×

Non-isothermal energy-momentum time integrations with drilling degrees of freedom of composites with viscoelastic fiber bundles and curvature-twist stiffness. (English) Zbl 1442.74015

Summary: 3D fiber-reinforced composites demand for a special simulation technique, because they consist of fiber bundles. Therefore, in the corresponding representative volume element of these metamaterials, secondary effects as a micro inertia and a curvature-twist stiffness have to bear in mind. The latter increases the strength-to-weight ratio of thin-walled lightweight structures due to a separate twisting and bending stiffness of the fiber bundles. In this paper, these secondary effects are introduced in a continuum formulation by means of independent drilling degrees of freedom. The resulting non-isothermal constrained micropolar continuum is derived by a principle of virtual power, which simultaneously generates in the discrete setting a mixed B-bar method and a Galerkin-based energy-momentum scheme of higher order. This work also takes into account viscoelastic material behavior in the fiber bundles, which arises from a mixture of organic and inorganic fibers. Here, the viscous evolution equation is solved elementwise by using a mixed field as viscous internal variable. Representative numerical examples demonstrate the inelastic material behavior, the effect of micro inertia on the physical properties of the continuum as well as on its space and time discretization and, finally, the twisting and bending stiffness of the fiber bundles. Further, non-standard boundary conditions are applied in the dynamic simulations performed by higher order energy-momentum schemes.

MSC:

74A40 Random materials and composite materials
74A35 Polar materials

Software:

Gmsh; FLagSHyP; CSparse
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Tong, L.; Mouritz, A. P.; Bannister, M. K., 3D Fibre Reinforced Polymer Composites (2002), Elsevier
[2] De Luycker, E.; Morestin, F.; Boisse, P.; Marsal, D., Simulation of 3D interlock composite preforming, Compos. Struct., 88, 4, 615-623 (2009)
[3] Uhlig, K., Beitrag zur Anwendung der Tailored Fiber Placement Technologie am Beispiel von Rotoren aus Kohlenstofffaserverstärktem Epoxidharz für den Einsatz in Turbomolekularpumpen (2018), Technische Universität Dresden, urn:nbn:de:bsz:14-qucosa-235151
[4] Steigmann, D. J.; dell’Isola, F., Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching, Acta Mech. Sin., 31, 373-382 (2015) · Zbl 1346.74128
[5] Green, A. E.; Adkins, J. E., Large Elastic Deformations and Non-Linear Continuum Mechanics (1970), Clarendon Press · Zbl 0090.17501
[6] Reese, S.; Raible, T.; Wriggers, P., Finite element modelling of orthotropic material behaviour in pneumatic membranes, Int. J. Solids Struct., 38, 52, 9525-9544 (2001) · Zbl 1016.74072
[7] Ferretti, M.; Madeo, A.; Dell’Isola, F.; Boisse, P., Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory, Z. Angew. Math. Phys., 65, 3, 587-612 (2014) · Zbl 1302.74008
[8] Madeo, A.; Ferretti, M.; Dell’Isola, F.; Boisse, P., Thick fibrous composite reinforcements behave as special second-gradient materials: three-point bending of 3D interlocks, Z. Angew. Math. Phys., 66, 4, 2041-2060 (2015) · Zbl 1329.74060
[9] Spencer, A. J.M.; Soldatos, K. P., Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness, Int. J. Non-Linear Mech., 42, 2, 355-368 (2007)
[10] Asmanoglo, T.; Menzel, A., A multi-field finite element approach for the modelling of fibre-reinforced composites with fibre-bending stiffness, Comput. Methods Appl. Mech. Engrg., 317, 1037-1067 (2017) · Zbl 1439.74091
[11] Rudrarajua, S.; Van der Venb, A.; Garikipati, K., Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains, Comput. Methods Appl. Mech. Engrg., 278, 705-728 (2014) · Zbl 1423.74105
[12] Franke, M.; Ortigosa, R.; Janz, A.; Gil, A. J.; Betsch, P., A mixed variational framework for the design of energy-momentum integration schemes based on convex multi-variable electro-elastodynamics, Comput. Methods Appl. Mech. Engrg., 351, 109-152 (2019) · Zbl 1441.74075
[13] Groß, M.; Dietzsch, J.; Röbiger, C., A mixed B-bar formulation derived by a principle of virtual power for energy-momentum time integrations of fiber-reinforced continua, Comput. Methods Appl. Mech. Engrg., 350, 595-640 (2019) · Zbl 1441.74243
[14] Groß, M.; Dietzsch, J., Variational-based locking-free energy-momentum schemes of higher-order for thermo-viscoelastic fiber-reinforced continua, Comput. Methods Appl. Mech. Engrg., 343, 631-671 (2019) · Zbl 1440.74095
[15] Groß, M.; Dietzsch, J., Variational-based energy-momentum schemes of higher-order for elastic fiber-reinforced continua, Comput. Methods Appl. Mech. Engrg., 320, 509-542 (2017) · Zbl 1439.74047
[16] Groß, M.; Dietzsch, J.; Bartelt, M., Variational-based higher-order accurate energy-momentum schemes for thermo-viscoelastic fiber-reinforced continua, Comput. Methods Appl. Mech. Engrg., 336, 353-418 (2018) · Zbl 1440.74103
[17] Nedjar, B., An anisotropic viscoelastic 3fibre-matrix model at finite strains: continuum formulation and computational aspects, Comput. Methods Appl. Mech. Engrg., 196, 1745-1756 (2007) · Zbl 1173.74322
[18] Reese, S.; Govindjee, S., A theory of finite viscoelasticity and numerical aspects, Int. J. Solids Struct., 35, 3455-3482 (1998) · Zbl 0918.73028
[19] Simo, J. C.; Hughes, T. J.R., Computational Inelasticity (Interdisciplinary Applied Mathematics) (Volume 7) (1998), Springer · Zbl 0934.74003
[20] Steinmann, P.; Stein, E., A unifying treatise of variational principles for two types of micropolar continua, Acta Mech., 121, 215-232 (1997) · Zbl 0878.73004
[21] Hughes, T. J.R.; Brezzi, F., On drilling degrees of freedom, Comput. Methods Appl. Mech. Engrg., 72, 1, 105-121 (1989) · Zbl 0691.73015
[22] Ibrahimbegovic, A.; Wilson, E. L., Thick shell and solid finite elements with independent rotation fields, Internat. J. Numer. Methods Engrg., 31, 7, 1393-1414 (1991) · Zbl 0758.73047
[23] Boujelben, A.; Ibrahimbegovic, A., Finite-strain three-dimensional solids with rotational degrees of freedom: non-linear statics and dynamics, Adv. Model. Simul. Eng. Sci., 4, 3, 1-24 (2017)
[24] Davis, T. A., Direct Methods for Sparse Linear Systems (2006), Siam · Zbl 1119.65021
[25] Balzani, D.; Neff, P.; Schröder, J.; Holzapfel, G. A., A polyconvex framework for soft biological tissues. Adjustment to experimental data, Int. J. Solids Struct., 43, 6052-6070 (2006) · Zbl 1120.74632
[26] Al-Kinani, R.; Hartmann, S.; Netz, T., Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p-version finite elements, Z. Angew. Math. Phys., 95, 7, 742-761 (2015) · Zbl 1326.74032
[27] Holzapfel, G. A., Nonlinear Solid Mechanics (2000), Wiley: Wiley Chichester · Zbl 0980.74001
[28] Simo, J. C.; Tarnow, N., The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics, Z. angew. Math. Phys., 43, 757-792 (1992) · Zbl 0758.73001
[29] Klinkel, S.; Sansour, C.; Wagner, W., An anisotropic fibre-matrix material model at finite elastic-plastic strains, Comput. Mech., 35, 6, 409-417 (2005) · Zbl 1096.74005
[30] Groß, M., Higher-order Accurate and Energy-Momentum Consistent Discretisation of Dynamic Finite Deformation Thermo-viscoelasticity, Series of the Chair for Computational Mechanics (2009), Department of Mechanical Engineering, University of Siegen, urn:nbn:de:hbz:467-3890
[31] Reese, S., Thermomechanische Modellierung gummiartiger Polymerstrukturen (2001), Institut für Baumechanik und Numerische Mechanik, Universität Hannover, Habilitationsschrift, F 01/4
[32] Stokes, V. K., Theories of Fluids with Microstructure: An Introduction (2012), Springer Science & Business Media
[33] Grbčić, S.; Ibrahimbegović, A.; Jelenić, G., Variational formulation of micropolar elasticity using 3D hexahedral finite-element interpolation with incompatible modes, Comput. Struct., 205, 1-14 (2018)
[34] Hackl, K., Generalized standard media and variational principles in classical and finite strain elastoplasticity, J. Mech. Phys. Solids, 45, 5, 667-688 (1997) · Zbl 0974.74512
[35] Zhilin, P. A., Rigid body oscillator: a general model and some results, Acta Mech., 142, 169-193 (2000) · Zbl 0970.70011
[36] Hughes, T. J.R., The Finite Element Method (2000), Dover: Dover Mineola · Zbl 1191.74002
[37] Groß, M.; Bartelt, M.; Betsch, P., Structure-preserving time integration of non-isothermal finite viscoelastic continua related to variational formulations of continuum dynamics, Comput. Mech., 62, 2, 123-150 (2018), appeared online in 2017 · Zbl 1433.74106
[38] Gonzalez, O., Exact energy and momentum conserving algorithms for general models in nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 190, 1763-1783 (2000) · Zbl 1005.74075
[39] Armero, F., Assumed strain finite element methods for conserving temporal integrations in non-linear solid dynamics, Internat. J. Numer. Methods Engrg., 74, 1795-1847 (2008) · Zbl 1195.74158
[40] Simo, J. C., Numerical analysis and simulation of plasticity, (P. G, Ciarlet; J. L., Lions, Handbook of Numerical Analysis, vol. VI (1998), Elsevier: Elsevier North Holland) · Zbl 0930.74001
[41] Groß, M.; Betsch, P., Galerkin-based energy-momentum consistent time-stepping algorithms for classical nonlinear thermo-dynamics, Math. Comput. Simulation, 82, 4, 718-770 (2011) · Zbl 1317.74086
[42] Wriggers, P., Nichtlineare Finite-Elemente-Methoden (2001), Springer · Zbl 0972.74002
[43] Felippa, C. A., Nonlinear Finite Element Methods (2001), University of Colorado: University of Colorado Boulder, Colorado, USA
[44] Felippa, C. A., A compendium of FEM integration formulas for symbolic work, Eng. Comput., 21, 8, 867-890 (2004) · Zbl 1134.74404
[45] Geuzaine, C.; Remacle, J.-F., Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Internat. J. Numer. Methods Engrg., 79, 11, 1309-1331 (2009) · Zbl 1176.74181
[46] Nguyen, Q.-T.; Thomas, J.-C., Inflation and bending of an orthotropic inflatable beam, Thin-Walled Structures, 88, 129-144 (2015)
[47] Bonet, J.; Gil, A. J.; Wood, R. D., Nonlinear Solid Mechanics for Finite Element Analysis: Statics (2016), Cambridge University Press · Zbl 1341.74001
[48] Weidenhammer, F., Das anlaufen eines ungedämpften Schwingers mit Massenkrafterregung, Z. Angew. Math. Mech., 38, 304-307 (1958) · Zbl 0094.18302
[49] Meng, X. N.; Laursen, T. A., Energy consistent algorithms for dynamic finite deformation plasticity, Comput. Methods Appl. Mech. Engrg., 191, 1639-1675 (2002) · Zbl 1141.74373
[50] Schröder, J.; Neff, P., Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions, Int. J. Solids Struct., 40, 2, 401-445 (2003) · Zbl 1033.74005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.