This paper deals with the equilibrium problem of slender beams inflexed under variable curvature in the framework of fully nonlinear elasticity. For the specific case of uniform flexion, the authors have recently proposed a mathematical model. In that analysis, the complete three-dimensional kinematics of the beam is taken into account and both deformations and displacements are considered large. In the present paper, the kinematics of the aforementioned model has been reformulated taking into account beams under variable curvature. Subsequently, focusing on the local determination of the curvature, new equilibrium conditions on cross sections are introduced in the mathematical formulation. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically through an iterative procedure. Therefore, for the generic class of hyperelastic and isotropic materials, explicit formulae for the displacement field, the stretches and stresses in every point of the beam, following both Lagrangian and Eulerian descriptions, are derived. The analysis allows studying a very wide class of equilibrium problems for nonlinear beams under different restraint conditions and subject to generic external load systems. By way of example, the Euler beam has been considered and the formulae obtained have been specialized for a specific neoprene rubber material, the constitutive constants of which have been determined experimentally. The shapes assumed by the beam as the load multiplier increases are shown through some graphs. The distributions of stretches and Cauchy stresses are plotted for the most stressed cross section. Some comparisons are made using a FE code. In addition, the accuracy of the obtained solution is estimated by evaluating a posteriori that the equilibrium equations are locally satisfied.

本文在全非线性弹性框架下研究了变曲率下弯曲的细长梁的平衡问题。对于均匀屈曲的具体情况，作者最近提出了一个数学模型。在该分析中，梁的完整三维运动学被考虑在内，并且变形和位移都被认为是大的。在本文中，考虑到可变曲率下的梁，重新制定了上述模型的运动学。随后，着重于曲率的局部确定，在数学公式中引入了横截面的新平衡条件。控制方程采用积分形式的三个方程的耦合系统的形式，通过迭代程序进行数值求解。因此，对于超弹性和各向同性材料的通用类，根据拉格朗日和欧拉描述，推导出位移场、梁每个点的拉伸和应力的显式公式。该分析允许研究非线性梁在不同约束条件下并受一般外部载荷系统影响的非常广泛的平衡问题。例如，考虑了欧拉梁，并且获得的公式专门用于特定的氯丁橡胶材料，其本构常数已通过实验确定。随着载荷倍增器的增加，梁假定的形状通过一些图表显示出来。拉伸和柯西应力的分布是针对应力最大的横截面绘制的。一些比较是使用 FE 代码进行的。此外，通过评估获得的解决方案的准确性来估计局部满足平衡方程的后验。