nonlinear elastic waves in a 1d layered composite material: some numerical results

Nonlinear Elastic Waves in a 1D LayeredComposite Material: Some Numerical Results

Igor V. Andrianov, Vladyslav V. Danishevs’kyy, DieterWeichert, Heiko Topol

ICNAAM 20119th International Conference of Numerical Analysis and Applied Mathematics

19. - 25. September 2011

Overview

I IntroductionI Given problem:

I Propagation of a long wave through a 1D compositeI Periodically repeated structure of the composite: Matrix with

inclusionsI Nonlinear elastic behavior of the components

I Application of theasymptotic homogenization method (AHM)

I Homogenized wave equation of different orders

I Stationary strain waves

I Example

I Conclusion

Introduction

Motivation

I Increasing number of applicationsfor composite materials

I civil engineeringI mechanical engineeringI biomechanicsI . . .

I Challenge: identification of the internal structure ofheterogeneous solids by means of measurements ofmacroscopic properties

I non-destructive testing of composite materialsI non-invasive physical examination (med.)I detection of the texture of soils and rocks (geo science)I . . .

I Popular model: The heterogeneous material is replaced by ahomogeneous one with the same mechanical properties

Governing Relations

Governing Relations

I One-dimensional wave propagation in x1 = x-directionI Periodically repeated structure of the composite:

I Ω(1): matrixI Ω(2): inclusion

I Nonlinear Problem

Governing Relations

Physical nonlinearity

Murnaghan potential1: Nonlinear elastic behavior of thecomponent Ω(k), k = 1, 2:

Φ(k) =

linear properties︷︸︸︷1

2λ(k)

(ε

(k)nn

)2+ µ(k)

(ε

(k)ij

)2

+1

3A(k)ε

(k)ij ε

(k)in ε

(k)jn + B(k)

(ε

(k)ij

)2εnn +

1

3C (k)

(ε

(k)nn

)3

︸︷︷︸physical nonlinearity

,

λ(k), µ(k) : Lame parameters

A(k),B(k),C (k) : Landau constants2

1Murnaghan, F. D. 1951 Finite deformation of an elastic solid. NY: Wiley.2Landau, L. D. & Lifshitz, E. M., 1953 Theory of elasticity, NY: Wiley.

Governing Relations

Geometrical nonlinearity

Cauchy-Green tensor:

Relation between strain ε(k)ij and displacement u

(k)i

ε(k)ij =

1

2

∂u(k)i

∂xj+∂u

(k)j

∂xi+∂u

(k)n

∂xj

∂u(k)n

∂xi

Governing Relations

The 1D wave equation for component Ω(k), k = 1, 2 is3:

α(k)∂2u(k)

∂x2+ β(k)∂u

(k)

∂x

∂2u(k)

∂x2= ρ(k)∂

2u(k)

∂t2,

where

α(k) = λ(k) + 2µ(k),

β(k) = 3[(λ(k) + 2µ(k)

)+ 2

(A(k) + 3B(k) + C (k)

)],

ρ(k) : mass density,

u(k) : longitudinal displacement,

3Cattani, C. & Rushchitsky, J. 2007 Wavelet and wave analysis applied tomaterials with micro or nanostructure, 2nd edn. Mineola, NY: Dover.

Governing Relations

Longitudinal stresses:

σ(k) = α(k)∂u(k)

∂x+β(k)

2

(∂u(k)

∂x

)2

Longitudinal stresses on the interface ∂Ω are equal:α(1) ∂u

(1)

∂x+β(1)

2

(∂u(1)

∂x

)2

= α(2) ∂u(2)

∂x+β(2)

2

(∂u(2)

∂x

)2∣∣∣∣∣

∂Ω

Perfect bonding on the interface ∂Ω:u(1) = u(2)

∣∣∣∂Ω

Asymptotic homogenizationmethod

Asymptotic homogenization method

We consider:

I wave of the wavelength L travelling through the composite

I length ` is comparable to the size of the heterogeneities

I ` L

I small parameter L = ε−1`.

We introduce:

I fast coodinate variable y , measuring the displacement withinthe periodically repeated unit cell

I slow coordinate variable x , measuring the displacement in thearea of interest

I y = ε−1x


The longitudinal displacement is now introduced as an asymptoticexpansion:

u(k) = u0(t, x) + εu(k)1 (t, x , y) + ε2u

(k)2 (t, x , y) + . . .

withu0 = u0(t, x) : Homogenized term depending only on the

slow coordinate and time

u(k)i = u

(k)i (t, x , y) : Term providing corrections on the order of

εi . Depends on both fast and slowcoordinates.


The spatial periodicity of the medium results in:

u(k)i (t, x , y) = u

(k)i (t, x , y + L)

The derivatives with respect to fast and slow variables are

∂

∂x→ ∂

∂x+ ε−1 ∂

∂y

∂2

∂x2→ ∂2

∂x2+ 2ε−1 ∂2

∂x∂y+ ε−2 ∂

2

∂y2


Bonding condition with respect to fast and slow variables:u(2) = u(1)

∣∣∣∂Ω

⇒u

(2)i = u

(1)i

∣∣∣∂Ω

Equality of stresses at the boundary with respect to fast and slow variables:

α(1) ∂u(1)

∂x+β(1)

2

(∂u(1)

∂x

)2

= α(2) ∂u(2)

∂x+β(2)

2

(∂u(2)

∂x

)2∣∣∣∣∣∣∂Ω

⇓⇓⇓α(1)

∂u(1)i−1

∂x+∂u

(1)i

∂y

+β(1)

2

i−1∑m=0

(∂u

(1)m

∂x+∂u

(1)m+1

∂y

)∂u(1)i−m−1

∂x+∂u

(1)i−m

∂y

= α(2)

∂u(2)i−1

∂x+∂u

(2)i

∂y

+β(2)

2

i−1∑m=0

(∂u

(2)m

∂x+∂u

(2)m+1

∂y

)∂u(2)i−m−1

∂x+∂u

(2)i−m

∂y

∣∣∣∣∣∣∂Ω


The macroscopic wave equation becomes4

α∂2u

∂x2+ β

∂u

∂x

∂2u

∂x2+ αγL2ε2∂

4u

∂x4= ρ

∂2u

∂t2

where

α =α(1)α(2)

c(1)α(2) + c(2)α(1), β =

c(1)β(1)(α(2)

)3+ c(2)β(2)

(α(1)

)3(c(1)α(2) + c(2)α(1)

)3,

ρ = c(1)ρ(1) + c(2)ρ(2), γ =

(c(1)c(2)

)2

12

ν40(

ν(1)ν(2))2

(z(1)

z(2)−

z(2)

z(1)

)2

c(n) : volume fractions of the components,

ν0 =√α/ρ : wave velocity at β = 0 and ε = 0

ν(n) =√α(n)/ρ(n) : wave velocity of component n ,

z(n) =√α(n)ρ(n) : acoustic impedance of component n

4Andrianov, I.V. et al. 2011: Homogenization of a 1D nonlinear dynamicalproblem for periodic composites. Z. Angew. Math. Mech. 91, 523–534

Stationary Strain Waves

Stationary solution of

α∂2u

∂x2+ β

∂u

∂x

∂2u

∂x2+ αγL2ε2∂

4u

∂x4= ρ

∂2u

∂t2?


Indroduction ofz = x − νt ⇒

displacement u(x , t) = u(z)

strain f =du

dz

α∂2u

∂x2+ β

∂u

∂x

∂2u

∂x2+ αγL2ε2∂

4u

∂x4= ρ

∂2u

∂t2

⇓⇓⇓d2f

dz2+ af + bf 2 + c = 0

witha =

(1− ν2/ν2

0

)/(γ`2), b = β/(2αγ`2),

c : integration constant


The here presented problem is now analysed for:

I initial conditions f (0) = −A0

2and

df (0)

dz= 0

I weak nonlinearity

I strong nonlinearity


Weak nonlinearity:Asymptotic solution by the Lindstedt-Pointcare technique5

Strain:f = −A0

[(12

+ 124η + 5

1152η2 + 5

13824η3 + 1

221184η4)

cos(kz)

−(

124η + 1

144η2 + 1

1152η3 + 1

13824η4)

cos(2kz)

+(

1384η2 + 1

1536η3 + 1

9216η4)

cos(3kz)

−(

16912

η3 + 120736

η4)

cos(4kz)

+ 5663552

η4 cos(5kz)

+ O(η5)]

;

Phase velocity:ν2

ν20

= 1− γk2`2

[1−

1

24η2 −

1

144η3 −

5

4608η4 + O(η5)

]

Wave number: k2 = a

[1 +

1

24ζ2 +

1

144ζ3 −

1

1536ζ4 + O(η5)

]

where η = −bA0/k2 and ζ = −bA0/a.

5Nayfeh, A. 2000 Perturbation Methods, NY: Wiley.


Comparison of the asymptotic solution (dashed line) with the exactone (solid line)


Strong nonlinearity: Exact integration in elliptic functions 6

Strain: f = −A0

2+

A0s2

2[1− E(s)/K(s)]sn2

(k0

2z, s

)

Phase velocity:ν2

ν20

= 1− γk20 `

2[3E(s)/K(s)− 2 + s2

]

Wave number: k =π

2K(s)k0, L =

2π

k=

4K(s)

k0

where

I sn(·): elliptic sine,

I K (s),E (s) complete elliptic integrals of the 1st and 2nd kind7,

6Erofeev, V.I. 2003 Wave Processes in Solids with Microstructure,Singapore: World Scientifiy.Porubov, A.V. 2003 Amplification of Nonlinear Strain Waves in Solids,Singapore: World Scientifiy

7Abramowitz, M., Stegun, I.A.,eds. 1965, Handbook of MathematicalFunctions, New York: Dover Publications


s (0 ≤ s ≤ 1): modulus of the elliptic functions/nonlinear factor.It can be calculated from the transcendental equation:8

2[1− E (s)/K (s)] = −bA0/k20

s → 0(purely harmonicwave)

f = A0

2 cos(kz),

u = A02k sin(kz),

ν2/ν20 = 1− γk2`2,

s → 1(solitary strainwave, shock disp.wave)

f = A0

2 sech2(z/h),

u = A0h2 tanh(z/h),

ν2/ν20 = 1 + 4γ(`/h)2

where h = 2/k0 and k0 = 2kK (s)/π

8Erofeev, V.I. 2003 Wave Processes in Solids with Microstructure,Singapore: World Scientifiy.


Comparison of the asymptotic solution (dashed line) with the exactone (solid line)

Example:Steel-aluminium composite

Example

α∂2u

∂x2+ β

∂u

∂x

∂2u

∂x2+ αγL2ε2∂

4u

∂x4= ρ

∂2u

∂t2

I ` = 10−3m, `(1) = `(2) = 0.5`

I α = 160 GPa, β = −2385 GPa, γ = 0.0171,ρ = 5250 kg/m3

Example

Nonlinear factor s as a function of the strain amplitude A0 and thefrequency ω.

Example

Phase velocity ν at different values of the nonlinear factor s and ofthe dispersion parameter ε = `/L = k`/(2π).ν exceeds ν0 =

√α/ρ and the wave passes from a subsonic to a

supersonic mode at s = 0.9804

Example

Dispersion curves. Horizontal dashed lines correspond to the firstSB threshold determined from the condition dω/dk = 0. Hereω0 = kν0 is the frequency in the quasi-homogeneous limit ε = 0.

Example

Influence of the nonlinear factor s on the frequency ωSB of the firststop band.

Conclusion

Conclusion

In this work nonlinear waves are investigated. Wave characteristics:

− shape of the wave− velocity− attenuation/dispersion

depending on−−−−−−−−→− amplitude of the signal− parameters of the

structure

Conclusion

Open problems:

I study of nonlinear waves in 2D and 3D problems

I examination of nonlinear vibration of heterogeneous structuresof finite sizes

Thank you for your attention

nonlinear elastic waves in a 1d layered composite material: some numerical results

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