Poisson s ratio of rectangular anti-chiral lattices with disorder

Transcription

1 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, Poznań, 2 Institute of Molecular Physics PAS, ul. M. Smoluchowskiego 17, Poznań 17 th September 2014 AUXETICS 2014 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

2 Instead of the outline A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

3 Instead of the outline A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

4 Instead of the outline A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

5 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 / 23

6 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 / 23

7 Source: On the design of 1-3 piezocomposites using topology optimization, O. Sigmund, S. Torquato, I.A. Aksay, Journal of Materials Research 13, 4, (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 / 23

8 Source: On the design of 1-3 piezocomposites using topology optimization, O. Sigmund, S. Torquato, I.A. Aksay, Journal of Materials Research 13, 4, (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 / 23

9 Experimental setup Source: Elasticity of anti-tetrachiral anisotropic lattices, Y.J. Chen et al., International Journal of Solids and Structures 50, (2013) A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

10 Source: Composite chiral structures for morphing airfoils: Numerical analyses and development of a manufacturing process, P. Bettini et al., Compisites: Part B 41, 2, (2010) A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

11 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 / 23

12 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 / 23

13 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 / 23

14 Disorder A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

15 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 / 23

16 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 / 23

17 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 / 23

18 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 / 23

19 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 / 23

20 Convergence as a function of the mesh element size CPS3 νxy n m t = 0.05 t = 0.10 t = 0.15 t = 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 Figure: Mesh for n m = 7 (t = 0.1) A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

21 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 = A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

22 Convergence as a function of the size of the sample n x t = 0.05 t = νxy r = 0.3,δ = 0.19 l x = 1 (a) n x = /n x Figure: Averaged ν xy for samples of size 1 1 to 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 / 23

23 CPS3 B21, elastic RC B21, rigid RC νxy 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 / 23

24 Poisson s ratio as a function of the anisotropy νxy lx 1/lx δ l x (a) t = 0.05 νyx A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

25 Poisson s ratio as a function of the anisotropy νxy lx 1/lx δ l x (a) t = 0.05 νyx νxy lx 1/lx δ l x (b) t = 0.2 νyx A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

26 (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 / 23

27 Positional disorder introduction A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

28 Positional disorder introduction A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

29 Positional disorder introduction νxy n x δ/δmax = 0.5 δ/δmax = 1.0 lx = 1 νxy lx = 16 lx = 8 r = 0.1 lx = 4 lx = 2 νxy νxy νxy νxy lx = 2 lx = 4 lx = lx = /n x νxy νxy δ/δ max lx = 16 lx = 8 r = 0.2 lx = 4 lx = δ/δ max lx = 16 lx = 8 r = 0.4 lx = 4 lx = δ/δ max A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

30 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 / 23

31 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

32 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

33 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

34 A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

35 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/ 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 / 23

36 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/ 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 / 23

37 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/ 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 / 23

38 Daily inspirations melting snow is reentrant c Prof. K.W. Wojciechowski A.A. Poźniak, K.W. Wojciechowski ν of disordered anti-chirals 17 Sep / 23

39 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 / 23