Wave Propagation Through Pre-stressedInhomogeneous Nonlinear Media

GDR Workshop onWaves in Pre-stressed Nonlinear Media,

Losehill Hall, Castleton7th January 2009

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.1/42

Overview

• History, Context and Applications for pre-stressedinhomogeneous nonlinear media

• Longitudinal waves in a pre-stressed nonlinear, thin composite bar• Pre-stress of extension or compression• low frequency - effective incremental Young’s modulus• arbitrary frequency incremental waves - stop/pass bands

• Scattering from a cavity in a pre-stressed nonlinear host medium• Hydrostatic pre-stress• Low frequency SH wave scattering• Matched asymptotic expansions

W.J.Parnell, School of Mathematics, University of Manchester. – p.2/42

History, Context and Applications

W.J.Parnell, School of Mathematics, University of Manchester. – p.3/42

History and ContextWave scattering from a pre-strained region.

How does an incident longitudinal wave scatter from a spherical cavityin a pre-stressed medium?

W.J.Parnell, School of Mathematics, University of Manchester. – p.4/42

Required the use of the so-called Small-on-large theory.

Unsatisfactory outcome. Revisited recently.

W.J.Parnell, School of Mathematics, University of Manchester. – p.5/42

History and Context

If ever there was a book to put a student off solid mechanics on firstsight - this is it!

W.J.Parnell, School of Mathematics, University of Manchester. – p.6/42

The bigger picture

Effective dynamic behaviour of inhomogeneous rubbery compositeswhich have undergone pre-stress (small-on-large).

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

governed by W = W (λ1, λ2, λ3) and Σj = λj/JdW/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• Composite modelling

W.J.Parnell, School of Mathematics, University of Manchester. – p.7/42

Literature

Some knowledge of the statics problem(effective nonlinear behaviour of a rubbery composite)

Hashin (1970’s)Ogden (1970’s)Ponte-Castaneda and Willis (1990’s/2000’s)The “rubber community” (numerous contributors) (1990’s/2000’s)

For small-on-large (wave propagation after pre-stress):Great strides have been made for homogeneous mediaBut what about inhomogeneous media? It appears very littleindeed..........

W.J.Parnell, School of Mathematics, University of Manchester. – p.8/42

Applications• 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”

Specific e.g. Thales - acoustic cladding on submarines.

In particular the understanding of pre-stress is imperative (diving, etc.)

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 and frequency

Recent development - Quantum Tunnelling Composites - QTC’s

W.J.Parnell, School of Mathematics, University of Manchester. – p.9/42

Wave propagation along a pre-stressedcomposite bar

W.J.Parnell, School of Mathematics, University of Manchester. – p.10/42

Thin Composite Bar - no pre-stress

(Periodic) microscale of bar is O(a).Cross-section of bar is O(q).Choose η = q/a ≪ 1 so that dispersive effects are negligible.

q

q

aφa

(1 − φ)a

Linear (time harmonic) wave propagation governed by:

d

dx

(

E(x)du

dx

)

+ ρ(x)ω2u = 0

W.J.Parnell, School of Mathematics, University of Manchester. – p.11/42

Consider low frequency propagation, i.e. ǫ = ak ≪ 1.

Using method of asymptotic homogenization, introduce twolengthscales

ξ = x z = L(ǫ)x

and use the expansion

u(x) = u0(z) + ǫu1(ξ, z) + O(eps2)

It is simple to show that:

d

dz

(

E∗

du0

dz

)

+ ρ∗u0 = 0

whereE∗ =

E1E0

(1 − φ)E1 + φE0

is the effective Young’s modulus of the bar.

W.J.Parnell, School of Mathematics, University of Manchester. – p.12/42

Pre-stress

Suppose the bar is nonlinearly elastic (and compressible) - each phaserepresented by a strain energy function W .

Impose the longitudinal pre-stress

Σ11 = Σ Σ22 = Σ33 = 0

Thin bar - ignore “necking” effects close to interfaces of phases:

x1 = ΛrX1 + γnr x2 = LrX2 x3 = LrX3

for r = 0, 1. (γnr reqd for continuity of disp).

Principal stretches determined by solving:

Σ =λ1

J

∂Wr

∂λ1

∣

∣

∣

λ1=Λr,λ2=Lr,λ3=Lr

0 =λ2

J

∂Wr

∂λ2

∣

∣

∣

λ1=Λr,λ2=Lr,λ3=Lr

where J = λ1λ2λ3.W.J.Parnell, School of Mathematics, University of Manchester. – p.13/42

Incremental DeformationSuppose now that an incremental longitudinal wave propagatesthrough the deformed structure such that

σ11 = σ σ22 = σ33 = 0

Form of these incremental stresses in terms of the incrementaldisplacements are (Ogden 1997)

σ = σ11 = A11∂w1

∂x1+ A12

∂w2

∂x2+ A13

∂w3

∂x3

0 = σ22 = A21∂w1

∂x1+ A22

∂w2

∂x2+ A23

∂w3

∂x3

0 = σ33 = A31∂w1

∂x1+ A32

∂w2

∂x2+ A33

∂w3

∂x3

where Aij are piecewise constant and are defined by

Aij(x) =λiλj

J

∂2W

∂λi∂λj, (no sum over i, j).

W.J.Parnell, School of Mathematics, University of Manchester. – p.14/42

Therefore we can write

σ =(A2

13A22 + A212A33 + A2

23A11 − 2A12A13A23 −A11A22A33)

(A223 −A22A33)

∂w1

∂x1

= A(x1)∂w1

∂x1

which defines the incremental Young’s modulus A1(x1).

No pre-stress gives,A11 = A22 = A33 = λ + 2µ

A12 = A23 = A13 = λ

so thatA = E = µ

(3λ + 2µ)

(λ + µ).

The governing equation for incremental longitudinal waves is therefore

∂

∂x1

(

A(x)∂w1

∂x1

)

+ ρ(x)ω2w1,

Note the form of this equation.W.J.Parnell, School of Mathematics, University of Manchester. – p.15/42

Effective incremental waves - Low frequency

Assume now that the incremental waves satisfy ak0, ak1 ≪ 1.

Since the form of the governing equation is identical to that in theunstressed case we can use asymptotic homogenization in the sameway to find that the effective incremental equation:

d

dz1

(

A∗

dw0

dz1

)

+ ρ∗w0 = 0

whereA∗ =

A0A1

(1 − φ)A1 + φA0

Some strain energy functions of use: Compressible Neo-Hookean:

µ

2(I1 − 3 − log[I3]) +

µν

(1 − 2ν)(I

1/23 − 1)2

W.J.Parnell, School of Mathematics, University of Manchester. – p.16/42

Results - Rubber/Steel BarRubber, µ = 105, ν = 0.49, Neo-HookeanSteel, µ = 109, ν = 0.25, linearly elastic

-0.003 -0.002 -0.001 0.000 0.001 0.002 0.003

0.001

0.01

0.1

1

10

100

W.J.Parnell, School of Mathematics, University of Manchester. – p.17/42

Stop and pass bands

Now consider waves of arbitrary frequency.On scaling x1 on a, can write e.o.m as

d

dx1

(

A(x1)dw1

dx1

)

+ ǫ2d(x1)w1 = 0

and thus in the form

d2w1n1

dx2+ ǫ2β2

1w1n1 = 0, n ≤ x ≤ n + φ,

d2w0n1

dx2+ ǫ2β2

0w0n1 = 0, n + φ ≤ x ≤ n + 1

where

β20 =

d0

A0β2

1 =d1

A1

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

W.J.Parnell, School of Mathematics, University of Manchester. – p.18/42

Pose quasi-periodic ansatz

zn(x) = einǫ∗(Ceiǫβ1(x−n) + De−iβ1ǫ(x−n))

wn(x) = einǫ∗(Aeiǫβ0(x−n) + Be−iβ0ǫ(x−n))

Satisfaction of BCs requires

cos ǫ∗ =1

4αβ

(

(1 + αβ)2 cos [ǫ(βφ + (1 − φ))]

− (1 − αβ)2 cos [ǫ(βφ − (1 − φ))])

where

α =A1

A0, β =

β1

β0.

W.J.Parnell, School of Mathematics, University of Manchester. – p.19/42

Results

Compressible Neo-Hookean phases with Σ scaled on µ0,µ1 = 2µ0, ν0 = 0.1, ν1 = 0.2

2 4 6 8

0.2

0.4

0.6

0.8

1

1.2

1.4

Green = no pre-stressBlue = pre-stress with Σ = 4µ0

Purple = pre-stress with Σ = −4µ0

W.J.Parnell, School of Mathematics, University of Manchester. – p.20/42

Results, Rubber/steel composite barRubber, µ = 105, ν = 0.49, Neo-HookeanSteel, µ = 109, ν = 0.25, linearly elastic

0.05 0.1

1

2

3

4

Green = no pre-stressBlue = pre-stress with Σ = µrubber

Purple = pre-stress with Σ = −µrubber

W.J.Parnell, School of Mathematics, University of Manchester. – p.21/42

Thus the pre-stress can act as a tuning device in order to allow wavesto propagate or not depending on frequency.

Parnell, 2007, IMA J. Appl. Math.

Note that a similar effect (i.e. tuning of a material by nonlinearpre-stress) was found in the following paper:

Bigoni, Mei, Movchan, 2008, JMPS

See Davide’s talk after this!

W.J.Parnell, School of Mathematics, University of Manchester. – p.22/42

Scattering from a cylindrical void inpre-stressed rubber

W.J.Parnell, School of Mathematics, University of Manchester. – p.23/42

Problem description

Wijewickerama and Leungvicharoen 2007: "Scattering from a hole in apre-stressed medium"However, this paper relates only to homogeneous stretches around thehole.

W.J.Parnell, School of Mathematics, University of Manchester. – p.24/42

No pre-stressNon-dimensionalize lengthscales on reciprocal wavenumber 1/Γ:

(∇2 + 1)w = 0,∂w

∂R= 0 on R = ǫ0 = AΓ

Incident plane wave:

winc = eiX1 = J0(R) + 2∞∑

n=1

inJn(R) cosnΘ

Separation of variables gives

ws = A0H0(R) + 2∞∑

n=1

inAnHn(R) cosnΘ with An = −J ′

n(ǫ0)

H ′

n(ǫ0)

and for ǫ0 = AΓ ≪ 1,

A0 =π

4iǫ20 + o(ǫ20), A1 = −

π

4iǫ20 + o(ǫ20)

W.J.Parnell, School of Mathematics, University of Manchester. – p.25/42

Incompressibility

Assume host medium is incompressible.

Impose a hydrostatic pressure Σrr = −p∞ at infinityand a fixed stretch L axially:

R = rQ(r) Θ = θ Z =z

L

Incompressibility gives

Q

(

Q + rdQ

dr

)

= L

so that Q2 =L(r2 + K)

r2, K =

A2

L− a2.

W.J.Parnell, School of Mathematics, University of Manchester. – p.26/42

EquilibriumSymmetry: dΣrr

dr+

1

r(Σrr − Σθθ) = 0

which on writing down stresses in terms of (derivatives of) the strainenergy function W (I1, I2) and the Lagrangian multiplier p associatedwith incompressibility, can be integrated to give

Σrr(r) + pinner = −2

∫ r

a

[(

Q2(r′)

L2−

1

Q2(r′)

)

W1(r′)

+

(

Q2(r′) −L2

Q2(r′)

)

W2(r′)

]

dr′

r′,

where W1 = ∂W/∂I1, W2 = ∂W/∂I2.

Mooney-Rivlin medium:

W =µ

2(S1(I1 − 3) + S2(I2 − 3)), S1, S2 ∈ R, S1 + S2 = 1.

W.J.Parnell, School of Mathematics, University of Manchester. – p.27/42

Deformed radius

Together with Σrr → −p∞ as r → ∞ this gives

p∞µ

= −1

2

(

S1

L+ LS2

)(

1 −A2

La2− log

(

A2

La2

))

,

so that a can be determined in terms of −p∞/µ, L, A, S1, S2 = 1 − S1.

-3 -2 -1 1 2 3

2

4

6

8

a

p∞/µ

L = 0.7

L = 1

L = 1.5

W.J.Parnell, School of Mathematics, University of Manchester. – p.28/42

Radial stress,L = 1

2 4 6 8 10

-5

-4

-3

-2

-1

rΣrr

p∞µ

= 1

p∞µ

= 2

p∞µ

= 4

W.J.Parnell, School of Mathematics, University of Manchester. – p.29/42

Incremental deformation

Consider superimposing small amplitude waves on the finitedeformation above:

u = u + ηu′

where u is the displacement associated with the finite deformation,η ≪ 1 and

u′ = (0, 0, w(r, θ)) exp(iωt).

For general W , using standard small-on-large theory we find that

∂

∂r

[

2(

Q2W1 + W2

) ∂w

∂r

]

+2

r

(

Q2W1 + W2

) ∂w

∂r

+2

r2

(

L2

Q2W1 + W2

)

∂2w

∂θ2+ L2ρω2w = 0

W.J.Parnell, School of Mathematics, University of Manchester. – p.30/42

Mooney-Rivlin

(

1 +k

r2

)

∂2w

∂r2+

1

r

(

1 −k

r2

)

∂w

∂r+

1

r2

(

1 −k

r2 + K

)

∂2w

∂θ2+ γ2w = 0,

where k = KLS1/S, K = A2/L − a2 and S = 1 + (L − 1)S1.

Modified wavenumber at infinity is γ2 = L2Γ2/S, where Γ2 = ρω2/µ.

Aside: ∂

∂r

(

µr(r)∂w

∂r

)

+µr(r)

r

∂w

∂r+

µθ(r)

r2

∂2w

∂θ2+ γ2w = 0

where we have defined the anisotropic shear moduli (scaled on µS) as

µr(r) = 1 +k

r2, µθ(r) = 1 −

k

r2 + K.

Derived from: ∂σrz

∂r+

1

r

∂σθz

∂θ+

1

rσrz + γ2w = 0

σrz = 2µr(r)erz , σθz = 2µθ(r)eθz .

W.J.Parnell, School of Mathematics, University of Manchester. – p.31/42

Low Frequency Scattering

Once again assume a plane incident wave incoming from infinity:

winc = eiγr cos θ = J0(γr) + 2∞∑

n=1

inJn(γr) cosnθ

so that the scattered wave is described by:

∂

∂r

[(

1 +k

r2

)

∂ws

∂r

]

+1

r

(

1 +k

r2

)

∂ws

∂r+

1

r2

(

1 −k

r2 + K

)

∂2ws

∂θ2+γ2ws

+k

r2

(

2γ

rJ1(γr) − γ2J0(γr)

)

+2k

r2

∞∑

n=1

in cosnθ

[

(

2γ

rJn+1(γr) +

(

n(n − 2)

r2− γ2

)

Jn(γr)

)

+n2

r2 + KJn(γr)

]

= 0.

W.J.Parnell, School of Mathematics, University of Manchester. – p.32/42

Matched Asymptotics (Focus on Monopole)By scaling this governing equation on either γ or a we obtain the outerand inner problems respectively. These are then linked using VanDyke’s principle of asymptotic matching.

Outer problem

Introducing ro = γr, we seek (separable) outer solutions of the form

wouts = F0(ro) + 2

∞∑

n=1

inFn(ro) cosnθ

where

F0(ro) = ǫ2F(2)0 (ro) + o(ǫ2),

This leading order term satisfies (k = k/a2):

d2F(2)0

dr2o

+1

ro

dF(2)0

dro+ F

(2)0 +

k

r2o

(

2

roJ1(ro) − J0(ro)

)

= 0

W.J.Parnell, School of Mathematics, University of Manchester. – p.33/42

Outer solution is:

F(2)0 (ro) = B0H

(1)0 (ro) + E0(ro)

where B0 is a constant and E0(r0) is the inhomogeneous solution:

W.J.Parnell, School of Mathematics, University of Manchester. – p.34/42

Outer solution is:

F(2)0 (ro) = B0H

(1)0 (ro) + E0(ro)

where B0 is a constant and E0(r0) is the inhomogeneous solution:

E0(ro) =kπ

32

[

4J0(ro)(

2J2(ro)Y0(ro) − roJ3(ro)Y0(ro)

+ roJ2(ro)Y1(ro) −2

π

)

− r2oY0(ro)

(

4 3F4

(

{1, 1, 3/2}; {2, 2, 2, 2};−r2o

)

− 3 3F4

(

{1, 1, 5/2}; {2, 2, 3, 3};−r2o

)

)]

.

W.J.Parnell, School of Mathematics, University of Manchester. – p.34/42

Outer solution is:

F(2)0 (ro) = B0H

(1)0 (ro) + E0(ro)

where B0 is a constant and E0(r0) is the inhomogeneous solution:

E0(ro) =kπ

32

[

4J0(ro)(

2J2(ro)Y0(ro) − roJ3(ro)Y0(ro)

+ roJ2(ro)Y1(ro) −2

π

)

− r2oY0(ro)

(

4 3F4

(

{1, 1, 3/2}; {2, 2, 2, 2};−r2o

)

− 3 3F4

(

{1, 1, 5/2}; {2, 2, 3, 3};−r2o

)

)]

.

E0(r0) = −k

4+ O(r2

0) r0 → 0

E0(r0) = O

(

1

r3/20

)

r0 → ∞

W.J.Parnell, School of Mathematics, University of Manchester. – p.34/42

Inner Problem

Full inner solution can also be written in separable form (ri = r/a):

wins (ri) = f0(ri) + 2

∞∑

n=1

infn(ri) cosnθ

and the monopole term f0 is written

f0(ri) = ǫ2f(2)0 (ri) + o(ǫ2)

and satisfies(

1 +k

r2i

)

d2f(2)0

dr2i

+1

ri

(

1 −k

r2i

)

df(2)0

dri= 0

which has explicit solution (stress free on ri = ǫ)

f(2)0 (ri) =

1

4(1 + k) log(r2

i + k) + C0 log ǫ + D0

W.J.Parnell, School of Mathematics, University of Manchester. – p.35/42

Matching

Outer solution:F0(ro) = ǫ2

(

B0H(1)0 (ro) + E0(ro)

)

+ o(r2o)

Inner solution:

f0(ri) = ǫ2(1

4(1 + k) log(r2

i + k) + C0 log ǫ + D0

)

+ o(ǫ2)

Van Dyke’s matching principle says that f(1,1)0 = F

(1,1)0 which gives:

B0 = ǫ2B0 = ǫ2π

4i(1 + k)

Modified Monopole Scattering Coefficient due to the pre-stress:

B0 =πǫ204i

[

a2L

A2(1 + (L − 1)S1)

(

1 +(A2 − a2)LS1

a2(1 + (L − 1)S1)

)]

Neo-Hookean:B0 =

πǫ204i

W.J.Parnell, School of Mathematics, University of Manchester. – p.36/42

Monopole Scattering CoefficientB0

L = 1:

-3 -2 -1 1 2 3

1.0

1.5

2.0

p∞/µ

B0/Γ2

S1 = 1

S1 = 0.9

S2 = 0.8

L = 1.5:

-3 -2 -1 1 2 3

1.0

1.5

2.0

p∞/µ

B0/Γ2

S1 = 1

S1 = 0.9

S2 = 0.8

W.J.Parnell, School of Mathematics, University of Manchester. – p.37/42

Composite Solution

F0(γr) = ǫ2

(

π(1 + k)

4iH

(1)0 (γr) + E0(γr) +

(1 + k)

4log

(

1 +ǫ2k

(γr)2

))

+ o(ǫ2)

With ǫ = 0.2:

0.5 1.0 1.5 2.0 2.5

-2.0

-1.5

-1.0

-0.5

0.5

r

f0(γr)

F0(γr)

F0(γr)

W.J.Parnell, School of Mathematics, University of Manchester. – p.38/42

Dipole Solution

This is much harder! Asymptotic solution for small S2 = 1 − S1 gives

B1 = −πǫ2

4i

(

A2

La2+

(1 − S1)

L

((

A2

La2− 1

)

+ 2E1

))

+ O(S22)

Again, for Neo-Hookean, S1 = 1:

B1 = −πǫ204i

W.J.Parnell, School of Mathematics, University of Manchester. – p.39/42

Dipole Scattering Coefficient:B1:L = 1:

-3 -2 -1 0 1 2 3

0.5

1.0

1.5

2.0

p∞/µ

|B1|/Γ2

S1 = 1

S1 = 0.9

S1 = 0.8

L = 1.5:

-3 -2 -1 0 1 2 3

0.5

1.0

1.5

2.0

p∞/µ

|B1|/Γ2

S1 = 1

S1 = 0.9

S1 = 0.8

W.J.Parnell, School of Mathematics, University of Manchester. – p.40/42

Summary

• Pre-stress of inhomogeneous media can act as a useful tuningmechanism

• Scattering where imposed pre-stress is homogeneous merelychanges anisotropy of host

• Scattering problems involving inhomogeneous pre-stress becomev. complicated!

• Interesting effects result:• Modified wave equation is equivalent to an anisotropic

material with r dependent moduli• Inhomogeneous terms in wave equation scatter incident wave

as soon as it is not at “infinity”• Matched asymptotics allows us to determine the low

frequency scattering effects analytically• When constitutive behaviour is Neo-Hookean, scattering

coefficients B0 and B1 remain unchanged

• Scattering interaction between 2 or N inclusions.......in progress!W.J.Parnell, School of Mathematics, University of Manchester. – p.41/42

Thankyou

• To EPSRC and Thales Underwater Systems for funding

• To David Abrahams (Manchester) andDavid Allwright, Jon Chapman and John Ockendon (Oxford)for v. helpful discussion

W.J.Parnell, School of Mathematics, University of Manchester. – p.42/42