at cooling rate of 10-40 K/s, or by slow cooling down from the melt through cold crucible casting, which has been investigated in several experimental works [15, 65]. The eutectic produced in arc-melted ingots exhibits direct solidification in the direction of heat transfer, Fig. 4.1(a), however equiaxed eutectic colonies are homogeneously distributed in the cross section of cast rod produced in cold crucible casting, Fig. 4.1(b). In the latter case the growth of the FeTi **phase** (brighter color) starts from a common center into different direction and have an average eutectic cell size and lamellar spacing (λ) of the order of 20 ± 5µm and 525 nm base on the work of Das et al. 2005 [15]. They have also measured the lattice mismatch owing to the difference between lattice parameters of β-Ti and FeTi phases, which introduces co- herency strain at the interface. This coherent interface induces mechanical stresses and elastic interactions in the system, therefore the elastic **phase**- **field** **model** is required to assess the formation of eutectic structures in Ti-Fe alloy.

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c
2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
1 Isogeometric Reissner-Mindlin **phase**-**field** **model**
An isogeometric Reissner-Mindlin shell formulation is combined with a **phase**-**field** **model** for the description of brittle fracture in thin and thick plates and shells. The shell is described only by its midsurface, where also the **phase**-**field** is defined. A director vector **field** is used for the description of the thickness direction. Since only linear problems are considered, an additive update of the director vector is used. The total strain tensor E of the shell can be split into the membrane strains ε m ,

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In order to achieve a better understanding of fracking processes, Rubin [182] studied the behaviour of hard rocks while Bohloli [30] carried out experiments on the unconsolidated soft rocks. Furthermore, experiments concerning the influence of confining stresses and natural fractures were conducted by Blanton [24], Warpinski & Teufel [197] and Zhou et al. [208]. Besides these experimental studies, the earlier theoretical investigation was usually based on very limited simple cases, cf. Rice & Cleary [180], Boone & Ingra↵ea [31], Boone & Detournay [32] and Detournay [55]. Thanks to the high-performance computers, more complicated and general scenarios have been under consideration. De Borst and Keschavarzi turned to the XFEM for a description of the fractured solid while Schrefler introduced a cohesive interface element. Proceeding from the successful application of the **phase**-**field** **model** to pure solid, some researchers attempted to combine it with Biot’s theory in modelling saturated soil, cf. Bourdin et al. [38], Mikelic et al. [159, 160], Wheeler et al. [199] and Miehe et al. [157, 158]. However, owing to the missing individual fluid balance equation, the transition from an in-pore fluid to a bulk flow requires an alternative treatment, for example, a substitution by an enlarging permeability. On the other hand, researchers started from the TPM and embedded the **phase** variable di↵usive crack, cf. Markert & Heider [113, 148] and Luo & Ehlers [141, 142]. In the work of Markert & Heider, the **model** was based partially on the standard form of the TPM, mainly for the balance equations, and partially on the variational formulations, mainly for the evolution of the

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We consider the optimal control of a droplet on a solid by means of the static contact angle between the contact line and the solid. The droplet is described by a thermodynamically consistent **phase** **field** **model** from [Abels et al., Math. Mod. Meth. Appl. Sc., 22(3), 2012] together with boundary data for the moving contact line from [Qian et al., J. Fluid Mech., 564, 2006]. We state an energy stable time discrete scheme for the forward problem based on known results, and pose an optimal control problem with tracking type objective.

As indicated in Chapter 1, the study of the **phase** change between water and ice in porous media plays a vital role in the understanding of important phenomena like the frost attack on concrete or the thawing of permafrost soil. In this chapter, we will present a mathematical **model** of this process. To this end, we start by introducing a **model** for the **phase** change on the pore scale of a porous medium. The porous medium is modeled with a periodic microstructure that consists of a pore space and a solid matrix. As usual in the theory of periodic homogenization, the characteristic size of the microstructure is given by a parameter 𝜀 > 0. The pore space is assumed to be filled by a mixture of water and ice that undergoes **phase** transitions due to changes in the temperature. The **phase** transition is modeled by a standard Caginalp **phase** **field** **model**, which is an approximation of a sharp interface **model** for **phase** transitions, namely of the classical Stefan problem. 1 Using time discretization, the existence of weak solutions of the microscale **model** is established.

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• We also want to give a comparison between the **model** of this thesis and the ther- mistor **model** studied in [46]. The models deal with variables (u, φ, P ) and (ϕ, θ) respectively. The strategies for both models are the same, namely, we reduce the problem to a semilinear parabolic equation with single variable in P (and in θ in the latter case), and then solve the semilinear parabolic equation using the result from [10]. The main difference is that the variable (u, φ) in this thesis is given as the solution of the piezo-system (vector-wise), while the variable ϕ in [46] is given as the solution of an elliptic equation (scalar-wise), thus the analysis of this thesis is significantly more complicated than the one given in [46]. We also point out that certain continuity arguments are applied in [46] to obtain the contraction condition in the Banach fixed point theorem from [10]. However, since the external loadings related to the piezo-system given in this thesis will also depend on the variable P , a direct application of the continuity arguments given in [46] to our case is difficult (notice that the external loadings related to the elliptic equation corresponding to the variable ϕ in [46] are independent of θ). Instead, we will use a direct difference comparison method to conclude such contraction condition, see Section 4.2 for de- tails. We should also point out that it is still possible to modify the method in [46] to fit our setting here, but the calculation is not getting simpler due to the complicated expressions of the functionals given in this thesis.

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There are many **phase**-**field** models for fracture based on a non-conserved order parameter that encompass much of the expected behavior of cracks [62, 52, 2, 30]. But as long as the scale of the growing patterns is set by the **phase**-**field** interface width ξ, these models do not possess a quantitatively valid sharp interface limit. While **phase**-**field** models are often used to **model** physical processes associated with microstructural evolution, we want to emphasize that this is not our goal here. The diffuse interface **model** we present in this section is instead used primarily as a numerical tool to solve the sharp interface equations Eqs. (5.3)-(5.6). This means that the **phase**-**field** **model** permits a strict physical interpretation only in the limit of vanishing interface thickness, ξ → 0, requiring that ξ is much smaller than any characteristic lengthscale in the problem. In order to succeed in this undertaking, our **phase**-**field** **model** has to fulfill two fundamental requirements: it has to have a valid sharp interface limit for the case ξ → 0, and the results must not depend on the **phase**-**field** width ξ. Both requirements are directly connected to the fact that ξ is a purely numerical parameter and not directly connected to physical properties. Alternative descriptions, which are intended to investigate the influence of elastic stresses on the morphological deformation of surfaces due to **phase** transition processes, are also based on macroscopic equations of motion. But they suffer from inherent finite time singularities which do not allow steady state crack growth unless the tip radius is again limited by the **phase**-**field** interface width [69].

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Numerische Simulationen von Kristallen mit multiplen Orientierungen sind im letzten Jahrzehnt zum Gegenstand intensiven Interesses der Indus- trie geworden, da eine große Bandbreite von industriellen Materialien - von Polykristallen bis zu Nanopartikeln - in diese Kategorie fallen. In dieser Ar- beit stelle ich mehrere detaillierte Studien vor, die das Potential von drei verschiedenen Phasenfeldmodellen f¨ ur die Beschreibung solcher Systeme un- tersuchen. Als erstes erweitere ich einen existierenden gekoppeltes Phasen- feld/Monte Carlo-Ansatz [H. Assadi, A **Phase**-**Field** **Model** for Crystalliza- tion into Multiple Grain Structures, in Solidification and Crystallization (2004), ed. von D. Herlach], um Gittereffekte zu eliminieren und verwende diesen Ansatz in Kombination mit dem zuvor entwickelten erweiterten Monte Carlo-Algoritmus, um Kristall-Wachstumswettbewerb in Gesteinskluften zu beschreiben. Anschliessend stelle ich ein neues Modell f¨ ur das Wachstum von metallischen Nanopartikeln in ionischen L¨osungen vor, welches auf einem klassischen Phasenfeldmodell basiert [Wheeler et al., Phys. Rev. A 45 (1992) 7424] und verwenden es f¨ ur erste qualitative Studien. Danach ver- wende ich die neue Phasenfeld-Kristall-Methode [K. R. Elder et al., Phys. Rev. Lett. 88 (2002) 235702-2 ] um die Korellation zwischen thermalem

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order parameter **field** Φ (r, t) takes the value 1 wherever and whenever the object is present. A diffuse continuous interface marks the transition from the object to the “non-object”, as shown here for a solid object in a liquid.
This initial shape is, thus, expressed as a scalar **field** of an “order parameter”, which is alternatively named “feature indicator” or, most commonly, a “**phase**–**field** variable”. For the present objective, it is sufficient to restrict further discussion to rigid objects. Neither the evolution of their shape nor their motion in vacuum space will be addressed here. The most basic use of a **phase**-**field** **model** is to just draw on the definition of the “**phase**-**field**”-variable. The **phase** **field** description of a static object—here a sphere—placed in a vacuum is schematically depicted in Figure 4.

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In this thesis, based on the random **field** theory, an electromechanically fully coupled **phase**- **field** **model** is proposed to simulate the peculiar behavior of the relaxor ferroelectrics. The **model** introduces a quenched local random **field** to characterize the effect of the chemical disorder. By treating the spontaneous polarization as an independent order parameter and the random **field** as an internal microforce, a thermodynamic analysis is performed. The deduced nonlin- ear constitutive and evolution equations are further discretized by the finite element method. Numerical examples show that the **model** can reproduce typical relaxor features, such as the miniaturization of domain size, the reduction of remanent polarization, and the enhancement of large-signal piezoresponse. The influence of the random **field** strength on the domain struc- ture and the hysteresis loops is also revealed and validated with the related experimental results. Subsequently, the **phase**-**field** **model** of relaxors, in combination with the conventional ferro- electric **model** is applied to analyze the large-signal piezoresponse for relaxor-based composites. More specifically, a series of simulations are presented for the relaxor/ferroelectric layer com- posites with different types of interfaces. The results confirm that the lateral strain coupling, in addition to the polarization coupling, contributes considerably to the large-signal piezoelectric coefficient. The lateral strain mismatch lowers the remanent strain in the ferroelectric layer and thus increases the macroscopic piezoelectric response. It is worth to be highlighted that the composition ratio of the relaxor constituent is optimized for different electric loadings. The composites with higher relaxor content are inclined to obtain higher large-signal piezoresponse with the increase of the applied electric **field**. These results can be referred in the future design of high-performance relaxor-based composites.

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In this contribution, we firstly provided a brief overview of the **phase**-**field** **model** for crack propagation in isotropic and anisotropic materials, respectively. Next, we presented the **phase**-**field** **model** for anisotropic fracture which can distinguish the loading under tension and compression. The fully coupled monolithic solution scheme within the finite element framework was formulated. Comprehensive parameter studies for the proposed **phase**-**field** **model** have been done and the results have been analyzed. Simulations of the anisotropic fracture within the lower value of the crack orientation θ are also considered, though the resulting crack path is no longer in alignment with the predefined crack orientation, and the opposite direction with fluctuating crack paths can be observed in the results. Furthermore, it is necessary to account for such phenomena that the widely used anisotropic materials (e.g. woods) consist of the lower value of the crack orientation in the structure and it can play an important role in material design processes. This topics will be addressed elsewhere. Representative numerical examples of the crack propagation in solar-grade polycrystalline silicon are carried out which can validate its capability of modelling of inter- and transgranular fracture process. Last but not least, the damage and failure analysis of solar-grade polycrystalline silicon using **phase**-**field** method will also be compared with experimental results in future.

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To tackle the moving boundary problem of dynamic crack propagation, we properly extended the above mentioned fully dynamic **phase** **field** **model** for solid-state transfor- mations to a **phase** **field** description of crack propagation by first order **phase** transforma- tion processes. Apart from that, also efficient steady state sharp interface methods based on the expansion of the elastic state in eigenfunctions of the straight mathematical cut were used. In contrast to the **phase** **field** technique this method easily could cover both the **phase** transformation kinetics as well as surface diffusion. However, it provides only accurate results in either the limit of vanishing viscosity or the limit of static elasticity (slow crack motion). Both quite complementary methods together then offer a profound insight into the phenomenon of fracture within the frame work of the present continuous description. Considering mode I cracks our theory predicts three generic features of frac- ture: The saturation of the steady state velocity appreciably below the Rayleigh speed; parameter regimes of normal crack behavior where the steady state velocity increases with increasing driving force; and the tip splitting instability for high applied tensions. However, concerning mixtures of mode I and mode III loadings the situation changed quite significantly. In contrast to pure mode I loadings, the cracks undergo now the tip splitting instability even without the inclusion of dynamic effects due to a finite mode III loading contribution. Furthermore, it turns out that the propagation via **phase** transforma- tions does not lead to a valid description of fracture due to a weak logarithmic increase of the fracture opening, which is revealed by an asymptotic analysis of the chemical poten- tial far behind the tip. However, the **model** of crack propagation by surface diffusion does not suffer from this problem, since in this case the material conservation condition leads to a suppression of this effect.

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Foams are complex and challenging materials. The damage process of the foam materials takes place on multiple scales changing several physical and structural properties of the material. In this study, the topology-based vari- able describing the connectivity state of a cell is introduced to formulate a non-variational **phase**-**field** **model** for the damage evolution in an open-cell foam. The material is considered consisting of the damaged and unimpaired **phase** with the proposed **phase**-**field** variable describing the separation of phases. The performance of the compu- tational **model** is examined by means of the standard benchmarks such as tensile and simple shear test. The results show a qualitative correspondence with the two-dimensional artificial foam **model** used as a reference. Further- more, the influence of the directional data extracted from the microstructure is investigated. The utilisation of the connectivity-based damage variable turns out to be a suitable choice for the simulation of the damage evolution in open-cell foam materials.

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In the last chapter a phenomenological two-dimensional Landau-type **phase** **field** representation of the nonpolar matrix material is suggested, which can be easily combined with existing **phase** **field** models for polar materials. In contrast to the equivalent circuit **model**, it is possible to simulate parts of the microstructure in the **model** to study the influence of the grain boundaries, the grain sizes, the grain shapes, the grain distribution, and the inter- and intra-granular electrical and mechanical interactions on the macroscopic polarization and strain. The **phase** separation energy, i.e., the part of the free energy density that depends on the polarization, is phenomenologically chosen to create the typical double loop polarization hysteresis curve of a nonpolar material. Necessary material parameters were determined from measurements or reports on similar materials. With this **model** it was possible to compare the behavior of a two layer composite, which is a representation of the equivalent circuit, to a simple grain **model** of an individual polar grain surrounded by matrix material. It was found that the boundary condi- tions at the grain boundary between both materials have a significant influence on the domain structure inside the polar seed material and a direct influence on the macroscopic strain and polarization of the composite. However the presented studies are only a proof of concept. Future studies have to address the influence of the character and shape of the grain boundary in more detail. The implementation of a parallelized solver will allow the simulation of more complex microstructures and the expansion of the **model** to three dimensions. This will allow for the study of interactions between different seed grains. An important point that should be addressed in future studies is the accuracy of the **model** parameters. Moreover, recent in situ transmission electron microscopy measurements revealed that the electric fields required for the nonpolar material to transform into its polar **phase** are actually smaller than the coercive **field** of the polar **phase** [271]. At the **phase** transition **field** the material transforms from the nonergodic relaxor state to a poly-domain polar state similar to that of a ferroelectric material. With further increas- ing electric fields these domains begin switching and continue until the material is fully poled. This is not predicted by the presented **phase** **field** **model**. Only transitions from nonpolar to a poled polar state are possible, domain walls can only occur at grain boundaries or highly spatially fluctuating electric fields. More advanced approaches are necessary to consider this experimental observation in the **phase** **field** **model**, for example the introduction of a new order parameter that describes polar or nonpolar **phase** of the material and couples to the electric **field**.

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PLL is a feedback loop which mainly consists of a **phase** detector, a loop filter and an oscillator (See Fig. 1). The **phase** detector compares the input **phase** with the reference **phase** synthesized by the oscillator, and generates an error measure which depends on the **phase** difference. The loop filter integrates these errors and the oscillator updates the reference **phase** accordingly.

S := K F (p) 0 S with K F as the fluid compressibility parameter and p as the fluid pressure [4]. Following the kinematics of multiphase materials, a Lagrangian description of the solid matrix via the solid displacement u S and velocity v S is considered. The pore-fluid flow is expressed either in an Eulerian description using the fluid velocity v F or by modified Eulerian settings via the seepage velocity w F := v F − v S . Within a geometrically-linear framework, the solid small strain tensor is defined as ε S := 1 2 (grad u S + grad T u S ). The onset and propagation of brittle fractures are modeled based on the diffusive interface **phase**-**field** modeling (PFM), which uses a scalar **phase**-**field** variable d S to determine the material state, i.e. d S = 1 for the cracked state and d S = 0 for the intact state. Within brittle fracture mechanics, the total potential energy is expressed as the

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The inversion procedure takes 7 images and generates 15 (= 6 × 5/2) independent closure phases. We adopt the simple **model** of Eq. (4) and consider the mean square er- ror as the figure of merit to be minimized. The minimiz- ing procedure (a steepest descent) is started many times with different random inizializations, since we know that the landscape has many local minima. Each solution cor- responds to a certain order of the acquisitions accord- ing to their inverted moisture level and solutions which are "circularly equivalent" are considered together. We select then the "correct" circular ordering based on the number of solutions that converged to it and the corre- ponding mean square error.

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evidence in favor of an isotropic fluid-to-isotropic fluid **phase** transition. Pshenichnikov and Mekhonoshin [30] utilized the Monte Carlo method to simulate dipolar hard sphere with open boundaries. They applied an extra **field** which confines the particles to a spherical region and observed a gas-like distribution within this region, or a pronounced clustering, depending on the strength of dipolar interaction. They interpret this as an indication for **phase** separation in the dipolar hard sphere bulk system. Ganzenm¨ uller and Camp [31] used a fluid of charged hard dumbbells, each made up of two oppositely charged hard spheres, separated by the distance d, to track the gas-liquid coexistence towards the dipolar hard sphere limit d → 0. Via extrapolation of their grand-canonical Monte Carlo results obtained for finite dumbbell length, they found a gas-liquid critical point in the dipolar hard sphere limit. Almarza et al. [32] confirm the results of Ganzenm¨ uller and Camp [31] using Monte Carlo to analyze a mixture of hard spheres and dipolar hard sphere. The critical parameters for the gas-liquid equilibrium, extrapolated based on their mixture results to the limit of vanishing neutral hard sphere concentration, are in accord with the extrapolation for the dumbbells approaching the dipolar hard sphere limit. Kalyuzhnyi et al. [33] examine the **phase** behaviour of the dipolar Yukawa hard sphere fluid using Monte Carlo simulations. Again the critical point may be tracked as the dipolar hard sphere limit is approached by decreasing the strength of the attractive Yukawa potential. They find a critical point for values of the Yukawa potential well depth which is used as control parameter and representing the ”distance” from the dipolar hard sphere limit, which are far lower than the limit set by the earlier study by Szalai et al. [19]. Continuation of this work in Ref. [34], however, results in the conclusion that **phase** separation is not observable beyond a critical value of the Yukawa energy parameter. Similar to the aforementioned work by Stevens and Grest [18], our colleague Jia [35] tracked the critical point for the dipolar soft sphere **model** as a parameter of the **field** strength and found critical values in the limit of vanishing **field**. The results were also confirmed as part of this work at zero **field** strength. A graphical representation to summarize the results on the localization of the gas-liquid critical point on dipolar systems is shown in Figure 1.1.

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After performing investigations utilizing isothermal sim- ulations for the Fe–Mn system, simulations of directional solidiﬁcation were carried out with the numerical **model** for the same system to examine the growth characteristics of the new solid **phase** (the ␥-**phase**) on the ﬁrst solid **phase** ( ␦- **phase**). This growth of the ␥-**phase** on the ␦-**phase** is known as peritectic reaction, or the ﬁrst stage of peritectic **phase** transformation. According to Fredriksson and Nylén [18] , dur- ing the peritectic reaction, the ␥-**phase** thickness close to the triple point would be constant and equal to twice radius of the ␥ tip. In addition to that, this ␥-**phase** thickness would be inﬂuenced by the equilibrium concentrations of the **phase** diagram and by the interfacial tensions between the phases. Fig. 1 shows a sketch adapted from Ref. [21] to clarify the growth of the ␥-**phase** on the ␦-**phase** during the peritec- tic reaction. These authors [21] suggested also an analytical theory to estimate the ␥-**phase** thickness during this stage. Therefore, numerical simulations were performed varying the ␦–␥ interfacial tension and the results for the ␥-**phase** thick- ness under steady-state growth were utilized for comparison with the analytical theory.

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