Minimization Of Elastic Energy Research Articles

Viscoelasticity estimation of sports prosthesis by energy-minimizing inverse kinematics and its validation by forward dynamics

In this study, we present a method for estimating the viscoelasticity of a leaf-spring sports prosthesis using advanced energy minimizing inverse kinematics based on the Piece-wise Constant Strain (PCS) model to reconstruct the three-dimensional dynamic behavior. Dynamic motion analysis of the athlete and prosthesis is important to clarify the effect of prosthesis characteristics on foot function. However, three-dimensional deformation calculations of the prosthesis and viscoelasticity have rarely been investigated. In this letter, we apply the PCS model to a prosthesis deformation, which can calculate flexible deformation with low computational cost and handle kinematics and dynamics. In addition, we propose an inverse kinematics calculation method that is consistent with the material properties of the prosthesis by considering the minimization of elastic energy. Furthermore, we propose a method to estimate the viscoelasticity by solving a quadratic programming based on the measured motion capture data. The calculated strains are more reasonable than the results obtained by conventional inverse kinematics calculation. From the result of the viscoelasticity estimation, we simulate the prosthetic motion by forward dynamics calculation and confirm that this result corresponds to the measured motion. These results indicate that our approach adequately models the dynamic phenomena, including the viscoelasticity of the prosthesis.

On the evolution of β1/β′ coupled-structures in Mg–Y–Nd alloys: A simulation study

In magnesium alloys based on the Mg–Y–Nd system, the key strengthening precipitate phases, β1 and β′, often form coupled-structures, with β′ precipitates invariably attached to both ends of the β1. A question here is how the attached β′ precipitates evolve as the β1 lengthens within the solid magnesium matrix. To investigate this, this study uses a multi-scale model based on a three-dimensional phase field approach to simulate the evolution of β1 and β′ precipitates in a β1/β′ coupled-structure. The simulation results reveal that, during the evolution process, the β1 precipitate lengthens, while the attached β′ precipitates grow only at the side located along the direction of β1 extension and dissolves at the opposite side. This results in a visual effect where the β′ precipitates appear to “move” within the solid matrix. This phenomenon is governed by the interplay between chemical free energy and elastic strain energy. Chemical free energy promotes uniform growth of the β′ precipitates. However, the elastic strain energy, generated by the β′ precipitates and the stress field concentrated at the end of the β1 plate to which they are attached, favours growth of only one side of the β′ precipitates while causing dissolution at the other side. When the evolution of a coupled-structure is influenced by neighbouring coupled-structures, the evolution of β′ precipitates is strongly driven by the minimization of elastic strain energy. In general, the β′ precipitates tend to grow under tensile stress fields and dissolve under compressive stress fields.

Cooperative dislocations for pressure-dependent sequential deformation of multi-principal element alloys under shock loading

Multi-principal element alloys (MPEAs) are promising materials for structural applications under extreme conditions. Their outstanding mechanical properties are closely related to the activation of multiple deformation modes of dislocation gliding, twinning, and phase transformation that appear in sequence during deformation at low temperatures, high pressures, or high strain rates. However, the inherent correlations among these deformation modes and, thus, underlying deformation mechanisms of MPEAs remain largely unknown. We report soft-recovery plate impact experiments of face-centered-cubic (FCC) CrCoNi MPEAs, demonstrating pressure-dependent deformation modes from low-pressure stacking faults to medium-pressure twinning and high-pressure FCC to hexagonal-close-packed (HCP) phase transformation. Atomic-scale characterizations unveil that the sequential deformation is manipulated by the cooperation of 90° and 30° Shockley partial dislocations at deformation fronts, which is facilitated by low stacking fault energy and pressure-dependent phase stability of the MPEAs. Moreover, the cooperative dislocation behavior can also be observed at twin fronts of shock-loaded CrMnFeCoNi MPEA, validating the universality of the cooperative deformation mode in FCC alloys with a low stacking fault energy. Theoretical analyses suggest that the distinctive cooperative dislocation behavior results in the self-compensation of dislocation strain fields and the minimization of interfacial elastic energy at incoherent twin and FCC/HCP interfaces.

Defect pairs in nematic cell alignment on doubly connected domains

We consider the orientational order that arises from the alignment of anisotropic biological cells confined in bounded domains. Especially, it is known that topological defects, where the orientation angles are not well-defined, play an important role in the global pattern formation of cell alignment. Hence, it is important to investigate the relationship between the shape of the region and the location of the defects. The orientational order of cells is theoretically modelled as a harmonic function achieving the minimum elastic energy state of the nematic liquid crystal, and an analytical formula in simply connected domains is given by Miyazako & Nara (Miyazako, Nara 2022 R. Soc. Open Sci . 9, 211663 ( doi:10.1098/rsos.211663 )). In this paper, we extend this study to construct an analytic formula of cell angle in the doubly connected domains, where topological defects and boundary shapes are represented as the positions of logarithmic singularities and as conformal maps from a given region to a standard annular region, respectively. In addition, using this analytical formula, we propose a numerical algorithm to minimize elastic energy, from which we investigate cell alignment in doubly connected domains with anisotropic quadrilateral boundaries. Our numerical simulations suggest that pairs of topological defects tend to be located around corners of the boundaries, which will be a design principle of the boundary geometry for controlling the defect positions and cell alignment.

Experimental determination and mathematical modeling of standard shapes of forming autophagosomes

The formation of autophagosomes involves dynamic morphological changes of a phagophore from a flat membrane cisterna into a cup-shaped intermediate and a spherical autophagosome. However, the physical mechanism behind these morphological changes remains elusive. Here, we determine the average shapes of phagophores by statistically investigating three-dimensional electron micrographs of more than 100 phagophores. The results show that the cup-shaped structures adopt a characteristic morphology; they are longitudinally elongated, and the rim is catenoidal with an outwardly recurved shape. To understand these characteristic shapes, we establish a theoretical model of the shape of entire phagophores. The model quantitatively reproduces the average morphology and reveals that the characteristic shape of phagophores is primarily determined by the relative size of the open rim to the total surface area. These results suggest that the seemingly complex morphological changes during autophagosome formation follow a stable path determined by elastic bending energy minimization.

Investigating the preferential growth of Bi grains in Sn-Bi based solder under thermal aging

SnBi based solders have attracted widespread attention due to their promising applications in heterogeneous integration and three-dimensional microelectronic packaging. However, the low melting point of SnBi based solder means their high homologous temperature during service, leading to fast coarsening of the microstructure, which can significantly impair the reliability of electron devices. It has been found that Bi grains in SnBi based solder interconnects grow significantly and exhibit a preferred orientation under thermal loading conditions, but the mechanisms underneath the preferential growth of Bi grains is yet to be understood. In this work, a phase field model incorporating the thermal stress effect is developed and employed to investigate the dynamic evolution of the microstructure of SnBi solder under thermal aging, and to capture the morphology changes of the grains with different orientations. It is demonstrated that Bi grains with c-axis parallel to the z-axis preferentially exist in eutectic SnBi solder under thermal aging, and Bi grains with a large angle between the c-axis and the z-axis are swallowed by the preferred grains. Moreover, there is a competitive evolution between grains with different orientations in the polycrystalline eutectic SnBi solder, due to the minimization of system elastic strain energy. Further mechanism analysis elucidates that the Bi grain orientation preference in SnBi solders is caused by the dependence of their Young’s modulus and coefficient of thermal expansion on the grain orientation, the favored grains possess a lower strain energy compared to other grains.

Two energies for conjoining boron nitride nanotorus and nanotube

AbstractOwing to a variety in nanoscale material applications, the conjunction of two nanostructures is frequently researched for potential new applications. Numerous methods are used to model this conjunction process. One such method, the minimization of elastic energy, only considers the axial curvature when modeling conjoined structures. Another method minimizes the Willmore energy, which depends on both the axial and rotational curvatures. In particular, because the catenoid is an absolute minimizer of Willomre energy, a catenoid section can be utilized to conjoin nanostrucrures. Owing to the similarities among carbon nanostructures, we expanded the use of two different energies to join a boron nitride nanotube with a boron nitride nanotorus. The primary objective of this study was to formulate a basic underlying structure from which any small perturbations can be viewed as departures from an ideal model. Accordingly, elastic energy was used to determine the conjunction region for two-dimensional structures, whereas Willmore energy was used to determine the conjunction region for three-dimensional structures. This approach may be extended to produce other hybrid nanoscale structures.

C-Shells: Deployable Gridshells with Curved Beams

We introduce a computational pipeline for simulating and designing C-shells , a new class of planar-to-spatial deployable linkage structures. A C-shell is composed of curved flexible beams connected at rotational joints that can be assembled in a stress-free planar configuration. When actuated, the elastic beams deform and the assembly deploys towards the target 3D shape. We propose two alternative computational design approaches for C-shells: (i) Forward exploration simulates the deployed shape from a planar beam layout provided by the user. Once a satisfactory overall shape is found, a subsequent design optimization adapts the beam geometry to reduce the elastic energy of the linkage while preserving the target shape. (ii) Inverse design is facilitated by a new geometric flattening method that takes a design surface as input and computes an initial layout of piecewise straight linkage beams. Our design optimization algorithm then calculates the smooth curved beams to best reproduce the target shape at minimal elastic energy. We find that C-shells offer a rich space for design and show several studies that highlight new shape topologies that cannot be achieved with existing deployable linkage structures.

Platonic and Archimedean bodies as the basis of the structure of self-accommodating complexes of martensite crystals in alloys with shape memory effects

The aim of the work is to analyze the relationship of the architecture of self-accommodation complexes (SC) with the lattice syngony of martensite crystals. Self-accommodating complexes consist of a set of pairwise twinned domains — crystals of martensite belonging to crystallographically equivalent variants of the orientation relationship between the lattices of austenite and martensite. The simplest SC are calculated for tetragonal, orthorhombic, rhombohedral and monoclinic distortion of the cubic lattice of austenite. It is shown that complete self-accommodation is possible only in complexes containing simultaneously all variants of the orientation relation. The issue of external faceting of complexes is discussed. The reason for the formation of SC is the minimization of elastic energy, i.e. the appearance regulated by the energy of the interphase boundary. On the other hand, if the outer surface of the SC is a polyhedron, then its symmetry should "fit"into the anisotropy of the elastic properties of austenite. For reasons of symmetry, it is clear that the polyhedron must be correct and have the same symmetry elements as the cubic lattice of austenite, while the axes of symmetry of the cubic lattice of austenite must coincide with the axes of symmetry of the polyhedron. Similar polyhedra are some of the bodies of Platon and Archimedes, which have axes of symmetry of the 2nd, 3rd and 4th order. A number of examples calculated in the work confirms the possibility of the existence of complexes in the form of these polyhedra.

THE ROLE OF SELFACCOMMODATION COMPLEXES IN THE CRYSTALLOGRAPHIC REVERSIBILITY OF NONTHERMOELASTIC MARTENSITE TRANSFORMATIONS AND CONJUGATION OF ANISOTROPIC STRUCTURES

The possible structure of self-accommodation complexes of martensite crystals in alloys based on g-Mn and titanium nickelide has been analyzed. The analysis is based on the calculation of shape deformation, averaged over an ensemble of domains (equivalent versions of orientation relationships); for complete self-accommodation this deformation should be described by a unit matrix. In this case the compensation of the shape variation and minimization of elastic energy occur on the microscopic level of individual complexes of twinned martensite crystals. It is shown that complete self-accommodation of rhombohedral martensite R, as well as tetragonal and orthorhombic martensites in alloys based on y-Mn is implemented only in the complexes containing either all crystallographically equivalent versions of orientation relationships or their doubled number

Compression Characteristics and Fracture Simulation of Gluten Pellet.

Gluten pellets are readily broken on packaging and transportation. This study aimed to research mechanical properties (elastic modulus, compressive strength, failure energy) with different moisture contents and aspect ratios under different compressive directions. The mechanical properties were examined with a texture analyzer. The results revealed that the material properties of the gluten pellet are anisotropic, and it was more likely to cause crushing during radial compression. The mechanical properties were positively correlated with the moisture content. The aspect ratio had no significant effect (p > 0.05) on the compressive strength. The statistical function model (p < 0.01; R2 ≥ 0.774) for mechanical properties and moisture content fitted well with the test data. The minimum elastic modulus, compressive strength, and failure energy of standards-compliant pellets (with moisture content less than 12.5% d.b.) were 340.65 MPa, 6.25 MPa, and 64.77 mJ, respectively. Moreover, a finite element model with cohesive elements was established using Abaqus software (Version 2020, Dassault Systèmes, Paris, France) to simulate the compression rupture form of gluten pellets. The relative error of the fracture stress in the axial and radial directions between the simulation results and the experimental value was within 4-7%.

An improved load distribution model for involute helical gear pairs considering an actual contact ratio

With the increasing requirements of helical gears for high strength and low noise, it is significant to improve the accuracy of the load distribution ratio depending on the minimum elastic potential energy (MEPE) model. In this paper, an improved distribution model is developed to replace the theoretical contact ratio with the actual contact ratio and introduce the potential energy of the fillet foundation for comparison. Then, the operation process of the helical gear is divided into seven cases. Finally, combined with the finite element method, the accuracy of the improved model is confirmed. Results indicate that the load distribution model with fillet foundation energy is only effective under the front or the full tooth meshing. When the gear meshing includes the rear part of a tooth, the inclusion of the actual contact ratio and fillet foundation energy should be considered.

THE ROLE OF ELASTIC STRESS DURING THERMOMIGRATION OF LIQUID ZONES BASED ON ALUMINUM IN SILICON

The method of thermomigration of liquid zones based on aluminum makes it possible to create complex structures of closed channels with boundaries formed from p-n junctions in single-crystal silicon wafers. The channels are characterized by the uniformity of their properties, and the pn junctions are characterized by their sharpness. Such structures are used in high-current electronics, photovoltaics, and microelectromechanical converters. Volumetric deformation inside and outside the channel due to alloying of silicon with aluminum leads to the formation of mechanical stresses. The equilibrium shape of the channels formed at high temperatures is determined by the minimum elastic energy and depends on the material parameters and geometry of the structure. Within the framework of the linear theory of elasticity, the behavior of elastic energy during the formation of the structure of thermomigration channels at high temperatures doped with aluminum in a single-crystalline (001) cut disk was studied. The simulation was performed using the finite element method in the COMSOL Multiphysics mathematical package. The study was carried out for practically important structures in which the direction of the edges is oriented along the diagonal of the square. Only such structures do not have breaks during the process of thermomigration. Based on the calculation results, it was revealed that the minimum elastic energy corresponds to structures with different crystalline orientations inside the channel and outside it – in the main silicon matrix. The direction of the crystalline axes inside the channel, corresponding to the minimum elastic energy, is rotated by 45 degrees in the plane of the disk relative to the direction of the axes of the main matrix of the silicon crystal. In addition, calculations showed that with such a turn, the shape of the channels changes. The minimum elastic energy corresponds not to vertical structures, but to inclined ones. The angle of inclination of the pyramids depends on the width of the channels and the distance between them.

An Energy Minimization Approach to Twinning with Variable Volume Fraction

In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example involving a single-crystal slab made from two martensite variants. When the slab is made to bend, the two variants form a characteristic microstructure that we like to call “twinning with variable volume fraction.” Two 1996 papers by Chopra et al. explored this example using bars made from InTl, providing considerable detail about the microstructures they observed. Here we offer an energy-minimization-based model that is motivated by their account. It uses geometrically linear elasticity, and treats the phase boundaries as sharp interfaces. For simplicity, rather than model the experimental forces and boundary conditions exactly, we consider certain Dirichlet or Neumann boundary conditions whose effect is to require bending. This leads to certain nonlinear (and nonconvex) variational problems that represent the minimization of elastic plus surface energy (and the work done by the load, in the case of a Neumann boundary condition). Our results identify how the minimum value of each variational problem scales with respect to the surface energy density. The results are established by proving upper and lower bounds that scale the same way. The upper bounds are ansatz-based, providing full details about some (nearly) optimal microstructures. The lower bounds are ansatz-free, so they explain why no other arrangement of the two phases could be significantly better.

The Elastic Effect of Evolving Precipitate Shapes on the Ripening Kinetics of Tetragonal Phases

Coherent tetragonal precipitates, such as the Ni3Nb phase γ″ found in Ni-base superalloys, appear as plate-shaped particles. These shapes are the result of anisotropic elastic misfit strains. We present 3D sharp phase-field simulations that capture this circumstance well due to the inclusion of the elastic effects from the misfit. These simulations reveal that the ripening behavior of γ″ precipitates deviates significantly from the classical LSW theory of Ostwald ripening. A ripening exponent of 2 rather than 3 describes the simulated γ″ size evolution at temperatures between 700 °C and 760 °C best. Employing a quantitative distinction argument, we show that 60 pct of this deviation is attributed to the elastically induced size dependence of the precipitate shapes. With increasing precipitate size, the minimization of elastic energy leads to steadily increasing plate aspect ratios. The precipitate ripening kinetics accelerate with increasing aspect ratio. Fitting the newly received square root time dependence to experimental data yields a physically conclusive activation energy of ripening close to the activation energy of Nb diffusion in the alloy.

Metallic Mimosa pudica: A 3D biomimetic buckling structure made of metallic glasses.

Metallic Mimosa pudica, a three-dimensional (3D) biomimetic structure made of metallic glass, is formed via laser patterning: Blooming, closing, and reversing of the metallic M. pudica can be controlled by an applied magnetic field or by manual reshaping. An array of laser-crystallized lines is written in a metallic glass ribbon. Changes in density and/or elastic modulus due to laser patterning result in an appropriate size mismatch between the shrunken crystalline regions and the glassy matrix. The residual stress and elastic distortion energy make the composite material to buckle within the elastic limit and to obey the minimum elastic energy criterion. This work not only provides a programming route for constructing buckling structures of metallic glasses but also provides clues for the study of materials with automatic functions desired in robotics, electronic devices, and, especially, medical devices in the field of medicine, such as vessel scaffolds and vascular filters, which require contactless expansion and contraction functions.

Energy minimizing twinning with variable volume fraction, for two nonlinear elastic phases with a single rank-one connection

In materials that undergo martensitic phase transformation, distinct elastic phases often form layered microstructures — a phenomenon known as twinning. In some settings the volume fractions of the phases vary macroscopically; this has been seen, in particular, in experiments involving the bending of a bar. We study a two-dimensional (2D) model problem of this type, involving two geometrically nonlinear phases with a single rank-one connection. We adopt a variational perspective, focusing on the minimization of elastic plus surface energy. To get started, we show that twinning with variable volume fraction must occur when bending is imposed by a Dirichlet-type boundary condition. We then turn to paper’s main goal, which is to determine how the minimum energy scales with respect to the surface energy density and the transformation strain. Our analysis combines ansatz-based upper bounds with ansatz-free lower bounds. For the upper bounds we consider two very different candidates for the microstructure: one that involves self-similar refinement of its length scale near the boundary, and another based on piecewise-linear approximation with a single length scale. Our lower bounds adapt methods previously introduced by Chan and Conti to address a problem involving twinning with constant volume fraction. The energy minimization problem considered in this paper is not intended to model twinning with variable volume fraction involving two martensite variants; rather, it provides a convenient starting point for the development of a mathematical toolkit for the study of twinning with variable volume fraction.

Formula omitted][formula omitted] morphology prediction of Zr hydride precipitates using atomistically informed Eshelby’s ellipsoidal inclusion

We energetically predicted the morphology of Zr hydride precipitates in a hexagonal close-packed (HCP) Zr matrix. Considering Zr hydride precipitates as ellipsoids, we used Eshelby’s ellipsoidal inclusions to calculate the elastic energy increment due to the presence of Zr hydride precipitates in the Zr matrix, in which the elastic anisotropy and inhomogeneity of the elastic constants between Zr and Zr hydride were considered. We compared the difference in the elastic energy increment between the ellipsoidal inclusions with different shapes: plates (mimicked by penny-shape ellipsoids), needles (mimicked by longitudinal ellipsoids) and sphere, and orientations to detect the stable structure with the minimum elastic energy increment. Eigenstrains of each Zr hydride and elastic constants of Zr hydrides and HCP Zr for Eshelby’s ellipsoidal inclusion analysis were determined using atomistic simulations based on a density functional theory calculation, achieving a parameter free abinitio morphology prediction. The morphology predictions were implemented for two cases: with and without shear components of eigenstrain (w/ and w/o shear). The 〈12̄10〉 longitudinal needle for the γ hydride (w/o shear) and plate (or disk) on the plane, which is 20° to 30° tilted about 〈12̄10〉-axis from basal plane (0001), for δ and ε hydrides (w/ shear) were successfully predicted as stable shapes and orientations of the precipitates under zero external stress conditions, qualitatively consistent with experimental observations. The external circumferential tensile stress on the basal plane reduces the elastic energy of [0001] parallel Zr hydride plates, which is also qualitatively consistent with the reoriented δ hydride precipitates observed in the experiment. On the other hand, predicted external stress for the reorientation of Zr hydride is quite high, around 10 GPa. This is inconsistent with experimental observation and further investigation is necessary. Generally, our predictions based on elasticity theory appear qualitatively consistent with experimental observations, suggesting an elastic origin of the morphology of Zr hydride precipitates in the HCP Zr matrix.

Elastic interaction energy analysis during twin plane formation in the martensitic transformation process of α"-martensite in Ti-Nb-Al

To understand the twin selection mechanism that occurs in the initial stage of the martensitic transformation of shape memory alloys, the stress state during twin plane formation was numerically analyzed by micromechanics. The self-accommodation microstructure of α''-martensite of the Ti-23Nb-3Al (at%) shape memory alloy, in which the lattice parameter was adjusted to eliminate the formation of internal twinning due to lattice-invariant deformation, was adopted for analysis. The two lattice correspondence variants (CVi and CVj) formed a twin plane. CVi was assumed to be spheroid, and the elastic strain energy during twin plane formation was evaluated by analyzing the elastic interaction energy when CVj formed around CVi. There are three types of twins ({011}α" compound twin, {111}α" type I twin, and <211>α" type II twin) in the self-accommodation microstructure of α"-martensite. Regardless of the CVi geometry (i.e., aspect ratio (1 to 0.01) and minor axis orientation), the {111}α" type I twin exhibited minimum elastic interaction energy, indicating that minimum elastic strain energy was generated during this twin plane formation. This result is consistent with the experimental results, which showed the selective formation of {111}α" type I twins in the initial stage of martensitic transformation.

Formation of oxide crystallites on the porous GaAs surface by electrochemical deposition

We demonstrate how the formation of octahedral microcrystals of arsenic oxide As2O3in the form of arsenolite with a size of 200 nm to 10 μm can be initiated by the electrochemical etching method with simultaneous deposition on the surface of substrates with n-GaAs (111). Crystallites were formed on a previously synthesized porous layer of GaAs. To explain the behavior of formation on the surface of the monocrystalline GaAs porous layer and As2O3crystallites in the electrochemical reaction, we propose a qualitative model based on the decomposition of binary semiconductors in contact with electrolytes. Under this model, the crystallization of precipitated oxides occurs as a result of the transfer of ions to the crystal surface as a result of the electrolysis process. The formation of the composite structure takes place on the surface of the semiconductor and is characterized by the minimization of elastic energy. XRD analysis showed the formation of a complex compound of As2O3and As0.172Sb0.570O1.113. The appearance of antimony is explained in terms of the formation of new centers when the As atom is replaced by an Sb doping atom in the crystal. Directed controlled oxidation technologies make it possible to synthesize a reliable passivating layer consisting of one type of oxide, namely As2O3in the cubic phase of arsenolite. In addition, such structures can be used in photonics devices and as photocatalysts.