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**A finite strain beam formulation. The three-dimensional dynamic problem. I.**
*(English)*
Zbl 0583.73037

In the context of a director type of approach, S. S. Antman has developed a three-dimensional extension of the classical Kirchhoff-Love rod model including finite extension and finite shearing [Q. Appl. Math. 32, 221-240 (1974; Zbl 0302.73031)]. This paper should be regarded as a convenient parametrization of this extension, realized by constraining the 3-dimensional theory with the introduction of the kinematic assumption.

In section 1, the formulation of the basic kinematics of the beam is given in terms of a 3-dimensional orthogonal moving frame defined so that one of its vectors, denoted by n, remains normal to a typical cross- section in any configuration. Corresponding expressions for the linear and angular momentum are obtained in section 2. The former is associated with the acceleration of the centroid and the latter with the time derivative of the vorticity vector of the moving frame. Next, the basic laws of motion are summarized in section 3, and the spatial and material descriptions of the resultant force acting on a cross-section are discussed.

In section 4, starting with the general 3-dimensional expression for the internal power, the author develops spatial and material reduced expressions involvig the resultant force and torque and their conjugate strain rates. In the material description the reduced expression for the internal power enables one to identify the appropriate strain measures. In particular, the strain measure conjugate to the resultant torque has a simple geometric interpretation as the axial vector of a skew-symmetric tensor associated with the moving frame. In the spatial description, in addition to the appropriate strain measures, the reduced expression of the internal power determines the appropriate (objective) strain rate as the rate of change relative to an observer spinning with the moving frame. Finally, in section 5, i) it is shown that for the plane problem the present development reduces to that proposed by E. Reissner [Z. Angew. Math. Phys. 23, 795-804 (1972; Zbl 0248.73022)]; ii) a remark is given on the significance of the parametrization employed in this paper in a finite element solution procedure.

Issues concerning the resulting geometric stiffness and the numerical implementation of the formulation discussed here in the context of the finite element method will be addressed in a forthcoming paper.

In section 1, the formulation of the basic kinematics of the beam is given in terms of a 3-dimensional orthogonal moving frame defined so that one of its vectors, denoted by n, remains normal to a typical cross- section in any configuration. Corresponding expressions for the linear and angular momentum are obtained in section 2. The former is associated with the acceleration of the centroid and the latter with the time derivative of the vorticity vector of the moving frame. Next, the basic laws of motion are summarized in section 3, and the spatial and material descriptions of the resultant force acting on a cross-section are discussed.

In section 4, starting with the general 3-dimensional expression for the internal power, the author develops spatial and material reduced expressions involvig the resultant force and torque and their conjugate strain rates. In the material description the reduced expression for the internal power enables one to identify the appropriate strain measures. In particular, the strain measure conjugate to the resultant torque has a simple geometric interpretation as the axial vector of a skew-symmetric tensor associated with the moving frame. In the spatial description, in addition to the appropriate strain measures, the reduced expression of the internal power determines the appropriate (objective) strain rate as the rate of change relative to an observer spinning with the moving frame. Finally, in section 5, i) it is shown that for the plane problem the present development reduces to that proposed by E. Reissner [Z. Angew. Math. Phys. 23, 795-804 (1972; Zbl 0248.73022)]; ii) a remark is given on the significance of the parametrization employed in this paper in a finite element solution procedure.

Issues concerning the resulting geometric stiffness and the numerical implementation of the formulation discussed here in the context of the finite element method will be addressed in a forthcoming paper.

Reviewer: M.Bernadou

### MSC:

74B20 | Nonlinear elasticity |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74S05 | Finite element methods applied to problems in solid mechanics |

### Keywords:

director type of approach; finite extension; finite shearing; constraining the 3-dimensional theory; basic kinematics; 3-dimensional orthogonal moving frame; linear and angular momentum; acceleration of the centroid; time derivative of the vorticity vector; basic laws of motion; spatial and material descriptions; resultant force; internal power; conjugate strain rates; reduced expression for the internal power; appropriate strain measures; plane problem; parametrization; resulting geometric stiffness; numerical implementation
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\textit{J. C. Simo}, Comput. Methods Appl. Mech. Eng. 49, 55--70 (1985; Zbl 0583.73037)

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### References:

[1] | Antman, S. S., The Theory of Rod, (Handbuch der Physik, Vol. VIa/2 (1972), Springer: Springer Berlin) · Zbl 0188.57501 |

[2] | Argyris, J. H., An excursion into large rotations, Comput. Meths. Appl. Mech. Engrg., 32, 85-155 (1982) · Zbl 0505.73064 |

[3] | Love, A. E.H., The Mathematical Theory of Elasticity (1944), Dover: Dover New York · Zbl 0063.03651 |

[4] | Naghdi, P. M., Finite deformations of elastic rods shells, (Proc. IUTAM Symposium on Finite Elasticity (1980), Lehigh University) · Zbl 0523.73033 |

[5] | Reissner, E., On one-dimensional finite strain beam theory: The plane problem, J. Appl. Math. Phys., 23, 795-804 (1972) · Zbl 0248.73022 |

[6] | Reissner, E., Some remarks on the problem of column buckling, Ingenieur-Archiv., 52, 115-119 (1982) · Zbl 0487.73047 |

[7] | Simo, J. C.; Hjelmstad, K. H.; Taylor, R. L., Numerical formulations for the elasto-viscoplastic response of beams accounting for the effect of shear, Comput. Meths. Appl. Mech. Engrg., 42, 301-330 (1984) · Zbl 0517.73074 |

[8] | Abraham, R.; Marsden, J. E.; Ratiu, T., Manifolds, Tensor Analysis and Applications (1983), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0875.58002 |

[9] | Antman, S. S., Kirchhoff’s problem for nonlinearly elastic rods, Quart. J. Appl. Math., 32, 221-240 (1974) · Zbl 0302.73031 |

[10] | Argyris, J. H.; Balmer, H.; Doltsinis, J. St.; Dunne, P. C.; Haase, M.; Kleiber, M.; Malejannakis, G. A.; Mlejnek, H. P.; Muller, M.; Scharpf, D. W., Finite element method-The natural approach, Comput. Meths. Appl. Mech. Engrg., 17/18, 1-106 (1979) · Zbl 0407.73058 |

[11] | Argyris, J. H.; Symeonidis, Sp., Nonlinear finite element analysis of elastic systems under nonconservative loading — natural formulation. Part I. Quasistatic problems, Comput. Meths. Appl. Mech. Engrg., 26, 75-123 (1981) · Zbl 0463.73073 |

[12] | Argyris, J. H.; Symeonidis, Sp., A sequel to: Nonlinear finite element analysis of elastic systems under nonconservative loading — natural formulation. Part I. Quasistatic problems, Comput. Meths. Appl. Mech. Engrg., 26, 377-383 (1981) · Zbl 0484.73059 |

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