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板状结构自发大变形问题的三维数值分析

THREE-DIMENSIONAL NUMERICAL ANALYSIS OF SPONTANEOUS LARGE-DEFORMATION OF PLATE-LIKE STRUCTURES

  • 摘要: 花朵、树叶等自然界的板状结构因发生大变形而具有婀娜多姿的形状, 工程实际的板状结构也会出现类似现象. 板状结构是指完全相同的面状结构在厚度方向堆砌而形成的厚度尺寸比面内尺寸相比较小的一类特殊三维结构, 在生长或外部环境等因素产生的不协调变形激励下, 这类结构会形成内部应力, 本文研究因之而发生的自发大变形行为. 首先, 将板状结构的变形能分离为类伸缩变形能和剩余变形能两部分, 并提出基于三维大变形有限元分析的能量计算方法; 然后, 建立板状结构的屈曲失稳条件为剩余变形能由小到超越类伸缩变形能的跨越点, 进而提出转变厚度概念, 通过与简支方板失稳的经典理论解比较, 验证三维大变形有限元分析结果及屈曲失稳条件; 最后, 运用三维大变形有限元方法, 研究几种典型自发大变形问题, 分析不协调变形因素对内部应力场拉压特性和模式转变厚度的影响规律. 本文工作表明, 板状结构的大变形过程是弹性变形能中剩余变形能从零开始增加、直至超过类伸缩变形能引起屈曲的一个自发现象. 特别地, 三维大变形有限元分析是求解复杂内部应力场激发的板状结构屈曲失稳问题的一条有效途径.

     

    Abstract: Plate-like structures in nature, such as flowers and leaves, tend to have a curvaceous shape as a result of a large deformation process. Similar phenomena can also be observed in plate-like structures in various engineering fields. Here, plate-like structure signifies a special three-dimensional structure that is the stack of identical planar structures and the thickness size is hence far less than the planar ones. Stimulated by the incompatible deformation due to factors such as growth and external stimuli, a plate-like structure possesses internal stresses. In this paper, a spontaneous deformation behavior produced by the existing internal stresses is studied. To this end, the strain energy of the plate-like structure is firstly decomposed into two contents, that is, the stretch-like deformation energy and the remaining deformation energy, respectively. To evaluate these two different energies, we suggest a numerical approach based on a three-dimensional (3D) large-deformation finite element analysis (FEA). A condition for buckling instability of such a plate-like structure is then proposed to be the crossover point at which the remaining deformation energy goes beyond the stretch-like deformation energy from none. With this condition, the concept of threshold thickness is further introduced in order to characterize the crossover point, which is verified by comparing the values from the FEA with those from the classical plate theory for a simply-supported square plate. Finally, several spontaneous large-deformation problems modeled by typical power-law thermal expansions are studied through a 3D large-deformation FEA, and the effects of incompatible factors such as the magnitude and the power index on internal stress fields and the threshold thickness are also examined. Our present work shows that the large deformation of plate-like structures is a spontaneous process such that the remaining deformation energy increases from zero to a value larger than the stretch-like deformation energy. In particular, the 3D large-deformation FEA is an effective method to solve the buckling instability of plate-like structures stimulated by a complex internal stress field.

     

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