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首页> 外文期刊>International journal of non-linear mechanics >On the large deflections of linear viscoelastic beams
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On the large deflections of linear viscoelastic beams

机译：关于线性粘弹性梁的大挠度

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The quasi-static response and the stored and dissipated energies due to large deflections of a slender inextensible beam made of a linear viscoelastic material and subjected to a time-dependent inclined concentrated load at the free end are investigated. The beam cross-section is considered prismatic, the self-weight is disregarded and the material is initially stress free. The set of four first-order non-linear partial integro-differential equations obtained from geometrical compatibility, equilibrium of forces and moments, and linear viscoelastic constitutive relation is numerically solved using a one-parameter shooting method combined with a fourth-order Runge-Kutta algorithm. An analytical expression is derived to divide the energy supplied by the external load into conserved and dissipated parts. For the case study presented, a three-parameter solid linear viscoelastic constitutive model is employed and a step load is applied. The variables are made non-dimensional, so four parameters govern the problem: the ratio between the final and initial relaxation moduli, the load magnitude, the angle of inclination and the unloading time. A finite-element model is also performed to compare and validate the analytical and numerical formulations. Results are presented for encastre curvature and tip displacement versus time, geometrical configuration, load versus tip displacement, total work done by the external force, stored and dissipated energies versus time, energy per unit length along arc length for three times and versus time for two beam sections.
机译：研究了由线性粘弹性材料制成的细长不可伸展梁的大挠度引起的准静态响应以及所存储和耗散的能量，该细长挠性梁在自由端受到时间依赖的倾斜集中载荷。梁的横截面被认为是棱柱形的，自重被忽略，材料最初没有应力。结合几何相容性，力和力矩的平衡以及线性粘弹性本构关系获得的四个一阶非线性偏积分-微分方程组，是通过单参数射击方法与四阶Runge-Kutta组合数值求解的算法。导出一个解析表达式，将外部负载提供的能量分为守恒和耗散部分。对于提出的案例研究，采用三参数实体线性粘弹性本构模型，并施加阶跃载荷。变量是无量纲的，因此有四个参数控制着这个问题：最终松弛模量与初始松弛模量之比，载荷大小，倾斜角度和卸载时间。还执行了有限元模型来比较和验证分析和数值公式。给出了铸件曲率和尖端位移与时间的关系，几何构型，载荷与尖端位移，由外力完成的总功，存储和耗散的能量与时间，沿弧长的单位长度能量的三倍以及相对于时间的两倍的结果梁截面。

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