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Strain localization is a common phenomenon existing in various multiscale materials or structures, e.g., the bulking bands of thin-walled structures, the local collapse of porous materials, and the crack in solid materials, etc. However, this phenomenon cannot be captured by the conventional homogenization methods, such as mean-field homogenization (e.g., Mori–Tanaka method), first-order or high-order computational homogenization (e.g., the Finite Element Square method), etc., due to the high strain gradient associated with strain localization. Aiming at this, a hybrid Direct FE2 method is proposed by combining the D-FE2 (Direct FE2) and the traditional FE methods, while the multiscale structure is modeled using the D-FE2 method in the region exhibiting low deformation gradient, and the other region displaying high deformation gradient is modeled using the traditional FE method. Moreover, a node displacement constraint and an overall node displacement constraint derived from the multilevel equilibrium equations using the Gauss–Ostrogradsky theorem are respectively prescribed to the interface between the D-FE2 model and the FE model of the multiscale structure, to enforce the energy equilibrium and deformation continuity. The proposed hybrid D-FE2 method is then applied to predict the strain localization behavior of multiscale materials or structures, including local bulking of honeycomb structures, in-situ crack propagation, and localized plastic deformation in fiber reinforced composites, etc. Comparison of the simulation results obtained from the hybrid D-FE2 method and the traditional FE method validates the accuracy, efficiency and ease of numerical implementation of the proposed hybrid D-FE2 method. • A hybrid D-FE2 method was proposed for m ultiscale modeling of strain localization. • Displacement constraints were derived for d eformation continuity at the interface. • The hybrid D-FE2 method is accurate and more e fficient compared to DNS. [ABSTRACT FROM AUTHOR]

Multiscale piezoelectric materials and structures have shown their promising potential of applications in sensor, energy harvester, actuator and so on. However, the large complexity of such materials and structures poses a significant challenge to the accurate and efficient prediction of their piezoelectric behavior which is significant for guiding and optimizing their applications. In this paper, a Direct FE 2 method was proposed for concurrent multilevel modeling of the coupled electromechanical behavior of multiscale piezoelectric materials and structures. The energy equilibrium and kinematic constraints between macro- and meso-scales were satisfied by prescribing periodic boundary conditions for both displacement and electric potential, which were derived from the governing equation of FEM. Another boundary condition was prescribed to the center node of each RVE, to prevent rigid motion and shifting of electric potential of meso-scale RVEs. All these boundary conditions can be easily prescribed to the Direct FE 2 using multiple points constraints (MPCs), a commonly used feature in most commercial software. The accuracy, computational efficiency and ease of numerical implementation of the proposed Direct FE 2 method were validated using a series of multiscale simulations, including the quasi-static response of homogeneous piezoelectric panels, an arc honeycomb structure, a regular honeycomb structure and a piezoelectric composite panel, as well as the dynamic response of the piezoelectric composite panel. It shows that the proposed Direct FE 2 method will be valuable for large-scale simulations of complicated multiscale piezoelectric materials and structures. • A Direct FE 2 method is proposed for concurrent multilevel modeling of multiscale piezoelectric materials and structures. • The proposed methods can accurately capture the coupled electromechanical response of piezoelectric materials and structures. • The computational efficiency of the proposed method is 9 times higher than traditional FE simulations. [ABSTRACT FROM AUTHOR]