Dynamics of harmonically excited irregular cellular metamaterials S. Adhikari 1 , T. Mukhopadhyay 2 , A. Al` u 3 1 Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Bay Campus, Swansea, Wales, UK, Email: S.Adhikari@swansea.ac.uk , Twitter: @ProfAdhikari , Web: http://engweb.swan.ac.uk/~adhikaris 1 Department of Engineering Science, University of Oxford, Oxford, UK 3 Cockrell School of Engineering, The University of Texas at Austin, Austin, USA META’17, the 8th International Conference on Metamaterials, Photonic Crystals and Plasmonics Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 1
Outline 1 Introduction Static homogenised properties 2 Unit cell deformation using the stiffness matrix 3 Dynamic homogenised properties 4 5 Results Conclusions 6 Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 2
Introduction Lattice based metamaterials Metamaterials are artificial materials designed to outperform naturally occurring materials in various fronts. We are interested in mechanical metamaterials - here the main concern is in mechanical response of a material due to applied forces Lattice based metamaterials are abundant in man-made and natural systems at various length scales Among various lattice geometries (triangle, square, rectangle, pentagon, hexagon), hexagonal lattice is most common This talk is about in-plane elastic properties of 2D hexagonal lattice materials - commonly known as “honeycombs” Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 3
Introduction Mechanics of lattice materials Honeycombs have been modelled as a continuous solid with an equivalent elastic moduli throughout its domain. This approach eliminates the need of detail numerical (finite element) modelling in complex structural systems like sandwich structures. Extensive amount of research has been carried out to predict the equivalent elastic properties of regular honeycombs consisting of perfectly periodic hexagonal cells (Gibson and Ashby, 1999). Analysis of two dimensional honeycombs dealing with in-plane elastic properties are commonly based on the assumption of static forces In this work, we are interested in in-plane elastic properties when the applied forces are dynamic in nature Dynamic forcing is relevant, for example, in helicopter/wind turbine blades, aircraft wings where light weight and high stiffness is necessary Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 4
Static homogenised properties Equivalent elastic properties of regular honeycombs Unit cell approach - Gibson and Ashby (1999) (a) Regular hexagon ( θ = 30 ◦ ) (b) Unit cell We are interested in homogenised equivalent in-plane elastic properties This way, we can avoid a detailed structural analysis considering all the beams and treat it as a material Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 5
Static homogenised properties Equivalent elastic properties of regular honeycombs The cell walls are treated as beams of thickness t , depth b and Young’s modulus E s . l and h are the lengths of inclined cell walls having inclination angle θ and the vertical cell walls respectively. The equivalent elastic properties are: � 3 � t cos θ E 1 = E s (1) l + sin θ ) sin 2 θ l ( h � 3 ( h l + sin θ ) � t E 2 = E s (2) cos 3 θ l cos 2 θ ν 12 = (3) ( h l + sin θ ) sin θ ν 21 = ( h l + sin θ ) sin θ (4) cos 2 θ � h � 3 � l + sin θ � t G 12 = E s (5) � h � 2 ( 1 + 2 h l l ) cos θ l Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 6
Static homogenised properties Finite element modelling and verification A finite element code has been developed to obtain the in-plane elastic moduli numerically for honeycombs. Each cell wall has been modelled as an Euler-Bernoulli beam element having three degrees of freedom at each node. For E 1 and ν 12 : two opposite edges parallel to direction-2 of the entire honeycomb structure are considered. Along one of these two edges, uniform stress parallel to direction-1 is applied while the opposite edge is restrained against translation in direction-1. Remaining two edges (parallel to direction-1) are kept free. For E 2 and ν 21 : two opposite edges parallel to direction-1 of the entire honeycomb structure are considered. Along one of these two edges, uniform stress parallel to direction-2 is applied while the opposite edge is restrained against translation in direction-2. Remaining two edges (parallel to direction-2) are kept free. For G 12 : uniform shear stress is applied along one edge keeping the opposite edge restrained against translation in direction-1 and 2, while the remaining two edges are kept free. Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 7
Static homogenised properties Finite element modelling and verification 1.2 E 1 E 2 1.15 ν 12 Ratio of elastic modulus 1.1 ν 21 G 12 1.05 1 0.95 0.9 0 500 1000 1500 2000 Number of unit cells θ = 30 ◦ , h / l = 1 . 5. FE results converge to analytical predictions after 1681 cells. Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 8
Unit cell deformation using the stiffness matrix The deformation of a unit cell (a) Deformed shape and free body diagram under the application of stress in direction - 1 (b) Deformed shape and free body diagram under the application of stress in direction - 2 (c) Deformed shape and free body diagram under the application of shear stress (The undeformed shapes of the hexagonal cell are indicated using blue colour for each of the loading conditions. Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 9
Unit cell deformation using the stiffness matrix The element stiffness matrix of a beam The equation governing the transverse deflection V ( x ) of the beam can be expressed as EI d 4 V ( x ) = f ( x ) (6) dx 4 It is assumed that the behaviour of the beam follows the Euler-Bernoulli hypotheses Using the finite element method, the element stiffness matrix of a beam can be expressed as A 11 A 12 A 13 A 14 12 6 L − 12 6 L 4 L 2 2 L 2 A 21 A 22 A 23 A 24 = EI 6 L − 6 L A = (7) − 6 L 2 A 31 A 32 A 33 A 34 − 12 − 6 L 12 L 3 2 L 2 4 L 2 A 41 A 42 A 43 A 44 6 L − 6 L ¯ bt 3 The area moment of inertia I = 12 Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 10
Unit cell deformation using the stiffness matrix Equivalent elastic properties Young’s modulus E 1 : E 1 = σ 1 A 33 l cos θ = A 33 cos θ = (8) b sin 2 θ l + sin θ ) sin 2 θ ¯ ( h + l sin θ )¯ ǫ 11 ( h b Young’s modulus E 2 : ( h l + sin θ ) E 2 = σ 2 = A 33 ( h + l sin θ ) = A 33 (9) b cos 3 θ cos 3 θ l ¯ ¯ ǫ 22 b Shear modulus G 12 : G 12 = τ 1 γ = 2 (10) 2 l ¯ b cos θ h 2 43 + 4 lA s 33 − A v 34 A v ( h + l sin θ ) � � A v 43 A v 44 ( • ) v = vertical element; ( • ) s = slant element Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 11
Dynamic homogenised properties Dynamic equivalent proerties (a) Typical representation of a hexagonal lattice structure in a dynamic environment (e.g., the honeycomb as part of a host structure experiencing wave propagation). (b) One hexagonal unit cell under dynamic environment (c) A dynamic element for the bending vibration of a damped beam with length L . It has two nodes and four degrees of freedom. Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 12
Dynamic homogenised properties Dynamic stiffness matrix Individual elements of the lattice have been considered as damped Euler-Bernoulli beams with the equation of motion EI ∂ 4 V ( x , t ) ∂ 5 V ( x , t ) + m ∂ 2 V ( x , t ) ∂ V ( x , t ) + c 1 + c 2 = 0 (11) ∂ x 4 ∂ x 4 ∂ t ∂ t 2 ∂ t Using the dynamic finite element method, the element stiffness matrix of a beam can be expressed as − b 2 ( cS + Cs ) b 2 ( S + s ) − b ( C − c ) − sbS − sbS − Cs + cS b ( C − c ) − S + s EIb A = b 2 ( S + s ) − b 2 ( cS + Cs ) ( cC − 1 ) b ( C − c ) sbS − b ( C − c ) − S + s sbS − Cs + cS (12) where C = cosh ( bL ) , c = cos ( bL ) , S = sinh ( bL ) and s = sin ( bL ) (13) b 4 = m ω 2 ( 1 − i ζ m /ω ) ; ζ k = c 1 / ( EI ) , ζ m = c 2 / m (14) EI ( 1 + i ωζ k ) Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 13
Dynamic homogenised properties Equivalent dynamic elastic properties Young’s modulus E 1 : Et 3 ( cS + Cs ) b 3 l cos θ E 1 = (15) 12 ( h + l sin θ ) sin 2 θ ( − 1 + cC ) Young’s modulus E 2 : E 2 = Et 3 ( cS + Cs ) b 3 ( h + l sin θ ) (16) 12 l cos 3 θ ( − 1 + cC ) Shear modulus G 12 : Et 3 ( h + l sin θ ) G 12 = � (17) � h 2 ( c s C s − 1 ) ( c v C v − 1 ) ( c v S v − C v s v ) 48 l cos θ + b 3 ( C v 2 s v 2 − c v 2 S v 2 − s v 2 S v 2 ) 8 ls s S s b 2 C s = cosh ( bl ) , c s = cos ( bl ) , S s = sinh ( bl ) , s s = sin ( bl ) , C v = � bh � � bh � � bh � � bh � c v = cos S v = sinh s v = sin cosh , , and 2 2 2 2 Adhikari (Swansea) Dynamics of harmonically excited cellular metamaterials July 26, 2017 14
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