The pull-out test is one of the common experiments to determine the bond strength. When the problem is modeled in the context of linear elasticity for a cylindrical reinforced concrete block, the resulting simplified 1-D model yields so-called pull-out paradox Rezaei et al. (Mech Res Commun 126:104015, 2022) due to extreme concentration of energy near the bar. Since the standard linear elasticity is not able to consider this high values of energy, the problem was investigated by strain-gradient elasticity solution in the work of Rezaei et al. (Mech Res Commun 126:104015, 2022) . In this study, to resolve the paradoxical solution, classical finite (i.e., St.-Venant–Kirchhoff model) and strain-gradient finite elasticity solutions are presented. Each mathematical model, assuming that the material is isotropic, is derived with the principle of minimum potential energy introducing appropriate strain energy. The numerical simulations are performed by the finite element method, and it is showed that numerical solution of each model converges well.

拉拔试验是测定粘结强度的常用实验之一。当在圆柱形钢筋混凝土块的线性弹性背景下对问题进行建模时，所得的简化一维模型产生了所谓的拉出悖论 Rezaei 等人。 （Mech Res Commun 126:104015, 2022）由于酒吧附近能量极度集中。由于标准线性弹性无法考虑如此高的能量值，因此 Rezaei 等人的工作中通过应变梯度弹性解决方案研究了该问题。 （机械研究通讯 126：104015，2022）。在本研究中，为了解决矛盾解，提出了经典有限（即圣维南-基尔霍夫模型）和应变梯度有限弹性解。每个数学模型都假设材料是各向同性的，是根据引入适当应变能的最小势能原理推导出来的。采用有限元方法进行数值模拟，结果表明各模型的数值解收敛良好。