Poisson s ratio of rectangular anti-chiral lattices with disorder Artur A. Poźniak 1 Krzysztof W. Wojciechowski 2 1 Institute of Physics, Poznań Universty of Technology, ul. Nieszawska 13A, 60-695 Poznań, 2 Institute of Molecular Physics PAS, ul. M. Smoluchowskiego 17, 60-179 Poznań 17 th September 2014 AUXETICS 2014 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 1 / 23
Instead of the outline A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 2 / 23
Instead of the outline A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 2 / 23
Instead of the outline A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 2 / 23
Motivation What is an auxetic? ν < 0 (Negtative Poisson s Ratio) is for mechanics what negative refractive index is for optics n < 0. Auxetics are metamaterials. Beneficial features from NPR Resistance to shape change and indentation; crack resistance; better vibration absorption (including acoustic one); synclastic curvature, different dynamics. Applications of NPR materials Medicine (stents, bandages, implants), defence (energy absorption), furniture industry (better mattresses indentation), automotive industry and sports (safety belts). A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 3 / 23
Poisson s ratio (stretching along x axis) W H Formal definition ν xy = ǫ yy ǫ xx W 0 H 0 Engineering definition ν = H H 0 W W 0 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 4 / 23
Source: On the design of 1-3 piezocomposites using topology optimization, O. Sigmund, S. Torquato, I.A. Aksay, Journal of Materials Research 13, 4, 1040-1048 (1998) Source: Hydrophone. (2014, February 28). In Wikipedia, The Free Encyclopedia. Retrieved 10:37, April 9, 2014 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 5 / 23
Source: On the design of 1-3 piezocomposites using topology optimization, O. Sigmund, S. Torquato, I.A. Aksay, Journal of Materials Research 13, 4, 1040-1048 (1998) The first article describing anti-chiral structures as NPR material. Cross sections different mechanisms. A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 5 / 23
Experimental setup Source: Elasticity of anti-tetrachiral anisotropic lattices, Y.J. Chen et al., International Journal of Solids and Structures 50, 996-1004 (2013) A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 6 / 23
Source: Composite chiral structures for morphing airfoils: Numerical analyses and development of a manufacturing process, P. Bettini et al., Compisites: Part B 41, 2, 133-147 (2010) A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 7 / 23
Geometry L y is a unit length. L x /L y l y, T/L y t; taking L y = 1 l x = L x being the anisotropy parameter. A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 8 / 23
Finite Element Method Approximate method for solving PDEs. Physical discretization. Various shape functions available. A system of PDEs very large system of algebraic equations. Arbitrary precision depending on t and computational resources ($). Variety of libraries (C++, Python) and proprietary software. Here Abaqus/STANDARD was employed (linear static elasticity). A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 9 / 23
Finite Element Method Approximate method for solving PDEs. Physical discretization. Various shape functions available. A system of PDEs very large system of algebraic equations. Arbitrary precision depending on t and computational resources ($). Variety of libraries (C++, Python) and proprietary software. Here Abaqus/STANDARD was employed (linear static elasticity). planar elements (CPS3) Timoshenko beam-type element(b21) A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 9 / 23
Disorder A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 10 / 23
Disorder r(= 0.3) r + r δ, r δ δ, δ, δ δ max (r, L y ) topology remains unchanged! A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 10 / 23
Disorder r(= 0.3) r + r δ, r δ δ, δ, δ δ max (r, L y ) topology remains unchanged! A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 10 / 23
Disorder r(= 0.3) r + r δ, r δ δ, δ, δ δ max (r, L y ) topology remains unchanged! A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 10 / 23
Disorder Minor topology changes are possible r(= 0.3) r + r δ, r δ δ, δ, δ δ max (r, L y ) topology remains unchanged! A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 10 / 23
Boundary conditions U x is a placement-type, orange x zero stress condition PBC easier for B21, CPS3 elems. require more equations no rotational DOFs A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 11 / 23
Convergence as a function of the mesh element size CPS3 νxy -0.6-0.7-0.8 n m 171513 11 9 7 5 t = 0.05 t = 0.10 t = 0.15 t = 0.20 1 2 3 4 5 6 7-0.9-1.0 0.00 0.05 0.10 0.15 0.20 1 n m Figure: l x L x/l y = 1.0 Filled symbol denotes mesh size chosen for further calculations corresponding to n x = 9. The figure on the right shows the case for n x = 7 for clarity reasons. 1 2 3 4 5 6 7 Figure: Mesh for n m = 7 (t = 0.1) A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 12 / 23
Convergence as a function of the mesh element size B21 Figure: Mesh for d m = 0.01 (t = 0.1) Figure: l x L x/l y = 1.0 Filled symbol denotes mesh size chosen for further calculations corresponding to d m = 0.01. A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 13 / 23
Convergence as a function of the size of the sample n x -0.4 16 876 5 4 3 2 1-0.5 t = 0.05 t = 0.20-0.6 νxy -0.7-0.8 r = 0.3,δ = 0.19 l x = 1 (a) n x = 4-0.9-1 0 0.2 0.4 0.6 0.8 1 1/n x Figure: Averaged ν xy for samples of size 1 1 to 16 16 of elementary units. The anisotropy parameter l x = 1, r = 0.3 with δ = 0.19 (almost maximal possible disorder). 5 samples for each average (b) n x = 16 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 14 / 23
0.2 0.0-0.2 CPS3 B21, elastic RC B21, rigid RC νxy -0.4-0.6-0.8-1.0 0 0.05 0.1 0.15 0.2 0.25 0.3 t (a) r = 0.15 (b) r = 0.40 Figure: The influence of rib s thickness on the Poiison s ratio for 2 radii of RC: r = 0.15 (a) and r = 0.4 (b) CPS3 planar elements, B21 elastic RC Timoshenko beams with deformable RCs, B21 rigid RC Timoshenko beams with rigid RCs, A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 15 / 23
Poisson s ratio as a function of the anisotropy νxy -1-2 -3-4 lx 1/lx δ -0.2-0.4-0.6-0.8 0.01 0.01 0.10 0.10 0.19 0.19-1.0 1.0 1.5 2.0 3.0 4.0 l x (a) t = 0.05 νyx A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 16 / 23
Poisson s ratio as a function of the anisotropy -1-0.2-1 -0.2-0.4-0.4 νxy -2-3 -4 lx 1/lx δ -0.6-0.8 0.01 0.01 0.10 0.10 0.19 0.19-1.0 1.0 1.5 2.0 3.0 4.0 l x (a) t = 0.05 νyx νxy -2-3 -4 lx 1/lx δ -0.6-0.8 0.01 0.01 0.10 0.10 0.19 0.19-1.0 1.0 1.5 2.0 3.0 4.0 l x (b) t = 0.2 νyx A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 16 / 23
(a) Reference state (b) Sample stretched along the X axis Figure: l x = 4, ν 4 1 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 17 / 23
Positional disorder introduction A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 18 / 23
Positional disorder introduction A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 18 / 23
Positional disorder introduction νxy -0.70-0.75-0.80-0.85-0.90-0.95-1.00 n x 8 4 2 1 δ/δmax = 0.5 δ/δmax = 1.0 lx = 1 νxy 100 10 lx = 16 lx = 8 r = 0.1 lx = 4 lx = 2 νxy νxy νxy νxy -1.40-1.60-1.80-2.00-2.20-3.50-4.50-5.50-6.50-7.0-11.0-15.0-19.0-23.0-20.0-40.0-60.0 lx = 2 lx = 4 lx = 8-80.0 lx = 16-100.0 0 0.2 0.4 1/n 0.6 0.8 1 x νxy νxy 1 0.01 0.1 1 δ/δ max 100 10 lx = 16 lx = 8 r = 0.2 lx = 4 lx = 2 1 0.01 0.1 1 δ/δ max 100 10 lx = 16 lx = 8 r = 0.4 lx = 4 lx = 2 1 0.01 0.1 1 δ/δ max A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 19 / 23
Technical issues Periodic mesh! Generation of the mesh when l x 1 in the case of planar elements(cps3) requires an introduction of artificial cuts. This is time-consuming. A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 20 / 23
A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 21 / 23
A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 21 / 23
A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 21 / 23
A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 21 / 23
Conclusions Changing the anisotropy parameter (l x ) one can tune ν. ν can reach any negative value. For thin ribs the Timoshenko beam elements approximation gives satisfactory convergence. The disorder introduced by random RC radii has a negligible impact on ν. Anti chiral structures with rectangual symmetry are effective strain amplifiers with ν 1. Thiner ribs give lower ν but the effective structure s stiffness is also decreased. This work was supported by the (Polish) National Centre for Science under the grant NCN 2012/05/N/ST5/01476. Part of the simulations was performed at the Poznań Supercomputing and Networking Center (PCSS). Special thanks to prof. K.W. Wojciechowski A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 22 / 23
Conclusions Changing the anisotropy parameter (l x ) one can tune ν. ν can reach any negative value. For thin ribs the Timoshenko beam elements approximation gives satisfactory convergence. The disorder introduced by random RC radii has a negligible impact on ν. Anti chiral structures with rectangual symmetry are effective strain amplifiers with ν 1. Thiner ribs give lower ν but the effective structure s stiffness is also decreased. This work was supported by the (Polish) National Centre for Science under the grant NCN 2012/05/N/ST5/01476. Part of the simulations was performed at the Poznań Supercomputing and Networking Center (PCSS). Special thanks to prof. K.W. Wojciechowski A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 22 / 23
Conclusions Changing the anisotropy parameter (l x ) one can tune ν. ν can reach any negative value. For thin ribs the Timoshenko beam elements approximation gives satisfactory convergence. The disorder introduced by random RC radii has a negligible impact on ν. Anti chiral structures with rectangual symmetry are effective strain amplifiers with ν 1. Thiner ribs give lower ν but the effective structure s stiffness is also decreased. This work was supported by the (Polish) National Centre for Science under the grant NCN 2012/05/N/ST5/01476. Part of the simulations was performed at the Poznań Supercomputing and Networking Center (PCSS). Special thanks to prof. K.W. Wojciechowski A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 22 / 23
Daily inspirations melting snow is reentrant c Prof. K.W. Wojciechowski A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 23 / 23
Daily inspirations melting snow is reentrant c Prof. K.W. Wojciechowski Thank you A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep 2014 23 / 23