In this paper we model the size-effects of metamaterial beams under bending with the aid of the relaxed micromorphic continuum. We analyze first the size-dependent bending stiffness of heterogeneous fully discretized metamaterial beams subjected to pure bending loads. Two equivalent loading schemes are introduced which lead to a constant moment along the beam length with no shear force. The relaxed micromorphic model is employed then to retrieve the size-effects. We present a procedure for the determination of the material parameters of the relaxed micromorphic model based on the fact that the model operates between two well-defined scales. These scales are given by linear elasticity with micro and macro elasticity tensors which bound the relaxed micromorphic continuum from above and below, respectively. The micro elasticity tensor is specified as the maximum possible stiffness that is exhibited by the assumed metamaterial while the macro elasticity tensor is given by standard periodic first-order homogenization. For the identification of the micro elasticity tensor, two different approaches are shown which rely on affine and non-affine Dirichlet boundary conditions of candidate unit cell variants with the possible stiffest response. The consistent coupling condition is shown to allow the model to act on the whole intended range between macro and micro elasticity tensors for both loading cases. We fit the relaxed micromorphic model against the fully resolved metamaterial solution by controlling the curvature magnitude after linking it with the specimen’s size. The obtained parameters of the relaxed micromorphic model are tested for two additional loading scenarios.

Residual stresses in components are a central issue in almost every manufacturing process, as they influence the performance of the final part. Regarding hot forming processes, there is a great potential for defining a targeted residual stress state, as many adjustment parameters, such as deformation state or temperature profile, are available that influence residual stresses. To ensure appropriate numerical modeling of residual stresses in hot forming processes, comprehensive material characterization and suitable multiscale Finite Element (FE) simulations are required. In this paper, experimental and numerical investigations of thermo-mechanically processed steel alloy 1.3505 (DIN 100Cr6) are presented that serve as a basis for further optimization of numerically modeled residual stresses. For this purpose, cylindrical upsetting tests at high temperature with subsequently cooling of the parts in the media air or water are carried out. Additionally, the process is simulated on the macroscale and compared to the results based on the experimental investigations. Therefore, the experimentally processed specimens are examined regarding the resulting microstructure, distortions, and residual stresses. For the investigation on a smaller scale, a numerical model is set up based on the state-data of the macroscopic simulation and experiments, simulating the transformation of the microstructure using phase-field theory and FE analysis on micro- and meso-scopic level.

The relaxed micromorphic model is a generalized continuum model that reduces the complexity of the general micromorphic theory [1] and shows many advantages such as the bounded stiffness for small sizes [2–4]. It keeps the full kinematics of the micromorphic theory but employs the matrix Curl operator of a second‐order micro‐distortion field for the curvature measurement. The solution of the micro‐distortion exists in H(curl) while the displacement is still in H¹. In this work, we introduce an H¹ × H(curl) finite element formulation of the relaxed micromorphic model. The presented mixed formulation satisfies the tangential continuity of the micro‐distortion field on the element boundaries. We compare the convergence behavior of the mixed formulation with the classical H¹ × H¹ finite element formulation using numerical examples. Finally, we show the model's main characteristics and the scale‐dependency of the model's components where the relaxed micromorphic model gives the different Cauchy elastic limit cases with determined elasticity tensors.

For modeling materials with size‐effects, the classical Cauchy linear elasticity theory and first‐order homogenization procedures are insufficient. Enriched continua are able to capture such phenomena in principle. In this work, we model the size‐effects of metamaterial beams via the relaxed micromorphic continuum. The material parameters of the relaxed micromorphic model are evaluated. We present a simple procedure for the determination of the material parameters based on the fact that the model operates between two well‐defined scales that are the classical continuum scales of macroscopic linear elasticity, whose stiffness is given by a known macroscopic elasticity tensor, and a microscopic scale with a microscopic elasticity tensor yet to be identified. The size‐effects of metamaterial cantilevers under traction load are captured successfully.