arXiv subjects

Kai Richter

Publications and source records attributed to Kai Richter.

3 recordsLinked to original sources

arXiv2025

Commutativity and non-commutativity of limits in the nonlinear bending theory for prestrained microheterogeneous plates

In this paper we study the derivation of nonlinear bending models for prestrained elastic plates from three-dimensional non-linear elasticity via homogenization and dimension reduction. We compare effective models obtained by either simultaneously or consecutively passing to the $Γ$-limits as the thickness $h\ll1$ and the size of the material microstructure $\e\ll1$ vanish. In the regime $\e\ll h$ we show that the consecutive and simultaneous limit are equivalent, and also analyze the rate of convergence. In contrast, we observe that there are several different limit models in the case $h\ll \e$.

math.AP↗

arXiv2024

Dimension reduction for elastoplastic rods in the bending regime

We rigorously derive an effective bending model for elastoplastic rods starting from three-dimensional finite plasticity. For the derivation we lean on a framework of evolutionary $Γ$-convergence for rate-independent systems. The main contribution of this paper is an ansatz for the mutual recovery sequence in the bending regime. In difference to previous works that deal with infinitesimal deformations in the limit, in the bending regime we are concerned with finite deformations that admit infinitesimally small strain but large rotations. To overcome these difficulties we provide new ideas based on a multiplicative decomposition for the Cosserat rod ansatz. As regularizing terms, we introduce strain gradient terms into the energy that vanish as the thickness of the rod tends to zero.

math.AP↗

arXiv2024

Linearization and Homogenization of nonlinear elasticity close to stress-free joints

In this paper, we study a hyperelastic composite material with a periodic microstructure and a prestrain close to a stress-free joint. We consider two limits associated with linearization and homogenization. Unlike previous studies that focus on composites with a stress-free reference configuration, the minimizers of the elastic energy functional in the prestrained case are not explicitly known. Consequently, it is initially unclear at which deformation to perform the linearization. Our main result shows that both the consecutive and simultaneous limits converge to a single homogenized model of linearized elasticity. This model features a homogenized prestrain and provides first-order information about the minimizers of the original nonlinear model. We find that the homogenization of the material and the homogenization of the prestrain are generally coupled and cannot be considered separately. Additionally, we establish an asymptotic quadratic expansion of the homogenized stored energy function and present a detailed analysis of the effective model for laminate composite materials. A key analytical contribution of our paper is a new mixed-growth version of the geometric rigidity estimate for Jones domains. The proof of this result relies on the construction of an extension operator for Jones domains adapted to geometric rigidity.

math.AP↗