We propose deducing from three-dimensional elasticity a one dimensional model of a beam when the lateral boundary is not free of traction. Thus the simplification induced by the order of magnitude of transverse shearing and transverse normal stress must be removed. For the sake of simplicity we consider a beam with rectangular cross section. The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod and we truncate the potential energy at the fourth order. By considering exact equilibrium equations, the highest-order displacement field can be removed and the Euler–Lagrange equations are simplified.
RigolotA.Sur la théorie asymptotique des poutres. PhD Thesis, Université Pierre et Marie Curie, Paris, North-Holland, 1976.
2.
TrabuchoLVianoJM.Mathematical modelling of rods. In: CiarletPGLionsJL (eds) Handbook of Numerical Analysis. Vol. IV. North-Holland: Elsevier, 1996, pp.487–979.
3.
MeunierN.Recursive derivation of one-dimensional models from three-dimensional nonlinear elasticity. Math Mech Solids2008; 13: 172–194.
4.
MarigoJJMeunierN.Hierarchy of one-dimensional models in nonlinear elasticity. J Elast2006; 83: 1–28.
5.
MoraMGMüllerS.Derivation of the nonlinear bending torsion theory for inextensible rods by Γ-convergence. Calc Var2003; 18: 287–305.
6.
PruchnickiE.One-dimensional models of fourth and sixth orders for rods derived from three-dimensional elasticity. Math Mech Solids2015. doi: 10.1177/1081286515577121.
7.
CarreraEGiuntaG.Refined beam theories based on a unified formulation. Int J Appl Mech2010; 2: 117–143.
FicheraG.Existence theorems in elasticity. In: TruesdellC (ed.) Handbusch der Physik. Vol. VIa/2. Berlin: Springer, 1972, pp.347–389.
10.
PruchnickiE.One-dimensional model for the combined bending, stretching, shearing and torsion of rods derived from three-dimensional elasticity. Math Mech Solids2012; 17: 378–392.
11.
DiCarloAPodio-GuidugliPWilliamsWO.Shells with thickness distension. Int J Solids Struct2001; 38: 1201–1225.
