63
\Modeling the Dynamic Mechanical Behavior of Elastomers"
H. T. Banks N. J. Lybeck
Center for Research in Scientic Computation Department of Mathematics
North Carolina State University Raleigh, NC 27695-8205
M. J. Gaitens B. C. Mu~noz
L. C. Yanyo Lord Corporation Thomas Lord Research Center
405 Gregson Drive Cary, NC 27511
Presented at a meeting of the
Rubber Division, American Chemical Society Louisville, Kentucky
October 8{11, 1996
Abstract
Accurate modeling of the dynamic mechanical behavior of elastomers presents many challenges, including the nonlinear relationship between stress and strain, the loss of kinetic energy (damping), and the loss of potential energy (hysteresis). Currently available software packages for studying the stress-strain laws in rubber-like materials assume a form of the strain energy function (SEF), such as a cubic Mooney-Rivlin form or an Ogden form. While these methods can produce good results, they are only applicable to static behavior, and they ignore hysteresis and damping.
We present a dynamic partial dierential equation (PDE) formulation, with a Kelvin-Voigt damping term, as an alternative approach to the SEF formulation. Constitutive laws are estimated using data from simple ex-tension experiments, leading to static results which compare favorably with results achieved by estimating a cubic Mooney-Rivlin SEF, and dynamic results which oer new insight. A neo-Hookean model for generalized sim-ple shear is presented in the PDE framework, and techniques for modeling hysteresis are discussed.
1 Introduction
The engineering uses of rubber have, in recent years, expanded well beyond traditional products such as tires and seals. Today rubber (and, more gener-ally, elastomers) can be found in a diverse set of components including engine mounts, building foundations, belts, and fenders (see [1, 2]). Increasingly, the applications of rubber are becoming more sophisticated, as exempliedby the use of rubber bearings in bridges which allow for thermal expansions of the deck without placing excessive loads on the bridge supports (see [3]).
Elastomers in current engineering roles are typically lled with inactive particles such as carbon black and silica. If active llers were used, such as piezoelectric, magnetic, or conductive particles, the resulting controllable elastomer could be used in products such as active vibration suppression devices. As these new materials are developed, the role of design will increase in both complexity and importance. In particular, the capability to predict the dynamic mechanicalresponse of the components will become increasingly valuable.
The many desirable characteristics of rubber as a design component, which include an ability to undergo large elastic deformations, good damp-ing properties, and near incompressibility, are also contributdamp-ing factors to the complications arising in the process of formulating models. Innitesimal based strain theory, for example, is not appropriate for modeling the large deformations often seen in rubber. The constitutive laws are nonlinear (thus Hooke's law is not applicable), and damping is highly signicant. The strain history, rate of loading, environmental temperature, and amount and type of ller aect the mechanical response in a nontrivial manner. Additionally, many elastomers, particularly those with a synthetic rubber base, exhibit strong hysteresis characteristics similar to those found in shape memory al-loys and piezoceramic actuators.
In spite of the many complex issues, researchers have made substantial progress in developing tools for modeling elastomers (see [4, 5, 6] for ba-sic texts). These models are predominantly phenomenological, based on strain energy function (SEF) and nite strain (FS) theories. SEF theo-ries contain information about the elastic properties of elastomers, but do not describe either damping or hysteresis, and hence are typically used for
static nite element analysis (see [7]). The SEF material models use the principal extension ratios
i to represent the deformed length of unit vec-tors parallel to the principal axes (the axes of zero shear stress). The
SEF models of Mooney and Rivlin are based on Rivlin's proposal ([8]) that the SEF should depend only on the strain invariants I
1 = 2 1 + 2 2 + 2 3 ; I 2 = 2 2 2 3 + 2 1 2 3 + 2 1 2 2 and I 3 = 2 1 2 2 2
3. For example, the Mooney SEF is given by U = C
1( I
1
?3) +C 2(
I 2
?3), or more generally, the modied expression U =C
1( I
1
?3) +f(I 2
?3), where f has certain qualitative prop-erties, and is most appropriate for components where the rubber is not tightly conned and where the assumption of absolute incompressibility (implying
1
2
3 = 1 or I
3 = 1) is a reasonable approximation. The more general Rivlin SEF U =
P N i+j1 C ij( I 1 ?3) i( I 2 ?3)
j and its generalization for near incompressibility (see [7]) allow higher order dependence of the SEF on the invariants. The works of Ogden, as well as Valanis and Landel, represent an important departure from Rivlin's proposal, presenting SEF's that depend only on the extension ratios.
The nite strain elastic theory of Rivlin [6, 8] is developed with a gener-alized Hooke's law in an analogy to innitesimal strain elasticity but makes no \small deformation" assumption and includes higher order exact terms in its formulation. Moreover, nite stresses are dened relative to the deformed body and hence are the \true stresses" as opposed to the \nominal" or \engi-neering" stresses (relative to the undeformed body) one usually encounters in the innitesimal 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 sys-tem for the undeformed body) - is an important feature of any development of models for use in analytical/computation/experimental investigations of rubber-like material bodies. The nite strain elasticity of Rivlin can be di-rectly related to the strain energy function formulations through equations relating the nite strains ~e
xx ;e~
y y ;e~
z z to the extension 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.
The nite strain approach can be put in a somewhat more general per-spective in the context of classical modeling of elastic solids and uids (elas-tomers and lled rubbers do not t exactly into either category). In classical approaches one frequently encounters an Eulerian formulation in dynamics of uids where large deformations or displacements are common whereas a Lagrangian formulation is employed for solids undergoing small elastic de-formations. In both formulations, momentum balance laws along with con-stitutive laws relating stress and strain are employed to develop theories of dynamics (see [9], [10]). In the general Lagrangian formulations, quantities
(such as stress, strain) are dened relative to an original or reference con-guration B
0 of the body or structure in terms of a xed coordinate system ~
X = fX i
g. For an Eulerian formulation one denes quantities relative to a \current" or deformed conguration B with coordinates ~x =fx
i
g relative to the deformed conguration. A fundamental role in discussing the rela-tionship between these formulations is a \conguration" map ~x = (
~ X) or \motion" ~x(t) = (t;
~
X) if the deformations are changing in time from an original conguration(0;
~ X) =
~
X. The deformations (in the usual elasticity terminology) are then given by
u(t; ~
X) =(t; ~
X)?(0; ~
X) =(t; ~ X)?
~ X :
The conguration gradient (also called the \deformation gradient" in an unfortunate misnomer) is dened by
A = @(
~ X) @
~ X
= @~x @
~ X and is used to dene the right (A
T
A) and left (AA
T) Cauchy{Green \de-formation tensors" and the usual (in elasticity theory) Green{ St. Venant strain
E = 12(A T
A?I) = 12(D T
D+D+D T) where D =
@u @
~ X =
A ? I. These denitions, along with momentum bal-ance laws, can be used to write dynamic equations in either the Lagrangian (reference) or Eulerian (current) coordinate systems. If one uses the Eu-lerian system, stresses are given in terms of the Cauchy or true stress
T
and constitutive (stress{strain) laws are expressed in terms of current co-ordinates ~x. However for computational and experimental purposes, it is often more desirable to use a Lagrangian coordinate system and then the Cauchy or true stress must be converted to an expression for the nominal or engineering stressS
= JA?1
T
, whereJ = detA and
S
T is called the First Piola{Kirchho stress tensor.
2 A Neo-Hookean Model for Extension
One can begin with a choice of the SEF or with Rivlin's nite strain for-mulation, and use these along with standard material independent force and
x=l M
F
x=0
Figure 1: Rod with tip mass under tension
moment balance derivations (the Timoshenko theory [11, 12]) as the basis of dynamic models. To illustrate this we take the simplest example: an isotropic, incompressible (
1
2
3 = 1) rubber-like rod (with a tip mass) under simple elongation with a nite applied stress in the principal axis di-rection x
1 =
x, as seen in Figure 1. (Here, following standard convention, we use lower case letters to denote the Lagrangian coordinates.) The nite stress theory (or the Mooney theory with SEF U = C
1( I
1
?3)) leads for what are termed neo-Hookean materials to a true stress T =
E 3(
2 1
? 1
1) or an engineering or nominal stress
S = T
1 = E
3 ( 1
? 1
2 1
) (1)
where in terms of deformation u in the x direction we have (since deforma-tions 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 :
Note in this case the stress tensors
T
,S
reduce to one nontrivial component T =T
11,
S =
S
11. Here
E is a generalized modulus of elasticity and we note these formulations are restricted to
1
>?1.
This can be used in the Timoshenko theory for longitudinal vibrations of a rubber bar with a tip mass to obtain ( = mass density, F(t) = applied
external force, A
c is the cross sectional area,
M is the tip mass, g is the gravitational constant, is the air damping coecient)
A c @ 2 u @t 2 + @u @t ? @S @x
= 0 0<x<l (3)
M @ 2 u @t 2(
t;l) =?S
x=l
+ F(t)+Mg
where S, the internal (engineering) stress resultant, is given by
S = A c E 3 ( 1 ? 1 2 1
) +C D A c @ 1 @t (4) = A c E 3 ~g
@u @x ! +A c C D @ 2 u @t@x (5) with ~g() = 1+?(1+)
?2. Here we have included a Kelvin-Voigt damping term (C
D is the Kelvin-Voigt damping coecient) in the stress
S as a rst attempt to model damping. This leads to the nonlinear partial dierential equation A c @ 2 u @t 2 + @u @t ? @ @x EA c 3 ~g
@u @x ! +A c C D @ 2 u @t@x !
= 0 0<x<l(6)
M @ 2 u @t 2(
t;l) =? A
c E 3 ~g
@u @x ! +C D A c @ 2 u @t@x ! x=l
+ F(t) +Mg for dynamic longitudinal displacements of a neo-Hookean material rod in extension. Since a series expansion 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.
In general, one does not expect the initial boundary value problem asso-ciated with (6) to have a classical (smooth) solution. Equations such as (6) are innite dimensional, and must be discretized before the solution can be approximated. We have chosen to use a Galerkin method with linear splines for the spatial discretization. The second order equation (6) is then written as a rst order system in time, and Gear's method (or a similar sti equation solver) can be used for the time integration. Details for a similar problem can be found in [13].
3 Approximation of Constitutive Laws
The neo-Hookean model (6) presented in the previous section provides a nat-ural example of a nonlinear PDE in the modeling of elastomers, but has only limited practical application since it is inadequate in describing most lled elastomers. Typically, one would employ equations such as (5) with a more general nonlinearity ~g which should be estimated from experiments. One does not expect such a general ~g to admit a SEF as a function of either the strain invariants or the extension ratios. Comparisons with SEF methods can be made by using the (approximate) SEF to derive the expected stress-strain relationship, and comparing results in the stress-strain (or, equivalently, the load-deection) plane. With this goal in mind, we now proceed to discuss the numerical estimation of ~g.
3.1 Static Inverse Problem
Although our ultimate goal is to use the results of dynamic experiments to determine the nonlinearity ~g(), a reasonable rst step is to estimate ~g() using data from static pull tests. While this type of experiment cannot be used to study damping, it can be used to study the nonlinearity ~g. Since methods currently exist to estimate an SEF from static testing results, we have a basis for comparison of our methods with commonly accepted tech-niques. In the static pull test, a slender rod is suspended vertically with the top end (x = 0) xed, as shown above in Figure 1. Let u
i(
x) denote the deformation (under a constant load f
i) of the cross section that was located at a distancexfrom the top when the rod was free hanging (with no applied load or mass). Thus u
i(0) = 0, and the end
x = l of the rod (with no tip mass) is subjected to a constant force f
i, with a resulting displacement i at x=l. The sample satises the steady state equation
@ @x
A c^
g @u
i @x
!!
= 0 0<x<l
A c^
g @u
i @x
!
(l) = f
i (7)
u
i(0) = 0 u
i( l) =
i
where the nonlinearity ^g = E
3~
g is unknown. We seek to nd ^g minimizing J(^g) =
k X i=1 j i ?u i( l;^g)j
2 (8)
over some class of admissible functions ^g 2 G, where f i
;f i
g k
i=1 are data from a series of static pull experiments.
In general, problems such as those involving (8) are innite dimensional in both state and parameter space and hence, for computational purposes, nite dimensional approximations must be made. For state approximation, one typically uses Galerkin techniques. However, in the case that ^g is monotone, the system of equations (7) satised by the rod can be simplied to
u i(
l) = Z l 0 ^ g ?1 f i A c !
dx; (9)
and thus one may avoid making a state approximation. For parameter space discretization (i.e., approximation of ^g), one may use a nite dimensional parameterization or representation. In light of (9), it is appropriate to ap-proximate ^g
?1 using
M approximating piecewise linear elements (e.g., linear splines)
^ g
?1 M (
x) = M X j=1 c j j( x) :
The least squares spline inverse problem (LSSIP) is then equivalent to: nd ~c2R
M minimizing
J(~c) = k X i=1 j i ?u i( l;~c)j
2
: (10)
(Since we expect ^g to be monotone, estimating the inverse function is an acceptable technique.)
We have used our methods (with 15 linear splines parameterizing ^g ?1) to t the data from static tensile strain experiments. One of the standard engineering techniques is to use data to estimate a cubic Mooney{Rivlin SEF (see [5], [6]). As seen in [6], the engineering stress in the x direction which arises from the estimated SEF U is given by the equation
Figure 2 demonstrates that the results from the two methods, viewed in the load{deection plane, are nearly identical. The material used for the sample is a common elastomer formulation, similar to those used for tire sidewalls. It is a blend of natural rubber (cis-polyisoprene) and polybutadiene. It includes approximately 30 percent carbon black by weight, and is crosslinked with sulfur and accelerators. The data is not included on the graph in Figure 2, as both ts are so close that the curves would be indistinguishable.
0 2.5 5.1 7.5 10.2 12.7 15.2 17.8 20.3 22.9 25.4 0
22.2 44.5 66.7 89.0 111.2 133.4 155.7
Displacement (centimeters)
Load (newtons)
Cubic MR LSSIP
Figure 2: Load vs. displacement for static pull experiment
A more common practice is to view the results in the reduced stress plane (see [5], pp. 95{99, [6], pp. 51{52), instead of the load-deection plane. For a rod in uniaxial tension, the reduced stress (or Mooney stress) is given by
M =
S 2(
1 ?
?2 1 ) =
@U @I
1 ?
1
1 @U @I
2 : The reduced stress
M is then plotted as a function of 1 =
1. Historically, this was used to gain some insight into the dependence of the SEF U on I
1 and on I
2. As seen in Figure 3, the reduced stress curve generated from our method approximates the curve generated from the data more closely than does the curve resulting from the standard SEF method. Thus the ap-proach proposed in this note oers the potential for improvement on existing industrial methods.
3.2 Dynamic Inverse Problem
To determine the nonlinearity ~g(), one has, for a given input (t) (orF(t)), observations z
i which are proportional to the strain @u @x(
t i
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 414
483 552 621 689 758 827 896
1/lambda
Reduced stress (KPa)
Cubic MR LSSIP Data
Figure 3: Reduced stress vs. 1
1
for static pull experiment
end. The estimation problem of interest consists of minimizing over some admissible class ~g 2G
J(~g) = M X
i=1 jz
i ?
@u @x
(t i
;0; ~g)j
2 (11)
whereu is the solution of (6) corresponding to ~g. One may also be fortunate enough to have observations ^u
i of deformations u(t
i
;x; ~g) at some pointx=
x, 0 < x < l, and then the optimization criterion (11) can be modied accordingly (although this is not the case for the experimental results that we are about to present).
Current dynamic testing for elastomers often takes the form of a cyclic deformation, such as a sinusoidal end displacement or a sinusoidal applied force. Our initial numerical simulations suggested that the dierence be-tween a Hookean material and a neo{Hookean was most readily seen under free vibration testing conditions (see [13, 14]). The free vibration tests also provide a natural setting for the study of damping.
For these experiments, the rod should have a tip mass that guarantees that the rod remains in simple extension, and compression (which leads to nontrivial shear) is not present during the course of the experiments. For the following experiments, a slender rod composed ofunlled natural rubber was used, as seen in Figure 4. The rod length wasl = 5:4356 cm, with ange height 0:3048 cm, inner diameter ID = 0:4572 cm, outer diameterOD = 1:905 cm, and the metal tabs were 1:27 cm high. The frame (which was used both as a mass and as a housing to protect the accelerometer) had mass 262:7
Frame
l ID
OD
Impedence Head
Metal Tabs Accelerometers
Figure 4: Rod with tip mass under tension
g, and the sample (including the bonded metal tabs) had mass 52:1 g. The impedance head calibration used was 978:4 mV/lb, and the accelerometer calibration used was 103:9 mv/lb. Data was collected using three Hewlett Packard HP 35652A analyzers. The top accelerometer was used to check the clamped boundary condition (i.e., to verify that we physically obtained a reasonable approximation to a \clamped end" boundary condition).
For the rst set of experiments, the rod with mass was allowed to hang until equilibrium was reached, and then a hammer was used to excite the system. The actual dynamic displacement was quite small for these experi-ments, although the overall strain was approximately 17 %. As seen in Figure 5, the linear Kelvin-Voigt constitutive law provides a nice t, although it does miss the higher frequency (which is approximately 350 hz). The program was run using up to a 8 term piecewise linear t, with and without a viscous (air) damping term, which seemed to make very little dierence in the results.
In the second set of experiments, the rod was lifted so that the rod itself was at its natural length (i.e., no compression or extension). The support was then removed, allowing the mass to fall freely. The rod achieved approx-imately 34 % dynamic strain during this test. As seen in Figures 6, 7, the 8 term piecewise linear constitutive approximation gives a much better t to the data than does the linear Kelvin-Voigt constitutive law. The air damping term was used only because it increased the consistency in the Kelvin-Voigt damping coecient between the hammer hit experiments and the free re-lease experiments. In Figure 8 the static constitutive relationship is shown for both the linear case and the 8-term piecewise linear case.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
Time (seconds)
Force at top (newtons)
Kelvin−Voigt fit to hammer hit data
Data Aprx
10−1 100 101 102 103 10−2
10−1 100 101 102 103 104
Log of frequency Kelvin−Voigt fit to hammer hit data
Data Aprx
Figure 5: Kelvin-Voigt t for hammerhit data in timeand frequency domains
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0
1 2 3 4 5 6
Time (seconds)
Force at top (newtons)
KV fit to free release data
Data Aprx
0 2 4 6 8 10 12 14 16 18 20 0
20 40 60 80 100 120 140 160 180 200
Frequency (hz) KV fit to free release data
Data Aprx
Figure 6: Kelvin-Voigt t for free release data in time and frequency domains
0 1 2 3 4 5 6 7 8 0
1 2 3 4 5 6
Time (seconds)
Force at top (newtons)
8 term fit to free release data
Data Aprx
0 2 4 6 8 10 12 14 16 18 20 0
20 40 60 80 100 120 140 160 180 200
Frequency (hz) 8 term fit to free release data
Data Aprx
Figure 7: 8-term t, air and KV damping for free release data in time and frequency domains
0 0.5 1 1.5 2 2.5
0 50 100 150 200 250 300 350
Strain
Stress (Kpa)
Estimated stress−strain curves for free release data
Linear
8 Term
Figure 8: Static linear strain relationship vs 8-term nonlinear stress-strain relationship
4 Generalized Simple Shear
Consider an object in simple shear as seen below in Figure 9. True simple shear is rarely achieved in laboratory situations, because for most samples the angle =
@u
@y is not truly constant, but actually depends on
y. It is useful, however, to see how a generalization of simple shear might be represented in the PDE model. As in the simple extension example above, we use x,y, and z for the Lagrangian coordinates.
(y)
Reference Configuration Deformed Configuration x
y
γ
Figure 9: Simple Shear Conguration
Ogden ([9]) argues that an incompressible, isotropic neo{Hookean mate-rial undergoing simple shear has Cauchy stress tensor
T
=2 6 4
2 +
?p 0
?p 0
0 0 ?p
3 7 5
;
whereis a material-dependent constant,pis an arbitrary hydrostatic stress (necessary due to assumed incompressibility). The engineering stress tensor is given by
S
=2 6 4
?p p 0
?p 0
0 0 ?p
3 7 5
:
The Lagrangian equations of motion then reduce to the linear wave equation
@
2 u @t
2 =
@ 2
u @y
2
: (12)
This is consistent with the fact that the neo{Hookean SEF (U =C 1(
I 1
?3)) is based upon a Hookean assumption in shear (see [5], [6]). The simple shear model as presented in (12) can be generalized to a nonlinear equation by including damping terms and assuming that the Cauchy stress tensor includes appropriate terms such as
T
12 =g() for some nonlinear function
g.
0 5 10 15 20 25 0
20 40 60 80 100 120 140 160
Extension (centimeters)
Load (newtons)
Hysteresis loop for sample with 250% maximum strain
Figure 10: Hysteresis loop
5 Modeling Hysteresis
In the above discussions we have totally ignored one important physical phe-nomenon experimentally observed in most elastomers and lled rubber prod-ucts. As depicted in Figure 10, one observes signicant hysteresis in the load-deection (i.e., stress-strain) curves. (This gure contains a plot of data obtained from a simple extension experiment with a rubber rod as discussed in Section 2.) The modeling implications of such results are fairly obvious: In models such as (6), one cannot represent the stress-strain constitutive law by a simple nonlinearity ~g. Rather one must have some type of memory mechanism involving a family of stress-strain laws f~gg to use as the \con-stitutive law." While investigations in this area of model development are in their infancy, the generalized hysteresis measure ideas (based on extended Preisach-Krasnoselskii-Pokrovskii formulations) developed for shape memory alloy actuators in [15, 16] oer promise in developing hysteretic constitutive laws.
6 Summary
The PDE based modeling method presented here provides a promising new avenue for modeling rubber. The experimental results for an elastomer rod in uniform extension suggest that the piecewise linear approximation of the constitutive laws provides an improvement over a linear approximation. Dif-ferent damping models are currently under investigation, using data from
more highly damped materials. Experiments have been conducted to collect hysteresis data, and the analysis and modeling eorts have begun. Eventu-ally, these eorts should converge to yield a highly accurate model for the dynamic mechanical behavior of elastomers.
References
[1] R. J. Crawford, Plastics and Rubbers: Engineering Design and Applica-tions, Mechanical Engineering Publications, Ltd., London, 1985.
[2] K. Nagdi, Rubber as an Engineering Material: Guideline for Users, Hanser Publishers, New York, 1993.
[3] A. N. Gent, Engineering with Rubber: How to Design Rubber Compo-nents, Hanser Publishers, New York, 1992.
[4] J. D. Ferry, Viscoelastic Properties of Polymers, John Wiley & Sons, New York, 1980.
[5] L. R. G. Treloar, The Physics of Rubber Elasticity, Clarendon Press, Oxford, 3rd ed., 1975.
[6] I. M. Ward, Mechanical Properties of Solid Polymers, John Wiley & Sons, New York, 2nd ed., 1983.
[7] D. J. Charlton, J. Yang, and K. K. Teh,A review of methods to charac-terize rubber elastic behavior for use in nite element analysis, Rubber Chemistry & Technology, 67 (1994), pp. 481{503.
[8] R. S. Rivlin,Large elastic deformations of isotropic materials I, II, III, Phil. Trans. Roy. Soc. A, 240 (1948), pp. 459{490, 491{508, 509{525. [9] R. W. Ogden, Non{Linear Elastic Deformations, Ellis Horwood
Lim-ited, Chichester, 1984.
[10] J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elas-ticity, Prentice{Hall, Englewood Clis, NJ, 1983.
[11] S. Timoshenko, D. H. Young, and W. Weaver, Jr., Vibration Problems in Engineering, J. Wiley & Sons, New York, 1974.
[12] R. W. Clough and J. Penzien, Dynamics of Structures, McGraw{Hill, New York, 1975.
[13] H. T. Banks, N. J. Lybeck, M. J. Gaitens, B. C. Mu~noz, and L. C. Yanyo, Computational methods for estimation in the modeling of non-linear elastomers, Tech. Rep. CRSC-TR95-40, NCSU, 1995; to appear in Kybernetika, (1997).
[14] H. T. Banks, N. J. Lybeck, B. C. Mu~noz, and L. C. Yanyo, Nonlinear elastomers: modeling and estimation, in Proceedings of the 3rd IEEE Mediterranean Symposium on New Directions in Control and Automa-tion, vol. 1, Limassol, Cyprus, 1995, pp. 1{7.
[15] H. T. Banks, A. J. Kurdila, and G. Webb, Identication of hysteretic inuence operators in smart actuators, Part I: Formulation, Tech. Rep. CRSC-TR96-14, NCSU, 1996; Mathematical Problems in Engineering, submitted.
[16] H. T. Banks and A. J. Kurdila, Hysteretic control inuence operators representing smart material actuators: identication and approximation, in Tech. Rep. CRSC-TR96-23, NCSU, August, 1996; Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan, December 1996, to appear.
Acknowledgments.
The authors gratefully acknowledge that this researchwas carried out with support in part by the U. S. Air Force Oce of Scientic Research under grant AFOSR F49620-95-1-0236, and the National Science Foundation under grant NSF DMS-9508617 (with matching funds for N.J.L. from the Lord Corporation).
