Minimization Of Elastic Energy Research Articles

Time‐Resolved Evolution of the 3D Nanoporous Structure of Sintered Ag by X‐Ray Nanotomography: Role of the Interface with a Copper Substrate

The evolution of the nanoporous structure of cylindrical sintered silver samples during high temperature aging, ranging from 200 to 350 °C for 350 min, is studied through in situ computed X‐ray tomography. Investigations are carried out for two types of specimens: pure sintered silver and specimens containing a silver‐copper interface. It is shown that the overall pore evolution is driven by the evolution of very few large ones. The smaller pores, although being more numerous, do not really evolve before being absorbed by the few bigger ones. In pure silver, pore evolution is driven by diffusion (Ostwald ripening) but the presence of an interface promotes faster growth kinetics until the aging time reaches a threshold value, after which a deviation from Ostwald ripening occurs. The transition is a function of the aging temperature. This behavior is associated with the competition between elastic relaxation and surface energy minimization.

Relevance of processing parameters for grain growth of metal halide perovskites with nanoimprint

The quality and the stability of devices prepared from polycrystalline layers of organic–inorganic perovskites highly depend on the grain sizes prevailing. Tuning of the grain size is either done during layer preparation or in a post-processing step. Our investigation refers to thermal imprint as the post-processing step to induce grain growth in perovskite layers, offering the additional benefit of providing a flat surface for multi-layer devices. The material studied is MAPbBr3; we investigate grain growth at a pressure of 100 bar and temperatures of up to 150 °C, a temperature range where the pressurized stamp is beneficial to avoid thermal degradation. Grain coarsening develops in a self-similar way, featuring a log-normal grain size distribution; categories like ‘normal’ or ‘secondary’ growth are less applicable as the layers feature a preferential orientation already before imprint-induced grain growth. The experiments are simulated with a capillary-based growth law; the respective parameters are determined experimentally, with an activation energy of Q ≈ 0.3 eV. It turns out that with imprint as well the main parameter relevant to grain growth is temperature; to induce grain growth in MAPbBr3 within a reasonable processing time a temperature of 120 °C and beyond is advised. An analysis of the mechanical situation during imprint indicates a dominance of thermal stress. The minimization of elastic energy and surface energy together favours the development of grains with (100)-orientation in MaPbBr3 layers. Furthermore, the experiments indicate that the purity of the materials used for layer preparation is a major factor to achieve large grains; however, a diligent and always similar preparation of the layer is equally important as it defines the pureness of the resulting perovskite layer, intimately connected with its capability to grow. The results are not only of interest to assess the potential of a layer with respect to grain growth when specific temperatures and times are chosen; they also help to rate the long-term stability of a layer under temperature loading, e.g. during the operation of a device.

Effective toughness of disordered brittle solids: A homogenization framework

This paper addresses the question of the homogenization of fracture properties for three-dimensional disordered brittle solids. The effective toughness, identified as the minimum elastic energy release rate required to ensure crack growth, is predicted from a semi-analytical framework inspired by both micromechanics and statistical physics, that encompasses the decisive influences of both the material disorder and the mechanisms of interaction between a crack and heterogeneities. Theoretical predictions are compared to numerical values of the effective toughness that are computed with the fracture-mechanics-based semi-analytical method of Lebihain et al. (2020). Based on a perturbative approach of Linear Elastic Fracture Mechanics, this method allows for the efficient computation of crack propagation under tensile Mode I loading in composite materials containing several millions of inclusions, where the crack interacts with them through two mechanisms : crossing, wherein the crack penetrates the inclusion, and by-pass, wherein the crack wanders out-of-plane and follows the inclusion/matrix interface. We show that our homogenization procedure provides an accurate prediction of the homogenized fracture properties for a broad range of microstructural parameters such as the inclusion toughness, density or shape. This original theoretical framework constitutes a powerful mean to connect the microstructural parameters of materials to their crack growth resistance, beyond the particular cases considered in the simulations performed. As a result, it provides new strategies for the rational design of optimized brittle composites with tailored fracture properties.

Insights into the nucleation, grain growth and phase transformation behaviours of sputtered metastable β-W films

Metastable phase in tungsten film is of great interests in recent years due to its giant spin Hall effects, however, little information has been known on its nucleation, growth and phase transformation. In this paper, a 900 nm-thick tungsten film with double-layer structure (α-W underlayer and β-W above it) was produced on SiO2/Si substrate by high vacuum magnetron sputtering at room temperature. The structural properties of β-W were systemically investigated by X-ray diffraction, transmission electron microscopy, thermodynamic calculation, first-principle and phase-field simulations. It is found that the β-W nucleation is energetically favoured on the SiO2 surface compared to the α-W one. As the film thickening proceeds, β-W[211] turns to be preferred direction of growth owing to the elastic strain energy minimization, which is verified by phase-field simulations. Moreover, the β→α phase transformation takes place near the film-substrate interface while the rest of the film keeps the β-W phase, leading to a double-layer structure. This localized phase transition is induced by lower Gibbs free energy of α-W phase at larger grain sizes, which can be confirmed by thermodynamic calculation. Further in-situ heating TEM analysis of the as-deposited film reveals that the β→α phase transformation is fulfilled by α/β interface propagation rather than local atomic rearrangements. Our findings offer valuable insights into the intrinsic properties of metastable phase in tungsten.

Tooth Contact Analysis of Herringbone Rack Gears of an Impulse Continuously Variable Transmission

Finite element based tooth contact analysis of a rack and pinion system of an impulse continuously variable transmission is performed in this study. The rack and pinion system are designed with involute profile herringbone tooth profiles. The involute profile herringbone tooth profile is a type of concave-convex pro-files, which can be used for heavy load conditions. A contact load distribution model of the rack gear with involute profile herringbone tooth profiles is devel-oped to analyze normal loads of any mashing position of the rack pinion system based on the minimum elastic potential energy theory. With the aim of improving the rack and pinion system design, the actual operation of gears under the terms of the three-dimensional tooth contact analysis is conducted. A rack gear with herringbone tooth profiles and a pinion are used to tooth contact analysis. Based on comparing the results of this analysis, a new type of rack gears with concave-convex involute tooth profile are advantages and disadvantages in terms of the contact stress.

Influence of substructure inhomogeneity on the anisotropy of physical properties of textured materials

Calculation of the physical properties anisotropy is based on single crystal properties and information about material texture, i.e. orientation distribution function (ODF). At the same time, the grains of different orientations are known to have different substructure parameters including the lattice distortion and microstrains. Determination of these parameters is possible with the X-ray method of Generalized Pole Figures (GPF), which consists of registration of the whole X-ray reflection profile under the pole figure recording. As a result, one can obtain the FWHMs of X-ray line profiles and microstrains distributed on the stereographic projection. Despite the observable equilibrium of tensile and compressive microstrains of differently oriented grains within the studied material volume, they can affect the product properties anisotropy. In this study the calculation method for elastic properties and thermal expansion coefficients taking into account substructure inhomogeneity is offered on the basis of elastic energy minimization. Its efficiency is demonstrated for the samples of zirconium alloys.

Strain Relaxation in "2D/2D and 2D/3D Systems": Highly Textured Mica/Bi2Te3, Sb2Te3/Bi2Te3, and Bi2Te3/GeTe Heterostructures.

Strain engineering as a method to control functional properties has seen in the last decades a surge of interest. Heterostructures comprising 2D-materials and containing van der Waals(-like) gaps were considered unsuitable for strain engineering. However, recent work on heterostructures based on Bi2Te3, Sb2Te3, and GeTe showed the potential of a different type of strain engineering due to long-range mutual straining. Still, a comprehensive understanding of the strain relaxation mechanism in these telluride heterostructures is lacking due to limitations of the earlier analyses performed. Here, we present a detailed study of strain in two-dimensional (2D/2D) and mixed dimensional (2D/3D) systems derived from mica/Bi2Te3, Sb2Te3/Bi2Te3, and Bi2Te3/GeTe heterostructures, respectively. We first clearly show the fast relaxation process in the mica/Bi2Te3 system where the strain was generally transferred and confined up to the second or third van der Waals block and then abruptly relaxed. Then we show, using three independent techniques, that the long-range exponentially decaying strain in GeTe and Sb2Te3 grown on the relaxed Bi2Te3 and Bi2Te3 on relaxed Sb2Te3 as directly observed at the growth surface is still present within these three different top layers a long time after growth. The observed behavior points at immediate strain relaxation by plastic deformation without any later relaxation and rules out an elastic (energy minimization) model as was proposed recently. Our work advances the understanding of strain tuning in textured heterostructures or superlattices governed by anisotropic bonding.

Comprehensive Phase Field Study on Directionally-Solidified MoSi<sub>2</sub>/Mo<sub>5</sub>Si<sub>3</sub> Eutectic Alloy

MoSi2/Mo5Si3 eutectic composites have been considered as one of the promising candidates for ultra-high temperature structural applications owing to their high melting point, good oxidation resistance, and low mass density. Their mechanical properties can be improved by controlling the eutectic structure (i.e. script lamellar structure) in directional solidification. It is important to elucidate the dominant factors underlining the unique pattern formation. We conducted a comprehensive phase field study to examine the influence of various factors on the MoSi2/Mo5Si3 eutectic microstructure with complicated morphology. First, the inclined lamellae have been attributed to the minimization of elastic strain energy due to the lattice mismatch between MoSi2 and Mo5Si3, which are partially relaxed by forming semi-coherent phase boundaries. Second, the maze-like pattern on the horizontal cross-section appeared when a two-fold anisotropy of interfacial energy is superimposed on the MoSi2/Mo5Si3 boundary. Third, the random and intersected lamellae have been obtained by assuming the instability of the solid-liquid interface and introducing successive nucleation of Mo5Si3 phase. These findings provide guidance for manipulating the eutectic structure and act as footsteps for further theoretical investigation.

Programmable shape-shifting 3D structures via frontal photopolymerization

Shape-shifting structures have gained growing interest recently and found wide applications in areas such as soft robotics, biomedical devices and self-folding origami, attributed to their ability to construct complicated shapes directly from simple structures. However, an efficient method to design and fabricate programmable 3D shape-shifting structures from 2D polymer films still lacks. In this work, we design programmable shape-shifting 3D structures via the release of internal gradient stress using the frontal photopolymerization (FPP) method. First, the relation between the non-uniformly distributed material and loading parameters, and the geometric and fabrication parameters are established theoretically. The finite element (FE) model is then developed based on the theoretically obtained material and loading parameters. Next, the elastic instability in the shape-shifting behaviors of a cured film is captured through an elastic energy minimization. Furthermore, by using grayscale light patterns, it is shown that we can selectively manipulate the geometric and fabrication parameters to improve the design freedom of various complex 3D structures.

Inhomogeneous deformation in hyperspace

Geometrical approach to continuum mechanics allowing an atomistic treatment is developed. It suggests a body as a surface immersed into ambient space. Four kinds of surfaces inherent to basic deformations determine classical dynamics of the atoms alongside their spatial coordinates. For uniaxial stretch, this surface is isomorphic to the cylinder. For a simple shear, it is the twisted cylinder, for a bending — a cone, and for a torsion — a helicoid. Scalar parameters of the metric of these surfaces that are stretch ratios, shear, cone, and torsion angles used instead of the strain tensor constitute extra coordinates of the particles in hypersurface. Correspondingly, vector-valued tensions are used instead of stress. In the case of pure deformation, the normal and extra coordinates of atoms obey the classical equation of motion admitting periodic boundary conditions. It is shown how the employing of artificial orders of freedom leads to a minimum of elastic energy. In boundary problems, the system of governing equations unambiguously reproduce the stress–strain–displacement relations with the help of using the tension boundary theorem instead of the Cauchy traction principle. The derived direct method of elastostatic makes the system of governing equations closed and the compatibility condition unnecessary. Commonest examples of bending problem normally following from the Airy stress function illustrate the new concept.

Grain boundary space charge modulation in BaZr0.8Y0.2-xMxO3−δ with transition metal (M= Ni, Co, Fe, and Zn) co-doping

The higher grain boundary resistance of proton-conducting ceramic electrolytes such as BaZrO3 drops the total conductivity. In the present study, BaZr0.8Y0.2-xMxO3−δ electrolyte with various transition metals (M= Ni, Co, Fe, and Zn) with molar concentrations (x = 0.01, 0.03 & 0.05) were prepared by hydrothermal assisted precipitation method. The structural analysis, such as XRD and Raman, revealed the presence of the cubic perovskite phase of barium zirconate. The existence of space charge layer was confirmed by nonlinear behaviour of the grain boundary resistance on applying DC bias. The Mott–Schottky analysis performed at 200 °C in reducing atmosphere showed a reduction in barrier potential from 0.24 to 0.13 V upon the addition of the secondary dopants. The elastic energy minimization by dopant segregation and electrostatic interaction between the charged defects and dopants suppress the space charge layer. The transition metal dopant, especially Ni and Fe, segregates along the grain boundary and acts as the counterbalance for charged defects. Besides, Fe act as perfect sintering aid which leads to higher densification (>99%) as well as higher protonic conductivity of 21.2 mScm−1 at 600 °C. These results imply that the transition metal ions as co-dopant influence the space charge effect and the density of the barium zirconate electrolyte.

Electrostrain Enhancement at Tricritical Point for BaTi1−xHfxO3 Ceramics

The maximum electromechanical property of piezoelectric ceramic is normally achieved at morphotropic phase boundary. Herein, we found that a maximum electrostrain is obtained in single rhombohedral phase region at the tricritical point (which is a single rhombohedral phase at room temperature) for BaTi1−xHfxO3 ceramics. The mechanism for electrostrain enhancement at tricritical point (TTP) was uncovered by analyzing its crystallographic feature. The results show that TTP (x = 0.11) exhibits largest electrostrain (S = 0.063%) at room temperature. And the TTP reveals a maximum crystallite size (66.3 nm) and minimum lattice strain (0.7 × 10−4). Further Raman results demonstrate a lowest Raman intensity occurs at TTP, which confirmed the specific crystallographic behavior. Therefore, a minimum lattice strain-induced low elastic energy could lower landau free energy, which is responsible for electrostrain enhancement at TTP. This work may provide a new insight for understanding and designing large electromechanical ceramics.

Confined elasticae and the buckling of cylindrical shells

Abstract For curves of prescribed length embedded into the unit disk in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds 2 ⁢ π {2\pi} and in the large length limit. In the small excess length case, we prove convergence to a fourth-order obstacle-type problem with integral constraint on the real line which we then solve. From the solution, we obtain the energy expansion 2 ⁢ π + Θ ⁢ δ 1 3 + o ⁢ ( δ 1 3 ) {2\pi+\Theta\delta^{\frac{1}{3}}+o(\delta^{\frac{1}{3}})} when a curve has length 2 ⁢ π + δ {2\pi+\delta} and determine first order coefficient Θ ≈ 37 {\Theta\approx 37} . We present an application of the scaling result to buckling in two-layer cylindrical shells where we can determine an explicit bifurcation point between compression and buckling in terms of universal constants and material parameters scaling with the thickness of the inner shell.

Exploration of Configurational and Topological Properties of Minichromosomes using Elastic Energy Optimizations at the DNA Base-Pair Level

Advances in experimental biophysics continue to broaden our understanding of protein-DNA interactions, highlighting structural characteristics that may account for modes of genetic expression and regulation. These interactions influence the configuration of DNA that enters and exits the protein and may lead to DNA supercoiling. Previous studies of minichromosomes showcased discrepancies in linking number and topology across short circular chains containing nucleosomes with ∼145 base-pairs of DNA wrapped less than two superhelical turns around a core of eight histone proteins. The increasing collection of high-resolution nucleosome structures has been valuable in developing computational models to survey differences in minichromosomes with different histone binding surfaces or DNA sequences. The models offer an approximation of the energetically favorable configurations of protein-free DNA, which depend on the spatial arrangements of the entry and exit base-pairs and/or the nucleotide sequence. Elastic energy minimization calculations performed at the level of base-pair steps starting from approximately closed circular DNA models generated by Monte Carlo sampling have revealed the sensitivity of minichromosomal configurations to free-DNA chain length, entry-exit spatial arrangements, and imposed base-pair twist. Producing optimized circular structures on a mesoscale level may shed light on certain topological behaviors, such as strand crossings at the entry and exit regions of linker DNA, and may reconcile the linking number paradox that surround minichromosomes.

Active curved surface deforming of flexible conformal electronics by multi-fingered actuator

Flexible conformal electronics can be applied in many novel fields, such as wearable sensors, curved display and epidermal electronics, owing to their good adaptability to curved surfaces. Currently, the manufacturability of flexible electronics has been greatly enhanced with the development of various printing technologies. In this paper, the active conformal deformation model and device are presented under the framework of transfer printing. In particular, we present a practical deformation process of a multi-fingered actuator based on the minimization of elastic potential energy of the flexible membrane. The whole transfer printing platform as well as the multi-fingered manipulator are manufactured and the feasibility of them are verified with multiple tests on different target surfaces.

Revisit of the shape and orientation of precipitates with tetragonal transformation strains that minimise the elastic energy

ABSTRACT The selection of the shape and orientation of precipitates with tetragonal misfit (transformation) strains that minimise the elastic energy is revisited. The complex map of preferred shapes suggested by Kaganova and Roitburd for inhomogeneous isotropic materials is extended to materials with cubic symmetry and for systems that compromise with the surface energy of close-packed planes. Several shape transitions are explained from the kinematics of precipitates with tetragonal transformation strains in solid matrices. Plate shaped precipitates are associated with minimum elastic energy in homogenous systems and for soft precipitates, in accordance with the theorem of Khachaturyan. We explain how compact shapes like rods and spheres arise when the matrix is softer than the precipitate. Analytic estimations are derived to determine the preferred shapes based on the difference between the elastic stiffness of the matrix and the precipitate. When the transformation strains differ in sign, they give rise to a large lattice rotation. This lattice rotation is necessary for the formation of twins by equivalent variants of the transformation strain.

Parallel 3d shape optimization for cellular composites on large distributed-memory clusters

Skin modeling is an ongoing research area that highly benefits from modern parallel algorithms. This article aims at applying shape optimization to compute cell size and arrangement for elastic energy minimization of a cellular composite material model for the upper layer of the human skin. A gradient-penalized shape optimization algorithm is employed and tested on the distributed-memory cluster Hazel Hen, HLRS, Germany. The performance of the algorithm is studied in two benchmark tests. First, cell structures are optimized with respect to purely geometric aspects. The model is then extended such that the composite is optimized to withstand applied deformations. In both settings, the algorithm is investigated in terms of weak and strong scalability. The results for the geometric test reflect Kelvin's conjecture that the optimal space-filling design of cells with minimal surface is given by tetrakaidecahedrons. The PDE-constrained test case is chosen in order to demonstrate the influence of the deformation gradient penalization on fine inter-cellular channels in the composite and its influence on the multigrid convergence. A scaling study is presented for up to 12,288 cores and 3 billion DoFs.

Enhanced numerical design of two-barrier infrared detectors with III\u2013V compounds heterostructures considering the influence of lattice strain and misfit dislocations on the band gap

The goal of this work is the design of efficient infrared radiation detectors based on InAsSb compounds with two energy barriers around the absorber region. Two types of two-barrier detectors that work in the 3–5.5 μm wavelength range at 230 K is designed. Using our computer program to iteratively solve the Poisson equation, the spatial distribution of energy band edges in III–V heterostructures was calculated The influence of lattice stress, bending of the structure and doping on the energy shift of the edge of the bands is considered. Lattice strain is the cause of the formation of the misfit dislocations at the boundaries of the individual layers of the heterostructure. These dislocations partially relax the resulting stress and affect the lattice deformation in individual layers. From the minimum elastic energy condition, the density of these dislocations is determined. It has been shown that the band offset can be eliminated in the areas of both barriers in the designed two-barrier detectors.

Estimation of influence of lattice strain, bending and doping on the width of energy gap in InAsSb heterostructures

In this paper influence of lattice strains, bending and doping on the energy gap width in the InAs1−xSbx structures was determined numerically. Due to this fact, by energy gap measurements (e.g. from photoluminescence, or photo conductance) we can determine the amount of lattice strain. The experimental results of other researchers were used in the work to verify our theoretical estimates. Lattice stress resulting from lattice mismatch between the substrate and the epitaxial layer and bending of the heterostructure is partially relaxed by misfit dislocations forming matrixes in the interfaces' areas. Average distances between dislocation lines in individual interfaces were calculated from the condition of minimum elastic energy, taking into account interaction of stress fields and strains caused by lattice misfit, bending and presence of dislocations. Doping influence on the band gap width change (band gap narrowing effect) is insignificant in comparison to the influence of lattice mismatch and generated dislocations.

Branching of {110) twin boundaries in five-layered Ni-Mn-Ga bent single crystals

Two bending tests around two perpendicular axes were applied to 10M Ni-Mn-Ga single crystals with five-layered modulated structure. The crystal structure and microstructure evolution were examined using synchrotron radiation and electron backscatter diffraction, respectively. The bend stress results in pseudoelasto-plastic strain due to {101) twins tapering. A close examination of the microstructure reveals an additional pattern indicating microstructural changes in the form of {110) twins. As bending proceeds the {110) twins branch undergoing a significant twin refinement. Additionally, an elastic change of lattice parameters is confirmed yielding a higher total pseudoelastic strain. Unloading restores the initial twin configuration removing a large amount of the {110) twin boundaries, however, this process is followed by incomplete recovery since the samples do not retain its original shape entirely. The paper underlines the differences in mechanism for bending around two perpendicular axes explaining the amount of strain observed under pseudoelastic deformation. Additionally, the results are discussed with respect to minimization of elastic energy due to twin refinement and branching as well as mobility of the {101) and {110) twin boundaries.