differential maxwell

8/9/2019 Differential Maxwell

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works in this field can be found in Ross,Ungar and Kerwin, Rao and Plun-

kett and Lee. Johnson and Kienholz[8] introduced the concept of the useof modal strain energy(MSE) and FE formulations to be used to calculatemodal loss factors for a system with viscoelastic dampers. MSE concept iswidely used to quantify the damping capacity of structures.

However, in above mentioned analyses the base material was consideredperfectly elastic which is reasonable assumption for metals. Usually there isa trade off between damping capacity and structural strength. Compositematerials have excellent structural properties and often an order of mag-nitude higher damping capacities than metals. The intrinsic damping incomposite is primarily due to viscoelastic nature of the matrix and interface

between fibre and matrix. This study will be focused understanding hystericdamping which is dominant mechanism in undamaged composites at smallamplitudes.

Due to anisotropic nature of fibre reincofrced polymers, damping is negligiblein direction of fibres is much higher in tranverse direction. The proeprtiesof composites are usually calculated using voight and reuss rules of mix-tures. Hashin-shritkman dervided bounds based on variational principleswhich provide a more accurate prediction of composite materials. Hashinfurther developed analytical equations to calculate complex modulus of uni-directional FRP based on properties of matrix and fibres. Numerical worksby [Hwang and Gibson, Chandra have been presented based on strain en-

ergy method to predict damping in a composite ply. The strain energymethod states that of any system of linaer viscoelastic elements the lossfactor can be expressed as a rati of summation of the product of individ-ual element loss factors and strain enrgy stored in each elemnt to the toalstrain energy. Damping and moduli of composites with short aligned fibrescan be predicted using Cox. Bert and chang used micromchanics and corre-spondence principle to characterize damping in a unidirectional composite.Saravanos and chamis developed integrated micromechanics methodologyto completely characterize damping in comosites via six damping coeffients.Kaliske and Rothert developed a modeol for material damping using a con-sistent micromechanical theory of a representative fibre matrix cell. Tsai

and chai explored the effects of packing array of fibres on overall dampingusing method of cells. Brinson and fisher workers have compared the me-chanical property predictions for a three phase viscoelastic coposites by useof bensivite and mori tanaka method.

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Given the damping properties of a lamina, there are several macromechani-

cal theories to predict damping of a composite. Adams and bacon developedmacromechanical model to predict damping in compostes. They character-ized overall damping as a summation of damping capasities due to longitu-dinal, tranverse and shear stresses. Adams and maheri used these resultsto predict flexural damping and strength with respect to fibre orientation.They showed that at lower angles(0-15), material behaved as hookean elas-tic material and damping can be predicted usinghysteric loops. However athigher angles of (30-90) damping is governed by interaction between fibresand matrix. CT SUN AND GIBSON used classical laminate theory (CLT)and correspondence principle to predict damping in composite materials. Astate of plane stress can only be assumed if the lamina is thin. However for

prediction of damping in composites with integrated compliant viscoelasticlayers, first order shear deformation theory gives better results. A com-paritve review of major macromechanical theories is given by Billups andcavalli. Crane and Gillespie used CLT to calculate A-B-D matrix of com-posites integrating the frequency dependence of damping. In the above the-ories, an assumption fibres being elastic and non-dissipative has been made.Saravanos and Chamis further developed their theory to predict dampingin composites using discrete layer damping mechanics for thick laminates.Gibson and Guan used CLT to predict damping in woven fabric-reinforcepolymer matrix composites using saravanos and chamiss equations. Thereare several studies in lieterature which further extend the findings are macro-scopic scale into structural level. However, they are not considered at this

stage of in this project.

3 Introduction

3.1 Damping

Damping is defined as the ability of a material or structure to dissipate en-ergy under oscillatory motion. It is broadly classified into Active damping-inwhich feeback forces in a control loop are used to limit deflections. Piezoelec-tic actuators are being widely researched and used for this purpose. Passivedamping-which will be the focus of this research, is related to the iherit abil-

ity of the material or system to dissipate energy under oscillatory loading.Passive damping is further divided in two branches Material or Hystericdamping and Structural damping.

Material damping is related to the energy dissipation inherent to the ma-

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terial. Materials such as elastomers have higher damping capacity than

metals at the expense of stiffness. Polymer matrix composites are known tohave significantly higher material damping due to viscoelastic nature of thepolymer matrix and heterogeneity. The major mechanisms which cause thisdamping are classified as internal friction include phenomenon like move-ment of polymeric chains, internal defects and relative movement of domainwall. The polymeric chains causes elastomers to disspate more energy inshear than in dilational deformation. Structural damping relates to inclusionof damping devices such as dashpots, viscoelastic constrained layers, frictionin riveted joints etc. This research will be focused on the understanding thematerial damping characteristics of the fibre reinforces composites(FRP) interms of its constituents usually Epoxy matrix and glass/carbon fibres.

3.2 Measurement of damping

There are several ways to quantify damping in material or structure. Forfurther description and derivation refer to THE relationship of traditionaldamping measures for materials

• Specific Damping Capacity(SDC) Ψ: SDC is defined as the ratio of the energy dissipated to the energy stored in the system.

Ψ = ∆W

W (1)

Where,

– ∆ W:Energy dissipated in one cycle

– W:Total Energy Stored in one cycle

There are several choices of W which have been used in literature.[Lazan, B.L., Damping of Materials and Members in Structural Me-chanics, Permagon Press Ltd., New York, 1968.] defined W to be totalstrain energy for the entire specimen at maximum deformation. Otherdefinitions include, peak potential energy solely of the elastic compo-nent of the model. However, [Ungars] definition of stored energy equalto the totalstrain energy at maximum displacement is widely used. Asfor any structure, the total strain energy and hence the damping will

be related to the global deformation. The concept of Modal damp-ing Capacity is widely used to simply characterization of damping instructures undergoing small deflections. Modal SDC with n mode isaccording to Chamis/saravanos 1991:

Psi n

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Figure 1

• Loss Factor Eta Loss factor is defined as the tangent of the loss anglePhi. It is also related to the SDC as follows. Loss factors are alsorelated to the complex modulus which will be discussed in more detailin subsequent sectons

• Logarithmic Decrement This method is useful to characterize dampingfor free vibrating structures in time domain. Logarithmic decrement is

defined as the natural log of the ratio of amplitudes of any successivepeaks.

δ = 1

n ln

x1

xn+1(2)

4 Linear Viscoelastic Theory

4.1 Introduction

In order to fully characterize damping in composite its essential to under-stand the theory of linear viscoelasticity and the constituitive models that

allow mathematical characterization of damping. The constituitive relation-ship between stress and strain for elastic materials in one dimensiom is givenby hookes law as follows.

σ = E (3)

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However, it is well known that real materials exhibit both elastic as well

as some viscous properties. These Viscoelastic materials exhibit proper-ties such as relaxation, creep, frequency dependent stiffness and dissipativecharacteristics as well as strain rate dependent hysteric behavious. Severaltexts such as Ferry, Christensen and aklonis and macknight describe thefundamental theory of viscoelasticity. Readers are also recommended textby hal and catherine brinson for an introduction to viscoelasticity and moreinformation about microscopic properties. The dissipative property of vis-coelastic polymer matrix arises from its long chained polymer structure. Themovement of polymer are highly dependent on temperature and frequency.

In order to develop constitutive relations for viscoelastic materials, it isimportant to understand relaxation and creep phenomena.

4.2 Stress Relaxation

On application of constant strain applied quasi-statically(to ignore intertialeffects) viscoelastic materials show gradual reduction in stress. As the Stressis function of time, corresponding relaxation modulus will also vary withtime.

E (t) = σ(t)

0= 0E (t) (4)

Where,

• E (t) = RelaxationModulus

The latter equation is analogous to Hookes law for material that is timedependent but only at constant strain. Similar relaxation functions can beobtained for Shear G(t) as well as Bulk Moduli(t). It is also helpful to definethe Instantaneous modulus, Eo at t=0 and Equilibirium modulus when t=.

4.3 Creep

Another fundamental characterization test for viscoelastic material is doneby appling constant stress on the viscoelastic material under which materialshows slow, progressive deformation i.e. Creep.

D(t) = (t)

σ0or (t) = σ0D(t) (5)

Where,

• D(t) = CreepCompliance

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Figure 2

Figure 3

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4.4 Linearity

In order to determine whether under specific loading conditions, the vis-coelastic response of the material is linear or non-linear. This can be doneby performing creep or relaxation tests at three different stress(or strain)levels and plotting the isochronous Stress-Strain diagrams as shown below.If the isochronous graph is linear for any given time, the material responseis also linear.

4.5 Boltzmann Superposition Principle

So far, the response of viscoelastic materials to step stress or strain have beendefined. To develop constituitive relationship at any given stress/strain we

use the Boltzman superposition principle. Boltzman superposition principlestates that the effect of a compound cause is the sum of the effects of theindividual causes.Consider any arbitary strain history as given in FIGURE. The idea here isto decompose entire strain history into small pulses. The stress at time tis the summation of the stress effects of each of these pulses. The principleof causality which states that the cause should precede the effect allows usto consider oly the pulses upto present time. The Boltzman superpositionintegral can be shown to be, where t is time and T is time variable of integration.

σ(t) = t

0

E (t − T )d(T )

dT dT (6)

4.6 Correspondence Principle

Correspondence principle first suggested by Alfrey, states that if a solutionto a linear elasticity problem is known, the solution to the corresponsingproblem for the viscoelastic material can be be obtained by replacing eachquantity which can depend on time by its laplace tranform mutiplied bythe transform variable(s or w),and then by transforming back into the timedomain. It can be shown that for viscoelastic materials, a simple substitutionof complex properties is sufficient.

5 Viscoelastic Material Models

In order to mathematically model viscoelasticity and gain a better under-standing of the physical relation between stress and strain, combination of spring and dash-pots are used. Maxwell model consists of a purely elastic

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Figure 4

spring and purely viscous damper in series. Kelvin model consist of a springand dashpot connected in parallel.

5.1 Differential Equation for Kelvin Model

For a Kelvin solid the equilibrium condition is,

σ = σs + σd (7)

however the stress is same in the model i.e.

= s = d (8)

For spring,σs = E ss (9)

For dashpot,σd = ηd ̇d (10)

Hence, the differential equation is given by,

σ = E ss + ηd ̇d (11)

Considering Creep loading i.e. step stress input, The Strain is given as,

(t) = σ

0E

1 − e− t

τ

(12)

and the corresponding creep compliance is,

D (t) = 1

E

1 − e−

tτ

(13)

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The dashpot doesn’t allow the spring to deform instantaneously. Hence

in order to obtain a step strain, infinite stress is required. Making Kelvinmodel unsuitable for the modelling real materials. Similarly it can be shownthat Maxwell model predicts correct Relaxation response but not Creep.However, these make good building blocks and combinations of both modelsare widely used to model viscoelastic response.

5.2 Standard Linear Solid

Standard linear Solid (SLS) model in the simplest model which can predictboth relaxation as well as damping phenomena effectively. The equilibiriumcondition and kinematic conditions are,

σ = σmaxwell + σ1

= maxwell = 1(14)

Considering the Maxwell arm, the equilibrium and kinematic conditions are,

maxwell = s + d (15a)

σmaxwell = σs = σd (15b)

Differentiating 15a and substituting equations for spring 9 and dashpot 10,the constituitive relationship for the maxwell element can be obtained,

.σ

E 2 +

σ

η =

.

(16)

Using 16 and 14 the constitutive equation for the SLS is as given below,

.σ +

1

τ σ = (E 1 + E 2)

. +

E 1

t (17)

Where,

• Relaxation time τ = ηE 2

In Rheological texts, the differential equations are written in standardform with accending derivatives from left to right.

a0σ + a1.

σ= b0 + b1. (18)

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5.2.1 Harmonic Response of SLS Solid

We are interested in understanding hysterisis i.e. damping properties of viscoelastic materials under oscillatory loading i.e. 0 cos or 0 sin. Usuallycomplex exponentials are used for derivations due to their simpler alge-bra.Let us consider a oscillatory input of,

(t) = 0eiωt (19)

Substituting 19 in 17 leads to,

σ̇ + σ

τ = iw0 (E 1 + E 2) e

iωt + E 1

τ 0e

iωt (20)

Solving the above differential equation and ignoring the transient terms, Forsteady state oscillations stress as a function of loading frequency is given by,

σ (t) = 0eiωt

E 1 + E 2

ω2τ 2

1 + ω2τ 2

+

iE 2ωτ

1 + w2τ 2

(21)

Where;

• Storage Modulus=E : E 1 + E 2( ω2t2

1+ω2τ 2)

• Loss Modulus:E = iE 2ωτ 1+w2τ 2

• Complex Modulus: E ∗ = E + iE

The storage modulus(Young’s modulus) is defined as the component in phaseof the applied strain while other component, Loss modulus is related to theenergy lost by the material and hence is out of phase by 90 deg

. (t) =

iw0eiωt.From 21 it is clear the the stress lags behind strain by certain angle,

called loss angle δ . The storage modulus represents purely elastic responseof the material, while loss modulus represents purely viscous response whichis out of phase by 90 deg. Similar to Storage and Loss Modulus, the Storageand Loss compliance can be shown to be

StorageCompliance, J = E 1 + ω

2t2 (E 1 + E 2)

E 2I

+ ω2τ 2(E 1

+ E 2

)2 (22)

LossCompliance, J = −E 2ωτ

E 21 + ω2t2(E 1 + E 2)

2 (23)

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Storage and Loss modulus are directly related to the energy dissipation

of the material at certain loading frequency via previously defined quantity,loss factor as follows,

LossFactor : η(ω) = tan δ (ω) = E

E (24)

Where,

• δ :Loss angle or phase angle between stress and strain

The Steady state reponse of viscoelastic material to oscillatory strain isgiven via 25

σ (t) = 0eiωt E + iE (25)

σ (t) = e0E ∗eiωt (26)

The above equation is for any general sinusoidal loading. For sin straininput, the stress function can be extracted as follows,

= 0sin(ωt) (27)

σ (t) = Im

0E ∗eiwt

(28)

using Euler’s Identity we get,

σ (t) = 0Im

E

+ iE

[cos(wt) + isin(ωt)]

(29)

σ (t) = 0

E sin(wt) + E cos(ωt)

(30)

5.2.2 Variation with Frequency

Using 21, storage modulus, loss modulus and loss factor can be plotted forcertain SLS solid at various frequency.

5.3 Wiechert Model

As seen from the spectrum of storage and loss modulus, a single Maxwellelement does not capture behaviour of real materials as the storage modulusdecays too rapdly. This can be overcome by using multiple Maxwell models

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Figure 5

in the SLS, also called as Wiechert model. The solution for stress relaxationand the relaxation modulus is given by,

σ(t) = 0

E ∞ +

ni=1

E ie−tτ i

(31)

Where,E ∞ is the equilibrium modulus. Hence,modulus can be defined as,

E (t) = E ∞ +n

i=1

E ie−tτ i (32)

The above series is called Prony series via which, experimental curve canbe fitted with many such maxwell models. An alternate form which relateselastic modulus to equilibrium modulus is also useful,

E (0) = E ∞ +n

i=1

E i (33)

Therefore,

E (t) = E 0 −n

i=1

E i

1 − e

−tτ i

(34)

identical series can be given for shear and bulk modulus.

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5.4 Viscoelasticity in ABAQUS

Since Abaqus FE is the primary FE tool used for this study, it is importantto understand how it handles viscoelasticity. Abaqus assumes time-domainviscoelasticity to be defined by Prony series expansion. Prony series termscan be indiviudally defined, or can be caluculated by Abaqus based on givenCreep test data, Relaxation test data or Frequency dependent data. It alsoallows the definition of instantaneous or equilibrium young’s modulus. Theinpur parameters of prony series are given by normalized tems Abaqus usesthe time domain viscoelastic data only in the Dynamic Implicit analysisstep and frequency domain data in Steady State Direct dynamic analy-sis, subspace-based steady state, nautral and complex frequency extractionsteps. Dynamic implicit analysis step was chosen for this study to make the

anaylsis very general, allowing inclusion of study of heat generation due tovibration in composite structures at a later stage. The prony series input forabaqus is required in normalized form as follows from 34 for Shear modulus,

G(t) = G0

g∞ +

N i=1

gie−tτ

(35)

Where, gi is the relative moduli termThe primary objective of this research is to establish a multi scale modelin through which damping and strength prediction of composites can bedirectly from properties of individual constituents. Accurate representation

of fibre and matrix is essential at microscale to obtrain accurate homogenizedproperty of lamina at meso scale.

6 Micromechnical Theory

Micromechanical analysis is essential in developing macroscopic constitu-tive equations for heterogeneous materials. Micromechanical analysis helpsto understand effect of fibre size, shape arrangement, volume fraction aswell as give information about the average and localized deformation. Thefundamental problem in micromechanical of heteogenous materials is thedetermination of stress and strain concentration tensors averaged over the

the volume.The ’average’ or homogenized stress and strain fields are extracted as

below,

σ̄ij = 1

V

V

σij (x , y , z) dV (36)

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Figure 6: Geometric representation of a)Statistically heterogenous mi-crostructure charactirized by RVE b)Periodic Microstructure Characterizedby RUC

and,

̄ij = 1

V

V

ij (x , y , z) dV (37)

The method of explicitly taking in account of heterogeniety of compositemicro structure can be classified into statistical homogeniety and periodicity.The concept of Repeating Unit Cell(RUC) is based on assumption of peri-odic arrangement of fibres in a matrix, usually square or hexagonal packing.RUC is the smallest building block of such arrangement and the response of entire macroscopic is equiavalent to that of RUC under any given loading.Inreality however, fibres are randomly distributed in a matrix at microscopic

level. The concept of statiscal homogenity explicitly takes into account of the actual stochastic nature of actual microstrucutre. Analysis of statiscallyhomogenous materials is based on the representative volume element(RVE)concept. Kanit defines RVE as ”a volume V of heterogeneous material that issufficiently large to be statistically representative of the composite, i.e. to ef-fectively include a sampling of all the microstructural heterogeneities(fibres,voids) that occur in the composite.” The number of inclusions required tobe modelled in a RVE needs to be determined statistically.

Boundary conditions heavily influence the homogenized properties andconvergence of the microscopic properties. Three boundary conditions usedfor homogenization are mentioned below.

• Kinematic uniform boundary condition (Displacement boundary con-diton): Displacement u is imposed at point x belonging to a boundary∂V such that

ui = E ijx j ∀x ∈ V (38)

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Figure 7

This enforces certain displament on every point on a boundary.

• Static uniform boundary condition (Traction boundary condition):The traction vector is prescribed at the boundary

tij = σijn j ∀x ∈ V (39)

where, n is the vector normal to ∂V

Since, the periodic array of RUC and RVE represents a continuous physicalbody, two continuity conditions must by satisfied at the boundary of neigh-bouring RUCs. Displacement over the opposite edges must be continuous,

meaning the RVE’s cannot overlap or separate from neighbours. Also thetraction on the opposite sides must be zero. These periodic boundary con-ditions allows the RUC to be assembled making a continuous physical body.For two points on opposite boundaries

ui (x0 + d) = ui (x0) + ̄ijd j and ti (x0 + d) + ti (x0) = 0 (40)

where d is the characteristic length scale of the microstructire. Periodicboundary conditions have shown to give better results and convergence andwill be used in this study.

7 Finite Element Analysis7.1 Square and Hexagonal Array

Aim of this study was to understand variation of damping and elastic modu-lus in a two phase composite with respect to volume fraction and geometrical

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Figure 8

Material E GPa v Density (kg/m3)

Fibre 50 0.1 2580

Matrix 2.76 0.38 1200

arrangement of fibres subject to loading at particular frequency and peri-odic boundary conditions. Only Axial and transverse damping and elasticmoduli were studied with an aim to extend to shear loading in future.

7.1.1 Model and Materials

A FE model was created in Abaqus CAE with shown in ??. Modelling wasdone with help of python scripts, via which the effect of volume fractioncould be modelled easily. The volume fractions of 0.2,0.4,0.6 and 0.7 werestudied for both square and hexagonal array. Glass fibre of diameter 10 µwas modelled as being elastic. A hypothetical material was considered asmatrix with a single term prony series.

As previously discussed, Abaqus requires normalized prony terms as in-put. Viscoelastic Property of matrix was input in time domain as follows

gi ki τ

0.6 0.6 20

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Volume Fraction Square Hexagonal

0.2 1860 187000.4 1680 7500

0.6 1680 4200

0.7 1680 3421

Figure 9

7.1.2 Mesh

8-node linear brick 3D stress elements were used for meshing. A mesh con-vergence study was performed and the required elements are given in thetable ??

7.1.3 Boundary Conditions and Loading

A harmonic sin strain at a circular frequency of 0.02 rad/s with an ampli-tude of 1%, applied on Z+ and Y+ face (??) were considered in two separateanalysis. In each analysis, the Y- face was constrained in Y direction. Peri-odic Boundary conditions were employed in the X+ and X- face for each pairof node opposite to each other. This was done via python script, given inAppendix. Nodal reaction forces at the Z+(or Y+) faces were summed anddivided by the area to obtain the average stress response. Given the strain

response, the storage and loss modulus were calculated using the hysteresisloop approach 10. Damping was obtained in terms of loss factor 24.

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Figure 10: Hysterisis plot

7.2 Stochastic Fibre Model

Since results of periodic array show minimal variation of axial properties,only tranverse properties were studied in stochastic fibre model.Creation of randomly placed fibre in a matrix is not a trivial problem, especially athigher volume fractions of 0.6 and 0.7. A python script was created usingSCIPY package. The length of the square RVE was fixed at 0.125mm i.e.typical thickness of a lamina. The algorithm is as follows,

1. For a given volume fraction, calculate number of fibres required in theRVE

2. Randomly create required amount of circles of a given radius 10µ inthe RVE

3. Define constraints between two circles and between RVE boundary and

circles for e.g. minimum distance between fibres to avoid touching

4. Use scipy optimize package to fit the circles subject to defined con-straints

5. Extract the coordinates of the circles to be used as an input in Abaqus

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This resulted in average of 54, 92 and 130 fibres for 0.3,0.5 and 0.7 volume

fractions respectively. Four noded Plain strain elements were used for the2D simulations. Based on mesh convergence study results, the global meshsize was set at 1.2 and 39 nodes were created along the circumference of each fibre. Also a minimum size control fraction was set at 0.01. Thisresulted in 29869, 31250, 32295 elements for volume fractions of 0.3,0.5 and0.7 respectively.

Two different realization of model at volume fractions of 0.3,0.5 and 0.7are as given below.

7.3 Checkerboard Model

Explicitly modelling fibres is more accurate but computationally very ex-

pensive. A binary voxel model is proposed in which the RVE is discretizedinto square grid, where each element represents either matrix or fibre. Thismodel is not expected to predict correct localizations due to square inclu-sions. However, the homogenized properties are of interest and would resultin substantial decrease in computational requirements. The matrix and fi-bres are considered perfectly bonded. RVE of 0.125mm was discretized intosquare grid. Fibres were disperesed randomly using in the RVE based onvolume fraction. The model is as shown below,

At higher volume fraction, substantial clustering of fibres is evidentwhich doesn’t reflect true geometry of the composite material.

8 Results and Discussion

8.1 Analytical results

FE simulation results will be compared with the theoretical calcullations viaRules of Mixtures and Hashin-Shitrikmann bounds.

8.1.1 Rules of Mixtures

The rules of mixtures are widely used to predict strength of compositesin axial and traverse direction. By using correspondence principle, theycan be extended to viscoelastic materials. Rules of mixtures are based on

assumptions of isostress and isostrain and do not include information re-garding fibre microstructure or interaction between fibres. However, ROMand IROM would provide absolute upper and lower bounds for elastic anddamping properties to validate FE simulations.

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(a) Vf=0.3 (b) Vf=0.3

(c) Vf=0.5 (d) Vf=0.5

(e) Vf=0.7 (f) Vf=0.7

Figure 11: 3 x 2

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(a) Vf=0.3 (b) Vf=0.5 (c) Vf=0.7

Figure 12: Binary Voxel Mesh Green-Fibres, Cream-Matrix

In axial directon the rules of mixtures,

E ∗axial = vf E ∗

f + vmE ∗

m (41)

substituting for complex terms,

E ∗composite +iE

axial = vf (E

f + iE

f ) + vm(E

m + iE

m) (42)

Where,

• E ∗axial = E

axial + iE

axial

• LossFactor = ηaxial = E axialE axial

For Tranverse direction, using inverse rules of mixtures,

1

E ∗tranverse=

vf

E f +

vm

E m(43)

again the complex transverse moduli can be found by substituion of complexterms in above equations. Below is the example given for transverse storagemodulus,

E transverse =

vf E

m + vmE

f

E

f E

m − E

f E

m

+

E

f E

m + E

f E

m

vf E

m + vmE

f

vf E m + vmE

F

2+

vf E m + vmE

f

2(44)

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8.1.2 Hashin and Hashin-Shtrikman Bounds

Hashin-Shtrikman developed tightest possible bounds for elastic propertiesof two phase composite materials, using variational principles for arbitaryphase geometry. Hashin 1972 further developed predictions of various mod-uli for composites which are as given below.

bounds using cocentric cylinder arrangement(CCA) model . Using vis-coelastic correspondence principles, these formulations can be used to obtainpredict axial and transverse elastic moduli and damping. The bounds forShear and Bulk modulus for a composite with cylindrical inclusions are asgiven below,

K ∗L = K m + vf 1

K f −K m+ 3vm3K m+4Gm

(45)

K ∗H = K f + vm1

K m−K f +

3vf 3K f +4Gf

(46)

G∗L = Gm + vf

1Gf −Gm

+ 6vm(K m+2Gm)5Gm(3K m+4Gm)

(47)

G∗H = Gf + vm

1Gm−Gf

+ 6vf (K f +2Gf )5Gf (3K f +4Gf )

(48)

Where subscripts L and H correspond to lower and higher bounds and m andf subscripts refere to matrix and fibres respectively. Corresponding boundson complex Young’s modulus were obtain by using correspondence principleand 9KG3K +G

9 FE Results

The FE models were used to study the effect of volume fraction and fibrearrangement on axial and transverse elastic modulus and damping. In thissection, the stress plots for σyy and σxy for square, hexagonal and randomarray for different volume fractions are as given below are given below. TheFE results are then compared to Rules of mixtures and Hashin prediction.

From figure , there is no significant influence of fibre arrangement

on the axial storage modulus. The FE results are in excellent agreementwith ROM and Hashin prediction. In the direction of fibres, the property of composite is highly dependent on fibres even at low volume fractions. Sincethe fibres are modelled as elastic, axial loss factor falls quickly to about 10 %that of the matrix at volume fraction of 0.2. Negligible damping is expected

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in the direction of fibres. In order to achieve high damping in axial direction,

lossy fibres like aramid or kevlar must be considered. Again, no influence of the packing arrangement is observed on the axial damping of composite.

Graph 9 show significant difference between theoretical and FE results.Rules of mixtures are unsuitable to predict loss modulus and damping intransverse direction. Hashin theory provides better predicition of transverseproperties due to the fact that it is based on CCA model and carries in-formation regarding fibre packing. Fibre arrangement affects the modulusand damping at higher volume fractions volume fractions (vf >0.4). Graphs9and 9 show that square array predicts higher modulus and lower dampingcompared to theory and hexagonal array. At higher volume fractions it isimportant model information regarding fibre interactions, which is absent

in the square unit cell. It is also known that viscoelastic damping is moreaffected by shear stresses. It is evident from figure??b and figure??, thatshear stress zones are much larger in hexagonal cell than in square cell.Hencesquare cell produces more ’stiffer’ response, this matches with findings of Brinson & Lin and Tsai &Chi. Figure ?? again shows formation of shearbands and strong fibre interactions. However at higher volume fraction thiseffect reduces and hence reduction in damping.

The checkerboard model gives good prediction for vf


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(a) Vf=0.4 (b) Vf=0.4

(c) Vf=0.6 (d) Vf=0.6

Figure 13: Stress Plots for Square Array

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(a) Vf=0.4 (b) Vf=0.4

(c) Vf=0.6 (d) Vf=0.6

Figure 14: Stress Plots for Hexagonal Array

26


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(a) Vf=0.3 (b) Vf=0.3

(c) Vf=0.5 (d) Vf=0.5

(e) Vf=0.7 (f) Vf=0.7

Figure 15: Stress Plots for Randomly distributed fibres

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(a) Vf=0.3 (b) Vf=0.3

(c) Vf=0.5 (d) Vf=0.5

(e) Vf=0.7 (f) Vf=0.7

Figure 16: Stress Plots for Randomly distributed fibres

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0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Volume Fraction

N o r m a l i z e d S t o r a g e M o d u l u s

E 1 1

E f

ROM

HashinSquare

Hex

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Volume Fraction

N o r m a l i z e d L o s s F a c t o r η 2 2

η m

ROMHashinSquare

Hex

29


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0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Volume Fraction N o r m a l i z e d S t o r a g e M o d u l u s T r a n s v e r s e

E 2 2

E m

ROMHashinSquare

HexRandom

Checkerboard

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Volume Fraction

N o r m a l i z e d L o s s F a c t o r T r a n s v e r s e η 2 2

η m

ROMHashinSquare

HexCheckerboard

30

differential maxwell

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