Effect of Stress-softening on the Dynamics of a Load Supported by a Rubber String

Effect of Stress-softening on the Dynamics of a LoadSupported by a Rubber String

S. Sarangi & R. Bhattacharyya & M. F. Beatty

Received: 20 September 2007 /Accepted: 23 December 2007 /Published online: 16 February 2008# Springer Science + Business Media B.V. 2008

Abstract The Mullins effect in the oscillatory motion of a load under gravity and attached toa stress-softening, neo-Hookean rubber string is investigated. Equations for the smallamplitude vertical oscillations of the load superimposed on the finite static stretch of both thevirgin and stress-softened cords, the latter subjected to varying degrees of preconditioning, arederived. The vibrational frequency of the small motion exhibits behavior similar to thatobserved in experiments by others on postmortem, human aortic tissue for which no stress-softening is reported. Standard numerical methods are applied to study the finite amplitudemotion of the load in the stress-softened case. The resultant motions and their variousphysical aspects under free-fall and general initial conditions are described in severalexamples. Oscillations that engage all three phases of motion consisting of the suspension, thefree-flight, and the retraction of the load in its general vertical motion are illustrated. Effectsdue to the degree of stress-softening are discussed; and the motion response for two values ofthe model softening parameter is compared in several examples. All results are illustratedgraphically and numerous tabulated numerical results are provided.

Keywords Mullins effect . Stress-softening phenomena . Neo-Hookean model .

Small amplitude frequencies . Finite amplitude oscillations . Nonlinear vibrations

AMS Classifications 74B20 Nonlinear elasticity . 34C15 Nonlinear oscillations . 74D10Nonlinear constitutive equations . 76E30 Nonlinear effects

J Elasticity (2008) 92:115–149DOI 10.1007/s10659-007-9154-9

S. Sarangi : R. BhattacharyyaDepartment of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India

S. Sarangie-mail: [email protected]

R. Bhattacharyyae-mail: [email protected]

M. F. Beatty (*)University of Nebraska-Lincoln, P.O. Box 910215, Lexington, KY 40591-0215, USAe-mail: [email protected]

1 Introduction

Over the past 60 years, the stress-softening phenomenon known as the Mullins effect thatoccurs in various incompressible elastomers and biological tissue has been a subject ofstudy by many researchers. See, for example, [1–9] among numerous other worksreferenced therein. Both experimental and theoretical studies leading to characterization ofthe stress-softening constitutive behavior of these kinds of rubberlike materials arepresented. These works mainly focus on phenomenological models and some comparisonswith experimental results from static tests. A few articles deal with dynamical problems.The Mullins effect on the natural frequency of small transverse oscillations of a stretchedrubber cord is studied in [3, 8]. The frequency of small amplitude longitudinal oscillationsof a load superimposed on a large static stretch or pre-stretch of a rubber band or cordwithout stress-softening is reported in [10, 11]. Some parallel results from dynamical testson biological tissues are explored in [12], and small amplitude frequency effects due tolimited extensibility of molecular networks without stress-softening are characterized in[13] for the Fung and Gent material models.

In this paper we study the Mullins effect in both the small and the finite amplitudeoscillations of a load supported by a homogeneous, incompressible, perfectly flexible,stress-softening rubber cord characterized by the Zúñiga and Beatty [7] constitutive model.This model involves only two material constants and it has demonstrated results that are infair agreement1 with both static [7] and dynamic [8] experimental data involving theuniaxial deformation of rubber strings. For further simplicity, the virgin material response ischaracterized by the classical neo-Hookean model. We recall that Beatty and Chow [14]provide an analytical, closed-form solution of the same dynamical problem for the neo-Hookean material without the stress-softening effect. Therefore, their solution is used tocompare the neo-Hookean, parent material response with results obtained for the stress-softening neo-Hookean model.

The dynamical problem of a loaded stress-softening, neo-Hookean rubber string isformulated in Section 2. Static equilibrium stretches for both the virgin and the stress-softened cases are discussed. The linearized oscillation problem about a static equilibriumstretch of the virgin and stress-softened materials is studied in Section 3, and expressionsfor the superimposed small amplitude oscillational frequencies are derived and discussedin some graphical illustrations. The finite amplitude oscillation problem is studied inSection 4. First, the simpler free-fall oscillatory motion is explored. The problem forgeneral initial conditions in the vertical uniaxial motion that leads to a three phase motionconsisting of the suspension, free-flight and retraction phases is investigated. Severalillustrative examples that describe the variety of possible motions of the load and thevariety of effects due to the extent of the stress-softening and the size of the softeningparameter are presented. All results are illustrated graphically and numerous tabulatednumerical results are provided throughout. Some concluding remarks are provided atthe end.

1 Other complex molecular based models studied in [8] show superior agreement with test data for thetransverse vibrational frequency versus cord stretch, whereas the neo-Hookean model provides only fairagreement in corresponding cases illustrated there. The analytical simplicity of the latter, however, is moreadvantageous in our studies here. Of course, the comparison also depends on the particular softening modelused in the study, and the simple Zúñiga–Beatty model is merely one example among other more complexstress-softening material models [5, 6, 9, 15]. We shall return to this later on.

116 S. Sarangi et al.

2 Formulation of the Problem

Let us consider a perfectly flexible, incompressible, isotropic, and hyperelastic rubber stringof undeformed length L, cross-sectional area A and shear modulus G, fixed at one end,fastened to a mass M at the other, and suspended vertically. At time t the length is l(t) andthe stretch l(t)≡l(t)/L. We shall assume that the mass of the string is negligible comparedwith that of the load M, as usual; each motion phase of the system is conservative; thestretch of the rubber cord is uniform and homogeneous so that the uniaxial tension T=T(l(t)); and the string is characterized by the phenomenological Zúñiga–Beatty [7]constitutive model for incompressible, stress-softening rubberlike materials, the virginmaterial being neo-Hookean. Wave propagation effects are ignored.

We shall follow the formulation by Beatty and Chow [14] and introduce a Cartesianframe with origin at the fixed end of the string and oriented vertically such that thegravitational vector g=gj, j being the unit vector directed downward. The verticalrectilinear motion of the load M is defined by its position vector y(t)j; and when thestring is stretched y(t)= l(t). There are three phases of the oscillation; (a) the suspensionphase (y(t)≥L), (b) the free-flight phase (−L≤y(t)≤L), and (c) the (slingshot-like) retractionphase (y(t)≤−L). Depending on the initial conditions, the mass may oscillate in thesuspension phase alone, or the overall motion may include the other two phases as well.The governing equation of motion for the load in the suspension and retraction phases isthus given by:

l00 þ T lð ÞML

¼ �1ð Þαp20; ð1Þ

in which a prime denotes the derivative with respect to time t: l0 tð Þ � dl tð Þ=dt; T(l) is thetension in the string at time t; α=0 or 1 for the suspension and retraction phases of motion,respectively; and:

p0 �ffiffiffiffig

L

r: ð2Þ

After a convenient shift of the origin to y=L and with lf � 1� y=L, the free-flight phase isdescribed by:

l0 0f tð Þ ¼ �p20; T lf

� � � 0 for lf 2 0; 2½ �: ð3Þ

For the neo-Hookean virgin and stress-softened materials the string tension is givenrespectively by (see [7, 14, 16]):

T lð Þ ¼ AG l� 1

l2

� �; ð4Þ

and (in accord with the Zúñiga–Beatty [7] constitutive model)

T lð Þ ¼ AG l� 1

l2

� �e�b

ffiffiffiffiffiffiS�s

p; ð5Þ

Effect of stress-softening on the dynamics of a load 117

with A being the undeformed cross-sectional area. The constant b>0 is called the softeningparameter, s ¼ l4 þ 2

�l2

� �1=2is the current magnitude of the strain (the strain intensity) to

which the string is subjected in a uniaxial extension at time t, and

S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛ4 þ 2

Λ2

rð5aÞ

is its maximum previous value. The maximum previous uniaxial stretch (see [6, 7]) isdenoted by Λ.

In the suspension phase for which α=0, the (generic) static equilibrium stretch le undera given weight Mg is obtained from the equilibrium equation T leð Þ ¼ Mg. The respectiveequilibrium equations for the virgin and stress-softened neo-Hookean materials thus followfrom (4) and (5) as

l3ev �1

2m l2ev � 1 ¼ 0; ð6Þ

and

l3es �1

2m l2ese

bffiffiffiffiffiffiffiffiS�se

p� 1 ¼ 0; ð7Þ

wherein lev, les are the respective virgin and stress-softened equilibrium stretches,se ¼ ðl4es þ 2=l2esÞ1=2, and μ is a dimensionless load parameter defined by:

m � 2Mg

AG: ð8Þ

Hereafter, when no confusion with the load M itself may ensue, the proportional loadparameter μ is briefly called the load. Its reciprocal k � 1=μ is the material stiffnessparameter2 adopted in [14].

Figure 1, constructed from the solutions of (6) and (7), shows the variation of the staticequilibrium stretch le with μ for the neo-Hookean virgin and stress-softened materialresponse for various values of the maximum previous stretch Λ of a buna-N elastomer [7]for which b=0.527. Note that the stress-softened equilibrium curves originate from thevirgin curve at the corresponding values lev ¼ les ¼ Λ. It is then evident from the diagramthat for a given value of μ (i.e., essentially for a given value of the load M), the staticequilibrium stretch of the stress-softened material for les<Λ exceeds that of the virginmaterial: les>lev; and under the same load their difference grows as the value of themaximum previous stretch Λ is increased. Similar results are exhibited in the engineeringstress-stretch curve in Fig. 4 and discussed later on. We now return to (1) to explore thesmall amplitude oscillation problem.

2 The 1/2 factor in k is introduced by Beatty and Chow [14] to avoid appearance of an additional factor of 2in the closed-form analytical solution. Because their result is used later for comparison with the stress-softened case, we here retain their definition for k.


3 Small Superimposed Oscillations about a Finite Static Extension

Let l ¼ le þ d, where |δ|≪1, and for the suspension mode set α=0 in (1), recall theequilibrium relation and thus obtain the linearized equation of motion of the load as:

δ00 þ AG

ML

d T lð Þ=AGð Þdl

� l¼le

δ ¼ 0; ð9Þ

with δ0 tð Þ � dδ tð Þ=dt, as before. Note that AG is used to normalize the tension. For thevirgin and stress-softened material response the following linearized equations of motionare obtained with the use of (4) and (5) in (9):

1. for the virgin case,

δ00 þ w2vδ ¼ 0; ð10Þ

2. for the stress-softened case,

δ00 þ w2s δ ¼ 0; ð11Þ

wherein wv , ws denote the corresponding vibrational frequencies of the load. Then, with:

AG

ML¼ 2

p20m

ð12Þ

from (2) and (8), the respective normalized natural frequencies are defined by:

eωv �ωv

p0¼ 2 13ev þ 2

� �μ13ev

" #1=2

; eωs � ωs

p0¼ 2

μ

b 16es � 1� �

13es � 1� �

14es

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi16es þ 2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiS � se

p þ 13es þ 2

13es

0B@1CA exp �b

ffiffiffiffiffiffiffiffiffiffiffiffiffiS � se

p �2643751=2

:

ð13Þ

Fig. 1 Static equilibrium stretchvariation with the load parameterμ for a stress-softeningneo-Hookean string subject tovarious values of maximumprevious stretch Λ for a buna-Nelastomer with b=0.527


These equations hold for the same load M and virgin properties of the undeformedstring. But the same load will produce different static stretches in accordance with (6) and(7). The respective load parameter μ, however, for possibly different loads may beeliminated by use of (6) and (7) to recast these respectively in terms of lev and les alone; wethus obtain:

ewv ¼l3ev þ 2� �

lev l3ev � 1� �" #1=2

; ews ¼ l3es þ 2

les l3es � 1� � þ b l6es � 1

� �l2es

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil6es þ 2� �

S � seð Þq

2643751=2

: ð14Þ

Notice that (14)1, first reported by Lawton and King [12], is a universal relation [10, 11]; itis independent of any material constants, whereas due to the presence of the softeningparameter b, (14)2 is not.

The relations (6) and (7) show that for the same static stretch, the respective values of μfor the corresponding applied loads, namely, μv and μs, are related by mv ¼ mse


p. This

difference is taken into account in writing (14); and hence for the same static stretch, thesmall amplitude, normalized frequency of the load for a stress-softened string is alwaysgreater than that for the virgin neo-Hookean string. Precisely, with lev ¼ les ¼ le in(14), we have:

ew2s ¼ ew2

v þb l6e � 1� �

l2e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil6e þ 2� �

S � seð Þq : ð15Þ

The first relation in (14) shows that the normalized vibrational frequency for the neo-Hookean virgin material is a monotone decreasing function of the static stretchlev 2 1;1ð Þ, accordingly varying from ∞ to 0. The second of (14) for the normalizedstress-softened material frequency of the load goes to ∞ at both end points les∈(1,Λ) foreach value of the maximum previous stretch Λ of the virgin material; and hence thisfunction must have a minimum at some interior point. These effects are illustrated in Fig. 2for a buna-N material with b=0.527 and for various values of Λ. Moreover, notice in Fig. 2that for the same static stretch ews > ewv, in accordance with (15). To maintain the same staticstretch, however, the virgin material string must be subjected to a greater static load for le∈(1,Λ), that is, mv ¼ mse


p, as remarked above. Therefore, an alternative comparison of

normalized frequencies for the same static load of identical strings, i.e., for the same valuesof μ ¼ 2Mg

�AG, is provided in Fig. 3, for various values of the maximum previous stretch

Λ∈[1.5, 4] of the virgin buna-N cord.We see from (6) and (7) that as μ→0, le→1; consequently, from (13), both normalized

frequencies ewv and ews tend to infinity, as shown in Fig. 3. Moreover, for each value of Λ inthe stress-softened case, there exists a value μ=μm for which the frequency is least. Forexample, for Λ=4, μm=0.925 and the corresponding normalized frequency for the stress-softened material is ewsð Þm¼ 0:949 in Fig. 3.

Figure 3 also shows that for each value of Λ, there exists a value of μ=μ*, i.e., a loadM* for which the normalized, small amplitude frequencies for the virgin and stress-softenedneo-Hookean materials are equal. Specifically, for Λ=1.5 and 2.0 these values are noted inFig. 3. Hence, for μ<μ*, the normalized frequency of the load for the stress-softenedmaterial is smaller than that for the virgin material, whereas for μ>μ* the opposite effectobtains. Moreover, as the value of Λ grows, the load vibrational frequency ews of the stress-


softened string decreases in comparison with the frequency for any previous value of Λover the latter's range of μ.

These stress-softening frequency effects in the superimposed small amplitude vibrationproblem may be further characterized geometrically in terms of the stress-stretch responsecurves shown in Fig. 4. First, let eT � T=AG denote the normalized engineering stress, andconsider the difference of the monotone normalized stresses as functions of l for the virginand stress-softened materials defined by y lð Þ � eTv lð Þ � eTs lð Þ, which vanishes at the endpoints of the interval: y 1ð Þ ¼ y Λð Þ ¼ 0. It is evident from the geometry in Fig. 4, and itfollows as well from Rolle's theorem, that there exists a unique value of l=l*∈[1,Λ] forwhich the difference function is greatest, and hence dy lð Þ=dl

l*

�� ¼ 0. Consequently, theslopes of the two stress-stretch response graphs at l=l* are equal; deTv lð Þ=dljl*¼deTs lð Þ=dljl*. With the aid of the generic frequency relation Mw2 ¼ AG=Lð ÞdeT lð Þ=dl��l¼leevident from (9), we find that this equality yields for the neo-Hookean class of Zúñiga-Beatty materials a universal relation for the normalized frequencies given by Mvew2

v ¼ Msew2s

Fig. 3 Variation of normalizedfrequencies with μ for variousvalues of Λ for a buna-N cordwith b=0.527

Fig. 2 Variation of normalizedfrequencies with le for variousvalues of Λ for a buna-Nelastomer with b=0.527


at the stretch l=l*. That is, independent of any properties of the simple neo-Hookeansupport and depending only on the appropriate loads, the normalized frequency of a loadon the stress-softened neo-Hookean string is proportional to and greater than that for anappropriate load on the virgin material string at the stretch l*:

ews ¼ffiffiffiffiffiffiffiMv

Ms

r ewv > ewv: ð16Þ

Here we recall that to produce the same static stretch Mv>Ms, equivalently μv>μs, wherefrom (12), Mv=Ms ¼ μv=μs . The inequality in (16) is consistent with our earlier resultews > ewv for the normalized frequencies based on (15) for the same stretch, hence differentload parameters μ. Obviously, the same relation (16) holds for the frequencies as well:ws>wv. In fact, more generally, based only on the equality of the tangents at the extremestretch l=l* on the virgin and stress-softened response curves in a simple uniaxialextension for which the greatest previous stretch is Λ, it follows that the same universal rule(16) holds for all stress-softening Zúñiga-Beatty materials for which the virgin response ofthe normalized engineering stress is a monotone increasing function of the uniaxial stretch.

For the data in Fig. 4, for example, we find l*=2.449; and hence eTv 2:449ð Þ ¼ 2:282,eTs 2:449ð Þ ¼ 0:918, and deTv lð Þ=dljl*¼ deTs lð Þ=dljl*¼ 1:136. With the aid of (8), thisimplies that the normalized small amplitude, longitudinal frequencies at l*=2.449 arerelated by:

μv

2ew 2

v

��l*¼ μs

2ew 2

s

��l*¼ 1:136: ð17Þ

The physical relevance of this geometrical result is thus provided by (16).The monotone normalized stress-stretch response functions being continuous and one-

to-one are plainly invertible for all l∈[1,Λ]; therefore, we may write lv ¼ fvðeTÞ and ls ¼

Fig. 4 Normalized engineeringstress eT lð Þ versus stretch l forthe neo-Hookean virgin andstress-softened materials unloadedfrom the softening point at Λ=3.0.The dash-dot line is the stress-softened tangent line at l=1


gsðeTÞ for the corresponding stretch-stress response of the virgin and stress-softenedmaterial. Then the difference of the stretch functions defined by χðeTÞ � gsðeTÞ � fvðeTÞvanishes at the end points χ 0ð Þ ¼ χ

~T Λð Þ� � ¼ 0. Hence, as before, there exists a unique

value of the normalized stress eT*¼ M*g=AG � μ*=2 2 ð0; eT Λð ÞÞ at which the slopes ofthe stretch-stress response curves are the same: dgsðeTÞ=deT ��eT *¼ dfvðeTÞ=deT ��eT * , and atwhich lev ¼ fvðeT*Þ and les ¼ gsðeT*Þ. The unique value 2eT* ¼ μ* is the point in Fig. 3at which the normalized frequency of vibration of the load for the stress-softened stringews and for the virgin material string ewv are the same.3 For the case Λ=3 in Fig. 4 forexample, eT* ¼ 1:125, μ*=2.250 and dgsðeTÞ=deT ��eT*¼ dfvðeTÞ=deT ��eT*¼ 0:648. Thecorresponding static stretches under the same static load M*, hence μ*, from (6) and(7), are lev=1.544 and les=2.606; and in this instance the corresponding normalizednatural frequencies are ewv ¼ ews ¼ 1:171. Plainly, as demonstrated in Fig. 3, points ofcoincident vibrational frequency for the virgin and stress-softened cord exist for eachvalue of the maximum previous stretch to which the virgin material is subjected, all arepoints at which slopes of the stretch-stress curves are equal and the difference of theequilibrium stretches is greatest.

These tangents provide a direct measure of local stiffness effects at each point on thecurves in Fig. 4. In the region μ<μ*, i.e., eT < eT*, the local stiffness for the stress-softenedpath is less than that for the virgin path. Accordingly, the stress-softened natural frequencyfor each Λ is smaller than that of the virgin material; and the greater the softening (i.e., thelarger the maximum previous stretch of the virgin material), the smaller the frequency, atevery value of μ<μ*. The picture is reversed in the region μ>μ*, i.e., where eT > eT*. Thelocal stiffness at every value of μ for the stress-softened material now is larger than itsvirgin counterpart; and hence the stress-softened natural frequency of the load for each Λexceeds that for the virgin material.

We should also observe (see Fig. 1) that for each Λ there exists a value of μ at whichlev ¼ Λ ¼ les S ¼ seð Þ and from (13)2 the frequency is undefined. For example, in Fig. 3,at μ=4.680 and 5.778 corresponding to Λ=2.5 and 3.0, respectively, the normalizedfrequencies are indeterminate. These points are considered to lie on the virgin materialcurve, in as much as we assume that the stress in loading along the virgin response curvedoes not change very much in small amplitude motion. Moreover, it should be emphasizedthat all of the foregoing numerical results adopt a softening parameter b=0.527. It isexpected, of course, that parallel numerical results may be readily obtained for any assignedsoftening parameter. We shall return to this at the end.

Finally, we recall the early experiments of Lawton and King [12]. They studied the smallamplitude longitudinal vibration of a load superimposed on a static equilibrium state forseveral varieties of rubber (for stretch le<6) and aortic tissue strips extracted postmortemfrom young and old human males (for stretch le<2). The specimens were statically loadedcontinuously without unloading, so the measured superimposed small amplitude oscilla-tional frequencies are essentially characteristic of virgin materials only. It was found that allof the experimental response curves for the normalized frequency versus static stretch, forthe most part, fell above and approached the universal neo-Hookean frequency (14)1vertically as le→1, and definite frequency minima were exhibited in all cases, very muchlike that shown in Fig. 2. Preconditioning the rubber specimens to a stretch le=4, however,did not appreciably alter their frequency response for stretches le<2, the range shown in

3 The two equations in (13) then yield an expression relating the corresponding static stretches for anassigned b>0; but we shall make no use of it.


[12]. For the same static stretch of the virgin tissues, the old human aorta exhibited stifferresponse characterized by higher frequency values than the young one.

Lawton and King [12] attribute the U-shaped, upturn effect in the frequency responsecurves to the limiting molecular chain extensibility of the materials, based on the non-Gaussian James–Guth model [17], which accounts for the increased stiffening effect ofshort molecular chain structures. The same stiffening effect in the small longitudinalfrequency response is illustrated by Horgan and Saccomandi [13] for the simpler Gentmodel [18]. In our work here, however, the upturn and stiffening effects are due solely tothe stress-softening function, which reflects some sort of molecular chain structuredegradation, the extent of which will vary with the value of the softening parameter andthe maximum previous stretch. A major open problem in the theory of stress-softeningentails the analytical characterization of the role and evolution of chain damage, and thedetermination of how the material damage function enters the general constitutive equationfor the material. To circumvent this complexity, the simple front factor model used here andin [6–8], for example, has been adopted to explore some physical aspects of stress-softeningand to provide a mechanism by which softening effects may be examined, predicted andfurther understood. While more complex stress-softening material models, including non-Gaussian molecular based models, may yield more accurate predictions, the overall effects,perhaps more or less intense, are expected to be similar to those reported here. A generaltheory of stress-softening in the uniaxial extension of elastomers having limiting molecularchain extensibility characterized by the Gent model is presented in [15]. The combinedeffects of limiting molecular chain extensibility and stress-softening in the small amplitudelongitudinal vibrations of a load on a rubber cord or biological tissue strip will be reportedelsewhere.

4 Finite Amplitude Motion

We now investigate the finite amplitude oscillation of a load supported by a neo-Hookean,stress-softening rubber string. In the absence of stress-softening effects, this problem hasbeen studied by Beatty and Chow [14], wherein closed form analytical solutions areprovided for a parent, or purely neo-Hookean string. In the current context, therefore, someof their results will provide a basis of comparison.

Let et � p0t denote the dimensionless time and define�l � dl

�det. Then substitution of

T(l) from (5) into (1) and use of (12) yields the governing equation of motion of the load inthe suspension (α=0) and retraction (α=1) phases for a neo-Hookean stress-softeningmaterial:

��l tð Þ þ 2

μl� 1

l2

� �e�b

ffiffiffiffiffiffiS�s

p¼ �1ð Þα: ð18Þ

We recall that (3), now written as:

��lf ¼ �1; T lf

� � � 0; for lf 2 0; 2½ �; ð19Þmay be used to obtain the motion in the free-flight phase. We recognize that (18) fallswithin the general class of twice integrable nonlinear differential equations described in[20] and from which a general formula for the small amplitude period, hence the frequencyof oscillations of the load about a relative equilibrium state may be readily obtained. Thisclass includes the equations of motion for the finite amplitude oscillations of shearing


oscillators and thick-walled, circular tubes and spherical shells. And hence our foregoing(13) for the normalized frequency of the small amplitude motions may be readily derivedfrom (18) following this unified structure based on equations of motion of the generalautonomous type

��x ¼ f x;

�x

� �.

Here we focus on the finite amplitude motion. Thus, for the general initial conditionsl 0ð Þ; �l 0ð Þ

�¼ l0;

�l0

�, the first integral of (18) may be expressed as:

1

2

�l2 þ 2

μ

Z l

1ζ � 1

ζ2

� �e�b

ffiffiffiffiffiffiffiffiffiffiS�s ζð Þ

pdζ � �1ð Þalþ 1 ¼ E l0;

�l0

�; ð20Þ

where the constant of integration, given by:

E l0;�l0

�¼ 1

2

�l20 þ

2

μ

Z l0

1ζ � 1

ζ2

� �e�b


pdζ � �1ð Þal0 þ 1; ð21Þ

is a normalized measure of the total energy of the motion such that for α=0, E(1, 0)=0.With v lð Þ � �

l et� �for the normalized velocity, a second integration yields the normalized

travel time of the motion: et ¼ R ll0dl=v lð Þ . Therefore, when the motion is periodic with

turning points χ={c, d} at which v(χ)=0, the normalized period et ¼ p0t and hence thenormalized frequency ew ¼ w=p0 of the finite amplitude motion are given by:

et ¼ 2πew ¼ 2

Z d

c

dlv lð Þ: ð22Þ

The integral in (20), however, cannot be reduced to a closed-form. Moreover, stress-softening introduces additional difficulties in the characterization of the motion. Therefore,(18) is solved numerically to obtain the finite amplitude response of the load for the stress-softening case. On the other hand, (20) may be used to determine the turning points of themotion and to explain certain aspects of the response behavior. Upon setting b≡0, we seethat (18) reduces to the equation of motion of the load supported by a parent neo-Hookeanstring for which the exact solution is given in [14]. The same equation, however,characterizes the non-oscillatory motion throughout the loading phase of the virgin, neo-Hookean softening material for which b≠0. In this case, the maximum previous stretch isits current value, i.e., S=s, and the motion on the virgin path of the stress-softening string isthus described by the parent neo-Hookean oscillator equation only until the current stretch lreaches its ultimate value Λ under the assigned load, after which the load retreats along astress-softened path where s<S. Plainly, the virgin material motion is not periodic. We thuscompare the stress-softened periodic motions with those of the parent neo-Hookeanoscillator developed in [14].

4.1 Free-fall Motion

We begin with the free-fall motion of the load released from rest at the undeformed virginstate of the neo-Hookean string so that l0=1,

�l0 ¼ 0 initially. We then explore the free-fall

motion from a stress-softened state. It is seen that the equation of motion (18) for the virginneo-Hookean material (S=s) can be integrated, as shown in [14], to derive the velocityequation from which the turning points in the motion of the load on the virgin string arethen obtained as Ξ={1, Λ}. Both the maximum previous stretch Λ and the softeningparameter b influence the subsequent response in the stress-softened motion. The resultsthat follow for various examples of motion for the stress-softened neo-Hookean material are


obtained from the numerical solution of (18). For the virgin material, the analyticalsolutions obtained in [14] are used.

4.1.1 Free-fall from the Undeformed Virgin State

The phase plane motions�l ¼ v lð Þ versus l for the parent, virgin and stress-softened neo-

Hookean materials are shown in Fig. 5 for several values of the load μ. The closed, dashline ovals trace the periodic motion of the load for the parent neo-Hookean material.However, for the stress-softening material the primary motion of the load on the virginstring starting from l=1 follows only the small dash curve in the region

�l > 0 as the load

falls to its maximum deflection at the stretch Λ, the extreme turning point in the subsequentstress-softened motion. The load then rebounds and oscillates with turning points χ={c, Λ}in the stress-softened suspension phase, as shown by the closed, solid oval-shaped curves inFig. 5.

The value of Λ for which�l ¼ 0 in (20), with b≡0 for the parent neo-Hookean material,

is determined by:

l3 � ml2 þ m� 3ð Þlþ 2 ¼ 0: ð23ÞClearly, this is satisfied identically for the primary state at l=1; and, for example, withμ=3.333 in (23), we find the ultimate stretch Λ=3.0. The subsequent phase plane motion of

Fig. 5 Phase plane plots in the free-fall case from the undeformed virgin state for various values of the load parameterμ. Small dashed curve virgin material response; solid curve stress-softened material response for buna-N rubber


the mass is then governed by (20) with b=0.527 and S(Λ) given by (5a). The turning pointsχ={c, d} of the load for the stress-softened material thus follow from (20) with 1

�¼ 0 . In

the present example, this yields the stretches c=2.643 and, of course, d ¼ Λ ¼ 3:0 . It isseen from (7) that the stress-softened motion occurs about the relative equilibrium stretchles=2.850; and the normalized period determined numerically from (22) is ets ¼ 4:270 . Wefind that in the absence of stress-softening the normalized period of the finite motion of theload on the parent string is etp ¼ 6:924 about the relative equilibrium state at lev=1.934.

It is evident in Fig. 5 that the amplitudes of the stress-softened oscillations are small incomparison with those for the parent material under the same load. Specifically, for μ=3.333, the stress-softened stroke (d−c)s=0.357 is much smaller than the stroke (Λ−1)p=2.0for the parent material, and hence the displacement (Λ−1)v of the load on the virginmaterial string. For the stress-softened material, we are thus led to compare the normalizedperiod for the finite motion with the corresponding period of the superimposed smallmotion determined from (13); we find setsmall ¼ 4:443. This agrees roughly with thenormalized period ets ¼ 4:270 for the finite motion obtained above. Similar results may beobtained for other values of μ. These are given in the table of Fig. 5 for the severalexamples illustrated there. Of course, for other values of the softening parameter, the phaseplane characterization of the motion will vary, perhaps significantly for small values ofb for which the motion for the stress-softened material will differ but little from that for theparent neo-Hookean material.

Because of deformation induced microstructural damage in the virgin material, l=1 cannever be the smaller turning point in the stress-softened motion— it may only approach thisvalue as μ→0, i.e., for very small loads. Obviously, the free-fall motion for the parent neo-Hookean material model always occurs with turning points Ξ={1, Λ}; and, as expected, themaximum previous stretch increases significantly with the load μ, as seen in Fig. 5. Plainly,the phase plane plot for motion in the stress-softened path is symmetric about

�l ¼ 0, in

accordance with (20); and for a fixed value of μ, the maximum velocity in the stress-softened motion is always less than that in the corresponding motion for the parent neo-Hookean, and hence the virgin material string.

The table in Fig. 5 reveals that as μ increases from 1.0, the stroke d � cð Þs¼ Λ� cð Þsand both the normalized small and finite amplitude stress-softened periods, first increaseand then decrease. Hence, there exists a value of μ for which the amplitude and bothperiods in the stress-softened motion are greatest, whereas both the period and the finiteamplitude for the parent model grow continuously with the load.

The extreme amplitude phenomenon is illustrated in Fig. 6 wherein variations of thestatic stretch les, the turning points c and d=Λ, and the stroke (Λ−c)s, all continuouslyvarying with μ, are shown. For a given value of μ, the static stretch les is readily evaluatedfrom (7) for each extreme stretch Λ obtained from (23). Numerically, for our buna-N model,the maximum stroke d � cð Þs¼ 0:402 and the greatest values of the periods ets ¼ 4:271 and

setsmall ¼ 4:461 occur for μ=2.067. The asymmetric oscillation occurs about les=1.871 withturning points (c, d)=(1.643, 2.045), hence with maximum stroke Λ� c ¼ 0:402, all notedin Fig. 6. The diagram also shows that for sufficiently small values of μ, the amplitudes ofthe stress-softened motion decrease and the motion is virtually symmetric with respect toles. For example, for μ=0.167, les =1.030, Λ=1.0576 and c=1.0022, henceles � cþ Λð Þ=2 ¼ 1:0299; and from the solution of (22), numerically the normalizedperiod of oscillation is found to be 1.085, whereas (13)2 yields 1.094. Thus, in the range ofsmall loads μ, the stress-softened motion is very closely described by the linearized (11), asexpected. For intermediate and physically meaningful greater loads μ, however, the finiteamplitude oscillation solution characterizes the motion.


4.1.2 Free-fall from a Stress-Softened State

It is seen in Fig. 5 that the load μ=2 in its free-fall motion on a virgin stress-softeningcord eventually moves on a stress-softened path with turning points (c, Λ)=(1.5983, 2)following the dynamical extension of the virgin string to its maximum stretch Λ=2. Theenergy in the stress-softened rebound motion is readily determined from (20) asE Λ; 0ð Þ ¼ E 2; 0ð Þ ¼ �0:400. Now, suppose this softened cord is unloaded and the sameload is released from rest at the unstretched state for which the constant energy E(1, 0)=0.Since this energy exceeds that in the previous motion on the same stress-softened cord, wemay expect that when the load reaches the maximum previous stretch Λ=2, it will leave thestress-softened path with a velocity v(Λ) and move again on the virgin response path until itreaches a new greatest previous stretch Λ* in a second-stage softening motion. In theexample for μ=2, the exit velocity is v(2)=0.8950, the new maximum stretch is Λ*=2.4031, and the load now moves on a new stress-softened, periodic trajectory with turningpoints (c, Λ*) and constant total rebound energy E

�Λ*; 0

� ¼ �0:476 < 0. Perhaps contraryto one's intuition, if the same load in this example is now released in free-fall as before, thesame stress-softening effect will be repeated over and over, indefinitely until the cord isfully softened, essentially at Λ* ¼ 1. It may be shown that the load μ0 that will justreach the new maximum previous stretch Λ*=2.4031 in its free-fall motion ism0 ¼ 1:321 < m ¼ 2; and hence the same original load μ=2 in its free-fall will once againmove in the virgin path further softening the cord. This interesting phenomenon willcontinue indefinitely in free-fall motions under the same load μ=2 until the cord is fullysoftened at Λ* ¼ 1.

The same effect occurs under any load μ>μ0, where μ0 is the load for which a free-fallmotion with l∈[1, Λ] is possible on a given preconditioned string. It is shown below forour special buna-N model, for example, that the smallest value of the load parameter forwhich a free-fall periodic motion on a preconditioned cord is possible occurs with μ0=1.397 for Λ*=3.261; and hence any larger load m > m0 ¼ 1:397 will continue indefinitelyto soften the cord upon repeated free-fall motions of that load. If the load is reduced so thatμ<μ0, the extreme stretch l=d in the free-fall motion will be smaller than the maximum

Fig. 6 Static stretch les, turningpoint stretches χ={c, Λ}, and thestroke (Λ−c)s as functions of μfor the stress-softened response inthe free-fall motion of M from theundeformed virgin state of abuna-N rubber string withb=0.527. Dash-dot line staticstretch; solid line turning pointstretches


previous stretch Λ, and the periodic, stress-softened motion will remain within thecorresponding preconditioned region with l∈(1, Λ).

Similarly and more generally, let us consider a load μ on a stress-softened cordpreconditioned to a maximum previous stretch Λ and subjected to general initial conditions10;

I10

� �for which E0 ¼ E 10;

I10

� �is the corresponding constant initial energy of the

ensuing stress-softened motion. Thus, as before, there exists a load μ0 that will just attainthe maximum previous stretch in the motion for the assigned energy E 10;

I10

� �. If the load is

smaller so that μ≤μ0, it remains on the stress-softened path for Λ; otherwise, the stretchunder a greater load μ>μ0 will eventually exceed its greatest previous value, and the loadwill then continue along a new virgin path that leads to a secondary stress-softened motion.An equation for the special limiting load μ0 for a constant initial energy motion, includingthe free-fall case, is presented farther on.

With the foregoing physical description in mind, let μ0 denote the load that will justattain the greatest previous stretch Λ of the stress-softened material. For a larger load μ>μ0,the motion will leave the current stress-softened state in the suspension mode with avelocity v*=v(Λ), say, determined by:

1

2v*2 ¼ E0 � 2

μ

Z Λ

1ξ � 1

ξ2

� �e�b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS Λð Þ�s ξð Þ

pdξ þ Λ� 1ð Þ; ð24Þ

in accordance with (20), E0 ¼ E 10;I10

� �denoting the aforementioned initial constant

energy of the ensuing stress-softened motion. The load thus leaves the currentpreconditioned state at (Λ, v*) and moves along the virgin material response path to anew maximum previous stretch Λ* at which

I1 ¼ 0. In accord with (20), Λ* is given by

2

μ

Z Λ*

1ξ � 1

ξ2

� �dξ � Λ* � 1

�¼ Ev Λ; v*

�; ð25Þ

in which Ev(Λ, v*) is provided by (21) for the virgin material. Finally, the load in itsrebound on the newly stress-softened cord now has a periodic motion for which theconstant energy E*=E(Λ*, 0), from (21), is

E* ¼ 2

μ

Z Λ*

1ξ � 1

ξ2

� �e�b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS Λ* �

�s ξð Þr

dξ � Λ* � 1 �

: ð26Þ

The subsequent turning point at l=c in the new stress-softened motion, by (21), is thenobtained from E(c, 0)=E*. In fact, either this or (26) may be used in the primary free-fallmotion of the load on the virgin string to determine the constant rebound energy from theprimary maximum stretch Λ*=Λ. Specifically, for the free-fall motion with μ=2 and Λ=2, orwith c=1.5983, illustrated in Fig. 5, for example, we find E=−0.400, as mentioned previously.

In addition, the foregoing relations show that for the entire motion in the suspensionphase for a load μ>μ0 moving consecutively through the former softened and new virginstates with initial energy E0 ¼ E 10;

I10

� �the following relation holds:

E0 ¼ 2

μ

Z Λ

1ξ � 1

ξ2

� �e�b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS Λð Þ�s ξð Þ

pdξ þ 2

μ

Z Λ*

Λξ2 � 1

ξ2

� �dξ � Λ* � 1

�: ð27Þ

Of course, in a free-fall motion from the unstretched, softened state E0=0.


We now return to some examples. First, consider the value of μ0 that produces aspecified value of the maximum previous stretch Λ in the free-fall motion of a load on astress-softened cord. This is readily determined from (24) with v*=0, specifically, for Λ=3,μ0=1.391. Accordingly, the phase-plane diagrams provided by (20) for the free-fall motionon a stress-softening string preconditioned to a stretch Λ=3 and for several values of m �m0 ¼ 1:391 are shown by the solid closed curves in Fig. 7, each starting at l0=1. It is seenthat the extreme stretch and maximum velocity in the free-fall motion for each case increaseas μ increases from 0.333 to 1.391. Consequently, in this example for our buna-N cord themotion remains on the stress-softened path in Fig. 4 shown earlier for Λ=3. Moreover, inany motion from a stretch l0>1 and a load m � m0 ¼ 1:391, the motion continues on thesame stress-softened path of Fig. 4 but with smaller amplitude. This effect is illustrated bythe dash-dot-dot path in Fig. 7 for the case l0=1.250,

I1 ¼ 0, μ=1 in which the new

extreme stretch is ls ¼ d ¼ 2:507. But for the same initial data and a greater loadm ¼ 1:147 < m0, the extreme stretch is ls=2.679, as shown by the dash-dot path. Thetabulated data in Fig. 7 show that for d≤Λ both the normalized periods and amplitudes ofthe finite motions on the preconditioned string increase with the load μ∈[0.333, 1.391]) inthe same manner described previously for the parent neo-Hookean material.

Now, for m ¼ 2 > m0 ¼ 1:391, the load in its free-fall motion leaves the current,preconditioned stress-softened path in Fig. 4 at the stretch l ¼ Λ ¼ 3 with velocity v*=1.104 determined from (24). It then follows the virgin response path in accordance with(20) in which we set s=S for b=0.527, as shown in Fig. 7 by the dashed curve with doublearrows extending from the transition point (3, 1.104) to the new extreme stretch at Λ*=3.298. The virgin string material is thus subjected to second-stage softening for which thegreatest previous stretch is now Λ*=3.298. In the rebound motion from this state of stretchfor μ=2, the load moves on the new stress-softened trajectory shown by the solid, closed

Fig. 7 Phase plot in free-fallcases for various loads μ withinitial maximum previous stretchΛ=3. Dashed curve virginmaterial response; solid curvestress-softened material responsefor buna-N rubber


oval-shaped curve having single directional arrows. We find numerically that this periodicmotion occurs in the range 1.866≤l≤3.298 shown in Fig. 7, and the load does not return tothe undeformed state. Some additional numerical values, including the periods of vibrationfor the stress-softened cord, are provided in the table of Fig. 7. The constant total energy(26) of the new stress-softened periodic motion is E ¼ E ¼ �0:694. The example forμ=2 is discussed in a different context later on.

We recall that in the vertical free-fall motion of a mass attached to a linear spring themaximum dynamic deflection is always twice the static deflection. It is evident from theenergy curves and the tables in Figs. 5 and 7 that this is not the case for the nonlinear parent(hence virgin) and stress-softened neo-Hookean materials. Variation of the maximumdynamic deflection in the free-fall motion of the load in the suspension phase and twice thestatic deflection are mapped in Fig. 8 against the load μ for both the neo-Hookean virginand stress-softened buna-N rubber strings. It is seen that as the effective load μ increasesfrom zero, its maximum dynamic deflection on the virgin neo-Hookean cord is alwaysgreater than twice its static deflection 2(lev−1); and the difference grows somewhat largerwith the increasing load. A similar result is reported by Beatty [10] for the free-fall motionof a load supported by a purely neo-Hookean rubber spring capable of withstanding axialcompression without buckling.

For the stress-softened material, however, the situation is different. Although it may notbe visibly clear in Fig. 8, we find that corresponding to each Λ, the maximum dynamicdeflection (the solid) curve crosses the doubled static deflection (the dash) curve at aparticular value of the load μ=μi, say; and hence the dynamic deflection under the load μiin its free-fall motion is exactly twice its static deflection. We also observe that the slope ofthe stress-softened engineering stress-stretch curve for Λ=3 in Fig. 4 at first decreases fromthe origin and then increases again, so the slope of the stress-softened graph attains aminimum value as the load increases. In fact, it turns out numerically that for each value ofΛ, the intersection point at μi in Fig. 8 corresponds to the load at which the slope of thestress-softened, normalized engineering stress-stretch curve is least. So, for smaller loadsμ<μi, the maximum dynamic deflection of the load on the stress-softened cord is somewhatgreater than twice the static deflection, whereas the response for larger loads μ>μi isreversed. Thus, in physical terms, since the slope of the stress-stretch curve corresponds toa local material stiffness, the string behaves as a soft spring in the region μ<μi for which

Fig. 8 Extreme dynamicdeflection and twice the staticdeflection as functions of the loadin the free-fall motion on theloaded buna-N rubber cord


the maximum dynamic deflection exceeds twice the static deflection. The slope beyond μi,and hence the local material stiffness, increases significantly, leading to a hard spring typebehavior for which the maximum dynamic deflection is less, possibly much less, than twicethe static deflection.

Figure 8 also shows that for a given value of Λ and μ>μi, the difference between thedashed and solid curves increases with μ; and for a fixed load μ, the difference increaseswith Λ. Of course, the curves for each case terminate at the load μ0 corresponding to therespective value of the greatest previous stretch Λ in the free-fall motion. We have alsomapped in Fig. 8 the maximum dynamic and corresponding static deflections for Λ=2.679.Notice that the plots of the maximum dynamic deflection for Λ=4 and 2.679 reveal thesame values of the maximum extensional load μ0=1.365. Moreover, the value of μ0=1.391for Λ=3 is greater than its value μ0=1.365 for Λ=4. We shall return to this anomaly later on.

4.1.3 Constant Energy Motions of a Load on a Preconditioned Cord

Here we consider a load on a stress-softened cord preconditioned to a maximum previousstretch Λ and subjected to general initial conditions 10;

I10

� �, hence assigned constant

energy E 10;I10

� �given by (21). The stress-softened motion is then described by (20) so

long as the maximum dynamic deflection d≤Λ in either the suspension or retraction phasesof the load's motion. Therefore, the value of the load μ0 that will just attain the maximumprevious stretch in the motion for the assigned energy E 10;

I10

� �is readily determined from

(20) as:

μ0 ¼2R 0

1 ζ � 1ζ2

�e�b


pdζ

E l0;�l0

�þ �1ð Þa0� 1

: ð28Þ

If μ≤μ0, in its free-fall motion with E(1, 0)=0 the load remains on the stress-softened pathfor Λ; otherwise, the stretch under a greater load with μ>μ0 will eventually exceed itsgreatest previous value, and the load will then continue along a new virgin path that leads toa new stress-softened motion, as described earlier. Notice, however, that if the energyE 10;

I10

� � ¼ 2Λ in the retraction mode (α=1) of a cord preconditioned to a stretch Λ, thenthe load μ0 given by (28) is the same as that required to produce the maximum deflectiond=Λ in the free-fall motion with E(1, 0)=0, and conversely.

These results may be visualized graphically. The general nature of the variation of μ0with Λ for various values of the energy E is obtained numerically from (28) anddiagrammed in Fig. 9. The points of intersection of the dotted retraction energy curves withthe solid free-fall null energy curve are those points at which the retraction mode energyE 10;

I10

� � ¼ 2Λ (plainly always >2), the maximum deflection in the free-fall motion isd=Λ, and the load parameter μ0 is the same for both. The retraction curve for E=4, forexample, intersects the free-fall curve E=0 with a maximum deflection Λ=2 at the commonload μ0=1.199 given by (28) but not noted in Fig. 9. Similar effects are evident for theother retraction mode curves. Motions in the retraction phase will be discussed in the nextsection for general initial data. We continue with some important observations in the free-fall suspension phase.

The free-fall curve E=0 in Fig. 9, however, exhibits a perplexing effect. It is foundnumerically, as shown in the diagram, that the free-fall curve μ0(Λ) has a maximum valueμ0m(Λ0)=1.397 at a preconditioned stretch Λ ¼ Λ0 ¼ 3:261, very nearly. Consequently, forany larger load for which μ>1.397, its free-fall motion will leave the stress-softened path


with velocity I1 ¼ v Λð Þ at the stretch l=Λ, irrespective of the current value of Λ. Moreover,

for a somewhat smaller load μ0<μ0m, there exist two stress-softened states in a free-fallmotion of the same load for which the extreme dynamic deflections of the same cord maydiffer! In fact, this effect results in the anomalous, non-unique behavior exhibited earlier inFig. 8. There we saw that in a free-fall motion of the same load for which m0 ¼ 1:365 <m0m there exist two values, Λ=2.679 and Λ=4, of the extreme dynamical deflection of thesame cord. A similar anomaly prevails for the load parameter m0 ¼ 1:391 < m0m for which,as shown later in the table of Fig. 10, Λ=3.0 and Λ=3.551 are the extreme dynamicaldeflections.

It is useful to view this differently in terms of energy. Let us consider the maximumprevious extension in the suspension mode α=0 and rewrite (20) in the form:

V 0ð Þ � Vg 0; kð Þ ¼ 1

2μE l0;

�l0

�; ð29Þ

where

V 0ð Þ ¼Z 0

1ζ � 1

ζ2

� �e�b


pdζ; Vg 0;μð Þ ¼ 1

2μ 0� 1ð Þ: ð30Þ

Thus, V(Λ) is a stored potential energy of deformation and Vg(Λ; μ) is a gravitationalpotential energy function. Then, for the free-fall case for which Eðl0; �l0Þ ¼ E 0; 0ð Þ ¼ 0and μ=μ0, it follows from (29) that V(Λ)=Vg(Λ; μ0). The graphical representations ofV (the solid curve) and Vg (the dash curves) as functions of Λ in the free-fall case are shownin Fig. 10 for a few values of μ, the map of Vg being linear and μ=μ0 at their points ofintersection. The point coordinates in Fig. 10 are thus given as (Λ, V=Vg) for the notedloads μ, and the accompanying table records the multiple values of Λ at these intersectionpoints. Notice that for m ¼ m0m ¼ 1:397 , the straight line representing Vg is tangent to thecurve for V at Λ ¼ Λ0 ¼ 3:261, and V ¼ Vg ¼ 1:578 . This is the puzzling maximum point(Λ0, μ0m) of the E=0 curve in Fig. 9.

This aside for the moment, when μ>1.397, say for μ=2, the straight line denoting Vg

does not intersect the V curve in Fig. 10. In fact, for each value of Λ>1, the graphs ofFig. 10 show that Vg>V for all m > m0m ¼ 1:397; and hence the total potential energy

Fig. 9 Variation of μ0 withmaximum previous stretch Λ forvarious initial energy levels in thestress-softened case for b=0.527.Solid curve suspension phase;dotted curve retraction phase


V−Vg associated with the resulting stress-softened oscillation is now negative, andadditional kinetic energy is required to render the total energy zero in the current free-fallmotion of the load. Consequently, for m > m0m ¼ 1:397, at 1 = Λ, I1 6¼ 0 and the motion ofthe load evolves finally into a new stress-softened state. This physical scenario wasdescribed earlier in the phase plane diagram of Fig. 7. For a load withm ¼ 2 > m0m ¼ 1:397, when the extensional stretch reaches its maximum previous valueat l ¼ Λ ¼ 3 in Fig. 7, the load now has the normalized velocity v 1ð Þ ¼ I

1 ¼ 1:104. So, itsmotion continues beyond the current stress-softened path to a new virgin path until the loadreaches a new maximum deflection at the turning point Λ=3.298 from which a second-stage, stress-softened periodic motion takes place. And this softening effect continuesindefinitely in all subsequent free-fall motions of the same load from l0=1 on the continuouslypreconditioned string. Recall that this phenomenon was described earlier with additionaldetails for a load with μ=2 in its free-fall motion on a cord initially preconditioned to a lessermaximum previous stretch Λ=2.

Although the energy curves in Figs. 9 and 10 aid our perception of the motion responseof the dynamical system, the disquieting non-uniqueness phenomenon encountered aboveinvites our further consideration. Returning to our earlier difficulty, however, we see inFig. 10 for m ¼ 1:333 < m0m ¼ 1:397 that the line graph Vg intersects V at two points Λ=2.467 and 4.380. Also, for m ¼ 1 < m0m ¼ 1:397, while the line Vg intersects V at only onepoint Λ=1.64 in the range of Λ shown here, actually their extensions intersect again at Λ=7.411, certainly physically reasonable. On the other hand, if Λ is chosen outside of therange Λ∈[2.467, 4.380] for μ=1.333, say, we see from Fig. 10 that V−Vg<0 and hence theload in its free-fall motion leaves the stress-softened path as described generally above.Specifically, for the case Λ=2, μ=1.333, say, we find easily V � Vg ¼ �0:067, and hencethe load leaves its stress-softened path. But for a given maximum previous stretch0 ¼ b0 2 2:467; 4:380ð Þ, we see in Fig. 10 that V−Vg>0, and the same load in its free-fall

Fig. 10 Variation of strain andgravitational potential energieswith maximum previous stretch Λfor various loads μ in the free-fallcase of the loaded stress-softened,neo-Hookean string withb=0.527


motion on the cord preconditioned to b0 continues in a periodic stress-softened motion.With b0 ¼ 3, μ=1.333, for example, we find V � Vg ¼ 0:569. We thus reach the sameconclusions demonstrated numerically by different methods.

Now, plainly, the general stress-stretch curves for our model in Fig. 4 show noirregularities that suggest a possible transition from one stress-softening path to anotherwithout first going through a virgin transitional deformation, and there is one and only onevalue of the greatest previous stretch for any assigned load. We recall, however, that ourneo-Hookean stress-softening model characterizes a certain buna-N elastomer with aspecific stress-softening parameter b=0.527. It is shown4 in [7] that this material modelsuffers much of its damage, roughly 86% softening, for an ultimate stretch of about 4;roughly 79% for Λ0=3.261 and very nearly 98% for Λ0=7.411! Therefore, this suggeststhat the point of maximum loading on the free-fall energy curve is a reflection of theextreme degree of softening damage incurred by our special buna-N model in maximumprevious stretches that exceed Λ0=3.261 in Fig. 9.

To picture these free-fall motion effects, let us consider a preconditioned string stretchedbeyond Λ0=3.261 at which μ0m=1.397 represents the greatest load Mm in the progressivefree-fall softening of the elastomer with b=0.527. First, let us recall the stress-softenedperiodic motion of the same cord preconditioned to a maximum previous stretch Λ=3 undera smaller load M<Mm for which m ¼ 1:391 < m0m in Fig. 7. This motion, with periodets ¼ 5:407, is described separately in Fig. 11 by the corresponding solid, oval-shapedcurve. The extent of the stress-softening damage for Λ=3 is about 76%. The samepreconditioned string is then subjected to a greater load for which m ¼ 4:630 > m0m andhence the load in its free-fall motion leaves its previous stress-softened path withnormalized velocity v(3)=1.673 at Λ=3. From this transitional point in Fig. 11, the cordundergoes additional stress-softening along its virgin response path (the dash curve withsingle directional arrows) to a new extreme stretch Λ*=5 at which the degree of softeningdamage is now 92.1%. The rebound periodic motion of the load evolves along the small,solid oval-shaped trajectory with turning points (4.523, 5).

The greatly softened cord is then relaxed to l0=1 and subjected to a smaller load bM withbμ ¼ 1:269 < μ, just sufficient for bM in its free-fall motion to reach the previous maximumstretch Λ*=5. The subsequent stress-softened motion of M̂ thus continues indefinitelyalong the periodic trajectory shown by the large, oval-shaped curve with double directionalarrows. This large amplitude, reduced load response, which arises from the extreme degreeof softening damage, occurs much beyond the extremum point (Λ0, μ0m) of greatest loadingMm possible in a free-fall motion from the unstretched state of a stress-softened, neo-Hookean string.

Continuing, we next suppose that the string, preconditioned to Λ*=5, is subjected to aslightly larger load M for which μ=1.370, but such that μ0m > μ ¼ 1:370 > bμ, i.e., bM <M < Mm (where μ0m=1.397,μ̂ ¼ 1:269), and released from rest at its unstretched state. Wenote that V � Vg ¼ �0:200. The load leaves the stress-softened trajectory with normalizedvelocity v(5)=0.774 at Λ*=5, as illustrated in Fig. 12. The stress-softening cord thenresumes its virgin response with only slight additional softening damage of 0.2% to a newextreme stretch Λ*=5.046, and afterwards the load rebounds significantly on the greatlysoftened string in its periodic motion with turning points (1.325, 5.046) and E(5.046, 0)=

4 The extent or degree of stress-softening in the Zúñiga–Beatty model [7], denoted by α, is defined by

α ¼ 1� exp �bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS Λð Þ � ffiffiffi

3pq� �

. Softening damage thus evolves from the undeformed, virgin state where

S 1ð Þ ¼ ffiffiffi3

p. Complete softening is achieved only as Λ ! 1, but a high degree of softening also may occur

for large, physically reasonable stretch.


−0.310. The total extent of stress-softened damage is now 92.3%. Of course, these physicalattributes depend on the constitutive behavior and characteristics of the simple stress-softening model studied here, and specifically the value of the softening parameterconsidered typical of a certain buna-N elastomer. Other models may yield different effects.

In sum, therefore, the extremum phenomenon is plainly a model effect that will varywith b. For smaller values of b, the greatest load that the material can physically support inthe free-fall motion without additional softening is the maximum point at the stretch Λ0 onthe free-fall energy curve. Therefore, for smaller values of b, the extremum point movestoward larger values of the greatest previous stretch Λ0 and under increasingly larger loadsμ0m. Specifically, we find that for b=0.25 and b=0.1 the corresponding maxima occur forloads μ0m=2.703 and μ0m=6.289 at the respective stretches Λ0=6.468 and Λ0=16.862,very nearly, the latter falling well beyond the range of physically possible stretch and wherethe degree of damage is 99.99%. Still, in the physical range Λ0=6.468 the extent of thesoftening is great at 96.5%. Thus, as b diminishes the non-uniqueness effect of stress-

Fig. 11 Phase plane map for thestress-softened response in thefree-fall motion of the load fordifferent preconditioned states ofa buna-N rubber string withb=0.527. Dashed curvevirgin response; solid curvestress-softened response

Fig. 12 Phase plane graph forthe stress-softened response in thefree-fall motion of the loadμ=1.370 for the preconditionedstate Λ*=5 of a buna-N rubberstring with b=0.527. Dashedcurve virgin response; solid curvestress-softened response


softening becomes less important to the overall physical aspects of the stress-softeningmotion for the current model. Further discussion of the curves drawn for general initial datawith E 10;

I10

� �> 0 shown in Fig. 9 is provided below.

4.2 Examples of Motion with Arbitrary Initial Velocity

We shall now study the motion of a load attached to a stress-softened neo-Hookean rubberstring preconditioned to a maximum stretch Λ, and then released vertically from thepreconditioned, unstretched state with an arbitrary initial velocity. The finite amplitudemotion of the load described by (18) is studied numerically for several cases. First, weinvestigate the effects of various initial velocities. We then explore the effects of differentloads and the extent of preconditioning (essentially the degree of stress-softening) for afixed initial velocity.

4.2.1 Motion on a Preconditioned Cord for Various Initial Velocities

Let us consider a fixed load μ=0.5 supported by a buna-N cord preconditioned to a stretchΛ=4. The results for various initial velocities starting from the unstretched state of thestress-softened cord are presented in Fig. 13. To uniformly describe all three phases of themotion, we adopt the abscissa ratio y(t)/L in place of the stretch l. Of course, fory tð Þ=Lj j 1, l tð Þ ¼ y tð Þ=Lj j defines the stretch in both the suspension and retractionphases of the motion. Similar phase plane graphs obtained from results in [14] for theparent neo-Hookean material are presented in Fig. 14 for the same load. The heavy dots at(1, 0), (1, 1), (1, 2) and (1, 3) in Figs. 13 and 14 mark the initial conditions under which theload is released. For these assigned initial data, the resultant motions for the two models arethen compared on the basis of their identical energy E 1; I10

� �in (20) with I

10 ¼ v 1ð Þ. Wethus focus on motions starting from the unstretched state of the preconditioned string.

If the load is released from the unstretched state of the softened cord with a normalizedvelocity ν 1ð Þ ¼ I

10 ¼ 3:222 shown in Fig. 13, the extreme stretch in the suspension mode isthe maximum previous stretch Λ=4 and the energy E(1, 3.222)=5.19. But if v(1)>3.222,the cord will undergo additional deformation along its virgin path to a new extreme stretchΛ*>4; and afterwards the load will move on a new stress-softened path. The initial energy,therefore, is reduced due to further damage incurred by the virgin material in its additionaldeformation; and the subsequent motion never returns to the unstretched state from which itbegan. Of course, for the parent neo-Hookean material in Fig. 14, the initial normalizedvelocity of the load would have to be much greater to just reach an extreme stretch Λ=4 inthe suspension phase.

For general initial data for which E 1; I10� �

> E 1; 0ð Þ ¼ 0, the load in its rebound motionon the preconditioned cord will enter the free-flight phase where the motion is described by(19). Notice that the motion in the free-flight phase is independent of the constitutive natureof the otherwise perfectly flexible rubber string. The first integral of (19), in accordancewith [14], is the free-flight energy equation 1

2I12

f¼ h� 1f , where h is the altitude at the

free-flight turning point I1f ¼ 0. Moreover, at the onset of free-flight, lf=0,I1f ¼ v 1ð Þ.

Now, at the unstretched state l0=1 with the initial velocity v(1)=v0 such that l(t)≤Λ underthe load μ, the constant energy of the suspension phase, by (20), is E 1; v0ð Þ ¼ 1

2 v20. We thus

have E(1, v0)=h, the altitude in the free-flight phase, independently of any materialproperties of the string. Therefore, for the same initial conditions, the motion in the free-flight phase is necessarily the same for both the parent and preconditioned strings. Thiseffect is illustrated by the trajectories starting from (1, 1) and (1, 2) in Figs. 13 and 14. In


the latter instance, in its motion for 10;I10

� � ¼ 1; 2ð Þ with constant total energy E(1, 2)=2,the load climbs to the ultimate height h=2 at y=L ¼ �1.

From the results illustrated in Fig. 13 for the preconditioned cord, it may be inferred thatif the load is released from rest at an extreme stretch l0=d0 which is greater than theextreme stretch d for the free-fall motion but less than or equal to the maximum previousstretch Λ of the preconditioned string, the resultant motion includes the suspension andfree-flight phases (with initial energy 0<E≤2) or all three phases (with initial energy E>2).In Fig. 13, for example, the free-fall motion with E(1, 0)=0 has an extreme deflectiond=2.685; and hence a motion from rest at d0 ¼ 3:018 > d with energy E(1, 1)=0.5 willcontinue into the free-flight phase to a height y=L ¼ h ¼ 0:5. Similarly, for d0=3.538 withE(1, 2)=2, the load flies to the ultimate free-flight altitude h=2. The motion that begins atd0=3.939 with E 1; 3ð Þ ¼ 4:5 > h in Fig. 13, however, will continue in all three phases, as itwill also for d0 ¼ Λ ¼ 4 with E 1; 3:222ð Þ ¼ 5:190 > h.

Similar effects arise for the parent neo-Hookean string shown in Fig. 14, but theamplitude of the motion is much smaller for the same initial circumstances. In particular,the free-fall motion from rest at (1, 0) is described by the familiar closed suspension phasetrajectory, the extreme stretch being 1.186 for the parent model in Fig. 14 and 2.685 for thepreconditioned example in Fig. 13, both with energy E(1, 0)=0. In the same way, themotion with 10;

I10

� � ¼ 1; 3ð Þ in the stress-softened case in Fig. 13 may be compared withthe corresponding motion for the parent model in Fig. 14, the normalized total energy inboth instances being obtained from (20) as E 10;

I10

� � ¼ 4:5. This exceeds the greatest free-flight energy E 1; 2ð Þ ¼ h ¼ 2 and hence the load must pass through the free-flight region

Fig. 13 Phase plane plots of thefinite amplitude motion of a loadμ=0.5 on a stress-softened neo-Hookean cord with maximumprevious stretch Λ=4 andb=0.527


into the retraction phase. It is seen that for the same initial energy conditions in Figs. 13 and14, the dynamical response of the load for the two materials, though clearly similar, differsin ways that may be intuitively anticipated. For example, with 10;

I10

� � ¼ 1; 3ð Þ, therespective stress-softened amplitudes in the suspension and retraction motions are 3.939and 2.382, while the corresponding parent amplitudes are 2.229 and 1.647. Other examplesare accordingly noted in Figs. 13 and 14. In general, for the same initial energy conditions,the amplitudes of the motion of the load on the stress-softened string in both the suspensionand retraction phases are always greater than those of the same load on the parent cord.

4.2.2 Motion on a Preconditioned String under Various Loads

Now let us consider the motion effects for different loads and degrees of stress-softening fora specified initial velocity from the unstretched state of the preconditioned cord. First, let usexamine the characteristics of the motion of a smaller load μ=0.333 for a virgin neo-Hookean string and for several of its preconditioned states, as illustrated in Fig. 15, forfixed initial conditions 10;

I10

� � ¼ 1; 2ð Þ with E(1, 2)=2. The dashed line trajectory startingat (1, 2) traces the motion of the load from the undeformed virgin state. After reaching theextreme stretch Λ=1.627, the subsequent stress-softened response, shown by the solidclosed curve with single arrows, completes the suspension mode as the load passes theundeformed stress-softened state at 10;

I10

� � ¼ 1;j1:564ð Þ in Fig. 15; and, in consequence,the load is now projected into the free-flight zone to the point at y tð Þ=L ¼ �0:223, i.e., ath=1.223. The load in its stress-softened downward return flight reaches the undeformedstate with the normalized velocity ν(1)=1.564, which is less than its initial velocity ν(1)=2.The load thus returns to its maximum previous deflection d ¼ Λ ¼ 1:627; and the stress-softened motion continues ad infinitum with constant energy E(1, 1.564)=1.223, asindicated by the closed stress-softened phase plane diagram.

In the absence of the Mullins effect, the dashed and solid line trajectories with doublearrows in Fig. 15 trace the perpetual, closed cycle motion of the load on the parent neo-

Fig. 14 Phase plane plots of thefinite amplitude motion of a loadμ=0.5 (κ=2) on a parent neo-Hookean string, from Beatty andChow [14]


Hookean cord, the motion comprising the suspension phase with deflection Λ=1.627 andthe free-flight phase to the ultimate altitude at y tð Þ=L ¼ �1, as shown. For the same loadand initial conditions, the extreme stretch in the suspension phase will increase with theextent of preconditioning, but the free-flight phase is unchanged. In preconditioned caseswith Λ={2, 3, 4} of the same stress-softening string under the same load, starting from theundeformed state l0=1 of the corresponding preconditioned material with the samenormalized initial velocity

�l0 ¼ 2, the load in each case completes the free-flight phase

with the extreme altitude h=2, clearly independent of the extent of preconditioning, asshown by the several solid curve trajectories in Fig. 15.

We next investigate the motion for various values of the load parameter μ for a stress-softening buna-N cord preconditioned to a stretch Λ=3 and for fixed initial conditions10;

I10

� � ¼ 1; 1ð Þ with constant energy E 1; 1ð Þ ¼ E0 ¼ 0:5. The results are presented inFig. 16. It is seen that the dynamical response here is similar to the free-fall responseillustrated in Fig. 7, but now the motion also occurs in the free-flight phase. For smallerloads with μ={0.333, 0.5} the motion in the suspension phase in Fig. 16 is along the stress-softened response curves with respective extreme deflections d={1.934, 2.264}, both lessthan the maximum previous stretch Λ=3; and, of course, in the free-flight phase both loadsreach the same highest point h=0.5. We find from (28) that the effective load that will justreach the assigned maximum previous stretch Λ=3 with the assigned energy E(1, 1)=0.5 isμ0=1.112; and Fig. 9 shows that for E(1, 1)=0.5, the maximum point on that energy curveis μ0m=1.171. Therefore, a larger load m ¼ 2 > m0m > m0 ¼ 1:112, regardless of theprevious degree of softening to a stretch Λ, in its resultant motion will leave the currentstress-softened path for which Λ=3 with velocity v*=1.4898, given by (24), i.e., at thepoint ðl; �lÞ ¼ 3; 1:490ð Þ in Fig. 16. The load subsequently moves along the virgin pathshown by the dashed curve with constant total energy E(3, 1.490)=2.443, determined from(21), to a new extreme stretch at Λ*=3.513, given by (25). Thereafter, the reboundmotion follows the closed stress-softened path with the constant total energy E 3:513; 0ð Þ ¼E* ¼ �0:765 calculated from (26); and the load executes a periodic motion only in the

Fig. 15 Phase plane plots show-ing the effect of preconditioningto a stretch Λ for a load μ=0.333on a buna-N rubber cord withb=0.527. Dashed curve virginmaterial response; solid curvestress-softened material response


suspension phase with period ets ¼ 5:662 and turning points (1.941, 3.513) shown inFig. 16.

It is evident in Fig. 13 that the motion of a load μ=0.5 on a buna-N cord preconditionedto a maximum previous stretch Λ=4 and started with 10;

I10

� � ¼ 1; 3ð Þ in the suspensionphase or with 10;

I10

� � ¼ 2:382; 0ð Þ in the retraction phase, both having the same initialenergy E0=4.5, will continue on the stress-softened path for which the greatest deflectiond=3.939<Λ. And the response in the motion of the same load on this preconditioned stringstarted at (1, 3.222) in the suspension mode or at (2.627, 0) in the retraction mode, bothhaving the same constant initial energy E0=5.190, is similar. But if the load is increased,additional softening will occur; and, depending on the initial conditions and the extent ofpreconditioning, stress-softening effects may be more complex. While the various stress-softening effects may be visualized in the energy diagrams in Fig. 9, it is simpler toillustrate the potential effects in an example.

We thus consider the motion of a larger load μ=2.5 on a string preconditioned to astretch Λ=2 and started in the upward direction with the initial data 10;

I10

� � ¼ 1;�3ð Þ atpoint A in Fig. 17 (this is the point A in Fig. 9), which, as seen from (20), corresponds to aninitial energy E0=4.5. Because the initial speed |I1 0|=3 here exceeds the initial speed|I10|=2 needed to just attain the height h=2, the load's motion continues into the retractionphase. We then find in this case that additional stress-softening occurs in both the retractionand suspension modes prior to the load's settling into its final periodic motion in thesuspension phase.

The resultant phase plane trajectory AA′BB′C and the closed terminal path of the motionare traced in Fig. 17. Starting at A with Λ=2, the trajectory AA′ with constant total energyE=4.5 describes the motion during the primary preconditioned stage as the load first movesthrough the free-flight phase to a height h=2 at the point �1;� ffiffiffi

5p� �

where the string thenextends into the retraction phase to its maximum previous stretch Λ=2 atA0 ¼ �2;�1:429ð Þ, these being initial data for the subsequent motion. The load continuesupward but now moves on the virgin material path A′B with constant energy E=4.820leading to second-stage softening at the extreme stretch Λ*=2.394 (this is the point B inFig. 9). At B the newly softened string retracts under tension in the ensuing motion of theload under gravity along BB′ with decreased, constant total energy E=4.129, entering the

Fig. 16 Phase plane plots for apreconditioned string with Λ=3,b=0.527, and several values ofthe load μ. Dashed curve virginmaterial response; solid curvestress-softened response


free-flight phase at (−1, 2.063), leaving it at (1, 2.873) (both normalized speeds beingsmaller than their values at �1;� ffiffiffi

5p� �

and (1, −3), respectively), and continuing in thesuspension phase as the string is elongated to its maximum previous stretch Λ*=2.394 atB′=(2.394, 3.094), these serving as initial data for the subsequent motion of the load. Thestring continues its elongation at B′ and begins loading again on the virgin path B′C withconstant total energy E=4.820 until the cord reaches its ultimate extension in thesuspension phase with stretch

~Λ ¼ 4:953 at the turning point C (this is the point C in

Fig. 9). The load now rebounds along a new stress-softened path on which the constanttotal energy E=−1.937, the load settling into a perpetual periodic motion of period ets ¼5:690 with turning points c; d ¼ ~

Λ� � ¼ 3:308; 4:953ð Þ, as shown in Fig. 17. Notice that the

energy on the virgin paths A′B and B′C is the same: E=4.820, because the load returns tothe virgin path at B′ with the same stretch Λ*=2.394 at which it left it at B following theunloading and reloading of the cord along the stress-softened response curve corresponding tothe trajectory BB′ in Fig. 16. The energy is thus reduced only in the subsequent stress-softenedmotions, reflecting the damage incurred in the continuing deformation of the virgin cord.

Finally, let us observe that motions for which E<0 are possible only in the suspensionphase, and these are of two types. Both are illustrated in Fig. 7. In one variety the stress-softened, periodic motion evolves from a second-stage softening at the maximum previousstretch Λ, and the other arises in a free-fall from a stretch 1<10<Λ. The first type occurs inFig. 7 at a stretch Λ=3.298, energy E ¼ E* ¼ �0:694 and turning points (c, d)=(1.866,3.298). The second kind arises in Fig. 7 in two free-fall motions starting at 10=1.25, thefirst under a load μ=1 having turning points (c, d)=(1.25, 2.507) with energy E=−0.210,the other for a load μ=1.147 having extreme stretches (c, d)=(1.25, 2.679) with energyE=−0.216. Similar examples of the first kind with second-stage softening are shown inFigs. 16 and 17, and the free-fall variety is seen in Fig. 13.

4.3 Motion with Arbitrary Positive Initial Energy

We have seen that if the initial energy for the motion on a loaded stress-softened string issufficiently great, the load may rise into the free-flight or retraction phases. So, we may

Fig. 17 Phase plane plot show-ing the finite amplitude motionof a load in the free-flight,retraction and suspension phasesfor (Λ, μ)=(2, 2.5) correspondingto an initial energy E0=4.5. Initial

conditions ð10; 1�

0Þ ¼ ð1;�3Þ.Dashed curve virgin materialresponse; solid curvestress-softened response


further imagine appropriate values of the load and initial energy for which a preconditionedcord (or even a virgin stress-softening cord) will undergo additional softening (or merelyprimary softening) to a new maximum stretch from which the load in its rebound motion onthe newly softened string is a free-fall periodic motion with zero total energy. In fact, it ispossible under appropriate conditions to drive the load to any assigned terminal energystate. These interesting situations are studied next.

Let us consider a stress-softening cord subjected to a primary preconditioned stretchΛ = Λi . For the virgin stress-softening material, we have Λi = 1; otherwise, Λi > 1. LetE0 > 0 denote the initial energy corresponding to specified initial data 10;

I10

� �. Then, the

subsequent motion of the load, depending upon its intensity, may exceed the maximumprevious stretch on its primary softened path and continue along the virgin path to a newgreatest previous stretch Λ*, or the motion might begin from the virgin state. The energy inthe newly softened state is denoted by E*.

For every assigned initial energy E0>0 of a load suspended by either a virgin or apreconditioned cord for which Λi≥1, there exists a specific value of the load parameter bμsuch that upon rebound the final periodic motion occurs with energy E*=0, and hence thefinal motion evolves in the free-fall mode with a new maximum stretch Λ*. In fact, the finalperiodic motion may be generated with any assigned value of positive energy 0≤E*<E0.

Specifically, for the terminal free-fall case we set E*=0 in (26), and recall (27) for theassigned total initial energy E ¼ E0 > 0 for a string preconditioned to a stretch Λ=Λi andsubjected to secondary softening to a new maximum previous stretch Λ* under a load bμ.This yields two equations for the values of the terminal stretch Λ* and the load bμ. For aprimary motion from the virgin state, we set Λ ¼ Λi ¼ 1 in (27). The resultant terminalmotion is then a free-fall, periodic motion of the load with turning points (1, Λ*).

For illustration, consider a cord preconditioned to a stretch Λi=3.0 and subjected to aninitial motion with energy E0=4.5 and required rebound energy E*=0. The solution of (26)and (27) yields bμ ¼ 1:359 and Λ*=4.087. Hence, the terminal motion for the load bμ ¼1:359 is equivalent to a free-fall periodic motion with turning points (1, 4.087) and energyE*=0, as shown by the closed orbit in Fig. 18. Now suppose the desired rebound energy onthe same preconditioned string is E*=3.0, with the same initial energy. We then find from(26) and (27) the required smaller load bμ ¼ 0:590 and the new maximum previous stretchΛ*=3.195. In this case, however, the load in its rebound periodic orbit in Fig. 18 movesthrough all three phases of motion, for which we shall omit the details. These exampleshave the same initial energy, but different degrees of second-stage softening occur for thetwo loads. Similar behavior for a cord preconditioned to Λi=3 and with initial energy E0=0.5 is shown in Fig. 19. The equivalent free-fall rebound motion with E*=0 occurs for aload bμ ¼ 1:395 and the cord undergoes second stage softening with Λ*=3.155.

Finally, consider a virgin stress-softening cord for which Λi=1 with the same previousinitial energy E0=0.5. We find that the rebound stress-softened equivalent free-fall periodicmotion for which E*=0 occurs under a load bμ ¼ 1:136. The primary motion thus followsonly the virgin path until the load reaches the maximum stretch Λ*=1.865 at which itrebounds on the newly stress-softened cord in the free-fall motion diagrammed in Fig. 19.

Let us define a stress-softening “damage energy” ED ¼ E0 � E as the differencebetween the assigned initial energy E0>0 in the motion of the load from an unstretchedstate and its final energy E* in its subsequent stress-softened motion of the load. Then, ourforegoing result implies that for any positive initial energy, it is always possible to find theappropriate load bμ such that after second-stage softening in which E*=0, the “damageenergy” is equal to the energy with which the load is released from its initial unstretchedstate, i.e., ED=E0.


4.4 The Stress-softening Effect for a Reduced Softening Parameter

We mentioned earlier that the value of the softening parameter for our model obviously willaffect the phase plane characterization of the stress-softened motion, so that the amplitudesof the stress-softened oscillations may be more or less intense than those exhibited for ourbuna-N material, for example. Therefore, to explore the effect of different stress-softeningparameters on the dynamic response of the load, the examples presented in Figs. 5, 7 and13 are here reexamined in Figs. 20, 21 and 22, respectively, for a smaller softeningparameter b=0.25.

Free-fall motions of the same load on a virgin material cord and the subsequent stress-softening effects for the two softening models are compared in Figs. 5 and 20. Of course,all results for the virgin material response are independent of b. Therefore, the maximumdeflection of a given load in its free-fall motion on a virgin neo-Hookean string is

Fig. 18 Phase plane plotsshowing the effect of two loadsbμ ¼ 1:359 and bμ ¼ 0:590 on themotion response followingsecond-stage softening of thebuna-N cord for E0=4.5 transi-tioning to the respective energiesE*=0 and E*=3.0. The initialpreconditioned stretch Λi=3; andΛ*=4.087 and Λ*=3.195 arethe respective secondarystretches. Dashed curve virginresponse; solid curvestress-softened response

Fig. 19 Phase plane plotsshowing the effect of two loadsbμ ¼ 1:395 and bμ ¼ 1:136respectively applied to a precon-ditioned string with Λi=3 and toan identical virgin cord withΛi=1 subject to the same initialenergy E0=0.5 andtransitioning to motions with thesame final energy E*=0. Therespective terminal stretches areΛ*=3.155 and Λ*=1.865.Dashed curve virgin response;solid curve stress-softenedresponse


necessarily the same for both stress-softening models, as shown in Figs. 5 and 20. Thesubsequent stress-softened response for corresponding values of the load μ and themaximum previous stretch Λ for the two models, on the other hand, is substantiallydifferent. The rebound stress-softened amplitude d � cð Þs¼ Λ� cð Þs, hence the period ofthe oscillation, and the maximum velocity in the motion for the smaller softening parameterb=0.25 in Fig. 20 are, in general, much greater than those for b=0.527 in Fig. 5. Weperceive that these results are more in line with our expectations. Moreover, similar to theextremum amplitude effect shown in Fig. 6 for b=0.527 and μ=2.067, we find that forthe softening parameter b=0.25 a maximum amplitude Λ� cð Þs¼ 0:972 occurs in theoscillation about the stress-softened static equilibrium stretch les=3.398 under a larger loadwith μ=4.292. The stress-softened periods for b=0.25, however, do not exhibit a maximumvalue under this load. Rather, the period for both the finite and small amplitude, stress-softened motions increase with the load, similar to the increase observed for the parentmaterial, which is independent of b. The periods for both stress-softened materials, on theother hand, are generally smaller than those for the parent material, though not always.Additional results for comparison of the two models are summarized in the tables of Figs. 5and 20.

Now consider Figs. 7 and 21 in which our two model cords are preconditioned tothe same maximum previous stretch Λ=3. We then find from (28) that μ0=2.169 is

Fig. 20 Phase plane plots of the free-fall motion from the undeformed virgin state of a neo-Hookean stress-softening string for various loads μ. Dashed curve with arrows virgin material response; solid curve stress-softened material response with b=0.25


the applied load that in its free-fall motion just attains the maximum previous stretch of thecord with softening parameter b=0.25, nearly twice the load μ0=1.391 needed for the samestretch of the buna-N string with b=0.527 in Fig. 7. Also, for the same maximum previousstretch, in accordance with footnote 4, the degree of softening of the material for b=0.527 is75.9%, whereas the extent of softening damage for b=0.25 is only 49.1%. As aconsequence of its greater degree of softening damage, the extreme stretch d, and hencethe amplitude (d−c)s in a free-fall motion of the same load μ≤1.391 on the preconditionedstring, as seen in comparison of Figs. 7 and 21, are much greater than those for b=0.25.Specifically, from the tabulated free-fall data for μ=1 and E=0, the ratio of thedisplacements is d0:527=d0:25 ¼ 1:40, or 40% greater; said differently, the ratio of theamplitudes is d � cð Þ0:527

�d � cð Þ0:25 ¼ 1:84, or 84% larger. Similar effects may be seen in

the other tabulated data. For the same load μ=1.391, the amplitude of its free-fall motionfor b=0.527 is nearly 51% greater than for b=0.25. For a load 2.169>μ>1.391, however,the situation is even more dramatic, for now the buna-N cord having the greater softeningdamage suffers additional stress-softening, while the other elastomer does not. In particular,we see in Fig. 7 that for μ=2 and b=0.527, the load leaves the current stress-softened pathand moves on the virgin path to a new greatest previous stretch Λ*=3.298 at which thedegree of softening is now 79.7%. The same load on the string with b=0.25, however,continues without additional damage on its previous stress-softened response path with alarger extreme stretch d ¼ 2:885 < Λ and amplitude d � 1ð Þs¼ 1:885. Some variations inthe periods of oscillation and other results also are apparent in the diagrams and tables ofFigs. 7 and 21, but we shall omit further discussion of these.

Fig. 21 Phase plot in free-fallmotions for various loads μwith initial maximum previousstretch Λ=3 and b=0.25


Finally, let us revisit the slingshot effect diagrammed in Figs. 13 and 22 for our twomodel strings preconditioned to a maximum previous stretch Λ=4. The degree of softeningof the buna-N cord is 86.3%, or 41.2% greater than that of our model string with b=0.25 at61.1%. Consequently, under a fixed load μ=0.5 and for various assigned initial conditionsnoted in the diagrams, the effects of the greater softening damage of the buna-N cord areexhibited by the greater amplitudes of the periodic motion of the load in both thesuspension and retraction phases. Of course, the degree of softening has no impact on thematerial independent free-flight phase. Notice that for b=0.527 and (l0, v0)=(1, 3.222) withthe initial energy E0=5.190, the load in Fig. 13 passes through the maximum previousstretch Λ=4 in the suspension phase, whereas considerably greater initial energy E0=11.878 for which (l0, v0)=(1, 4.874) in Fig. 22 is required for the same load to reach thesame greatest stretch of the cord with b=0.25. Similar effects arise in the retraction phase.Additional details of the motions for the two model strings may be read from all of thecorresponding phase plane maps and the tabulated data.

5 Concluding Remarks

In primary work by Beatty and Chow [14], the closed-form, analytical solution of both thesmall and the finite amplitude oscillations of a loaded (parent) neo-Hookean rubber stringare presented. In contrast, the present study includes the Mullins stress-softening effect in

Fig. 22 Phase plane plots of thefinite amplitude motion of a loadμ=0.5 on a stress-softenedneo-Hookean cord withmaximum previous stretch Λ=4and b=0.25


characterizing the neo-Hookean constitutive behavior of the string. In this constitutivemodel, the virgin material response is that of the parent model only until the stretch attainsits greatest value Λ under a given load. Thereafter, throughout any subsequent motion forwhich the stretch l∈[1, Λ], the deformation traces the stress-softened response. Within thisframework, the small oscillation problem of a load on a stress-softened cord is readilystudied. Because of the additional constitutive complexity of even a simple stress-softeningmodel, however, it is not possible to obtain a corresponding closed form analytical solutionfor the finite amplitude, nonlinear vibration problem. Therefore, numerical methods areadopted throughout to explore the salient features of the vertical, finite oscillatory responseof a load attached to a perfectly flexible, stress-softening neo-Hookean cord.

In our analytical study of the small oscillations of a load superimposed on the finitestatic stretch of a stress-softened cord characterized by the simple Zúñiga–Beatty model [7],we find the same frequency versus static stretch response first reported by Lawton and King[12] for aortic tissues and certain elastomers. Similar behavior is described by Horgan andSaccomandi [15] for elastomers with finite molecular chain extensibility. Needless to say,the same procedures used here may be extended generally to include stress-softening effectsin other materials, including biological tissues and similar materials that exhibit limitedextensibility, such as the Gent [18] and Arruda–Boyce [19] models, among others. And, ofcourse, other stress-softening material models [5, 15] may deliver different effects. Wereserve the study of these additional modeling issues for future works.

We have found numerically that in the free-fall motion of the load from the unstretchedstate of our preconditioned cord, the extreme finite dynamical deflection d−1 of the loadmay be more or less than twice the static deflection les−1, depending on the extent of theloading. Although such results are known to hold separately for a hard or a soft spring, theobservance of both features in a stress-softened rubber spring appears unprecedented.

The graph in Fig. 9 and a similar map that may be constructed for negative energies butnot shown here, provide a qualitative way of conceptualizing the motion without oursolving the equations. This property has been illustrated here in a few simple situations.However, the characterization of more complex softened motions for both positive andnegative stress-softened energies requires further study.

Although energy is conserved in the motions of the load on its separate virgin and thestress-softened trajectories, overall the microstructural damage effects of stress-softeninglead to an energy loss in transition from the virgin to the softened state, and in anysubsequent motion that involves the secondary deformation of the virgin material to a newmaximum stretch. We thus find that the phenomenological stress-softening functionintroduced by Zúñiga and Beatty [7] results in a reduction of elastic potential energy whena stress-softening rubberlike material is deformed along its virgin path. The cumulativedamage, however, is evident only when the material is unloaded. This energy loss isexhibited in several examples. Although we anticipate some sort of energy loss due tomicrostructural damage in loading the virgin stress-softening material, the Zúñiga-Beattystress-softening model is not energy based so it does not show clearly that strain energy isdepleted during loading on the virgin path. Other more complex models [5, 15], however,are energy based, and their application may reveal more clearly the nature of the evidentenergy loss seen here. Moreover, the extent of the energy loss exhibited in our examples forthe buna-N model appears greater than we might anticipate. By our reducing the softeningparameter, we have shown that this loss is reduced and the amplitudes seem to be closer tothe sort of results that one might expect to find in laboratory trials of rubber materials.Nevertheless, we know of no current experimental results that could lend support to thisconjecture. This is plainly an issue that invites additional study.


References

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Effect of Stress-softening on the Dynamics of a Load Supported by a Rubber String

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