Elastic Metamaterials with Simultaneously Negative Effective Shear Modulus and Mass Density

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Elastic Metamaterials with Simultaneously

Negative Effective Shear Modulus and Mass Density

Item Type Article

Authors Wu, Ying;Lai, Yun;Zhang, Zhao-Qing

Citation Elastic Metamaterials with Simultaneously Negative Effective Shear Modulus and Mass Density 2011, 107 (10) Physical Review Letters

Eprint version Publisher's Version/PDF

DOI 10.1103/PhysRevLett.107.105506 Publisher American Physical Society (APS) Journal Physical Review Letters

Rights Archived with thanks to Physical Review Letters Download date 2024-01-26 19:27:33

Link to Item http://hdl.handle.net/10754/553009

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Supplemental Material for “Elastic metamaterials with simultaneously negative effective shear modulus and mass density”

Ying Wu^1,2, Yun Lai^1,3 and Zhao-Qing Zhang^1*

1 Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

2 Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia

3 Department of Physics, Soochow University, 1 Shizi Street, Suzhou 215006, People’s Republic of China

Fig. S1. A schematic graph of s-p mode conversion during the negative refraction on the interface of a double-negative elastic metamaterial of Case 1 below.

There can be four cases of total s-p or p-s mode conversions on the interface of a double- negative metamaterial and a normal solid under negative refraction:

Case 1: Incident shear waves from a double negative metamaterial.

Case 2: Incident longitudinal waves from a double negative metamaterial.

Case 3: Incident shear waves from a normal solid.

Case 4: Incident longitudinal waves from a normal solid.

It is obvious that Case 3 can be obtained from the time reversal of Case 2 and Case 4 can be obtained from the time reversal of Case 1. Below we derive the general conditions of total mode conversions under negative refraction.

uGl

uGt

kGt

kGl

x Interface

α β

Incident

transverse wave

Refracted

longitudinal wave

1, 1 0, 1 0

κ μ < ρ <

Double negative elastic metamaterial

2, 2, 2

κ μ ρ

Normal positive elastic medium Flux

Flux

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For s waves, the displacement is ^u^G^t ⁼^u^t

(

^sin^α^x^ˆ⁺^cos^α^{y e}^ˆ

)

^ik^t⁽⁻^cos^α^x⁺^sin^α^y⁾

For p waves, the displacement is ^u^G^l ⁼^u^l

(

^cos^β^x^ˆ⁺^sin^β^{y e}^ˆ

)

^ik^l⁽^cos^β^x⁺^sin^β^y⁾

The displacements must match on the interface x=0, i.e.,

sin sin

sin cos ,

cos sin .

t l

ik y ik y

t l

ik y ik y

t l

u e u e

α β

=

= (S1)

From Eq. (S1), we find sinu_t α =u_lcosβ and u_tcosα =u_lsinβ , which means that

l t

u =u and α β π+ = 2. In the case of normal refraction, for incident angle 0< <α π 2 we have β <0, thus α β π+ < 2, the displacements are not possible to match on the interface. Total conversion is only possible in the case of negative refraction.

From Eq. (S1), we also find sink_t α =k_lsinβ, which means the wave vector parallel to the interface must conserve. Substituting α β π+ = 2, we obtain sink_t α =k_lcosα. From

1 1

t t

k v

ω ω

= = μ ρ and

(

² ²

)

²

l l

k v

ω ω

κ μ ρ

= =

+ , we find

( )

1 1

2 2 2

tan ^l

t

k k α μ ρ

κ μ ρ

= =

+ .

We also need to consider the matching condition of stresses.

For s waves, ^u^G^t ⁼^u^t

(

^sin^α^x^ˆ⁺^cos^α^{y e}^ˆ

)

^ik^t⁽⁻^cos^α^x⁺^sin^α^y⁾. Thus, the strains are

( )

⁽ ⁾ ⁽ ⁾

( )

⁽ ⁾ ⁽ ⁾

cos sin cos sin

sin cos sin cos ,

cos sin sin cos ,

t t

ik x y ik x y

t tx

xx t t t t

ty ik x y ik x y

t

yy t t t t

S u u ik e iu k e

x

S u u ik e iu k e

y

α α α α

− + − +

=∂ = − = −

∂

=∂ = =

∂

( )

⁽ ⁾

( )

⁽ ⁾

( )

⁽ ⁾

cos sin cos sin

cos sin

2 2

1 2

1 sin sin cos cos

2

1 sin cos .

2

t t

t

t tx ty

xy

ik x y ik x y

t t t t

ik x y

t t t t

u u

S y x

u ik e u ik e

iu k iu k e

α α α α

α α

α α α α

α α

− + − +

− +

∂

⎛∂ ⎞

= ⎜⎝ ∂ + ∂ ⎟⎠

= + −

= −

And the stress can be written as

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( ) ( )

( )

⁽ ⁾

( )

⁽ ⁾

( )

( ) ( )

( )

1 1 1 1

cos sin cos sin

1 1 1 1

cos sin 1

1 1 1 1

1 1

sin cos sin cos

2 sin cos

sin 2 ,

sin c

t t

t

t t t

xx xx yy

ik x y ik x y

t t t t

ik x y

t t

ik x y

t t

t t t

yy yy xx

t t

S S

iu k e iu k e

i u k e

S S

iu k

α α α α

α α

σ κ μ κ μ

κ μ α α κ μ α α

μ α α

μ α

σ κ μ κ μ

κ μ α

− + − +

− +

= + + −

= − + + −

= −

= + + −

= + ⁽ ⁾

( )

⁽ ⁾

( )

⁽ ⁾

( )

cos sin cos sin

1 1

cos sin 1

1

cos sin

2 2

1

cos sin 1

os sin cos

2 sin cos

sin 2 ,

2

sin cos

cos 2 .

t t

t

ik x y ik x y

t t

ik x y

t t

ik x y

t t

t t

xy xy

ik x y

t t t t

ik x y

t t

e iu k e

i u k e

S

iu k iu k e

i u k e

α α α α

α α

α κ μ α α

μ α α

μ α

σ μ

μ α α

μ α

− + − +

− +

− −

=

= −

For p waves, ^u^G^l ⁼^u^l

(

^cos^β^x^ˆ⁺^sin^β^{y e}^ˆ

)

^ik^l⁽^cos^β^x⁺^sin^β^y⁾. Thus, the strains are

( )

⁽ ⁾ ⁽ ⁾

( )

⁽ ⁾ ⁽ ⁾

cos sin 2 cos sin

cos cos cos ,

sin sin sin ,

l l

ik x y ik x y

l lx

xx l l l l

ly ik x y ik x y

l

yy l l l l

S u u ik e iu k e

x

S u u ik e iu k e

y

β β β β

β β β

+ +

=∂ = =

∂

=∂ = =

∂

( )

⁽ ⁾

( )

⁽ ⁾

( )

cos sin cos sin

cos sin

1 2

1 cos sin sin cos

2

sin cos .

l l

l

l lx ly

xy

ik x y ik x y

l l l l

ik x y

l l

u u

S y x

u ik e u ik e

iu k e

β β β β

β β

β β β β

β β

+ +

+

∂

⎛∂ ⎞

= ⎜⎝ ∂ + ∂ ⎟⎠

= +

=

And the stress can be written as

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( ) ( )

( )

⁽ ⁾

( )

⁽ ⁾

( ) ( )

( )

⁽ ⁾

( )

⁽ ⁾

( ) ( )

( )

2 2 2 2

cos sin cos sin

2 2

2 2 2 2

cos sin

2 2 2 2

2 2

cos sin

2 2

2 2 2 2

2 2

cos sin

cos sin cos sin

cos 2 ,

l l

l

l l l

xx xx yy

ik x y ik x y

l l l l

ik x y

l l

ik x y

l l

l l l

yy yy xx

l

S S

iu k e iu k e

i u k e

S S

iu k

β β β β

β β

σ κ μ κ μ

κ μ β κ μ β

κ β β μ β β

κ μ β

σ κ μ κ μ

κ μ

+ +

= + + −

= +

= + + −

= + ⁽ ⁾

( )

⁽ ⁾

( ) ( )

( )

⁽ ⁾

( )

⁽ ⁾

( )

cos sin cos sin

2 2

cos sin

2 2 2 2

2 2

cos sin

2 2

2

cos sin 2

cos 2

sin cos

cos sin cos sin

cos 2 ,

2

2 sin cos

sin 2

l l

l

ik x y ik x y

l l l

ik x y

l l

ik x y

l l

l l

xy xy

ik x y

l l

ik x

l l

e iu k e

i u k e

S

iu k e

i u k e

β β β β

β β

β

β κ μ β

κ β β μ β β

κ μ β

σ μ

μ β β

μ β

+ +

+ −

= + + − +

= −

=

= ⁽ ^sin^β^y⁾.

σxx^andσxy must also be continuous on the interface x=0, i.e.,

¹ ^sin

(

² ²

)

^sin

sin sin

1 2

sin 2 cos 2 ,

cos 2 sin 2 .

t l

ik y ik y

t t l l

ik y ik y

t t l l

i u k e i u k e

i u k e i u k e

α β

μ α κ μ β

μ α μ β

− = +

− = (S2)

By substituting α β π+ = 2, u_l =u_t and sink_t α =k_lcosα in Eq. (S2), we find

2 2cos 2 1sin 2 tan

κ =μ α μ− α α and − =μ μ₁ ₂tan 2 tanα α .

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Conclusion: total conversion is possible under the case of negative refraction.

The total conversion matching conditions are:

( )

2 2 1

1 2

1 1

2 2 2

2,

cos 2 sin 2 tan , tan 2 tan ,

tan ^l .

t

k k α β π

κ μ α μ α α

μ μ α α

α μ ρ

κ μ ρ + =

= −

− =

= =

+

. (S3)

Since Case 3 can be obtained from the time reversal of Case 2 and Case 4 can be obtained from the time reversal of Case 1, here we discuss only Cases 1 and 3.

Case 1: we consider a transverse plane wave incident from the left medium of

1 0, 1 0

μ < ρ < with an incident angle of 0< <α π 2as shown in Fig. S1. First, we can obtain μ₂ ^{by using} − =μ μ1 2tan 2 tan .α α Then, we can obtain κ₂ ^by usingκ₂ =μ₂cos 2α μ− ₁sin 2α tanα μ= ₂ cos 2α . At last, we can obtain ρ₂ by using

( )

1 1

2 2 2

tan ^l

t

k k α μ ρ

κ μ ρ

= =

+ . Therefore, μ₂ ^,κ2^andρ2 are all obtained. It is worth mentioning that when α π> 4 , we have μ₂ <0 , but κ₂ >0 and

( )

2 2 2 1 cos 2 cos 2 0

κ +μ =μ + α α > , which indicates a double positive medium for refracted longitudinal waves on the right (together with ρ₂ >0). In this case the medium on the right is not a normal solid.

Case 3: we consider a transverse plane wave incident from the left medium of

1 0, 1 0

μ > ρ > with an incident angle of 0< <α π 2. μ₂^,κ2^andρ2 can also be obtained from Eq. (S3). Note that we have ^κ²⁺^μ² ⁼^μ²

(

^{1 cos 2}⁺ ^α

)

^{cos 2}^α ^<⁰^and^ρ² ^<⁰^{, which}

indicate a double negative medium for refracted longitudinal waves on the right.