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Theoretical Analysis

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The Bending Theory of Fully Nonlinear Beams

Abstract

This chapter deals with the equilibrium problem of fully nonlinear beams in bending by extending the model for the anticlastic flexion of solids recently proposed in the context of finite elasticity by Lanzoni and Tarantino (J Elast 131:137–170, 2018, [1]). Initially, kinematics is reformulated and, subsequently, a nonlinear theory for the bending of slender beams has been developed. In detail, no hypothesis of smallness is introduced for the deformation and displacement fields, the constitutive law is considered nonlinear and the equilibrium is imposed in the deformed configuration. Explicit formulas are obtained which describe the displacement field, stretches and stresses for each point of the beam using both the Lagrangian and Eulerian descriptions.

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Notes

  1. 1.

    Recent contributions on the nonlinear damage theory can be found in [33,34,35,36].

  2. 2.

    \(\mathcal {V}\) is the vector space associated with \(\mathcal {E}\).

  3. 3.

    Lin is the set of all (second order) tensors whereas \(Lin^{+}\) is the subset of tensors with positive determinant.

  4. 4.

    For slender beams, the kinematic model contains only one unknown geometrical parameter: the anticlastic radius r, while for a block the geometrical parameters are three [1].

  5. 5.

    The existence of at least one arc, for which \(\lambda _{Z}=1\), is ensured by the continuity of deformation. Uniqueness is guaranteed by the hypothesis 2 of conservation of the planarity of the cross section.

  6. 6.

    In the case of a block, the two arcs, for which \(\lambda _{Z}=1\) and \(\lambda _{Y}=1\), are distinct and do not pass through the point \(O'\). Instead when a slender beam is considered, the distance between the two arcs vanishes [1].

  7. 7.

    In the sequel, it will be found the relationship between this angle \(\alpha _{0}\) and the pair of self-balanced bending moment to apply to the end faces of the beam.

  8. 8.

    It is assumed that the isotropy property is preserved in the deformed configuration.

  9. 9.

    The other components of tensor C are zero.

  10. 10.

    Or equivalently by the principal invariants of the right Cauchy-Green strain tensor \(\mathbf {C}\).

  11. 11.

    The following notations: \(\parallel \mathbf {A}\parallel =\left( \mathrm {tr}\mathbf {A}^{\mathrm {T}}\mathbf {A}\right) ^{1/2}\) for the tensor norm in the linear tensor space Lin and \(\mathbf {A}^{\star }=(\mathrm {det}\mathbf {A})\mathbf {A}^{\mathrm {-T}}\) for the cofactor of the tensor A (if A is invertible) are used.

  12. 12.

    This function is polyconvex and satisfies the growth conditions: \(\omega \rightarrow \infty \) as \(\lambda \rightarrow 0^{+}\) or \(\lambda \rightarrow +\infty \). It was used, for example, in [57,58,59,60].

  13. 13.

    Similar positions can be found in [61,62,63,64].

  14. 14.

    The first equation of system (1.22) is also verified for all points of the vertical plane \(X=0\).

  15. 15.

    The same symbols are used for normalized and non-normalized constants.

  16. 16.

    Using (1.25), it can be promptly verified that, in the absence of deformation, \(\lambda =\lambda _{Z}=1\), is \(S_{Z}=N=0\).

  17. 17.

    From (1.32)\(_{3}\) the quantity in square brackets is attained and then replaced into (1.32)\(_{2}\), obtaining (1.33)\(_{3}\). Similarly, from (1.32)\(_{1}\), \(r\,e^{-\frac{Y}{r}}\) is evaluated and then substituted into (1.32)\(_{2}\), obtaining (1.33)\(_{1}\). Expression (1.33)\(_{2}\) is evaluated directly from (1.32)\(_{2}\) using (1.33)\(_{1}\) and (1.33)\(_{3}\).

  18. 18.

    It can be see that, by taking \(B=2\beta _{0}r\), (1.41) reduces to BH as \(r\rightarrow \infty \). Similarly, by taking \(L=2\alpha _{0}R_{0}\), (1.43) becomes \(V'=BHL\) as \((R_{0},\,r)\rightarrow \infty \).

  19. 19.

    The Landau symbols are used.

  20. 20.

    Using the Taylor series expansions, the following approximations are employed:

    $$ e^{-\frac{Y}{r}}\simeq 1-\frac{Y}{r}+\frac{Y^{2}}{2\,r^{2}}+o(r^{-2}),\quad \sin \frac{X}{r}\simeq \frac{X}{r}+o(r^{-2}),\quad \cos \frac{X}{r}\simeq 1-\frac{X^{2}}{2\,r^{2}}+o(r^{-3}), $$
    $$ \sin \frac{Z}{R_{0}}\simeq \frac{Z}{R_{0}}+o(R_{0}^{-2}),\quad \cos \frac{Z}{R_{0}}\simeq 1-\frac{Z^{2}}{2\,R_{0}^{2}}+o(R_{0}^{-3}). $$
  21. 21.

    In the sequel, the infinitesimal terms of higher order are omitted definitively.

  22. 22.

    After linearization, the following relationships hold: \(\mathrm {\mathbf {R}}=\mathbf {I}+\mathbf {W}\), \(\mathbf {U}=\mathbf {I}+\mathbf {E}\).

  23. 23.

    Using the Taylor series expansions, the following approximation is employed:

    $$ \sinh \frac{H}{2r}\simeq \frac{H}{2r}+o(r^{-2}), $$

    as well as similar expressions for different arguments of hyperbolic sine function.

  24. 24.

    Using the Taylor series expansions, the following approximation is employed:

    $$ \frac{1}{\lambda }\simeq 1+\frac{Y}{r}+o(r^{-1}), $$

    and the relationship among the constitutive constants (1.25) has been used to obtain (1.63).

  25. 25.

    Using the Taylor series expansions, the following approximation is employed:

    $$ \frac{1}{\lambda ^{2}\lambda _{Z}}\simeq 1+\frac{2Y}{r}-\frac{Y}{R_{0}}+o(r^{-1})+o(R_{0}^{-1}). $$

    .

  26. 26.

    Of course, the same result can be achieved for a compressible Mooney-Rivlin material that satisfies the conditions (1.69). In effect, replacing (1.70) into (1.63), it is found

    $$ S=\left[ -(4a+12b+8c)+(4b+4c)\,\frac{1}{\nu }\right] \frac{Y}{r}=0, $$
    $$ S_{Z}=\left[ -(8b+8c)\,\nu +(4a+8b+4c)\right] \,\frac{Y}{R_{0}}=E\,\varepsilon _{z}. $$

    .

References

  1. L. Lanzoni, A.M. Tarantino, Finite anticlastic bending of hyperelastic solids and beams. J. Elast. 131, 137–170 (2018). https://doi.org/10.1007/s10659-017-9649-y

  2. J. Bernoulli, Specimen alterum calculi differentialis in dimetienda spirali logarithmica, loxodromiis nautarum et areis triangulorum sphaericorum. Una cum additamento quodam ad problema funicularium, aliisque. Acta Eruditorum, Junii 282–290–Opera, 442–453

    Google Scholar 

  3. J. Bernoulli, Véritable hypothèse de la résistance des solides, avec la démonstration de la courbure des corps qui font ressort. Académie Royale des Sciences, Paris (1705)

    Google Scholar 

  4. A. Parent, Essais et Recherches de Mathématique et de Physique, Nouv. Ed., Paris (1713)

    Google Scholar 

  5. L. Euler, Mechanica, sive, Motus scientia analytice exposita (Ex typographia Academiae Scientiarum, Petropoli, 1736)

    Google Scholar 

  6. L. Euler, Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes (Bousquent, Lausanne, 1744)

    Google Scholar 

  7. L. Euler, Genuina principia doctrinae de statu aequilibrii et motu corporum tam perfecte flexibilium quam elasticorum. Opera Omnia II 11, 37–61 (1771)

    Google Scholar 

  8. L. Euler, De gemina methodo tam aequilibrium quam motum corporum flexibilium determinandi et utriusque egregio consensu. Novi Commentarii academiae scientiarum Petropolitanae 20, 286-303 (1776)

    Google Scholar 

  9. C.L.M.H. Navier, Mémoire sur les lois de l’équilibre et du mouvement des corps solides élastiques. Mémoires de l’Académie des Sciences de l’ Institut de France, s. 2 7 375-393

    Google Scholar 

  10. A.-J.-C. Barré de Saint-Venant, Memoire sur la torsion des prismes. Comptes rendus de l’ Académie des Sci. 37 (1853)

    Google Scholar 

  11. J.A.C. Bresse, Recherches analytiques sur la flexion et la résistance des pièces courbes (Carilian-Goeury et VrDalmont Libraires, Paris, 1854)

    Google Scholar 

  12. H. Lamb, Sur la flexion d’un ressort élastique plat. Philos. Mag. 31, 182–188 (1891)

    Article  Google Scholar 

  13. W. Thomson (Lord Kelvin), P.G. Tait, Treatise on Natural Philosophy (Cambridge University Press, Cambridge, 1867)

    Google Scholar 

  14. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. (Cambridge University Press, Cambridge, 1927)

    MATH  Google Scholar 

  15. B.R. Seth, Finite strain in elastic problems. Proc. R. Soc. Lond. A 234, 231–264 (1935)

    MATH  Google Scholar 

  16. R.S. Rivlin, Large elastic deformations of isotropic materials. V. The problem of flexure. Proc. R. Soc. Lond. A 195, 463–473 (1949)

    Google Scholar 

  17. R.S. Rivlin, Large elastic deformations of isotropic materials. VI. Further results in the theory of torsion, shear and flexure. Proc. R. Soc. Lond. A 242, 173–195 (1949)

    Google Scholar 

  18. J.L. Ericksen, Deformations possible in every isotropic, incompressible, perfectly elastic body. ZAMP J. Appl. Math. Phys 5, 466–489 (1954)

    Google Scholar 

  19. M.M. Carroll, Finite deformations of incompressible simple solids I. Isotropic solids. Quart. J. Mech. Appl. Math. 21, 148–170 (1968)

    Google Scholar 

  20. N. Triantafyllidis, Bifurcation phenomena in pure bending. J. Mech. Phys. Solids. 28, 221–245 (1980)

    Article  MathSciNet  Google Scholar 

  21. O.T. Bruhns, N.K. Gupta, A.T.M. Meyers, H. Xiao, Bending of an elastoplastic strip with isotropic kinematic hardening. Arch. Appl. Mech. 72, 759–778 (2003)

    Article  Google Scholar 

  22. C.-C. Wang, Normal configurations and the nonlinear elastoplastic problems of bending, torsion, expansion, and eversion for compressible bodies. Arch. Ration. Mech. Anal. 114, 195–236 (1991)

    Google Scholar 

  23. R.T. Shield, Bending of a beam or wide strip. Quart. J. Mech. Appl. Math. 45, 567–573 (1992)

    Article  Google Scholar 

  24. M. Aron, Y. Wang, On deformations with constant modified stretches describing the bending of rectangular blocks. Quart. J. Mech. Appl. Math. 48, 375–387 (1995)

    Article  MathSciNet  Google Scholar 

  25. R.W. Ogden, Non-linear Elastic Deformations (Ellis Horwood, Chichester, 1984 and Dover Publications 1997)

    Google Scholar 

  26. O.T. Bruhns, H. Xiao, A. Meyers, Finite bending of a rectangular block of an elastic Hencky material. J. Elast. 66, 237–256 (2002)

    Article  MathSciNet  Google Scholar 

  27. D.M. Haughton, Flexure and compression of incompressible elastic plates. Int. J. Eng. Sci. 37, 1693–1708 (1999)

    Article  MathSciNet  Google Scholar 

  28. C. Coman, M. Destrade, Asymptotic results for bifurcations in pure bending of rubber blocks. Quart. J. Mech. Appl. Math. 61, 395–414 (2008)

    Article  MathSciNet  Google Scholar 

  29. S. Roccabianca, M. Gei, D. Bigoni, Plane strain bifurcations of elastic layered structures subject to finite bending: theory versus experiments. IMA J. Appl. Math. 75, 525–548 (2010)

    Article  MathSciNet  Google Scholar 

  30. A.N. Gent, I.S. Cho, Surface instabilities in compressed or bent rubber blocks. Rubber Chem. Tech. 72, 253–262 (1999)

    Article  Google Scholar 

  31. F. Kassianidis, R.W. Ogden, On large bending deformations of transversely isotropic rectangular elastic blocks. Note di Matematica 27, 131–154 (2007)

    MathSciNet  MATH  Google Scholar 

  32. K.R. Rajagopal, A.R. Srinivasa, A.S. Wineman, On the shear and bending of a degrading polymer beam. Int. J. Plast. 23, 1618–1636 (2007)

    Article  Google Scholar 

  33. A.M. Tarantino, Equilibrium paths of a hyperelastic body under progressive damage. J. Elast. 114 225–250 (2014)

    Google Scholar 

  34. L. Lanzoni, A.M. Tarantino, Damaged hyperelastic membranes. Inter. J. Nonlinear Mech. 60, 9–22 (2014)

    Article  Google Scholar 

  35. L. Lanzoni, A.M. Tarantino, Equilibrium configurations and stability of a damaged body under uniaxial tractions. ZAMP J. Appl. Math. Phys. 66 171–190 (2015)

    Google Scholar 

  36. L. Lanzoni, A.M. Tarantino, A simple nonlinear model to simulate the localized necking and neck propagation. Inter. J. Nonlinear Mech. 84, 94–104 (2016)

    Article  Google Scholar 

  37. L.M. Kanner, C.O. Horgan, Plane strain bending of strain-stiffening rubber-like rectangular beams. Inter. J. Solid. Struct. 45, 1713–1729 (2008)

    Article  Google Scholar 

  38. M. Destrade, A.N. Annaidh, C.D. Coman, Bending instabilities of soft biological tissues. Inter. J. Solid. Struct. 46, 4322–4330 (2009)

    Article  Google Scholar 

  39. T.M. Wang, S.L. Lee, O.C. Zienkiewicz, Numerical analysis of large deflections of beams. Inter. J. Mech. Sci. 3, 219–228 (1961)

    Article  Google Scholar 

  40. R. Frisch-Fay, Flexible Bars (Butterworths, London, 1962)

    MATH  Google Scholar 

  41. T.M. Wang, Non-linear bending of beams with uniformly distributed loads. Inter. J. Nonlinear Mech. 4, 389–395 (1969)

    Article  Google Scholar 

  42. J.T. Holden, On the finite deflections of thin beams. Inter. J. Solid. Struct. 8, 1051–11055 (1972)

    Article  Google Scholar 

  43. E. Reissner, On one-dimensional finite-strain beam theory: the plane problem. ZAMP J. Appl. Math. Phys. 23 795–804 (1972)

    Google Scholar 

  44. E. Reissner, On one-dimensional large-displacement finite-strain beam theory. Stud. Appl. Math. 52, 87–95 (1973)

    Article  Google Scholar 

  45. E. Reissner, On finite deformations of space-curved beams. ZAMP J. Appl. Math. Phys. 32, 734–744 (1981)

    Google Scholar 

  46. K.-J. Bathe, S. Bolourchi, Large displacement analysis of three-dimensional beam structures. Num. Meth. Eng. 14, 961–986 (1979)

    Article  Google Scholar 

  47. J.C. Simo, A finite strain beam formulation. The three -dimensional beam structures. Part I. Comput. Meth. Appl. Mech. Eng. 49, 55–70 (1985)

    Google Scholar 

  48. J.C. Simo, L. Vu-Quoc, A three-dimensional finite-strain rod model. Part II: Computational aspects. Comput. Meth. Appl. Mech. Eng. 58, 79–116 (1986)

    Google Scholar 

  49. J.C. Simo, L. Vu-Quoc, On the dynamics of flexible beams under large overall motions-the plane case: Part I. J. Appl. Mech. 53, 849–854 (1986)

    Article  Google Scholar 

  50. J.C. Simo, L. Vu-Quoc, On the dynamics of flexible beams under large overall motions-the plane case: Part II. J. Appl. Mech. 53, 854–855 (1986)

    MATH  Google Scholar 

  51. A. Cardona, M. Geradin, A beam finite element non-linear theory with finite rotations. Int. J. Numer. Meth. Eng. 26, 2403–2438 (1988)

    Article  Google Scholar 

  52. J.C. Simo, L. Vu-Quoc, On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput. Meth. Appl. Mech. Eng. 66, 125–161 (1988)

    Article  MathSciNet  Google Scholar 

  53. A. Ibrahimbegovič, On finite element implementation of geometrically nonlinear Reissner’ s beam theory: three-dimensional curved beam elements. Comput. Methods Appl. Mech. Eng. 122, 11–26 (1995)

    Article  Google Scholar 

  54. F. Auricchio, P. Carotenuto, A. Reali, On the geometrically exact beam model: a consistent, effective and simple derivation from three-dimensional finite-elasticity. Inter. J. Solid. Struct. 45, 4366–4781 (2008)

    Article  Google Scholar 

  55. A.K. Nallathambi, C.L. Rao, S.M. Srinivasan, Large deflection of constant curvature cantilever beam under follower load. Int. J. Mech. Sci. 52, 440–445 (2010)

    Article  Google Scholar 

  56. K. Lee, Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int. J. Nonlinear Mech. 37, 439–443 (2002)

    Article  Google Scholar 

  57. A.M. Tarantino, Thin hyperelastic sheets of compressible material: field equations, Airy stress function and an application in fracture mechanics. J. Elast. 44, 37–59 (1996)

    Google Scholar 

  58. A.M. Tarantino, The singular equilibrium field at the notch-tip of a compressible material in finite elastostatics. ZAMP J. Appl. Math. Phys. 48, 370–388 (1997)

    Google Scholar 

  59. A.M. Tarantino, On extreme thinning at the notch-tip of a neo-Hookean sheet. Quart. J. Mech. Appl. Mech. 51(2), 179–190 (1998)

    Google Scholar 

  60. A.M. Tarantino, On the finite motions generated by a mode I propagating crack. J. Elast. 57, 85–103 (1999)

    Google Scholar 

  61. A.M. Tarantino, Crack propagation in finite elastodynamics. Math. Mech. Solids 10 577–601 (2005)

    Google Scholar 

  62. A.M. Tarantino, Nonlinear fracture mechanics for an elastic Bell material. Quart. J. Mech. Appl. Math. 50, 435–456 (1997)

    Article  MathSciNet  Google Scholar 

  63. A.M. Tarantino, A. Nobili, Finite homogeneous deformations of symmetrically loaded compressible membranes. ZAMP J. Appl. Math. Phys. 58, 659–678 (2006)

    Google Scholar 

  64. A.M. Tarantino, Homogeneous equilibrium configurations of a hyperelastic compressible cube under equitriaxial dead-load tractions. J. Elast. 92, 227–254 (2008)

    Article  MathSciNet  Google Scholar 

  65. A.M. Tarantino, Scienza delle Costruzioni (Pitagora Editrice Bologna, Bologna, 2005). in Italian

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia, Modena, Italy

    Angelo Marcello Tarantino, Luca Lanzoni & Federico Oyedeji Falope

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  1. Angelo Marcello Tarantino
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  2. Luca Lanzoni
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  3. Federico Oyedeji Falope
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Correspondence to Angelo Marcello Tarantino .

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Tarantino, A.M., Lanzoni, L., Falope, F.O. (2019). Theoretical Analysis. In: The Bending Theory of Fully Nonlinear Beams. Springer, Cham. https://doi.org/10.1007/978-3-030-14676-4_1

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  • DOI: https://doi.org/10.1007/978-3-030-14676-4_1

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