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MAP123: a data-driven approach to use 1D data for 3D nonlinear elastic materials modeling. (English) Zbl 1442.65417
Summary: Solving three-dimensional boundary-value engineering problems numerically requires material laws. However, it is difficult to build the material laws in three dimension, since the material behaviors are usually measured by one-dimensional uniaxial tension/compression experiments. In this way, the material behavior in the three-dimension is ‘compressed’ into one-dimensional data. Here, we propose a new method, coined MAP123 (map data from one-dimension to three-dimension), to decompress the one-dimensional data into three dimension for nonlinear elastic material modeling without the construction of analytic mathematical function for the material law. The decomposition of stress and strain into deviatoric and spherical parts for isotropic nonlinear elastic materials at finite deformation makes this data-driven approach work quite well. Several examples are used to demonstrate the capability of MAP123, such as a rectangular plate with a circular hole under uniaxial tension. Corresponding experiments are also carried out to further verify the MAP123 method. Based on the proposed approach, uniaxial experiment is suggested to measure the deformation in three directions not only the force and extension along the loading direction. Limitation of the proposed MAP123 approach is also discussed.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
74B20 Nonlinear elasticity
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[1] Simo, J. C.; Hughes, T. J., Computational Inelasticity, Vol. 7 (2006), Springer Science & Business Media
[2] Marsden, J. E.; Hughes, T. J., Mathematical Foundations of Elasticity (1994), Courier Corporation
[3] Hughes, T. J., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2012), Courier Corporation
[4] Belytschko, T.; Liu, W. K.; Moran, B.; Elkhodary, K., Nonlinear Finite Elements for Continua and Structures (2013), John wiley & sons · Zbl 1279.74002
[5] Kirchdoerfer, T.; Ortiz, M., Data driven computational mechanics, Comput. Methods Appl. Mech. Engrg., 304, 81-101 (2016) · Zbl 1425.74503
[6] Kirchdoerfer, T.; Ortiz, M., Data-driven computing in dynamics, Internat. J. Numer. Methods Engrg., 113, 11, 1697-1710 (2017) · Zbl 07186068
[7] Kirchdoerfer, T.; Ortiz, M., Data driven computing with noisy material data sets, Comput. Methods Appl. Mech. Engrg., 326, 622-641 (2017) · Zbl 07186068
[8] Conti, S.; Müller, S.; Ortiz, M., Data driven problems in elasticity, Arch. Ration. Mech. Anal., 229, 1, 79-123 (2017) · Zbl 1402.35276
[9] Leygue, A.; Coret, M.; Rthor, J.; Stainier, L.; Verron, E., Data based derivation of material response, Comput. Methods Appl. Mech. Engrg., 331, 184-196 (2018) · Zbl 1439.74050
[10] Chinesta, F.; Ladeveze, P.; Ibanez, R.; Aguado, J. V.; Abisset-Chavanne, E.; Cueto, E., Data-driven computational plasticity, Procedia Eng., 207, 209-214 (2017)
[11] Bower, A. F., Applied Mechanics of Solids (2009), CRC press
[12] Mooney, M., A theory of large elastic deformation, J. Appl. Phys., 11, 9, 582-592 (1940) · JFM 66.1021.04
[13] Ogden, R. W., Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubberlike solids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 326, 1567, 565-584 (1972) · Zbl 0257.73034
[14] Arruda, E. M.; Boyce, M. C., A 3-dimensional constitutive model for the large stretch behavior of rubber elastic materials, J. Mech. Phys. Solids, 41, 2, 389-412 (1993) · Zbl 1355.74020
[15] Li, Z.; Zhou, Z.; Li, Y.; Tang, S., Effect of cyclic loading on surface instability of silicone rubber under compression, Polymers, 9, 4, 148 (2017)
[16] Li, Y.; Tang, S.; Abberton, B. C.; Kröger, M.; Burkhart, C.; Jiang, B.; Papakonstantopoulos, G. J.; Poldneff, M.; Liu, W. K., A predictive multiscale computational framework for viscoelastic properties of linear polymers, Polymer, 53, 25, 5935-5952 (2012)
[17] Li, Y.; Tang, S.; Kröger, M.; Liu, W. K., Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers, J. Mech. Phys. Solids, 88, 204-226 (2016)
[18] Liu, Z.; Moore, J. A.; Liu, W. K., An extended micromechanics method for probing interphase properties in polymer nanocomposites, J. Mech. Phys. Solids, 95, 663-680 (2016)
[19] He, C.; Ge, J.; Qi, D.; Gao, J.; Chen, Y.; Liang, J.; Fang, D., A multiscale elasto-plastic damage model for the nonlinear behavior of 3d braided composites, Compos. Sci. Technol., 171, 21-33 (2019)
[20] Wang, Q.; Gossweiler, G. R.; Craig, S. L.; Zhao, X., Mechanics of mechanochemically responsive elastomers, J. Mech. Phys. Solids, 82, 320-344 (2015)
[21] Liu, Z.; Wu, C.; Koishi, M., A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials, Comput. Methods Appl. Mech. Engrg., 345, 1138-1168 (2019) · Zbl 1440.74340
[22] Bessa, M. A.; Bostanabad, R.; Liu, Z.; Hu, A.; Apley, D. W.; Brinson, C.; Chen, W.; Liu, W. K., A framework for data-driven analysis of materials under uncertainty: countering the curse of dimensionality, Comput. Methods Appl. Mech. Engrg., 320, 633-667 (2017) · Zbl 1439.74014
[23] Oishi, A.; Yagawa, G., Computational mechanics enhanced by deep learning, Comput. Methods Appl. Mech. Engrg., 327, 327-351 (2017) · Zbl 1439.74458
[24] Bessa, M.; Pellegrino, S., Design of ultra-thin shell structures in the stochastic post-buckling range using bayesian machine learning and optimization, Int. J. Solids Struct., 139, 174-188 (2018)
[25] Wang, K.; Sun, W., A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning, Comput. Methods Appl. Mech. Engrg., 334, 337-380 (2018) · Zbl 1440.74130
[26] Wang, K.; Sun, W., Meta-modeling game for deriving theory-consistent, microstructure-based traction-separation laws via deep reinforcement learning, Comput. Methods Appl. Mech. Engrg., 346, 216-241 (2019) · Zbl 1440.74016
[27] Lei, X.; Liu, C.; Du, Z.; Zhang, W.; Guo, X., Machine learning-driven real-time topology optimization under moving morphable component-based framework, J. Appl. Mech., 86, 1, 011004 (2019)
[28] Jolliffe, I., Principal component analysis, (International Encyclopedia of Statistical Science (2011), Springer), 1094-1096
[29] Wold, S.; Esbensen, K.; Geladi, P., Principal component analysis, Chemometr. Intell. Lab. Syst., 2, 1-3, 37-52 (1987)
[30] Abdi, H.; Williams, L. J., Principal component analysis, Wiley Interdiscip. Rev. Comput. Stat., 2, 4, 433-459 (2010)
[32] Hill, R., The Mathematical Theory of Plasticity (1998), OXford University Press · Zbl 0923.73001
[33] Hill, R.; Rice, J. R., Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids, 20, 6, 401-413 (1972) · Zbl 0254.73031
[34] Tang, S.; Guo, T. F.; Cheng, L., Rate effects on toughness in elastic nonlinear viscous solids, J. Mech. Phys. Solids, 56, 3, 974-992 (2008) · Zbl 1419.74272
[35] Wong, W. H.; Guo, T. F., On the energetics of tensile and shear void coalescences, J. Mech. Phys. Solids, 82, 259-286 (2015)
[36] Liu, Z. G.; Wong, W. H.; Guo, T. F., Void behaviors from low to high triaxialities: Transition from void collapse to void coalescence, Int. J. Plast., 84, 183-202 (2016)
[37] Berg, M.d.; Cheong, O.; Kreveld, M.v.; Overmars, M., Computational Geometry: Algorithms and Applications (2008), Springer-Verlag TELOS
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