Understanding metamaterials through models
Transcript of Understanding metamaterials through models
![Page 1: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/1.jpg)
Understanding metamaterials through
models
Graeme Milton, University of Utah
![Page 2: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/2.jpg)
Outline:
• Introduction• Models for negative, anisotropic, and complex densities• Models for the Willis equations• Unimode, and Bimode Affine Materials• Complete Charaterization of the linear dynamics of
mass-spring networks• Field Patterns: a new type of wave
![Page 3: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/3.jpg)
Hall Voltage
![Page 4: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/4.jpg)
![Page 5: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/5.jpg)
![Page 6: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/6.jpg)
![Page 7: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/7.jpg)
![Page 8: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/8.jpg)
![Page 9: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/9.jpg)
![Page 10: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/10.jpg)
![Page 11: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/11.jpg)
Experiment: Liu et. al (2000)
![Page 12: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/12.jpg)
(With John Willis, 2007)
![Page 13: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/13.jpg)
“The monochromatic macroscopic behavior is elastic, but with an effective density of tensorial character and depending on the pulsation”
"hatched areas correspond to negative densities , i.e., to stopping bands."
Aurialt and Bonnet (1984); Aurialt (1994)
Early work recognizing anisotropic and negative densities
![Page 14: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/14.jpg)
Anisotropic Density
Anisotropic density in layered materials:Schoenberg and Sen (1983)
(With John Willis,2007)
![Page 15: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/15.jpg)
(With John Willis)
![Page 16: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/16.jpg)
(With John Willis,2007)
![Page 17: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/17.jpg)
Models for the Willis equations
Analog of the bianisotropic equations of electromagnetism
![Page 18: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/18.jpg)
![Page 19: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/19.jpg)
![Page 20: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/20.jpg)
![Page 21: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/21.jpg)
“Transformation Optics” that datesto Dolin (1961)
See also Nassar, Xu, Norris, Huang (2017)
![Page 22: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/22.jpg)
Unimode and BimodeAffine Materials
![Page 23: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/23.jpg)
![Page 24: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/24.jpg)
![Page 25: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/25.jpg)
(Answer: any trajectory!)
![Page 26: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/26.jpg)
Hart’s A Frame
![Page 27: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/27.jpg)
![Page 28: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/28.jpg)
![Page 29: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/29.jpg)
Bimode material for which the only easy modes of deformationsare affine ones
![Page 30: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/30.jpg)
Characteristic Feature of Affine Materials:
They dislike strain gradients
Example of Pierre SeppecherLike a Pantograph:
![Page 31: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/31.jpg)
Dynamic Response of Mass-Spring Networks
![Page 32: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/32.jpg)
In some sense any linear elastic metamaterial can be approximated by a mass-spring network.So what responses can these have? First lets look at statics- no masses.
![Page 33: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/33.jpg)
Relation between the displacements and forces at the terminal nodes
![Page 34: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/34.jpg)
(With Guevara-Vasquez and Onofrei)
![Page 35: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/35.jpg)
![Page 36: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/36.jpg)
![Page 37: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/37.jpg)
![Page 38: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/38.jpg)
![Page 39: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/39.jpg)
Realizing networks that only support one loading.
Inductive proof of Camar-Eddine and Seppecher (2003)
Here we introduce an alternative approach.
Pentamodes: Introduced with Cherkaev 1995
Key observation: It only supports a singlestress because 4 elements meetat each junction- the tension in onedetermines the tension in the others,by balance of forces.
![Page 40: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/40.jpg)
The Airy Stress Function must have non-negative curvature!
The Inverse Problem for inextensible Wire Webs under Tension
![Page 41: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/41.jpg)
The net anticlockwise torque going clockwise around the boundary must be non-negative
![Page 42: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/42.jpg)
Combining Networks under Compression and Tension to support any balancedset of forces
![Page 43: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/43.jpg)
![Page 44: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/44.jpg)
![Page 45: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/45.jpg)
![Page 46: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/46.jpg)
![Page 47: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/47.jpg)
![Page 48: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/48.jpg)
![Page 49: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/49.jpg)
![Page 50: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/50.jpg)
![Page 51: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/51.jpg)
![Page 52: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/52.jpg)
![Page 53: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/53.jpg)
![Page 54: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/54.jpg)
Bloch Wave:Infinitely Degenerate!
![Page 55: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/55.jpg)
![Page 56: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/56.jpg)
![Page 57: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/57.jpg)
A New Wave
![Page 58: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/58.jpg)
![Page 59: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/59.jpg)
![Page 60: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/60.jpg)
Bloch Waves are:Infinitely Degenerate!
![Page 61: Understanding metamaterials through models](https://reader035.fdocuments.us/reader035/viewer/2022081407/6295c6ce9d4cf32bdc560928/html5/thumbnails/61.jpg)
Available on my website
Only $80.