In the present work, we combine Mindlin’s strain gradient elasticity theory and Gudmundson–Gurtin–Anand strain gradient plasticity theory to form a unified framework. The gradient plasticity model is enriched by including the gradient of elastic strains into the expression of the internal virtual work and free energy. This augments the modelling capabilities by incorporating elasticity-related length scales along with plasticity-related energetic and dissipative ones. The strong form governing equations are derived via the principle of virtual work addressing a complete set of boundary conditions. The fourth-order boundary value problem of the gradient elasto-plasticity model is then formulated in a variational form within an $H^2$ Sobolev space setting. Conforming Galerkin discretizations for numerical results are obtained utilizing an isogeometric approach with NURBS basis functions of degree $p\ge 2$ providing $C^{p-1}$-continuity. The implementation follows a viscoplastic constitutive framework and adopts the backward Euler time integration scheme. A set of benchmark examples is considered to illustrate convergence properties and to accomplish parameter studies. It is shown that the elastic length scale parameter controls the slope of the elastic part and causes an additional hardening in the plastic part of the material response curves. Finally, an illustrative example is considered in order to demonstrate the applicability of both the continuum model and the numerical method in capturing the size-dependent torsion response of cellular structures.

在目前的工作中，我们结合 Mindlin 的应变梯度弹性理论和 Gudmundson-Gurtin-Anand 应变梯度塑性理论，形成一个统一的框架。通过将弹性应变梯度包含在内部虚功和自由能的表达式中，丰富了梯度塑性模型。这通过结合与弹性相关的长度尺度以及与塑性相关的能量和耗散尺度来增强建模能力。强形式控制方程是通过解决完整边界条件集的虚功原理推导出来的。然后将梯度弹塑性模型的四阶边值问题以变分形式在$H^2$索博列夫空间设置。数值结果的符合 Galerkin 离散化是利用具有 NURBS 度数基函数的等几何方法获得的$磷\ge 2$ 提供 $C^{磷-1}$- 连续性。实现遵循粘塑性本构框架，并采用后向欧拉时间积分方案。一组基准示例被认为是为了说明收敛特性并完成参数研究。结果表明，弹性长度尺度参数控制弹性部分的斜率，并在材料响应曲线的塑性部分引起额外硬化。最后，考虑了一个说明性示例，以证明连续介质模型和数值方法在捕获细胞结构的尺寸相关扭转响应方面的适用性。