Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

We prove that microstructures in shape-memory alloys have a self-similar refinement pattern close to austenite-martensite interfaces, working within the scalar Kohn-Müller model. The latter is based on nonlinear elasticity and includes a singular perturbation representing the energy of the interfaces between martensitic variants. Our results include the case of low-hysteresis materials in which one variant has a small volume fraction. Precisely, we prove asymptotic self-similarity in the sense of strong convergence of blow-ups around points at the austenite-martensite interface. Key ingredients in the proof are pointwise estimates and local energy bounds. This generalizes previous results by one of us to various boundary conditions, arbitrary rectangular domains, and arbitrary volume fractions of the martensitic variants, including the regime in which the energy scales as $\varepsilon ^{2/3}$ ε 2 / 3 as well as the one where the energy scales as $\varepsilon ^{1/2}$ ε 1 / 2 .

Counting quantum states and field modes

“Counting quantum states and field modes” deals with wave modes identified as terms in a Fourier expansion made within a large arbitrary volume. Travelling-wave modes are preferred as they are eigenstates of momentum; counting modes is also made straightforward, whence the density of states. This is in contrast to a fashion that works instead with standing waves.

Analysis of the Double-Bounce Interaction between a Random Volume and an Underlying Ground, Using a Controlled High-Resolution PolTomoSAR Experiment

The radar response of vegetated environments, and forested areas in particular, are usually modeled using a very simple structure made of a random volume, representing a cloud of vegetation particles, lying over a semi-infinite medium with a rough interface, associated with the underlying ground. This Random Volume over Ground model can efficiently handle double-bounce scattering mechanisms, or arbitrary volume reflectivity profiles. This paper proposes to analyze a specific component of the Random Volume over Ground simplified scattering model, which concerns the double-bounce interaction between the ground and the volume. This specific contribution is not considered by classical characterization techniques and is studied in this work using a controlled experiment involving a Synthetic Aperture Radar operated in a Polarimetric and Tomographic configuration in order to image in 3D a controlled miniaturized scene composed of volume lying over a ground. It is shown that ground/volume double-bounce scattering, which remains focused at the ground level even in 3D imaging mode, and has polarimetric patterns that differ largely from those usually expected from double-bounce reflections, with volume-like features, such as a strong cross-polarized reflectivity or decorrelation between co-polarized channels. Moreover, it is shown that the full rank polarimetric patterns of the ground-volume mechanism are tightly linked to the reflectivity of the volume and may mask the ground response. As a consequence, isolating the ground response using 3D imaging does not permit to avoid a generally very strong distortion of the soil response by the double-bounce reflection, and the estimation of different geophysical parameters of the ground, such as its humidity or roughness are significantly altered.

A two-stage homogenization for modelling of elastic-plastic functionally graded composites

Purpose The purpose of this study is to develop a homogenization approach that ensures both high accuracy and time-efficient solution for elastic-plastic functionally graded composites. Design/methodology/approach The paper presents a novel two-stage hybrid homogenization approach that combines advantages of the mean field homogenization and homogenization based on the finite element method (FEM). The groundbreaking nature of the developed approach is associated with division of the hybrid homogenization procedure into two stages, which allows to very efficiently determine the solution for arbitrary volume fraction of the reinforcement. This paper concerns also on modelling of composites with randomly distributed prolate and oblate particles. For this purpose, the hybrid homogenization was implemented in the framework of the discrete orientation averaging procedure involving pseudo-grain discretization method. Findings Agreement between the results obtained using the proposed approach and the standard FEM-based homogenization is very good (up to the volume fraction of 0.3). Originality/value The proposed two-stage homogenization approach allows to obtain the solution for materials with arbitrary volume fraction of the reinforcement very efficiently; therefore, it is highly beneficial for the two-scale modeling of nonlinear functionally graded materials and structures.

The Fourth State of Matter

It is shown that quantum entanglement is the only force able to maintain the fourth state of matter, possessing fixed shape at an arbitrary volume. Accordingly, a new relativistic Schrödinger equation is derived and transformed further to the relativistic Bohmian mechanics via the Madelung transformation. Three dissipative models are proposed as extensions of the quantum relativistic Hamilton-Jacobi equation. The corresponding dispersion relations are obtained.

A dislocation-based solution for stress introduced by arbitrary volume expansion in cylinders

A cylindrical structure undergoing volume expansion and contraction is common in engineering practice. For example, the charging (discharging) process of axisymmetrical batteries will give rise to volume expansion (shrinkage). The nitriding process of axles for better fatigue performance also introduces volume expansion. Here, by taking the equivalence of volume expansion (or shrinkage) as continuous insertion (or distraction) of infinitesimal dislocations, we supply a framework to solve the stress field of a cylinder with arbitrary insertion (distraction) profile of materials along the radial direction. Under the assumptions that the volume expansion profile along the axis of a cylinder is uniform and the deformation is small so that the current configuration is regarded as the original, we supply analytical solutions of stress fields to several typical volume expansion or shrinkage profiles. Our analysis shows that different volume variation gives rise to either high tensile stress in the surface or hydrostatic tension in the core, and supplies distinct failure mechanisms in cylindrical batteries.

Helicity is the only integral invariant of volume-preserving transformations

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional ℐ defined on exact divergence-free vector fields of class C1 on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that ℐ is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.