Elastic Curves Research Articles

Ultrasonic characterization of localized passive elastic properties of human pennate muscle with a single-probe setup

Quantitative evaluation of passive elastic properties of an individual skeletal muscle in vivo is among the major challenges of biomechanics, and its clinical application is severely limited. By combining shear-wave elastography (SWE) and B-mode imaging techniques, this study develops a novel non-invasive method to measure the local elastic modulus-fascicle strain curve of human pennation muscle during passive stretching using a single probe. Physiologically meaningful parameters are estimated and compared in subjects with different ages or pathological conditions. The in vivo experimental group comprised 12 healthy subjects (four children, four adults, and four seniors) and eight patients (four suffering from pseudohypertrophy, four from atrophy). Their gastrocnemius muscles were passively stretched using an ankle joint motion instrument. Local elastic moduli of the muscle were measured using SWE imaging frames and a built-in ‘F-ROI’ tool. The corresponding fascicle strains were simultaneously obtained using B-mode imaging frames and a gradient Radon transform. Three parameters (η, μ, G0) were estimated from a normalized elastic modulus-strain curve using the Gauss–Newton method. The measured elastic modulus-strain curves all agreed with models of the estimated parameters (0.910 < R2 < 0.999) and presented different patterns among normal and diseased subjects. η values were lower for pseudohypertrophies (1.93 ± 0.12), but higher for atrophies (63.40 ± 98.89), compared with normal ones (6.02 ± 2.53). In addition, μ values were higher for pseudohypertrophies (22.65 ± 16.40), but lower for atrophies (0.28 ± 0.41), compared with normal ones (1.07 ± 1.22). The proposed method may provide novel insight into the biomechanics of pennate muscle and has the potential to serve for clinical musculoskeletal medical diagnosis, as the single-probe scanning setup is broadly accepted.

First integrals for elastic curves: twisting instabilities of helices

We put forward a variational framework suitable for the study of curves whose energies depend on their bend and twist degrees of freedom. By employing the material curvatures to describe such elastic deformation modes, we derive the equilibrium equations representing the balance of forces and torques on the curve. The conservation laws of the force and torque on the curve, stemming from the Euclidean invariance of the energy, allow us to obtain first integrals of the equilibrium equations. To illustrate this framework, we apply it to determine the first integrals for isotropic and anisotropic Kirchhoff elastic rods, whose energies are quadratic in the material curvatures. We use them to analyze perturbatively the deformations of helices resulting from their twisting. We examine three kinds of twisting instabilities on unstretchable helices, characterized by their wavenumbers, depending on whether their boundaries are fixed, displaced along the radial direction or orthogonally to it. We also analyze perturbatively the effect of the bending anisotropy on the deformed states, which introduces a coupling between deformation modes with different wavenumbers.

Stress in ordered systems: Ginzburg-Landau-type density field theory

We present a theoretical method for deriving the stress tensor and elastic response of ordered systems within a Ginzburg-Landau type density field theory in the linear regime. This is based on spatially coarse graining the microscopic stress which is determined by the variation of a free energy with respect to mass displacements. We find simple expressions for the stress tensor for phase field crystal (PFC) models for different crystal symmetries in two and three dimensions. Using tetradic product sums of reciprocal lattice vectors, we calculate elastic constants and show that they are directly related to the symmetries of the reciprocal lattices. We also show that except for bcc lattices, there are regions of model parameters for which the elastic response is isotropic. The predicted elastic stress-strain curves are verified by numerical strain-controlled bulk and shear deformations. Since the method is independent of a reference state, it extends also to defected crystals. We exemplify this by considering an edge and screw dislocation in the simple cubic lattice.

Tensile creep mechanical behavior of periodontal ligament: A hyper-viscoelastic constitutive model

ObjectiveIn orthodontic treatment, the biomechanical response of periodontal ligament (PDL) induces tooth movement. Coupling modeling of PDL can effectively reflect its biomechanical response. The nonlinear creep mechanical behavior of PDL was studied by uniaxial tensile creep test and a new hyper-viscoelastic constitutive model. Two coupling modeling methods with limitations were excluded. MethodsPDL specimens were prepared from the central incisors of pig mandible. The theoretical step function was replaced by static loading with a total loading time of 1 s. The creep loading with the constant stresses of 0.05, 0.1, and 0.15 MPa was selected and kept unchanged for 1000 s. The instantaneous hyperelastic mechanical behavior and time-dependent nonlinear viscoelastic mechanical behavior of PDL were characterized by coupled instantaneous third-order Ogden hyperelastic and time-dependent nonlinear creep models. ResultsThe results showed that the instantaneous elastic curve of PDL increases in the form of hyperelastic index. The creep strain and creep compliance curves increase rapidly before 200s, and then increase slowly in steady state. The creep strain increased with an increase in the constant stress; conversely, the creep compliance decreased with an increase in the constant stress. The results showed that the experimental data were highly consistent with the hyper-viscoelastic constitutive model (R2>0.97). SignificanceWe normalize the framework of hyper-viscoelastic coupling modeling (Instantaneous hyperelastic model + time-dependent nonlinear viscoelastic model). Which can be extended to other nonlinear viscoelastic biomaterials.

МОДЕЛИРОВАНИЕ ПРИРОДОПОДОБНЫХ КРИВЫХ

A physical spline is called an elastic rod the cross- section dimensions of which are rather small as compared with the length and radius of its axis curvature. Such a rod when passing through specified points obtains in natural way a nature-like shape characterized with minimum energy of inner stresses and minimum mean curvature. A search for the equation of elastic line is a difficult mathematical problem having no elementary solution. &#x0D; The work purpose: the development of the experimental-rated procedure for modeling a nature-like elastic curve passing through complanar points specified in advance. &#x0D; The investigation methods: methods of piece-cubic interpolation based on the application of polynomial splines and compound curves specified by parametric equations. In the paper there are considered polynomial and parametric methods of the geometric modeling of the physical spline passing through the points specified in advance. The elastic line of the physical spline is obtained experimentally. &#x0D; The investigation results: it is shown that unlike a polynomial model a parametrized model on the basis of Fergusson curve gives high accuracy of approximation if in basic points there are specified tangents to the elastic line of the physical spline with large deflections. &#x0D; Novelty: there is offered a simplified method for the computation of factors of an approximating spline allowing the substitution of the 2n system of nonlinear equations (smoothness conditions) by the successive solution of n systems of two equations. &#x0D; Conclusions: for the modeling of nature-like curves with large deflections there is offered the application of Fergusson cubic spline passing through specified points and touching the specified straight lines in these points. The error of the modeling of the natural elastic line with free ends at n=5 does not exceed 0.4%.

Experimental and Numerical Investigation of PA66 Fabrics Coated/Uncoated PVC by Biaxial Tensile Tests

Automobile airbags have become mandatory safety components in many countries. In this study, the PA66 (polyamide 66) airbag fabric was tested with and without a surface coating of PVC (polyvinyl chloride). Cyclic loading and unloading experimental conditions were defined according to the loading characteristics of the airbag deployment process. To determine mechanical properties such as elastic modulus and stress-strain curves for the two materials, biaxial loading/unloading cyclic tensile tests were carried out using a DIC (Digital Inspection Camera) to monitor the field strain process. Quadratic polynomial regression was used to fit material constitutive models of each loading and unloading process to the test data. The material subroutine VUMAT was encoded and applied to the simulation of the biaxial tensile testing of the fabrics. There was good agreement between the experimental and simulation results, which verifies that the quadratic polynomial constitutive material model is suitable for the study of coated and uncoated PA66 automobile airbag fabric. Hence, the material behaviour of automobile airbag fabrics can be modelled accurately.

Small-strain behaviour of unsaturated silty clay: experiments and model interpretation

The experimental data dealing with the so-called small strain stiffness of soils are indispensable in developing and calibrating advanced numerical models. A literature review revealed a lack of such data for unsaturated soils. In this study, the laboratory measurements of very small and small strain stiffness are carried out in a double-walled triaxial cell using bender elements and LVDT transducers. The tested soil is a reconstituted silty clay, representing a typical soil encountered in engineering practice. A so-called effective stress for unsaturated soil is used in interpreting and numerically simulating the effects of suction magnitude, net stress magnitude and suction history on the initial elastic shear modulus, and on its decrease with increasing strain. The increase of both net stress and suction leads to increase in stiffness. This is observed by both elastic shear modulus and shear modulus reduction curve measurements. The “overconsolidation” by suction leads to the increase of stiffness, too. The experimental data are used for calibrating constitutive parameters and for assessing the capabilities of an advanced hypoplastic model for unsaturated soils. The model predictions are consistent with the laboratory measurements and the model can capture the increase of stiffness with respect to the increase of suction and net stress reasonably well. Moreover, the model is able to predict the increase of stiffness due to suction “overconsolidation”.

Shape programming of a magnetic elastica

We consider a cantilever beam which possesses a possibly non-uniform permanent magnetization, and whose shape is controlled by an applied magnetic field. We model the beam as a plane elastic curve and we suppose that the magnetic field acts upon the beam by means of a distributed couple that pulls the magnetization towards its direction. Given a list of target shapes, we look for a design of the magnetization profile and for a list of controls such that the shapes assumed by the beam when acted upon by the controls are as close as possible to the targets, in an averaged sense. To this effect, we formulate and solve an optimal design and control problem leading to the minimization of a functional which we study by both direct and indirect methods. In particular, we prove that minimizers exist, solve the associated Lagrange-multiplier formulation (besides non-generic cases), and are unique at least for sufficiently low intensities of the controlling magnetic fields. To achieve the latter result, we use two nested fixed-point arguments relying on the Lagrange-multiplier formulation of the problem, a method which also suggests a numerical scheme. Various relevant open question are also discussed.

Simulation of constrained elastic curves and application to a conical sheet indentation problem

Abstract We consider variational problems that model the bending behavior of curves that are constrained to belong to given hypersurfaces. Finite element discretizations of corresponding functionals are justified rigorously via $\varGamma $-convergence. The stability of semi-implicit discretizations of gradient flows is investigated, which provide a practical method to determine stationary configurations. A particular application of the considered models arises in the description of conical sheet deformations.

Rotation capacity assessment in developed non prismatic flanged reinforced concrete Tee beams

Flanged reinforced concrete beam of Tee section is an optimal member and has special importance in civil engineering for its efficiency in structural fields to maximize strength capacity. The flexural ductility in reinforced concrete beams is a complex nonlinear phenomenon dominated by plastic rotation capacity. This paper introduces and assesses rotation capacity improving in Tee- section reinforced concrete beams of a proposed tapered flange configuration. The special section area distribution in proposed reinforced concrete Tee beams intends to introduce smart configuration regards tapering benefits besides flange section benefits to upgrade the rotational capacity besides other structural characteristics. Twelve reinforced concrete beams have been fabricated and tested; which are classified into nine specimens of non-prismatic flanges, one specimen of rectangular section, two specimens of T- section of upper and lower tapered flange thickness of considered nine sections. The main experimented parameters are non- prismatic flange domain, and web thickness. The results confirmed the positive effectiveness of considered in plane non-prismatic flanges within T-section upon elastic and ultimate deflection curves, plastic rotational capacity upgrading and plastic hinge development without any significant effect upon ultimate load capacity and flexural stiffness.

Design of a Series of Metallic BxNx+1 with Tunable Mechanical Properties.

Here a series of sp2-sp3 BxNx+1 (x = 1, 2, 3, 4, 5, 6) structures was constructed. These structures can be viewed as diamond-like BN blocks connected by single N-N bonds. Elastic constants and phonon dispersion curves confirm that all of the proposed structures are mechanically and dynamically stable. These structures all possess metallicity originating from the conductive channels formed by sp2-hybridized N atoms and adjacent sp3-hybridized B and N atoms. These structures exhibit tunable mechanical properties with a regular change in the sp2/sp3 ratio. The theoretical Vickers hardness increases and the ductility decreases as the number of diamond-like BN blocks increases, gradually approaching those of c-BN. Moreover, the convex hull at ambient pressure and 50 GPa indicates that high pressure is beneficial in the synthesis of these B-N phases. The simulated X-ray diffraction patterns of these structures were also calculated to provide more information for further experiments.

Elastic magnetic curves of ferromagnetic and superparamagnetic models

In this paper, new energy functionals are described via the variational approach method. By doing this, a new class of elastic curves is described in the three‐dimensional ordinary space. An alternative method is also investigated to compute critical points of the bending energy functionals acting on a class of magnetic curves. Then, classifications of these critical curves are presented as elastic magnetic curves of the Frenet–Serret vectors family. Moreover, the dynamics of the elastic magnetic curves of ferromagnetic and superparamagnetic models are discussed with their numerical and analytical analysis.

Hyperelastic Curves in 3-dimensional Lightlike Cone

We study hyperelastic curves known as a generalization of elastic curves in $3-$dimensional lightlike cone which is a degenerate hypersurface in Minkowski $4-$space as critical points of the cone curvature energy functional constructed with the $r-$th power of the cone curvature depending on the given boundary conditions for the natural number $r \geq 2$. We derive the Euler-Lagrange equations for the critical points of this functional that is namely the hyperelastic curves and solve completely the Euler-Lagrange equations by quadratures. Then, we construct Killing vector fields along the hyperelastic curves. Lastly, we give explicitly the hyperelastic curves by integral according to the selected cylindrical coordinate systems in $3-$dimensional lightlike cone using these Killing vector fields.

Elastic magnetic curves of ferromagnetic and superparamagnetic models on the surface

We are interested in defining new energy functionals and solving them by using the variational approach method and Darboux equations. That is, we aim to define a new class of elastic curves on the regular surface [Formula: see text]. We further improve an alternative method to find critical points of the bending energy functionals acting on a class of magnetic curves on [Formula: see text]. As a result, we classify these critical curves as elastic magnetic curves of the Darboux vector family.

Quantitative Estimation of Passive Elastic Properties of Individual Skeletal Muscle in Vivo Using Normalized Elastic Modulus-Length Curve.

Precise characterization of the passive elastic properties of human skeletal muscle could provide valuable information for functional assessment and medical diagnosis. Using normalized elastic modulus-length curve based on a piecewise exponential model, a non-invasive ultrasonography (US) method was developed for estimating three physiologically meaningful parameters, termed as passive-elastic coefficient k, slack length l0 and slack elastic modulus G0, to quantify the passive elastic properties of human gastrocnemius (GM) muscle, and correlate different findings with healthy and atrophic subjects. Dynamic B-mode US and shear wave elastography (SWE) videos of right GM muscles were performed in 32 healthy subjects and 18 atrophic patients with their ankle angles from 40° plantarflexion to 30° dorsiflexion by an isokinetic dynamometer. During muscle passive stretching induced by ankle rotation, consecutive muscle lengths were measured by automatically tracking the distal muscle-tendon junction positions in B-mode imaging frames using Hough transform and an active contour method. The corresponding elastic moduli were obtained using SWE imaging frames. The Gauss-Newton algorithm was used to estimate the parameters k, l0 and G0 from normalized elastic modulus-length curve. In-vivo experimental results showed that the measured muscle elastic modulus-length data fitted well to the proposed model with three estimated parameters for all subjects (0.942 < R2 < 0.997). The passive-elastic coefficient k was significantly higher for atrophic subjects compared with normal subjects (105.5 ± 45.3 versus 48.4 ± 16.0, p < 0.001). The proposed parameters allow further characterization of muscle essential mechanical properties and have a potential to become effective indexes for muscular disease diagnosis.

A new approach to the bienergy and biangle of a moving particle lying in a surface of lorentzian space

Abstract In this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

Conjugate-beam analogy for inflexed laminates

A conjugate beam analogy is here proposed to evaluate the shear coupling in three-layered laminates formed by two external elastic plies bonded by a compliant interlayer. Based on the formal analogy of the governing equations, the method allows the evaluation of the shear coupling in the external plies by studying the response of a monolithic conjugate beam under a tensile constant axial force and a transversal load with the same shape of that acting on the parent laminate. Remarkably, the boundary conditions for the conjugate beam depend not only upon the external constraints of the laminate, but also upon possible internal constraints that limit the relative sliding of the plies at the end points. The axial force at a point of each ply corresponds to the displacement of the elastic curve of the conjugate beam at the same point, while the flux of shear stress from the interlayer is related to the corresponding slope. The static state of the laminate can finally be determined by analyzing a simple structure under an effective transversal load, which accounts for the contribution of the shear coupling. Since, in most cases, the deformation of the conjugate beam can be intuitively guessed, the proposed analogy permits a synthetic view of the interaction between the various layers of the laminate. This can be very useful especially in preliminary design.

Prediction of three-dimensional B3N5 with one-dimensional metallicity

In this paper, a layer-like sp2-/sp3-hybridized orthorhombic phase of B3N5, named as oA-B3N5, is theoretically predicted to be stable at ambient pressure by first-principles calculations. The mechanical and dynamical stability has been confirmed by the elastic constants and phonon frequency curves calculations. The analyses of electronic properties show the metallicity of oA-B3N5 originates from the delocalized 2p electrons of sp2-hybridized N atoms, forming several independent 1D conductive channels, giving the 1D metallicity in 3D framework. Furthermore, the incompressibility of oA-B3N5 exhibits highly anisotropic, and higher axial incompressibility along the N-N bonds direction comparing to the predicted semiconductor C2221-B3N5.

The dependence of the annealed cold-packed hearth mass elasticity from the degree of its compaction

The impact of the cold-packed hearth mass density on the annealed mass dynamic elastic modulus is investigated. It is shown that the analysis of the elastic modulus curves allows to determine the area of the hearth mass optimal compaction. In this area, the annealed hearth mass has a level of mechanical properties comparable to hearth blocks.

Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space

UDC 515.1 We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space and investigate its relationship with the energy of the same curve in Bishop vector fields in . Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space .