of the Pure Stretching Tensor

o0

Figure 5.1: Normalized thermal conductivity tensor as a function of the pure stretch-
ing tensor for silicone

Therefore, for this crosslinked silicone rubber, the normalized thermal conductivity

tensor can be expressed in terms of U as

k}:— = —0.01581 + 0.132U‘1 + 0.879U‘2 (5-44)
69

When the object is undeformed, the pure stretching tensor becomes U = I. Therefore,

3 = 0.9951 (5.45)

keg

which means that K is an isotropic tensor, with a coefficient of thermal conductivity
0.995/seq. In this experiment, the specimen is initially isotropic in thermal conductiv-

ity, and its initial coefficient of thermal conductivity is keg according to the definition.

65

Therefore, the description of equation 5.44 is in good agreement with thermal conduc-

tion of crosslinked silicone rubber, introducing 0.5% error at the undeformed state.

5.4.2 Analysis Of biaxially stretched elastomers

LeGall [23] measured the thermal diffusivity of several polymers that were subject
to biaxial loading. He measured the three orthogonal components of diffusivity si-
multaneously during biaxial stretching of polyurethane, natural gum, and neoprene
rubbers. Here, we examine the polyurethane, RTV data.

For polyurethane RTV that is subject to biaxial loading, assume that the stretch
ratios of the in-plane directions are A1 and A2, and that the out-of-plane direction is
A3. Assume A3 = 1/(A1A2), since polyurethane RTV is volume preserving during de-

formation, and A1 = A2 since it is equibiaxial loading. Therefore, the pure stretching

 

 

 

 

tensor U is _ _
A1 0 0
U = 0 A2 0 (5-46)
0 0 1/A1A2 .
and U”1 and U"2 are _ q
A;1 0 0
11-1: 0 12-1 0 (5.47)
0 0 A1A2
. A;2 0 0
11-2 = 0 1;? 0 (5.48)
0 0 A'f’Agz

 

 

.3
o

NW‘UIQNQD

A
1'

Normalized Thermal conductivity tensor Components

 

0.5 1 1.5 2 2.5
The Components of the Pure Stretching Tensor

Go

Figure 5.2: Normalized thermal conductivity tensor as a function of pure stretching
tensor for polyurethane RTV

Again the thermal diffusivity is transformed from the current (measurement) config-
uration to the referential (undeformed) configuration by using equation 3.5, and U,
UT], and U’2 are calculated using equations 5.46 and 5.48. In order to get com-
parable results with those Of the uniaxial stretching, define the referential thermal
conductivity as keg to normalized the thermal conductivity. Using a least squares
fitting yields the coefficients 1121 = 0.372, 1,02 = —0.858, and 1123 = —0.501. Figure 5.2
shows that the experimental data fit the curve well, with an error of R2 = 0.998.

Therefore, for this biaxial stretched polyurethane RTV, the normalized thermal con-

ductivity tensor can be expressed in terms of the pure stretching tensor as

E. = —0.37241 + 0.8584U—l + 0.5008U"2 (5-49)

67

Table 5.1: Polymers subject to biaxial loading with form a/aeq = —(¢1I + ng‘l +

 

 

 

 

 

$3U’2)
Sample 1111 we «is “/0611 82
for U = I
Polyurethane A1 0.372 -0.858 -0.501 0.987 0.998
Polyurethane A2 0.349 -0.724 -0.609 0.984 0.999
NeOprene B3 0.582 -1.140 -0.411 0.970 0.999
Neoprene B4 0.435 -0.964 -0.437 0.967 0.999

 

 

 

 

 

 

 

 

when the specimen is undeformed, the pure stretching tensor becomes U = I. There-
fore,

3— = 0.9881

(5.50)
which means that a is an isotropic tensor, with a coefficient of thermal conductivity
09880.3(]. The polyurethane RTV specimen was initially isotropic in thermal diffusiv-
ity, and its initial coefficient of thermal diffusivity is aeq according to the definition.
Therefore, the description of equation 5.49 is in good agreement with thermal dif-
fusivity of polyurethane RTV. LeGall [22] also measured the thermal diffusivity of
commercially available neoprene rubber that was anisotropic in its undeformed state,
as a result of manufacturing process. Table 5.1 lists coefficients for two neoprene rub-
ber specimens, as well as an additional polyurethane specimen. In the special case
as U = I, all of the results predict the undeformed diffusivity within 3% of initial
condition of the thermal diffusivity. Particularly, for specimens Polyurethane A1 and
Polyurethane A2, the coefficients 1121, 1/22, and 1123 are within about 3%, and for spec-

imens Neoprene B3 and Neoprene B4, the coefficients close, although are different

from polyurethane RTV specimens.

68

Chapter 6

Discussion

Considering that heat transfer can only occur from material at a higher temperature
to material at a lower temperature, due to restrictions imposed by the second law of
thermodynamics, Fourier’s inequality must be satisfied. The Fourier inequality may

be written,

—KVT- VT 3 0 (6.1)

which means that

K 2 0 (6.2)

thus requiring K to be a semi-positive definite tensor. Equation 5.37 demonstrates
that the thermal conductivity tensor can be written in terms of U as K = —(1p11 +

ng‘l + 1/22U'2). Therefore,

-(¢1I + «WU—1 + (AU-2) 2 0 (6.3)

69

In the case of silicone rubber undergoing uniaxial loading, a specific formula (equation
5.44) has been found to demonstrate the relationship between the normalized thermal
conductivity and the deformation tensor U. It is necessary to Show that the scaled
conductivity K / Iraq is positive.

Again assuming that the stretch ratio in the direction of stretching is A1 and the
stretch ratio in the normal direction Of stretching is A2. The normalized thermal

conductivity in the direction of stretching K11 /keq is

K11

 

= f1(A1) = —0.0158 + 0.1321A1’1 + 0.8786A;2 (6-4)

and the normalized thermal conductivity in the perpendicular direction of stretching

K22 /keq is
K 22
keq

 

= f2(A2) = —0.0158 + 0.1321A;1 + 0.8786A52 (6.5)

where f1(A1) and f2(A2) are introduced for convenience of description. For equations

6.4 and 6.5, we do the first derivative and have

f{(A1) = —0.1321A;’2 — 1.7572A1’3 (6.6)

f§(A2) = —0.1321A2‘2 — 1.7572A2-3 (6.7)

Because A1 and A2 are always positive, we have

film < 0 (6.8)

70

and

f2(/\2) < 0 (6.9)

Therefore, f1(A1) is a monotonically decreasing function of A1 and f2(A2) is a monoton-
ically decreasing function of A2. Notice that when A1 3 12.7, K 11 / keg = 0. Therefore,
when A1 < 12.7, K11 /keq > 0. Treloar [30] states that the maximum stretch ratio of
silicone rubber generally falls below 10 suggesting that streches over 12.7 are unlikely,
which ensures that the K11 > 0 since keg > 0. These values would be expected to
vary with specific material formulation.

Considering that the physical meaning of A1 makes A1 2 1.0 and A2 = 1 / JA— , we

have A2 S 1.0. Therefore, f2(A2) has the minimum value when A2 = 1, which is that

f2(A2) = —0.0158 + 0.132 + 0.879 = 0.995 (6.10)

therefore, Kgg/keq > 0.995 > 0. Since keg > 0, then K22 > 0. This guarantees that
K is a positive definite tensor.

In deriving a general constitutive relation q as in Chapter 5, three linearly in-
dependent vectors, VOT, U’lVoT, and U’zVoT, were carefully chosen based on
the analysis of the existing models in Section 2.2.2 of Chapter 2 and the correla-
tion from Chapter 4. When another set of three linearly independent vectors (i. e.
VoT, UVOT, and U2V0T) was chosen as the basis of heat flux vector function q,
one component of thermal conductivity tensor was a parabolic function of U, which
yields inaccurate representation of practical situation. Special cases of fluid flow of

polymers may have this situation [33, 34]. Wang derived a generalized Fourier law

71

to extend the classical Fourier law to the processes of heat transfer in fluids that
involves macroscopic relative motion between two sides exchanging heat, in which q
is a function of velocity gradient tensor L. By using material frame indifference and

the second law of thermodynamics, an explicit form of K is obtained as

K = —(¢OI+¢1D+¢2D2) (6.11)

where D is a symmetric part of L.

A special form of D such that

c(t) 0 0

D= 0 0 0 (6.12)
0 0 0
. .l

 

 

where c(t) is a scalar function of time t, describes the Situation for a convective velocity
changing only in one direction. The thermal conductivity can be found by referring
to equation 6.11. The nonzero components of K are the diagonal components, and

kn is related to c(t) as

kn = -(¢0+¢16(t) +¢26(t)2) (613)

Therefore, (1:11 is a parabolic function of c(t). As c(t) varies, kn reaches its extreme
value when c(t) = -q>1/(2 :1: (02).
In the current study, by referring the theoretical and data analysis in chapter 4, one

set of linearly independent vectors, VT, U-lVT, and U'2VT, is chosen carefully

72

to express q. We developed an explicit form of K as a function of U for solids
in a systematic, rigorous way by using the principle of material frame indifference,
second law of thermodynamics, and properties of material symmetry in a reference
configuration. By examining the form of equation 5.37, which is derived for both
uniaxial and biaxial situations, the form is seen to fulfill the initial condition of those
specimens, and fits the experiment data with small error.

Chapter 4 develops a correlation by using phenomenological arguments and least
squares fitting of experimental data. How a or K is related to the deformation ten-
sor is determined by the curve fitting of the results from uniaxial stretching [2] and
biaxial stretching [23]. This method is taken in their respective reference configu-
rations (undeformed configuration) to Simplify the constitutive function since those
Specimens possess material symmetry in their reference configuration. After exam-
ining those data in the reference configuration, figures 4.1 and 4.2 show that each
component of a takes the same functional form of its respective stretch ratio, which
verifies the material symmetry in the reference configuration and leads to a simple
constitutive relation for which (10 is approximately inversely proportional C. These
results demonstrate that for an initially isotropic material, if the constitutive relation
is developed in the reference frame, one functional form can be sufficient. It is, there-
fore, unnecessary to separate the diffusivity in the direction of the situation from the
component of the diffusivity in the direction orthogonal to the stretching so that two
different linear relations are used for those two directions.

In the correlation of (10 with respect to 0'1, two parameters are introduced as in

equation 4.8 and demonstrated in Table 4.1 and 4.2, where a is 1 for the uniaxial data

73

but deviates from 1 by as much as 10% for the biaxial data and b is approximately —1
with differences for the different materials, as expected. By considering the special
case of equation 4.8 with C = I, meaning that a specimen is undeformed, and a
is essentially 1. The deviation may come from the scatter in the measurements.
On the other hand, transformation to the reference configuration may cause hide
experimental scatter, which needs to be carefully examined. Other possible factors
that influence the thermal diffusivity or thermal conductivity are the molecular weight
and cross-linking density of the material, which have been examined experimentally
in the studies of Washo and Hansen [36] and Rivlin and Saunders [27]. These factors
may contribute to the change of b, which warrants further study.

Based on this phenomenological study, it becomes possible for the work of chapter
5, in which one general functional form is suggested and developed in a systematic,
theoretical framework. An explicit form of K is derived in a mathematical way. The
coefficients 1/21, 1,02, and 1,03 are decided by those invariants as described in equation
5.36. Although these coefficients can be decided by curve-fitting the experimental
data, the physical meaning and an explicit form of Jk(V()T, U) still need further

examination.

74

APPENDICES

75

Constitutive functions for elastic materials

In Chapter 1, we cited the thermodynamic analysis of Coleman and Gurtin [5] to
note that the thermomechanical response of an elastic material is generally described
by four constitutive functions. We also noted that the four equations can be reduced
to a smaller set by using the principle of material frame indifference and restrictions
due to the second law of thermodynamics. This analysis of these simplifications is
now considered routine enough that a specific reference need not be cited, but it is
included here for completeness.

Recall equations 1.1, 1.2, 1.3, 4 from Chapter 1, but now written in terms of a

reference configuration as

e = é(F, T, VOT) (1)
n = 6(F. T. VOT) (2)
T = ‘ (F,T,V0T) (3)
q = 609‘. T. VOT) (4)

These four constitutive functions must satisfy the principle of material frame indif-
ference, meaning that the functional form of the constitutive relations for different
Observers should be the same. A description of Observer transformation is demon-

strated as Figure 3.1 in Chapter 3.

76

To demonstrate how the principle of material frame indifference may be used to
develop functional forms, consider the example of the Cauchy stress function T. The

Cauchy stress is introduced by Cauchy’s stress theorem, which states that

t = Tn (5)

where t is the traction force and n is the unit direction associated with the traction
force. For a second Observer, whose current frame is rotated by Q with respect to
that of the first observer, we have t“ = Qt and n“ = Qn. Therefore, with t“ = T’n"

and t* = Qt we can write

T*n* = QTn (6)
and due to 11* = Qn

T'Qn = QTn (7)
Therefore,

1* = QTQT (8)

' If the two observers have the same reference frame, the parameteric variables are
related by T" = T, VEST = VoT, and F“ = QF. Material flame indifference requires

the constitutive functional forms to be the same, which leads to

T* = T(F*, T*, V5T)

TlQF.T.voT) (9)

77

Substituting flom equations 9 and 3 into equation 8 leads to

'i‘(QF.T. mm = 0T(F.T. voT)0T (10)

Similar substitutions for the other constitutive functions yield the following relations

é(QF, T, VoT) = 5(F, T, VoT) (11)
fi(QF. T. VOT) = 1“7(1".T.'\70T) (12)
Q(QF. T. VoT) = QQ(F.T. VOT) (13)

In addition to satisfying the principle of material flame indifference, the constitu-
tive functions must satisfy the second law of thermodynamics. The second law may
be written as the Clausius-Duhem inequality [19] expressed as

F-la - VOT

110(5‘ — T13) — JFTT -FF'1 + JF T

S 0 (14)

where p0 is the mass density described in the reference configuration and J = det F.
Introducing the specific Helmholtz free energy, ii) = e — T7), the Clausius-Duhem
inequality may be rewritten as

q - F-TVOT

p(15+T'fi) —TT.FF‘1 + T

S 0 (15)

 

78

where p = p0 / J is the mass density in the current configuration.

Assume that at an initial time to, a fixed point X0 in the reference configuration
has following state: T(X0,t0) = a, T(X0,t()) = fl, VT(X0, to) = a, VT(X0,t()) = b,
F(X0,t()) = A, F(X0,t0) = B, where a is a positive number and A is a positive
tensor. Otherwise, a, [3, a, b, A, B are arbitrary. Fields of x and T(X, t) that could

satisfy the initial state are

x(X,t) = X0+F(t)(X—X0) (16)
with F(t0) = A

and F(t0) = B

and

T(X, t) = aexp [EU — to) + (a + (t — t0)(b — a?” - (X — X0)/a] (17)

The initial state of the point must also satisfy the second law of thermodynamics.

Substituting flom the initial conditions above into equation 15 results in

fi(x01t0)l1/3T(ai a) A)fl + 113V0T(aa 3) A) ' b + 523F011 31 A) ° B

+9002. a, A)] — TT(o, a, A) . BA-1 + $6191. a, A) . A-Ta g 0 (18)

Equation 18 must hold for all a, B, a, b, A, B. First fix a, B, a, A, B. Then dividing

through by p(X0, to), equation 18 may be rewritten as

¢V0T(0181A) + f0(a.fl.a.A.B.P0(X0)) S 0 (19)

79

which must hold for all b. Equation 19 can be violated unless
¢V0T(ar 8) A) ' b = O (20)

which shows that

TVOTWJ. A) = 0 (21)

which is true for all a,a, and A. It follows that the constitutive response function

A

1M0, a, A) is independent of the temperature gradient VOT and we may write

8 = T(F. T) (22)

Returning to the inequality in relation 18 and making used of equation 20, we may

write

8x6. to) [1131(0. Am + (318. A) - B + 968. a. A)]

-TT(o,a, A) . 13A-1 + $618. a, A) . A-T a g 0 (23)

which must hold for all a, B, a, A, and B. By assigning fixed values to a, a, A, and

B, equation 23 requires

(3(X0. to) [$18.19) + 6(a.a. A)] 9 + go(a.a. A. B) s o (24)

80

for all values of 6. Equation 24 can be violated unless
iffla. A) + (761.8124) = 0 (25)

This says that that constitutive response function 1'7(T, VOT, F) is independent Of

VOT. Moreover,

A

fi(T1V0T) F) = _wT(Tv F) = fi(T) F) (26)

Again returning to the equation 23, and employing this new result, we have
p(X(), t0)rLF(a, A) - B — TT(a,a, A) - BA"1 + imam, A) - A—Ta _<_ 0 (27)

which must hold for all a, a, A, and B. Now assigning fixed values to a, a, and A

requires, in order not to violate the inequality by the choice of B, that
[p(xo. to)(i1.~(a. A) — TT(a.a. A)A'T] - B = 0 (28)

and that

$901. a, A) - A-Ta s 0 (29)

Equation 28 requires the constitutive response function T(T, VOT, F) to be indepen-

dent of VOT and given by

T(T, VOT, F) = pFr/3F(T, F) = T(T, F) (30)

81

A more general result flom inequality 29 may be developed by assuming that

61(0, a, A) is a continuous function of VOT and writing

where 0(a) means that lim|al_,0 O(a)/|a| is bounded. Then, from inequality 29 we

have

[A‘19(a. 0. A) + 0(a)] - a s o (32)

which must hold for all a, a, and A. If we divide through by la] and take lim |a| —> 0,
we then get

41-16(81). A) - e = 0 (33)

for all unit vectors e. Thus, A’lffia, 0, A) = 0, which implies that (“1(0, 0, A) =
0. This shows that the constitutive response function q(T, VoT, F) must have the
property that it vanishes when the temperature gradient vanishes. Moreover, the
final residual inequality

6(T, VoT, F) . F‘TVOT g 0 (34)

must hold for all T, VOT, and F. This represents the classical Fourier inequality.

82

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