Thematic Einstein Semester on
Energy-based mathematical methods for reactive multiphase flows

3-day Workshop

MA4M: Mathematical Analysis for Mechanics

November 23–25, 2020

This workshop is devoted to recent aspects regarding the analysis of mathematical problems arising in the continuum mechanics of solids.
It features 20 presentations by invited speakers and will be held as an online format. Registration is required and open till November 20, 2020.
In particular, please note that the links to the online-presentation sessions will be available for registered participants, only.

Contact: If you have questions, please feel free to contact us using MA4M@wias-berlin.de.

Registration

Speakers

Program

Manuel Friedrich (Westfalische Wilhelms-Universität Münster)

Equilibrium configurations for epitaxially strained crystalline films

We revisit results obtained by Bonnetier, Chambolle, and Solci about the existence of minimizers and relaxation for energies related to epitaxially strained crystalline films. Our goal is to extend their analysis to the framework of three-dimensional linear elasticity. A major obstacle to overcome is the lack of any a priori assumption on the integrability properties of displacements which leads to a formulation in the space GSBD^p of generalized special functions of bounded deformation. Based on joint work with Vito Crismale.

Petr Pelech (WIAS Berlin)

Separately global solutions to rate-independent systems – applications to large-strain deformations of damageable solids

Rate independent systems (RIS) are characterized by the lack of any internal time length scale: rescaling the input of the system in time leads to the very same rescaling of its solution. In continuum mechanics, rate-independent models represent a reasonable approximation whenever the external conditions change slowly enough so that the system can always reach its equilibrium. This applies if inertial, viscous, and thermal eects are neglected. Rate independent systems have proven to be useful in modeling of hysteresis, phase transitions in solids, elastoplasticity, damage, or fracture in small and large strain regimes.
The talk introduces the notion of separately global solutions for large-strain rateindependent systems and explains an existence result for a model describing bulk damage. The analysis covers non-convex energies blowing up for extreme compression, yields solutions excluding interpenetration of matter, and allows for handling nonlinear couplings of the deformation and the internal variable, which emerges e.g. from the interplay between Eulerian and Lagrangian description. It extends the theory developed so far in the small strain setting.

Sven Tornquist (WIAS Berlin)

Temporal regularity of solutions to a dynamic phase-field fracture model in visco-elastic materials

A model for the description of cracks in visco-elastic materials is investigated using a phase-field approach. While respecting also inertial effects, we examine for the damage a viscous evolution with an quadratic approach and also a rate-independent law with a 1-homogeneous dissipation potential. Material healing is prevented in both cases by incorporating a non-smooth constraint that ensures a unidirectionality of the damage propagation. As a consequence of the viscous damping of the displacement together with convexity properties of the system energy, the regularity of the internal variable can be improved to a continuous evolution in time with values in the state space that guarantees finite values of the energy in the viscous case and even Holder-continuity in the rate-independent setting. This work is financially supported by the German Research Foundation (DFG) within the priority programme “Reliable Simulation Techniques in Solid Mechanics” (SPP1748).

Mario Varga (Technische Universitat Dresden)

Representative volume element approximations for laminated nonlinearly elastic random materials

The representative volume element (RVE) method is a crucial procedure for approximation of homogenized properties of random materials. In recent times, a significant progress has been made in understanding the approximation error for linear elliptic equations and convex integral functionals. In this context, we study nonlinearly elastic randomly laminated composite materials, and thus homogenization of a nonconvex integral functional. In particular, the integrand is assumed to be minimized at the set of rotations. Under the assumption of a rapid decay of correlations of the random material, we establish an error estimate of order L^-1/2 for the RVE approximation and its first two derivatives in a neighborhood of the set of rotations. The estimates are given in terms of the representative volume size L>>1. The presentation is based on a joint work with Stefan Neukamm and Mathias Schäffner.

Sourav Mitra (Julius-Maximilians-Universität Würzburg)

Global existence of weak solutions for a magnetic fluid model

In this talk I will present some resent results we obtained on the global existence of weak solutions of a system of partial dierential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids. The system consists of the incompressible Navier-Stokes equations coupled with an evolutionary equation for the magnetization vector and the Cahn-Hilliard equations. We will consider both the cases of matched and unmatched densities.

Barbara Zwicknagl (Humboldt-Universität zu Berlin)

On variational models for the geometry of martensite needles

Around macrointerfaces in martensites one often observes microstructures that involve characteristic needle-shaped domains. The specific needle shape is largely determined by the bending and tapering of the needle towards the interface. We show that the tapering length can be understood within nonlinear elasticity but not within linearized elasticity. Our analytical results are in good agreement with experiments and supported by numerical simulations.
This is based on joint work with S. Conti, M. Lenz, N. Lüthen, and M. Rumpf.

Francesco De Anna (Julius-Maximilians-Universität Würzburg)

On classical solutions for some Oldroyd-B model of viscoelastic fluids

This talk is devoted to the analysis of an Oldroyd-B system of PDEs which models the evolution of certain viscoelastic fluids. A particular emphasis is spent on the so called “corotational” model. We are interested in the well-posedness theory of classical solutions. We show in a bidimensional setting that, without any restriction on the initial data, the solutions exist globally in time and they are unique. This result is due to the particular structure of the system which allows to propagate high regularities of the solutions, in particular a Lipschitz regularity of the velocity field. A specific toolbox of Fourier Analysis is presented to address the mentioned result.

Thomas Eiter (WIAS Berlin)

Spatially asymptotic structure of time-periodic Navier-Stokes flows

Consider the time-periodic flow of a viscous incompressible fluid past an obstacle. To study its asymptotic behavior in space, the notion of time-periodic fundamental solutions to the associated linearized Navier-Stokes equations is introduced. This allows to derive an asymptotic expansion for the time-periodic velocity field, from which sharp results on the behavior at spatial infinity are concluded in terms of integrability and pointwise estimates. Moreover, pointwise estimates of the associated vorticity field are established.

Martin Heida (WIAS Berlin)

Elasticity on randomly perforated domains

We consider homogenization of elasticity on randomly perforated and locally Lipschitz domains. The domains we consider are not of Jones-type and hence need different methodologies. We discuss the arising difficulties in proving sufficiently strong Korn inequalities based on results of the recent arXiv-preprint 2001.10373.

We furthermore explain how to apply the resulting Korn-inequality in two-scale Gamma-convergence.

Carolin Kreisbeck (KU Eichstatt-Ingolstadt)

Nonlocal variational problems: Structure-preservation during relaxation?

Nonlocal variational problems arise in various applications, such as in continuum mechanics through peridynamics, the theory of phase transitions, or image processing. Naturally, the presence of nonlocalities leads to new eects, and the standard methods in the calculus of variations, which tend to rely intrinsically on localization arguments, do not apply. This talk addresses the relaxation of two classes of functionals – double-integrals and nonlocal supremals. Our focus lies on the question of whether the resulting relaxed functionals preserve their structure. We give an armative answer for nonlocal supremals in the scalar setting, along with a closed representation formula in terms of separate level convexification of a suitably diagonalized supremand, and discuss results in the vectorial case. As for double-integrals, a full understanding of the problem is still missing. Wepresent the first counterexample showing that weak lower semicontinuous envelopes fail to be double-integrals in general. On a technical level, both findings rely on a characterization of the asymptotic behavior of (approximate) nonlocal inclusions via Young measures, a theoretical result of independent interest. This is joint work with Elvira Zappale (Sapienza University of Rome) and Antonella Ritorto (Utrecht University).

Marek Tyburec (Czech Technical University in Prague)

Global optimality in minimum-compliance topology optimization by moment-sum-of-squares hierarchy

Designing minimum-compliance bending-resistant structures with continuous cross-section parameters has been a challenging task because of its non-convexity. We develop a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow at extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sucient condition of global $latex epsilon$-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. We illustrate these theoretical findings on examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.

Max Griehl (Technische Universitat Dresden)

Derivation of a bending plate model for nematic liquid-crystal elastomers via Gamma-convergence

Liquid-crystal elastomers (LCEs) are a class of materials, whose shape can be controlled via external stimulation. In the talk we consider a three-dimensional model that consists of a hyperelastic energy and the liquid-crystal’s Oseen-Frank energy. Using Gamma convergence, we then derive and examine a dimension-reduced model, effectively describing the bending behaviour and director fields for thin LCE-plates. The talk is based on joint work with S. Neukamm.

Sergio Conti (Rheinische Friedrich-Wilhelms-Universität Bonn)

Variational modeling of ductile fracture

I will discuss recent work on the variational modeling of ductile fracture in solids. The first theme is the approximation of static ductile fracture models via phase-field approximations. Approximation in the sense of Gamma-convergence can be proven for scalar models with quadratic energies, in some cases the result can be extended to nonquadratic energy densities. The second theme is the modeling of irreversible evolution. I will discuss an irreversibility constraint for the static phase-field model, and its sharp-interface limit. For the sharp interface limit in one dimension, existence of a quasi-static evolution can be proven. This talk is based on joint work with Marco Bonacini, Matteo Focardi, Flaviana Iurlano and Jorn Mosler.

Angkana Rüland (Ruprecht-Karls Universität Heidelberg)

Symmetry, (ir-)regularity and avalanching dynamics in shape-memory alloys

In this talk I discuss three aspects of solutions to differential inclusions modelling shapememory alloys: First I classify certain highly symmetric solutions. Next, I consider a dichotomy between rigidity and flexibility of solutions depending on the prescribed regularity. The highly symmetric solutions here saturate the “phase transformation” between rigidity and flexibility. Finally, I present results on a geometrically constrained, probabilistic model for avalanching in shape-memory alloys upon nucleation.

Antoine Gloria (Sorbonne Universite)

Random suspensions in a Stokes flow

In this talk I will consider the effect of randomly distributed rigid particles on the viscosity of a Stokes fluid. Assuming first that the particles have the same mass density as the fluid I will give a homogenisation result and define the notion of effective viscosity. In the regime when the density of particles is small, one can make a Taylor expansion of that effective viscosity, and rigorously recover the celebrated Einstein formula. Then I will address the much more subtle case when particles are heavier than the fluid, and consider sedimentation. The definition of the eective sedimentation speed of the particles will require quantitative mixing conditions. I will discuss in particular the Caflisch-Luke paradox, and how it can be ruled out if we assume long-range order, eg in form of hyperuniformity. This is joint work with Mitia Duerinckx (CNRS, Orsay).

Joshua Kortum (Julius-Maximilians-Universität Würzburg)

Concentration-cancellation in the simplified Ericksen-Leslie model

We investigate the existence and weak stability of global weak solutions for the simplified Ericksen-Leslie system. The latter is a model for uniaxial liquid crystal flows and comprises a suitably adapted version of the Navier-Stokes equations and a harmonic map heat flow-like equation. The construction of weak solutions relies on the Ginzburg-Landau approximation where the main problem consists of the limit passage in the Navier-Stokes equation. This issue is handled by invoking partial regularity techniques and the method of concentration-cancellation originally introduced for the incompressible Euler equations.

Daniel Campbell (University of Hradec Kralove)

Planar models for elastic deformations which allow for cavitation and fractures

Motivated by a relaxation result of Kristensen and Rindler for BV maps we study the strict limits of BV homeomorphisms. Such maps need not be injective and fail to be continuous on almost every line. We show that the class has reasonable behaviour expected of deformations such as the INV condition, we characterize cavitation points and fractures and we prove a characterization result for such limits.