Dynamics of Lattice Materials E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
- Provides a comprehensive introduction to the dynamic response of lattice materials, covering the fundamental theory and applications in engineering practice
- Offers comprehensive treatment of dynamics of lattice materials and periodic materials in general, including phononic crystals and elastic metamaterials
- Provides an in depth introduction to elastostatics and elastodynamics of lattice materials
- Covers advanced topics such as damping, nonlinearity, instability, impact and nanoscale systems
- Introduces contemporary concepts including pentamodes, local resonance and inertial amplification
- Includes chapters on fast computation and design optimization tools
- Topics are introduced using simple systems and generalized to more complex structures with a focus on dispersion characteristics
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Veröffentlichungsjahr: 2017
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Table of Contents
Cover
Title Page
Copyright
Dedication
List of Contributors
Foreword
References
Preface
Chapter 1: Introduction to Lattice Materials
1.1 Introduction
1.2 Lattice Materials and Structures
1.3 Overview of Chapters
Acknowledgment
References
Chapter 2: Elastostatics of Lattice Materials
2.1 Introduction
2.2 The RVE
2.3 Surface Average Approach
2.4 Volume Average Approach
2.5 Force-based Approach
2.6 Asymptotic Homogenization Method
2.7 Generalized Continuum Theory
2.8 Homogenization via Bloch Wave Analysis and the Cauchy–Born Hypothesis
2.9 Multiscale Matrix-based Computational Technique
2.10 Homogenization based on the Equation of Motion
2.11 Case Study: Property Predictions for a Hexagonal Lattice
2.12 Conclusions
References
Chapter 3: Elastodynamics of Lattice Materials
3.1 Introduction
3.2 One-dimensional Lattices
3.3 Two-dimensional Lattice Materials
3.4 Lattice Materials
3.5 Tunneling and Evanescent Waves
3.6 Concluding Remarks
3.7 Acknowledgments
References
Chapter 4: Wave Propagation in Damped Lattice Materials
4.1 Introduction
4.2 One-dimensional Mass–Spring–Damper Model
4.3 Two-dimensional Plate–Plate Lattice Model
References
Chapter 5: Wave Propagation in Nonlinear Lattice Materials
5.1 Overview
5.2 Weakly Nonlinear Dispersion Analysis
5.3 Application to a 1D Monoatomic Chain
5.4 Application to a 2D Monoatomic Lattice
Summary
Acknowledgements
References
Chapter 6: Stability of Lattice Materials
6.1 Introduction
6.2 Geometry, Material, and Loading Conditions
6.3 Stability of Finite-sized Specimens
6.4 Stability of Infinite Periodic Specimens
6.5 Post-buckling Analysis
6.6 Effect of Buckling and Large Deformation on the Propagation Of Elastic Waves
6.7 Conclusions
References
Chapter 7: Impact and Blast Response of Lattice Materials
7.1 Introduction
7.2 Literature Review
7.3 Manufacturing Process
7.4 Dynamic and Blast Loading of Lattice Materials
7.5 Results and Discussion
Concluding Remarks
Acknowledgements
References
Chapter 8: Pentamode Lattice Structures
8.1 Introduction
8.2 Pentamode Materials
8.3 Lattice Models for PM
8.4 Quasi-static Pentamode Properties of a Lattice in 2D and 3D
8.5 Conclusion
Acknowledgements
References
Chapter 9: Modal Reduction of Lattice Material Models
9.1 Introduction
9.2 Plate Model
9.3 Reduced Bloch Mode Expansion
9.4 Bloch Mode Synthesis
9.5 Comparison of RBME and BMS
References
Chapter 10: Topology Optimization of Lattice Materials
10.1 Introduction
10.2 Unit-cell Optimization
10.3 Plate-based Lattice Material Unit Cell
10.4 Genetic Algorithm
10.5 Appendix
References
Chapter 11: Dynamics of Locally Resonant and Inertially Amplified Lattice Materials
11.1 Introduction
11.2 Locally Resonant Lattice Materials
11.3 Inertially Amplified Lattice Materials
11.4 Conclusions
References
Chapter 12: Dynamics of Nanolattices: Polymer-Nanometal Lattices
12.1 Introduction
12.2 Fabrication
12.3 Lattice Dynamics
12.4 Conclusions
12.5 Appendix: Shape Functions for a Timoshenko Beam with Six Nodal Degrees of Freedom
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Foreword
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction to Lattice Materials
Figure 1.1 Periodic materials and structures across different length scales and disciplines. (MEMS: microelectromechanical systems.)
Figure 1.2 Lattice materials formed from a periodic network of beams: (a) ultralight nanometal truss hybrid lattice; (b) pentamode lattice.
Figure 1.3 Unit-cell geometry-dependent effective in-plane elastic moduli of planar lattices for the same mass: Young's modulus tensor component (left) and shear modulus tensor component (right) along the Cartesian (positive to the right) and (positive upwards) axes. The Young's modulus of the isotropic parent solid is denoted .
Chapter 2: Elastostatics of Lattice Materials
Figure 2.1 A schematic of the homogenization process of a square lattice material (left) into an equivalent homogeneous medium (right) with effective properties obtained from the RVE analysis.
Figure 2.2 Initial (
IJ
) and deformed (
ij
) geometry of a typical cell member of a cellular solid.
Figure 2.3 Multiscale scheme [51, 52]. VWP, virtual work principle.
Figure 2.4 Lattice (transformation) matrix: (a) mapping a unit cell into the lattice space; (b) lattice topology in the lattice space.
Figure 2.6 Effective Young's modulus, , as function of relative density for the hexagonal cell. Results obtained with the DBC and NBC bound those calculated with the PBC.
Figure 2.5 Effective elastic constants obtained with: force-based approach [27], asymptotic homogenization method [103], generalized continuum theory [16, 108], Bloch's theorem [120], matrix-based approach [51], and homogenization based on the equations of motion [50].
Figure 2.7 Relative difference of the effective elastic constants obtained with the force-based approach [27], generalized continuum theory [16, 108], Bloch's theorem [120], matrix-based approach [51], and homogenization based on equation of motion [50]. AH is used as the baseline.
Figure 2.8 Effective yield strength obtained with: force-based approach [27], asymptotic homogenization method [103], generalized continuum theory [16, 108], Bloch's theorem [120], and matrix-based approach [51].
Figure 2.9 Relative difference of the yield strength obtained with force-based approach [27], generalized continuum theory [16, 108], Bloch's theorem [120], and matrix-based approach [51]. Effective property values are normalized by AH results.
Chapter 3: Elastodynamics of Lattice Materials
Figure 3.1 A monoatomic lattice is shown in (a), along with two possible unit cell choices in (b) and (c). Note that other choices exist for both symmetric and asymmetric unit cells. All masses are equidistant.
Figure 3.2 A diatomic lattice is shown in (a), along with two possible unit cell choices in (b) and (c). Note that other choices exist for both symmetric and asymmetric unit cells. The spacing between all masses is equal. .
Figure 3.3 Dispersion curves for (a) monoatomic and (b) diatomic discrete spring–mass lattices. Natural frequencies of the unit cell under free–free boundary conditions and pinned–pinned boundary conditions are shown by the horizontal dashed lines, and dash-dot lines, respectively. Note that the resonances of the unit cell bound the band edges.
Figure 3.4 Continuous beam lattices: (a) beam with uniformly spaced masses, (b) beam with uniformly spaced resonators, (c) beam with uniformly spaced masses and resonators, (d) unit cell for the general case (c), with pinned and guided boundary conditions shown in (e) and (f), respectively.
Figure 3.5 Dispersion curves for beam lattices in Figure 3.4. The top row corresponds to (a) beam with periodic masses, (b) beam with periodic resonators, and (c) beam with periodic masses and resonators. The bottom row is a magnified version of the top-row Figure in a lower frequency range. The sub-Bragg band gap in (e) around is due to local resonance. Notice how the width of the Bragg band around decreases in (e) compared to (d), but is restored in (f) when periodic masses are introduced in the beam with periodic resonators. is the nondimensional frequency , where is the pinned–pinned resonance frequency of the unit cell of the beam with periodic masses. is the phase constant. The dashed and dashed-dotted horizontal lines are the resonant frequencies of the unit cell under pinned–pinned and free–free (guided) boundary conditions shown in Figure 3.4e and f, respectively.
Figure 3.6 The concept of receptance coupling: the vertical displacement of the beam, mass, and resonator are coupled to obtain the respective unit cells for (a) the pinned and (b) the guided boundary conditions. Here coupling is viewed as a parallel coupling, in which the displacements of the connected units are the same but the forces add up.
Figure 3.7 Sub-Bragg band gap viewed from a receptance perspective. The lower and upper band edges are, respectively, the resonances of the unit cell under pinned and guided boundary conditions. These resonant frequencies are shown as the vertical dashed (pinned–pinned supports) and dashed-dotted lines (guided supports). Resonant frequencies of the pinned–pinned unit cell (lower edge of the sub-Bragg band gap) lies at the intersection of receptance of the resonator () and the pinned–pinned beam (). The same holds for the guided boundary conditions: the resonance (upper edge of the band gap) is the intersection of and . The deep anti-resonance of the attached resonator is evident in the minimum of . Notice the agreement with Figure 3.5e for the sub-Bragg band gap.
Figure 3.8 A generic unit cell for a 2D periodic structure, showing the degrees of freedom shared with the neighboring unit cells.
Figure 3.9 A square lattice with possible unit-cell choices. Note that other choices exist for both symmetric and asymmetric unit cells.
Figure 3.10 Out-of-plane wave dispersion in a square lattice: (a) dispersion surface with the IBZ shown as a square in the interval ; notice the repetition of the dispersion surface outside the full Brillouin zone (not shown) defined by the interval . The dispersion surface in (a) can also be represented as dispersion curves obtained by following different paths along the edges of the IBZ. Shown in the second row are the dispersion curves for the path in (b), in (c) and in (d). Notice how the normal mode (zero group velocity) associated with is clearly evident in (c) but not so in (d). Also, note that the curve in (c) misses the maximum frequency evident in (b) and (d). It should be observed that the band edges are the natural frequencies of the unit cell under free and fixed boundary conditions.
Figure 3.11 Five representative 2D beam lattice topologies (left column), along with the unit cell (middle column) and the first Brillouin zone (right column): (a) hexagonal lattice; (b) Kagome lattice; (c) triangular lattice; (d) square lattice; (e) pentamode lattice.
Figure 3.12 Beam element with nodes numbered 1 and 2. The three nodal degrees of freedom are shown, together with the local element coordinate axes with origin located at the middle of the beam. The nondimensional coordinate is , where is the length of the beam.
Figure 3.13 Band structure of a hexagonal honeycomb with slenderness ratio equal to 50. The eigenwaves of a typical cell are shown in tabular form. The three rows correspond to the three points , and in -space, while the four columns correspond to the first four dispersion branches in ascending order.
Figure 3.14 Band structure of a Kagome lattice with slenderness ratio equal to 50. The eigenwaves of a typical cell are shown in tabular form. The three rows correspond to the three points , and in -space, while the four columns correspond to the first four dispersion branches in ascending order.
Figure 3.15 Band structure of a triangular honeycomb with slenderness ratio equal to 50. The eigenwaves of a typical cell are shown in tabular form. The three rows correspond to the three points , and in -space, while the four columns correspond to the first four dispersion branches in ascending order.
Figure 3.16 Band structure of a hexagonal honeycomb with slenderness ratio equal to 50. The eigenwaves of a typical cell are shown in tabular form. The three rows correspond to the three points , and in -space, while the four columns correspond to the first four dispersion branches in ascending order.
Figure 3.17 Band structure of a pentamode lattice with slenderness ratio equal to 50 in the interests of brevity. The eigenwaves of a typical cell are shown in tabular form. The three rows correspond to the three points , and in -space, while the four columns correspond to the first four dispersion branches in ascending order.
Figure 3.18 Directionality of plane-ave propagation in the four lattice topologies with slenderness ratio equal to 50: (a) hexagonal honeycomb; (b) Kagome lattice; (c) triangular honeycomb; (d) square honeycomb. The nondimensional frequency () associated with each contour is labelled. At each frequency, energy flow direction is given by the normal to the contour and is in the direction of maximum rate of change of frequencies.
Chapter 4: Wave Propagation in Damped Lattice Materials
Figure 4.1 (a) One-dimensional mass–spring–damper chain with unit cell highlighted, and (b) freed unit cell with position of the first mass from an adjacent unit cell indicated.
Figure 4.2 Damped dispersion for 1D spring–mass–damper chain: free-wave frequency band structure (top left), corresponding damping ratio diagram (bottom left), and driven-wave band structure (top right).
Figure 4.3 Plate lattice unit cell with finite-element mesh shown.
Figure 4.4 Damped dispersion for plate lattice material: free-wave frequency band structure (top left), corresponding damping ratio diagram (bottom left), and driven-wave band structure (top right). Frequencies are normalized as follows: .
Figure 4.5 Damped dispersion for plate lattice material: iso-frequency plot (top left) and iso-damping plot (bottom left) for free waves; iso-frequency plots for imaginary (top middle) and real (top right) wavenumbers for driven waves.
Figure 4.6 Iso-frequency plots for imaginary (left) and real (right) wavenumbers for driven waves for the 2D plate lattice. The top Figure show the damped wavenumber surfaces, the middle plots show the undamped wavenumber surfaces, and the bottom plots show both damped and undamped wavenumbers at select frequencies.
Chapter 5: Wave Propagation in Nonlinear Lattice Materials
Figure 5.1 Central unit cell (darker color) surrounded by neighboring unit cells for 1D, 2D, and 3D periodic systems.
Figure 5.2 Monoatomic mass-spring chain with cubic stiffness and lattice vector .
Figure 5.3 Multi-wave corrected dispersion curves compared with the linear curve and the nonlinear single-wave corrected curve .
Figure 5.4 Typical solutions for , , , and .
Figure 5.5 A 10% shift in frequency at can be achieved by injecting high-amplitude, low frequency waves, or low-amplitude, high-frequency waves.
Figure 5.6 Monoatomic lattice configuration with lattice vectors and . Dashed lines indicate boundaries for the unit cell.
Figure 5.7 Three cases of wave–wave interaction in the monoatomic lattice.
Figure 5.8 Brillouin zone symmetry is retained by dispersion shifts resulting from wave interactions. (a) Linear dispersion relationship with the FBZ identified by points , . Wave vectors and corresponding to horizontal and oblique control waves used for frequency corrections plotted in (b) and (c), respectively.
Figure 5.9 Negative group velocity corrections are a unique result of wave interactions. A linear wave beam (black, dashed) receives a negative group velocity correction (black arrow) to produce beam shifting (black, solid). Control waves that achieve are found in region (I).
Figure 5.10 Initial wave field and corresponding displacement probe located centrally in the field . Markers denote a nonlinear least-squares fit while solid lines indicate the numerical simulation time signal.
Figure 5.11 Numerical simulation results for frequency–amplitude relationship using a least-squares curve fitting method.
Figure 5.12 Initial wave field for orthogonal and oblique wave interaction (for ) and corresponding displacement probe located centrally at . Symbols demarcate the time series corresponding to the nonlinear least-squares curve fit for frequency, phase, and amplitude.
Figure 5.13 Numerical simulation results for orthogonal and oblique wave interactions. Theoretical results (line) are validated by numerical time-domain simulations (markers). Unlike orthogonal interactions (a), oblique interactions (b) result in nonzero frequency shifts for low-amplitude
A
waves.
Figure 5.14 Wave beam steering using a control wave with and a point-source excitation at . Increasing levels of control-wave amplitude , shown in (a)–(c), vary the beam angle. Solid lines indicate a theoretical beam path; dashed lines indicate the low-amplitude beam path for comparison.
Figure 5.15 (a) Schematic of a tunable focus device that utilizes constructive interference on a central focal plane. (b) Power distributions calculated along the focal plane from numerical simulation results reveal a sharpening of the focal point (see also Figure 5.16) and tunable distance.
Figure 5.16 Time-domain simulation results for a tunable focus device. In the presence of a control-wave field (), dynamic anisotropy introduced into the lattice alters the focal point (FP) distance and sharpness. Dashed lines indicate the focal plane.
Chapter 6: Stability of Lattice Materials
Figure 6.1 The period square lattice considered: (a) schematic of a finite-sized square lattice; (b) schematic of the unit cell and 22 and 15 enlarged unit cells.
Figure 6.2 Stability of finite-sized specimens: (a) critical strains for equibiaxial and uniaxial loading as function of the sample size; (b) critical buckling mode under equibiaxial loading obtained for a finite-sized sample of 1010 unit cells; (c) critical buckling mode under unibiaxial loading obtained for a finite-sized sample of 1010 unit cells.
Figure 6.3 Evolution of the frequency parameter as a function of the applied strain for: (a) biaxial and (b) uniaxial loading conditions. Mode shapes associated with: (c) , (d) and (e) obtained at their corresponding critical load under equibiaxial compression.
Figure 6.4 Macroscopic stress–strain curves for the square lattice under (a) equibiaxial and (b) uniaxial loading conditions. The departure from linearity indicates the onset of instability.
Figure 6.5 Deformation field obtained for different values of the global equibiaxial strain: (a)–(d) finite-sized sample; (e)–(h) unit cell of the infinite periodic system.
Figure 6.6 Deformation filed obtained for different values of the global uniaxial strain: (a)–(d) finite-sized sample; (e)–(h) unit cell of the infinite periodic system.
Figure 6.7 Square periodic lattice: (a) lattice vectors; (b) first Brillouin Zone.
Figure 6.9 Comparison between the dispersion relations (a,c,e) and frequency response functions (b,d,f) of the system obtained for different values of the applied equibiaxial strain.
Figure 6.8 Comparison between the dispersion relations (a,c,e) and frequency response functions (b,d,f) of the system obtained for different values of the applied uniaxial strain.
Chapter 7: Impact and Blast Response of Lattice Materials
Figure 7.1 The MCP Realizer II machine.
Figure 7.2 Microlattice geometries used in this study: (a) BCC and (b) BCC-Z.
Figure 7.3 316L stainless steel lattice blocks with (a) BCC unit cells and (b) BCC-Z unit cells.
Figure 7.4 A 100 × 100 × 20-mm sandwich panel with BCC lattice core.
Figure 7.5 Drop-hammer impact tower test rig.
Figure 7.6 Ballistic pendulum used for the blast tests.
Figure 7.7 Schematic of the experimental set-up for the blast tests.
Figure 7.8 Schematic of the blast tube used in the blast tests on the sandwich panel.
Figure 7.9 Stress–strain curves for the lattice structures following compression tests on the drop-hammer impact tower.
Figure 7.10 Stress–strain curves for BCC (lattice D) structures following compression tests on the drop-hammer impact tower.
Figure 7.11 Variation of plateau stress with increasing strain rate for (a) lattice structures A and B and (b) lattice structures D and H.
Figure 7.12 Images of blast loaded specimen: (a) lattice B2 (BCC); (b) lattice F1 (BCC-Z).
Figure 7.13 Percentage crush vs applied impulse: (a) BCC lattices; (b) BCC-Z lattices.
Figure 7.14 Sandwich panels based on the BCC unit cell: (a) specimen SP4 (impulse = 4 Ns); (b) specimen SP2 (impulse = 7 Ns); (c) specimen SP3 (impulse = 5.2 Ns); (d) specimen SP1 (impulse = 14 Ns).
Figure 7.15 Contour plot showing the residual thickness of sandwich panel SP3 (impulse = 5.2 Ns).
Figure 7.16 Percent crush versus applied impulse for the BCC lattice CFRP sandwich panels.
Chapter 8: Pentamode Lattice Structures
Figure 8.1 The fundamental cell for a diamond-like lattice of thin beams. The effective elasticity is cubic, with principal axes shown by the dashed lines. The effective properties approximate pentamodal when the bending compliance of the rod is much greater than the axial compliance, which is achieved by long slender beams and/or junctions with large bending compliance.
Figure 8.2 An aluminum honeycomb structure with density and in-plane bulk modulus equal to that of water. The thin struts have large bending compliance, which minimizes the shear modulus relative to the bulk modulus. The “islands” of metal provide matched density, with little extra stiffness. Phononic properties of a similar metal water structures are given in Hladky–Hennion et al. [21].
Figure 8.3 Diamond-like lattice of thin steel rods designed to have PM wave speed equal to that in water, 1500 m/s: (a) unit cell; (b) dispersion curves, from a simulation by A.J. Nagy.
Figure 8.4 A rectangular block of PM is in static equilibrium under the action of surface tractions. The two orthogonal arrows inside the rectangle indicate the principal directions of (30 from horizontal and vertical) and the relative magnitude of its eigenvalues (2:1). The equispaced arrows represent the surface loads to scale.
Figure 8.5 The diamond-like structure for the transversely isotropic PM lattice. The longer members are of length , the others of length .
Figure 8.6 Each of these 2D PM lattices has isotropic quasi-static properties. Vertical members are of length ; the other members are of length . The ratio of to is determined by Eq. (8.27). The pure honeycomb structure is .
Figure 8.7 The tetrakaidecahedral unit cell [37] has low-density PM behavior similar to the diamond lattice: effective bulk modulus given by Eq. (8.29) and shear moduli of relative magnitude .
Figure 8.8 The elastic moduli for 2D and 3D PM lattices with rods of equal length and stiffness as a function of the junction angle . Note that the 2D (3D) moduli are identical at the isotropy angle 60 (70.53). The axial stiffness vanishes at . Since it follows that also vanishes at .
Figure 8.9 (a) The solid curves show for the type of lattice structures considered in Figure 8.6, with and . The Poisson ratio describes the lateral contraction for loading along the axial -direction. The related Poisson's ratio is shown by the dashed curves. (b) The 2D lattice for (top) and for (bottom).
Figure 8.10 The principal stiffnesses (a) and Poisson's ratios (b) for a diamond lattice with the central “atom” shifted along the cube diagonal. The four vertices of the unit cell at , , , , and the center junction (atom) lies at . Isotropy is .
Chapter 9: Modal Reduction of Lattice Material Models
Figure 9.1 Plate schematic.
Figure 9.2 Demonstration of (a) a pure bending state and (b) a linear element under bending load.
Figure 9.3 (a) Portion of a periodic lattice material with a single unit cell outlined; (b) finite-element mesh of the unit cell for this material. This model is featured on the cover of the book.
Figure 9.4 Expansion point locations for 2-point, 3-point, and 5-point RBME schemes.
Figure 9.5 Comparison of full system dispersion with 2-point RBME dispersion (left) and 3-point RBME dispersion (right). Frequencies are normalized as follows: .
Figure 9.6 Error in 6th dispersion branch computed with 2-point (left) and 3-point (right) RBME models as the number of modes, , per expansion point is increased.
Figure 9.7 Maximum error in 6th dispersion branch for increasing number of modes, , per expansion point.
Figure 9.8 A selection of constraint modes (left), and fixed-interface modes (right).
Figure 9.9 Comparison of full-system dispersion with BMS dispersion using 6 fixed-interface modes (left), and 24 fixed-interface modes (right). Frequencies are normalized as follows: .
Figure 9.10 Error in 6th dispersion branch computed with BMS models for increasing numbers of fixed-interface modes, .
Figure 9.11 Maximum error in 6th dispersion branch computed with BMS models as the number of fixed-interface modes is increased.
Figure 9.12 Flowcharts showing computation steps for full-dispersion evaluation compared to reduced methods. The computationally limiting step in each section is highlighted.
Chapter 10: Topology Optimization of Lattice Materials
Figure 10.1 Realization of PnCs and MMs in one-, two-, and three-dimensions.
Figure 10.2 Unit-cell optimization approaches.
Figure 10.3 Search-space size for a simple
n
×
n
2D unit cell.
Figure 10.4 Example of disconnected unit cells.
Figure 10.5 Schematic representation of the most common lattices—(a) square; (b) hexagonal—in real and reciprocal space. The first Brilliouin zone of each is shown; the irreducible Brilliouin zone is also highlighted.
Figure 10.6 Single-node connectivity when using four-node quadratic elements versus hexagonal elements. The gray pixels represent “material” and the white pixels represent “void”.
Figure 10.7 Unit-cell construction.
Figure 10.8 Optimized unit-cell design.
Figure 10.9 Schematic models for PnCs and MMs.
Figure 10.10 Dispersion curves for PnC and MM systems.
Chapter 11: Dynamics of Locally Resonant and Inertially Amplified Lattice Materials
Figure 11.1 Longitudinal (axial) vibrations in rods: (a) infinite periodic elastic rod with resonators, with longitudinal waves propagating along the
x-
direction considered here; (b) finite periodic rod with eight resonators used in the experiment. Source: Wang et al. [13], with permission of ASME.
Figure 11.2 Infinite 1D locally resonant lattice made up of organic glass rod: (a) calculated attenuation (μ) and phase (ϵ) constants; (b) calculated and measured FRF for six and eight unit cells. Source: Wang et al. [13], with permission of ASME.
Figure 11.3 Infinite periodic elastic shaft with torsional resonators: (a) schematic of the shaft, with torsional waves propagating along the
x-
direction considered here; (b) cross-section. Source: Yu et al. [14], with permission of Elsevier.
Figure 11.4 Complex band structure of the elastic shaft with torsional resonators. (a) Real wave vector. (b) Absolute value of the imaginary part of the complex wave vector. Source: Yu et al. [14], with permission of Elsevier.
Figure 11.5 Frequency response function of the locally resonant shaft with eight unit cells. Source: Yu et al.[14], with permission of Elsevier.
Figure 11.6 Taut string with periodically attached resonators. Source: Xiao et al. [12], with permission of Elsevier.
Figure 11.7 Complex band structure of the string with local resonators for and : (a) real wave vector; (b) absolute value of the imaginary part of the complex wave vector. Source: Xiao et al. [12], with permission of Elsevier.
Figure 11.8 Beam with periodically attached resonators. Source: Wang et al. [20], with permission of APS.
Figure 11.9 (a) Band structure of the locally resonant beam; (b) calculated frequency response function of the locally resonant beam with five unit cells; (c) measured frequency response function of the finite periodic locally resonant beams. The solid, dash-dot, dashed and dotted lines represent the measured results corresponding to samples of five, three, one, and no oscillators, respectively. Source: Wang et al. [20], with permission of APS.
Figure 11.10 Square array of rubber coated lead cylinders in glass matrix: (a) cross-section and the Brillouin zone of the square lattice; (b) band structure of the lattice for a filling fraction of 0.4 and
r
a
/
r
b
= 0.722. Source: Zhang et al. [25], with permission of Elsevier.
Figure 11.11 Normalized gap width (Δω/ω
g
) versus radius ratio of the core and the coating layer (
r
a
/
r
b
). Here, the numbers on each curve represent the filling fraction for that curve. Source: Zhang et al. [25], with permission of Elsevier.
Figure 11.12 Thin plate with 2D periodic array of attached mass-spring resonators. Source: Xiao et al. [11], with permission of IOP.
Figure 11.13 (a) Real band structures of the locally resonant plate (solid lines) and the bare plate without resonators (dotted lines). (b) Real and imaginary parts of the complex band structure of the locally resonant plate along the ΓX direction (ϕ = 0). The imaginary part of the plot also contains solutions along other directions. Here, ϕ = π/4 corresponds to the ΓM direction. Source: Xiao et al. [11], with permission of IOP.
Figure 11.14 Silicone rubber coated lead sphere in an epoxy matrix: (a) cross section; (b) 8×8×8 ternary locally resonant lattice; (c) calculated (solid line) and measured (circles) amplitude transmission coefficient along the [100] direction; (d) band structure of the 3D lattice. Source Liu et al. [8], with permission of AAAS.
Figure 11.15 Calculated displacements of the unit cell at the first (a) and second (b) antiresonance frequencies. The displacement shown is for a cross-section through the center of one coated sphere, located at the front surface. The arrows indicate the direction of the incident wave. Source Liu et al. [8], with permission of AAAS.
Figure 11.16 3D structure with SC arrangement of holes: (a) unit cell; (b) corresponding first Brillouin zone of the periodic lattice; (c) band structure of the 3D lattice. Source: Wang and Wang [36], with permission of ASME.
Figure 11.17 Vibration modes of the unit cell at the band edges in Figure 11.16(c). Here, (a)–(c) show the modes corresponding to the lower-edge of the band gap (point A in Figure 11.16c), whereas (d)–(f) show the modes corresponding to the upper-edge of the band gap (point B in Figure 11.16c). Source: Wang and Wang [36], with permission of ASME.
Figure 11.18 1D mass-spring lattice with inertial amplification: (a) the unit cell; (b) unit cells connected in series to form the lattice.
Figure 11.19 1D locally resonant mass-spring lattice: (a) the unit cell; (b) unit cells connected in series to form the lattice.
Figure 11.20 1D alternating mass-spring lattice: (a) the unit cell; (b) unit cells connected in series to form the lattice.
Figure 11.21 Complex band structures of the alternating mass-spring, inertially amplified, and locally resonant lattices: (a) real wave vector; (b) absolute value of the imaginary part.
Figure 11.22 2D mass-spring lattice with inertial amplification: (a) the lattice; (b) the unit cell; (c) 6×6 finite periodic lattice. Source: Yilmaz and Hulbert [17], with permission of Elsevier.
Figure 11.23 2D mass-spring lattice with inertial amplification: (a) band structure; (b) FRF plot of the 6×6 lattice. Source: Yilmaz and Hulbert [17], with permission of Elsevier.
Figure 11.24 Contour plots of acceleration (dB): (a) at the inertial amplification induced antiresonance frequency, ω = 1.09; (b) at the local resonance induced antiresonance frequency, ω = 3.47. Source: Yilmaz and Hulbert [17], with permission of Elsevier.
Figure 11.25 Distributed parameter model of the inertial amplification mechanism that is formed by combining different-sized beam sections. Source: Acar and Yilmaz [38], with permission of Elsevier.
Figure 11.26 First two mode shapes of the optimized inertial amplification mechanism. Source: Acar and Yilmaz [38], with permission of Elsevier.
Figure 11.27 Two-dimensional distributed parameter lattice with embedded inertial amplification mechanisms. Here, four small mechanisms are connected to each small node and four small and four large mechanisms are connected to each large node. Source: Acar and Yilmaz [38], with permission of Elsevier.
Figure 11.28 2D distributed parameter lattice with embedded inertial amplification mechanisms: (a) 43rd mode shape (282.3 Hz); (b) 44th mode shape (619.2 Hz). Source: Acar and Yilmaz [38], with permission of Elsevier.
Figure 11.29 2D inertially amplified lattice: (a) experimental and numerical FRF plots for the longitudinal direction: |
x
1
(ω)/
y
(ω)|; (b) experimental FRF plots for the longitudinal (
x
1
) and transverse (
x
2
) directions. Source: Acar and Yilmaz [38], with permission of Elsevier.
Figure 11.30 3D inertially amplified lattices: (a) BCC unit cell; (b) FCC unit cell; (c) inertial amplification mechanism. Source: Taniker and Yilmaz [39], with permission of Elsevier.
Figure 11.31 Phononic band structures without amplification mechanisms: (a) BCC lattice for , ; (b) FCC lattice for , . Source: Taniker and Yilmaz [39], with permission of Elsevier.
Figure 11.32 Phononic band structures with inertial amplification mechanisms: (a) BCC lattice, for , , , , ; (b) FCC lattice, for , , , , . Source: Taniker and Yilmaz [39], with permission of Elsevier.
Figure 11.33 FRF plots of the 8×8×8 lattices with inertial amplification: (a) BCC, for , , , , ; (b) FCC, for , , , , . Source: Taniker and Yilmaz [39], with permission of Elsevier.
Figure 11.34 Inertial amplification in SC lattices: (a) SC unit cell; (b) SC unit cell with embedded inertial amplification mechanisms; (c) phononic band structures of the SC lattices with and without embedded inertial amplification mechanisms. For the SC lattice,
k
= 1 and
m
= 1; for the SC lattice with inertial amplification mechanisms,
k
= 1,
m
= 0.5,
m
a
= 0.5/6, and θ = π/18. Source: Taniker and Yilmaz [40], with permission of ASME.
Chapter 12: Dynamics of Nanolattices: Polymer-Nanometal Lattices
Figure 12.1 Multi-scale breakdown of metal–polymer microtruss lattices. The starting beam can be divided into unit cells which are composed of a varying number of struts based on geometry choice. Finally, the grain size of the coating allows for modifications to the mechanical properties of the hybrid structure.
Figure 12.2 Tensile stress–strain curves for as-deposited 20 nm, 40 nm, and 10 m electrodeposited Ni. As a comparison, the tensile stress–strain curve of the as-printed polymer material is included on the secondary ordinate axis.
Figure 12.3 As-printed polymer 3D printer material properties juxtaposed with electrodeposited nickel and nickel-alloy materials on a log–log scale for (a) Young's modulus vs density and (b) tensile strength vs density.
Figure 12.4 Cross-sections of struts in a tetrahedral microtruss beam measuring 1 0.5 m, optimally designed to support a central load of 700 N in three-point bending for the cases of: (a) 10-m Ni on a polymer microtruss; (b) hollow 10-m Ni microtruss; (c) 21-nm Ni on a polymer microtruss; (d) hollow 21-nm Ni microtruss.
Figure 12.5 Octet lattice: (a) reference cells; (b) unit cell and direct basis vectors; (c) first Brillouin zone and reciprocal basis vectors.
Figure 12.6 Notation for the nodal displacements of a Timoshenko beam element.
Figure 12.7 Forces applied to the reference cell of a generic 2D lattice.
Figure 12.8 Dispersion curve for an octet lattice with a radius-to-length ratio of 10.
Figure 12.9 Dispersion curve for an octet lattice with a radius-to-length ratio of 50.
List of Tables
Chapter 3: Elastodynamics of Lattice Materials
Table 3.1 Unit-cell resonant frequencies under free and fixed boundary conditions that bound the pass-band edges of discrete 1D lattices with a symmetric unit cell
Table 3.2 Resonant frequencies of the unit cell in Figure 3.9b. These are , (repeated twice), for free boundary condition, and for fixed boundary condition. Note that the natural frequencies of the unit cell are normalized using
Table 3.3 Effective elastic properties of the microstructures in Figure 3.11
Chapter 7: Impact and Blast Response of Lattice Materials
Table 7.1 Details of the manufactured lattice structures used in the blast tests
Table 7.4 Summary of the lattice specimens used in the drop-hammer tests
Table 7.2 Properties of the CFRP used in the sandwich panels
Table 7.3 Properties of the strikers used in the blast tests
Table 7.5 Summary of the blast conditions adopted during the lattice tests
Table 7.6 Summary of the blast conditions during the sandwich panel tests
Chapter 10: Topology Optimization of Lattice Materials
Table 10.1 Possible design segments before and after mutation
Table 10.2 Summary of parameters for PnC and MM unit cells
Dynamics of Lattice Materials
Edited by
A. Srikantha Phani
University of British Columbia, Canada
Mahmoud I. Hussein
University of Colorado Boulder, USA
This edition first published 2017
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Library of Congress Cataloging-in-Publication Data
Names: Phani, A. Srikantha, editor. | Hussein, Mahmoud I., editor.
Title: Dynamics of lattice materials / [edited by] A. Srikantha Phani, Mahmoud I. Hussein.
Description: Chichester, West Sussex, United Kingdom : John Wiley & Sons, Inc., 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2016042860 | ISBN 9781118729595 (cloth) | ISBN 9781118729571 (epub) | ISBN 9781118729564 (Adobe PDF)
Subjects: LCSH: Lattice dynamics.
Classification: LCC QC176.8.L3 D85 2017 | DDC 530.4/11-dc23 LC record available at https://lccn.loc.gov/2016042860
Cover design by Wiley
Cover image: Courtesy of the author
To Ananya and Krishna To Alaa and Ismail
List of Contributors
M. Arya
University of Toronto
Ontario
Canada
S. Arabnejad
McGill University
Montreal
Quebec
Canada
K. Bertoldi
Harvard University
Cambridge
Massachusetts
USA
O.R. Bilal
University of Colorado Boulder
Colorado
USA
W.J. Cantwell
Khalifa University of Science
Technology and Research
Abu Dhabi
UAE
F. Casadei
Harvard University
Cambridge
Massachusetts
USA
Z.W. Guan
University of Liverpool
UK
G. Hibbard
University of Toronto
Ontario
Canada
G.M. Hulbert
University of Michigan
Ann Arbor
USA
M.I. Hussein
University of Colorado Boulder
Colorado
USA
D. Krattiger
University of Colorado Boulder
Colorado
USA
A.T. Lausic
University of Toronto
Ontario
Canada
M.J. Leamy
Georgia Institute of Technology
Atlanta
USA
K. Manktelow
Georgia Institute of Technology
Atlanta
USA
A.N. Norris
Rutgers University
Piscataway
New Jersey
USA
D. Pasini
McGill University
Montreal
Quebec
Canada
M. Ruzzene
Georgia Institute of Technology
Atlanta
USA
M. Smith
University of Sheffield
Rotherham
UK
A.S. Phani
University of British Columbia
Vancouver
Canada
C.A. Steeves
University of Toronto
Ontario
Canada
C. Yilmaz
Bogazici University
Istanbul
Turkey
P. Wang
Harvard University
Cambridge
Massachusetts
USA
Foreword
When Srikantha Phani and Mahmoud Hussein asked me if I would write a foreword to their book, I was at first a bit hesitant as I was very busy working on a new book (Extending the Theory of Composites to Other Areas of Science, now submitted for publication with chapters coauthored by Maxence Cassier, Ornella Mattei, Mordehai Milgrom, Aaron Welters and myself). But then, when I saw the high quality of the chapters submitted by various people, I was happy to agree.
In 1928 the doctoral thesis of Felix Bloch established the quantum theory of solids, using Bloch waves to describe the electrons. Following this, in 1931–1932, Alan Herries Wilson explained how energy bands of electrons can make a material a conductor, a semiconductor or an insulator. Subsequently there was a tremendous effort directed towards calculating the electronic properties of crystals by calculating their band structure; that is, through solving Schrödinger's equation in a periodic system. So it is rather surprising that it took until the late 1980s for similar calculations to be done for wave equations in man-made periodic structures (with the exception of the layered materials that Lord Rayleigh in 1887 had shown exhibited a band gap). Subsequently there was exponentially growing interest in the subject, as illustrated by the graph in the extensive “Photonic and sonic band gap and metamaterial bibliography” of Jonathan Dowling [1], which he maintained until 2008. Now there seems to be a similar migration of ideas from people who have studied topological insulators in the context of the quantum 2D Hall effect to the study of similar effects in man-made periodic structures where there is some time-symmetry breaking. In the context of elasticity this time-symmetry breaking can be achieved with gyroscopic metamaterials [2] and, most significantly, waves can only travel in one direction around the boundary.
This book sheds light on the dynamics of lattice materials from different perspectives. As I read through it, connections with other work (sometimes mine) came to mind. I suspect this is probably a reflection of my background, as the writer (or writers) may have been exposed to different schools of thought than myself, but I believe the cross-pollination of ideas is always beneficial to the advancement of science. Therefore I hope the collection of remarks I have made here will lead the reader (if they have the time to explore the references I have given) on some excursions of the mind that complement those provided by the authors of the individual chapters.
Chapter 1 provides the setting for the book, giving a brief but excellent introduction to lattice materials. Maxwell's rule for determining the stiffness of a structure is discussed and, as the authors mention, Maxwell realized this is only a necessary condition for a structure to be stiff. The exact condition is nontrivial to determine, but in a 2D structure one can play the “pebble game” to resolve thequestion [3]. Maxwell's counting rule has been generalized for periodic lattices [4]. Perhaps there is a generalization (maybe an obvious one) of the “pebble game” to periodic 2D lattices, but I have not fully explored the literature. The possible motions of kinematically indeterminant periodic arrays of rigid rods with flexible joints are of considerable interest to me, and the case in which the macroscopic motions are affine is described in the literature [5, 6], and references therein.
Pasini and Arabnejad's Chapter 2 provides an excellent survey of homogenization methods for the elastostatics of lattice materials. This is still very much an active area of research. It is an important one, because not only do the homogenized equations govern the macroscopic response but also, as emphasized by Pasini and Arabnejad, the solution of the so-called cell-problem (that is needed to calculate the effective moduli) can provide useful estimates of the maximum fields in the material, which are helpful in knowing if plastic yielding or cracking might occur. While Pasini and Arabnejad's review concentrates on periodic lattice materials, it is worth mentioning that, curiously, for random composites the justification of successive terms in the asymptotic expansions, such as their Eq. (2.12), requires successively higher dimensions of space [[7], and references therein]; while 2D and 3D composites are those of practical interest, one may of course think of composites in higher dimensions too. Also, it is important to remember that with high-contrast linear elastic materials one can theoretically achieve almost any homogenized response compatible with the natural constraint of positivity of the elastic energy [8]: non-local interactions in the homogenized equations can be achieved with dumbbell shaped inclusions where the diameter of the bar is so small that it does not couple with the surrounding medium except in the near vicinity of the bar. These results are only in the framework of linear elasticity, because such bars can easily buckle when the dumbbell is under compression. Some beautiful examples of exotic elastic behavior, which go beyond that of Cosserat theory, are given by Seppecher, Alibert, and Dell Isola [9].
Chapter 3, by Phani, gives a great introduction to the elastodynamics of lattice materials. I especially like their use of simple mass-spring models. My coauthors and I find mass-spring models, with the addition of rigid elements, to be very helpful in explaining concepts such as negative effective mass, anisotropic mass density, and (when the springs have some viscous damping) complex effective mass density [10, 11]. In fact it is possible (with the framework of linear elasticity) to give a complete characterization of the possible dynamic responses of multiterminal mass-spring networks [12]. The presentation by Phani of the deformation modes associated with the branches in the dispersion diagram in Figures 3.13–3.17 is beautiful, and sheds a lot more light on the behavior than is contained in dispersion curves, which frequently is all most scientists present. Also, I would mention that a dramatic illustration of the directionality of wave propagation is in phonon focussing [13]. If at low temperatures one heats acrystal from below by directing a laser at a point on the surface, then the distribution of heat on the top surface (as seen by the height of liquid helium on the surface that, due to the fountain effect, flows towards the heat) has amazing patterns, due to caustics in the “slowness” surface associated with the direction of elastic wave propagation in crystals that is governed simply by the elasticity tensor of the crystal. The elastic waves carry the heat (phonons). It is worth remarking that, subsequent to pioneering work by Bensoussan, Lions, and Papanicolaou in Chapter 4 of their book [14], there has been a resurgence of interest in high-frequency homogenization at stationary points in the dispersion diagram, which may be local minima or maxima, or even saddle points [15–21]. The wave is a modulated Bloch wave and modulation satisfies appropriate effective equations. The most interesting effects occur when one has a saddle point: then the effective equation is hyperbolic and there are associated characteristic directions. One may also employ homogenization techniques for travelling waves at other points in the dispersion diagram [22–26].
Chapter 4, by Krattiger, Phani, and Hussein examines wave propagation in damped lattice materials, both for passive waves and driven waves. One rarely sees dispersion diagrams with damping, but of course for many materials damping is a significant factor. Their dispersion diagrams with driven waves (Figures 4.2 and 4.4) have an interesting and complex structure. It is interesting that some periodic materials with damping can have trivial dispersion relations, with a dispersion diagram equivalent to that of a homogeneous damped material [27, 28]: this happens when the moduli are analytic functions, not of the frequency, but of the complex variable , where and for a 2D material and are the Cartesian spatial coordinates. Closely related materials were discovered by Horsley, Artoni, and La Rocca, who realized they would not reflect radiation incident from one side, whatever the angle of incidence [29].
In Chapter 5 by Manktelow, Ruzzene, and Leamy we encounter the exciting topic of wave propagation in nonlinear lattice materials. The study of nonlinear effect in composites is largely a wide-open area of research: there are so many interesting and novel directions that could be explored, and it is a certainty that surprises await. One surprise we found is as follows [30]. When one mixes linear conducting composites in fixed proportions, if one wants to maximize the current in the direction of the electric field then it is best to layer the materials with the layer boundaries parallel to the applied field; by contrast, in some nonlinear materials we found that the maximum current sometimes occurs when the layer interfaces are normal to the applied field. Manktelow, Ruzzene, and Leamy talk about higher harmonic generation in nonlinear materials. Anyone who has used an inexpensive green laser may be interested to know that the green light comes from frequency doubling the infrared light from a neodymium-ion oscillator as it passes through a nonlinear crystal, and this can pose adanger if the conversion is faulty because the infrared light can easily damage eyes [31].
Chapter 6 by Casadei, Wang and Bertoldi also deals with nonlinearity, but in the context of buckling creating a pattern transformation that can be used to tune the propagation of elastic waves. This is fantastic work, and in an entirely new direction. Buckling instabilities are well known in Bertoldi's group: they created the Buckliball a structured sphere that remains approximately spherical, but much reduced in size, as it buckles [32]. Much remains to be explored in this area: one especially significant result that I have found is that materials that combine a stable phase with an unstable one could have a stiffness greater than diamond in dynamic bending experiments [33]. It had been hoped that one could get stiffnesses dramatically higher than that of the components in stable static materials too [34], but this was ruled out when it was realized that the well-known elastic variational principles still hold even when some of the components are in isolation unstable (that is, they have negative elastic moduli) [35] .
I found interesting the work in Chapter 7 of Smith, Cantwell, and Guan on the impact and blast response of lattice materials. A feature of their experiments is that the stress has a plateau as the lattice structure is crumpled. This is exactly what one needs if the aim is to minimize the maximum force felt by an object colliding with the structure, subject to the constraint that the object should decelerate over a fixed distance. We recently encountered similar questions when trying to determine the optimal non-linear rope for a falling climber [36]. The answer turned out to be a “rope” with a stress plateau, like a shape memory wire (and with a big hysterisis loop to absorb the energy). It is pretty amazing to see the progress that has been made recently with impact-resistant composites: a good example is the composite metal foam of Afsaneh Rabiei, which literally obliterates bullets [37].
Pentamode materials, as discussed by Norris in Chapter 8, are a class of materials close to my heart. When we invented them, back in 1995 [38], we never dreamed they would actually be made, but that is exactly what the group of Martin Wegener did, in an amazing feat of 3D lithography [39]. Their lattice structure is similar to diamond, with a stiff double-cone structure replacing each carbon bond. This structure ensures that the tips of four double-cone structures meet at each vertex. This is the essential feature: treating the double-cone structures as struts, the tension in one determines uniquely the tension in the other three. This is simply balance of forces. Thus the structure as a whole can essentially only support one stress, but that stress can be any desired symmetric matrix if the pentamode lattice structure is appropriately tailored. Water is a bit like a pentamode, but unlike water, which can only support a hydrostatic stress, pentamodes can support any desired stress matrix, in other words, a desired mixture of shear and compression. They are the building blocks for constructing any desired elasticity matrix that is positive definite. Elasticity tensors of 3D materials are actually fourth-order tensors, specifically linear maps on the space of symmetric matrices, but using a basis on the 6D space of symmetric matrices, they can be represented by a 6-by-6 matrix as is common in engineering notation. Expressing in terms of its eigenvectors and eigenvalues,
The idea, roughly speaking, is to find six pentamode structures, each supporting a stress represented by the vector , . The stiffness of the material and the necks of the junction regions at the vertices need to be adjusted so each pentamode structure has an effective elasticity tensor close to
Then one successively superimposes all these six pentamode structures, with their lattice structures being offset to avoid collisions. Additionally, one may need to deform the structures appropriately to avoid these collisions [38], and when one does this it is necessary to readjust the stiffness of the material in the structure to maintain the value of . Then the remaining void in the structure is replaced by an extremely compliant material. Its presence is just needed for technical reasons, to ensure that the assumptions of homogenization theory are valid so that the elastic properties can be described by an effective tensor. But it is so compliant that essentially the effective elasticity tensor is just a sum of the effective elasticity tensors of the six pentamodes; in other words, the elastic interaction between the six pentamodes is neglible. In this way we arrive at a material with (approximately) the desired elasticity tensor . Now, Andrew Norris and the group of Martin Wegener have become the leading experts on pentamodes and their 2D equivalents, which strictly speaking should be called bimodes. One important observation that Norris makes (see his Eq. (8.5)) is that if a pentamode is macroscopically inhomogeneous then the stress field it supports should be divergence-free in the absence of body forces such as gravitational forces. The new and important ingredient in the chapter of Norris is the analytic inclusion of bending effects, to better analyse the elements of the effective elasticity tensor.
Chapter 9, by Krattinger and Hussein, uses a reduced number of modes in a Bloch mode expansion to treat the vibration of plates within a frequency range ofinterest. Expanding on the ideas of structural mechanics, where one splits a structure into substructures, conducts a modal analysis on each of these, and then links the modes through interface boundary conditions, they develop a similar procedure at the unit-cell level for very efficiently calculating the band structure, which they call “Bloch mode synthesis.” I very much like the word “platonic crystal” [40] – crafted after the terms photonic crystals, phononic crystals, and plasmonic crystals – which Ross McPhedran coined for such studies of the propagation of flexural waves through plates with periodic structure. The term has caught on in Australia, France, New Zealand and the UK (where Ross is a frequent visitor) but not yet in the U.S.
Chapter 10 by Bilal and Hussein deals with topology optimization of lattice materials. Their pixel-based designs remind me very much of the digital metamaterials of my colleague Rajesh Menon (also produced by topology optimization, but in the context of electromagnetism rather than elasticity), which have been incredibly successful, for example resulting in the world's smallest polarization beam-splitter [41]. The field of topology optimization has seen some amazing achievements, producing stuctures with fascinating and sometimes unexpected geometries that optimize performance in some respect. In particular, the group of Ole Sigmund in Denmark is well known for mastering this art, and recently they have used it for acoustic design [42]; the next wave of symphony halls will probably use the technique in their designs.
Chapter 11
