Static magnetic and spin wave properties of square lattices of permalloy micron dots with thicknesses of 500 Å and 1000 Å and with varying dot separations have been investigated. A magnetic fourfold anisotropy was found for the lattice with dot diameters of 1 micrometer and a dot separation of 0.1 micrometer. The anisotropy is attributed to an anisotropic dipole-dipole interaction between magnetically unsaturated parts of the dots. The anisotropy strength (order of 100000 erg/cm^3 ) decreases with increasing in-plane applied magnetic field.
Static magnetic and spin wave properties of square lattices of permalloy micron dots with thicknesses of 500 Å and 1000 Å and with varying dot separations have been investigated. The spin wave frequencies can be well described taking into account the demagnetization factor of each single dot. A magnetic four-fold anisotropy was found for the lattice with dot diameters of 1 micrometer and a dot separation of 0.1 micrometer. The anisotropy is attributed to an anisotropic dipole-dipole interaction between magnetically unsaturated parts of the dots. The anisotropy strength (order of 100000 erg/cm^3 ) decreases with increasing in-plane applied magnetic field.
In this paper we present a renormalizability proof for spontaneously broken SU (2) gauge theory. It is based on Flow Equations, i.e. on the Wilson renormalization group adapted to perturbation theory. The power counting part of the proof, which is conceptually and technically simple, follows the same lines as that for any other renormalizable theory. The main difficulty stems from the fact that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the Slavnov-Taylor identities of SU (2) Yang-Mills theory on which the gauge invariance of the renormalized theory is based.
Abstract: We analyse 4-dimensional massive "phi" ^ 4 theory at finite temperature T in the imaginary-time formalism. We present a rigorous proof that this quantum field theory is renormalizable, to all orders of the loop expansion. Our main point is to show that the counterterms can be chosen temperature independent, so that the temperature flow of the relevant parameters as a function of T can be followed. Our result confirms the experience from explicit calculations to the leading orders. The proof is based on flow equations, i.e. on the (perturbative) Wilson renormalization group. In fact we will show that the difference between the theories at T > 0 and at T = 0 contains no relevant terms. Contrary to BPHZ type formalisms our approach permits to lay hand on renormalization conditions and counterterms at the same time, since both appear as boundary terms of the renormalization group flow. This is crucial for the proof.
Abstract: In this paper we present a renormalizability proof for spontaneously broken SU (2) gauge theory. It is based on Flow Equations, i.e. on the Wilson renormalization group adapted to perturbation theory. The power counting part of the proof, which is conceptually and technically simple, follows the same lines as that for any other renormalizable theory. The main difficulty stems from the fact that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the Slavnov-Taylor identities of SU (2) Yang-Mills theory on which the gauge invariance of the renormalized theory is based.
Recently, phase field modeling of fatigue fracture has gained a lot of attention from many researches and studies, since the fatigue damage of structures is a crucial issue in mechanical design. Differing from traditional phase field fracture models, our approach considers not only the elastic strain energy and crack surface energy, additionally, we introduce a fatigue energy contribution into the regularized energy density function caused by cyclic load. Comparing to other type of fracture phenomenon, fatigue damage occurs only after a large number of load cycles. It requires a large computing effort in a computer simulation. Furthermore, the choice of the cycle number increment is usually determined by a compromise between simulation time and accuracy. In this work, we propose an efficient phase field method for cyclic fatigue propagation that only requires moderate computational cost without sacrificing accuracy. We divide the entire fatigue fracture simulation into three stages and apply different cycle number increments in each damage stage. The basic concept of the algorithm is to associate the cycle number increment with the damage increment of each simulation iteration. Numerical examples show that our method can effectively predict the phenomenon of fatigue crack growth and reproduce fracture patterns.
In this work we illustrate the ability of a phase field model for fatigue crack growth in terms of extension capability to various amplitude loading and mean stress effects. The additional energy density contribution accounting for the energy associated with fatigue is modified in order to provide a more general model. Results obtained from numerical fatigue crack growth simulations are briefly presented and discussed.
We report on the resonant excitation of spin waves in micro-structured magnetic thin films by short-wavelength surface acoustic waves (SAWs). The spin waves as well as the acoustic waves are studied by micro-focused Brillouin light scattering spectroscopy. At low magnetic bias fields, a resonant phonon–magnon conversion is possible, which results in the excitation of short-wavelength spin waves. Using micromagnetic simulations, we verify that during this excitation both energy and linear momentum are conserved and fully transferred from the SAW to the spin wave. This conversion can already be detected after an interaction length of a few micrometers. Thus, our findings pave the way for miniaturized magneto-elastic spin-wave emitters for magnon computing.
Phase field modeling of fracture has been in the focus of research for over a decade now. The field has gained attention properly due to its benefiting features for the numerical simulations even for complex crack problems. The framework was so far applied to quasi static and dynamic fracture for brittle as well as for ductile materials with isotropic and also with anisotropic fracture resistance. However, fracture due to cyclic mechanical fatigue, which is a very important phenomenon regarding a safe, durable and also economical design of structures, is considered only recently in terms of phase field modeling. While in first phase field models the material’s fracture toughness becomes degraded to simulate fatigue crack growth, we present an alternative method within this work, where the driving force for the fatigue mechanism increases due to cyclic loading. This new contribution is governed by the evolution of fatigue damage, which can be approximated by a linear law, namely the Miner’s rule, for damage accumulation. The proposed model is able to predict nucleation as well as growth of a fatigue crack. Furthermore, by an assessment of crack growth rates obtained from several numerical simulations by a conventional approach for the description of fatigue crack growth, it is shown that the presented model is able to predict realistic behavior.
Within this work, we utilize the framework of phase field modeling for fracture in order to handle a very crucial issue in terms of designing technical structures, namely the phenomenon of fatigue crack growth. So far, phase field fracture models were applied to a number of problems in the field of fracture mechanics and were proven to yield reliable results even for complex crack problems. For crack growth due to cyclic fatigue, our basic approach considers an additional energy contribution entering the regularized energy density function accounting for crack driving forces associated with fatigue damage. With other words, the crack surface energy is not solely in competition with the time-dependent elastic strain energy but also with a contribution consisting of accumulated energies, which enables crack extension even for small maximum loads. The load time function applied to a certain structure has an essential effect on its fatigue life. Besides the pure magnitude of a certain load cycle, it is highly decisive at which point of the fatigue life a certain load cycle is applied. Furthermore, the level of the mean load has a significant effect. We show that the model developed within this study is able to predict realistic fatigue crack growth behavior in terms of accurate growth rates and also to account for mean stress effects and different stress ratios. These are important properties that must be treated accurately in order to yield an accurate model for arbitrary load sequences, where various amplitude loading occurs.