Ingo Münch

Linear Cosserat Elasticity, Conformal Curvature and Bounded Stiffness

We describe a principle of bounded stiffness and show that bounded stiffness in torsion and bending implies a reduction of the curvature energy in linear isotropic Cosserat models leading to the so called conformal curvature case \(\frac{1}{2}\mu L_{c}^{2}\Vert{\operatorname{dev}\operatorname{sym}\nabla \operatorname{axl}\overline{A}}\Vert^{2}\) where \(\overline{A}\in\mathfrak{so}(3)\) is the Cosserat microrotation. Imposing bounded stiffness greatly facilitates the Cosserat parameter identification and allows a well-posed, stable determination of the one remaining length scale parameter L c and the Cosserat couple modulus μ c .

A variant of the linear isotropic indeterminate couple-stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and orthogonal boundary conditions

In this paper we venture a new look at the linear isotropic indeterminate couple-stress model in the general framework of second-gradient elasticity and we propose a new alternative formulation which obeys Cauchy–Boltzmann’s axiom of the symmetry of the force-stress tensor. For this model we prove the existence of solutions for the equilibrium problem. Relations with other gradient elastic theories and the possibility of switching from a fourth-order (gradient elastic) problem to a second-order micromorphic model are also discussed with the view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple-stress model can be written entirely with symmetric force-stress and symmetric couple-stress. The difference of the alternative models rests in specifying traction boundary conditions of either rotational type or strain type. If rotational-type boundary conditions are used in the integration by parts, the classical anti-symmetric nonlocal force-stress tensor formulation is obtained. Otherwise, the difference in both formulations is only a divergence-free second-order stress field such that the field equations are the same, but the traction boundary conditions are different. For these results we employ an integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the orthogonal boundary conditions for both models.

Rediscovering GF Becker’s early axiomatic deduction of a multiaxial nonlinear stress–strain relation based on logarithmic strain

We discuss a completely forgotten work of the geologist GF Becker on the ideal isotropic nonlinear stress–strain function (Am J Sci 1893; 46: 337–356). Due to the fact that the mathematical modelling of elastic deformations has evolved greatly since the original publication we give a modern reinterpretation of Becker’s work, combining his approach with the current framework of the theory of nonlinear elasticity. Interestingly, Becker introduces a multiaxial constitutive law incorporating the logarithmic strain tensor, more than 35 years before the quadratic Hencky strain energy was introduced by Heinrich Hencky in 1929. Becker’s deduction is purely axiomatic in nature. He considers the finite strain response to applied shear stresses and spherical stresses, formulated in terms of the principal strains and stresses, and postulates a principle of superposition for principal forces which leads, in a straightforward way, to a unique invertible constitutive relation, which in today’s notation can be written as T Biot = 2 G ⋅ dev ⁡ 3 log ⁡ ( U ) + K ⋅ tr [ log ⁡ ( U ) ] ⋅ 11 = 2 G ⋅ log ⁡ ( U ) + Λ ⋅ tr [ log ⁡ ( U ) ] ⋅ 11 , where TBiot is the Biot stress tensor, log(U) is the principal matrix logarithm of the right Biot stretch tensor U = F T F , X = ∑ i = 1 3 X i , i denotes the trace and dev3 X = X − (1/3) tr (X) · 11 denotes the deviatoric part of a matrix X ∈ ℝ 3 × 3 . Here, G is the shear modulus and K is the bulk modulus. For Poisson’s number ν = 0 the formulation is hyperelastic and the corresponding strain energy W Becker ν = 0 ( U ) = 2 G [ < U , log ( U ) − 11 > + 3 ] has the form of the maximum entropy function.

A new view on boundary conditions in the Grioli–Koiter–Mindlin–Toupin indeterminate couple stress model

Abstract In this paper we consider the Grioli–Koiter–Mindlin–Toupin linear isotropic indeterminate couple stress model. Our main aim is to show that, up to now, the boundary conditions have not been completely understood for this model. As it turns out, and to our own surprise, restricting the well known boundary conditions stemming from the strain gradient or second gradient models to the particular case of the indeterminate couple stress model, does not always reduce to the Grioli–Koiter–Mindlin–Toupin set of accepted boundary conditions. We present, therefore, a proof of the fact that when specific “mixed” kinematical and traction boundary conditions are assigned on the boundary, no “a priori” equivalence can be established between Mindlins and our approach.

Ferroelectric nanogenerators coupled to an electric circuit for energy harvesting

The direct transformation of ambient mechanical energy into electricity using ferroelectric nanogenerators is discussed within the context of usability for self-sustaining microelectronics. Thus, it is essential to store the generated electric energy within an accumulator or capacitor. However, the contact and charge status of the electric storage medium strongly influences the performance of the generator. This necessitates coupling of the generator and the electric circuit to determine working points. Therefore, a phase field model for the ferroelectric generator is coupled with the response of a standard full-wave rectifier and a capacitor. Nonlinear diode characteristics as well as energy losses are under consideration. The amount and the type of connections for the nanogenerators in the harvesting field are discussed to bridge from the nanoscale to electrical quantities for microelectronics.

Domain engineered ferroelectric energy harvesters on a substrate

Phase-field modeling is used to study the domain evolution of nano-scaled ferroelectric devices influenced by the mechanical strain of an underlying substrate. The investigations focus on the design of the energy harvesting systems to convert mechanical into electrical energy. Mechanical energy is provided by an alternating in-plane strain in the substrate through bending or unidirectional stretching. Additionally, lattice mismatch between the substrate and the ferroelectric material induces epitaxial strain and controls the polarization behavior within the system. Further, electrical boundary conditions are used to stabilize the domain topology. Finite element simulations are employed to explore the performance of the engineered domain topologies in delivering electrical charge from mechanical deformation.

Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy

For homogeneous higher-gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain ε = symGrad [ u ] , the new invariance condition is equivalent to the isotropy of the linear formulation. Therefore, it may also be used in higher-gradient elasticity models for a simple check of isotropy and for extensions to anisotropy. In this respect we consider in more detail variational formulations of the linear indeterminate couple-stress model, a new variant of it with symmetric force stresses and general linear gradient elasticity.

Phase-field simulation and design of a ferroelectric nano-generator

We study the behavior of ferroelectric material (BaTiO3) for the design of a nano-generator to convert mechanical into electrical energy. The investigations consider an electro-mechanical phase-field model with polarization as state variable. This widely accepted model has its origins in the work of1-3 and is fully developed by Landis and coworkers.4,5 We use a finite element model to simulate tetragonal regions of ferroelectric material sputtered on substrate. Different geometries as well as various mechanical and electrical boundary conditions are considered. The model parameters are normalized to achieve better computational conditions within the stiffness matrix. The major objective of this contribution is the fundamental understanding of domain switching caused by a cyclic electrical field. The corresponding hysteresis loops of the overall polarization cannot be achieved by using a two-dimensional model because the domain topologies evolve in three dimensions. The three-dimensional nature of the domain structure evolution is even true for flat regions or thin films.6 We show some examples of three-dimensional domain topologies, which are able to break energetically unfavorable symmetries. Finally, the computational model of a tetragonal nano-generator with dimensions 10 x 60 x 10 nm is presented. The specific ratio of height to width and the mounting on substrate is essential for its performance and principle of energy harvesting. We discuss the challenges and scopes of such a system.

Low Energy Periodic Microstructure in Ferroelectric Single Crystals

Two distinct modelling approaches are used to find minimum energy (equilibrium) microstructural states in tetragonal ferroelectric single crystals. The first approach treats domain walls as sharp interfaces and uses analytical solutions of the compatibility conditions at domain walls to identify multi-rank laminate microstructures that are free of residual stress and electric field. The second method treats domain walls as diffuse interfaces, using a phase-field model in 3-dimensions. This is computationally intensive, but takes the full field equations into account and allows a more general class of periodic microstructure to be explored. By searching for minimum energy configurations of a cube of tetragonal material, candidate unit cells of a periodic microstructure are identified. Adding periodic boundary conditions allows the assembly of the unit cells into a macro-structure of low energy. A noteworthy structure identified in this way is a “hexadomain” vortex consisting of six tetragonal domains meeting along the major diagonal of a cube. Several of the structures identified by the phase-field model are found to be special cases of multi-rank laminate structure. Thus the analytical approach offers a fast method for finding equilibrium microstructures, while the phase-field model provides a validation of these solutions.

Form- und Systemoptimierung für Rahmentragwerke

Eine optimale Tragwerkskonstruktion soll bei niedrigem Materialverbrauch eine moglichst hohe Traglast aufweisen. Die ganzheitliche Optimierung einer bereits geplanten oder aber die Neuplanung einer Rahmenkonstruktion agiert hierbei auf zwei Ebenen. Die Topologieoptimierung liefert zunachst das grundlegende Konzept und legt die masgeblichen Charakteristiken der Struktur fest wie z.B.