The higher order two-phase local/nonlocal elasticity model and the higher order strain gradient theory are unified via an abstract variational scheme. The higher order constitutive integral convolution is established in a consistent variational framework governed by ad hoc functional space of test fields. Equivalent differential constitutive law equipped with nonclassical boundary conditions of constitutive type is determined. The proposed higher order elasticity theory provides as special cases a range of well-known size-dependent elasticity models such as nonlocal, two-phase local/nonlocal, strain gradient, modified nonlocal strain gradient, and nonlocal strain-driven gradient models. Evidences of well-posedness of the introduced higher order two-phase local/nonlocal gradient problems are elucidated by rigorous examination of the elastostatic torsional response of structural schemes of applicative interest in nano-mechanics. The exact analytical solution of the torsion problem of elastic nano-beams is derived, graphically demonstrated, and compared with analogous outcomes in the literature. The conceived higher order elasticity theory can efficiently characterize advanced nano-materials and structural elements of modern nano-systems.

中文翻译：

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弹性扭转的两相局部/非局部梯度力学

高阶两相局部/非局部弹性模型和高阶应变梯度理论通过抽象变分方案统一起来。高阶本构积分卷积建立在由测试场的特设功能空间控制的一致变分框架中。确定了具有本构型非经典边界条件的等价微分本构律。所提出的高阶弹性理论作为特殊情况提供了一系列众所周知的尺寸相关弹性模型，例如非局部、两相局部/非局部、应变梯度、修正的非局部应变梯度和非局部应变驱动梯度模型。引入的高阶两相局部/非局部梯度问题的适定性证据通过严格检查纳米力学中具有应用兴趣的结构方案的弹性扭转响应来阐明。推导出弹性纳米梁的扭转问题的精确解析解，并以图形方式证明，并与文献中的类似结果进行比较。构想的高阶弹性理论可以有效地表征先进的纳米材料和现代纳米系统的结构元素。