Isotropic Deformation Research Articles

Optimization of the Magnetic Properties of Hot Deformed Nd-Fe-B Magnets

We investigated the effects of density of Nd-Fe-B isotropic MQ2 counterparts and hot deformation parameters, including temperature and time, on the microstructure, (0 0 L) texture, and magnetic properties of anisotropic MQ3 magnets. A higher MQ2 density yields a more homogeneous density distribution throughout the MQ2 magnet. When the MQ2 density reaches 7.58 g/cm 3 , the grain size is as fine as 70 nm. After hot deformation, the length of the platelet-like grains in the MQ3 magnet can be as small as 300 nm when deformed at 780 °C for 120 s. An increase in the deformation time to 240 s gives more uniform MQ3 grain sizes throughout the magnet, resulting in higher values of remanent induction B r = 13.8 kG, intrinsic coercivity i H c = 15.7 kOe, and maximum energy product (B·H) max = 46.5 MG·Oe. Inhomogeneity in (0 0 L) texture and microstructure along the MQ3 thickness axis is inevitable after hot deformation. However, if 12.5% of the upper and lower surfaces of the MQ3 magnet, deformed at 780 °C for 120 s, are polished, outstanding magnetic properties can be achieved: B r = 14.7 kG, i H c = 15.8 kOe, and (B·H) max = 51.5 MG·Oe.

Theory of heterogeneities in polymer networks

Following Edwards’ ideas we present main experimental results and the theory of random heterogeneities in neutral and charged networks obtained by instantaneous as well as chemical cross-linking of a melt and semidilute solution of linear chains. We study how random monomer density patterns in such networks change after swelling and stretching. We also describe main features of monomer density correlation functions, which determine the neutron and light scattering on spatial heterogeneities. We show that largescale cross-link density patterns written into network structure in the melt or semidilute state, can be revealed upon swelling by monitoring the monomer density patterns. We demonstrate that while isotropic deformations in good solvent yield magnified images of the original pattern, anisotropic deformations distort the image. We study how the monomer density image changes under different solvent conditions and discuss the difference between deformations of the density images in gels and ordinary solids. Possible tests of our predictions and some potential applications are proposed.

Anisotropy characteristics of coal shrinkage/swelling and its impact on coal permeability evolution with CO2 injection

AbstractCoal deforms with gas depletion during coal bed methane (CBM) primary recovery and CO2 storage for purposes of sequestration or enhanced recovery. The sorption‐induced deformation associated with gas adsorption and desorption has been experimentally and theoretically studied and is considered to be a unique feature of coal. The results of a series of experimental measurements of coal deformation with gas injection and depletion using helium, methane, CO2, and various mixtures of methane and CO2 revealed that the coal sorption‐induced deformation exhibits anisotropy, with larger deformation in direction perpendicular to bedding than those parallel to the bedding planes. Furthermore, the deformation of coal is reversible for helium and methane with injection/depletion, but not for CO2. Aiming at improving the understanding of enhanced CBM process, a coal deformation experimental study on methane displacement with CO2 was performed as well, where incremental swelling was measured for each CO2 displacement step. The total induced swelling strain (1.68%) with CO2 saturation was less than that for pure CO2 injection‐induced strain (1.87%) at same pressure of 5.9 MPa (850 psi). The measured deformation with CO2 injection was applied to a constant‐volume based analytical coal permeability model, which is capable of accommodating the anisotropic sorption‐induced deformation. Based on the modeling results, it was found that application of isotropic deformation in permeability model can overestimate the permeability loss compared to anisotropic deformation. This demonstrates that the anisotropic coal deformation should be considered to predict the permeability behavior of CBM as well as CO2 sequestration/ECBM projects. © 2016 Society of Chemical Industry and John Wiley & Sons, Ltd

Origin of reduced anisotropic deformation in hexagonal close packed Ti-Al alloy

The origin of low ductility in the hexagonal close-packed (hcp) metal system like Ti is high anisotropic critical resolved shear stress (CRSS) of different slip systems. To reduce the anisotropic deformation is a key for improving the mechanical performance. In the present study, Al alloying shows it can effectively weaken the anisotropic deformation of the hcp system. The relationship between Al alloying and anisotropic deformation was studied by X-ray diffraction, geometric phase analysis, and ab initio calculations. The atomic-scale mechanism behind the isotropic deformation of the Ti-Al alloy was shown that {10−10} prismatic planes were the preferred substitution planes for the Al atoms. The hardening trend of prismatic slip was faster compared to that of basal slip. A high CRSS rate and low CRSS starting point of prismatic slip contributed to the reduction in anisotropic deformation of the Ti-Al alloy.

Strain Rate Dependant Material Model for Orthotropic Metals

In manufacturing processes anisotropic metals are often exposed to the loading with high strain rates in the range from 102 s-1 to 106 s-1 (e.g. stamping, cold spraying and explosive forming). These types of loading often involve generation and propagation of shock waves within the material. The material behaviour under such a complex loading needs to be accurately modelled, in order to optimise the manufacturing process and achieve appropriate properties of the manufactured component.The presented research is related to development and validation of a thermodynamically consistent physically based constitutive model for metals under high rate loading. The model is capable of modelling damage, failure and formation and propagation of shock waves in anisotropic metals. The model has two main parts: the strength part which defines the material response to shear deformation and an equation of state (EOS) which defines the material response to isotropic volumetric deformation [1]. The constitutive model was implemented into the transient nonlinear finite element code DYNA3D [2] and our in house SPH code. Limited model validation was performed by simulating a number of high velocity material characterisation and validation impact tests.The new damage model was developed in the framework of configurational continuum mechanics and irreversible thermodynamics with internal state variables. The use of the multiplicative decomposition of deformation gradient makes the model applicable to arbitrary plastic and damage deformations.To account for the physical mechanisms of failure, the concept of thermally activated damage initially proposed by Tuller and Bucher [3], Klepaczko [4] was adopted as the basis for the new damage evolution model. This makes the proposed damage/failure model compatible with the Mechanical Threshold Strength (MTS) model Follansbee and Kocks [5], 1988; Chen and Gray [6] which was used to control evolution of flow stress during plastic deformation. In addition the constitutive model is coupled with a vector shock equation of state which allows for modelling of shock wave propagation in orthotropic the material.Parameters for the new constitutive model are typically derived on the basis of the tensile tests (performed over a range of temperatures and strain rates), plate impact tests and Taylor anvil tests.The model was applied to simulate explosively driven fragmentation, blast loading and cold spraying impacts.

Determination of α/β phase boundaries and mechanical characterization of Mg-Sc binary alloys

In this study, α(hcp)/β(bcc) phase boundaries of the Mg-Sc binary phase system were determined by Mg/Sc diffusion couple and conventional equilibrated alloy methods. It was confirmed that α/β phase boundaries exist in regions with a higher Mg content than those in reported diagram. The Mg-Sc alloy with a β single-phase showed tensile strength of 254MPa and elongation of 25.4%. The tensile strength of the Mg-Sc alloy increased with increasing volume fraction of the α phase (fα) and the alloy with a fα of 34% showed high UTS of 310MPa and elongation of 28.8%, properties which are better balanced than those of conventional Mg alloys. This improvement was suggested to be due to isotropic deformation of the β phase, small grain size and small c/a of the α phase.

SU-G-TeP1-01: A Simulation Study to Investigate Maximum Allowable Deformations of Implant Geometry Before Plan Objectives Are Violated in Prostate HDR Brachytherapy

Purpose: In prostate HDR brachytherapy dose distributions are highly sensitive to changes in prostate volume and catheter displacements. We investigate the maximum deformations in implant geometry before planning objectives are violated. Methods: A typical prostate Ir-192 HDR brachytherapy reference plan was calculated on the Oncentra planning system, which used CT images from a tissue equivalent prostate phantom (CIRS Model 053S) embedded inside a pelvis wax phantom. The prostate was deformed and catheters were displaced in simulations using a code written in MATLAB. For each deformation dose distributions were calculated, based on TG43 methods, using the MATLAB code. The calculations were validated through comparison with Oncentra calculations for the reference plan, and agreed within 0.12%SD and 0.3%SD for dose and volume, respectively. Isotropic prostate volume deformations of up to +34% to −27% relative to its original volume, and longitudinal catheter displacements of 7.5 mm in superior and inferior directions were simulated. Planning objectives were based on American Brachytherapy Society guidelines for prostate and urethra volumes. A plan violated the planning objectives when less than 90% of the prostate volume received the prescribed dose or higher (V100), or the urethral volume receiving 125% of prescribed dose or higher was more than 1 cc (U125). Lastly, the dose homogeneity index (DHI=1-V150/V100) was evaluated; a plan was considered sub-optimal when the DHI fell below 0.62. Results and Conclusion: Planning objectives were violated when the prostate expanded by 10.7±0.5% or contracted by 11.0±0.2%; objectives were also violated when catheters were displaced by 4.15±0.15 mm and 3.70±0.15 mm in the superior and inferior directions, respectively. The DHI changes did not affect the plan optimality, except in the case of prostate compression. In general, catheter displacements have a significantly larger impact on plan optimality than prostate volume changes.

Mathematical review on source-type diagrams

A source-type diagram is a visualization tool used to display earthquake sources, including double-couples, compensated linear vector dipoles, and isotropic deformation. Together with recent observations of non-double-couple events in a variety of tectonic settings, it is important to be able to recognize the source type intuitively from a representative diagram. Since previous works have proposed diagrams created using a range of projections, we review these diagrams in the framework of the moment tensor eigenvalue space. For further applications, we also provide complete formulas for conversion between moment tensor representation and the coordinate system of each diagram style. Using both a global catalog and synthetic data, we discuss differences between types of diagrams and the relative effectiveness of each.

Anisotropic separable free energy functions for elastic and non-elastic solids

In incompressible isotropic elasticity, the Valanis and Landel strain energy function has certain attractive features from both the mathematical and physical view points. This separable form of strain energy has been widely and successfully used in predicting isotropic elastic deformations. We prove that the Valanis–Landel hypothesis is part of a general form of the isotropic strain energy function. The Valanis–Landel form is extended to take anisotropy into account and used to construct constitutive equations for anisotropic problems including stress-softening Mullins materials. The anisotropic separable forms are expressed in terms of spectral invariants that have clear physical meanings. The elegance and attractive features of the extended form are demonstrated, and its simplicity in analysing anisotropic and stress-softening materials is expressed. The extended anisotropic separable form is able to predict, and compares well with, numerous experimental data available in the literature for different types of materials, such as soft tissues, magneto-sensitive materials and (stress-softening) Mullins materials. The simplicity in handling some constitutive inequalities is demonstrated. The work here sets an alternative direction in formulating anisotropic solids in the sense that it does not explicitly use the standard classical invariants (or their variants) in the governing equations.

Effects of Aluminum on Hydrogen Solubility and Diffusion in Deformed Fe-Mn Alloys

We discuss hydrogen diffusion and solubility in aluminum alloyed Fe-Mn alloys. The systems of interest are subjected to tetragonal and isotropic deformations. Based onab initiomodelling, we calculate solution energies and then employ Oriani’s theory which reflects the influence of Al alloying via trap site diffusion. This local equilibrium model is complemented by qualitative considerations of Einstein diffusion. Therefore, we apply the climbing image nudged elastic band method to compute the minimum energy paths and energy barriers for hydrogen diffusion. Both for diffusivity and solubility of hydrogen, we find that the influence of the substitutional Al atom has both local chemical and nonlocal volumetric contributions.

All-loop anomalous dimensions in integrable λ-deformed σ-models

We calculate the all-loop anomalous dimensions of current operators in λ-deformed σ-models. For the isotropic integrable deformation and for a semi-simple group G we compute the anomalous dimensions using two different methods. In the first we use the all-loop effective action and in the second we employ perturbation theory along with the Callan–Symanzik equation and in conjunction with a duality-type symmetry shared by these models. Furthermore, using CFT techniques we compute the all-loop anomalous dimension of bilinear currents for the isotropic deformation case and a general G. Finally we work out the anomalous dimension matrix for the cases of anisotropic SU(2) and the two couplings, corresponding to the symmetric coset G/H and a subgroup H, splitting of a group G.

Cross-Linking Patterns and Their Images in Swollen and Deformed Gels

Using the theory of elasticity of polymer gels we show that large-scale cross-link density patterns written into the structure of the network in the melt state, can be revealed upon swelling by monitoring the monomer density patterns. We find that while isotropic deformations in good solvent yield magnified images of the original pattern, anisotropic deformations distort the image (both types of deformation yield affinely stretched images in $\theta$ solvents). We show that in ordinary solids with spatially inhomogeneous profile of the shear modulus, isotropic stretching leads to distorted density image of this profile under isotropic deformation. Using simple physical arguments we demonstrate that the different response to isotropic stretching stems from fundamental differences between the theory of elasticity of solids and that of gels. Possible tests of our predictions and some potential applications are discussed.

Understanding the effects of inter-particle contact friction on the elastic moduli of granular materials

Understanding the mechanical stiffness of closely packed, dense granular systems is of interest in many fields, such as soil mechanics, material science and physics. The main difficulty arises due to discreteness and disorder in granular materials at the microscopic scale which requires a multi-scale approach. The Discrete Element Method (DEM) is a powerful tool to inspect the influence of the microscopic contact properties of its individual constituents on the bulk behavior of granular assemblies. In this study, the isotropic deformation mode of polydisperse packings of frictionless and frictional spheres are modeled by using DEM, to investigate the effective stiffness of the granular assembly. At various volume fractions, for every sample, we determine the stress and fabric incremental response that result from the application of strain-probes. As we are interested first in the reversible, elastic response, the amplitude of the applied perturbations has to be small enough to avoid opening and closing of too many contacts, which would lead to irreversible rearrangements in the sample. Counterintuitively, with increasing inter-particle contact friction, the bulk modulus decreases systematically with the coefficient of friction for samples with the same volume fraction. We explain this by the difference in microstructure (isotropic fabric) the samples get when compressed to the same density.

Constitutive modeling for polymer hydrogels: A new perspective and applications to anisotropic hydrogels in free swelling

Abstract The paper presents a modified theoretical framework for isotropic and anisotropic deformation of hydrogels in both steady and transient states. By applying the kinematic constraint between the mechanical and chemical fields in the second law of thermodynamics, a unified constitutive equation could be obtained, which is capable to describe the isotropic or anisotropic deformation of hydrogels at the same time. Also, the diffusion equation for solvent molecules is modified based on the constraint and the Lagrange multiplier used to force the constraint is avoided being introduced in the free energy function. The theoretical framework is specialized to describe the steady swelling behaviors of anisotropic hydrogels and the corresponding solution properties for the equations of free swelling are discussed. The anisotropic-swelling behaviors for several kinds of configurations are investigated, including hydrogel blocks and a cylindrical tube with one or two families of reinforced fibers. We find that the swelling of anisotropic hydrogels shows rich behaviors even for the simple configurations in free states, such as selective expansion/shrinkage and anomalous residual stress. An agreement between the results obtained by the constitutive equations and experiments for the transversally isotropic swelling demonstrates the effectiveness of the constitutive model.

Piezoelectric effect in chemical vapour deposition-grown atomic-monolayer triangular molybdenum disulfide piezotronics.

High-performance piezoelectricity in monolayer semiconducting transition metal dichalcogenides is highly desirable for the development of nanosensors, piezotronics and photo-piezotransistors. Here we report the experimental study of the theoretically predicted piezoelectric effect in triangle monolayer MoS2 devices under isotropic mechanical deformation. The experimental observation indicates that the conductivity of MoS2 devices can be actively modulated by the piezoelectric charge polarization-induced built-in electric field under strain variation. These polarization charges alter the Schottky barrier height on both contacts, resulting in a barrier height increase with increasing compressive strain and decrease with increasing tensile strain. The underlying mechanism of strain-induced in-plane charge polarization is proposed and discussed using energy band diagrams. In addition, a new type of MoS2 strain/force sensor built using a monolayer MoS2 triangle is also demonstrated. Our results provide evidence for strain-gating monolayer MoS2 piezotronics, a promising avenue for achieving augmented functionalities in next-generation electronic and mechanical–electronic nanodevices.

Anisotropic Time-Dependent Modeling of Tunnel Excavation in Squeezing Ground

Most of the viscoplastic models used to describe the constitutive behavior of squeezing grounds assume isotropic deformation. However, it is commonly observed that squeezing behavior is characterized not only by large time-dependent but also often by anisotropic deformations. This study uses a semi-empirical approach based on the analysis of convergence measurements and a numerical model that takes into account the time-dependent and the anisotropic response of the rock mass to investigate the squeezing behavior of the Saint-Martin-la-Porte access gallery, excavated within the Lyon–Turin railway project. We first show how the semi-empirical convergence law of Sulem et al. (Int J Rock Mech Min Sci Geomech Abstr 24(3):155–164, 1987a; Int J Rock Mech Min Sci Geomech Abstr 24(3):145–154, 1987b) can be extended to anisotropic tunnel closure by considering an elliptical deformation of the rock mass and by fitting the convergence data along the principal axes of deformation. A new anisotropic time-dependent constitutive model is then proposed. This model includes ubiquitous joints of specific orientation embedded in an isotropic viscoplastic medium. This model is implemented in FLAC3D and numerical simulations are performed to back-analyze the anisotropic closure of the Saint-Martin-la-Porte access gallery. An efficient two-step procedure for calibrating the model parameters is proposed: the parameters of the isotropic solid matrix are first estimated by performing axisymmetric numerical simulations. The parameters of the ubiquitous joints are then calibrated by performing 3D computations. It is shown that the numerical results reproduce very well the convergence measurements of the studied sections of Saint-Martin-la-Porte gallery.

Ferrogels cross-linked by magnetic nanoparticles—Deformation mechanisms in two and three dimensions studied by means of computer simulations

Ferrogels, i.e., magnetic nanoparticles immersed into a swollen polymer gel, are materials which combine the magnetic properties of a ferrofluid with the elastic properties of a hydrogel. This makes it possible to control their shape and elasticity by means of external magnetic fields. Magnetic gels are therefore considered for medical and technical applications such as actuation and drug delivery systems. In this paper, we focus on a special class of magnetic gels in which the magnetic nanoparticles act as the cross-linkers of the polymer network. Experimentally this is achieved by functionalizing the nanoparticles' surface such that the polymers can covalently bind to it. In this way, a coupling between the orientation of the nanoparticles and the polymer matrix is created. When an external magnetic field is applied to such a system, the magnetic particles align to the field, and due to the coupling between the nanoparticles' orientation and the gel matrix, a strain is exerted on the polymers. This leads to a shrinking of the gel as a whole. In this paper, we explain how to extend a two-dimensional simulation model for such a system (Soft Matter 8 (2012) 9923 [18]) to three dimensions. We will show that, while an isotropic deformation was observed in the two-dimensional case, in three dimensions the gel shrinks anisotropically. This is due to the fact that orientational fluctuations of the nanoparticles are hindered in the direction perpendicular to an external magnetic field.

Radó-Kneser-Choquet theorem

We present a new approach to the celebrated theorem of Radó–Kneser–Choquet (RKC) on univalence of planar harmonic mappings. The novelty lies in establishing a continuous path (isotopy) from the given harmonic map to a conformal one. Along this path the mappings retain positive Jacobian determinant by virtue of so-called Minimum Principle. These ideas extend to nonlinear uncoupled systems of partial differential equations, as in Iwaniec, Koski and Onninen [‘Isotropic p-harmonic systems in 2D, Jacobian estimates and univalent solutions’, Rev. Mat. Iberoam, to appear]. Unfortunately, details of such digression would lead us too far afield. Nonetheless, one gains (in particular) the RKC-Theorem for the isotropic p-harmonic deformations.

The outliers of a deformed Wigner matrix

We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1–47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (1013) 107–133; Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Preprint]. We also show that, under suitable conditions, outliers may be strongly correlated even if they are far from each other. Our proof relies on the isotropic local semicircle law established in [The isotropic semicircle law and deformation of Wigner matrices. Preprint]. The main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.

Mechanical Response of the Cross-Linked Pericardial Tissue

The significance of studying of human pericardial mechanics arises from a wide range of possible applications. For instance, tissue engineers study potential use of a human pericardium in a design of homograft valve bioprosthesis. In cooperation with the Institute of the Physiology of Academy of Sciences of the Czech Republic, human pericardium tissue has been modified by different cross-linking agents. The function and performance of the potential future bioprostheses depends closely on sources material. Thus characterization of the mechanical properties of the human pericardial tissue is an important problem. Several samples of pericardial tissue as well as cross-linked tissue underwent uniaxial loading and unloading. Their nonlinear mechanical response was modeled using hyperelastic theory, assuming incompressible isotropic deformation. Considering future possible tissue application, mechanical properties of the material under stress corresponding to the loading in aortic valve were evaluated.