A phenomenological constitutive theory for polycrystalline ferroelectric ceramics based on orientation distribution functions
A phenomenological constitutive theory for the ferro-electro-elastic response of polycrystalline ceramics with tetragonal perovskite structure is proposed. The state of a material point is characterized by an intrinsic polarization vector, an infinitesimal deformation tensor, and an internal variable representing the orientation distribution of c-axes of the ferroelectric domains at that point. A class of convex thermodynamic potentials in terms of these state variables is posited, and constitutive relations within the framework of generalized standard materials are then derived. The functional form of the dissipation is selected in such a way that it effects an order reduction of the constitutive description whereby the infinite-dimensional internal variable is reduced to finite-dimensional internal variables representing polarization and deformation due to ferroelectric switching, preserving at the same time the generalized standard structure of the theory. The resulting constitutive theory is able to emulate most essential features of ferroelectric and ferroelastic behavior with minimal computational cost and, furthermore, generates stable predictions in contrast to current phenomenological theories in common use.
Thorsten Bartel, Andreas Menzel
Relaxed energy potentials based on perturbations of divergence-free fields
The combined physically and mathematically sound modelling and simulation of the inelastic constitutive behaviour of heterogeneous materials is still a major challenge in continuum mechanics. In this context, the concept of quasiconvexification is not only a suitable mathematical tool to proof the existence of solutions. From the viewpoint of mechanics, it might as well be regarded as the optimal homogenisation method. Being achievable only in rare cases, however, the quasiconvex energy hull of an underlying non-quasiconvex energy landscape (e.g. a multi-well potential) is approximated in terms of upper and lower bounds. The perturbations applied in order to achieve such bounds are commonly defined with respect to curl-free fields, e.g. deformation gradients derived from compatible displacement fields. Such approaches, for example in the context of laminate-based perturbation fields, a priori fulfil the Dirichlet-type as well as the Neumann-type continuity conditions at underlying interfaces. The drawback of this concept lies in the fact, that in general upper bounds to the quasiconvex hull are obtained. Thus, the effective energy potential may violate the Cauchy-Hadamard condition and numerical simulations may suffer from instabilities. In the present contribution, relaxed energy potentials are derived based on the perturbation of divergence-free fields (such as stresses) with the aim of finding laminate-based energy hulls which may preserve the beneficial features of quasiconvexity. The numerical examples will focus on simulations of the behaviour of conventional and magnetic shape memory alloys.
Bernhard Eidel, Andreas Fischer
Error Analysis for Adaptive Mesh-Coarsening of Pixel-Based Microstructure Representation in Homogenization
For pixel-based microstructure representations we propose two
different variants of adaptive, quadtree-based mesh-coarsening algorithms
that serve the purpose of a preprocessor for two-scale finite element anal-
yses in the context of computational homogenization. High resolution is pre-
served at interfaces for accuracy, coarse-graining in the defect-free interior
of phases for efficiency. Error analysis is carried out on the microscale by
reconstruction-based error estimation which itself is assessed by true error
computation. A modified stress recovery scheme for a superconvergent error
estimator is proposed which overcomes the deficits of the standard recovery
scheme for nodal stress computation in cases of interfaces with stiffness jump
. By virtue of error analysis the improved efficiency by the reduction of
unknowns is put into relation to the increase of the discretization error and
thereby sets a rational basis for decisions on favorable meshes having the best
trade-off between accuracy and efficiency. The performance of the proposed
methodology is demonstrated by simulation results employing a recently in-
troduced formulation of the Finite Element Heterogeneous Multiscale Method
(FE-HMM) for elastic solids  for different, energetically-consistent micro-
coupling conditions .
 A. Fischer and B. Eidel (2019): Error analysis for quadtree-type mesh-
coarsening algorithms adapted to pixelized heterogeneous microstructures.
 A. Fischer and B. Eidel (2019): Convergence and error analysis of FE-
HMM/FE2 for energetically consistent micro-coupling conditions in linear
elastic solids. Eur. J. Mech. A. Solids 77, 103735.
 B. Eidel and A. Fischer (2018): The heterogeneous multiscale finite ele-
ment method for the homogenization of linear elastic solids and a comparison
with the FE2 method. Comput. Methods Appl. Mech. Eng. 329, 332–368.
On the Stability of Objective Structures
We discuss the stability of objective structures of interacting atoms. To this end, we use methods from harmonic analysis to set up a second derivative test of the configurational energy at equilibria, a key point being the identification of a carefully chosen seminorm with respect to which coercivity is estimated. We also provide an efficient algorithm to check for atomistic stability which we illustrate numerically for a general carbon nanotube.
Georges Griso, Julia Orlik, Stephan Wackerle
Homogenization and Dimension Reduction of Textiles
In this work, we investigate periodic structures made of fibers or yarns, like
textiles and derive their macroscopic properties via simultaneous homogeniza-
tion and dimension reduction. As reference domain we consider a canvas struc-
ture, which we assume to consist of periodically oscillating and isotropic beams
with periodicity and radius of the same order. The beams are in contact and
thereby the elasticity problem is restricted on a cone fulfilling non-penetration
and gap conditions. To obtain different compactness results for all components
of the displacement, we apply the decomposition of displacements for thin struc-
tures, introduced by G. Griso in 2008. The derived estimates depend on the
small parameters, the elastic energy, and the contact. Further we extend the
fields into the full plate domain. We introduce an adapted unfolding operator
with an incorporated dimension reduction from three to two dimensions. The
properties of the unfolding operator together with the compactness results lead
to its weak convergence, equivalent to the two-scale convergence. Consequently,
the unfolded limits of the displacements, the strain tensor and contact condition
yield homogenized 2D-model for a textile.
A cohesive crack model for plates
We introduce a variational model for elastic plates with cracks that
accounts for the cohesive forces acting between the crack faces. The model
is based on the Kirchhoff—Love theory and coupled in the sense that the
crack faces are subjected to nonpenertation conditions including both the
vertical deflection and horizontal displacements. The density of the
energy spent by the cohesive forces depends on the crack opening and is
not convex, in general. Following the basic idea behind the Griffith
theory, we focus on the study of the differentiability of the deformation
energy with respect to the crack length.
Towards the engineering design of metamaterials’ structures through micromorphic enriched continuum modeling
In this talk, I will show how the relaxed micromorphic model, which I have contributed to pioneer, can be used to describe the dynamical behavior of anisotropic mechanical metamaterials. I will show to which extent the proposed model is able to capture all the main macroscopic dynamical characteristics of the targeted metamaterials, namely, stiffness, anisotropy, dispersion and band-gaps. The simple structure of our material model,
which simultaneously lives on a micro-, a meso- and a macroscopic scale, requires only the identification of a limited number of frequency-independent parameters, thus allowing the introduction of pertinent boundary conditions to be imposed at macroscopic metamaterials’ boundaries when the model is framed in the context of Variational Principles. I will show how this modelling approach can be applied to the study of the scattering properties of finite-size metamaterials’ structures thus opening new perspectives for metastructural engineering design.
Random tessellations – Stochastic models for cellular materials
andom tessellation models from stochastic geometry are powerful tools for studying relations between the microstructure of a cellular material and its macroscopic properties.
In practice, a tessellation model can be fit to a real microstructure using geometric characteristics which are estimated from image data. In particular, three dimensional CT images are an important source of information on the microstructure of materials. We will introduce methods for the estimation of geometric characteristics of cellular materials from image data and describe how these characteristics can be used to fit tessellation models to the observed structure.
Finally, we will present some applications where tessellation models are used to investigate relations between the microstructure and macroscopic properties of foams.
Laurent Guin, L. Benoit-Maréchal, L. Shaabani Ardali, M. E. Jabbour, N. Triantafyllidis
Stability of Vicinal Surfaces: Beyond the Quasistatic Approximation
Atomic steps are observed on vicinal surfaces—crystal surfaces slightly mis-
oriented from the high-symmetry crystallographic planes—of crystalline mate-
rials where they form periodic arrays separating atomic terraces. Under appro-
priate experimental conditions, the epitaxial growth of crystals occurs through
the propagation of these atomic steps.
The deposited atoms diffuse on the terraces before attaching to the steps
where they crystallize, with each step propagating at the same speed, providing a
uniform crystal growth. A frequently observed instability is the step bunching: a
bifurcation away from the above-mentioned uniform speed mechanism by which
differences in step velocities result in the bunching of atomic steps, thereby
forming “macrosteps” separated by wide terraces.
With the pioneering work of Burton, Cabrera and Frank (BCF) the dynam-
ics of steps has been modeled by accounting for adatom adsorption, desorption
and diffusion on terraces and their attachement/detachement (a/d) to steps.
Since then, several physical mechanisms (elasticity, asymmetry in a/d rates at
steps, adatom diffusion along steps) have been added to the original model with
the aim of understanding occurences of step bunching. However, all existing
analyses have been carried out in the framework of the quasistatic approxima-
tion, a mathematical assumption usually considered appropriate in regimes of
low deposition rate.
In this work, we develop a linear stability analysis of the general governing
equations without resorting to that simplification and surprisingly find that the
stability results are significantly modified, even in the regime of slow deposition.
While this result questions the scope of the quasistatic approximation it also
provides a new candidate mechanism for step bunching, pertinent for the under-
standing of its occurrence in growth of GaAs(001) and Si(111)-7x7. That the
proposed mechanism is indeed involved in some instances of bunching is sup-
ported by numerical computations of the long term evolutions of step bunches,
which show that the patterns formed by the step bunches are consistent with
those observed in Si(111)-7x7 at 750◦C.
L. Guin, M. E. Jabbour, L. Shaabani Ardali, L. Benoit-Maréchal,
and N. Triantafyllidis, “Stability of vicinal surfaces: Beyond the quasistatic
approximation”, Physical Review Letters (In press).
Dominik Brands, S. Uebing, L. Scheunemann, J. Schröder
Towards a multiscale analysis of residual stresses during cooling of hot forming parts
In actual research, more and more attention is paid to the understanding of residual stress states as
well as the application of targeted residual stresses to expand liftime or stiffness among other
things. In course of that, the numerical simulation and analysis of the forming process of
components, which goes along with the evolution of residual stresses, plays an important role.
Temperature dependent forming processes, such as hot bulk forming, offer the opportunity to adjust
material parameters, e.g. deformation state, temperature profile or cooling media, independently.
Hence, the upsetting test of a cylinder with an eccentric hole at high temperatures on different scales
is examined. In this contribution, we focus on the mircoscopic and mesoscopic level. This
multiscale view enables a detailed description of phenoma on the microscale such as the lattice
shearing from face-centered cubic austenite unit cells to body-centered tetragonal martensite cells
and is directly related to the typical classification of residual stresses following .
A combination of a Multi-Phase-Field model, see , and a twoscale Finite Element method, see 
is utilized for numerical analysis. A first microscopic simulation considers the lattice change, such
that the results can be homogenized and applied on the mesoscale. Based on this result, a
polycrystal consisting of a certain number of austenitic grains is built and the phase transformation
from austenite to martensite is described with respect to the mesoscale. Afterwards, a twoscale
Finite Element simulation is applied to introduce plastic effects and compute resulting residual
stress states. In this contribution, the work flow will be explained and some first results will be
discussed. For details we also refer to .
 E. Macherauch, H. Wohlfahrt and U. Wolfstied, Härterei-Technische Mitteilungen – Zeitschrift für
Werkstoffe, Wärmbehandlung, Fertigung, 28(3), 201-211, (1973).
 I. Steinbach, Phase-field models in materials science, Modelling and Simulation in Materials Science
and Engineering, 17(7):073001, (2009).
 J. Schröder, Plasticity and Beyond – Microstructures, Crystal Plasticity and Phase Transitions, CISM
Lecture Notes 550, Eds. J. Schröder, K. Hackl, chapter A numerical two-scale homogenization scheme:
the FE2-method, 1-64, (2014).
 Behrens, B.-A., Schröder, J., Brands, D., Scheunemann, L., Niekamp, R., Chugreev, A., Sarhil, M.,
Uebing, S., Kock, C., Experimental and Numerical Investigations of the Development of Residual
Stresses in Thermo-Mechanically Processed Cr-Alloyed Steel 1.3505. Metals. 9, 480 (28 pages) (2019)
Claude Le Bris
Elliptic homogenization theory: Addressing defects in the microstructure
We overview a series of mathematical works that introduce new modeling
and computational approaches for non-periodic materials and media. The approaches
consider various types of defects embedded in a periodic structure, which can be
either deterministic or random in nature. A portfolio of possible theoretical tools and computational techniques addressing the identification of the homogenized properties of the material is presented.
Bounds on Precipitate Hardening of Line and Surface Defects in Solids
In this talk, we investigate how line and surface defects in solids are affected by a nearly periodic arrangement of precipitates. We assume that the defects can be described as graphs of functions and that their motion is described by a driving equation. The driving equation depends on three terms. The first term is a penalty for the deviation of the geometry of the defect from a flat state,
the second term describes the interaction of the graph with the precipitates, and the last term constitutes the external driving force. In this setting, we derive an abstract result for graphical interfaces that allows us to investigate how the critical pinning force scales with the radius of the precipitates. We show how this result can be applied to a local and linear model and discuss the extension to the aforementioned models. Our main result states that the critical pinning force depends mainly on the geometry of the problem and therefore surface defects remain largely unaffected by precipitates compared to line defects which is in agreement with experimental observations.
This is joint work with Kaushik Bhattacharya (Caltech) and Patrick Dondl (University of Freiburg)
Peter Lewintan, Stefan Müller, Patrizio Neff
$L^p$-versions of the generalized Korn’s inequalities for incompatible tensor fields
Chiral magnetic skyrmions and computational micromagnetism
We consider the numerical approximation of the Landau-Lifshitz-Gilbert equation (LLG),
which describes the dynamics of the magnetization in ferromagnetic materials. In addition
to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii-Moriya
interaction (DMI), which is the most important ingredient for the enucleation and the
stabilization of chiral magnetic skyrmions. Besides convergent tangent plane integrators,
the talk also discusses weak-strong uniqueness of solutions as well as a reduced thin-film
model, which is particularly interesting for computations.
The talk is based on joint work, in particular, with Giovanni Di Fratta, Michael Innerberger,
Carl-Martin Pfeiler, and Michele Ruggeri.
Hrkac, Pfeiler, Praetorius, Ruggeri, Segatti, Stiftner:
Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics,
Advances in Computational Mathematics 45 (2019), 1329-1368.
Di Fratta, Innerberger, Praetorius:
Weak-strong uniqueness for the Landau-Lifshitz-Gilbert equation in micromagnetics,
preprint arXiv:1910.04630, submitted for publication.
Davoli, Di Fratta, Praetorius, Ruggeri:
Micromagnetics of thin films in the presence of the Dzyaloshinskii-Moriya interaction,
in preparation 2019/20.
Robert J. Martin, Patrizio Neff
The spin modulus in the context of Riemannian geometry on the general linear group
Simulation of nonlinear bending phenomena for bilayer plates in the presence of contact
The bending behaviour of plates is usually described using dimensionally reduced models. We propose a practical method for the numerical simulation of bilayer plate bending that is based on a nonlinear two-dimensional plate model. Our method employs a discretization of the resulting energy using DKT (discrete Kirchhof
triangle) elements in space and a discrete gradient flow restricted to appropriate tangent spaces for the minimization of the energy functional. We conclude the presentation by discussing the simulation of (self)-contact.
Siddhant Kumar, A. Vidyasagar, Dennis M. Kochmann
An assessment of numerical techniques to find energy-minimizing microstructures associated with nonconvex potentials
Microstructural patterns emerge ubiquitously during phase transformations, deforma-
tion twinning, or crystal plasticity. Challenges are the prediction of such microstructural
patterns and the resulting effective material behavior. Mathematically speaking, the exper-
imentally observed patterns are energy-minimizing sequences produced by an underlying
non-(quasi)convex strain energy. Therefore, identifying the microstructure and effective
response is linked to finding the quasiconvex, relaxed energy. Due to its nonlocal nature,
quasiconvexification has traditionally been limited to (semi-)analytical techniques or has
been dealt with by numerical techniques such as the finite element method (FEM). Both
have been restricted to primarily simple material models. We here contrast three numerical
techniques – FEM, an FFT spectral scheme, and a meshless maximum-entropy (max-ent)
method. We demonstrate their performances by minimizing the energy of a representative
volume element for both hyperelasticity and finite-strain phase transformations. Unlike
FEM, which fails to converge in most scenarios, the FFT scheme captures microstructures
of intriguingly high resolution, whereas max-ent is superior at approximating the relaxed
energy. None of the methods are capable of accurately predicting both microstructures
and relaxed energy; yet, both FFT and max-ent show significant advantages over FEM.
Numerical errors are explained by the energy associated with microstructural interfaces in
the numerical techniques compared here.
Kumar, S., Vidyasagar, A. and Kochmann, D.M. (2019), An assessment of numerical tech-
niques to find energy-minimizing microstructures associated with nonconvex potentials. Int
J Numer Methods Eng. Accepted Author Manuscript. doi:10.1002/nme.6280.
Isogeometric multi-scale modeling of 3D lattice metamaterials and structures with strain-gradient elasticity
Additive manufacturing now enables fabrication of heterogeneous structures with tailored complexity at topological, microstructural and material scales. In particular, metamaterials such as cellular, foam- and lattice-type microstructures facilitate variation and grading of their topology, density, or material constitution in order to locally adapt structural and functional properties. Though multi-scale modeling methods have already been applied for the simulation and optimization of lattice structures and metamaterials, practical 3D printed structures often only have an insufficient separation of scales, i.e., the size of the macrostructure is only an order of magnitude or so larger than the characteristic size of the microstructure. To overcome this issue, we propose a multiscale simulation method for such lattice structures based on strain-gradient elasticity, which considers size effects. Macroscopic strain-gradient constitutive models are obtained from homogenization of the unit cells of lattice metamaterials, which are modelled as 3D beam structures. Both, the discretization of the microscopic beam model as well as the macroscopic strain-gradient elasticity are implemented using spline-based, isogeometric methods. The methods are applied to various lattice topologies such as BCC, FCC, Octahedron, etc., and the influence of cell size and aspect ratio on the constitutive parameters of the strain-gradient model is investigated. Furthermore, the increased accuracy of the strain-gradient model compared to classical linear elasticity is demonstrated.
Richard D. James
The direct conversion of heat to electricity in the small temperature difference regime
We describe recent progress on materials, devices and theory for the direct conversion of
heat to electricity applicable to the small temperature difference regime, 10-200 C.
We are pursuing an idea based on the use of first order phase transformations
in materials that have either a jump of magnetization or of polarization at the phase transformation.
It is a “direct” method in the sense that there is no separate electrical generator: the material
itself does the energy conversion. We develop a mathematical theory of this method and
apply it to the design of the materials and devices and the analysis of cycles. We compare
theoretical predictions and the behavior of a prototype under cyclic heating/cooling.
These devices provide interesting possible ways to recover the vast amounts of energy
stored on earth at small temperature difference. They move heat produced by natural
and man-made sources from higher to lower temperature and therefore contribute
negatively to global warming.
Hörsaal II, 2nd floor, Albertstraße 23b, 79104 Freiburg (map).