Continuum Mechanics and Thermodynamics

, Volume 30, Issue 3, pp 573–592 | Cite as

Mechanics of an elastic solid reinforced with bidirectional fiber in finite plane elastostatics: complete analysis

Original Article


A continuum-based model is presented for the mechanics of bidirectional composites subjected to finite plane deformations. This is framed in the development of a constitutive relation within which the constraint of material incompressibility is augmented. The elastic resistance of the fibers is accounted for directly via the computation of variational derivatives along the lengths of bidirectional fibers. The equilibrium equation and necessary boundary conditions are derived by virtue of the principles of virtual work statement. A rigorous derivation of the corresponding linear theory is developed and used to obtain a complete analytical solution for small deformations superposed on large. The proposed model can serve as an alternative 2D Cosserat theory of nonlinear elasticity.


Finite plane deformations Fiber-reinforced material Bidirectional fiber Superposed incremental deformations Strain-gradient theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Spencer, A.J.M.: Deformations of Fibre-Reinforced Materials. Oxford University Press, Oxford (1972)MATHGoogle Scholar
  2. 2.
    Pipkin, A.C.: Stress analysis for fiber-reinforced materials. Adv. Appl. Mech. 19, 1–51 (1979)CrossRefMATHGoogle Scholar
  3. 3.
    Boutin, C.: Microstructural effects in elastic composites. Int. J. Solids and Struct. 33(7), 1023–1051 (1996)CrossRefMATHGoogle Scholar
  4. 4.
    Forest, S.: Homogenization methods and the mechanics of generalised continua part 2. Theor. Appl. Mech. 28, 113–143 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Mulhern, J.F., Rogers, T.G., Spencer, A.J.M.: A continuum theory of a plastic–elastic fibre-reinforced material. Int. J. Eng. Sci. 7, 129–152 (1969)CrossRefMATHGoogle Scholar
  6. 6.
    Pipkin, A.C., Rogers, T.G.: Plane deformations of incompressible fiber-reinforced materials. ASME J. Appl. Mech. 38(8), 634–640 (1971)ADSCrossRefMATHGoogle Scholar
  7. 7.
    Spencer, A.J.M., Soldatos, K.P.: Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. Int. J. Non-Linear Mech. 42, 355–368 (2007)ADSCrossRefGoogle Scholar
  8. 8.
    Toupin, R.A.: Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Koiter, W.T.: Couple-stresses in the theory of elasticity. Proc. K. Ned. Akad. Wetensc. B 67, 17–44 (1964)MathSciNetMATHGoogle Scholar
  11. 11.
    Park, H.C., Lakes, R.S.: Torsion of a micropolar elastic prism of square cross section. Int. J. Solids Struct. 23, 485–503 (1987)CrossRefMATHGoogle Scholar
  12. 12.
    Maugin, G.A., Metrikine, A.V. (eds.): Mechanics of Generalized Continua: One Hundred Years After the Cosserats. Springer, New York (2010)MATHGoogle Scholar
  13. 13.
    Neff, P.: A finite-strain elastic–plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci. 44, 574–594 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Munch, I., Neff, P., Wagner, W.: Transversely isotropic material: nonlinear Cosserat vs. classical approach. Cont. Mech. Therm. 23, 27–34 (2011)CrossRefMATHGoogle Scholar
  15. 15.
    Neff, P.: Existence of minimizers for a finite-strain micro-morphic elastic solid. Proc. R. Soc. Edinb. A 136, 997–1012 (2006)CrossRefMATHGoogle Scholar
  16. 16.
    Park, S.K., Gao, X.L.: Variational formulation of a modified couple-stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 59, 904–917 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fried, E., Gurtin, M.E.: Gradient nanoscale polycrystalline elasticity: intergrain interactions and triple-junction conditions. J. Mech. Phys. Solids 57, 1749–1779 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Steigmann, D.J.: Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist. Int. J. Non-Linear Mech. 47, 734–742 (2012)ADSCrossRefGoogle Scholar
  19. 19.
    Steigmann, D.J., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech. Sin. 31(3), 373–382 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Z. Angw. Math. Phys. 67(5), 114 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.L.: A review on 2D models for the description of pantographic fabrics. Z. Angw. Math. Phys. 67(5), 121 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Cuomo, M., dell’Isola, F., Greco, L., Rizzi, N.L.: First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. Part B Eng. 115, 423–448 (2017). CrossRefGoogle Scholar
  23. 23.
    dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 103(1), 127–157 (2017). MathSciNetCrossRefGoogle Scholar
  24. 24.
    dell’Isola, F., Della Corte, A., Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)CrossRefGoogle Scholar
  25. 25.
    dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z. Angw. Math. Phys. ZAMP 66(6), 3473–3498 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. A 472(2185), 20150790 (2016)ADSCrossRefGoogle Scholar
  27. 27.
    Zeidi, M., Kim, C.: Mechanics of fiber composites with fibers resistant to extension and flexure. Math. Mech. Solids (2017). Google Scholar
  28. 28.
    Zeidi, M., Kim, C.: Finite plane deformations of elastic solids reinforced with fibers resistant to flexure: complete solution. Arch. Appl. Mech. (2017). Google Scholar
  29. 29.
    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Pergamon, Oxford (1986)MATHGoogle Scholar
  30. 30.
    Dill, E.H.: Kirchhoff’s theory of rods. Arch. Hist. Exact Sci. 44, 1–23 (1992)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, Berlin (2005)MATHGoogle Scholar
  32. 32.
    Dell’Isola, F., Steigmann, D.J.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118(1), 113–125 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Dong, C., Davies, I.J.: Flexural strength of bidirectional hybrid epoxy composites reinforced by E glass and T700S carbon fibres. Compos. Part B 72, 65–71 (2015)CrossRefGoogle Scholar
  34. 34.
    Ogden, R.W.: Non-linear Elastic Deformations. Ellis Horwood Ltd., Chichester (1984)MATHGoogle Scholar
  35. 35.
    Read, W.W.: Analytical solutions for a Helmholtz equation with Dirichlet boundary conditions and arbitrary boundaries. Math. Comput. Model. 24(2), 23–34 (1996)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Read, W.W.: Series solutions for Laplace’s equation with nonhomogeneous mixed boundary conditions and irregular boundaries. Math. Comput. Model. 17(12), 9–19 (1993)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Huang, Y., Zhang, X.: General analytical solution of transverse vibration for orthotropic rectangular thin plates. J. Mar. Sci. Appl. 1(2), 78–82 (2002)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Flugge, S. (ed.) Handbuch der Physik, vol. 3. Springer, Berlin (1965)Google Scholar
  39. 39.
    Reissner, E.: A further note on finite-strain force and moment stress elasticity. Z. Angew. Math. Phys. 38, 665–673 (1987)CrossRefMATHGoogle Scholar
  40. 40.
    Pietraszkiewicz, W., Eremeyev, V.A.: On natural strain measures of the non-linear micropolar continuum. Int. J. Solids Struct. 46, 774–787 (2009)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Germain, P.: The method of virtual power in continuum mechanics, part 2: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)CrossRefMATHGoogle Scholar
  42. 42.
    Turco, E., et al.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations. Z. Angw. Math. Phys 67(5), 122 (2016)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Turco, E., Barcz, K., Luigi Rizzi, N.: Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part II: comparison with experimental evidence. Z. Angw. Math. Phys. 67(5), 123 (2016)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics á Hamilton–Piola: least action principle for second gradient continua and capillary fluids. Mech. Math. Solids (MMS) 20(4), 375–417 (2015)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    dell’Isola, F., Andreaus, U., Placidi, L.: 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. Mech. Math. Solids (MMS) 20(8), 887–928 (2015)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro–macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017). MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)CrossRefGoogle Scholar
  48. 48.
    dell’Isola, F., Steigmann, D.J., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67 (6), 060804, 21 pages (2016)Google Scholar
  49. 49.
    Turco, E., Golaszewski, M., Cazzani, A., Rizzi, N.L.: Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model. Mech. Res. Commun. 76, 51–56 (2016)CrossRefGoogle Scholar
  50. 50.
    Abali, B.E., Muller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87, 1495–1510 (2017)ADSCrossRefGoogle Scholar
  51. 51.
    Cuomo, M., dell’Isola, F., Greco, L.: Simplified analysis of a generalized bias-test for fabrics with two families of inextensible fibres. Z. Angw. Math. Phys. 39 pages. (2016)
  52. 52.
    Abali, B.E.: Technical University of Berlin, Institute of Mechanics, Chair of Continuums Mechanics and Material Theory. Computational Reality. (2017)
  53. 53.
    Alnaes, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS Project Version 1.5. Arch. Numer. Softw. 3, 9–23 (2015). Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada

Personalised recommendations