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## Über dieses Buch

As the sequel to the proceedings of the International Conference of Continuum Mechanics Focusing on Singularities (CoMFoS15), the proceedings of CoMFoS16 present further advances and new topics in mathematical theory and numerical simulations related to various aspects of continuum mechanics. These include fracture mechanics, shape optimization, modeling of earthquakes, material structure, interface dynamics and complex systems.. The authors are leading researchers with a profound knowledge of mathematical analysis from the fields of applied mathematics, physics, seismology, engineering, and industry. The book helps readers to understand how mathematical theory can be applied to various industrial problems, and conversely, how industrial problems lead to new mathematical challenges.

## Inhaltsverzeichnis

### Mathematical Modeling of the Desiccation Cracking

Abstract
Desiccation cracks have a net-like appearance and tessellate the drying surface of the materials into polygonal cells with the typical size. The basic features of the crack pattern and the pattern formation process are conserved regardless of the choice of the materials. This implies the existence of the common governing mechanism behind the pattern formation in desiccation cracking. We propose the coupled model of desiccation, deformation, and fracture for the desiccation crack phenomenon in the framework of the continuum mechanics. By using this coupled model and the appropriate numerical analysis methods, the typical geometry and the typical length scale of the desiccation crack pattern are reproduced in the complete homogeneous field without any artificial length scale. These results indicate that the proposed coupled model captures the fundamental mechanism for the pattern formation in desiccation cracking.
Sayako Hirobe

### Fatigue Crack Growth Analysis of an Interfacial Crack in Heterogonous Material Using XIGA

Abstract
In the present work, the fatigue crack growth analysis of an interfacial cracked plate has been performed by extended isogeometric analysis (XIGA). In isogeometric analysis (IGA), non-uniform rational B-splines (NURBS) are employed for defining the geometry as well as the solution. In XIGA, the merits of isogeometric analysis and extended finite element method are combined together for analyzing the cracked geometries. The crack faces are modeled by discontinuous Heaviside jump function, whereas the singularity in the stress field at the crack tip is modeled by crack-tip enrichment functions. The values of stress intensity factors (SIFs) for the interface cracks are evaluated by XIGA and XFEM. Paris law is employed for computing the fatigue life of an interfacial cracked plate.
Indra Vir Singh, Gagandeep Bhardwaj

### A Comparison of Delamination Models: Modeling, Properties, and Applications

Abstract
This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed.
Marita Thomas

### Simulation of Ductile Fracture in Amorphous and Polycrystalline Materials by Multiscale Cohesive Zone Model

Abstract
A multiscale cohesive zone model (MCZM) that combines finite element method with atomistic modeling is applied to simulate fracture of amorphous materials and polycrystalline solids. In order to apply MCZM to model amorphous materials, the Cauchy–Born rule is linked with the Parrinello–Rahman MD method to associate atom configurations with material deformation by using molecular statics (MS). We found the algorithm allows us to simulate ductile fracture of amorphous materials successfully. In addition, the methodology is applied to model the amorphous grain boundaries of polycrystalline solids, and we show that it can capture ductile fracture of polycrystalline metals.
Shingo Urata, Shaofan Li

### Critical Scaling and Prediction of Snap-Through Buckling

Abstract
We examined predictability of catastrophic events in snap-through buckling. In our study, we applied load at the center of an elastic arch and monitored the fundamental frequency by hammering. We found that the frequency goes to zero as the load approaches the critical value (load limit point), suggesting that the occurrence of the snap-through buckling accompanying an abrupt collapse of the structure can be predicted before it happens. We described the behavior with a simple theoretical model, derived scaling relations, and confirmed reasonable agreements with experimental results.
Tetsuo Yamaguchi, Hiroshi Ohtsubo, Yoshinori Sawae

### Second Derivatives of Cost Functions and Newton Method in Shape Optimization Problems

Abstract
We derive the second-order shape derivatives (shape Hessians) of cost functions for shape optimization problems of domains in which boundary value problems of partial differential equations are defined, and propose an $$H^1$$ Newton method to solve the problems using the shape Hessians. In this paper, we formulate an abstract shape optimization problem and show the computations of the first- and second-order shape derivatives of cost functions under the abstract framework. Then, using the shape gradients and Hessians, we propose an $$H^1$$ Newton method to solve the given problem. As an illustration, the shape Hessians of a mean compliance and a domain measure are derived and then used for a numerical example.
Hideyuki Azegami

### Shape Optimization by Generalized J-Integral in Poisson’s Equation with a Mixed Boundary Condition

Abstract
Generalized J-integral is the tool for shape sensitivity analysis of singular points in boundary value problem for partial differential equations. We can solve shape optimization problems of singular points by using Generalized J-integral and $$H^{1}$$-gradient method (Azegami’s method). Here, the mathematical method is proposed to examine shape optimization in detail by dividing the sensitivity on sets of singular points, and apply the method to Poisson’s equation defined on a polygonal domain with mixed boundary condition. The boundary divides into the parts that Dirichlet boundary condition, Neumann boundary condition, and the joint of them are given. It is examined about each role of the parts of boundary in shape optimization process on a numerical example of finite element analysis.
Kohji Ohtsuka

### On Applications of Fast Domain Partitioning Method to Earthquake Simulations with Spatiotemporal Boundary Integral Equation Method

Abstract
This paper introduces the recent developments in the earthquake rupture simulations particularly focusing on our applications of the spatiotemporal domain boundary integral equation method (ST-BIEM) and the fast domain partitioning method (FDPM), which enable us to reduce the required memory storage and the computation time, respectively, to $$O(M^{2})$$ and $$O(MN^{2})$$ from the original values of $$O(M^{2}N)$$ and $$O(MN^{3})$$ for the given elements M and time steps N. FDPM utilizes the particular spatiotemporal dependence of the stress Green’s function (fundamental solutions) by partitioning the causality cone. FDPM can also seamlessly combine fully dynamic and quasi-dynamic simulation algorithms adapted in seismology. Related issues in seismological simulations are also discussed.
Ryosuke Ando

### Integral Representation and Its Applications in Earthquake Mechanics: A Review

Abstract
In seismology, a faulting process as a source is linked with an elastic wavefield as an observable not only via a partial differential equation (PDE) but also via an integral equation. We conduct a review of these links and focus on the latter in terms of forward/inverse analyses of kinematic/dynamic modeling, which are investigated by many seismologists. Difficulties in the analyses are also mentioned: estimation and hyper-singularity of an integration kernel, determination of the number of parameters for modeling, and assumed dynamic friction on faults.
Shiro Hirano

### Brief Introduction to Damage Mechanics and Its Relation to Deformations

Abstract
We discuss some principle concepts of damage mechanics and outline a possibility to address the open question of the damage-to-deformation relation by suggesting a parameter identification setting. To this end, we introduce a variable motivated by the physical damage phenomenon and comment on its accessibility through measurements. We give an extensive survey on analytic results and present an isotropic irreversible partial damage model in a dynamic mechanical setting in form of a second-order hyperbolic equation coupled with an ordinary differential equation for the damage evolution. We end with a note on a possible parameter identification setting.

### Structured Deformations of Continua: Theory and Applications

Abstract
The scope of this contribution is to present an overview of the theory of structured deformations of continua, together with some applications. Structured deformations aim at being a unified theory in which elastic and plastic behaviours, as well as fractures and defects can be described in a single setting. Since its introduction in the scientific community of rational mechanicists [10], the theory has been put in the framework of variational calculus [8], thus allowing for solution of problems via energy minimization. Some background, three problems and a discussion on future directions are presented.
Marco Morandotti

### Gradient Flows with Wiggly Potential: A Variational Approach to Dynamics

Abstract
Free energies with many small wiggles, arising from small-scale micro-structural changes, appear often in phase transformations, protein folding and friction problems. In this paper, we investigate gradient flows with energies $$E_\varepsilon$$ given by the superposition of a convex functional and fast small oscillations. We apply the time-discrete minimising-movement scheme to capture the effect of the local minimisers of $$E_\varepsilon$$ in the limit equation as $$\varepsilon$$ tends to zero. We perform a multiscale analysis according to the mutual vanishing behaviour of the spatial parameter $$\varepsilon$$ and the time step $$\tau$$, and we highlight three different regimes $$\tau \ll \varepsilon$$, $$\varepsilon \ll \tau$$ (Braides, Local Minimization, Variational Evolution and $$\varGamma$$-convergence. Springer, Cham (2014), [4]) and $$\tau \sim \varepsilon$$ (Ansini et al., Minimising movements for oscillating energies: the critical regime, [3]). We discuss for each case the existence of a pinning threshold, and we derive the limit equation describing the motion.

### Energy-Stable Numerical Schemes for Multiscale Simulations of Polymer–Solvent Mixtures

Abstract
We present a new second-order energy dissipative numerical scheme to treat macroscopic equations aiming at the modeling of the dynamics of complex polymer–solvent mixtures. These partial differential equations are the Cahn-Hilliard equation for diffuse interface phase fields and the Oldroyd-B equations for the hydrodynamics of the polymeric mixture. A second-order combined finite volume/finite difference method is applied for the spatial discretization. A complementary approach to study the same physical system is realized by simulations of a microscopic model based on a hybrid Lattice Boltzmann/Molecular Dynamics scheme. These latter simulations provide initial conditions for the numerical solution of the macroscopic equations. This procedure is intended as a first step toward the development of a multiscale method that aims at combining the two models.
Mária Lukáčová-Medvid’ová, Burkhard Dünweg, Paul Strasser, Nikita Tretyakov

### On Mathematical Modeling and Analysis of Brain Network

Abstract
In this article, we first formulate a functional equation-based modeling of the resting-state network, which is attracting attention in the research of the brain. Then, we discuss the local-in-time solvability of the model in a suitable function space.

### Convergence Rates for Discrete-to-Continuum Limits in 1D Particle Systems

Abstract
We contribute to a recent series of papers on discrete-to-continuum limits of one-dimensional particle systems governed by nonlocal and unbounded interactions. While convergence of the equilibrium positions of the particles to a limiting density profile is known, and while in several cases boundary layers have been characterised, any quantitative bound on the convergence rate to the limiting density is missing. Our main result guarantees such bounds. The proof method relies on quantitative versions of the compactness estimates in a recent paper on boundary layers, supplemented with new estimates, which give a precise meaning to the deviation from the particle positions in equilibrium to the equispaced configuration.
Patrick van Meurs
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