Nonlinear Elastomers: Modeling and Estimation
H. T. Banks y
N. J. Lybeck y
B. C. Mu~noz z
L. C. Yanyo z
Abstract
We report on our eorts to model nonlinear dynam-ics in elastomers. Our eorts include the development of computational techniques for simulationstudies and for use in inverse or system identication problems.
1 Introduction
A problem of fundamental interest and great impor-tance in modern material sciences is the development of both passive and active (\smart") vibration de-vices constructed from polymer (long molecular chains of covalently bonded atoms often having cross-linking chains) composites such as elastomers lled with car-bon black and/or silica or with active elements (i.e., piezoelectric, magnetic or conductive particles). These rubber based products (even without active elements) involve very complex viscoelastic materials that are not at all like metals (where large deformations lead to permanent material changes) and do not satisfy the usual, well-developed linear theory of (innites-imal) elasticity for deformable bodies. They typi-cally exhibit mechanical properties that combine elas-tic (purely reversible) and hystereelas-tic (irreversible) phe-nomena. In considering macroscopic behavior, one nds that the usual constitutive relationships (e.g., Hooke's law) or rheological equations of state for pure elastics are not applicable. Indeed, one nds a num-ber of factors that contribute to diculty in modeling mechanical behavior: (i) deformations are sensitive to environmental temperatures as well as the strain level, strain history and rate of loading; (ii) one observes nonlinearities in both material and geometric behavior - in general, there is a nonlinear relationship between stress and strain even for small strains; (iii) deforma-tions in the range of practical interest are large and innitesimal based theories break down; (iv)
Research supported in part by the Air Force Oce of
Scientic Research under grants AFOSR F49620-93-1-0198 and F49620-95-1-0236
yCenter for Research in Scientic Computation,
De-partment of Mathematics, North Carolina State Univer-sity, Raleigh, NC 27695
zLord Corporation, Thomas Lord Research Center, 405
Gregson Drive, Cary, NC 27511
tions are not reversible; (v) the anisotropic nature of polymers can be important in certain types of defor-mations - for example, the amount and type of ller aects mechanical response in a nontrivial manner.
In spite of all these diculties, there is a substantial literature on modeling of rubber-like elastomers (see [F, T, W] for basic texts), predominantly based on one of the two rather distinct approaches: (i) molec-ular (polymer chain) statistical thermodynamic for-mulations (ii) phenomenological (usually continuum) formulations involving stored energy or strain energy functions(SEF) and/or nite strain (FS) theories. In the phenomenological approach (which will be the ba-sis of our eorts) most investigators begin with an isotropic material under homogeneous strain.
Strain energy function theories typically embody only elastic properties of elastomers or rubbers and hence are mostly used in static (equilibrium) nite element analysis (e.g. see [CYT]) of materials (e.g. natural gum rubbers) that exhibit little or no hys-teretic behavior. SEF material models, such as those of Mooney-Rivlin, Ogden, Treloar and numerous
oth-ers, are based on strain invariants I
i, where
I
1 =
2
1+
2
2+
2
3 ;I
2=
2
2
2
3+
2
1
2
3+
2
1
2
2and I
3=
2
1
2
2
2
3
and the
i are the principal extension ratios
(de-formed length of unit vectors along directions paral-lel to the principal axes i.e. the axes of zero shear strain). For example, the Mooney SEF is given by
U =C 1(
I
1
?3) +C 2(
I
2
?3), or more generally, the
modied expressionU =C
1( I
1
?3)+f(I 2
?3), wheref
has certain qualitative properties, and is most appro-priate for components where the rubber is not tightly conned and where the assumption of absolute
incom-pressibility (implying
1
2
3 = 1 or
I
3 = 1) is a
rea-sonable approximation. The more general Rivlin SEF
U = P
N
i+j1 C
ij( I
1 ?3)
i( I
2 ?3)
jand its generalization
for near incompressibility (see [CYT]) allow higher or-der dependence of the SEF on the invariants.
undeformed body) one usually encounters in the in-nitesimal linear elasticity used with metals. This Eulerian measure of strain (relative to a coordinate system convected with the deformations) - as opposed to the usual Lagrangian measure (relative to a xed coordinate system for the undeformed body) - is an important feature of any development of models for use in analytical/computation/experimental investi-gations of rubber-like material bodies. The nite strain elasticity of Rivlin can be directly related to the strain energy function formulations through
equa-tions relating the nite strains ~e
xx ;~e
y y ;e~
z z to the
ex-tension ratios
1 ;
2 ;
3 used to dene the SEF. For
example, in homogeneous pure tensile strain we have
2
1 = 1 + 2~
e
xx ;
2
2 = 1 + 2~
e
y y ;
2
3 = 1 + 2~
e
z z and
~
e
y z= ~ e
z x= ~ e
xy = 0.
Whether one begins with a choice of the SEF or with Rivlin's nite strain formulation, one can use these along with standard material independent force and momentbalance derivations (the Timoshenko the-ory [TYW, CP]) as the basis of dynamical models. To illustrate this we take the simplest example: an
isotropic, incompressible (
1
2
3= 1) rubber-like rod
under simple elongation with a nite applied stress in
the principal axis directionx
1. The nite stress
the-ory (or the Mooney thethe-ory with SEFU =C
1( I
1 ?3))
leads to a true stress =
E
3(
2
1 ?
1
1) or an
engineer-ing or nominalstress for what are termed neo-Hookean materials
eng =
1
=E
3 ( 1
?
1
2
1
) (1)
where in terms of deformation u in the x
1 = x
di-rection we have (since deformations in the y and z
directions are negligible)
2
1 = 1 + 2~
e
xx= 1 + 2
@u
@x
+
@u
@x
2
(2) =
1 +@u
@x
2
:
HereE is a generalized modulus of elasticity.
This can be used in the Timoshenko theory for
lon-gitudinal vibrations of a rubber bar to obtain ( =
mass density,F = applied external force)
A @
2
u
@t 2
? @S
@x
=F (3)
whereS, the internal (engineering) stress resultant, is
given by
S= AE
3 (
1 ?
1
2
1
) =AE
3 g
@u
@x
(4)
withg() = 1+?(1+)
?2and
Ais the cross sectional
area. This leads to the nonlinear partial dierential
equation
A @
2
u
@t 2
? @
@x
EA
3 g
@u
@x
=F (5)
for dynamic longitudinal displacements of a neo-Hookean material rod in extension. Since a series
ex-pansion of g yields g() = 3?3
2+ 4
3
?:::, this
is readily seen, in the case of small displacements, to reduce to the usual longitudinal deformation equation for Hookean materials.
The neo-Hookean or simple Mooney expression for
the SEF yielding the form ofgin (4) has only limited
practical application since it is inadequate in describ-ing most lled elastomers. In general, one would em-ploy equations such as (5) with a more general
nonlin-earitygwhich should be estimated from dynamical
ex-periments. Moreover, one must also include hysteresis in the nonlinear integropartial dierential equations (see [F, SC, RHN, C, P]).
In this lecture we report on our eorts to develop such a class of nite strain dynamic models to be used in design of rubber based elastomers. We develop a mathematical framework for well-posedness and ap-proximation in the context of identication or inverse problems. Computational results as well as our use of these techniques with experiments will be discussed.
2 Variational Formulation
Our immediate interest is in the design of dynamic experiments to use in determining the strain function
g in models such as (5). In one type of dynamic
ex-periments a slender rod is suspended vertically with
the top end (x= 0) xed, as shown in Figure 1. Let
F(t)
x= 0
x=l
Figure 1: Experimental setup
u(t;x) denote the deformation at time t of the cross
section that was located at a distancexfrom the top
when the rod was free hanging (with no applied load).
Thus u(t;0) = 0 for allt. The end x= l of the rod
is attached to a motor which supplies a force F(t)
to produce some prescribed movement or deformation
(5) has the formF(t;x) = F(t)(x?l) and the
equa-tion (5) and its soluequa-tions must be interpreted in some generalized or distributional sense. With the bound-ary conditionsu(t;0) = 0, u(t;l) = (t) one also has
initial conditionsu(0;x) = (x), _u(0;x) = (x). In
general, one should not expect the resulting initial-boundary value problem (IBVP) for (5) to have clas-sical (smooth) solutions. For example, if we start the
system from rest 0, 0 (or 6= 0, = 0 in
the \preload" experiments described below) and ap-ply a periodic force of the form F(t) =asin! t, then
classical solutions will not exist due to incompatibility in boundary and initial conditions. It is thus useful to write (5) in some generalized sense,
Au(t) =Au(t) +F(t) in V
where A is an operator (in general, nonlinear) from
a space V of test functions to its dual (or conjugate
dual)V
. For the system (5), it is most convenient to
choose V = H
1
L(0
;l) f2 H 1(0
;l)j(0) = 0g and
treat the boundary conditionu(t;0) = 0 as an essential
boundary condition. HereH
1(0
;l) is the usual Sobolev
space ofL
2(0
;l) functions with derivatives inL
2(0 ;l).
If we take as pivot space H=L
2(0
;l) in the Gelfand
triple settingV ,!H,!V
, then it is readily shown
that the operatorA
@
@x ?
EA
3 g
?
@u
@x
has the form (Au)() =?h
EA
3 g
@u
@x
; 0
i
L2(0;l)
as a mapping fromV toV
and reduces to the usual
(and well-known) linear operator in L(V;V
) in the
case that g is linear. The corresponding weak or
variational form of (5) is given by (for F(t;x) =
F(t)(x?l))
hAu; i+h EA
3 g
@u
@x
; 0
i = hF(t)(x?l);i
(6) = F(t)(l)
for all 2V =H
1
L(0
;l). The associated IBVP
con-sists of nding u(t;) 2 V satisfying u(t;l) = (t), u(0;) = , _u(0;) = . To see that this is the correct
variational form, we argue as follows: Suppose uis a
solution of (6) that also satisesu(t)2H 2(0
;l). Then
the usual integration by parts arguments in (6) yield
(assuming smoothness ofg)
hAu? @
@x
EA
3 g
@u
@x
;i +
EA
3 g
@u
@x
(l) = F(t)(l)
for all2V =H
1
L(0
;l). But this is equivalent to
Au= @
@x
EA
3 g
@u
@x
; 0<x<l (7)
in the sense ofL
2(0 ;l) and
EA
3 g
@u
@x
(l) = F(t);t>0: (8)
We note that the left side of (8) is just the internal
stress (see (4)) at x=l which of course must match
the total external force applied at the endx=l. (Here
the \forces" are actually linear force densities { i.e.,
forces per unit length sinceis mass density.)
In the experiments being discussed here, the sys-tem will be preloaded, i.e., we take the initial state
of the rod so that u(0;l) = (and thus the initial
displacement is taken as the solution of the steady
state version of (7) subject to u(0) = 0, u(l) = ).
Then we take (t) = +d(t) where d(t) > ? is
some prescribed dynamics for the end x= l. In this
way one can guarantee that the rod remains in simple extension, and compression (which leads to nontrivial shear) is not present during the course of the experi-ments.
To determine the nonlinearityg, one has, for a given
input (t) (or F(t)), observations z
i which are
pro-portional to the strain @u
@x( t
i
;0) at the xed end. The
estimation problem of interest consists of minimizing
over some admissible classg2G
J(g) = M
X
i=1
z
i ?k
@u
@x
(t i
;0;g)
2
(9)
whereuis the solution of (6) corresponding tog. One
may also be fortunate enough to have observations ^u
i
of deformations u(t
i
;x;g) at some point x = x, 0 <
x<l, and then the optimization criterion (9) can be
modied accordingly.
Questions of well-posedness for the IBVP for (6) are nontrivial. In the event sucient damping is present (the equations (5) or (6) are undamped and hence are not realistic physically), one may use the tech-niques and results of [BGS1], [BGS2] to establish well-posedness for a rather generous class of nonlinearities,
even though we only have F(t;) in V
for the
prob-lems at hand. (For g linear, well-posedness follows
immediately from the results in [BIW].) These impor-tant, but more theoretical, questions will be discussed elsewhere.
3 Numerical Approximations
The ultimate goal of solving an inverse problem to
de-termine the nonlinearityg must be considered when
developing a numerical scheme to solve the forward problem (6). The presence of the prescribed
deforma-tionu(t;l) = +d(t) indicates that a Tau-Galerkin
method would be a convenient method for solving the weak formulation (6). The theory developed in [BW] shows further that a Tau-Galerkin method is an ap-propriate choice in the context of solving least-squares parameter estimation problems.
For a given g we use a Tau-Galerkin method with
linear splines for the spatial discretization, and a Fehlberg fourth-fth order Runge-Kutta method in time. We therefore seek an approximate solution to (6) of the form
u N(
t;x) = N
X
j=1 w
j( t)L
j( x)
where theL
j are linear splines (given below), and
w
j
are unknown functions of time. In the Tau method,
we impose the boundary conditionu(t;l) = +d(t)
on the trial solutions u
N, not on the individual
ba-sis elements. The boundary condition u(t;0) = 0 is
treated as an essential boundary condition and is
im-posed directly on each of the basis elementsL
j.
Let h=
l
N denote the grid spacing. Let
x
j = jh,
j = 1;:::;N be the grid points. The usual linear
splines are dened for 1jN?1 by
L
j( x) =
8 > > < > > :
0; 0x(j?1)h
x=h?(j?1); (j?1)hxjh
?x=h+j+ 1; jhx(j+ 1)h
0; x(j+ 1)h
;
with
L
N( x) =
0; 0x(N?1)h
x=h?N+ 1 ; x(N?1)h :
In order to use the Runge-Kutta method, (6) must be rewritten as a rst order system in time. To ac-complish this, let
u 1 u 2 = u _ u : Then _ u 1 _ u 2 = u 2 u :
Restate the weak formulation (6) as _
u
1= u
2 (10)
and for each2H
1
0(0
;l)V H 1
L(0 ;l) = Z l 0 _ u 2
Adx=? Z l 0 0 AE 3 g @u 1 @x
dx: (11)
The approximate solutions of (10) and (11) are given by
u N
1 ( t;x) =
N?1
X
j=1 w
j( t)L
j(
x) + ( +d(t))L N(
x);
and
u N
2( t;x) =
N?1
X
j=1 j(
t)L j(
x) + _d(t)L N(
x):
Substituting the approximate solutions into (10) we obtain equations
_
w
j( t) =
j(
t); j= 1;::: ;N?1:
Substituting the approximate solutions into (11), and
choosing=L
k, we nd
Z l 0 0 @ N?1 X j=1 _j L
j+ dL N 1 A AL k dx= ? Z l 0 L 0 k AE 3 g 0 @ N?1 X j=1 w j L 0
j+ ( +
d)L 0
N 1
A
dx:
For xed t the discrete system is achieved by
allow-ing k to vary between 1 and N ?1. The following
notation is necessary to write down the discrete sys-tem. Dene the (symmetric) tridiagonal mass matrix
M with entries
[M i;i] =
Z l 0 AL 2 i dx; and [M
i;i+1] = [ M
i+1;i] = Z l 0 AL i L i+1 dx:
Dene the vector ~
Gwith entries
~ G i= ? Z l 0 L 0 i AE 3 g 0 @ N?1 X j=1 w j L 0
j+ ( +
d)L 0
N 1
A
dx
i= 1;:::;N?2;
and ~ G N?1 = ? Z l 0 L 0 N?1 AE 3 g 0 @ N?1 X j=1 w j L 0
j+ ( +
d)L 0
N 1
A
?dAL N
L
N?1 dx:
The rst order discrete system is then given by
4 Numerical Studies
We have used the numerical scheme outlined above in numerical simulation studies to aid in design of exper-iments. We describe briey here some of our ndings. The rst question to be addressed in using the ap-proximation schemes is related to their accuracy. One
can prove convergence results: as N ! 1, the
ap-proximate solutions u
N converge to the solution
u
of (6). However, this does not answer the question
of the value of N that we x to use in our
calcula-tions. To aid in this matter, we performed a series of eigenvalue calculations with the approximate
sys-tem in the case of a Hookean material (g() = 3 in
(5)). For a rod of length l = 15 cm, and constant
cross section, one nds that the eigenvalues are given
by
n = n =l
p
E=. Choosing material parameters
E= 2:110 7dyn
=cm 2
,=:92 g=cm 3
, we obtain the
corresponding natural frequenciesf
n=
n
=(2) given
by f
n = 159
:25n. Eigenvalue calculations with the
approximate systems demonstrated good
approxima-tion of the rst eight natural frequencies atN = 64.
For example, at N = 48 and 64 we obtained f
48
1 =
159:28 ; f 64
1 = 159
:27 to approximate f
1 = 159
:25
while f
48
7 = 1124
:55 ; f 64
7 = 1120
:28 approximate
f
7= 1114
:75. Similar approximates are found for the
other frequencies. Based on these eigenvalue studies for the linear systems (of course, the ideas of eigen-values and modes are not useful for the neo-Hookean
system) we usedN = 48 in our calculations, including
those reported on here.
We considered a number of questions in support of our experimental design questions and we list several
of these. (i)What shape (l ;A=A(x), etc.) should be
used that will provide ample information about the material and yet be easily constructed? (ii) If addi-tional observations (say of displacement) are possible, where should these be taken to maximize the dier-ences between Hookean and neo-Hookean material re-sponses? ( We know, for example, that displacement sensors near nodal points for a linear system is not an intelligent choice!) (iii) What type of input signals will suciently distinguish the Hookean and neo-Hookean responses so as to enhance our possibilities for
identi-fyinggfrom vibration response data? (iv) How should
we test damping? The linear version of equation (6)
is undamped (does the nonlinearity g itself provide
some type of damping of input energy?), while the ex-perimental responses will denitely exhibit signicant damping.
We discuss briey some of our ndings related to two of these questions. First the question of damping is a dicult one that we are still pursuing. However, as the graphs in Figure 2 reveal, the Hookean model,
as expected, has no inherent damping. Each graph in Figure 2 corresponds to a simulation with zero initial displacement but with the same nonzero initial veloc-ity (this simulates a structure that has been excited
with the same energy input at t = 0 and afterwards
allowed to freely oscillate). The neo-Hookean response suggests that there is some inherent energy reduction in the nonlinear system but it clearly is not damping in the traditional sense we understand. It also does not correspond to the rapid dissipation we see in os-cillating rubber samples.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6
0.8 1 1.2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6
0.8 1 1.2
time (sec)
cm
Neo-Hookean model, displacement atl=4
time (sec)
cm
Hookean model, displacement atl=4
Figure 2: Damping experiment for both a Hook-ean and a neo-HookHook-ean model
The question of type of input signal to use has yielded quite nicely to our numerical investigations.
We have considered persistent sinusoidal inputsd(t) =
asin! twith the driving frequency off =! =(2) near
and far from resonance for the linear system. We also considered periodic triangular inputs (a \sawtoothed" sinusoidal that is between a square wave and a smooth sinusoidal). In many numerical experiments, it was observed that the driving frequency dominated the responses with varying levels of energy exciting the natural structural modes of the Hookean system. In FFT's of the neo-Hookean responses, one sees energy concentrations but not so sharply as in systems with modes. One important constraint on the periodic
in-put signals is the frequency range for f possible in
the experimental test equipment. While a frequency sweep (say between 0 and 500 Hz) would reveal signif-icant information, we are physically restricted to
exci-tations in the 0?25 Hz range. These (in the range of
in-put signal (a triangular displacement approximating an impulse input) and the FFT of the corresponding
displacement atx=l =4 in the rod for an input
possi-ble with frequency set at 15 Hz. Figure 3 contains the Hookean response while Figure 4 is the neo-Hookean. In Figures 5 and 6 we give similar plots for an input
signal corresponding tof = 25 Hz.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 3.5
4 4.5 5 5.5 6
0 50 100 150 200 250 300 350 400 0
10 20 30 40 50
time (sec)
cm
Position atl,f= 15
Hertz FFT of response atl=4
Figure 3: Hookean response at 15 Hz.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 3.5
4 4.5 5 5.5 6
position at l, f=25
0 50 100 150 200 250 300 350 400 0
5 10 15 20
time (sec)
cm
Hertz FFT of response atl=4
Figure 4: Neo-Hookean response at 15 Hz.
0 50 100 150 200 250 300 350 400 0
20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 3.5
4 4.5 5 5.5 6
Hertz FFT of response atl=4
time (sec)
cm
Position atl,f = 25
Figure 5: Hookean response at 25 Hz.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 3.5
4 4.5 5 5.5 6
position at l, f=15
0 50 100 150 200 250 300 350 400 0
10 20 30
time (sec)
cm
Hertz FFT of response atl=4
Figure 6: Neo-Hookean response at 25 Hz.
5 Concluding Remarks
In the discussions above we have outlined some of our initial eorts on the intellectually stimulatingbut di-cult problems related to the understanding of the dy-namics of lled elastomers. We are currently begin-ning experiments and developing computational
tech-niques for the estimation of g in (6) from vibration
response data. Moreover, in related studies [BMZ] we are developing models and computational schemes to treat hysteresis as well as damping in composite ma-terial structures containing viscoelastic components.
References
[R] R.S. Rivlin, Large elastic deformations of
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[W] I.M Ward, Mechanical Properties of Solid Polymers, J. Wiley & Sons, New York, 1983.
[F] J.D. Ferry, Viscoelastic Properties of
Poly-mers, J. Wiley & Sons, New York, 1980. [CYT] D.J. Charlton, J. Yang, and K.K. Teh, A
re-view of methods to characterize rubber elas-tic behavior for use in nite element analysis, Rubber Chemistry and Technology, 67 (1994), 481-503.
[TYW] S. Timoshenko, D.H. Young, and W. Weaver, Jr., Vibration Problems in Engineering, J. Wi-ley & Sons, 1974.
[CP] R.W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, New York, 1975. [SC] I.H. Shames and F.A. Cozzarelli, Elastic and
Inelastic Stress Analysis, Prentice Hall, 1992. [RHN] M. Renardy, W.J. Hrusa, and J.A. Nohel,
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[P] A.C. Pipkin, Lectures on Viscoelasticity
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[BGS2] H. T. Banks, D. S. Gilliam, V. I. Shubov, Global solvability for damped abstract nonlin-ear hyperbolic systems, preprint October 1994. [BIW] H. T. Banks, K. Ito, and Y. Wang, Well-posedness for damped second order systems with unbounded input operators, CRSC-TR93-10, June 1993, Dierential and Integral Equations, 8 (1995), 587{606.
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