The thesis presents an implementation including different applications of a variational-based approach for gradient type standard dissipative solids. Phasefield model for brittle fracture is an application of the variational-based framework for gradient type solids. This model allows the prediction of dif- ferent crack topologies and states. Of significant concern is the application of theoretical and numerical formulation of the phasefieldmodeling into the commercial finite element software Abaqus in 2D and 3D. The fully coupled incremental variational formulation of phasefield method is implemented by using the UEL and UMAT subroutines of Abaqus. The phasefield method considerably reduces the implementation complexity of fracture problems as it removes the need for numerical tracking of discontinuities in the displace- ment field that are characteristic of discrete crack methods. This is accom- plished by replacing the sharp discontinuities with a scalar damage phasefield representing the diffuse crack topology wherein the amount of diffusion is controlled by a regularization parameter. The nonlinear coupled system consisting of the linear momentum equation and a diffusion type equation governing the phasefield evolution is solved simultaneously via a Newton- Raphson approach. Post-processing of simulation results to be used as vi- sualization module is performed via an additional UMAT subroutine imple- mented in the standard Abaqus viewer.
et al. . These models may be considered as time dependent viscous regularizations of the above mentioned rate independent theories of energy minimization. In this context, the reader is referred to the work of Kuhn & M¨ uller  on particular aspects of the numerical implementation by means of the introduction of exponential shape functions. However, these approaches still have a few drawbacks, which limit their application to particular model problems. The model outlined in Hakim & Karma  is based on a Ginzburg-Landau-type evolution equation of the fracture phasefield, which does not differentiate between energy storage and dissipation. These are very strong simplifications of the physical mechanisms of brittle fracture. The formulation of rate independent diffu- sive fracture proposed by Bourdin et al.  models the irreversibility of the process only on a time-discrete level by setting hard Dirichlet-type conditions on the phasefield. Furthermore, both existing phasefield models consider energy release driven fracture in both tension and compression. Clearly, these fully symmetric formulations are unrealistic for most materials and restricted to the modeling of boundary value problems with tensile stresses in the full solid domain. These approaches are, in general, not thermodynami- cally consistent and applicable only in particular situations with monotonous loading of arbitrary sub-domains of a fracturing solid. Observe, that most existing phasefield ap- proaches to fracture are related to brittle crack propagation in elastic solids. The phasefield model, on the other hand, has an enormous potential with respect to the prediction of complex crack phenomena. Energy, stress or strain based criteria for brittle or ductile failure can be included in a modular form. Furthermore, crack growth in multi-physics problems, such as thermo-, electro- or chemo-mechanical problems can be directly mod- eled with a fracture phasefield. The key advantage is that the constitutive formulation of the crack propagation in regularized phasefield models is related to the three-dimensional bulk response. The recent works of Miehe et al. [161, 160], Miehe & Sch¨ anzel , Borden et al. [27, 28], Pham et al.  and Verhoosel & de Borst  out- line a general thermodynamically consistent framework for the phasefieldmodeling of crack propagation at small strains and serve as a basis for the development of phasefield fracture approach at large strains.
The phasefieldmodeling of ferroelectrics with static defect dipole is developed in Chapter 6 and computation is conducted in order to study the dipole effect of oxygen vacancies on the different domain structures. In the model, the oxygen vacancies are treated as electric dipoles and the ensuing interior electric field is taken into account as well. The numerical simulations demonstrate that the internal bias electric field caused by the oxygen vacancies does play a notable role in modifying the overall domain switching behavior of ferroelectric single crystals. The domain memory effect, in which the zero overall polarization and the initial domain pat- terns are recovered after electric unloading and thus the double hysteresis loops take place, are well reproduced in the modeling. Calculation shows that the recoverable strain in the rank-2 domain structure is more significant than that in the rank-1 domastructure, and thus one is led to the conclusion that the former is the main factor for the effect. The phenomenological frame has limitations because it is unable to capture all the mechanisms in the structural inhomo- geneities, thus neglecting some details of the interactions between defects and domain walls in ferroelectrics.
The double-well function represents an approximation of the Van der Waals [168, 169] near the critical point, and has been employed extensively in the phase-field models. How- ever, when the model is developed solely for interface tracking purposes, this has led to the frequently observed spontaneous volume shrinkage phenomenon. Whereby, the high vol- ume phase allows significant infiltration into the low volume phases and can ultimately cause the complete disappearance of the lower volume phases . In addition, the movement of the lower volume phase inside the higher one may further enhance this effect. Therefore, to minimize these losses over the duration of a simulation, it requires a reconsid- eration of the free energy function. An alternative energy function for interface tracking ap- plications in the form of the double-obstacle can be considered. The diffuse interface of the order parameter follows cosine function in the double-obstacle, while hyperbolic-tangent is expected in the former case, see Appendix A. This provides an extremely controlled diffuse interface width and less dispersion of inclusion volume. Consequently, lower change in in- clusion volume leads to a stoppage of the spontaneous volume shrinkage phenomena. In addition, this function is adequate to track larger movements of the lower volume phase. Therefore, the obstacle-type function may prove useful for interface tracking applications of the phase-field model where the nature of the simulated phenomena introduces phase continuity concerns like the inclusion migration in the metal conductors.
model to describe the rearrangement of a martensitic microstructure due to the application of an external magnetic or elastic stress ﬁeld. The work in this ﬁeld resulted in two major publications [4, 5] so far. The latter is benevolently cited in the literature (see e.g. [6, 7]). A third publication that discusses more recent results (which are also presented in this work) has been accepted for publication in the European Physical Journal B and will be published in the ﬁrst quarter of 2013 in a special issue called New trends in magnetism and magnetic materials . The author has been invited to submit this article by the organization committee of the Joint European Magnetic Symposia (JEMS) 2012 in Parma (Italy), where he had presented the phase-ﬁeld modeling approach for magnetic shape mem- ory alloys. The author tried to combine two tasks that are very diﬀerent at ﬁrst glance, but have in common the modeling approach on which they are based: The phase-ﬁeld modeling of the competitive growth of grains on thin ﬁlms and of eﬀects related to the microstructure rearrangement in magnetic shape memory materials.
The model developed in the context of this work has proven well in diﬀerent scenarios, and opens up for further applications. Special attention in this work was paid to the development of a model description that couples a phase-ﬁeld approach with micromagnetics and mechan- ical elasticity. The main focus was drawn on developing computation methods to make the micromagnetic problems feasible, on ﬁnding sound magnetic and elastic boundary conditions and on solving the elastic equations (the dynamic wave equation Eq. (7.4) and Eq. (7.6) for the mechanical equilibrium) in a general context, allowing, in principle, for arbitrarily oriented phases with diﬀering elastic properties. Nevertheless, when the elastic dynamic wave equation is solved that is implemented at the moment, it is hard to relate the damping mechanism to energy dissipation properties and match it with physical conditions (cp. the parameter κ in Eq. (7.2)). Finel et al. include the kinetic energy density and a Rayleigh dissipation density into their modeling approaches (see ). Applying theses ideas might improve the phase- ﬁeld simulation results that are achievable with the model presented in this work. Further, the restriction to linear elasticity when modeling the MSME, although often used, is consid- ered a severe limitation sometimes in the literature (cp. e.g. ): The disregard of large deformations is indicated as a source of non-physical behavior, because the giant strains at- tributed to pseudoplastic behavior are a characteristic of MSMAs. It has to be investigated if, in the context of ferromagnetic shape memory alloys, the geometric linearization of elasticity is justiﬁed, or if a model formulation based on non-linear elasticity leads to more appropriate results. The following sections brieﬂy sketch some possibilities of future applications for the developed models.
Lithium-ion batteries, with their high energy densities and light-weight designs, have found broad applications in portable electronics and electric vehicles. However, their mechanisms and operation are not yet fully understood, which has motivated a wide span of multi-physical models from different disciplines. In this thesis, a thermodynamically consistent phase-field framework is presented, to investigate the electro-chemo-mechanical behavior of lithium-ion battery electrode materials. Within this framework, a series of coupled models is developed sequentially towards the more realistic modeling. Firstly, a mechanically coupled two-phase model of a single particle is proposed, based on a thorough study of the chemical phase separa- tion of this particle. Thereby, the effect of large strains and the concentration-dependent elastic properties are considered, which has been proved in this thesis to have a great impact on the phase separation. A more comprehensive model is formulated, which deals additionally with the electrochemical reaction on the particle surface and the orthotropic phase separation. The reaction rate is governed by a modified Butler–Volmer equation, which takes both chemical and mechanical states into account. Based on this model, we further investigate the fracture in the particle by the phase-field approach, where the reaction on the newly cracked surfaces is also taken into consideration. Finally, the model of the particle embedded in a polymer matrix is presented to study the interaction between the particle and the surrounding materials.
a type of local or generalized gradient approximation in the context of classical density functional theory (e.g., Evans, 1979; Davis, 1996). For the current dislocation case, the homogeneous part of this energy is related to homogeneous atomic displacement across the slip plane during dislocation slip and so to the (generalized) SFE (Vitek, 1968); the ”gradient” part is due to deviations from homogeneity such as occurs in the dislocation core (something also predicted by DFT-based core modeling: Iyer et al., 2015). As shown in this work, interface energy modeling facilitates global energy scaling, which in turn can be exploited to identify the PF model completely using atomistic data. In particular, the resulting gradient energy is material specific and related to the physical core. To demonstrate the application of the current approach, PF models were identified for fcc Al and Cu with the help of atomistic data and applied to the case of dynamic tensile loading of a dislo- cation loop in these materials. In the process, the thermodynamic PF picture in terms of energetic favorability and thermodynamic driving forces was compared to, and shown to be consistent with, the more standard mechanical picture in terms of the Peach-K¨ohler force. The example cases of Al and Cu demonstrate the ease of calculating atomistic input data for the current PF model and its predicative capability in the simulation of dislocation processes which are difficult to investigate via MD. The use of an atomic potential and MS in this work to determine this input data was for simplicity only; indeed, if available, more exact DFT-based input data can (and should) be used.
variable ς chosen in what follows in order to obtain the simplest possible formulation. Lastly, ϕ = (ϕ 1 , . . . , ϕ p ) represents an array of p scalar phase fields of both conservative and non-conservative type. These will be specified in more detail later. On this basis,
x ς := (χ, m, ς, ϕ) ≡ (χ 1 , χ 2 , χ 3 , m 1 , m 2 , m 3 , ς, ϕ 1 , . . . , ϕ p ) (10) represents the complete set of GENERIC-based variables for the current constitutive class. All these fields are referential or Lagrangian with respect to the mixture. In this case, mixture mass conservation implies ˙ρ = 0 and v = ˙χ holds. In the context of (10), then, E[ x ς ] and S[ x ς ] take the forms
A number of further model developments are currently under investigation. One possibility in this regard is the application of the more general scaling method of Reina et al. (2014) in the current context. Rather than on thermodynamic equilibrium, their method is based on steady state, applies to both energetics and kinetics (i.e., mobility), and can be used to identify PF models using atomistic data from molecular dynamics (MD) simulations. As has been alluded to above, an- other further development involves the use of ab initio DFT-based methods (e.g., Woodward et al., 2008), instead of an interatomic potential and MS or MD. Going even further, the generalization of such methods to larger systems with the help of coarse-graining methods (e.g., quasi-continuum: Iyer et al., 2015) may offer the possibility of true ”concurrent” PF-DFT modeling of dislocation processes in the future.
based on a core-shell microstructure model. It revealed a strong gradient in the electric potential at the interfacial area, which causes the nucleation and propagation of an in-plane polarization The phase-field simulation demonstrated that the observed evolution of field-induced lateral domains via PFM is due to a compensation of polarization charges at the core-shell interface be- neath the sample surface. Transferring these local observations within an individual grain to the macroscopic scale offers new insights into the performance and functionality of actuators: actu- ator applications require materials with large, recoverable, electric field-induced strain outputs executed at low fields. Macroscopically, an actuator based on a relaxor runs from the remanent state (generally negligible) to a state where the entire material is held at the ferroelectric state by a sufficiently strong external electric field and returns back to the remanent state. Interpret- ing this cycle microscopically implies that the relaxed ergodic state of each shell is switched to a ferroelectric state and back. The core, in contrast, permanently remains in the ferroelectric state. Upon switching on the external field, the alignment of the “pivot points” at the inter- face from a polarization direction perpendicular to the applied field to a parallel configuration facilitates the domain orientation in the shell. These pivot points originate from the residual high potential at the interface and generate domain nuclei at the adjacent shell. These nuclei aggregate and the surrounding PNRs align with the external electric field. Consequently, the energy barrier for the completion of the phase transition in the shell is reduced. Conversely, upon switching off the external field, the pivot points reduce the mechanical stress at the in- terface that was established during the relaxation process between the polarized core and the arbitrarily distributed shell polarization. The formation of the lateral polarization thus plays the central role in the compensation of the polarization and strain mismatches between the core and the shell during the poling and relaxation processes.
A mixture between the behavior of the two scalar models is exhibited by the LCT model (Figs. 4.8). Convergence is also very slow, even though it always keeps a generally nicer profile. While the area around the contour line φ = 1 2 assumes the correct profile form, the phase-field approaches the wrong bulk values if the boundary values of φ are not fixed to be equal to zero or one. This can be seen by looking at Eq. (3.54): any constant value of φ satisfies the bulk equations of motion. This incorrect behavior is cured if elastic degrees of freedom are added to the chemical potential. For all geometrical situations where a phase extends to the boundaries of the system, the value of the respective constant is determined by the boundary conditions. Therefore, the model should always be run with Dirichlet boundary conditions for the phasefield. For inclusions of one phase in the other, the constant of the inner phase will be preserved by the conservation law if it is correctly initialized to zero or one. This is true for all inclusions with an inner volume much larger than that of the interface and remains true also when elasticity is added to the problem. The simulation from which Figs. 4.8 was obtained used Dirichlet boundary conditions on the right and left end, and we found relaxation to be comparably slow to the SM and RRV models but the interface profile to look more reasonable.
the capillary ﬂow on the thickness of the intermetallic phase is due to the following two factors: (i) The ﬁrst factor is the difference in the time scales of diffusion and convection. As shown in Fig. 7 (a), the convergence of the contact angle due to convection takes less time than for pure diffusion. In both cases, with and without capillary ﬂow, the growth of the intermetallic phase is dominated by diffusion. So, when the capillary ﬂow is considered, the inter- metallic layer has less time for the diffusion-dominated growth and the corresponding evolution seems to be inhibited. (ii) The second factor is the convection of the capillary ﬂow. The time evolution of the intermetallic phase with capillary ﬂow (C ¼ 1000) for the phase ﬁeld is illustrated in Fig. 8 (a). The corresponding temporal evolu- tion of the concentration of Au in Fig. 8 (b) shows that the Au species is convected from the bulk of the droplet to the triple point to enhance the kinetics near the region of the triple junction. Hence, the Au-component is accumulated near the triple junction and is reduced at the L-I interface. Regarding the fact that the predomi- nant diffusant for the growth of the intermetallic phase is Au [ 9 ],
S := K F (p) 0 S with K F as the fluid compressibility parameter and p as the fluid pressure . 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-fieldmodeling (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
The continuum mechanical background of phase transformations in solids is given by Fischer et al. (1994). Con- ventional models of phase transformations assume an infinitely sharp interface between the phases. Using this approach, energetic considerations yield the driving forces on the interfaces (Cherkaoui and Berveiller (2000)). However, tracking the interfaces can become difficult, so the numerical realization of this approach is cumber- some. The phasefield concept can be applied as an alternative. In mathematical terms it is a regularization of a sharp interface approach: By the introduction of a scalar valued order parameter, which indicates the present phase, the discontinuities are regularized so that the transition zone is diffuse. Based on Chen et al. (1992) and Wang and Khachaturyan (1997) many phasefield models on martensitic transformation have been developed, for example Artemev et al. (2000), Jin et al. (2001), which are based on the fast Fourier transformation (FFT) for- malism. Kundin et al. (2011) propose an FFT-based approach, too, in which additionally dislocation kinetics are considered. Yamanaka et al. (2008) use a finite differences scheme to solve the field equations for an elastoplastic phasefield model. Also Bartel et al. (2011) focus on the interaction of plasticity with martensitic phase transfor- mations, which is based on the concept of energy relaxation. For considering complicated boundary conditions or complex material laws, the finite element method is more effective. In the context of phasefieldmodeling of the martensitic transformation it is for example applied by Levitas et al. (2009), Hildebrand and Miehe (2011) or by Schmitt et al. (2013). Hildebrand and Miehe model two-variant martensitic laminates at large strains. The two martensitic variants are both stable states of the system. The thermomechanical model proposed by Levitas et al. (2009) considers stable and metastable phases of the martensitic transformation. However, the same elastic compliance tensor is used for both phases so that the different elastic properties of the phases cannot be taken into account.
In CeAuSb 2 , the second critical end point is at a temperature of 3.7 K. This is more than half the maximum T N , so quantum critical scaling to T ∼ 0 K is not expected. However, it would be a very compelling experiment to track the end points with pressure, and to attempt to drive them to 0 K. (This may also be achievable with very high in-plane field: the data in Fig. 8 show some reduction of the c-axis transition fields with 30-T-scale in-plane fields.) The antiferromagnetic order of both CeRh 2 Si 2 and CeNiGe 3 can be suppressed with pressure, with superconductivity appearing in a window of pressure around the antiferromagnetic QCP [ 36 , 37 ]. It would be interesting to determine whether metamagnetic quantum criticality is also involved in this superconductivity. Regarding CeAuSb 2 , a published pressure study at H = 0 showed that pressure initially increases the temperature of the first resistivity shoulder and decreases T N [ 26 ].
For the treatments I, II and IV signal periodicity is as- sumed and thus the ideal phase shift filter via DFT (Sec. 2.2) was applied. To create treatments V and III, no signal periodicity was assumed for the whole musical piece as well as for the generated pink noise raw material of 6 minutes duration. Thus FIR filtering according to Sec. 2.1 was realized. Considering the audio contents as rectangularly windowed signals of infinite duration, the filter order of 3 963 530 (≈ 90 s!) ensures that linear convolution of the chosen excerpt of Hotel California is complete. The resulting magnitude ripple of the Blackman windowed FIR is negligible for the relevant reproduction bandwidth. Since the pink noise length can be arbitrarily set, the same FIR filter was utilized for consistence.