The present manuscript delineates the derivation of strong solutions for the linear elasticity problem in a two dimensional rectangular beam. The materials under consideration can exhibit either isotropic or orthotropic properties. Additionally, the analysis is not restricted to slender beams because the ratio between the length and the height of the beam can be arbitrary. The boundary conditions fall into the Dirichlet category, implying that both horizontal and vertical displacements are specified on all four surfaces. The sole requirement is that these surface displacements are continuous functions, although they may not necessarily be smooth. Since the displacements at the surfaces can be arbitrary, there is no need to consider approximations, such as those concerning local (small) boundaries in slender beams. Nonetheless, it is demonstrated that an equivalent principle to the Saint-Venant principle exists for pure displacement boundary conditions. The partial differential equations are solved through the separation of variables method, leading to the identification of two solution types, encompassing both cosine and sine Fourier series. Particular emphasis is placed on evaluating the convergence of these solutions. For two distinct and general examples, it is confirmed that the solutions indeed exist.

中文翻译：

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二维梁问题的解析解：纯位移边界条件


本手稿描述了二维矩形梁中线弹性问题的强解的推导。所考虑的材料可以表现出各向同性或正交各向异性特性。此外，分析不限于细长梁，因为梁的长度和高度之间的比率可以是任意的。边界条件属于狄利克雷类别，这意味着在所有四个表面上都指定了水平和垂直位移。唯一的要求是这些表面位移是连续函数，尽管它们不一定是平滑的。由于表面处的位移可以是任意的，因此无需考虑近似值，例如涉及细长梁中的局部（小）边界的近似值。尽管如此，事实证明，对于纯位移边界条件，存在与圣维南原理等效的原理。偏微分方程通过变量分离方法求解，从而识别出两种解类型，包括余弦和正弦傅里叶级数。特别强调评估这些解决方案的收敛性。对于两个不同且一般的例子，证实解确实存在。