Conference Agenda

Every material in nature exhibits heterogeneous behaviour at a certain scale. In a system, defects such as pores, grain boundaries, phase boundaries, secondary phases and particles can be the reasons for heterogeneity. The effective behaviour of the materials is significantly influenced by the underlying microstructure. Interfaces, such as grain boundaries, can affect the overall response of the material under consideration. Experimental findings shows that grain boundaries have a critical influence on electrical properties [1] and in order to model the macroscopic behaviour realistically, interfaces at the microscale should be taken into account.

Motivated by the change of effective electrical properties due to interfaces, e.g. microcracks or grain boundaries, a computational multiscale framework for continua with interfaces at the microscale is proposed in this contribution. More specifically speaking, the computational multiscale formulation for electrical conductors [2] is extended to account for interfaces at the microscale. Cohesive-type interfaces are considered at the microscale, such that displacement and electrical potential jumps can be accounted for. The governing equations for the materials with interfaces under mechanical and electical loads are provided. Based on these, a computational multiscale formulation is established. In particular, averaging theorems for kinematic quantities and for their energetic duals are discussed and their consistency with an extended Hill-Mandel condition for suitable boundary conditions is shown. The coupling between the electrical and mechanical subproblem is established by the constitutive equations at the material interface. In order to investigate deformation-induced property changes at the microscale, evolution of interface damage is elaborated.

To show the capabilities of the proposed framework, different representative simulations are selected. In particular, the calculation of effective macroscopic conductivity tensors for given two-dimensional microstructures is discussed and the fully coupled effective electro-mechanical material response due to the damage evolution is presented.

References

[1] H. Bishara, S. Lee, T. Brink, M. Ghidelli, and G. Dehm, “Understanding grain boundary electrical resistivity in Cu: The effect of boundary structure,” ACS Nano, vol. 15, no. 10, pp. 16607–16615, 2021.

[2] T. Kaiser and A. Menzel, “An electro-mechanically coupled computational multiscale formulation for electrical conductors,” Arch. Appl. Mech., vol. 91, pp. 1–18, 2021.

The technical relevance of small-scale electromechanical systems is rapidly increasing today. For this reason, the flexoelectric effect, which occurs in all dielectrics, is increasingly getting into the focus of research. This size-dependent effect describes the linear coupling between the electric polarization in the material and an occurring strain gradient in, for example, bent cantilever beams. There also exists a converse flexoelectric effect defined as a mechanical stress response under the action of an electric field gradient especially noticeable at sharp electrode tips in microelectromechanical systems (MEMS). In order to make these coupling effects technically usable, suitable models are required to predict the resulting system response.

A continuum-based model approach that takes into account elastic, dielectric, piezoelectric and flexoelectric effects is presented. Different model variants will be discussed and suitable finite element formulations for solving the electromechanical boundary value problem will be presented. A mixed variation formulation is used here in order to reduce the higher continuity requirements due to the occurring gradient fields. When considering ferroelectric materials (e.g. PZT), microstructural domain switching processes must be taken into account in order to be able to predict the behavior realistically. A microscopically motivated material model representing these dissipative processes is introduced and fitted into the flexoelectric continuum approach. The influence of acting strain and electric field gradients on the domain switching processes in ferroelectrics when considering the flexoelectric effect is studied by numerical experiments.

Inspired by lattice structures that can be observed in nature, periodic unit cells and their mechanical properties have caused an ever increasing interest in recent years due to the growing performance of additive manufacturing methods. In order to incorporate cells with optimal properties into printed high-performance structures and devices that can respond to given macroscopic stress-strain states in an optimal manner, one has to provide anisotropic properties that can respond to these individual loads.

We discuss the performance of a phase-field approach for optimizing periodic micro-structures based on triply periodic minimal surface problems (TPMS) to obtain unit cells with an optimal homogenized stiffness response in the direction of the maximal principal stress direction. We show that different TPMS-types exhibit fundamental differences in the way they can respond to uni-axial or shear-dominated loads. An essential aspect in optimizing cells is, on the one hand, to maximize the compliance with external loads and, on the other hand, to limit the danger of failure due to local buckling which is achieved by preserving the connectivity of the cell grid.

Further aspects that are discussed include numerical strategies to handle linear systems of such high-resolution optimization problems in an efficient manner as well as strategies to verify the gain of the homogenized stiffness experimentally.

The Bouligand structure comprises twisted parallel fibers arranged in a helical pattern, which enables greater energy dissipation and fracture toughness, mainly through a large crack surface area and crack-bridging phenomenon compared to regular fiber-reinforced composites. Considering the complex nature of this structure, numerical models that accurately capture the propagation of cracks through its twisted fiber arrangement are limited. This is due to the most popular simulating approach, the finite element method (FEM), is based on the classic continuum mechanics that uses spatial differential equations to describe continuous material behaviors. In contrast, Peridynamics is a computational framework that has been developed to overcome the limitations of classical continuum mechanics in describing crack propagation. Unlike FEM, Peridynamics is a non-local continuum theory that utilizes integral equations instead of differential equations in space to simulate material behaviors. This characteristic makes it highly suitable for modeling the complex crack propagation behaviors in the Bouligand structure. In this study, we present a bond-based peridynamics model to accurately describe the fiber-reinforced composites with a small angle mismatch between adjacent layers in Bouligand structures. To investigate the fracture mechanisms of such a structure, we conduct comprehensive numerical simulations, including 3-point bending and low-velocity impact tests, to obtain detailed information on its deformation and failure behavior. This information is difficult to achieve solely through experimental and theoretical studies. Based on our insights into the toughening mechanisms of the Bouligand structure, we propose a novel approach to further enhance the material’s fracture toughness by combining the Bouligand structure with other toughening mechanisms. Overall, the current study provides important insights into the fracture behavior of Bouligand structures and presents new avenues for designing advanced materials with superior mechanical properties.