A cross-scale strain-gradient viscoelastic model bridging micron-sized adhesive layers and macroscopic bonded structures

Adhesive layers in macroscopic bonded structures typically have thicknesses ranging from several micrometers to hundreds of micrometers, where pronounced size effects, complex viscoelasticity, and cross-scale interactions severely challenge conventional mechanical modeling. This work proposes a cross-scale strain-gradient viscoelastic model grounded in strain gradient theory and the discontinuous Galerkin method. A higher-order generalized Maxwell model is formulated for adhesive bulks, while a higher-order compatible interface model is introduced to consistently capture gradient continuity across adhesive interfaces. A hybrid numerical scheme incorporating three-dimensional collocation and cross-interface elements is developed, and the resulting nonlinear system is solved using a Newton-Raphson scheme. The proposed model is verified through convergence and stability analyses and validated against analytical solutions for micron-sized bending beams and a representative cross-scale bonded structure. Size-viscoelastic interactions and model-dependent responses are quantified, and stress evolution in cross-scale bonded structures is examined. Results demonstrate that the mechanical behavior of micron-sized adhesive layers is governed by the competition between size-dependent strengthening and viscoelastic softening, the proposed model accounting for size effects predicts a faster relaxation rate than the classical viscoelastic model, and pronounced size effects induce near-elastic behavior and non-monotonic stress evolution in bonded structures.