Dynamic anisotropy and primitive waveforms in discrete elastic systems D.J. Colquitt 1 I.S. Jones 2 A.B. Movchan 1 N.V. Movchan 1 R.C. McPhedran 3 1 Department of Mathematical Sciences University of Liverpool 2 School of Engineering John Moores University 3 CUDOS, School of Physics University of Sydney British Applied Mathematics Colloquium 2012 DJC gratefully acknowledges the support of an EPSRC research studentship (EP/H018514/1). ABM and NVM acknowledge the financial support of the European Community’s Seven Framework Programme under contract number PIAPP-GA-284544-PARM-2. RCM acknowledges the support of the ARC through its Discovery Grants Scheme. Colquitt et al. | British Applied Mathematics Colloquium 2012 1 / 24
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Dynamic anisotropy and primitive waveforms in discreteelastic systems
1Department of Mathematical SciencesUniversity of Liverpool
2School of EngineeringJohn Moores University
3CUDOS, School of Physics
University of Sydney
British Applied Mathematics Colloquium 2012
DJC gratefully acknowledges the support of an EPSRC research studentship (EP/H018514/1). ABM and NVM acknowledgethe financial support of the European Community’s Seven Framework Programme under contract number
PIAPP-GA-284544-PARM-2. RCM acknowledges the support of the ARC through its Discovery Grants Scheme.
Colquitt et al. | British Applied Mathematics Colloquium 2012 1 / 24
Outline
1 Introduction
2 A uniform (scalar) square latticeGreen’s function & primitive wave forms
3 Triangular latticesScalar problem
Green’s function & primitive waveforms
Vector problem - planar elasticityGoverning EquationsDispersive propertiesPrimitive Waveforms
Colquitt et al. | British Applied Mathematics Colloquium 2012 2 / 24
| Introduction
Existing Literature
Dispersion & design of structures for controlling stop bands inconcentrated mass systems
P. Martinsson & A. Movchan, QJMAM 56 (2003), pp. 45–64.
Primitive waveforms in scalar lattices
Ayzenberg-Stepanenko & Slepyan, J Sound Vib 313 (2008), pp812–821.Osharovich et al., Continuum Mech Therm 22 (2010), pp. 599–616.Langley, J Sound Vib 197 (1996), pp. 447–469.Langley, J Sound Vib 201 (1997), pp. 235–253.
Dynamic homogenization of periodic media
Craster et al., QJMAM 63 (2010), pp.497–519.Craster et al., Proc R Soc A 466 (2010), pp. 2341–2362.Craster et al., JOSA A (2011), pp. 1032–1040.
Colquitt et al. | British Applied Mathematics Colloquium 2012 3 / 24
| A uniform (scalar) square lattice
A uniform square lattice
Label each node by n ∈ Z2
Introduce counting vectorsei = [δ1i , δ2i ]
T .
Lattice vectors: t1 = [`, 0]T ,t2 = [0, `]T .
Translation matrixT = [t1, t2].
Position of nth node:x = n⊗ T .
Colquitt et al. | British Applied Mathematics Colloquium 2012 4 / 24
| A uniform (scalar) square lattice
A uniform square lattice
(n)
t1
t2
Label each node by n ∈ Z2
Introduce counting vectorsei = [δ1i , δ2i ]
T .
Lattice vectors: t1 = [`, 0]T ,t2 = [0, `]T .
Translation matrixT = [t1, t2].
Position of nth node:x = n⊗ T .
Colquitt et al. | British Applied Mathematics Colloquium 2012 4 / 24
| A uniform (scalar) square lattice
A uniform square lattice
(n)
t1
t2
(n− e1) (n + e1)
(n + e1 + e2)
(n− e1 − e2)
(n− e2)
(n + e2)
(n− e1 − e2)
(n− e1 + e2)Label each node by n ∈ Z2
Introduce counting vectorsei = [δ1i , δ2i ]
T .
Lattice vectors: t1 = [`, 0]T ,t2 = [0, `]T .
Translation matrixT = [t1, t2].
Position of nth node:x = n⊗ T .
Colquitt et al. | British Applied Mathematics Colloquium 2012 4 / 24
| A uniform (scalar) square lattice | Green’s function & primitive wave forms
Uniform Square LatticeGreen’s function for time-haromic out-of-plane shear
Balance of linear momentum at node n connected to set of nodesP(n) = {n± e1,n± e2} (for time-harmonic deformations)∑
p∈P(n)
u(p)− |P(n)|u(n) + ω2u(n) = δm0δn0
Discrete Fourier transform:
F : u(m) 7→ U(ξ) =∑m∈Z2
u(m) exp {−iξ · (n⊗ T )}
Inverse transform:
Green’s Function
u(n) =1
π2
∫ π
0
∫ π
0
cos(n1ξ1) cos(n2ξ2)
ω2 − 4 + 2(cos ξ1 + cos ξ2)dξ1dξ2.
Colquitt et al. | British Applied Mathematics Colloquium 2012 5 / 24
| A uniform (scalar) square lattice | Green’s function & primitive wave forms
Primitive Wave Forms at a Saddle Point
Dispersion Diagram
−2
0
2
−2
0
2
0
0.5
1
1.5
2
2.5
3
ξ1
ξ2
ω
−A−B
BA
σ(ω, ξ) = 0
Slowness Contour
ξ1
ξ 2−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
σ(2, ξ) = 0
Colquitt et al. | British Applied Mathematics Colloquium 2012 6 / 24
| A uniform (scalar) square lattice | Green’s function & primitive wave forms
Two stationary points of different kinds
−10 −5 0 5 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
m
n
ω = 2
−10 −5 0 5 10
−10
−5
0
5
10
m
n
ω = 2√
2
Colquitt et al. | British Applied Mathematics Colloquium 2012 7 / 24
| Triangular lattices
Uniform Triangular Lattice
Colquitt et al. | British Applied Mathematics Colloquium 2012 8 / 24
| Triangular lattices
Uniform Triangular Lattice
(n)t2
t1
cell n = [n1, n2]T ∈ Z2
x(n) = n⊗ T , T = [t1, t2] , t1 = [`, 0]T , t2 = `[1,√
3]T/2
ei = [δ1i , δ2i ]T
Colquitt et al. | British Applied Mathematics Colloquium 2012 8 / 24
| Triangular lattices
Uniform Triangular Lattice
(n)t2
t1
(n + e1)(n− 2e1) (n− e1) (n + 2e1) (n + 3e1)
(n + e2) (n + e1 + e2)(n− 2e1 + e2)
(n− e1 + e2) (n + 2e1 + e2)
(n− e2)
(n + e1 − e2)
(n− 2e1 − e2)
(n− e1 − e2)
(n + 2e1 − e2, 1)
(n + 3e1 − e2, 2)
cell n = [n1, n2]T ∈ Z2
x(n) = n⊗ T , T = [t1, t2] , t1 = [`, 0]T , t2 = `[1,√
3]T/2
ei = [δ1i , δ2i ]T
Colquitt et al. | British Applied Mathematics Colloquium 2012 8 / 24
Horizontal Forcing Vertical Forcing Concentrated Moment
Shape of primitive waveform determined by slowness contour
Existence of primitive waveforms not associated with saddle points asin scalar case (Ayzenberg-Stepanenko et al.)
Frequency dependent “switching” of waveform orientations formonotonic lattice is a novel feature of elastic lattice
Elastic lattice “allows selection” of dominant orientation viatype/orientation of applied forcing
Colquitt et al. | British Applied Mathematics Colloquium 2012 23 / 24
| Summary
Summary
Statically isotropic lattices can exhibit very strong dynamic anisotropy
Qualitative behaviour of fundamental solution can be predicted fromthe Bloch-Floquet problem
Established the presence of primitive waveforms in elastic latticespreviously only demonstrated in scalar lattice
Offered an explanation for these effects via analysis of slownesscontours
Established the importance of the slowness contours in these primitivewaveforms
Inclusion of micropolar interactions allows for additional primitivewaveforms
Additional details can be found inColquitt et al. Dynamic anisotropy & localization in elastic latticesystems. Waves in Random & Complex Media. To appear (doi:10.1080/17455030.2011.633940).
Colquitt et al. | British Applied Mathematics Colloquium 2012 24 / 24