“A brief history of homogenization and micromechanics, with applications” School of Mathematics...

“A brief history of homogenization andmicromechanics, with applications”

School of MathematicsUniversité Paris 12, Mon 26th April 2010

William J. Parnell

Lecturer in Applied Mathematics

School of Mathematics, University of Manchester, UK

William.Parnell@manchester.ac.uk

http://www.maths.man.ac.uk/∼wparnell.

Work with David Abrahams (Manchester), Salah Naili and Mai-Ba Vu (Paris 12)

Quentin Grimal (Paris 6)

W.J.Parnell, School of Mathematics, University of Manchester. – p.1/50

Overview

• What is homogenization?• A little history• Some results as examples• Problems using homogenization (low-frequency limit of dynamic

problems):• Waves in bone• Waves in pre-stressed composites

• Summary and the “future”

What is homogenization?

Continuum Mechanics

��

Assign material properties, e.g.Young’s modulus for linear elasticity: σ = Ee,Viscosity for Newtonian fluids: σ = µe.

With conservation eqns, these gives pdes for deformation of body.

Continuum assumption works IF λ ≫ a, where,λ is characteristic lengthscale of applied loading,a is molecular or atomistic lengthscale.

Inhomogeneous media

For simple (homogeneous) media (e.g. water, steel, rubber, etc)these material properties are “constants”.

But what about inhomogeneous media when σ = E(x)e or σ = µ(x)e?

Using these in equations of motion give complicated pdes to solve...

E.g.s: Industrial Composites and biological media

Is it necessary to solve these complicated pdes?

Do we really need this level of detail?

Consider the following extension of the continuum assumption:If λ ≫ a where nowλ is characteristic lengthscale of applied loading,a is the microscale of the inhomogeneous material,

then we can HOMOGENIZE the material.

I.e.σ = E(x)e → σ = E∗e

and thus E∗ is seen as an effective Young’s modulus.

Extension to more general linear elasticity contexts:σij = Cijkl(x)ekl → σij = C∗

ijklekl,

or, to avoid clutter, omit the indices:σ = C(x)e → σ = C∗e

A little history and some results

Pre-1960s

Consider now a two-phase composite, of the host/inclusion type.Two materials perfectly bonded together

Phase 0: Host phase C = C0 so that σ = C0e

phase 1: Inclusion phase, C = C1 so that σ = C1e.

��

Early 1960s, only results:simple boundsdilute results when interactions between inclusions is negligible

We will now derive an important result in homogenization theory.

Effective properties in terms of strain concentrationStress in the composite can be written as:

σ(x) = χ0(x)σ(x) + χ1(x)σ(x),

= χ0(x)C0e(x) + χ1(x)C1e(x)

Integrate over the whole body V = V0 ∪ V1.

σ dV =C0

e dV0 +C1

and since V0 = V \V1,

σ =C0

e dV −∫

= C0e + (C1 − C0)|V1||V |

= C0e + (C1 − C0)φe1 [where φ = |V1|/|V |]= [C0 + φ(C1 − C0)A] e. [where e1 = Ae].

Concentration tensor

Consider a single-inclusion problem:

��

Given far-field displacement u = ex, (so far-field strain is e),we find e1(x) in the form (linear problem) e1 = Ae.

So, if we use A = A, then we get:

Cd∗ = C0 + φ(C1 − C0)A.

where the superscript d indicates dilute (well-separated inclusions).

Eshelby’s seminar paper of 1957Single, isolated ellipsoidal inclusion in an isotropic host phase.

Impose far field strain e, (u = ex).

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Given this, Eshelby showed that:the strain e1 inside the inclusion is uniform,

i.e. each component of e1 is constant.

Hence A is uniform for isolated ellipsoids.

Eshelby also hypothesized that:• this was also the case for anisotropic host phases (proved mid-60s)• this result holds only for ellipsoids (only proved in 2008)

Simple bounds

Note again:σ = [C0 + φ(C1 − C0)A]e, e1 = Ae

If we take A = I (Identity tensor), i.e. uniform strain, then

CV∗ = C0 + φ(C1 − C0) = (1 − φ)C0 + φC1

Similarly, can show that the assumption of uniform stress gives

CR∗ = [(1 − φ)C−1

0 + φC−11 ]−1

In 1951 Hill showed that these were strict bounds on effectiveproperties (using variational principles):

CR∗ ≤ C∗ ≤ CV

An example - spherical particles inside a host phaseTake Cr = κrI1 + µrI2, r = 0, 1and C∗ = κ∗I1 + µ∗I2, and so we have

(1 − φ)

≤ κ∗ ≤ (1 − φ)κ0 + φκ1,

(1 − φ)

,≤ µ∗ ≤ (1 − φ)µ0 + φµ1.

Tungsten-Carbide/Cobalt alloy:

Properties (1010N/m2)µ1 = 28.8,κ1 = 42,µ0 = 8,κ0 = 17.2,

0.0 0.2 0.4 0.6 0.8 1.015

κV∗

κR∗

A dilute dispersion of spheresFor an isolated sphere, it is (relatively) straightforward to show that

3κ0 + 4µ0

3κ1 + 4µ0

5µ0(3κ0 + 4µ0)

3κ0(3µ0 + 2µ1) + 4µ0(2µ0 + 3µ1)

κd∗ = κ0 + φ(κ1 − κ0)

3κ0 + 4µ0

3κ1 + 4µ0

µd∗ = µ0 + φ(µ1 − µ0)

5µ0(3κ0 + 4µ0)

3κ0(3µ0 + 2µ1) + 4µ0(2µ0 + 3µ1)

So pre-1962:

0.0 0.2 0.4 0.6 0.8 1.015

κV∗

κR∗

κd∗

1960s - an important decadeThis was a very important time for research in this area:

• Hill (Cambridge),• Hashin, Shtrikman, Dow, Rosen, Tsai and Halpin (NASA),• Budiansky (Harvard)

Important breakthroughs due to increased use of fibre reinforcedcomposites and need for knowledge of effective properties of suchmaterials:

Hashin-Shtrikman bounds

Hashin and Shtrikman developed bounds based on the following form:

σ(x) = Cme(x) + τ(x)

Cm is a reference materialτ(x) is known as the stress polarization tensor.

They developed a variational principle in terms of τ(x).

Using information about the macroscopic anisotropy of the compositeand taking τ(x) to be piecewise constant, they found the nowmuch-used Hashin-Shtrikman bounds.

Much work was done after this by Walpole (1966, 1969) and Willis(1970s, 1980s) in order to clarify and extend these early results.

Hashin-Shtrikman bounds

0.0 0.2 0.4 0.6 0.8 1.015

κV∗

κR∗

κd∗

Approximate schemes

1965 - Hill, Budiansky, developed self-consistent schemes to accountfor interaction between inhomogeneities (albeit approximately).

A is approximated in the following manner:

1) Embed the inclusion in the effective medium.2) Solve the single-inclusion problem with u = ex as before.3) This gives implicit equations for κ∗ and µ∗.

��

κSC∗ = κ0 + φ(κ1 − κ0)

3κSC∗ + 4µSC

3κ1 + 4µSC∗

µSC∗ = µ0 + .......

Self-consistent scheme

0.0 0.2 0.4 0.6 0.8 1.015

κSC∗

κV∗

κR∗

κd∗

1970s - Classical Asymptotic homogenizationDeveloped for composites whose microstructure is periodic.(Sanchez-Palencia, Bensoussan et al, Bakhvalov and Panasenko,......)

(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.

(1 − φ)a

Linear (time harmonic) wave propagation governed by:

E(x)du

+ ρ(x)ω2u = 0

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

Using method of asymptotic homogenization, introduce twolengthscales

ξ = x z = ǫx

and use the expansion

u(x) = u0(z) + ǫu1(ξ, z) + O(ǫ2)

It is simple to show that:

+ ρ∗u0 = 0

whereE∗ =

(1 − φ)E1 + φE0

is the effective Young’s modulus of the bar.

1980s - present day

• Huge growth in the subject area.• Better bounds using microstructure information and numerous

advances in approximate static and dynamic schemes.• development of design of materials, metamaterials, etc.• Applications in poroelasticity, piezoelectric materials,

reaction-diffusion problems, etc.• Ideas have touched almost every area of applied maths.• Multi-scale problems now a huge area of research.• Discuss modern ideas (the future) later.

Applications

• Effective elastic moduli of Cortical bone• Multiple scattering in pre-stressed (random) composites

Effective Elastic Moduli of Cortical Bone

Introduction

CancellousCancellous bonebone

CorticalCortical bonebone

CancellousCancellous boneboneCancellousCancellous bonebone

CorticalCortical bonebone

Cortical Bone - Osteonal Microstructure

100100 µµmm100100 µµmm

Clearly there are several levels ofmicrostructure.You will see any number of these scalesdepending upon your level of observation

Homogenization process

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Idealize

Homogenize

Can we replace

σ = C(z, ξ)e

σ = C∗(z)e?

Previous work: Crolet et al. (1993, 96), Hellmich (2002-06).

MAH and generalized Hooke’s law

Use the method of asymptotic homogenization (MAH) (bar problemabove).

Pores arranged on a hexagonal lattice.We can derive the effective elastic properties of such a material.

Difficulty comes in solving the so-called cell problem: here we usecomplex analysis.

Generalized Hooke’s law: σij = C∗ijklekl

σ11 = c∗11e11 + c∗12e22 + c∗13e33,

σ33 = c∗13e11 + c∗13e22 + c∗33e33,

σ12 = c∗66e12

Hexagonal lattice: transverse isotropy - 5 elastic properties.

Constituent properties

Consider results for the two-phase, hexagonal lattice case (Grimal(2010)).

Cmatrix1212 = 9.25, Cmatrix

1122 = 11,

Cmatrix1133 = 11.9, Cmatrix

3333 = 38.1,

Hard (isotropic) pore:

Epore = 0.13 νpore = 0.49

Soft pore:

Epore → 0 νpore → 0

Hashin-Rosen bounds onµ∗12 = c∗66

µ∗12

porosity

FEMAsymptotic schemeHashin Rosen +Hashin Rosen -

0.05 0.1 0.15 0.2

Anisotropy parameter

c∗33/c∗11

porosity

Asymptotic (hard)

Asymptotic (soft)

FEM (hard)

FEM (soft)

0.05 0.1 0.15 0.2

Multiple scattering (and homogenization)in a (random) pre-stressed medium

BackgroundEffective (incremental) dynamic behaviour of pre-stressed,inhomogeneous rubbery composites (small-on-large).

Difficult to characterize: large def., constitutive behaviour,...

BackgroundEffective (incremental) dynamic behaviour of pre-stressed,inhomogeneous rubbery composites (small-on-large).

Difficult to characterize: large def., constitutive behaviour,...

Literature: Ogden, 1984, Destrade and Saccomandi, 2007 CISMAlmost exclusively: homogeneous initial deformations.

What about inhomogeneous deformations? Recently:

Parnell, W.J. 2007, IMA JAMBigoni D., Gei, M. and Movchan, A.B. 2008, JMPSBertoldi, K. and Boyce, M. 2008, Phys Rev E

Influence of pre-stress on the pass-band structure in periodic solids.W.J.Parnell, School of Mathematics, University of Manchester. – p.33/50

Applications

• Oil and geophysical industry• NDT for 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”

Pre-stress and inhomogeneousdeformation

Problem description

(b)(a)

Incompressibility and Constitutive behaviour

Assume host medium is incompressible.

Impose a hydrostatic pressure Σrr = −p∞ at infinity (far-field)and a fixed stretch L axially:

R = R(r) Θ = θ Z =z

Constitutive behaviour:

Take a neo-Hookean medium:

WSE =µ

2(I1 − 3),

Stresses Σrr, Σθθ come from derivatives of this potential.

Radial stress,L = 1

Only 1 equilbrium equation:

r(Σrr − Σθθ) = 0

which can be integrated to give Σrr:

2 4 6 8 10

p∞µ

Deformed radius

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

p∞µ

La2− 1 + log

so that a/A can be determined in terms of p∞/µ and L.

-3 -2 -1 1 2 3

p∞/µ

L = 0.7

L = 1.5

Incremental deformationSuperpose small amplitude waves on the finite deformation:

u = u + ηu′

where u is the finite displacement, η ≪ 1 and

u′ = (0, 0, w(r, θ)) exp(iωt).

u = u + ηu′

u′ = (0, 0, w(r, θ)) exp(iωt).

Transpires that the modified wave equation is:

r2 + M

∂θ2+ k2w = 0.

where M = A2/L − a2, k2 = LK2.

u = u + ηu′

u′ = (0, 0, w(r, θ)) exp(iωt).

Transpires that the modified wave equation is:

r2 + M

∂θ2+ k2w = 0.

where M = A2/L − a2, k2 = LK2.

Can think of this as:

(∇2 + LM + k2)w = 0

where LM = 0 for no pre-stress.W.J.Parnell, School of Mathematics, University of Manchester. – p.40/50

Line source in the pre-stressed medium

Line source of strength c, at (r0, θ0) (c = C/L at (R0, Θ0)):

L(∇2 + LM + ρω2)w =

Lr0δ(r − r0)δ(θ − θ0).

L(∇2 + LM + ρω2)w =

Lr0δ(r − r0)δ(θ − θ0).

Introducing the mapping R2 = L(r2 + M), Θ = θ, we get

∇2W + K2W =C

R0δ(R − R0)δ(Θ − Θ0)

L(∇2 + LM + ρω2)w =

Lr0δ(r − r0)δ(θ − θ0).

∇2W + K2W =C

R0δ(R − R0)δ(Θ − Θ0)

So that

W (R, Θ) = Wi +∞∑

n=−∞

inanHn(KR)ein(Θ−Θ0)

L(∇2 + LM + ρω2)w =

Lr0δ(r − r0)δ(θ − θ0).

∇2W + K2W =C

R0δ(R − R0)δ(Θ − Θ0)

So that

W (R, Θ) = Wi +∞∑

n=−∞

inanHn(KR)ein(Θ−Θ0)

and therefore

w(r, θ) = wi +∞∑

n=−∞

inanHn

r2 + M)

ein(θ−θ0)

Multiple scattering

Multiple scatteringN well separated voids within a pre-stressed medium.What is the effect of the pre-stress on the effective wavenumber?

In the deformed configuration the solution is therefore

w = wi +

ajnZnHn

r2j + M

einθj

where (rj , θj) is the local coordinate system of the jth void.W.J.Parnell, School of Mathematics, University of Manchester. – p.43/50

Solution in the vicinity of the sth void is

w(rs, θs) = wi +∞∑

n=−∞

asnZnHn(k

r2s + M)einθs

j=1,j 6=s

∞∑

n=−∞

ajnZnHn(k

r2j + M)einθj

Solution in the vicinity of the sth void is

w(rs, θs) = wi +∞∑

n=−∞

asnZnHn(k

r2s + M)einθs

j=1,j 6=s

∞∑

n=−∞

ajnZnHn(k

r2j + M)einθj

Voids are well-separated, so

w(rs, θs) ≈ wi +

∞∑

n=−∞

asnZnHn(k

r2s + M)einθs

j=1,j 6=s

∞∑

n=−∞

ajnZnHn(krj)e

inθj .

Graf’s addition theorem implies

w(rs, θs) ≈ wi +∞∑

n=−∞

asnZnHn(k

r2s + M)einθs

j=1,j 6=s

∞∑

n=−∞

∞∑

m=−∞

Hn−m(kRjs)ei(n−m)θjsJm(krs)e

where Rjs is the distance between the jth and sth void and θjs is theangle between the jth and sth void.

Graf’s addition theorem implies

w(rs, θs) ≈ wi +∞∑

n=−∞

asnZnHn(k

r2s + M)einθs

j=1,j 6=s

∞∑

n=−∞

∞∑

m=−∞

Hn−m(kRjs)ei(n−m)θjsJm(krs)e

where Rjs is the distance between the jth and sth void and θjs is theangle between the jth and sth void.

Zero traction on rs = a gives

j=1,j 6=s

∞∑

n=−∞

ajnZnHn−m(kRjs)e

i(n−m)θjs = Fi

Zn = PJ ′

H′n(Pka)

Effective wavenumber

Take ensemble averages, use QCA and hole-correctionpair-correlation function, and assuming

〈asn〉 = Cneik∗xs

in the low frequency limit we obtain for the effective wavenumber k∗

k2∗/K

2 =ρ∗ρ0

= L(1 − LP 2φ)

1 + LP 3φ

1 − LP 3φ

Effective wavenumber

Take ensemble averages, use QCA and hole-correctionpair-correlation function, and assuming

〈asn〉 = Cneik∗xs

in the low frequency limit we obtain for the effective wavenumber k∗

k2∗/K

2 =ρ∗ρ0

= L(1 − LP 2φ)

1 + LP 3φ

1 − LP 3φ

Given that ρ∗ = (1 − φ)ρ0, this gives

(1 − φ)(1 − LP 3φ)

L(1 − LP 2φ)(1 + LP 3φ),

which for no pre-stress (P = L = 1, φ = φ0) is

1 − φ0

1 + φ0.

Effective Shear Modulus

-2 -1 1 2

0.8 φ0 = 0.01

φ0 = 0.1

φ0 = 0.2

µ∗/µ0

Effective Wavespeed

-2 -1 1 2

φ0 = 0.01

φ0 = 0.1

φ0 = 0.2

c∗/c0

Summary

• Homogenization - a classical subject but with much still to offer• History - Eshelby is a classical paper!• Applications:

• Waves in bone• Pre-stressed random composites• Flexural waves in thin plates (sea-ice?)• Design of “tunable” wave-filters

• Current “hot topics”• metamaterials - unnatural material properties.• cloaking materials• nanomaterials• fractal media - e.g. the gecko’s feet• soft tissue modelling

Related Publications

• Parnell, W.J. and Abrahams, I.D. 2011An introduction to homogenization in Continuum Mechanics,Cambridge University Press

• Parnell, W.J., Vu, M.-B., Naili, S. and Grimal, Q. 2010A comparison of analytical and experimental methods to determine theeffective mesoscopic elastic properties of cortical bone, In preparationfor submission to Mech. Mat. 2010.

• Parnell, W.J. and Abrahams, I.D.Scattering from a cylindrical void in a pre-stressed neo-Hookeanmaterial, Submitted to Comm. Comp. Phys., Dec. 2009

• Parnell, W.J. and Grimal, Q. 2009The influence of mesoscopic porosity on cortical bone anisotropy.Investigations via asymptotic homogenization, J. Roy. Soc. Interface 6,97-109

• Parnell, W.J. 2007Effective wave propagation in a pre-stressed nonlinear elasticcomposite bar, IMA J. Appl. Math. 72, 223-244.

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