arXiv subjects

Vivekanand Dabade

Publications and source records attributed to Vivekanand Dabade.

8 recordsLinked to original sources

arXiv2026

Spatial deformation of a ferromagnetic elastic rod

Ferromagnetic elastic slender structures offer the potential for large actuation displacements under modest external magnetic fields, due to the magneto-mechanical coupling. This paper investigates the phase portraits of the Hamiltonian governing the three-dimensional deformation of inextensible ferromagnetic elastic rods subjected to combined terminal tension and twisting moment in the presence of a longitudinal magnetic field. The total energy functional is formulated by combining the Kirchhoff elastic strain energy with micromagnetic energy contributions appropriate to soft and hard ferromagnetic materials: magnetostatic (demagnetization) energy for the former, and exchange and Zeeman energies for the latter. Exploiting the circular cross-sectional symmetry and the integrable structure of the governing equations, conserved Casimir invariants are identified and the Hamiltonian is reduced to a single-degree-of-freedom system in the Euler polar angle. Analysis of the resulting phase portraits reveals that purely elastic and hard ferromagnetic rods undergo a supercritical Hamiltonian Hopf pitchfork bifurcation, whereas soft ferromagnetic rods exhibit this bifurcation only within a restricted range of the magnetoelastic parameter, $0<\tilde{K}_{dM}<1/8$. Both helical and localized post-buckling configurations are analyzed, and the corresponding load-deformation relationships are systematically characterized across a range of loading scenarios. Localized buckling modes, corresponding to homoclinic orbits in the Hamiltonian phase space, are constructed numerically. In contrast to the purely elastic case, the localized configurations of soft ferromagnetic rods exhibit non-collinear extended straight segments, a geometrically distinctive feature arising directly from the magnetoelastic coupling.

cond-mat.mtrl-sci↗

arXiv2025

Revisiting the cofactor conditions: Elimination of transition layers in compound domains

This paper investigates the conditions necessary for the elimination of transition layers at interfaces involving compound domains, extending the classical framework of cofactor conditions. Although cofactor conditions enable stress-free phase boundaries between Type I/II domains and austenite, their applicability to compound domains has remained limited. Here, we present a comprehensive theoretical framework to characterize all compatible interfaces, highlighting the fundamental importance of the commutation property among martensitic variants. By establishing necessary and sufficient algebraic conditions, referred to as extreme compatibility conditions, we demonstrate the simultaneous elimination of transition layers at phase interfaces for both Type I/II and compound laminates, across all volume fractions of the martensitic variants. We also investigate the possibility of achieving supercompatibility in non-conventional twins, recently observed in the NiMnGa system. The focus of our work is on cubic-to-orthorhombic and cubic-to-monoclinic~II phase transformations, for which the extreme compatibility conditions are explicitly derived and systematically analyzed. The theory predicts novel zero-elastic-energy microstructures, including an increased number of triple clusters, spearhead-shaped martensitic nuclei, stress-free inclusions of austenite within martensite, and distinctive four-fold martensitic clusters. This significantly expands the possible modes of forming stress-free interfaces between phases and reveals new energy-minimizing microstructures that can facilitate the nucleation of martensite within austenite and vice versa. These configurations highlight significant enhancements in transformation reversibility and material durability, guiding the rational design of next-generation shape memory alloys with optimized functional properties.

cond-mat.mtrl-sci↗

arXiv2024

Remote Nucleation and Stationary Domain Walls via Transition Waves in Tristable Magnetoelastic Lattices

We present a magnetoelastic lattice in which a localized external magnetic field, generated by an assembly of fixed magnets, tunes the potential landscape to create monostable, bistable, and tristable configurations. Focusing on the tristable potential, we numerically and experimentally confirm the existence of three distinct types of transition waves, each characterized by unique amplitudes and velocities, and establish a scaling law that governs their behavior. We also examine how these transition waves interact with the system's finite boundaries. Furthermore, by adjusting the potential symmetry through the localized external field, we investigate wave collision dynamics. In lattices with asymmetric potentials, the collision of similar transition waves leads to the remote nucleation of a third phase. In symmetric potentials, the collision of dissimilar transition waves results in the formation of a stationary domain wall, with its width tuned by the shape of the tristable potential well.

nlin.PS↗

arXiv2024

Kirchhoff's analogy for a planar ferromagnetic rod

Kirchhoff's kinetic analogy relates the equilibrium solutions of an elastic rod or strip to the motion of a spinning top. In this analogy, time is replaced by the arc length parameter in the phase portrait to determine the equilibrium configurations of the rod. Predicted equilibrium solutions from the phase portrait for specific boundary value problems, as well as certain localized solutions, have been experimentally observed. In this study, we employ the kinetic analogy to investigate the equilibrium solutions of planar soft ferromagnetic rods subjected to transverse and longitudinal external magnetic fields. Our analysis reveals a subcritical pitchfork bifurcation in the phase portrait of a ferromagnetic rod subjected to transverse external magnetic field as the axial load is decreased continuously from a large compressive load. Similarly, a supercritical pitchfork bifurcation is observed in the case of longitudinal external magnetic field. We predict equilibrium configurations for a free-standing soft ferromagnetic elastic rod and the same subjected to canonical boundary conditions. Furthermore, we observe novel localized equilibrium solutions arising from homoclinic and heteroclinic orbits, which are absent in the phase portraits of purely elastic rods.

cond-mat.mtrl-sci↗

arXiv2024

Deformation of a planar ferromagnetic elastic ribbon

While extensive studies have been conducted on purely elastic ribbons, in this paper we explore the influence of magnetisation on the deformation of planar ferromagnetic elastic ribbons. We begin the investigation by deriving the leading-order magnetic energy associated with a curved planar ferromagnetic elastic ribbon. The sum of the magnetic and the elastic energy is the total energy of the ribbon. We derive the equilibrium equations by taking the first variation of the total energy. We then systematically determine and analyse solutions to these equilibrium equations under various canonical boundary conditions. We also analyse the stability of the equilibrium solutions. Comparing our findings with the well-studied Euler's Elastica provides insights into the magnetic effects on the deformation behaviour of elastic ribbons. Our analysis contributes to a deeper understanding of the interplay between magnetisation and the mechanical response of planar ferromagnetic structures, and offers valuable insights for both theoretical and practical applications.

cond-mat.mtrl-sci↗

arXiv2020

Bounds on the energy of a soft cubic ferromagnet with large magnetostriction

We complete the analysis initiated in [5] on the micromagnetics of cubic ferromagnets in which the role of magnetostriction is significant. We prove ansatz-free lower bounds for the scaling of the total micromagnetic energy including magnetostriction contribution, for a two-dimensional sample. This corresponds to the micromagnetic energy-per-unit-length of an infinitely thick sample. A consequence of our analysis is an explanation of the multi-scale zig-zag Landau state patterns recently reported in single crystal Galfenol disks from an energetic viewpoint. Our proofs combines a number of well-developed techniques in energy-driven pattern formation with a recent regularity result for the kinetic formulation of the Eikonal equation of Ghiraldin and Lamy.

math.AP↗

arXiv2019

Branching of twins in shape memory alloys revisited

We study the branching of twins appearing in shape memory alloys at the interface between austenite and martensite. In the framework of three-dimensional non-linear elasticity theory, we propose an explicit, low-energy construction of the branched microstructure, generally applicable to any shape memory material without restrictions on the symmetry class of martensite or on the geometric parameters of the interface. We show that the suggested construction follows the expected energy scaling law, i.e., that (for the surface energy of the twins being sufficiently small) the branching leads to energy reduction. Furthermore, the construction can be modified to capture different features of experimentally observed microstructures without violating this scaling law. By using a numerical procedure, we demonstrate that the proposed construction is able to predict realistically the twin width and the number of branching generations in a Cu-Al-Ni single crystal.

cond-mat.mtrl-sci↗

arXiv2013

Study of the cofactor conditions: conditions of supercompatibility between phases

The cofactor conditions, introduced in James and Zhang, are conditions of compatibility between phases in martensitic materials. They consist of three subconditions: i) the condition that the middle principal stretch of the transformation stretch tensor $\mathbf U$ is unity ($λ_2 = 1$), ii) the condition $\mathbf a \cdot \mathbf U\, \cof (\mathbf U^2 - \mathbf I)\mathbf n = 0$, where the vectors $\mathbf a$ and $\mathbf n$ are certain vectors arising in the specification of the twin system, and iii) the inequality ${\rm tr} \mathbf U^2 + \det \mathbf U^2 -(1/4) |\mathbf a|^2 |\mathbf n|^2\ge 2$. Together, these conditions are necessary and sufficient for the equations of the crystallographic theory of martensite to be satisfied for the given twin system but for any volume fraction f of the twins, $0 \le f \le 1$. This contrasts sharply with the generic solutions of the crystallographic theory which have at most two such volume fractions for a given twin system of the form $f^*$ and $1-f^*$. In this paper we simplify the form of the cofactor conditions, we give their specific forms for various symmetries and twin types, we clarify the extent to which the satisfaction of the cofactor conditions for one twin system implies its satisfaction for other twin systems. In particular, we prove that the satisfaction of the cofactor conditions for either Type I or Type II twins implies that there are solutions of the crystallographic theory using these twins that have no elastic transition layer. We show that the latter further implies macroscopically curved, transition-layer-free austenite/martensite interfaces for Type I twins, and planar transition-layer-free interfaces for Type II twins which nevertheless permit significant flexibility of the martensite. We identify some real material systems nearly satisfying the cofactor conditions.

math-ph↗