Research ArticleOn the Propagation of Longitudinal Stress Waves inSolids and Fluids by Unifying the Navier-Lame andNavier-Stokes Equations

Ahmad Barzkar and Hojatollah Adibi

Department of Applied Mathematics Faculty of Mathematics and Computer Sciences Amirkabir University of Technology424 Hafez Avenue PO Box 15875-4413 Tehran Iran

Correspondence should be addressed to Hojatollah Adibi adibihautacir

Received 2 September 2014 Accepted 14 December 2014

Academic Editor Florin Pop

Copyright copy 2015 A Barzkar and H Adibi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Propagation of mechanical wavesrsquo phenomenon is the result of infinitely small displacements of integrated individual particles inthe materials These displacements are governed by Navier-Lame and Navier-Stokes equations in solids and fluids respectively Inthe present work a generalized Kelvin-Voigt model of viscoelasticity has been proposed with the aim of bridging the gap betweensolids and fluids leading to a new concept of viscoelasticity which unifies the Navier-Lame and the Navier-Stokes equations Onsolving this equation in one dimension propagation of stress disturbance in the so-called ldquoKelvin-Voigt materialsrdquo will be studiedThe model of these materials involves all the elastic and viscoelastic solids as well as fluids and soft materials

1 Introduction

The subject matter of mechanics is the study of motionin how a physical object changes position with time andwhy [1] In continuum mechanics we are concerned withthe mechanical behavior and shape of materials under loadThe physical reasons for this behavior can be quite differentfor different materials Solid media will deform when forcesare applied on them These materials are called elastic ifthe object will return to its initial shape and size whenthese forces are removed Hence elasticity is the tendencyof solid materials to return to their original shape afterbeing deformed [2] In continuum mechanics we considerthe basic equations describing the physical effects created byexternal forces acting upon solids and fluids In addition tothe basic equations that are applicable to all continua thereare other equations called constitutive equations which areconstructed to take into account material characteristics Inthe study of solids the constitutive equations for a linearelastic material are a set of relations between stress andstrainHookrsquos law represents thematerial behavior and relatesthe Cauchy stress tensor 119879 = (120590

119894119895) and infinitesimal strain

tensor 119890 = (119890119894119895) The general form of Hookrsquos law in compo-

nents is

120590119894119895= 119862119894119895119897119896119890119897119896 (1)

where119862119894119895119897119896

is the fourth-order stiffness tensor [3] If the bodyis isotropic and homogenous then this law is simply writtenas [3]

119879 = 120582120579119868 + 2120583119890 (2)

in which 120579 is the trace of 119890 119868 is the unit tensor and 120582 and 120583are the Lame constants Perfect elasticity is an approximationof the real world and few materials remain purely elasticeven after very small deformations When an elastic materialis not stressed in tension or compression beyond its elasticlimit its individual particles perform elastic movement Thedisplacement of the particles mass center denoted by vectorfield 119906 = (119906

1 1199062 1199063) is related to the strain tensor by the

relation [4]

119890 =1

2[grad119906 + (grad119906)119879] (3)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 789238 9 pageshttpdxdoiorg1011552015789238

2 Mathematical Problems in Engineering

Newtonrsquos second law of motion is based on the conservationof momentum and expressed as [5]

div119879 = 1205881205972119906

1205971199052(4)

in which 120588 is the mass density of the body Now in an iso-tropic homogenous elastic solid by combining (2) (3) and(4) we have

119906119905119905=120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906 (5)

which is the vector form of the Navier-Lame equations andgoverns the infinitesimal movements of the bodyrsquos integratedindividual particles [5]

Viscousmaterials resist shear flow and strain linearly withtime when stress is applied [6] If a material exhibits a linearresponse it is categorized as a Newtonianmaterial [6] In thiscase the stress is linearly proportional to the strain rate If thematerial exhibits a nonlinear response to the strain rate itis categorized as non-Newtonian fluid To get the equationthat governs the small movement of integrated particles offluids let us first consider the continuity equation governingthe continuity of the integrated individual particles which arederived from the principle of conservation of mass and isgiven as

120597120588

120597119905+ div(120588119881) = 0 (6)

where the vector field119881 represents the velocity of the particlesand 120588 is the mass density of the fluid [7] In the absence ofbody forces conservation of momentum on which Newtonrsquossecond law is based is expressed as [7]

div119879 = 120588120597119881120597119905 (7)

In the study of fluids the constitutive equations consist of a setof relations between stress and rate of strain these equationsin tensor form are given as [8]

119879 = minus119901119868 + 120582 tr(119863)119868 + 2120583119863 (8)

where the pressure 119901 is induced by tension that is thedifference between the dynamical and thermodynamicalpressures 120582 and120583 are independent parameters characterizingviscosity 119868 is the unit tensor and119863 is the rate of strain whichis a symmetric tensor of order two

Finally the kinematical relation of fluids describing therelation between tensor119863 and velocity vector field 119881 is givenas

119863 =1

2[grad119881 + (grad119881)119879] (9)

Moreover the pressure can be written as [8]

119901 = minus(120582 +2

3120583) tr(119863) (10)

It can be deduced from (7) (8) (9) and (10) that

120597119881

120597119905=120582 + 120583

120588grad(div119881) +

120583

120588nabla2119881 (11)

However in terms of displacement vector field 119906

119906119905119905=120582 + 120583

120588grad div +

120583

120588nabla2 (12)

which is the set of Navier-Stokes equationsIn fluid continuum themotion of substances is described

by the Navier-Stokes equations These equations are strictlythe statement of conservation of momentum and are basedon the assumption that the fluid at the scale of interest isa continuum In other words it is not made up of discreteparticles but rather a continuous substance Another neces-sary assumption is that all the fields of interest like pressurevelocity density and temperature are differentiable weakly atleast [9]

There are materials for which a suddenly applied andmaintained state of uniform stress induces an instantaneousdeformation followed by a flowprocesswhichmay ormay notbe limited in magnitude as time grows [10] These materialsexhibit both solid and fluid characteristics Behavior of thesematerials clearly cannot be described by either elasticity orviscosity theories alone as it combines features of each andis called viscoelastic Viscoelasticity is a generalization ofelasticity and viscosity [11]

All materials exhibit some viscoelastic response In com-mon metals such as steel or aluminum as well as in quartzat room temperature and at small strain the behavior doesnot deviate much from linear elasticity which is the simplestresponse of a viscoelasticmaterial Synthetic polymers woodand human tissue as well as metals at high temperaturedisplay significant viscoelastic effects Some phenomena inviscoelastic materials are as follows

(i) If the stress is held constant the strain increases withtime (creep)

(ii) If the strain is held constant the stress decreases withtime (relaxation)

(iii) Acoustic waves experience attenuation

The material creeps that gives the prefix visco- and thematerial fully recovers which gives the suffix-elasticity [12]The viscoelastic materials play an important role in manyengineering structures Those materials such as polymers arebeing used for example to dissipate and to insulate vibrationcaused by rotating or reciprocal movements [13] They alsohave potential application in a new Hopkinson pressure bartesting apparatus [14ndash18]Therefore having the knowledge ofthe behavior of these materials particularly related to theirmechanical parameters is essential The theory which illus-trates this behavior is the viscoelasticity theory This theoryis used in many fields such as solid mechanics seismologyexploration geophysics acoustics and engineering [19]

Modeling and model parameter estimation are of greatimportance for a correct prediction of the foundation behav-ior [19] In many cases elastic constitutive models work well

Mathematical Problems in Engineering 3

when time dependent effects can be neglected However inthose cases when time dependent effects cannot be neglectedwe will need to utilize different constitutivemodels Basicallytime dependent effects indicate that the stress-strain behaviorof material will change with time The elastic material modelfor time dependent effects is viscoelasticity Many researchersas Alfrey [20] Barberan and Herrera [21] Achenbach andReddy [22] Bhattacharya and Sengupta [23] and Acharya etal [24] formulated and developed this theory [19] FurtherBert and Egle [25] Abd-Alla and Ahmed [26] and Batra[27] successfully applied this theory to wave propagationin homogenous elastic media [19] Murayama and Shibata[28] and Schiffman et al [29] have proposed higher orderviscoelastic models of five and seven parameters to representthe soil behavior Jankowski et al [30] discussed the linearviscoelastic model and the nonlinear viscoelastic model [19]

One of the most important classic models which isfocused on in this paper is Kelvin-Voigt modelThe constitu-tive relation of this model is expressed as a linear first-orderdifferential equation which can be derived as below

The most simple one-dimensional stress and strain ten-sors are of the form

119879 = [

[

120590 0 0

0 0 0

0 0 0

]

]

119890 = [

[

11989010 0

0 11989020

0 0 1198903

]

]

(13)

which according to the generalized Hookrsquos law are related by

120590 = 1198641198901 (14)

where 119864 is Youngrsquos modulus of elasticity [3] In the Kelvin-Voigt model for viscoelastic materials (14) is augmented toinclude viscosity leading to the generalized equation

120590 = 1198641198901+ 120578

1205971198901

120597119905 (15)

where 120578 is the constant of viscosity [31 32]This model represents a solid undergoing reversible

viscoelastic strain and is extremely goodwithmodeling creepin materials but with regard to relaxation the model is muchless accurate [33]

All material substances are comprised of particles Whena material is not stressed in tension or compression beyondits elastic limit its individual particles may be forced intovibrational motion about their equilibrium positions Thusthese particles perform elastic oscillations Elastic waves arefocused on particles that move in unison to produce amechanical wave In solids elastic waves can propagate infour principle modes that are based on the way the particlesoscillate These waves can propagate as longitudinal shearand surface waves and in the thin materials as plate waves

In longitudinal waves the oscillations occur in the lon-gitudinal direction or the direction of wave propagationSince compressional and dilatational forces are active inthese waves they are also called pressure or compressionalwaves They are also sometimes called density waves becausetheir particle density fluctuates as they move Compressionwaves can be generated in liquids as well as solids because

the energy travels through the atomic structure by a seriesof compressions and expansion movements [34] In thetransverse or shear wave the particles oscillate at a rightangle or perpendicular to the direction of propagation Shearwaves require a solid material for effective propagation andtherefore are not effectively propagated in materials such asliquids or gasses [34] Waves in an isotropic elastic solid aregoverned by the vector Navier-Lame equations

2 New Mathematical Model

21 Generalization of Kelvin-Voigt Model In Kelvin-Voigtmodel for viscoelastic materials the viscosity term is aug-mented to (14) leading to the generalized equation (15) Butthismodel is a generalization of one-dimensional constitutiveequation (14) In order to obtain a model for viscoelasticmaterials in most general form the constitutive equation (2)should be extended In this paper this equation is rewritten as

119879 = 120582120579119868 + 2120583119890 + 120582119891120579119868 (16)

in which 120582119891is called the first constant of viscosity Also by

symmetric of (16) this equation can be written as

119879 = 120582120579119868 + 2120583119890 + 120582119891120579119868 + 2120583

119891119890 (17)

in which 120583119891is the second constant of viscosity In the simple

case (13) (16) is reduced as

120590 = 1198641198901+

120583120582119891

120582 + 1205831198901minus

120583120572

120582 + 120583exp(minus

120582 + 120583

120582119891

119905) (18)

in which 120572 is a positive constant By comparing (18) and (15)it can be easily inferred that

120578 =

120583120582119891

120582 + 120583 (19)

In the next sections it has been shown that (16) and (17) canjustify the physics

22 Generalization of Navier-Lame Equation In the first sec-tion the Navier-Lame equation was deduced by combining(2) (3) and (4) Now (2) is replaced by (16) thereforeby combining (16) (3) and (4) a generalized Navier-Lameequation is obtained as follows

119906119905119905=120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906 +

120582119891

120588grad div (20)

where = 120597119906120597119905 Also if (2) is replaced by (17) then bycombining (17) (3) and (4) another generalization ofNavier-Lame equation can be deduced as follows

119906119905119905= (

120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906)

+ (

120582119891+ 120583119891

120588grad div +

120583119891

120588nabla2)

(21)

where 120582119891and 120583

119891resemble the form of equations for 120582 and 120583

respectively

4 Mathematical Problems in Engineering

23 Unifying Navier-Lame and Navier-Stokes Equations Atthis point by comparing (21) and (12) it is apparent that (21)can be written in two parts The first and second parts areconcerned with the solid and fluid properties of the bodyrespectively In fact (21) governs the displacements of theparticles in both solids and fluids and in bodies with bothsolid and fluid properties but not in materials involvingtwo or more different phases Equation (21) is a unificationof Navier-Lame and Navier-Stokes equations therefore (16)and (17) can be used as constitutive equation to all viscoelasticmaterials

3 Longitudinal Waves in Materials

Equation (21) governs displacement of integrated individualparticles of any continuum body To study the propagation oflongitudinal stress waves in a body we note that (21) can bewritten in one dimension as follows

119906119905119905= 1198882119906119909119909+ 1198892119906119905119909119909 infin lt 119909 lt +infin 119905 ge 0 (22)

in which

1198882=120582119904+ 2120583119904

120588 119889

2=

120582119891+ 2120583119891

120588 (23)

where the only initial condition is 119906(0 0) = 0 This equationcan be solved by separation of variables method so 119906(119909 119905) =119883(119909)119879(119905) Using this in (22) leads to

119883

11988310158401015840= 1198882 119879

11987910158401015840+ 1198892 1198791015840

11987910158401015840= 120574 (24)

in which 120574 is constant Now the following five cases areconsidered

Case 1 (120574 is positive) In this case one can assume 120574 = 11198962119896 = 0 hence (24) becomes

11988310158401015840minus 1198962119883 = 0

11987910158401015840minus 119896211988921198791015840minus 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881119890119896119909+ 1198882119890minus119896119909

119879(119905) = 11988911198901199031119905+ 11988921198901199032119905

(25)

in which

1199031=1

2(11989621198892+ 119896radic11989621198894 + 41198882)

1199032=1

2(11989621198892minus 119896radic11989621198894 + 41198882)

(26)

The solution therefore can be written as

119906(119909 119905) = (1198881119890119896119909+ 1198882119890minus119896119909)(11988911198901199031119905+ 11988921198901199032119905) (27)

By applying the initial condition one arrives at

119906(0 0)

= (1198881+ 1198882)(1198891+ 1198892) = 0 997888rarr 119888

2= minus1198881

or 1198892= minus1198891

(28)

If 1198882= minus1198881 then the velocity of the particle placed at the

origin is zero since

119906119905(0 119905) = (1198881 + 1198882)(11988911199031119890

1199031119905+ 119889211990321198901199032119905) (29)

Therefore in this case there is no motion in the medium

On the other hand since the medium is assumed to beelastic the movement of the particle located at the originstops after a while Hence if 119889

2= minus1198891 at time 119879 then (29)

can be written as

119906119905(0 119879) = 1198891(1198881 + 1198882)(1199031119890

1199031119879minus 11990321198901199032119879) = 0 (30)

Since 1199031is positive and 119903

2is negative the second part of (30)

is not zero and so 1198882= minus1198881 This means that there is no

motion in the medium Therefore in this case the equationdoes not have a nontrivial solution

Case 2 If 120574 = 0 then the solution is trivial

Case 3 If 120574 is negative and 11989621198894 minus 41198882 is positive then onecan assume 120574 = minus11198962 119896 = 0 so

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 1198891119890

1199031119905+ 11988921198901199032119905

(31)

in which

1199031=1

2(minus11989621198892+ 119896radic11989621198894 minus 41198882)

1199032=1

2(minus11989621198892minus 119896radic11989621198894 minus 41198882)

(32)

Thus the solution is as follows

119906(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11988911198901199031119905+ 11988921198901199032119905) (33)

Also the velocity is

119906119905(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11990311198891119890

1199031119905+ 119903211988921198901199032119905)

(34)

Now from the initial condition one obtains

119906(0 0) = 1198881(1198891 + 1198892) = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

2= minus1198891

(35)

If 1198881= 0 then 119906

119905(0 119905) = 0 which means that there is no

motion and the solution is trivial Otherwise 1198892= minus119889

1

which yields

119906119905(0 119905) = 11988811198891(1199031119890

1199031119905minus 11990321198901199032119905) (36)

Since after a time119879 the particle that is at the origin stops then

11990311198901199031119879minus 11990321198901199032119879= 0

yields997888997888997888997888rarr 119879 =

1

1199031minus 1199032

ln(11990321199031

) (37)

Mathematical Problems in Engineering 5

Consequently the solution can be written as

119906(119909 119905) = 1198891(1198881 cos 119896119909 + 1198882 sin 119896119909)(1198901199031119905minus 1198901199032119905) (38)

On the other hand it can be written as follows

119906(119909 119905) = (1205721 cos 119896119909 + 1205722 sin 119896119909)(1198901199031119905minus 1198901199032119905) (39)

Case 4 (120574 is negative and 11989621198894 minus 41198882 = 0) In this case thesolution is trivial

Case 5 (120574 is negative and 11989621198894 minus41198882 lt 0) In this case one canassume 120574 = minus11198962 119896 = 0 so that

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 119890

minus(11989621198892119905)2(1198891cos 119903119905 + 119889

2sin 119903119905)

(40)

in which

119903 =1

2119896radic41198882 minus 11989621198894 (41)

Hence the solution is

119906(119909 119905) = 119890minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times (1198891cos 119903119905 + 119889

2sin 119903119905)

(42)

Also the velocity is given as

119906119905(119909 119905) = 119890

minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times [(1199031198892minus1

2119896211988921198891) cos 119903119905

minus (1199031198891+1

2119896211988921198892) sin 119903119905]

(43)

From the initial condition one obtains

119906(0 0) = 11988811198891 = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

1= 0 (44)

If 1198881= 0 then 119906

119905(0 119905) = 0 and hence there is no motion in

the medium and the solution is trivial Therefore 1198881= 0 and

1198891must be zero so that

119906119905(0 119905) = 119890

minus(11989621198892119905)211988811198892[119903 cos 119903119905 minus 1

211989621198892 sin 119903119905] (45)

Since after a time 119879 the particle which is at the origin stopsthen

119903 cos 119903119879 minus 1211989621198892 sin 119903119879 = 0

997888rarr 119879119899=1

119903(tanminus1( 2119903

11989621198892) + 119899120587)

(46)

Consequently the solution can be written as

119906(119909 119905) = 211988811198892radic11988812 + 11988822

times 119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905)

(47)

in which

120593 = cosminus1( 1198881

radic11988812 + 11988822

) (48)

Therefore the solution can be written as

119906(119909 119905) = 120573119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905) (49)

4 Results

41 Longitudinal Waves in a Perfectly Elastic Solid In a metallike aluminum whose characteristics are listed in Table 1 thewave that propagates in the medium is of the form (49)

If one considers 119896 = 2 120573 = 0003 and 120593 = 1 thedepiction of the longitudinal wave that propagates from theorigin is shown in Figure 1 It is clear that it is a wave with noattenuation

42 LongitudinalWaves inViscoelastic Solids If the viscoelas-tic properties are characterized as in Table 2 the depiction ofthe longitudinal stress wave is shown in Figure 2 It is alsoclear that this wave is attenuated

43 Longitudinal Waves in a Viscoelastic Newtonian FluidIn a liquid with parameters shown in Table 3 the wavepropagating in the medium is of the form (39) So thenthe wave characteristics are discussed in the following formaterial with properties as mentioned in Table 3

If one assumes 119896 = 2 120572 = 001 and 120573 = 0003 then thedisplacement graph of particles in this medium is as shownin Figure 3 In Figure 3(a) the displacement of particles atthe origin without returning to their initial position and inFigure 3(b) the displacement of a particle in each position 119909without returning to its initial position are seen

44 Longitudinal Waves in an Ideal Newtonian Fluid If aliquid is viscoelastic with viscosity coefficients shown inTable 4 the graph of longitudinal stress wave in this mediumis as shown in Figure 4 In Figure 4(a) the distance betweenthe particle originally located in the origin and the sameparticle moved away from the origin in time 119879 is seen Thisdistance starts to decrease with the particles returning to theorigin In Figure 4(b) the distance between the particle atthe time 119879 and the same particle approaching its equilibriumposition at the origin is shown in Figure 4(c) the distancebetween the particles located in their respective equilibriumposition and the same particles moved away from the equi-librium position in time 119879 is shownThis distance then startsto decrease with the particles returning to their respectiveequilibrium position

6 Mathematical Problems in Engineering

u

0002

0001

0

minus0001

minus0002

02 04 06 08 1

t

(a) Particle at the origin

0002

0001

0

minus0001

minus0002

x

01

23 4

5

t

02 04 06 08 1

(b) Particle at each position 119909

Figure 1 The vibration of particles

u

0002

0001

0

minus0001

minus0002

1 2 3 4 5

t

(a) Particle at the origin

minus0002

minus0001

0

0001

0002

20 15 10 5 05

1015

(b) Particle at each position 119909

Figure 2 Vibration of a particle with attenuation

Table 1 Parameters of aluminum that is considered to be perfectlyelastic

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 0 276 times 10

6 KNm2 0 56000Kgm3

Table 2 Parameters of aluminum with fluid properties pertainingto viscoelasticity

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 3016 276 times 106 KNm2 2130 56000Kgm3

Table 3 Parameters of a Newtonian liquid like water which isperfectly fluid with no solid parameters

120582119904

120582119891

120583119904

120583119891

120588

0 3012 times 102 0 213 times 102 9963 Kgm3

Table 4 Parameters of a liquid like water which is not perfectly fluidand has solid parameters

120582119904

120582119891

120583119904

120583119891

120588

506 3012 times 102 276 213 times 102 9963 Kgm3

Mathematical Problems in Engineering 7

u

0012

0010

0008

0006

0004

0002

0

t

0 01 02 03 04 05

(a) Particle at the origin

0010

0005

0

minus0005

minus0010

020 015010 005 0

12

34

5

t

x

(b) Particle at each position 119909

Figure 3 The displacement of particles

u

4

3

2

1

0

t

0 02 04 06 08 1

(a) Thedistance between the particle originally located in the originand the same particle moved away from the origin in time 119879

u

4

3

2

1

0

t

0 20 40 60 80 100

(b) The distance between the particle at the time119879 and the same particleapproaching its equilibrium position at the origin

3

2

1

0

minus1

minus2

minus3

87

65

43

21

0 80 60 40 20 0

(c) The distance between the particles located in their respectiveequilibrium position and the same particles moved away from theequilibrium position in time 119879

Figure 4 The distance between the particles at the time 119879

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Newtonrsquos second law of motion is based on the conservationof momentum and expressed as [5]

div119879 = 1205881205972119906

1205971199052(4)

in which 120588 is the mass density of the body Now in an iso-tropic homogenous elastic solid by combining (2) (3) and(4) we have

119906119905119905=120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906 (5)

which is the vector form of the Navier-Lame equations andgoverns the infinitesimal movements of the bodyrsquos integratedindividual particles [5]

Viscousmaterials resist shear flow and strain linearly withtime when stress is applied [6] If a material exhibits a linearresponse it is categorized as a Newtonianmaterial [6] In thiscase the stress is linearly proportional to the strain rate If thematerial exhibits a nonlinear response to the strain rate itis categorized as non-Newtonian fluid To get the equationthat governs the small movement of integrated particles offluids let us first consider the continuity equation governingthe continuity of the integrated individual particles which arederived from the principle of conservation of mass and isgiven as

120597120588

120597119905+ div(120588119881) = 0 (6)

where the vector field119881 represents the velocity of the particlesand 120588 is the mass density of the fluid [7] In the absence ofbody forces conservation of momentum on which Newtonrsquossecond law is based is expressed as [7]

div119879 = 120588120597119881120597119905 (7)

In the study of fluids the constitutive equations consist of a setof relations between stress and rate of strain these equationsin tensor form are given as [8]

119879 = minus119901119868 + 120582 tr(119863)119868 + 2120583119863 (8)

where the pressure 119901 is induced by tension that is thedifference between the dynamical and thermodynamicalpressures 120582 and120583 are independent parameters characterizingviscosity 119868 is the unit tensor and119863 is the rate of strain whichis a symmetric tensor of order two

Finally the kinematical relation of fluids describing therelation between tensor119863 and velocity vector field 119881 is givenas

119863 =1

2[grad119881 + (grad119881)119879] (9)

Moreover the pressure can be written as [8]

119901 = minus(120582 +2

3120583) tr(119863) (10)

It can be deduced from (7) (8) (9) and (10) that

120597119881

120597119905=120582 + 120583

120588grad(div119881) +

120583

120588nabla2119881 (11)

However in terms of displacement vector field 119906

119906119905119905=120582 + 120583

120588grad div +

120583

120588nabla2 (12)

which is the set of Navier-Stokes equationsIn fluid continuum themotion of substances is described

by the Navier-Stokes equations These equations are strictlythe statement of conservation of momentum and are basedon the assumption that the fluid at the scale of interest isa continuum In other words it is not made up of discreteparticles but rather a continuous substance Another neces-sary assumption is that all the fields of interest like pressurevelocity density and temperature are differentiable weakly atleast [9]

There are materials for which a suddenly applied andmaintained state of uniform stress induces an instantaneousdeformation followed by a flowprocesswhichmay ormay notbe limited in magnitude as time grows [10] These materialsexhibit both solid and fluid characteristics Behavior of thesematerials clearly cannot be described by either elasticity orviscosity theories alone as it combines features of each andis called viscoelastic Viscoelasticity is a generalization ofelasticity and viscosity [11]

All materials exhibit some viscoelastic response In com-mon metals such as steel or aluminum as well as in quartzat room temperature and at small strain the behavior doesnot deviate much from linear elasticity which is the simplestresponse of a viscoelasticmaterial Synthetic polymers woodand human tissue as well as metals at high temperaturedisplay significant viscoelastic effects Some phenomena inviscoelastic materials are as follows

(i) If the stress is held constant the strain increases withtime (creep)

(ii) If the strain is held constant the stress decreases withtime (relaxation)

(iii) Acoustic waves experience attenuation

The material creeps that gives the prefix visco- and thematerial fully recovers which gives the suffix-elasticity [12]The viscoelastic materials play an important role in manyengineering structures Those materials such as polymers arebeing used for example to dissipate and to insulate vibrationcaused by rotating or reciprocal movements [13] They alsohave potential application in a new Hopkinson pressure bartesting apparatus [14ndash18]Therefore having the knowledge ofthe behavior of these materials particularly related to theirmechanical parameters is essential The theory which illus-trates this behavior is the viscoelasticity theory This theoryis used in many fields such as solid mechanics seismologyexploration geophysics acoustics and engineering [19]

Modeling and model parameter estimation are of greatimportance for a correct prediction of the foundation behav-ior [19] In many cases elastic constitutive models work well

Mathematical Problems in Engineering 3

when time dependent effects can be neglected However inthose cases when time dependent effects cannot be neglectedwe will need to utilize different constitutivemodels Basicallytime dependent effects indicate that the stress-strain behaviorof material will change with time The elastic material modelfor time dependent effects is viscoelasticity Many researchersas Alfrey [20] Barberan and Herrera [21] Achenbach andReddy [22] Bhattacharya and Sengupta [23] and Acharya etal [24] formulated and developed this theory [19] FurtherBert and Egle [25] Abd-Alla and Ahmed [26] and Batra[27] successfully applied this theory to wave propagationin homogenous elastic media [19] Murayama and Shibata[28] and Schiffman et al [29] have proposed higher orderviscoelastic models of five and seven parameters to representthe soil behavior Jankowski et al [30] discussed the linearviscoelastic model and the nonlinear viscoelastic model [19]

One of the most important classic models which isfocused on in this paper is Kelvin-Voigt modelThe constitu-tive relation of this model is expressed as a linear first-orderdifferential equation which can be derived as below

The most simple one-dimensional stress and strain ten-sors are of the form

119879 = [

[

120590 0 0

0 0 0

0 0 0

]

]

119890 = [

[

11989010 0

0 11989020

0 0 1198903

]

]

(13)

which according to the generalized Hookrsquos law are related by

120590 = 1198641198901 (14)

where 119864 is Youngrsquos modulus of elasticity [3] In the Kelvin-Voigt model for viscoelastic materials (14) is augmented toinclude viscosity leading to the generalized equation

120590 = 1198641198901+ 120578

1205971198901

120597119905 (15)

where 120578 is the constant of viscosity [31 32]This model represents a solid undergoing reversible

viscoelastic strain and is extremely goodwithmodeling creepin materials but with regard to relaxation the model is muchless accurate [33]

All material substances are comprised of particles Whena material is not stressed in tension or compression beyondits elastic limit its individual particles may be forced intovibrational motion about their equilibrium positions Thusthese particles perform elastic oscillations Elastic waves arefocused on particles that move in unison to produce amechanical wave In solids elastic waves can propagate infour principle modes that are based on the way the particlesoscillate These waves can propagate as longitudinal shearand surface waves and in the thin materials as plate waves

In longitudinal waves the oscillations occur in the lon-gitudinal direction or the direction of wave propagationSince compressional and dilatational forces are active inthese waves they are also called pressure or compressionalwaves They are also sometimes called density waves becausetheir particle density fluctuates as they move Compressionwaves can be generated in liquids as well as solids because

the energy travels through the atomic structure by a seriesof compressions and expansion movements [34] In thetransverse or shear wave the particles oscillate at a rightangle or perpendicular to the direction of propagation Shearwaves require a solid material for effective propagation andtherefore are not effectively propagated in materials such asliquids or gasses [34] Waves in an isotropic elastic solid aregoverned by the vector Navier-Lame equations

2 New Mathematical Model

21 Generalization of Kelvin-Voigt Model In Kelvin-Voigtmodel for viscoelastic materials the viscosity term is aug-mented to (14) leading to the generalized equation (15) Butthismodel is a generalization of one-dimensional constitutiveequation (14) In order to obtain a model for viscoelasticmaterials in most general form the constitutive equation (2)should be extended In this paper this equation is rewritten as

119879 = 120582120579119868 + 2120583119890 + 120582119891120579119868 (16)

in which 120582119891is called the first constant of viscosity Also by

symmetric of (16) this equation can be written as

119879 = 120582120579119868 + 2120583119890 + 120582119891120579119868 + 2120583

119891119890 (17)

in which 120583119891is the second constant of viscosity In the simple

case (13) (16) is reduced as

120590 = 1198641198901+

120583120582119891

120582 + 1205831198901minus

120583120572

120582 + 120583exp(minus

120582 + 120583

120582119891

119905) (18)

in which 120572 is a positive constant By comparing (18) and (15)it can be easily inferred that

120578 =

120583120582119891

120582 + 120583 (19)

In the next sections it has been shown that (16) and (17) canjustify the physics

22 Generalization of Navier-Lame Equation In the first sec-tion the Navier-Lame equation was deduced by combining(2) (3) and (4) Now (2) is replaced by (16) thereforeby combining (16) (3) and (4) a generalized Navier-Lameequation is obtained as follows

119906119905119905=120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906 +

120582119891

120588grad div (20)

where = 120597119906120597119905 Also if (2) is replaced by (17) then bycombining (17) (3) and (4) another generalization ofNavier-Lame equation can be deduced as follows

119906119905119905= (

120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906)

+ (

120582119891+ 120583119891

120588grad div +

120583119891

120588nabla2)

(21)

where 120582119891and 120583

119891resemble the form of equations for 120582 and 120583

respectively

4 Mathematical Problems in Engineering

23 Unifying Navier-Lame and Navier-Stokes Equations Atthis point by comparing (21) and (12) it is apparent that (21)can be written in two parts The first and second parts areconcerned with the solid and fluid properties of the bodyrespectively In fact (21) governs the displacements of theparticles in both solids and fluids and in bodies with bothsolid and fluid properties but not in materials involvingtwo or more different phases Equation (21) is a unificationof Navier-Lame and Navier-Stokes equations therefore (16)and (17) can be used as constitutive equation to all viscoelasticmaterials

3 Longitudinal Waves in Materials

Equation (21) governs displacement of integrated individualparticles of any continuum body To study the propagation oflongitudinal stress waves in a body we note that (21) can bewritten in one dimension as follows

119906119905119905= 1198882119906119909119909+ 1198892119906119905119909119909 infin lt 119909 lt +infin 119905 ge 0 (22)

in which

1198882=120582119904+ 2120583119904

120588 119889

2=

120582119891+ 2120583119891

120588 (23)

where the only initial condition is 119906(0 0) = 0 This equationcan be solved by separation of variables method so 119906(119909 119905) =119883(119909)119879(119905) Using this in (22) leads to

119883

11988310158401015840= 1198882 119879

11987910158401015840+ 1198892 1198791015840

11987910158401015840= 120574 (24)

in which 120574 is constant Now the following five cases areconsidered

Case 1 (120574 is positive) In this case one can assume 120574 = 11198962119896 = 0 hence (24) becomes

11988310158401015840minus 1198962119883 = 0

11987910158401015840minus 119896211988921198791015840minus 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881119890119896119909+ 1198882119890minus119896119909

119879(119905) = 11988911198901199031119905+ 11988921198901199032119905

(25)

in which

1199031=1

2(11989621198892+ 119896radic11989621198894 + 41198882)

1199032=1

2(11989621198892minus 119896radic11989621198894 + 41198882)

(26)

The solution therefore can be written as

119906(119909 119905) = (1198881119890119896119909+ 1198882119890minus119896119909)(11988911198901199031119905+ 11988921198901199032119905) (27)

By applying the initial condition one arrives at

119906(0 0)

= (1198881+ 1198882)(1198891+ 1198892) = 0 997888rarr 119888

2= minus1198881

or 1198892= minus1198891

(28)

If 1198882= minus1198881 then the velocity of the particle placed at the

origin is zero since

119906119905(0 119905) = (1198881 + 1198882)(11988911199031119890

1199031119905+ 119889211990321198901199032119905) (29)

Therefore in this case there is no motion in the medium

On the other hand since the medium is assumed to beelastic the movement of the particle located at the originstops after a while Hence if 119889

2= minus1198891 at time 119879 then (29)

can be written as

119906119905(0 119879) = 1198891(1198881 + 1198882)(1199031119890

1199031119879minus 11990321198901199032119879) = 0 (30)

Since 1199031is positive and 119903

2is negative the second part of (30)

is not zero and so 1198882= minus1198881 This means that there is no

motion in the medium Therefore in this case the equationdoes not have a nontrivial solution

Case 2 If 120574 = 0 then the solution is trivial

Case 3 If 120574 is negative and 11989621198894 minus 41198882 is positive then onecan assume 120574 = minus11198962 119896 = 0 so

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 1198891119890

1199031119905+ 11988921198901199032119905

(31)

in which

1199031=1

2(minus11989621198892+ 119896radic11989621198894 minus 41198882)

1199032=1

2(minus11989621198892minus 119896radic11989621198894 minus 41198882)

(32)

Thus the solution is as follows

119906(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11988911198901199031119905+ 11988921198901199032119905) (33)

Also the velocity is

119906119905(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11990311198891119890

1199031119905+ 119903211988921198901199032119905)

(34)

Now from the initial condition one obtains

119906(0 0) = 1198881(1198891 + 1198892) = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

2= minus1198891

(35)

If 1198881= 0 then 119906

119905(0 119905) = 0 which means that there is no

motion and the solution is trivial Otherwise 1198892= minus119889

1

which yields

119906119905(0 119905) = 11988811198891(1199031119890

1199031119905minus 11990321198901199032119905) (36)

Since after a time119879 the particle that is at the origin stops then

11990311198901199031119879minus 11990321198901199032119879= 0

yields997888997888997888997888rarr 119879 =

1

1199031minus 1199032

ln(11990321199031

) (37)

Mathematical Problems in Engineering 5

Consequently the solution can be written as

119906(119909 119905) = 1198891(1198881 cos 119896119909 + 1198882 sin 119896119909)(1198901199031119905minus 1198901199032119905) (38)

On the other hand it can be written as follows

119906(119909 119905) = (1205721 cos 119896119909 + 1205722 sin 119896119909)(1198901199031119905minus 1198901199032119905) (39)

Case 4 (120574 is negative and 11989621198894 minus 41198882 = 0) In this case thesolution is trivial

Case 5 (120574 is negative and 11989621198894 minus41198882 lt 0) In this case one canassume 120574 = minus11198962 119896 = 0 so that

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 119890

minus(11989621198892119905)2(1198891cos 119903119905 + 119889

2sin 119903119905)

(40)

in which

119903 =1

2119896radic41198882 minus 11989621198894 (41)

Hence the solution is

119906(119909 119905) = 119890minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times (1198891cos 119903119905 + 119889

2sin 119903119905)

(42)

Also the velocity is given as

119906119905(119909 119905) = 119890

minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times [(1199031198892minus1

2119896211988921198891) cos 119903119905

minus (1199031198891+1

2119896211988921198892) sin 119903119905]

(43)

From the initial condition one obtains

119906(0 0) = 11988811198891 = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

1= 0 (44)

If 1198881= 0 then 119906

119905(0 119905) = 0 and hence there is no motion in

the medium and the solution is trivial Therefore 1198881= 0 and

1198891must be zero so that

119906119905(0 119905) = 119890

minus(11989621198892119905)211988811198892[119903 cos 119903119905 minus 1

211989621198892 sin 119903119905] (45)

Since after a time 119879 the particle which is at the origin stopsthen

119903 cos 119903119879 minus 1211989621198892 sin 119903119879 = 0

997888rarr 119879119899=1

119903(tanminus1( 2119903

11989621198892) + 119899120587)

(46)

Consequently the solution can be written as

119906(119909 119905) = 211988811198892radic11988812 + 11988822

times 119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905)

(47)

in which

120593 = cosminus1( 1198881

radic11988812 + 11988822

) (48)

Therefore the solution can be written as

119906(119909 119905) = 120573119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905) (49)

4 Results

41 Longitudinal Waves in a Perfectly Elastic Solid In a metallike aluminum whose characteristics are listed in Table 1 thewave that propagates in the medium is of the form (49)

If one considers 119896 = 2 120573 = 0003 and 120593 = 1 thedepiction of the longitudinal wave that propagates from theorigin is shown in Figure 1 It is clear that it is a wave with noattenuation

42 LongitudinalWaves inViscoelastic Solids If the viscoelas-tic properties are characterized as in Table 2 the depiction ofthe longitudinal stress wave is shown in Figure 2 It is alsoclear that this wave is attenuated

43 Longitudinal Waves in a Viscoelastic Newtonian FluidIn a liquid with parameters shown in Table 3 the wavepropagating in the medium is of the form (39) So thenthe wave characteristics are discussed in the following formaterial with properties as mentioned in Table 3

If one assumes 119896 = 2 120572 = 001 and 120573 = 0003 then thedisplacement graph of particles in this medium is as shownin Figure 3 In Figure 3(a) the displacement of particles atthe origin without returning to their initial position and inFigure 3(b) the displacement of a particle in each position 119909without returning to its initial position are seen

44 Longitudinal Waves in an Ideal Newtonian Fluid If aliquid is viscoelastic with viscosity coefficients shown inTable 4 the graph of longitudinal stress wave in this mediumis as shown in Figure 4 In Figure 4(a) the distance betweenthe particle originally located in the origin and the sameparticle moved away from the origin in time 119879 is seen Thisdistance starts to decrease with the particles returning to theorigin In Figure 4(b) the distance between the particle atthe time 119879 and the same particle approaching its equilibriumposition at the origin is shown in Figure 4(c) the distancebetween the particles located in their respective equilibriumposition and the same particles moved away from the equi-librium position in time 119879 is shownThis distance then startsto decrease with the particles returning to their respectiveequilibrium position

6 Mathematical Problems in Engineering

u

0002

0001

0

minus0001

minus0002

02 04 06 08 1

t

(a) Particle at the origin

0002

0001

0

minus0001

minus0002

x

01

23 4

5

t

02 04 06 08 1

(b) Particle at each position 119909

Figure 1 The vibration of particles

u

0002

0001

0

minus0001

minus0002

1 2 3 4 5

t

(a) Particle at the origin

minus0002

minus0001

0

0001

0002

20 15 10 5 05

1015

(b) Particle at each position 119909

Figure 2 Vibration of a particle with attenuation

Table 1 Parameters of aluminum that is considered to be perfectlyelastic

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 0 276 times 10

6 KNm2 0 56000Kgm3

Table 2 Parameters of aluminum with fluid properties pertainingto viscoelasticity

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 3016 276 times 106 KNm2 2130 56000Kgm3

Table 3 Parameters of a Newtonian liquid like water which isperfectly fluid with no solid parameters

120582119904

120582119891

120583119904

120583119891

120588

0 3012 times 102 0 213 times 102 9963 Kgm3

Table 4 Parameters of a liquid like water which is not perfectly fluidand has solid parameters

120582119904

120582119891

120583119904

120583119891

120588

506 3012 times 102 276 213 times 102 9963 Kgm3

Mathematical Problems in Engineering 7

u

0012

0010

0008

0006

0004

0002

0

t

0 01 02 03 04 05

(a) Particle at the origin

0010

0005

0

minus0005

minus0010

020 015010 005 0

12

34

5

t

x

(b) Particle at each position 119909

Figure 3 The displacement of particles

u

4

3

2

1

0

t

0 02 04 06 08 1

(a) Thedistance between the particle originally located in the originand the same particle moved away from the origin in time 119879

u

4

3

2

1

0

t

0 20 40 60 80 100

(b) The distance between the particle at the time119879 and the same particleapproaching its equilibrium position at the origin

3

2

1

0

minus1

minus2

minus3

87

65

43

21

0 80 60 40 20 0

(c) The distance between the particles located in their respectiveequilibrium position and the same particles moved away from theequilibrium position in time 119879

Figure 4 The distance between the particles at the time 119879

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

when time dependent effects can be neglected However inthose cases when time dependent effects cannot be neglectedwe will need to utilize different constitutivemodels Basicallytime dependent effects indicate that the stress-strain behaviorof material will change with time The elastic material modelfor time dependent effects is viscoelasticity Many researchersas Alfrey [20] Barberan and Herrera [21] Achenbach andReddy [22] Bhattacharya and Sengupta [23] and Acharya etal [24] formulated and developed this theory [19] FurtherBert and Egle [25] Abd-Alla and Ahmed [26] and Batra[27] successfully applied this theory to wave propagationin homogenous elastic media [19] Murayama and Shibata[28] and Schiffman et al [29] have proposed higher orderviscoelastic models of five and seven parameters to representthe soil behavior Jankowski et al [30] discussed the linearviscoelastic model and the nonlinear viscoelastic model [19]

One of the most important classic models which isfocused on in this paper is Kelvin-Voigt modelThe constitu-tive relation of this model is expressed as a linear first-orderdifferential equation which can be derived as below

The most simple one-dimensional stress and strain ten-sors are of the form

119879 = [

[

120590 0 0

0 0 0

0 0 0

]

]

119890 = [

[

11989010 0

0 11989020

0 0 1198903

]

]

(13)

which according to the generalized Hookrsquos law are related by

120590 = 1198641198901 (14)

where 119864 is Youngrsquos modulus of elasticity [3] In the Kelvin-Voigt model for viscoelastic materials (14) is augmented toinclude viscosity leading to the generalized equation

120590 = 1198641198901+ 120578

1205971198901

120597119905 (15)

where 120578 is the constant of viscosity [31 32]This model represents a solid undergoing reversible

viscoelastic strain and is extremely goodwithmodeling creepin materials but with regard to relaxation the model is muchless accurate [33]

All material substances are comprised of particles Whena material is not stressed in tension or compression beyondits elastic limit its individual particles may be forced intovibrational motion about their equilibrium positions Thusthese particles perform elastic oscillations Elastic waves arefocused on particles that move in unison to produce amechanical wave In solids elastic waves can propagate infour principle modes that are based on the way the particlesoscillate These waves can propagate as longitudinal shearand surface waves and in the thin materials as plate waves

In longitudinal waves the oscillations occur in the lon-gitudinal direction or the direction of wave propagationSince compressional and dilatational forces are active inthese waves they are also called pressure or compressionalwaves They are also sometimes called density waves becausetheir particle density fluctuates as they move Compressionwaves can be generated in liquids as well as solids because

the energy travels through the atomic structure by a seriesof compressions and expansion movements [34] In thetransverse or shear wave the particles oscillate at a rightangle or perpendicular to the direction of propagation Shearwaves require a solid material for effective propagation andtherefore are not effectively propagated in materials such asliquids or gasses [34] Waves in an isotropic elastic solid aregoverned by the vector Navier-Lame equations

2 New Mathematical Model

21 Generalization of Kelvin-Voigt Model In Kelvin-Voigtmodel for viscoelastic materials the viscosity term is aug-mented to (14) leading to the generalized equation (15) Butthismodel is a generalization of one-dimensional constitutiveequation (14) In order to obtain a model for viscoelasticmaterials in most general form the constitutive equation (2)should be extended In this paper this equation is rewritten as

119879 = 120582120579119868 + 2120583119890 + 120582119891120579119868 (16)

in which 120582119891is called the first constant of viscosity Also by

symmetric of (16) this equation can be written as

119879 = 120582120579119868 + 2120583119890 + 120582119891120579119868 + 2120583

119891119890 (17)

in which 120583119891is the second constant of viscosity In the simple

case (13) (16) is reduced as

120590 = 1198641198901+

120583120582119891

120582 + 1205831198901minus

120583120572

120582 + 120583exp(minus

120582 + 120583

120582119891

119905) (18)

in which 120572 is a positive constant By comparing (18) and (15)it can be easily inferred that

120578 =

120583120582119891

120582 + 120583 (19)

In the next sections it has been shown that (16) and (17) canjustify the physics

22 Generalization of Navier-Lame Equation In the first sec-tion the Navier-Lame equation was deduced by combining(2) (3) and (4) Now (2) is replaced by (16) thereforeby combining (16) (3) and (4) a generalized Navier-Lameequation is obtained as follows

119906119905119905=120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906 +

120582119891

120588grad div (20)

where = 120597119906120597119905 Also if (2) is replaced by (17) then bycombining (17) (3) and (4) another generalization ofNavier-Lame equation can be deduced as follows

119906119905119905= (

120582 + 120583

120588grad div 119906 +

120583

120588nabla2119906)

+ (

120582119891+ 120583119891

120588grad div +

120583119891

120588nabla2)

(21)

where 120582119891and 120583

119891resemble the form of equations for 120582 and 120583

respectively

4 Mathematical Problems in Engineering

23 Unifying Navier-Lame and Navier-Stokes Equations Atthis point by comparing (21) and (12) it is apparent that (21)can be written in two parts The first and second parts areconcerned with the solid and fluid properties of the bodyrespectively In fact (21) governs the displacements of theparticles in both solids and fluids and in bodies with bothsolid and fluid properties but not in materials involvingtwo or more different phases Equation (21) is a unificationof Navier-Lame and Navier-Stokes equations therefore (16)and (17) can be used as constitutive equation to all viscoelasticmaterials

3 Longitudinal Waves in Materials

Equation (21) governs displacement of integrated individualparticles of any continuum body To study the propagation oflongitudinal stress waves in a body we note that (21) can bewritten in one dimension as follows

119906119905119905= 1198882119906119909119909+ 1198892119906119905119909119909 infin lt 119909 lt +infin 119905 ge 0 (22)

in which

1198882=120582119904+ 2120583119904

120588 119889

2=

120582119891+ 2120583119891

120588 (23)

where the only initial condition is 119906(0 0) = 0 This equationcan be solved by separation of variables method so 119906(119909 119905) =119883(119909)119879(119905) Using this in (22) leads to

119883

11988310158401015840= 1198882 119879

11987910158401015840+ 1198892 1198791015840

11987910158401015840= 120574 (24)

in which 120574 is constant Now the following five cases areconsidered

Case 1 (120574 is positive) In this case one can assume 120574 = 11198962119896 = 0 hence (24) becomes

11988310158401015840minus 1198962119883 = 0

11987910158401015840minus 119896211988921198791015840minus 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881119890119896119909+ 1198882119890minus119896119909

119879(119905) = 11988911198901199031119905+ 11988921198901199032119905

(25)

in which

1199031=1

2(11989621198892+ 119896radic11989621198894 + 41198882)

1199032=1

2(11989621198892minus 119896radic11989621198894 + 41198882)

(26)

The solution therefore can be written as

119906(119909 119905) = (1198881119890119896119909+ 1198882119890minus119896119909)(11988911198901199031119905+ 11988921198901199032119905) (27)

By applying the initial condition one arrives at

119906(0 0)

= (1198881+ 1198882)(1198891+ 1198892) = 0 997888rarr 119888

2= minus1198881

or 1198892= minus1198891

(28)

If 1198882= minus1198881 then the velocity of the particle placed at the

origin is zero since

119906119905(0 119905) = (1198881 + 1198882)(11988911199031119890

1199031119905+ 119889211990321198901199032119905) (29)

Therefore in this case there is no motion in the medium

On the other hand since the medium is assumed to beelastic the movement of the particle located at the originstops after a while Hence if 119889

2= minus1198891 at time 119879 then (29)

can be written as

119906119905(0 119879) = 1198891(1198881 + 1198882)(1199031119890

1199031119879minus 11990321198901199032119879) = 0 (30)

Since 1199031is positive and 119903

2is negative the second part of (30)

is not zero and so 1198882= minus1198881 This means that there is no

motion in the medium Therefore in this case the equationdoes not have a nontrivial solution

Case 2 If 120574 = 0 then the solution is trivial

Case 3 If 120574 is negative and 11989621198894 minus 41198882 is positive then onecan assume 120574 = minus11198962 119896 = 0 so

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 1198891119890

1199031119905+ 11988921198901199032119905

(31)

in which

1199031=1

2(minus11989621198892+ 119896radic11989621198894 minus 41198882)

1199032=1

2(minus11989621198892minus 119896radic11989621198894 minus 41198882)

(32)

Thus the solution is as follows

119906(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11988911198901199031119905+ 11988921198901199032119905) (33)

Also the velocity is

119906119905(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11990311198891119890

1199031119905+ 119903211988921198901199032119905)

(34)

Now from the initial condition one obtains

119906(0 0) = 1198881(1198891 + 1198892) = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

2= minus1198891

(35)

If 1198881= 0 then 119906

119905(0 119905) = 0 which means that there is no

motion and the solution is trivial Otherwise 1198892= minus119889

1

which yields

119906119905(0 119905) = 11988811198891(1199031119890

1199031119905minus 11990321198901199032119905) (36)

Since after a time119879 the particle that is at the origin stops then

11990311198901199031119879minus 11990321198901199032119879= 0

yields997888997888997888997888rarr 119879 =

1

1199031minus 1199032

ln(11990321199031

) (37)

Mathematical Problems in Engineering 5

Consequently the solution can be written as

119906(119909 119905) = 1198891(1198881 cos 119896119909 + 1198882 sin 119896119909)(1198901199031119905minus 1198901199032119905) (38)

On the other hand it can be written as follows

119906(119909 119905) = (1205721 cos 119896119909 + 1205722 sin 119896119909)(1198901199031119905minus 1198901199032119905) (39)

Case 4 (120574 is negative and 11989621198894 minus 41198882 = 0) In this case thesolution is trivial

Case 5 (120574 is negative and 11989621198894 minus41198882 lt 0) In this case one canassume 120574 = minus11198962 119896 = 0 so that

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 119890

minus(11989621198892119905)2(1198891cos 119903119905 + 119889

2sin 119903119905)

(40)

in which

119903 =1

2119896radic41198882 minus 11989621198894 (41)

Hence the solution is

119906(119909 119905) = 119890minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times (1198891cos 119903119905 + 119889

2sin 119903119905)

(42)

Also the velocity is given as

119906119905(119909 119905) = 119890

minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times [(1199031198892minus1

2119896211988921198891) cos 119903119905

minus (1199031198891+1

2119896211988921198892) sin 119903119905]

(43)

From the initial condition one obtains

119906(0 0) = 11988811198891 = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

1= 0 (44)

If 1198881= 0 then 119906

119905(0 119905) = 0 and hence there is no motion in

the medium and the solution is trivial Therefore 1198881= 0 and

1198891must be zero so that

119906119905(0 119905) = 119890

minus(11989621198892119905)211988811198892[119903 cos 119903119905 minus 1

211989621198892 sin 119903119905] (45)

Since after a time 119879 the particle which is at the origin stopsthen

119903 cos 119903119879 minus 1211989621198892 sin 119903119879 = 0

997888rarr 119879119899=1

119903(tanminus1( 2119903

11989621198892) + 119899120587)

(46)

Consequently the solution can be written as

119906(119909 119905) = 211988811198892radic11988812 + 11988822

times 119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905)

(47)

in which

120593 = cosminus1( 1198881

radic11988812 + 11988822

) (48)

Therefore the solution can be written as

119906(119909 119905) = 120573119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905) (49)

4 Results

41 Longitudinal Waves in a Perfectly Elastic Solid In a metallike aluminum whose characteristics are listed in Table 1 thewave that propagates in the medium is of the form (49)

If one considers 119896 = 2 120573 = 0003 and 120593 = 1 thedepiction of the longitudinal wave that propagates from theorigin is shown in Figure 1 It is clear that it is a wave with noattenuation

42 LongitudinalWaves inViscoelastic Solids If the viscoelas-tic properties are characterized as in Table 2 the depiction ofthe longitudinal stress wave is shown in Figure 2 It is alsoclear that this wave is attenuated

43 Longitudinal Waves in a Viscoelastic Newtonian FluidIn a liquid with parameters shown in Table 3 the wavepropagating in the medium is of the form (39) So thenthe wave characteristics are discussed in the following formaterial with properties as mentioned in Table 3

If one assumes 119896 = 2 120572 = 001 and 120573 = 0003 then thedisplacement graph of particles in this medium is as shownin Figure 3 In Figure 3(a) the displacement of particles atthe origin without returning to their initial position and inFigure 3(b) the displacement of a particle in each position 119909without returning to its initial position are seen

44 Longitudinal Waves in an Ideal Newtonian Fluid If aliquid is viscoelastic with viscosity coefficients shown inTable 4 the graph of longitudinal stress wave in this mediumis as shown in Figure 4 In Figure 4(a) the distance betweenthe particle originally located in the origin and the sameparticle moved away from the origin in time 119879 is seen Thisdistance starts to decrease with the particles returning to theorigin In Figure 4(b) the distance between the particle atthe time 119879 and the same particle approaching its equilibriumposition at the origin is shown in Figure 4(c) the distancebetween the particles located in their respective equilibriumposition and the same particles moved away from the equi-librium position in time 119879 is shownThis distance then startsto decrease with the particles returning to their respectiveequilibrium position

6 Mathematical Problems in Engineering

u

0002

0001

0

minus0001

minus0002

02 04 06 08 1

t

(a) Particle at the origin

0002

0001

0

minus0001

minus0002

x

01

23 4

5

t

02 04 06 08 1

(b) Particle at each position 119909

Figure 1 The vibration of particles

u

0002

0001

0

minus0001

minus0002

1 2 3 4 5

t

(a) Particle at the origin

minus0002

minus0001

0

0001

0002

20 15 10 5 05

1015

(b) Particle at each position 119909

Figure 2 Vibration of a particle with attenuation

Table 1 Parameters of aluminum that is considered to be perfectlyelastic

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 0 276 times 10

6 KNm2 0 56000Kgm3

Table 2 Parameters of aluminum with fluid properties pertainingto viscoelasticity

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 3016 276 times 106 KNm2 2130 56000Kgm3

Table 3 Parameters of a Newtonian liquid like water which isperfectly fluid with no solid parameters

120582119904

120582119891

120583119904

120583119891

120588

0 3012 times 102 0 213 times 102 9963 Kgm3

Table 4 Parameters of a liquid like water which is not perfectly fluidand has solid parameters

120582119904

120582119891

120583119904

120583119891

120588

506 3012 times 102 276 213 times 102 9963 Kgm3

Mathematical Problems in Engineering 7

u

0012

0010

0008

0006

0004

0002

0

t

0 01 02 03 04 05

(a) Particle at the origin

0010

0005

0

minus0005

minus0010

020 015010 005 0

12

34

5

t

x

(b) Particle at each position 119909

Figure 3 The displacement of particles

u

4

3

2

1

0

t

0 02 04 06 08 1

(a) Thedistance between the particle originally located in the originand the same particle moved away from the origin in time 119879

u

4

3

2

1

0

t

0 20 40 60 80 100

(b) The distance between the particle at the time119879 and the same particleapproaching its equilibrium position at the origin

3

2

1

0

minus1

minus2

minus3

87

65

43

21

0 80 60 40 20 0

(c) The distance between the particles located in their respectiveequilibrium position and the same particles moved away from theequilibrium position in time 119879

Figure 4 The distance between the particles at the time 119879

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

23 Unifying Navier-Lame and Navier-Stokes Equations Atthis point by comparing (21) and (12) it is apparent that (21)can be written in two parts The first and second parts areconcerned with the solid and fluid properties of the bodyrespectively In fact (21) governs the displacements of theparticles in both solids and fluids and in bodies with bothsolid and fluid properties but not in materials involvingtwo or more different phases Equation (21) is a unificationof Navier-Lame and Navier-Stokes equations therefore (16)and (17) can be used as constitutive equation to all viscoelasticmaterials

3 Longitudinal Waves in Materials

Equation (21) governs displacement of integrated individualparticles of any continuum body To study the propagation oflongitudinal stress waves in a body we note that (21) can bewritten in one dimension as follows

119906119905119905= 1198882119906119909119909+ 1198892119906119905119909119909 infin lt 119909 lt +infin 119905 ge 0 (22)

in which

1198882=120582119904+ 2120583119904

120588 119889

2=

120582119891+ 2120583119891

120588 (23)

where the only initial condition is 119906(0 0) = 0 This equationcan be solved by separation of variables method so 119906(119909 119905) =119883(119909)119879(119905) Using this in (22) leads to

119883

11988310158401015840= 1198882 119879

11987910158401015840+ 1198892 1198791015840

11987910158401015840= 120574 (24)

in which 120574 is constant Now the following five cases areconsidered

Case 1 (120574 is positive) In this case one can assume 120574 = 11198962119896 = 0 hence (24) becomes

11988310158401015840minus 1198962119883 = 0

11987910158401015840minus 119896211988921198791015840minus 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881119890119896119909+ 1198882119890minus119896119909

119879(119905) = 11988911198901199031119905+ 11988921198901199032119905

(25)

in which

1199031=1

2(11989621198892+ 119896radic11989621198894 + 41198882)

1199032=1

2(11989621198892minus 119896radic11989621198894 + 41198882)

(26)

The solution therefore can be written as

119906(119909 119905) = (1198881119890119896119909+ 1198882119890minus119896119909)(11988911198901199031119905+ 11988921198901199032119905) (27)

By applying the initial condition one arrives at

119906(0 0)

= (1198881+ 1198882)(1198891+ 1198892) = 0 997888rarr 119888

2= minus1198881

or 1198892= minus1198891

(28)

If 1198882= minus1198881 then the velocity of the particle placed at the

origin is zero since

119906119905(0 119905) = (1198881 + 1198882)(11988911199031119890

1199031119905+ 119889211990321198901199032119905) (29)

Therefore in this case there is no motion in the medium

On the other hand since the medium is assumed to beelastic the movement of the particle located at the originstops after a while Hence if 119889

2= minus1198891 at time 119879 then (29)

can be written as

119906119905(0 119879) = 1198891(1198881 + 1198882)(1199031119890

1199031119879minus 11990321198901199032119879) = 0 (30)

Since 1199031is positive and 119903

2is negative the second part of (30)

is not zero and so 1198882= minus1198881 This means that there is no

motion in the medium Therefore in this case the equationdoes not have a nontrivial solution

Case 2 If 120574 = 0 then the solution is trivial

Case 3 If 120574 is negative and 11989621198894 minus 41198882 is positive then onecan assume 120574 = minus11198962 119896 = 0 so

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 1198891119890

1199031119905+ 11988921198901199032119905

(31)

in which

1199031=1

2(minus11989621198892+ 119896radic11989621198894 minus 41198882)

1199032=1

2(minus11989621198892minus 119896radic11989621198894 minus 41198882)

(32)

Thus the solution is as follows

119906(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11988911198901199031119905+ 11988921198901199032119905) (33)

Also the velocity is

119906119905(119909 119905) = (1198881 cos 119896119909 + 1198882 sin 119896119909)(11990311198891119890

1199031119905+ 119903211988921198901199032119905)

(34)

Now from the initial condition one obtains

119906(0 0) = 1198881(1198891 + 1198892) = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

2= minus1198891

(35)

If 1198881= 0 then 119906

119905(0 119905) = 0 which means that there is no

motion and the solution is trivial Otherwise 1198892= minus119889

1

which yields

119906119905(0 119905) = 11988811198891(1199031119890

1199031119905minus 11990321198901199032119905) (36)

Since after a time119879 the particle that is at the origin stops then

11990311198901199031119879minus 11990321198901199032119879= 0

yields997888997888997888997888rarr 119879 =

1

1199031minus 1199032

ln(11990321199031

) (37)

Mathematical Problems in Engineering 5

Consequently the solution can be written as

119906(119909 119905) = 1198891(1198881 cos 119896119909 + 1198882 sin 119896119909)(1198901199031119905minus 1198901199032119905) (38)

On the other hand it can be written as follows

119906(119909 119905) = (1205721 cos 119896119909 + 1205722 sin 119896119909)(1198901199031119905minus 1198901199032119905) (39)

Case 4 (120574 is negative and 11989621198894 minus 41198882 = 0) In this case thesolution is trivial

Case 5 (120574 is negative and 11989621198894 minus41198882 lt 0) In this case one canassume 120574 = minus11198962 119896 = 0 so that

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 119890

minus(11989621198892119905)2(1198891cos 119903119905 + 119889

2sin 119903119905)

(40)

in which

119903 =1

2119896radic41198882 minus 11989621198894 (41)

Hence the solution is

119906(119909 119905) = 119890minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times (1198891cos 119903119905 + 119889

2sin 119903119905)

(42)

Also the velocity is given as

119906119905(119909 119905) = 119890

minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times [(1199031198892minus1

2119896211988921198891) cos 119903119905

minus (1199031198891+1

2119896211988921198892) sin 119903119905]

(43)

From the initial condition one obtains

119906(0 0) = 11988811198891 = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

1= 0 (44)

If 1198881= 0 then 119906

119905(0 119905) = 0 and hence there is no motion in

the medium and the solution is trivial Therefore 1198881= 0 and

1198891must be zero so that

119906119905(0 119905) = 119890

minus(11989621198892119905)211988811198892[119903 cos 119903119905 minus 1

211989621198892 sin 119903119905] (45)

Since after a time 119879 the particle which is at the origin stopsthen

119903 cos 119903119879 minus 1211989621198892 sin 119903119879 = 0

997888rarr 119879119899=1

119903(tanminus1( 2119903

11989621198892) + 119899120587)

(46)

Consequently the solution can be written as

119906(119909 119905) = 211988811198892radic11988812 + 11988822

times 119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905)

(47)

in which

120593 = cosminus1( 1198881

radic11988812 + 11988822

) (48)

Therefore the solution can be written as

119906(119909 119905) = 120573119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905) (49)

4 Results

41 Longitudinal Waves in a Perfectly Elastic Solid In a metallike aluminum whose characteristics are listed in Table 1 thewave that propagates in the medium is of the form (49)

If one considers 119896 = 2 120573 = 0003 and 120593 = 1 thedepiction of the longitudinal wave that propagates from theorigin is shown in Figure 1 It is clear that it is a wave with noattenuation

42 LongitudinalWaves inViscoelastic Solids If the viscoelas-tic properties are characterized as in Table 2 the depiction ofthe longitudinal stress wave is shown in Figure 2 It is alsoclear that this wave is attenuated

43 Longitudinal Waves in a Viscoelastic Newtonian FluidIn a liquid with parameters shown in Table 3 the wavepropagating in the medium is of the form (39) So thenthe wave characteristics are discussed in the following formaterial with properties as mentioned in Table 3

If one assumes 119896 = 2 120572 = 001 and 120573 = 0003 then thedisplacement graph of particles in this medium is as shownin Figure 3 In Figure 3(a) the displacement of particles atthe origin without returning to their initial position and inFigure 3(b) the displacement of a particle in each position 119909without returning to its initial position are seen

44 Longitudinal Waves in an Ideal Newtonian Fluid If aliquid is viscoelastic with viscosity coefficients shown inTable 4 the graph of longitudinal stress wave in this mediumis as shown in Figure 4 In Figure 4(a) the distance betweenthe particle originally located in the origin and the sameparticle moved away from the origin in time 119879 is seen Thisdistance starts to decrease with the particles returning to theorigin In Figure 4(b) the distance between the particle atthe time 119879 and the same particle approaching its equilibriumposition at the origin is shown in Figure 4(c) the distancebetween the particles located in their respective equilibriumposition and the same particles moved away from the equi-librium position in time 119879 is shownThis distance then startsto decrease with the particles returning to their respectiveequilibrium position

6 Mathematical Problems in Engineering

u

0002

0001

0

minus0001

minus0002

02 04 06 08 1

t

(a) Particle at the origin

0002

0001

0

minus0001

minus0002

x

01

23 4

5

t

02 04 06 08 1

(b) Particle at each position 119909

Figure 1 The vibration of particles

u

0002

0001

0

minus0001

minus0002

1 2 3 4 5

t

(a) Particle at the origin

minus0002

minus0001

0

0001

0002

20 15 10 5 05

1015

(b) Particle at each position 119909

Figure 2 Vibration of a particle with attenuation

Table 1 Parameters of aluminum that is considered to be perfectlyelastic

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 0 276 times 10

6 KNm2 0 56000Kgm3

Table 2 Parameters of aluminum with fluid properties pertainingto viscoelasticity

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 3016 276 times 106 KNm2 2130 56000Kgm3

Table 3 Parameters of a Newtonian liquid like water which isperfectly fluid with no solid parameters

120582119904

120582119891

120583119904

120583119891

120588

0 3012 times 102 0 213 times 102 9963 Kgm3

Table 4 Parameters of a liquid like water which is not perfectly fluidand has solid parameters

120582119904

120582119891

120583119904

120583119891

120588

506 3012 times 102 276 213 times 102 9963 Kgm3

Mathematical Problems in Engineering 7

u

0012

0010

0008

0006

0004

0002

0

t

0 01 02 03 04 05

(a) Particle at the origin

0010

0005

0

minus0005

minus0010

020 015010 005 0

12

34

5

t

x

(b) Particle at each position 119909

Figure 3 The displacement of particles

u

4

3

2

1

0

t

0 02 04 06 08 1

(a) Thedistance between the particle originally located in the originand the same particle moved away from the origin in time 119879

u

4

3

2

1

0

t

0 20 40 60 80 100

(b) The distance between the particle at the time119879 and the same particleapproaching its equilibrium position at the origin

3

2

1

0

minus1

minus2

minus3

87

65

43

21

0 80 60 40 20 0

(c) The distance between the particles located in their respectiveequilibrium position and the same particles moved away from theequilibrium position in time 119879

Figure 4 The distance between the particles at the time 119879

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Consequently the solution can be written as

119906(119909 119905) = 1198891(1198881 cos 119896119909 + 1198882 sin 119896119909)(1198901199031119905minus 1198901199032119905) (38)

On the other hand it can be written as follows

119906(119909 119905) = (1205721 cos 119896119909 + 1205722 sin 119896119909)(1198901199031119905minus 1198901199032119905) (39)

Case 4 (120574 is negative and 11989621198894 minus 41198882 = 0) In this case thesolution is trivial

Case 5 (120574 is negative and 11989621198894 minus41198882 lt 0) In this case one canassume 120574 = minus11198962 119896 = 0 so that

11988310158401015840+ 1198962119883 = 0

11987910158401015840+ 119896211988921198791015840+ 11989621198882119879 = 0

yields997888997888997888997888rarr

119883(119909) = 1198881 cos 119896119909 + 1198882 sin 119896119909119879(119905) = 119890

minus(11989621198892119905)2(1198891cos 119903119905 + 119889

2sin 119903119905)

(40)

in which

119903 =1

2119896radic41198882 minus 11989621198894 (41)

Hence the solution is

119906(119909 119905) = 119890minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times (1198891cos 119903119905 + 119889

2sin 119903119905)

(42)

Also the velocity is given as

119906119905(119909 119905) = 119890

minus(11989621198892119905)2(1198881cos 119896119909 + 119888

2sin 119896119909)

times [(1199031198892minus1

2119896211988921198891) cos 119903119905

minus (1199031198891+1

2119896211988921198892) sin 119903119905]

(43)

From the initial condition one obtains

119906(0 0) = 11988811198891 = 0yields997888997888997888997888rarr 119888

1= 0 or 119889

1= 0 (44)

If 1198881= 0 then 119906

119905(0 119905) = 0 and hence there is no motion in

the medium and the solution is trivial Therefore 1198881= 0 and

1198891must be zero so that

119906119905(0 119905) = 119890

minus(11989621198892119905)211988811198892[119903 cos 119903119905 minus 1

211989621198892 sin 119903119905] (45)

Since after a time 119879 the particle which is at the origin stopsthen

119903 cos 119903119879 minus 1211989621198892 sin 119903119879 = 0

997888rarr 119879119899=1

119903(tanminus1( 2119903

11989621198892) + 119899120587)

(46)

Consequently the solution can be written as

119906(119909 119905) = 211988811198892radic11988812 + 11988822

times 119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905)

(47)

in which

120593 = cosminus1( 1198881

radic11988812 + 11988822

) (48)

Therefore the solution can be written as

119906(119909 119905) = 120573119890minus(11989621198892119905)2 cos(119896119909 minus 120593) sin(119903119905) (49)

4 Results

41 Longitudinal Waves in a Perfectly Elastic Solid In a metallike aluminum whose characteristics are listed in Table 1 thewave that propagates in the medium is of the form (49)

If one considers 119896 = 2 120573 = 0003 and 120593 = 1 thedepiction of the longitudinal wave that propagates from theorigin is shown in Figure 1 It is clear that it is a wave with noattenuation

42 LongitudinalWaves inViscoelastic Solids If the viscoelas-tic properties are characterized as in Table 2 the depiction ofthe longitudinal stress wave is shown in Figure 2 It is alsoclear that this wave is attenuated

43 Longitudinal Waves in a Viscoelastic Newtonian FluidIn a liquid with parameters shown in Table 3 the wavepropagating in the medium is of the form (39) So thenthe wave characteristics are discussed in the following formaterial with properties as mentioned in Table 3

If one assumes 119896 = 2 120572 = 001 and 120573 = 0003 then thedisplacement graph of particles in this medium is as shownin Figure 3 In Figure 3(a) the displacement of particles atthe origin without returning to their initial position and inFigure 3(b) the displacement of a particle in each position 119909without returning to its initial position are seen

44 Longitudinal Waves in an Ideal Newtonian Fluid If aliquid is viscoelastic with viscosity coefficients shown inTable 4 the graph of longitudinal stress wave in this mediumis as shown in Figure 4 In Figure 4(a) the distance betweenthe particle originally located in the origin and the sameparticle moved away from the origin in time 119879 is seen Thisdistance starts to decrease with the particles returning to theorigin In Figure 4(b) the distance between the particle atthe time 119879 and the same particle approaching its equilibriumposition at the origin is shown in Figure 4(c) the distancebetween the particles located in their respective equilibriumposition and the same particles moved away from the equi-librium position in time 119879 is shownThis distance then startsto decrease with the particles returning to their respectiveequilibrium position

6 Mathematical Problems in Engineering

u

0002

0001

0

minus0001

minus0002

02 04 06 08 1

t

(a) Particle at the origin

0002

0001

0

minus0001

minus0002

x

01

23 4

5

t

02 04 06 08 1

(b) Particle at each position 119909

Figure 1 The vibration of particles

u

0002

0001

0

minus0001

minus0002

1 2 3 4 5

t

(a) Particle at the origin

minus0002

minus0001

0

0001

0002

20 15 10 5 05

1015

(b) Particle at each position 119909

Figure 2 Vibration of a particle with attenuation

Table 1 Parameters of aluminum that is considered to be perfectlyelastic

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 0 276 times 10

6 KNm2 0 56000Kgm3

Table 2 Parameters of aluminum with fluid properties pertainingto viscoelasticity

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 3016 276 times 106 KNm2 2130 56000Kgm3

Table 3 Parameters of a Newtonian liquid like water which isperfectly fluid with no solid parameters

120582119904

120582119891

120583119904

120583119891

120588

0 3012 times 102 0 213 times 102 9963 Kgm3

Table 4 Parameters of a liquid like water which is not perfectly fluidand has solid parameters

120582119904

120582119891

120583119904

120583119891

120588

506 3012 times 102 276 213 times 102 9963 Kgm3

Mathematical Problems in Engineering 7

u

0012

0010

0008

0006

0004

0002

0

t

0 01 02 03 04 05

(a) Particle at the origin

0010

0005

0

minus0005

minus0010

020 015010 005 0

12

34

5

t

x

(b) Particle at each position 119909

Figure 3 The displacement of particles

u

4

3

2

1

0

t

0 02 04 06 08 1

(a) Thedistance between the particle originally located in the originand the same particle moved away from the origin in time 119879

u

4

3

2

1

0

t

0 20 40 60 80 100

(b) The distance between the particle at the time119879 and the same particleapproaching its equilibrium position at the origin

3

2

1

0

minus1

minus2

minus3

87

65

43

21

0 80 60 40 20 0

(c) The distance between the particles located in their respectiveequilibrium position and the same particles moved away from theequilibrium position in time 119879

Figure 4 The distance between the particles at the time 119879

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

u

0002

0001

0

minus0001

minus0002

02 04 06 08 1

t

(a) Particle at the origin

0002

0001

0

minus0001

minus0002

x

01

23 4

5

t

02 04 06 08 1

(b) Particle at each position 119909

Figure 1 The vibration of particles

u

0002

0001

0

minus0001

minus0002

1 2 3 4 5

t

(a) Particle at the origin

minus0002

minus0001

0

0001

0002

20 15 10 5 05

1015

(b) Particle at each position 119909

Figure 2 Vibration of a particle with attenuation

Table 1 Parameters of aluminum that is considered to be perfectlyelastic

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 0 276 times 10

6 KNm2 0 56000Kgm3

Table 2 Parameters of aluminum with fluid properties pertainingto viscoelasticity

120582119904

120582119891

120583119904

120583119891

120588

506 times 106 KNm2 3016 276 times 106 KNm2 2130 56000Kgm3

Table 3 Parameters of a Newtonian liquid like water which isperfectly fluid with no solid parameters

120582119904

120582119891

120583119904

120583119891

120588

0 3012 times 102 0 213 times 102 9963 Kgm3

Table 4 Parameters of a liquid like water which is not perfectly fluidand has solid parameters

120582119904

120582119891

120583119904

120583119891

120588

506 3012 times 102 276 213 times 102 9963 Kgm3

Mathematical Problems in Engineering 7

u

0012

0010

0008

0006

0004

0002

0

t

0 01 02 03 04 05

(a) Particle at the origin

0010

0005

0

minus0005

minus0010

020 015010 005 0

12

34

5

t

x

(b) Particle at each position 119909

Figure 3 The displacement of particles

u

4

3

2

1

0

t

0 02 04 06 08 1

(a) Thedistance between the particle originally located in the originand the same particle moved away from the origin in time 119879

u

4

3

2

1

0

t

0 20 40 60 80 100

(b) The distance between the particle at the time119879 and the same particleapproaching its equilibrium position at the origin

3

2

1

0

minus1

minus2

minus3

87

65

43

21

0 80 60 40 20 0

(c) The distance between the particles located in their respectiveequilibrium position and the same particles moved away from theequilibrium position in time 119879

Figure 4 The distance between the particles at the time 119879

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

u

0012

0010

0008

0006

0004

0002

0

t

0 01 02 03 04 05

(a) Particle at the origin

0010

0005

0

minus0005

minus0010

020 015010 005 0

12

34

5

t

x

(b) Particle at each position 119909

Figure 3 The displacement of particles

u

4

3

2

1

0

t

0 02 04 06 08 1

(a) Thedistance between the particle originally located in the originand the same particle moved away from the origin in time 119879

u

4

3

2

1

0

t

0 20 40 60 80 100

(b) The distance between the particle at the time119879 and the same particleapproaching its equilibrium position at the origin

3

2

1

0

minus1

minus2

minus3

87

65

43

21

0 80 60 40 20 0

(c) The distance between the particles located in their respectiveequilibrium position and the same particles moved away from theequilibrium position in time 119879

Figure 4 The distance between the particles at the time 119879

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

5 Conclusions

When a material is subjected to an impulsive force thereoccur infinitesimal displacements of the integrated parti-cles of that body In solids and in fluids the Navier-Lameand the Navier-Stokes equations govern these displace-ments respectively In the present work it was shown thatthrough the use of a generalized form of Kelvin-Voigt modelof viscoelasticity the unification of these two equations ispossible and one obtains an equation that can represent solidmaterials fluids and soft materials A new concept of vis-coelasticity was defined in which every material has somedegree of both solid and fluid properties depending onthe particular parameters of viscosity This is of particularimportance in relation to the soft materials Using this gen-eralized equation propagation of stress disturbance pulseand waves in solids as well as fluids can be studied in onesingle frame

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K Hackl and M Goodarzi An Introduction to Linear Contin-uum Mechanics Ruhr-University Bochum 2010

[2] L Treloar The Physics of Rubber Elasticity Clarendon PressOxford UK 1975

[3] N Muskhilishvili Some Basic Problem of the MathematicalTheory of Elasticity Noordhoff Leiden The Netherlands 1975

[4] C TruesdellMechanics of Solids vol 2 Springer NewYork NYUSA 1972

[5] J Allard Propagation of Sound in Porous Media Elsevier Sci-ence Essex UK 1993

[6] M A Meyers and K K Chawla Mechanical Behavior ofMaterials Cambridge University Press London UK 2008

[7] C-e Foias Navier-Stokes Equation and Turbulence CambridgeUniversity Cambridge UK 2001

[8] ldquoOvertheadsrdquo 2014 httpwwwucdiemechengccmOver-heads 67pdf

[9] ldquoDerivation of Naveir Stokes equationsrdquo 2014 httpenwikipe-diaorgwikiDerivation of the NavierE28093Stokesequations

[10] R M Christensen Theory of Viscoelasticity Academic PressNew York NY USA 1971

[11] A Musa ldquoPredicting wave velocity and stress of impact ofelastic slug and viscoelastic rod using viscoelastic discotinuityanalysis (standard linear solid model)rdquo in Proceedings of theWorld Congress on Engineering London UK 2012

[12] N McCrum C Buckley and C Bucknall Principles of PolymerEngineering Oxford University Press Oxford UK 1997

[13] A D Drozdov and A Dorfmann ldquoThe nonlinear viscoelasticresponse of carbon black-filled natural rubbersrdquo InternationalJournal of Solids and Structures vol 39 no 23 pp 5699ndash57172002

[14] H Zhao and G Gary ldquoA three dimensional analytical solutionof the longitudinal wave propagation in an infinite linearviscoelastic cylindrical bar Application to experimental tech-niquesrdquo Journal of the Mechanics and Physics of Solids vol 43no 8 pp 1335ndash1348 1995

[15] H Zhao G Gary and J R Klepaczko ldquoOn the use of a visco-elastic split Hopkinson pressure barrdquo International Journal ofImpact Engineering vol 19 no 4 pp 319ndash330 1997

[16] C SalisburgWave propagation through a polymeric Hopkinsonbar [MS thesis] Department of Mechanical EngineeringUnivesity of Waterloo Waterloo Canada 2001

[17] A Benatar D Rittel and A L Yarin ldquoTheoretical and experi-mental analysis of longitudinal wave propagation in cylindricalviscoelastic rodsrdquo Journal of the Mechanics and Physics of Solidsvol 51 no 8 pp 1413ndash1431 2003

[18] D T Casem W Fourney and P Chang ldquoWave separation inviscoelastic pressure bars using single-point measurements ofstrain and velocityrdquo Polymer Testing vol 22 no 2 pp 155ndash1642003

[19] K Rajneesh and K Kanwaljeet ldquoMathematical analysis of fiveparameter model on the propagation of cylindrical shear wavesin non- homogeneous viscoelastic mediardquo International Journalof Physical and Mathematical Science vol 4 no 1 pp 45ndash522013

[20] T Alfrey ldquoNon-homogeneous stresses in visco-elastic mediardquoQuarterly of Applied Mathematics vol 2 pp 113ndash119 1944

[21] J Barberan and I Herrera ldquoUniqueness theorems and the speedof propagation of signals in viscoelastic materialsrdquo Archive forRational Mechanics and Analysis vol 23 pp 173ndash190 1966

[22] J D Achenbach and D P Reddy ldquoNote on wave propagation inlinearly viscoelastic mediardquo Zeitschrift fur angewandte Mathe-matik und Physik ZAMP vol 18 no 1 pp 141ndash144 1967

[23] S Bhattacharya and P Sengupta ldquoDisturbances in a generalviscoelastic medium due to impulsive forces on a sphericalcavityrdquo Gerlands Beitrage zur Geophysik vol 87 no 8 pp 57ndash62 1978

[24] D P Acharya I Roy and P K Biswas ldquoVibration of an infi-nite inhomogeneous transversely isotropic viscoelasticmediumwith cylindrical holerdquo Applied Mathematics and Mechanics(English Edition) vol 29 no 3 pp 367ndash378 2008

[25] C Bert and D Egle ldquoWave propagation in a finite lengthbar with variable area of cross-sectionrdquo Journal of AppliedMechanics vol 36 pp 908ndash909 1969

[26] A M Abd-Alla and S M Ahmed ldquoRayleigh waves in anorthotropic thermoelastic medium under gravity field andinitial stressrdquo Earth Moon and Planets vol 75 no 3 pp 185ndash197 1996

[27] R C Batra ldquoLinear constitutive relations in isotropic finiteelasticityrdquo Journal of Elasticity vol 51 pp 269ndash273 1998

[28] S Murayama and T Shibata ldquoRheological properties of claysrdquoin Proceedings of the 5th International Conference of SoilMechanics and Foundation Engineering Paris France 1961

[29] R Schiffman C Ladd and A Chen ldquoThe secondary consoli-dation of clay rheology and soil mechanicsrdquo in Proceedings ofthe International Union of Theoritical and Applied MechanivsSymposium Berlin Germany 1964

[30] R Jankowski KWilde and Y Fujino ldquoPounding of superstruc-ture segments in isolated elevated bridge during earthquakesrdquoEarthquake Engineering and Structural Dynamics vol 27 no 5pp 487ndash502 1998

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

[31] V N Nikolaevskij Mechanics of Porous and Fractured MediaWorld Scientific 1990

[32] N Thin Understanding Viscoelasticity Springer HeidelbergGermany 2002

[33] ldquoviscoelasticityrdquo 2014 httpenwikipediaorgwikiViscoelasticity[34] ldquoWavepropagationrdquo 2014 httpswwwnde-edorgEducation-

ResourcesCommunityCollegeUltrasonicsPhysicswaveprop-agationhtm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of