Journal of Theoretical
and Applied Mechanics
0, 0, pp. , Warsaw 0

Computation and experimental comparison of the deformation behavior of pantographic structures with different micro-geometry under shear and torsion

Hua Yang, Wolfgang H. Müller
Additive manufacturing methods, commonly known as 3D printing, allow more sophisticated designs to be created. Willingly designed substructures incorporated into the solid open up new possibilities for uncommon macroscopic deformation behavior. Such a man-made structure is also referred to as a metamaterial. A detailed simulation of a polymer-based metamaterial is challenging but accurately possible by means of the theory of elasticity. In this study we present experimental investigations of a metamaterial composed of pantographic substructures of different internal geometry. The pantographic structures show an unexpected type of deformation, which can be modeled via elasticity with the aid of a direct numerical simulation by using the Finite Element (FE) method. In other words, a detailed mesh is generated involving the substructure. The metamaterial is additively manufactured out of a common polymer showing nonlinear elastic deformation, and therefore hyperelastic material models are used. Specifically, analytical solutions of a circular cylinder are examined and compared in the case of extension and torsion in order to comprehend the effects of the coefficients inherent to the energy function of the hyperelastic model. Finally FE computations of pantographic structures are performed and compared with the experimental measurements.
Keywords: nonlinear elasticity; metamaterial; numerical simulation

References


ABALI, B. E., MÜLLER, W. H., AND DELL’ISOLA, F. 2017. Theory and computation of higher gradient

elasticity theories based on action principles. Archive of Applied Mechanics 87:1495–1510.

ALTENBACH, H. AND EREMEYEV, V. A. 2012. Generalized Continua-from the Theory to Engineering Applications, volume 541. Springer.

BAHREMAN, M. AND DARIJANI, H. 2015. New polynomial strain energy function; application to rubbery

circular cylinders under finite extension and torsion. Journal of Applied Polymer Science 132.

BARCHIESI, E., GANZOSCH, G., LIEBOLD, C., PLACIDI, L., GRYGORUK, R., AND MÜLLER, W. H. 2018. Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Continuum Mechanics and Thermodynamics pp. 1–13.

BOUTIN, C., DELL’ISOLA, F., GIORGIO, I., AND PLACIDI, L. 2017. Linear pantographic sheets: Asymptotic micro-macro models identification. Mathematics and Mechanics of Complex Systems 5:127–162.

CHEN, J.-S. AND WU, C.-T. 1997. On computational issues in large deformation analysis of rubber bushings. Journal of Structural Mechanics 25:287–309.

DEL VESCOVO, D. AND GIORGIO, I. 2014. Dynamic problems for metamaterials: review of existing models

and ideas for further research. International Journal of Engineering Science 80:153–172.

DELL’ISOLA, F., ANDREAUS, U., AND PLACIDI, L. 2014. At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of

gabrio piola. Mathematics and Mechanics of Solids 20:887–928.

DELL’ISOLA, F., GIORGIO, I., PAWLIKOWSKI, M., AND RIZZI, N. 2016. Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples

of equilibrium. Proc. R. Soc. A 472:20150790.

DELL’ISOLA, F., LEKSZYCKI, T., PAWLIKOWSKI, M., GRYGORUK, R., AND GRECO, L. 2015. Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental

evidence. Zeitschrift für angewandte Mathematik und Physik 66:3473–3498.

EREMEYEV, V. A. AND ALTENBACH, H. 2014. Equilibrium of a second-gradient fluid and an elastic solid with surface stresses. Meccanica 49:2635–2643.

ERINGEN, A. 1967. Theory of micropolar elasticity. Technical report, DTIC Document.

GANZOSCH, G., DELL’ISOLA, F., TURCO, E., LEKSZYCKI, T., AND MÜLLER, W. H. 2016. Shearing tests applied to pantographic structures. Acta Polytechnica CTU Proceedings 7:1–6.

GANZOSCH, G., HOSCHKE, K., LEKSZYCKI, T., GIORGIO, I., TURCO, E., AND MÜLLER, W. 2018. 3dmeasurements of 3d-deformations of pantographic structures. Technische Mechanik 38:233–245.

GIORGIO, I. 2016. Numerical identification procedure between a micro-cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik 67:95.

GIORGIO, I., RIZZI, N., AND TURCO, E. 2017. Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A 473:20170636.

HOFFMAN, J., JANSSON, J., JOHNSON, C., KNEPLEY, M., KIRBY, R., LOGG, A., SCOTT, L. R., AND WELLS, G. N. 2005. Fenics. http://www.fenicsproject.org/.

HOLZAPFEL, A. G. 2000. Nonlinear Solid Mechanics II. John Wiley & Sons, Inc.

KANNER, L. M. AND HORGAN, C. O. 2008. On extension and torsion of strain-stiffening rubber-like elastic circular cylinders. Journal of Elasticity 93:39.

LAM, D., YANG, F., CHONG, A., WANG, J., AND TONG, P. 2003. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids 51:1477–1508.

LAWLOR, M. G., O’DONNELL, M. R., O’CONNELL, B. M., AND WALSH, M. T. 2011. Experimental determination of circumferential properties of fresh carotid artery plaques. Journal of biomechanics 44:1709–1715.

LOGG, A., MARDAL, K. A., AND WELLS, G. N. 2011. Automated solution of differential equations by the finite element method, the FEniCS book, volume 84 of Lecture Notes in Computational Science and

Engineering. Springer.

MINDLIN, R. AND ESHEL, N. 1968. On first strain-gradient theories in linear elasticity. International Journal of Solids and Structures 4:109–124.

MINDLIN, R. D. AND TIERSTEN, H. F. 1962. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis 11:415–448. 10.1007/BF00253946.

MISRA, A., LEKSZYCKI, T., GIORGIO, I., GANZOSCH, G., MÜLLER, W. H., AND DELL’ISOLA, F. 2018. Pantographic metamaterials show atypical Poynting effect reversal. Mechanics Research Communications 89:6–10.

NEFF, P., GHIBA, I.-D., MADEO, A., PLACIDI, L., AND ROSI, G. 2014. A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mechanics and Thermodynamics 26:639–681.

PLACIDI, L., ANDREAUS, U., DELLA CORTE, A., AND LEKSZYCKI, T. 2015. Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für angewandte

Mathematik und Physik 66:3699–3725.

RIVLIN, R. 1948. Large elastic deformations of isotropic materials. iii. some simple problems in cyclindrical polar co-ordinates. Phil. Trans. R. Soc. Lond. A 240:509–525.

THAI, H.-T., VO, T. P., NGUYEN, T.-K., AND KIM, S.-E. 2017. A review of continuum mechanics models for size-dependent analysis of beams and plates. Composite Structures 177:196–219.

TOUPIN, R. A. 1964. Theories of elasticity with couple-stress. Archive for Rational Mechanics and Analysis 17:85–112.

TRIANTAFYLLIDIS, N. AND AIFANTIS, E. C. 1986. A gradient approach to localization of deformation. i. hyperelastic materials. Journal of Elasticity 16:225–237.

TURCO, E., DELL’ISOLA, F., CAZZANI, A., AND RIZZI, N. L. 2016. Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67:85.

YANG, H., GANZOSCH, G., GIORGIO, I., AND ABALI, B. E. 2018. Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Zeitschrift für angewandte Mathematik

und Physik 69:105.