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Modeling of fiber circumplacement around a hole using a streamline approach

Year 2018, , 17 - 28, 11.03.2018
https://doi.org/10.32323/ujma.398481

Abstract

The insertion of holes into laminates can be done by producing a fiber reinforced composite plate and, subsequently, drilling the borehole. Alternatively, we can bypass the fibers around the final hole before injecting the matrix material. In the first case, the spatial distribution of the axis of anisotropy, and the structural tensor concerned, are spatially constant. In the second case, i.e. the fiber circumplacement around the hole, a space-dependent anisotropy has to be considered. Instead of the common approach of defining region-wise constant fiber orientations, we propose a continuous formulation of fiber orientations using streamlines. To estimate the final stress and strain state for a unidirectional composite plate, three-dimensional finite element simulations are performed, where spatially constant transverse isotropy is compared to inhomogeneously distributed fiber orientation around the hole. It will turn out that the resulting stress states lead to both reduced stress amplitudes in loading direction as well as compressive strains in lateral direction. A detailed mathematical derivation of the basic equations accompanies the investigations.

References

  • [1] M. Kaliske. Zur Theorie und Numerik von Strukturen aus Faserverbundmaterial. Mitteilung nr. 49-99, Universit¨at Hannover, 1999.
  • [2] M. Fiolka. Theorie und Numerik volumetrischer Schalenelemente zur Delaminationsanalyse von Faserverbundlaminaten. Report no.2/2008, Institute of Mechanics, University of Kassel, 2008.
  • [3] M. Weise and A. Meyer. Grundgleichungen f ¨ur transversal isotropes Materialverhalten. Preprint CSC/10-03, TU Chemnitz, Fakult¨atf ¨ur Mathematik, Chemnitz (Germany), 2003.
  • [4] A. J. M. Spencer. Constitutive theory for strongly anisotropic solids. In A. J. M. Spencer, editor, Continuum theory of the mechanicsof fibre-reinforced composites, number 282 in Courses and lectures - International Centre for Mechanical Sciences, pages 1–32.Springer-Verlag, Wien, 1984.
  • [5] B.D. Agarwal, L.J. Broutman, and K. Chandrashekhara. Analysis and Performance of Fiber Composites. Wiley, 2006.
  • [6] R.F. Gibson. Principles of Composite Material Mechanics, Third Edition. Mechanical Engineering. CRC Press, 2011.
  • [7] V.Z. Parton and B.A. Kudryavtsev. Engineering Mechanics of Composite Structures. CRC Press, Inc., 1993.
  • [8] J. Aboudi, S.M. Arnold, and B.A. Bednarcyk. Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach.Elsevier Science, 2012.
  • [9] J. Aboudi. Mechanics of Composite Materials: A Unified Micromechanical Approach. Studies in Applied Mechanics. ElsevierScience, 2013.
  • [10] J. A. Weiss, B. N. Maker, and S. Govindjee. Finite element implementation of incompressible, transversely isotropic hyperelasticity.Computer Methods in Applied Mechanics and Engineering, 135:107–128, 1996.
  • [11] J. Schr ¨oder and P. Neff. Invariant formulation of hyperelasticity transverse isotropy based on polyconvex free energy functions.International Journal of Solids and Structures, 40:401–445, 2003.
  • [12] J. Schr ¨oder, P. Neff, and D. Balzani. A variational approach for materially stable anisotropic hyperelasticity. International Journal ofSolids and Structures, 42:4352–4371, 2005.
  • [13] M. Itskov and N. Aksel. A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvexstrain energy function. International Journal of Solids and Structures, 41:3833–3848, 2004.
  • [14] D. Balzani, P. Neff, J. Schr ¨oder, and G. A. Holzapfel. A polyconvex framework for soft biological tissues. adjustment to experimentaldata. International Journal of Solids and Structures, 43:6052–6070, 2006.
  • [15] G. A. Holzapfel, T. C. Gasser, and R. W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative studyof material models. Journal of Elasticity, 61:1–48, 2000.
  • [16] C. Sansour. On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. European Journal ofMechanics - A/Solids, 27:28–39, 2008.
  • [17] A. Zdunek, W. Rachowicz, and T. Eriksson. A novel computational formulation for nearly incompressible and nearly inextensiblefinite hyperelasticity. Computer Methods in Applied Mechanics and Engineering, 281:220–249, 2014.
  • [18] P. Wriggers, J. Schr ¨oder, and F. Auricchio. Finite element formulations for large strain anisotropic material with inextensible fibers.Advanced Modeling and Simulation in Engineering Sciences, 3:1–18, 2016.
  • [19] O. Sepahia, L. Radtke, S.E. Debus, and A. D¨uster. Anisotropic hierarchic solid finite elements for the simulation of passive-activearterial wall models. Computers & Mathematics with Applications, 2018.
  • [20] Gustavo Gonzalez Lozano, Ashutosh Tiwari, Christopher Turner, and Simon Astwood. A review on design for manufacture of variablestiffness composite laminates. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture,230(6):981–992, 2016.
  • [21] X. J. Niu, T. Yang, Y. Du, and Z. Q. Xue. Tensile properties of variable stiffness composite laminates with circular holes based onpotential flow functions. Archive of Applied Mechanics, 86(9):1551–1563, Sep 2016.
  • [22] Dani¨el M.J. Peeters, Simon Hesse, and Mostafa M. Abdalla. Stacking sequence optimisation of variable stiffness laminates withmanufacturing constraints. Composite Structures, 125:596 – 604, 2015.
  • [23] J. Huang and R.T. Haftka. Optimization of fiber orientations near a hole for increased load-carrying capacity of composite laminates.Structural and Multidisciplinary Optimization, 30(5):335–341, Nov 2005.
  • [24] Yingdan Zhu, Jiancai Liu, Dong Liu, Haibing Xu, Chun Yan, Bin Huang, and David Hui. Fiber path optimization based on a familyof curves in composite laminate with a center hole. Composites Part B: Engineering, 111:91 – 102, 2017.
  • [25] M. W. Hyer and R. F. Charette. Use of curvilinear fiber format in composite structure design. AIAA, 125:1011 – 1015, 1991.
  • [26] Pedro Ribeiro, Hamed Akhavan, Andrzej Teter, and Jerzy Warmi ´nski. A review on the mechanical behaviour of curvilinear fibrecomposite laminated panels. Journal of Composite Materials, 48(22):2761–2777, 2014.
  • [27] M.W. Hyer and H.H. Lee. The use of curvilinear fiber format to improve buckling resistance of composite plates with central circularholes. Composite Structures, 18(3):239 – 261, 1991.
  • [28] R. E. Rowlands, I. M. Daniel, and J. B. Whiteside. Stress and failure analysis of a glass-epoxy composite plate with a circular hole.Experimental Mechanics, 13(1):31–37, Jan 1973.
  • [29] Adriana W. Blom, Mostafa M. Abdalla, and Zafer G¨urdal. Optimization of course locations in fiber-placed panels for general fiberangle distributions. Composites Science and Technology, 70(4):564 – 570, 2010.
  • [30] Lotfi Toubal, Moussa Karama, and Bernard Lorrain. Stress concentration in a circular hole in composite plate. Composite Structures,68(1):31 – 36, 2005.
  • [31] A. V. Malakhov and A. N. Polilov. Construction of trajectories of the fibers which bypass a hole and their comparison with thestructure of wood in the vicinity of a knot. Journal of Machinery Manufacture and Reliability, 42(4):306–311, Jul 2013.
  • [32] A.V. Malakhov and A.N. Polilov. Design of composite structures reinforced curvilinear fibres using fem. Composites Part A: AppliedScience and Manufacturing, 87:23 – 28, 2016.
  • [33] A. Skordos, P. H. Chan, J. F. V. Vincent, and G. Jeronimidis. A novel strain sensor based on the campaniform sensillum of insects.Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 360(1791):239–253,2002.
  • [34] Ermias G. Koricho, Anton Khomenko, Tommy Fristedt, and Mahmoodul Haq. Innovative tailored fiber placement technique forenhanced damage resistance in notched composite laminate. Composite Structures, 120:378 – 385, 2015.
  • [35] M. Itskov. Tensor algebra and tensor analysis for engineers. Springer, Berlin, 2007.
  • [36] J. Casey. Approximate kinematical relations in plasticity. International Journal of Solids and Structures, 21:671–682, 1985.
  • [37] P. Haupt. Continuum Mechanics and Theory of Materials. Springer, Berlin, 2 edition, 2002.
  • [38] R. de Boer. Vektor- und Tensorrechnung f¨ur Ingenieure. Springer, Berlin, 1st edition, 1982.
  • [39] S. Hartmann. Finite-Elemente Berechnung inelastischer Kontinua. Interpretation als Algebro-Differentialgleichungssysteme. Habilitation,University of Kassel, Institute of Mechanics, 2003. Report No. 1/2003.
  • [40] F.M. White. Fluid Mechanics. McGraw-Hill, 2009.
Year 2018, , 17 - 28, 11.03.2018
https://doi.org/10.32323/ujma.398481

Abstract

References

  • [1] M. Kaliske. Zur Theorie und Numerik von Strukturen aus Faserverbundmaterial. Mitteilung nr. 49-99, Universit¨at Hannover, 1999.
  • [2] M. Fiolka. Theorie und Numerik volumetrischer Schalenelemente zur Delaminationsanalyse von Faserverbundlaminaten. Report no.2/2008, Institute of Mechanics, University of Kassel, 2008.
  • [3] M. Weise and A. Meyer. Grundgleichungen f ¨ur transversal isotropes Materialverhalten. Preprint CSC/10-03, TU Chemnitz, Fakult¨atf ¨ur Mathematik, Chemnitz (Germany), 2003.
  • [4] A. J. M. Spencer. Constitutive theory for strongly anisotropic solids. In A. J. M. Spencer, editor, Continuum theory of the mechanicsof fibre-reinforced composites, number 282 in Courses and lectures - International Centre for Mechanical Sciences, pages 1–32.Springer-Verlag, Wien, 1984.
  • [5] B.D. Agarwal, L.J. Broutman, and K. Chandrashekhara. Analysis and Performance of Fiber Composites. Wiley, 2006.
  • [6] R.F. Gibson. Principles of Composite Material Mechanics, Third Edition. Mechanical Engineering. CRC Press, 2011.
  • [7] V.Z. Parton and B.A. Kudryavtsev. Engineering Mechanics of Composite Structures. CRC Press, Inc., 1993.
  • [8] J. Aboudi, S.M. Arnold, and B.A. Bednarcyk. Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach.Elsevier Science, 2012.
  • [9] J. Aboudi. Mechanics of Composite Materials: A Unified Micromechanical Approach. Studies in Applied Mechanics. ElsevierScience, 2013.
  • [10] J. A. Weiss, B. N. Maker, and S. Govindjee. Finite element implementation of incompressible, transversely isotropic hyperelasticity.Computer Methods in Applied Mechanics and Engineering, 135:107–128, 1996.
  • [11] J. Schr ¨oder and P. Neff. Invariant formulation of hyperelasticity transverse isotropy based on polyconvex free energy functions.International Journal of Solids and Structures, 40:401–445, 2003.
  • [12] J. Schr ¨oder, P. Neff, and D. Balzani. A variational approach for materially stable anisotropic hyperelasticity. International Journal ofSolids and Structures, 42:4352–4371, 2005.
  • [13] M. Itskov and N. Aksel. A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvexstrain energy function. International Journal of Solids and Structures, 41:3833–3848, 2004.
  • [14] D. Balzani, P. Neff, J. Schr ¨oder, and G. A. Holzapfel. A polyconvex framework for soft biological tissues. adjustment to experimentaldata. International Journal of Solids and Structures, 43:6052–6070, 2006.
  • [15] G. A. Holzapfel, T. C. Gasser, and R. W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative studyof material models. Journal of Elasticity, 61:1–48, 2000.
  • [16] C. Sansour. On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. European Journal ofMechanics - A/Solids, 27:28–39, 2008.
  • [17] A. Zdunek, W. Rachowicz, and T. Eriksson. A novel computational formulation for nearly incompressible and nearly inextensiblefinite hyperelasticity. Computer Methods in Applied Mechanics and Engineering, 281:220–249, 2014.
  • [18] P. Wriggers, J. Schr ¨oder, and F. Auricchio. Finite element formulations for large strain anisotropic material with inextensible fibers.Advanced Modeling and Simulation in Engineering Sciences, 3:1–18, 2016.
  • [19] O. Sepahia, L. Radtke, S.E. Debus, and A. D¨uster. Anisotropic hierarchic solid finite elements for the simulation of passive-activearterial wall models. Computers & Mathematics with Applications, 2018.
  • [20] Gustavo Gonzalez Lozano, Ashutosh Tiwari, Christopher Turner, and Simon Astwood. A review on design for manufacture of variablestiffness composite laminates. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture,230(6):981–992, 2016.
  • [21] X. J. Niu, T. Yang, Y. Du, and Z. Q. Xue. Tensile properties of variable stiffness composite laminates with circular holes based onpotential flow functions. Archive of Applied Mechanics, 86(9):1551–1563, Sep 2016.
  • [22] Dani¨el M.J. Peeters, Simon Hesse, and Mostafa M. Abdalla. Stacking sequence optimisation of variable stiffness laminates withmanufacturing constraints. Composite Structures, 125:596 – 604, 2015.
  • [23] J. Huang and R.T. Haftka. Optimization of fiber orientations near a hole for increased load-carrying capacity of composite laminates.Structural and Multidisciplinary Optimization, 30(5):335–341, Nov 2005.
  • [24] Yingdan Zhu, Jiancai Liu, Dong Liu, Haibing Xu, Chun Yan, Bin Huang, and David Hui. Fiber path optimization based on a familyof curves in composite laminate with a center hole. Composites Part B: Engineering, 111:91 – 102, 2017.
  • [25] M. W. Hyer and R. F. Charette. Use of curvilinear fiber format in composite structure design. AIAA, 125:1011 – 1015, 1991.
  • [26] Pedro Ribeiro, Hamed Akhavan, Andrzej Teter, and Jerzy Warmi ´nski. A review on the mechanical behaviour of curvilinear fibrecomposite laminated panels. Journal of Composite Materials, 48(22):2761–2777, 2014.
  • [27] M.W. Hyer and H.H. Lee. The use of curvilinear fiber format to improve buckling resistance of composite plates with central circularholes. Composite Structures, 18(3):239 – 261, 1991.
  • [28] R. E. Rowlands, I. M. Daniel, and J. B. Whiteside. Stress and failure analysis of a glass-epoxy composite plate with a circular hole.Experimental Mechanics, 13(1):31–37, Jan 1973.
  • [29] Adriana W. Blom, Mostafa M. Abdalla, and Zafer G¨urdal. Optimization of course locations in fiber-placed panels for general fiberangle distributions. Composites Science and Technology, 70(4):564 – 570, 2010.
  • [30] Lotfi Toubal, Moussa Karama, and Bernard Lorrain. Stress concentration in a circular hole in composite plate. Composite Structures,68(1):31 – 36, 2005.
  • [31] A. V. Malakhov and A. N. Polilov. Construction of trajectories of the fibers which bypass a hole and their comparison with thestructure of wood in the vicinity of a knot. Journal of Machinery Manufacture and Reliability, 42(4):306–311, Jul 2013.
  • [32] A.V. Malakhov and A.N. Polilov. Design of composite structures reinforced curvilinear fibres using fem. Composites Part A: AppliedScience and Manufacturing, 87:23 – 28, 2016.
  • [33] A. Skordos, P. H. Chan, J. F. V. Vincent, and G. Jeronimidis. A novel strain sensor based on the campaniform sensillum of insects.Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 360(1791):239–253,2002.
  • [34] Ermias G. Koricho, Anton Khomenko, Tommy Fristedt, and Mahmoodul Haq. Innovative tailored fiber placement technique forenhanced damage resistance in notched composite laminate. Composite Structures, 120:378 – 385, 2015.
  • [35] M. Itskov. Tensor algebra and tensor analysis for engineers. Springer, Berlin, 2007.
  • [36] J. Casey. Approximate kinematical relations in plasticity. International Journal of Solids and Structures, 21:671–682, 1985.
  • [37] P. Haupt. Continuum Mechanics and Theory of Materials. Springer, Berlin, 2 edition, 2002.
  • [38] R. de Boer. Vektor- und Tensorrechnung f¨ur Ingenieure. Springer, Berlin, 1st edition, 1982.
  • [39] S. Hartmann. Finite-Elemente Berechnung inelastischer Kontinua. Interpretation als Algebro-Differentialgleichungssysteme. Habilitation,University of Kassel, Institute of Mechanics, 2003. Report No. 1/2003.
  • [40] F.M. White. Fluid Mechanics. McGraw-Hill, 2009.
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Stefan Hartmann

Ali Kheiri Marghzar This is me

Publication Date March 11, 2018
Submission Date February 25, 2018
Acceptance Date March 6, 2018
Published in Issue Year 2018

Cite

APA Hartmann, S., & Kheiri Marghzar, A. (2018). Modeling of fiber circumplacement around a hole using a streamline approach. Universal Journal of Mathematics and Applications, 1(1), 17-28. https://doi.org/10.32323/ujma.398481
AMA Hartmann S, Kheiri Marghzar A. Modeling of fiber circumplacement around a hole using a streamline approach. Univ. J. Math. Appl. March 2018;1(1):17-28. doi:10.32323/ujma.398481
Chicago Hartmann, Stefan, and Ali Kheiri Marghzar. “Modeling of Fiber Circumplacement Around a Hole Using a Streamline Approach”. Universal Journal of Mathematics and Applications 1, no. 1 (March 2018): 17-28. https://doi.org/10.32323/ujma.398481.
EndNote Hartmann S, Kheiri Marghzar A (March 1, 2018) Modeling of fiber circumplacement around a hole using a streamline approach. Universal Journal of Mathematics and Applications 1 1 17–28.
IEEE S. Hartmann and A. Kheiri Marghzar, “Modeling of fiber circumplacement around a hole using a streamline approach”, Univ. J. Math. Appl., vol. 1, no. 1, pp. 17–28, 2018, doi: 10.32323/ujma.398481.
ISNAD Hartmann, Stefan - Kheiri Marghzar, Ali. “Modeling of Fiber Circumplacement Around a Hole Using a Streamline Approach”. Universal Journal of Mathematics and Applications 1/1 (March 2018), 17-28. https://doi.org/10.32323/ujma.398481.
JAMA Hartmann S, Kheiri Marghzar A. Modeling of fiber circumplacement around a hole using a streamline approach. Univ. J. Math. Appl. 2018;1:17–28.
MLA Hartmann, Stefan and Ali Kheiri Marghzar. “Modeling of Fiber Circumplacement Around a Hole Using a Streamline Approach”. Universal Journal of Mathematics and Applications, vol. 1, no. 1, 2018, pp. 17-28, doi:10.32323/ujma.398481.
Vancouver Hartmann S, Kheiri Marghzar A. Modeling of fiber circumplacement around a hole using a streamline approach. Univ. J. Math. Appl. 2018;1(1):17-28.

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