A strain gradient linear viscoelasticity theory

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.