Planar auxeticity from various inclusions

Planar auxeticity from various inclusions Artur A. Poźniak 1 Krzysztof W. Wojciechowski 2 Joseph N. Grima 3 Luke Mizzi 4 1 Institute of Physics, Poznań Universty of Technology, Piotrowo 3, 6-695 Poznań 2 Institute of Molecular Physics PAS, M. Smoluchowskiego 17, 6-179 Poznań 3 Department of Chemistry, Faculty of Science, University of Malta, Msida, MSD 28 4 Metamaterials Unit, Faculty of Science, University of Malta, Msida, MSD 28 15 th September 215 AUXETICS 215 Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 1 / 23

Motivation What is an auxetic? ν < (Negtative Poisson s Ratio) is for mechanics what negative refractive index is for optics n <. 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). Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 2 / 23

Poisson s ratio (stretching along x axis) W H Formal definition ν xy = ǫ yy ǫ xx = s yyxx s xxxx W H Engineering definition ν = H H W W Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 3 / 23

Motivation rotating units Grima, J. N., Alderson, A., & Evans, K. E. (25). Auxetic behaviour from rotating rigid units. Physica Status Solidi (B), 242(3), 561 575. Grima, J. N., Chetcuti, E., Manicaro, E., Attard, D., Camilleri, M., Gatt, R., & Evans, K. E. (211). On the auxetic properties of generic rotating rigid triangles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2139), 81 83. Grima, J., Zammit, V., Gatt, R., Alderson, A., & Evans, K. (27). Auxetic behaviour from rotating semi-rigid units. Physica Status Solidi (B), 244(3), 866 882. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 4 / 23

Motivation rotating units Grima, J. N., Alderson, A., & Evans, K. E. (25). Auxetic behaviour from rotating rigid units. Physica Status Solidi (B), 242(3), 561 575. Grima, J. N., Chetcuti, E., Manicaro, E., Attard, D., Camilleri, M., Gatt, R., & Evans, K. E. (211). On the auxetic properties of generic rotating rigid triangles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2139), 81 83. Grima, J., Zammit, V., Gatt, R., Alderson, A., & Evans, K. (27). Auxetic behaviour from rotating semi-rigid units. Physica Status Solidi (B), 244(3), 866 882. Our result from 21 Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 4 / 23

Motivation rotating units Grima, J. N., Alderson, A., & Evans, K. E. (25). Auxetic behaviour from rotating rigid units. Physica Status Solidi (B), 242(3), 561 575. Grima, J. N., Chetcuti, E., Manicaro, E., Attard, D., Camilleri, M., Gatt, R., & Evans, K. E. (211). On the auxetic properties of generic rotating rigid triangles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2139), 81 83. Grima, J., Zammit, V., Gatt, R., Alderson, A., & Evans, K. (27). Auxetic behaviour from rotating semi-rigid units. Physica Status Solidi (B), 244(3), 866 882. Our result from 21 Taylor, M., Francesconi, L., Gerendás, M., Shanian, A., Carson, C., & Bertoldi, K. (214). Advanced Materials, 26(15), 2365 7. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 4 / 23

Motivation rotating units Grima, J. N., Alderson, A., & Evans, K. E. (25). Auxetic behaviour from rotating rigid units. Physica Status Solidi (B), 242(3), 561 575. Grima, J. N., Chetcuti, E., Manicaro, E., Attard, D., Camilleri, M., Gatt, R., & Evans, K. E. (211). On the auxetic properties of generic rotating rigid triangles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468(2139), 81 83. Grima, J., Zammit, V., Gatt, R., Alderson, A., & Evans, K. (27). Auxetic behaviour from rotating semi-rigid units. Physica Status Solidi (B), 244(3), 866 882. Our result from 21 Taylor, M., Francesconi, L., Gerendás, M., Shanian, A., Carson, C., & Bertoldi, K. (214). Advanced Materials, 26(15), 2365 7. Why not fill the gaps? Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 4 / 23

Structures a) Geometry: Periodically arranged ellipses Alternating axes (perpendicular) b) Repeating unit cell (RUC): 4 rotating units Periodic mesh c) Boundary conditions within reduced RUC All boundaries remain straight lines Less computational resources needed Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 5 / 23

Simulations Finite Elements 1 Abaqus/CAE via Python interface generates inp files (wc -l ellipses-uniform-mesh.py gives 64 lines), 2 Abaqus/STANDARD performs simulations generating odb files, 3 Abaqus/CAE extracts important information from each odb file. One run is enough to calculate ν as well as E in one direction: V y the displacement of (an arbitrary) top node E elas elastic deformation energy T denotes the plane-stress thickness. U x known deformation ν eff = V y/h U x /W. (1) E eff = 2 Eelas W T U 2 x H. (2) Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 6 / 23

Meshes: uniform vs. non-uniform The discretization (meshing) is an essential and most challenging step on FE analysis. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 7 / 23

Mesh & convergence Figure: Nonuniform exemplary mesh. Each edge was assigned a bias seed in order to obtain possibly dense mesh in the narrowest (bottleneck) regions. In this case L gap/d min = 8 was chosen in order to make he figure readable. In the simulations this ratio was 1. Fig. on the right shows that the convergence tests were performed up to the value d min = 1 4 Figure: Convergence of Poisson s ratio as a function of d min at fixed d max =.1. Here ν matr = ν incl =.99, E matr = 1. The geometric parameters are: a =.9, b =.89. The convergence displays various characteristics depending on the value of E incl. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 8 / 23

Convergence.6.4.2.2.4.6.8 5 1 2 5 1 2 4 8 1 4 1 2 1 1 2 1 4 1 8 1 6 1 4 1 2 1 1 2 1 4 1 61 6 E incl /E matr Figure: Convergence of ν eff and E eff /E matr as obtained within non-uniform meshing for a =.9, b =.89, ν incl = ν matr =.99 and increasing mesh density. The upper, monotonic curves refer to the right (Young s modulus, red) axis, while the non monotonic curves refer to the left (Poisson s ratio, blue) axis. The numbers in the legend stand for the L gap/d min ratio. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 9 / 23 Eeff/Ematr

Detailed convergence ν.45 1 5 4 3 2 1/ n l 1 5.5.55.6.65.7 E incl = 1 4 uniform mesh Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 1 / 23

Detailed convergence ν.45.5.55.6.65 1 5 4 3 2 E incl = 1 4 non-uniform mesh uniform mesh 1/ n l 1 5.7.5.1.15.2 d max PR PR.52.54.56.58.6.62.5.52.54.56.58..1.2.3.4.5 dmin..5.1.15.2.25.3.35 dmin PR.5.51.52.53.54..5.1.15 dmin PR.525.52.515.51.55.5.495..2.4.6.8.1.12 dmin Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 1 / 23

Detailed convergence ν.45.5.55.6.65 1 5 4 3 2 4 3 E incl = 1 4 non-uniform mesh uniform mesh 1/ n l 1 5.7.5.1.15.2 2 d max 1 PR PR.52.54.56.58.6.62.5.52.54.56.58 1..1.2.3.4.5 2 dmin..5.1.15.2.25.3.35 dmin PR.5.51.52.53 3.54..5.1.15 dmin PR.525.52.515.51.55.5 4.495..2.4.6.8.1.12 dmin Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 1 / 23

Detailed convergence E.75 1 5 4 3 2 1/n l 1 5.7 Eeff/Ematr.65.6.55 E incl = 1 4 non-uniform mesh uniform mesh.5.5.1.15.2 d max Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 11 / 23

Required time vs. nuber of FE layers UNIFORM N-UNIFORM, d max =.2 N-UNIFORM, d max =.1 N-UNIFORM, d max =.5 1 5 1 4 t [s] 1 3 1 2 1 1 2 4 6 1 2 32 64 128 L gap /d min or L gap /d u Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 12 / 23

Memory consumed vs. nuber of FE layers 1 9 UNIFORM N-UNIFORM, d max =.2 N-UNIFORM, d max =.1 N-UNIFORM, d max =.5 memory [MB] 1 7 1 5 1 3 1 1 2 4 6 1 2 32 64 128 L gap /d min or L gap /d u Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 13 / 23

Results Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 14 / 23

Results Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 15 / 23

E incl limits, anisotropy.2.2.4.6.8 L gap=.1 L gap=.2 L gap=.5 L gap=.1 L gap=.2.2.4.6.8 1 a/ b Void limit, E incl c) The dependence of Poisson s ratio as a function of a/b for various values of L gap. Here both ν matr and ν incl are equal. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 16 / 23

E incl limits, anisotropy.2.2.4.6.8 L gap=.1 L gap=.2 L gap=.5 L gap=.1 L gap=.2.2.4.6.8 1 a/ b Void limit, E incl.6.4.2.2.4.6 L gap =.1 L gap =.2 L gap =.5 L gap =.1 L gap =.2.2.4.6.8 1 a/b Rigid limit, E incl c) The dependence of Poisson s ratio as a function of a/b for various values of L gap. Here both ν matr and ν incl are equal. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 16 / 23

Rigid inclusions, f (L gap ).6.4.2.2.4.6.8 a/ b=.5 a/ b=.1 a/ b=.5 a/ b=.1 a/ b=.2 a/ b=.3 a/ b= 1..1.2.3.4.5.6.7.8.9 1 L gap Figure: The dependence of Poisson s ratio as a function of L gap for various values of a/b within the rigid inclusion limit. Here both ν matr and ν incl are equal. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 17 / 23

Figure: The dependence of effective Poisson s ratio as a function of the matrix Poisson s ratio in the limit of infinitely hard inclusions (E incl ). The value of ν incl in this limit has no impact on the effective properties since the inclusions do not deform. The inset figure helps in estimating of the a/b ratio, for which ν eff changes sign. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 18 / 23 Rigid inclusions, f (ν matr ).1.5.5 a =.9,b =.89 a =.14,b =.84 a =.19,b =.79 a =.24,b =.74 a =.29,b =.69 a =.34,b =.64 a =.39,b =.59 a =.44,b =.54 a =.49,b =.49 E incl νmatr =.1.1.1.15.2.3.1 log(a/ b) 1..2.99.49.49 ν matr

Figure: Effective Poisson s ratio, ν eff, (blue colour) and effective Young s modulus, E eff /E matr, (red colour) as functions of E incl /E matr for strongly auxetic matrices and inclusions. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 19 / 23 More and more auxetic constituents.6.4.2.2.4.6.8 ν matr =.9,ν incl =.9 ν matr =.99,ν incl =.99 ν matr =.999,ν incl =.999 ν matr =.9999,ν incl =.9999 1 4 1 2 1 1 2 1 4 1 6 1 8 1 6 1 4 1 2 1 1 2 1 4 1 6 1 8 E incl /E matr Eeff/Ematr

Large deformations.2.4.6.8 % 5% 1% ε 15% 2% 1 4 7 2 5 8 3 6 9 Poisson s ratio Effective Poisson s ratio, ν eff, and effective Young s modulus, E eff /E matr as functions of nominal strain for ν matr = and void inclusions (the material of inclusions is entirely removed). The numbers in the legend correspond to the following geometry parameter pairs: 1) a =.9,.89, 2) a =.14,.84, 3) a =.19,.79, 4) a =.24,.74, 5) a =.29,.69, 6) a =.34,.64, 7) a =.39,.59, 8) a =.44,.54, 9) a =.49,.49. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 2 / 23

Large deformations.2.4.6.8 % 5% 1% ε 15% 2% 1 4 7 2 5 8 3 6 9 Poisson s ratio Eeff/Ematr 1 1 2 1 3 % 5% 1% ε 15% 2% 1 4 7 2 5 8 3 6 9 Young s modulus Effective Poisson s ratio, ν eff, and effective Young s modulus, E eff /E matr as functions of nominal strain for ν matr = and void inclusions (the material of inclusions is entirely removed). The numbers in the legend correspond to the following geometry parameter pairs: 1) a =.9,.89, 2) a =.14,.84, 3) a =.19,.79, 4) a =.24,.74, 5) a =.29,.69, 6) a =.34,.64, 7) a =.39,.59, 8) a =.44,.54, 9) a =.49,.49. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 2 / 23

Star-like inclusions, parameters a and b a =.5, b =.945 Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 21 / 23

Conclusions For strongly anisotropic inclusions the effective Poisson s ratio, which is close to for very low E incl /E matr, grows and after reaching a maximum (of negative value) decays to the matrix values close to E incl /E matr 1 and then grows again to the large E incl /E matr limit. In the vicinity of E incl /E matr for which the effective PR shows its maximum, for the most auxetic inclusions the effective Young s modulus is essentially larger than for less auxetic inclusions. For highly anisotropic inclusions of very large Young s modulus, the effective PR of the composite can be negative for nonauxetic matrix and inclusions. These is a very simple example of an auxetic structure being not only entirely continuous, i.e. without empty regions (cuts, holes) introduced into the system, but with E incl E matr, i.e. inclusions (much) harder than the matrix..6.4.2.2.4.6 Lgap =.1 Lgap =.2 Lgap =.5 Lgap =.1 Lgap =.2.2.4.6.8 1 a/ b Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 22 / 23

The end Thank you for your attention This work was supported by the (Polish) National Centre for Science under the grant NCN 212/5/N/ST5/1476. Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 23 / 23

.6.4 1.8.4.2.2.4.6.8 1 1 2 1 4 6 14 1 16.4.2.2.4.6.8 1 8 1 6 1 4 1 2 1 1 14 1 16 6.4.2.2.4.6.8 1 8 1 6 1 4 1 2 1 1 6 14 1 16 Poźniak, Wojciechowski, Grima, Mizzi.4.2.2.4.6.8 1 6 1 8 1 6 1 4 1 2 1 14 1 16 1 6 14 1 16 1 8 1 6 1 4 1 2 1 1 1 2 1 4 14 1 6 16 1 Eeffν/ eff Ematr Eeffν/ eff Ematr Eeffν/ eff Ematr Eeffν/ eff Ematr.4.2.2.4.6.8 1 2 1 6 1.2 1 2.4 1 4 1 6.4.2 1 1 6 14 1 16 1 2.6 1 4 1 2 1 4.2.4.8 1 6.4.2.2.4.6.8 1 1 4 1 8 1 6 1 4 1 2 1 1 16 1 6 1 4 1 2.8 1 2 1 4.4.2.2.4.6.8 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.8 1 6.2.4.6 1 4 14 1 1 6.4.2 6.6 4.4 1 1 1 1 4.2 1 2 1 2 1 2.6 1 2 1.2.4.8 1 6.2 1 6.2.4.8 1 4.6 1 4 1 6.8 1 6 1 1 2.4 1 4 1 4 1.4 1 8 1 6 1 4 1 2 1.2 1.2.4.2.4.2.2.4.6.8.4 1 1 1 8 1 6 1 4 1 2 1.6.2.6 1 2 1 4 1 16 1 2.6 4.8 1 6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.4 1 1 2 1 4 14.2.6.6.2.4.4 1 6.8 1 6 1 2 1 1 4.8 1 6.2.4 1 2 1 4 1 8 1 6 1 4 1 2 1.6.2 1.4.2.2.4.6.8.4 1.2.8 1 2 1 4 1 16.6 4 1 6 14 1 2.4 1 1 2.4 1 4 1 2 1 4.6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.2.8.6.2.4 1 1 2 6.8 1 6.8.2.4 1 1 4 1 6 1 6.2 1.4 1 1 2 1 4 1 8 1 6 1 4 1 2 1.4 1 2.2.4 1 4.8 1 6 1 16.4.2.2.4.6.8.6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.6 4.2.6 1 4.6.8 14 1 1 2 1 1 2.6.2.4.8.2.4 6.8 1 6.2 1 1 2 1 1 4.6 4 1 6.4 1.2.2.2 1 2.8 1 6.4.2.4.4 1.4 1 1 2 1 4 1 8 1 6 1 4 1 2 1 1 2 1 16.4.2.2.4.6.8.6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.6.2.2.4.6 4 14.8 1 6.4 1 1 2 1 4.2.4.6 1 8 1 6 1 4 1 2 1.2 1 2.6.2.4 1.2.4.8 1 6 1 8 1 6 1 4 1 2 1.6 1 4.8 νincl =.99 νincl =.49 νincl =. νincl =.49 1 16 6 1 6 16.2 1 1 2 14 1 14 1 6.4.2.2.4.6.8 1 8 1 6 1 4 1 2 1 Planar auxeticity from various inclusions.4.2.4 1 2 1 4 1 1 2 1 8 1 6 1 4 1 2 1.6.4.2.2.4.6.8 1 1 8 1 6 1 4 1 2 1 1 6.8 1 2 νincl =.99 νincl =.49 νincl =. νincl =.49 1 16 6 1 2 1 4 1 2 1 6.2.4.6 14 1.4.2.2.4.6.8 1 4.4 1 4.8 1 2 1 4 1 8 1 6 1 4 1 2 1.6.4.2.2.4.6.8 1 6 16 1 6.2 1 6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.6.2 1 4 14 1 1 2.4 1.6.2.4 1 2 1.8 1 6 1 2 1 2 1 4.8 1 4 1 4.8 6.2 1 1 6.2.4 1 6.4.6 1 4.4.2.2.4.6.8.2.4.6 1 6 1 8 1 6 1 4 1 2 1.6.2.8 1 1 6 1 6 16 1 2.4 1 1 2.6 1 4 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.2.4 1 2 1 4 1 6.6.8 14.6.2 1 1 2.8 1 6.4.2.4 1 1 1 4 1 2 1 2.8 1 4 1 4 1 6.4 1 4.2.8.4.2.2.4.6.8 1 6.6.4 1 1 2 1 4.4.2.2.4.6.8.2.2.4.6 1 6 1 8 1 6 1 4 1 2 1.6.2.4.6.2.8 νincl =.99 νincl =.49 νincl =. νincl =.49 1 6 16.2 1 14.4 1 1 2.8 1 2.2.4.6 1 6.2.4 1.8 1 6.4 1 2.6 1 4 1 6.4 1 1 4.8 1 4.2 1 4.8 1 6.6 1 4.8.6 1 4.2 1 1 4.4.2.2.4.6.8.4.6 1 6 1 8 1 6 1 4 1 2 1.6 1 2.6.2 1 2 νincl =.99 νincl =.49 νincl =. νincl =.49.8 1 1 2.4 1.2.4 1 6 16.2.2.4.6 1 2 14.4 1 6.8.2.2.4 1 4 1 6.4 1 1 2.8.2.2 1.2 1.6 1 4.8 1 6.4.2.4.2 1 2 1 1 8 1 6 1 4 1 2 1.6 1 2.4 1.2.4 1 2 1 4 1 6.4 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.2.4 1 6 16.6 1 6.6 1 4.8 14.8.2 1 2 1 4.4 1.2.4.6 1 4.8.4.2 1 8 1 6 1 4 1 2 1.6 1 1 2.6.2 1 1 2 1 6 16.2.4.8 1 6.4 νincl =.99 νincl =.49 νincl =. νincl =.49 1 4 14 1 2 1 4.2.2.4 1 2.8 1 6.4 1 1 4.8 1 6.4.2.2.4.6.8.6.2.2.4.6.4 1 2.8 1 6.4.2 1.2.4 1.4 1 4 1 8 1 6 1 4 1 2 1.6 1 2 1 4.6.8 νincl =.99 νincl =.49 νincl =. νincl =.49 1 6 16.4.2.2.4.6.8.2 1 1 2 14 1.4.2.4 1 2 1 4 1 8 1 6 1 4 1 2 1.6.4.2.2.4.6.8 νincl =.99 νincl =.49 νincl =. νincl =.49 1 6 16 1 1 2.6 1 4.8.2.4 1.2.4.6 1 4.2 1 1 2.2 1 1 2.8 1 6.4.2.2.4.6 1 6 1 6.4 1 1 4 14 1 2 1 4 1 2.8 1 6.4.2.2.4.6.8.2.4.6 1.2 1 1 4.8 1 6 1 2 1 4 1 2 1 6.4.2.2.4.6.4.2.2.4.6.8 1 8 1 6 1 4 1 2 1.6 1 6.4 1 1 4.8 1 6.8 1 2.6 1 4.2 1.2.4 1 6.4.2.4.6 1 4 1.2.8 1 2 1 4.8 1 6.4 1 1 2 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.6 1 4.2.4.6 1 4.8.2 1 1 2.8.2.4.2.4.6 1 6.4 1 1 2.6 νincl =.99 νincl =.49 νincl =. νincl =.49 1 4.2.4.6 1 4.8.2 1 1 2.8.2 1 1 2.4.2.4.6 1 6.4.2.4.6 1 4.8 1 6.4.2 1 2.6.2 1 6.4 1 1 4.8 1 2.6 νincl =.99 νincl =.49 νincl =. νincl =.49 1 4.2 1.2.4.6.2.2.4.4.2 1 1 2.2 1 1 2.8 1 6.4.2.4.6 1 4.6 νincl =.99 νincl =.49 νincl =. νincl =.49 1 2.8 1 6.4.2.4.6 1 4.8 1 6.4 1 2.2.4.6 1 4.2 1 1 2.8.4.4.2 1.2.4.6.8 1 6.4.2.4.8 1 6.2 1.2.4.6 1 4.4.2 1 1 2.6 1 4.6 νincl =.99 νincl =.49 νincl =. νincl =.49.4.2.4.8 1 6.2 1.6 1 4.6 νincl =.99 νincl =.49 νincl =. νincl =.49.4 1 2.6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.8.2 1.6 1 4.4 1 2.6.6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.2 1.4 1 2.6 νincl =.99 νincl =.49 νincl =. νincl =.49.2.4.6.2.4 1 4 14 1 6 16 15 Sep 215 24 / 23

ν ν ν a=.9 b=.89, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 a=.24 b=.74, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 a=.39 b=.59, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 ν ν ν a=.14 b=.84, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 a=.29 b=.69, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 a=.44 b=.55, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 ν ν a=.19 b=.79, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 a=.34 b=.64, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49.6.4.6.4.6.4.2.2.2.2.4.2.4.2.4.6.6.6.8.8.8 1 8 1 6 1 4 1 2 1 14 16 1 8 1 6 1 4 1 2 1 14 16 1 8 1 6 1 4 1 2 1 14 16 νmat =.99 νmat =.49 νmat =. νmat =.49 νmat =.99 νmat =.49 νmat =. νmat =.49 νmat =.99 νmat =.49 νmat =. νmat =.49.6.4.6.4.6.4.2.2.2.2.4.2.4.2.4.6.6.6.8.8.8 1 8 1 6 1 4 1 2 1 14 16 1 8 1 6 1 4 1 2 1 14 16 1 8 1 6 1 4 1 2 1 14 16 νmat =.99 νmat =.49 νmat =. νmat =.49 νmat =.99 νmat =.49 νmat =. νmat =.49 νmat =.99 νmat =.49 νmat =. νmat =.49.6.4.6.4.6.4.2.2.2.2.4.6.2.4.6.8.8 1 8 1 6 1 4 1 2 1 14 16 1 8 1 6 1 4 1 2 1 14 16 νmat =.99 νmat =.49 νmat =. νmat =.49.2.4.6.8 1 8 1 6 1 4 1 2 1 14 16 νmat =.99 νmat =.49 νmat =. νmat =.49 ν a=.49 b=.49, gwiazdki νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νincl =.999 νincl =.99 νincl =.9 νincl =.49 νincl =. νincl =.49 νmat =.99 νmat =.49 νmat =. νmat =.49 Poźniak, Wojciechowski, Grima, Mizzi Planar auxeticity from various inclusions 15 Sep 215 25 / 23