A second strain gradient theory-based continuum model is presented for the mechanics of an elastic solid reinforced with extensible fibers in plane elastostatics. The extension and bending kinematics of fibers are formulated via the second and the third gradient of the continuum deformation. The Euler equations arising in the third gradient of virtual displacement are then formulated by means of iterated integration by parts and variational principles. A rigorous derivation of the associated boundary conditions is also presented from which the expressions of triple forces and stresses are obtained. The obtained triple forces are found to be in conjugation with the Piola-type triple stress and are necessary to determine energy contributions on edges and points of Cauchy cuts. In particular, a complete linear model including admissible boundary conditions is derived within the description of superposed incremental deformations. The obtained analytical solution predicts smooth deformation profiles and, more importantly, assimilate gradual and dilatational shear angle distributions throughout the domain of interest.

提出了基于第二应变梯度理论的连续体模型，该模型用于在平面弹性体中用可伸长纤维增强的弹性固体的力学。纤维的延伸和弯曲运动学是通过连续变形的第二和第三梯度来确定的。然后，通过零件和变分原理的迭代积分，可以得出在虚拟位移的第三梯度中出现的欧拉方程。还给出了相关边界条件的严格推导，从中可以得出三重力和应力的表达式。发现获得的三重力与Piola型三重应力共轭，并且对于确定Cauchy切口的边缘和点上的能量贡献是必需的。特别是，在叠加增量变形的描述中，得出了一个包含允许边界条件的完整线性模型。所获得的分析解决方案可预测平滑的变形轮廓，更重要的是，可在整个目标域中吸收渐变的和扩张的剪切角分布。