Abstract In this paper, a strain gradient viscoelastic theory is proposed strictly, which can be used to describe the cross-scale mechanical behavior of the quasi-brittle advanced materials. We also expect the theory to be applied to the description for the cross-scale mechanical behavior of advanced alloy metals in linear elastic deformation cases. In the micro-/nano-scale, the mechanical properties of advanced materials often show the competitive characteristics of strengthening and softening, such as: the strength and hardness of the thermal barrier coatings with nanoparticles and the nanostructured biological materials (shells), as well as the strength of nanocrystalline alloy metals which show the characteristics of positive-inverse Hall-Petch relationship, etc. In order to characterize these properties, a strain gradient viscoelastic theory is established by strictly deriving the correspondence principle. Through theoretical derivation, the equilibrium equations and complete boundary conditions based on stress and displacement are determined, and the correspondence principle of strain gradient viscoelasticity theory in Laplace phase space is obtained. With the help of the high-order viscoelastic model, the specific form of viscoelastic parameters is presented, and the time curve of material characteristic scale parameters in viscoelastic deformation is obtained. When viscoelasticity is neglected, the strain gradient viscoelasticity theory can be simplified to the classical strain gradient elasticity theory. When the strain gradient effect is neglected, it can be simplified to the classical viscoelastic theory. As an application example of strain gradient viscoelastic theory, the solution to the problem of cross-scale viscoelastic bending of the Bernoulli-Euler beam, is analyzed and presented.

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应变梯度线性粘弹性理论

摘要 本文严格提出了应变梯度粘弹性理论，可用于描述准脆性先进材料的跨尺度力学行为。我们还期望将该理论应用于描述线性弹性变形情况下高级合金金属的跨尺度力学行为。在微/纳米尺度上，先进材料的力学性能往往表现出增强和软化的竞争特性，例如：具有纳米颗粒的热障涂层和纳米结构生物材料（壳）的强度和硬度，以及作为纳米晶合金金属的强度，表现出正反霍尔-佩奇关系等特征。为了表征这些特性，严格推导对应原理，建立了应变梯度粘弹性理论。通过理论推导，确定了基于应力和位移的平衡方程和完备的边界条件，得到了拉普拉斯相空间应变梯度粘弹性理论的对应原理。借助高阶粘弹模型，给出了粘弹参数的具体形式，得到了粘弹变形中材料特征尺度参数的时间曲线。当忽略粘弹性时，应变梯度粘弹性理论可以简化为经典的应变梯度弹性理论。当忽略应变梯度效应时，可以简化为经典粘弹性理论。