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Effective wave propagation in pre-stressed compositesCardiff School of Mathematics,

Cardiff University,20th February 2008

William J. Parnell

Faculty Research Fellow

School of Mathematics, University of Manchester, UK

William.Parnell@manchester.ac.uk

http://www.maths.man.ac.uk/∼wparnell.

W.J.Parnell, School of Mathematics, University of Manchester. – p.1/57

Motivation – Composite Media

Waves in prestressed media• Thales Underwater Systems• Scattering from a single inclusion in pre-stressed nonlinear

materials.• Recent theoretical work by Gareth Jones (OCIAM) in his PhD• Recent experimental work at Queen Mary• The problem of wave propagation through non-dilute filled rubbers

(with pre-stress) has been initiated(Natasha Willoughby - CASE PhD studentship).

Inherent difficulties

• The constitutive behaviour of rubber is strange (see shortly)• Large deformation (geometric nonlinearity)• Even simplified, nonlinear elastic behaviour (i.e. not Hookes Law),

governed by W = W (λ1, λ2, λ3) and Tj = Aj(λ)dW/dλj is verydifficult.

• Microstructure evolution (shape, position tracking, buckling, etc.)• Changes in phase properties (density, elastic moduli, etc.)• Rubber is strongly temperature and frequency dependent• Natural rubber is pretty useless - additives - complicates the

constitutive behaviour further• Composite modelling

Constitutive behaviour of rubber

Behaviour of (Newtonian) fluids and (Hookean) solids is linear:

σ = µ∂e

∂tσ = Ee.

Rubber however is a nonlinearly elastic (thermo) visco-plastic materialSo how does σ depend on e or some other measure of deformation?

Note also that rubber is multiscale:* in its constituents* in its constitutive behaviour (time)

Idealized behaviour - nonlinearly elasticAs an idealized nonlinearly elastic solid it behaves like this:σ = measure of the stressλ = measure of the deformation (stretch)

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Clearly the constitutive behaviour is:* Nonlinear* Fully reversible

The shape of the curve σ = σ(λ) is complicated......more later.

Inelasticity - the idealized “Mullins effect”Let us first consider very slow (quasi-static) loading.This means we can (sort of) neglect viscous effects.

Rubber exhibits the Mullins effect(Mullins, L., J. Rubber Res., 1947)

The Idealized case:

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* Stress softening on unloading* Related to “damage” - initially bonds/cross-links are broken* On re-loading this energy is not needed again

(assuming all such bonds are broken)* After several cycles this influence is lost

The real Mullins effect - quasi-staticsAgain, load very slowly. What you really see is this:

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Unload

* Residual strain (plastic deformation - or is it?!)* Reloading curves don’t follow unloading (new bonds broken)* After about 6-7 cycles of loading we end up with a single (hysteretic)curve for loading/unloading.........

Removal of the Mullins effect

So in experiments on rubber, you have to cycle a few times first toremove the influence of the Mullins effect.

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Unload

Viscoelastic effects - experimentsFor slow (but finite) deformations you see the Mullins effect andviscoelastic effects.

Some data - first cycle

−15 −14 −13 −12 −11 −10 −9 −8 −7−1

−0.5

2s80−5x5−100%

Displacement (mm)

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LoadUnload

A second cycle

−14 −13 −12 −11 −10 −9 −8 −7 −6−0.5

2.5s80−5x5−100%−2Lo

Displacement (mm)

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LoadUnload

Residual Strain

We noted that some residual strain was left over from the cycling.

In fact if you return to the specimen in about 24 hours this residualstrain has usually disappeared!

We therefore conclude that this is a longer time viscoelastic effect andnot plasticity!

(Thus multi-scale constitutive behaviour).

Other thoughts

What about thermal effects?What about waves in rubbery materials?

There is a definite frequency dependence of rubber.For rubbers in a stress-free state we can apply the WLF transform.

The difficulty is for pre-stressed rubbers - the WLF transform doesn’twork!

Applications - the compromiseNatural rubber is pretty useless.It very quickly becomes a crumbling mess.Vulcanization is when sulphur is added to rubber. It forms extra bondsand cross links between the rubber molecules

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Rubber molecules

Rubber properties depend on the length of the cross links,Short links → very good heat resistanceLong links → better dynamic properties

Applications - the thermo bit is important!The challenger disaster of 1986,

Rockets stuffed with highly explosive propellant and each of thesections are joined together.

Each join is lined with an o-ring which should prevent leakage.

Bitter cold the night before launch, causing the rubber to become verybrittle and unable to expand properly, leading to catastrophic failure.

Submarines

Thales Underwater Systems use rubber composites (withmicrospherical inclusions) on the exterior of subs for acoustic cladding.

In particular the understanding of pre-strain is important.

However knowing how the material will behave when the sub dives,resurfaces, dives again, etc. is extremely difficult.

Thus the eventual aim is to have a detailed knowledge of the effectiveconstitutive response of the cladding material with dependence on

• phase properties and geometry• pre-strain/stress history• temperature• frequency

Elasticity: Several Problems of Interest Alone

Let us now simplify things and consider (nonlinear) elasticity alone.

• Linear homogenization (linear elastic waves or statics) and thedetermination of the effective tensor of elastic moduli

σ = C∗ : e

• Nonlinear homogenization (large deformation and/or nonlinearelastic behaviour) and the evaluation of an effective strain energyfunction (difficult)

W∗ = W∗(λ1, λ2, λ3)

• Nonlinear pre-stress (say Σ) with superimposed linear elasticwaves (incremental stress σ) and the determination of theeffective tensor of incremental elastic moduli

σ = A∗ : e

Incremental deformations - applications

Understanding incremental deformations given some initial nonlineardeformation in composites is common in many applications. E.g.

• Oil and geophysical industry• NDT in pre-stressed materials• Use of rubber composites in automotive, aerospace and defence

industries (often in pre-stressed states)• Biological tissues (lung, tendon, etc) are all nonlinear

(visco)elastic, pre-stressed in vivo and are natural “composites”

An Overview of Homogenization Theory

HomogenizationMaterials with microstructure

Effective Constitutive Laws

Fluid, etcViscoelasticElasticPhases:

Periodic Random

Statics Dynamics

Linear

Classical

Variationaltechniques.(Bounds)

Willis 1981

Directnumericalsimulations

Experimentaltesting andengineering

Nonlinear

Nonlocal

Rigorous proofand convergenceresults

Continuum Hypothesis

Allaire, 1992 Hashin 1962, 1963

Stochasticpartial

equationsdifferential

Formal

approximationsasymptotics and

Bakhvalov, 1990Sabina, Willis, 1988

Swan, 1994 Katz, 1970s

Homogenization

Geometry of inhomogeneities

Defines anisotropy

Beran, 1968

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ε � η � λ

One dimensional long thin Composite Bar

First - case of no pre-stress.

Longitudinal, low frequency wave propagation through the bar whereλ � a (separation of scales).

σ = E(x)e

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φa (1 − φ)aW.J.Parnell, School of Mathematics, University of Manchester. – p.21/57

Notion of separation of scales for statics

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Good e.g. of separation of scales and homogenization

Scale on a, divide by E0, time harmonic,

A(x)dw

+ ε2d(x)w = 0

where ε = ak0 � 1 and

A(x) =

α = E1/E0 x ∈ [n + φ]

1 x ∈ [n + φ, n + 1]

d(x) =

d = ρ1/ρ0 x ∈ [n + φ]

1 x ∈ [n + φ, n + 1]

Asymptotic homogenizationEquation with rapidly oscillating coefficients.Thus introduce two lengthscales,

x = ξ (ε + L2ε2 + ...)x = z

∂ξ+ (ε + L2ε

2 + ...)∂

and suppose

w = w0(z, ξ) + εw1(z, ξ) + ε2w2(z, ξ) + ...

where wr and derivatives are 1-periodic in ξ:

∂nwr

∂ξn(z, ξ) =

∂nwr

∂ξn(z, ξ + 1)

Hierarchy of problemsOne at each order in ε.O(1)It turns out that at leading order

w0(ξ, z) = w∗(z)

O(ε)Pose a (separable) solution of the form

wr1(ξ, z)

n= [(ξ − φ − n)Br + nC]

∂w∗

∂z(z)

where Br, C are constants (appropriately defined).

The homogenized equationO(ε2)

Integrate the governing equation over each phase and sum.Impose periodicity conditions and boundary conditions and we find:

+ ρ∗w∗ = 0

A∗ =α

(1 − φ)α + φ

ρ∗ = (1 − φ) + φd

A∗ is the effective Young’s modulus of the composite bar.ρ∗ is the effective density modulus of the composite bar.These are standard results.

Prestress - a little nonlinear elasticity

1D Strain energy function:

W (λ) = E∑

4m2Cm(λ2m − 1 − 2m log λ) +

2log2 λ,

Σ(λ) =dW

where λ is the principal stretch and∑

m Cm = 1.

The latter ensures in linear limit (λ = e + 1 ≈ 1):

2Ee2 + O(e3) Σ =

dλ= Ee

Homogeneous rubber

For homogeneous rubber for example we could have

C1 = 0.2 C2 = 0.8 Cm = 0 other m

0.5 1.5 2 2.5

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Imposed deformationAssume this is quasi-static (i.e. no viscoelastic deformation). This isbelieved to be a good approximation.

Use x as reference (current) configuration (with a periodic cell havingunit length) and X as the original (rest) configuration.To create a pre-stressed bar, take the nth cell and imposedisplacement BCs say

u(n) = −U u(n + 1) = U

u1(n + φ) = u0(n + φ)dW1

where φ is the volume fraction in the deformed composite.

This is satisfied by the homogeneous deformation

n= λrX + γn

r , r = 0, 1

with Eulerian displacement field

u(x)∣

n= x − X(x) =

(λr − 1)

λrx +

Stretches/stresses piecewise constant and thus eqm eqn satisfied:

where γnr is necessary for compatability (rigid body displacement).

Incremental displacements

Firstly, assume purely elastic

With deformed configuration as the reference, superimpose small (timeharmonic) incremental displacements v(x)eiωt,

displacement = u(x) + η(v(x)eiωt), η � |u|.

These induce incremental stresses, strain energy, etc.

Straightforward to show that these are governed by

A(x)dv

+ ω2ρ(x)v = 0

A(x) = λd2W

Thus for our problem, on nondimensionalizing

A(x)dv

+ ε2d(x)v = 0

A(x) =

α1 = λ1

, x ∈ [n, n + φ],

α0 = λ0

, x ∈ [n + φ, n + 1].

Similar form to original equation. Thus the effective Youngs modulusin the pre-stressed state is simply

A∗ =α1α0

(1 − φ)α1 + φα0

C1 = 1 (Mooney material), α0 = A01/A

02 = 10

-0.3 -0.2 -0.1 0.1 0.2 0.3

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A∗A∗

C1 = 0.2, C2 = 0.8 (Mooney-Rivlin), α0 = 10

-0.3 -0.2 -0.1 0.1 0.2

12PSfrag replacements

A∗A∗

Phase 1 Mooney, Phase 2 M-R, α0 = 10

-0.3 -0.2 -0.1 0.1 0.2

12PSfrag replacements

A∗A∗

Wave speed, Mooney, d0 = 5, α0 = 10

-0.3 -0.2 -0.1 0.1 0.2 0.3

1.8PSfrag replacements

c∗c∗

Wave speed, Mooney-Rivlin, d0 = 5, α0 = 10

-0.2 -0.1 0.1 0.2

1.4PSfrag replacements

c∗c∗

Stop and pass bands

Now consider waves of arbitrary frequency. Can write e.o.m

A(x)dw

+ ε2d(x)w = 0

dx2+ ε2β2

1zn = 0, n ≤ x ≤ n + φ,

dx2+ ε2β2

0wn = 0, n + φ ≤ x ≤ n + 1

β20 =

α0β2

How does the pre-stress affect the ability of waves to propagate?

Pose quasi-periodic ansatz

zn(x) = einε∗(Ceiεβ1(x−n) + De−iβ1ε(x−n))

wn(x) = einε∗(Aeiεβ0(x−n) + Be−iβ0ε(x−n))

Satisfaction of BCs requires

cos ε∗ =1

(1 + αβ)2 cos [ε(βφ + (1 − φ))]

− (1 − αβ)2 cos [ε(βφ − ε(1 − φ))])

α =α1

α0, β =

Results

Mooney material, C1 = 1, E1 = 10, E0 = 1, d = 10, φ0 = 0.4.

2 4 6 8 10 12

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<[ε∗]

U = −0.2

U = 0.2

Mooney material, C1 = 1, E1 = 10, E0 = 1, d = 10, φ0 = 0.4.

2 4 6 8 10 12

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=[ε∗]

U = −0.2

U = 0.2

Viscoelastic response

Consider first a homogeneous material.

Take the constitutive law of the form

Σ(x, t) =∂W

∂λ−

g(t − s)∂

where g(t − s) is the relaxation function.

Note that for small displacement gradients this gives

σ(x, t) = Ee(x, t) − E

g(t − s)∂e

∂s(x, s) ds

Also note the assumption above that the (finite) deformation does notaffect the relaxation function - this is thought to be reasonable forhomogeneous rubber.

Step stretchTake

λ(t) =

0 t < 0,

λ0, t ≥ 0,Σ(t) =

λ=λ0

(1 − g(t))

0.5 1 1.5 2 2.5 3 3.5 4

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Incremental equations

Apply the (finite) initial deformation quasi-statically and then superposesmall stretches

λ(t) = λ0 + δ(t), |δ| � λ0

As in the purely elastic case, linearize to find the incremental stress

σ(t) =∂2W

∂λ2

λ=λ0

δ(t) −

g(t − s)∂δ

∂s(s) ds

and take Laplace transforms which gives

σ(p) = A(p)δ(p)

where A(p) is the transform modulus (here relating stress to stretch inthe transformed domain).

Complex Incremental Modulus

Take oscillatory form δ(t) = δ(ω)eiωt and for simplicity

g(t) = g(1 − e−t/τ ).

After inverting LT and waiting for the transients to decay we find

σ(t) =∂2W

∂λ2

λ=λ0

1 −g

1 + iωτ

δ(ω)eiωt

= λ0∂2W

∂λ2

λ=λ0

1 −g

1 + iωτ

e(ω)eiωt

where we introduced the incremental strain e(ω) = δ(ω)/λ0 and thus

A(iω) = λ0∂2W

∂λ2

λ=λ0

1 −g

1 + iωτ

is the (complex) incremental modulus.

Viscoelastic effect at leading orderWe can write

A(iω) = λ0∂2W

∂λ2

λ=λ0

1 −g

1 + iετ

ε = qk =q

cω τ =

For homogenization, the non-dimensional frequency ε � 1.

Indeed, for rubber, c = O(10) − O(103)m/s, q = o(10−3)m so

may be large (i.e. 1/ε) even for small τ . Typically τ = O(10)s.

Viscoelastic behaviour therefore becomes a leading order (in ε) effect.

Effective Complex Incremental Modulus

Applying the same pre-strain as above for a two-phase composite, wetherefore conclude that

A∗(iε) =A1(iε)A0(iε)

(1 − φ)A1(iε) + φA0(iε)= A∗

1(ε) + iA∗2(ε)

What is the dependence of this on• volume fraction,• pre-strain,• frequency,• strain energy function.

For ease of exposition restrict to Mooney material, C1 = 1.

No pre-strain, E1 = E0, τ1 = 1000, τ0 = 10

0.00001 0.0001 0.001 0.01 0.1 1

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No pre-strain, E1 = E0, τ1 = 1000, τ0 = 10

0.00001 0.0001 0.001 0.01 0.1 1

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Pre-strain, φ0 = 0.2