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This paper aims to present a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the representative volume element (RVE) of discrete particle assembly in the frame of the second-order computational homogenization methods for granular materials.

The proposed scheme is based on the generalized Hill’s lemma of gradient Cosserat continuum and the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE of discrete particle assembly. The generalized Hill’s lemma conducts to downscale the macroscopic strain or stress measures and to impose the periodic boundary conditions on the RVE boundary so that the Hill-Mandel energy equivalence condition is ensured. Because of the incremental non-linear constitutive relation condensed to the peripheral particles of the RVE, the periodic boundary displacement and traction constraints together with the downscaled macroscopic strains and strain gradients, micro-rotations and curvatures are imposed in the point-wise sense without the need of introducing the Lagrange multipliers for enforcing the periodic boundary displacement and traction constraints in a weak sense.

Numerical results demonstrate that the applicability and effectiveness of the proposed scheme in imposing the periodic boundary conditions on the RVE. The results of the RVE subjected to the periodic boundary conditions together with the displacement boundary conditions in the second-order computational homogenization for granular materials provide the desired estimations, which lie between the upper and the lower bounds provided by the displacement and the traction boundary conditions imposed on the RVE respectively.

Each grain in the particulate system under consideration is assumed to be rigid and circular.

The proposed scheme for imposing periodic boundary conditions on the RVE can be adopted solely for estimating the effective mechanical properties of granular materials and/or integrated into the frame of the second-order computational homogenization method with a nested finite element method-discrete element method solution procedure for granular materials. It will tend to provide, at least theoretically, more reasonable results for effective material properties and solutions of a macroscopic boundary value problem simulated by the computational homogenization method.

This paper presents a novel scheme for imposing periodic boundary conditions with downscaled macroscopic strain measures of gradient Cosserat continuum on the RVE of discrete particle assembly for granular materials without need of introducing Lagrange multipliers for enforcing periodic boundary conditions in a weak (integration) sense.

In this paper, to obtain shear and bending performance of carbon fiber-reinforced polymer (CFRP)-strengthened beams bonded by geopolymers, the effects of impregnated adhesive types, strengthened scheme, CFRP layer and pre-cracked width are investigated, and the performance of CFRP-strengthened beams is validated by the establishment of Finite Element Models (FEMs).

In this paper, static loading test and finite element analysis of epoxy-CFRP-strengthened (ECS) and geopolymer-CFRP-strengthened (GCS) were carried out, and the bearing capacity and stiffness were compared, the results show that GCS reinforced concrete (RC) beam is feasible and effective.

The bearing capacity, crack distribution and development, load–deflection curves of GCS RC beams with different pre-crack widths were investigated. The reinforcement effect of geopolymer achieves the same as epoxy, effectively improving the ultimate bearing capacity of the beam, with a maximum increase rate of 28.9%. The failure mode of CFRP is broken in the yield failure stage of GCS RC beam with reasonable strengthening form, and the utilization rate of CFRP is improved. CFRP-strengthened layers, pre-cracked widths significantly affect the mechanical properties, and deformation properties of the strengthened beams.

Compared with ECS RC beams, the bearing capacity and stiffness of GCS RC beams are similar to or even better, indicating that GCS RC beam is feasible and effective. It is a new method for CFRP-strengthened beams, which not only conforms to the concept of national ecological civilization construction, but also provides an economical, environmentally friendly and excellent performance solution for structural reinforcement.