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On pure bending in non-linear elasticity: A circular closed-form 2D solution for semi-linear orthotropic material
European Journal of Mechanics - A/Solids ( IF 4.1 ) Pub Date : 2021-06-02 , DOI: 10.1016/j.euromechsol.2021.104289
Gordan Jelenić

Large-deformation pure bending of a straight beam of uniform rectangular cross-section in plane strain is re-visited and generalised to an orthotropic linear-elastic material and the case of plane stress. The beam is bent into a completely shearless state by means of normal surface traction of a specific profile. For a linear-elastic material expressed in terms of Biot strain and stress tensors (orthotropic generalisation of the so-called semi-linear material) the solution still exists in closed form and is expressed in terms of the applied cross-sectional traction. As such, it provides a convenient benchmark reference for numerical methods in non-linear elasticity.

In agreement with the isotropic case, through-the-thickness stresses develop in the form a hyperbolic cosine, but now with the magnitude actually depending on a measure of orthotropy in the material, while the longitudinal stresses change over the thickness in the form of a hyperbolic sine. In general, the unstrained axis is displaced from the centroidal axis.

For an ideal material with infinite yield strength, the theoretical maximum of the magnitude of the longitudinal traction is limited to the amount of the geometric mean of the effective Young's moduli in the longitudinal and the through-the-thickness directions while, for a brittle material, bending strength reduces by approximately 715 of the square of the ratio between the magnitude of the applied traction and this mean value.



中文翻译:

关于非线性弹性中的纯弯曲:半线性正交各向异性材料的圆形封闭形式二维解决方案

重新讨论了平面应变下均匀矩形截面直梁的大变形纯弯曲,并将其推广到正交各向异性线弹性材料和平面应力的情况。梁通过特定轮廓的法向表面牵引力弯曲成完全无剪切状态。对于以 Biot 应变和应力张量(所谓的半线性材料的正交各向异性推广)表示的线弹性材料,解仍然以封闭形式存在,并以施加的横截面牵引力表示。因此,它为非线性弹性中的数值方法提供了方便的基准参考。

与各向同性情况一致,全厚度应力以双曲余弦的形式发展,但现在其大小实际上取决于材料中正交各向异性的度量,而纵向应力随厚度变化的形式为双曲正弦。通常,无应变轴与质心轴发生位移。

对于具有无限屈服强度的理想材料,纵向牵引力的理论最大值限于纵向和厚度方向上有效杨氏模量的几何平均值,而对于脆性材料, 弯曲强度大约降低 715 所施加牵引力的大小与该平均值之比的平方。

更新日期:2021-06-10
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