Green Tensor Research Articles

Strain Response in the Facet Joint Capsule During Physiological Joint Rotation and Translation Following a Simulated Impact Exposure: An In Vitro Porcine Model.

Low back pain (LBP) is frequently reported following rear impact collisions. Knowledge of how the facet joint capsule (FJC) mechanically behaves before and after rear impact collisions may help explain LBP development despite negative radiographic evidence of gross tissue failure. This study quantified the Green strain tensor in the facet joint capsule during rotation and translation range-of-motion tests completed before and following an in vitro simulation of a rear impact collision. Eight FSUs (4 C3-C4, 4 C5-C6) were tested. Following a preload test, FSUs were flexed and extended at 0.5 deg/s until an ±8 N·m moment was achieved. Anterior and posterior joint translation was then applied at 0.2 mm/s until a target ±400 N shear load was imposed. Markers were drawn on the facet capsule surface and their coordinates were tracked during pre- and postimpact range-of-motion tests. Strain was defined as the change in point configuration relative to the determined neutral joint posture. There were no significant differences (p > 0.05) observed in all calculated FJC strain components in rotation and translation before and after the simulated impact. Our results suggest that LBP development resulting from the initiation of strain-induced mechanoreceptors and nociceptors with the facet joint capsule is unlikely following a severe rear impact collision within the boundaries of physiological joint motion.

A macroscopic approach for stress-driven anisotropic growth in bioengineered soft tissues

The simulation of growth processes within soft biological tissues is of utmost importance for many applications in the medical sector. Within this contribution, we propose a new macroscopic approach for modelling stress-driven volumetric growth occurring in soft tissues. Instead of using the standard approach of a-priori defining the structure of the growth tensor, we postulate the existence of a general growth potential. Such a potential describes all eligible homeostatic stress states that can ultimately be reached as a result of the growth process. Making use of well-established methods from visco-plasticity, the evolution of the growth-related right Cauchy–Green tensor is subsequently defined as a time-dependent associative evolution law with respect to the introduced potential. This approach naturally leads to a formulation that is able to cover both, isotropic and anisotropic growth-related changes in geometry. It furthermore allows the model to flexibly adapt to changing boundary and loading conditions. Besides the theoretical development, we also describe the algorithmic implementation and furthermore compare the newly derived model with a standard formulation of isotropic growth.

Green Tensor Analysis of Lattice Resonances in Periodic Arrays of Nanoparticles

When arranged in a periodic geometry, arrays of metallic nanostructures are capable of supporting collective modes known as lattice resonances. These modes, which originate from the coherent multiple scattering between the elements of the array, give rise to very strong and spectrally narrow optical responses. Here, we show that, thanks to their collective nature, the lattice resonances of a periodic array of metallic nanoparticles can mediate an efficient long-range coupling between dipole emitters placed near the array. Specifically, using a coupled dipole approach, we calculate the Green tensor of the array connecting two points and analyze its spectral and spatial characteristics. This quantity represents the electromagnetic field produced by the array at a given position when excited by a unit dipole emitter located at another one. We find that, when a lattice resonance is excited, the Green tensor is significantly larger and decays more slowly with distance than the Green tensor of vacuum. Therefore, in addition to advancing the fundamental understanding of lattice resonances, our results show that periodic arrays of nanostructures are capable of enhancing the long-range coupling between collections of dipole emitters, which makes them a promising platform for applications such as nanoscale energy transfer and quantum information processing.

Plane Boundary Value Problems of Thermoelasticity and Their Generalized Solutions

Nonstationary boundary value problems of uncoupled thermoelasticity are considered. A method of boundary integral equations in the initial space-time has been developed for solving boundary value problems of thermoelasticity by plane deformation by use general functions method (GFM). The generalized solutions of boundary value problems are constructed in the space of generalized vector functions. Fundamental Green tensor, stress tensor and their antiderivatives over time are used for construction their regular integral representations. By use analogue of Gauss formula for fundamental stress the singular boundary integral equations are constructed to determine the unknown boundary functions. The resulting formulas have important engineering application. They make possible to determine the thermal stressed state of the medium by the boundary values of stresses, displacements, temperature and heat flux, without solving singular boundary integral equations. Because for real engineering problems these physical characteristics can be experimentally measured at the boundary. Moreover, the formulas allow to calculate the influence of each of these characteristics of the process on its stressstrain state. The last one is very important in designing structures made of thermoelastic materials.

An inverted Rivlin-type universal relation for simple shear

The pure shear stress formulation for incompressible isotropic hyperelastic materials is generalised to include an in-plane normal traction. A new inverted Rivlin-type universal relation for simple shear is then obtained in terms of the left Cauchy–Green deformation tensor, the counterpart for the stress formulation of the classical Rivlin relation for simple shear that follows from a displacement formulation of the problem. Bounds are then obtained for the corresponding amount of shear and the triaxial stretches in terms of the applied stress. The general results obtained are then illustrated for the neo-Hookean material, for which it is shown that a compressive normal stress in conjunction with a shearing traction results in a non-monotonic relationship between the amount of shear and the shear stress, with a finite amount of shear predicted in the limit of infinite shear stress. However the relative displacement of the top and bottom faces of the sheared specimen is shown to be a monotonic function of the shear stress, with an infinite relative displacement predicted for an infinite shear stress.

Fourier Series Approximation of Tensor Green’s Function in Biaxial Anisotropic Media

A new approximation algorithm is developed to compute the electromagnetic (EM) tensor Green’s function in the biaxial anisotropic media based on the Fourier series expansion. First, a rectangular region is chosen as the computation region, in which the EM field can be expressed as a 2-D Fourier series. The Fourier coefficients can be regarded as the EM field in discrete wavenumber domain. We further obtain the EM solution in the spatial domain in the form of Fourier series. Then, we use the finite terms of the Fourier series to approximate the EM field for enhancement of computation efficiency. Because the new method avoids the numerical integration in the infinite wavenumber domain, it is more convenient to implement than other methods based on integral transform. Finally, the spatial distribution of the tensor Green’s function for a transversely isotropic medium is presented to verify the proposed approach. The agreements between the results obtained by the present method and the analytic solution demonstrate the validity and the robustness of our algorithm.

MODELING OF A LUNENBURG LENS ANTENNA WITH THE INNER METALIC SPHERE

One of the statutory principle of fifth generation mobile communications is the adaptive antenna system beam-forming technology that allow the radiation to be oriented to specific subscriber. Promising solution of the task can be using the passive antennas, which provide the direct beam-forming with no active component in the circuit, thus is low-cost. Lunenburg lens antenna is perspective solution of the task described above and allows the direct radiation to be formed in wide-angle range and keeps the required performance. Besides, it's slightly compact, so the antenna can be installed on various of city infrastructure elements, and the spherical form of device is advantage when the installation is performed on to the vehicles surface (train, car, aircraft, etc.) In the paper, the metallic sphere installation inside the spherical Lunenburg lens as a mean of attachment to constructions is described. The analysis results of the inner element impact to the antenna characteristics of the Lunenburg lens are provided. Analytical system model with the spherical conductor inside the antenna, which can also be used to calculate the presence of metallic particles impact to the average permittivity of the lens, is shown. The diffraction task is addressed by means of the Green's tensor functions application. The results of numerical modeling of the described antenna system using the Ansys Electronics Desktop (HFSS Design) are presented. Analytical and numerical calculations show that the limited-sized metallic sphere presence inside the lens does not significantly affect the focusing property of the device. The acquired formulas can be used in the future work to find the solution of the metallic element optimal radius approximation, using the neural networks as one of the mean to do it.

The influence of a metal rod inside a spherical Luneberg lens on its characteristics

The paper discusses the option of installing a load-bearing element inside a spherical Luneberg lens (LL). The presence of a metal element inside the LL will increase the structural rigidity of the spherical antenna and thereby expand the scope of use of the LL at various mobile communications and radar facilities with severe operating conditions. The influence of a metal rod inside the main spherical structure on the radiation pattern in two main planes in the linear polarization mode and on the diffraction pattern of the radiation field is estimated. The results of the analysis of the directional characteristics and the diffraction pattern of the LL radiation field were obtained using modeling in the Ansys Electronics Desktop (HFSS Design) working environment, as well as mathematical modeling using the Green tensor function method. At the same time, the obtained mathematical relations can be used in the future to solve the problem of finding the optimal size and position of a metal element in the sphere of LL.

On asymptotic rigidity and continuity problems in nonlinear elasticity on manifolds and hypersurfaces

Intrinsic nonlinear elasticity deals with the deformations of elastic bodies as isometric immersions of Riemannian manifolds into the Euclidean spaces (see Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties of elastic bodies for the intrinsic approach to nonlinear elasticity. We first establish a geometric rigidity estimate for mappings from Riemannian manifolds to spheres (in the spirit of Friesecke–James–Müller [23]), which is the first result of this type for the non-Euclidean case as far as we know. Then we prove the asymptotic rigidity of elastic membranes under suitable geometric conditions. Finally, we provide a simplified geometric proof of the continuous dependence of deformations of elastic bodies on the Cauchy–Green tensors and second fundamental forms, which extends the Ciarlet–Mardare theorem in [18] to arbitrary dimensions and co-dimensions.

A method for long-time integration of Lyapunov exponent and vectors along fluid particle trajectories

Finite-time Lyapunov exponents (FTLEs) and Lyapunov vectors (LVs) are powerful tools to illustrate Lagrangian coherent structures (LCSs) in experiments and numerical simulations of fluid flows. To obtain the FTLEs and LVs with the flow simulation simultaneously, we computed the eigenvectors and eigenvalues of the left Cauchy–Green tensor along the trajectories of fluid particles separately instead of computing deformation gradient tensor directly. The method proposed in the present study not only avoids solving the eigenvalue problem of the singular matrix at each time step but also guarantees a stable simulation for a long time. The method is applied in the computation of FTLEs and LVs in two-/three-dimensional (2D/3D) compressible/incompressible cases. In 2D cases, we found that LCSs are folded as fine filaments induced by vortices, while LCSs are sheet-like structures among the vortices for 3D cases. Meanwhile, the directions of stretching and compression of LVs are tangent and normal to the FTLE ridges (2D)/iso-surfaces (3D), respectively.

Quantum surface effects in strong coupling dynamics

Plasmons in nanostructured metals are widely utilized to trigger strong light--matter interactions with quantum light sources. While the nonclassical behavior of such quantum emitters (QEs) is well-understood in this context, the role of quantum and surface effects in the plasmonic resonator is usually neglected. Here, we combine the Green's tensor approach with the Feibelman $d$-parameter formalism to theoretically explore the influence of quantum surface effects in metal-dielectric layered nanostructures on the relaxation dynamics of a proximal two-level QE. Having identified electron spill-out as the dominant source of quantum effects in jellium-like metals, we focus our study on sodium. Our results reveal a clear splitting in the emission spectrum, indicative of having reached the strong-coupling regime, and, more importantly, non-Markovian relaxation dynamics of the emitter. Our findings establish that strong light--matter coupling is not suppressed by the emergence of nonclassical surface effects in the optical response of the metal.

Anisotropic hyperelastic/plastic behavior on stress-constrained thin structures by iterating on the elastic Cauchy–Green tensor

Thin-walled structures impose zero-stress constraints at free faces. These are in general nonlinear functions of strain and, in the elasto-plastic case, non-smooth. Under these circumstances, the conjugate strain components become constitutive unknowns. By iterating for the elastic Cauchy–Green tensor in the 6−dimensional space, we directly obtain a system that includes the zero-stress constraints. Coupling of the stress and strain constitutive unknowns is possible due to the use of a convenient equation arrangement. This contrasts with the standard decoupling schemes. Herein, we use the Kröner–Lee decomposition of the deformation gradient to obtain a differential–algebraic system which includes stress constraints in analogous form to additional yield conditions. The source is the right Cauchy–Green tensor. The main contribution is the development of a thin-walled implicit integrator for any hyperelastic case and any yield function. Integration is based on a backward-Euler method for the flow law complemented by the solution of a yield condition and the stress constraints. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Two complete fully orthotropic worked examples are presented.

Displacements and Stress Functions of Straight Dislocations and Line Forces in Anisotropic Elasticity: A New Derivation and Its Relation to the Integral Formalism

The displacement and stress function fields of straight dislocations and lines forces are derived based on three-dimensional anisotropic incompatible elasticity. Using the two-dimensional anisotropic Green tensor of generalized plane strain, a Burgers-like formula for straight dislocations and body forces is derived and its relation to the solution of the displacement and stress function fields in the integral formalism is given. Moreover, the stress functions of a point force are calculated and the relation to the potential of a Dirac string is pointed out.

Buckling of an elastic layer based on implicit constitution: Incremental theory and numerical framework

A general class of implicit bodies was proposed to describe elastic response of solids, which contains the Cauchy–Green tensor as a function of Cauchy stress. Here, we consider the buckling of solids described by such a subclass of implicit constitutive relation. We present a general linear incremental theory and carry out bifurcation analysis of a uniaxially compressed rectangular layer described by an implicit constitution. We then provide general governing equations regarding the mixed unknowns, i.e., displacement and stress fields, within the framework of finite strain deformation. Thereby, the combination of Asymptotic Numerical Method (ANM) and spectral collocation method is applied to solve the resulting nonlinear equations. We validate our numerical framework by comparing the computational results with analytical ones, and then explore effects of width-to-length ratio and material nonlinearity on the buckling and post-buckling behavior. The larger the width-to-length ratio is, the larger the critical buckling load is. The material parameter, η, has a significant impact up to 28% on the critical buckling load and influence up to 50% on the post-buckling behavior. Our model based on the implicit constitutive relation well predicts the buckling and post-buckling of Gum metal alloy.

“pyGDM” - new functionalities and major improvements to the python toolkit for nano-optics full-field simulations

pyGDM is a python toolkit for electro-dynamical simulations of individual nano-structures, based on the Green Dyadic Method (GDM). pyGDM uses the concept of a generalized propagator, which allows to solve cost-efficiently monochromatic problems with a large number of varying illumination conditions such as incident angle scans or focused beam raster-scan simulations. We provide an overview of new features added since the initial publication [Wiecha, Comput. Phys. Commun. 233 (2018) 167–192]. The updated version of pyGDM is implemented in pure python, removing the former dependency on fortran-based binaries. In the course of this re-write, the toolkit's internal architecture has been completely redesigned to offer a much wider range of possibilities to the user such as the choice of the dyadic Green's functions describing the environment. A new class of dyads allows to perform 2D simulations of infinitely long nanostructures. While the Green's dyads in pyGDM are based on a quasistatic description for interfaces, we also provide as new external python package “pyGDM2_retard” a module with retarded Green's tensors for an environment with two interfaces. We have furthermore added functionalities for simulations using fast-electron excitation, namely electron energy loss spectroscopy and cathodoluminescence. Along with several further new tools and improvements, the update includes also the possibility to calculate the magnetic field and the magnetic LDOS inside nanostructures, field-gradients in- and outside a nanoparticle, optical forces or the chirality of nearfields. All new functionalities remain compatible with the evolutionary optimization module of pyGDM for nano-photonics inverse design. Program summaryProgram Title: pyGDM2CPC Library link to program files:https://doi.org/10.17632/5zdrppdc3j.1Licensing provisions: GPLv3Programming language: pythonNature of problem: Full-field electrodynamical simulations of photonic nanostructures. This includes calculations of the optical extinction, scattering and absorption, as well as the near-field distribution or the interaction of quantum emitters with nanostructures as well as fast electron beam simulations. The toolkit includes a module for automated evolutionary optimization of nanostructure geometries to obtain a user-defined optical response.Solution method:: The optical response of photonic nanostructures is calculated using field susceptibilities (“Green Dyadic Method”, GDM) via a volume discretization. The approach is formally similar to the coupled dipole approximation.Additional comments including restrictions and unusual features: 2D and 3D nanostructures. On typical office PCs (8-16GB RAM) the discretization is limited to about 10000-15000 meshpoints, therefore it applies best to single, small nanostructures.

Microscopic characterisation of local strain field in healing tissue in the central third defect of mouse patellar tendon at early-phase of healing

Tendons exhibit a hierarchical collagen structure, wherein higher-level components, such as collagen fibres and fascicles, are elongated, slid, and rotated during macroscopic stretching. These mechanical behaviours of collagen fibres play important roles in stimulating tenocytes, imposing stretching, compression, and shear deformation. It was hypothesised that a lack of local fibre behaviours in healing tendon tissue may result in a limited application of mechanical stimuli to cells within the tissue, leading to incomplete recovery of tissue structure and functions in regenerated tendons. Therefore, the present study aimed to measure the microscopic strain field in the healing tendon tissue. A central third defect was created in the patellar tendon of mice, and the regenerated tissue in the defect was examined by tensile testing, collagen fibre analysis, and local strain measurement using confocal microscopy at 3 and 6 weeks after surgery. Healing tissue at 3 weeks exhibited a significantly lower strength and disorganised collagen fibre structure compared with the normal tendon. These characteristics at 6 weeks remained significantly different from those of the normal tendon. Moreover, the magnitude of local shear strain in the healing tissue under 4% tissue strain was significantly smaller than that in the normal tendon. Differences in the local strain field may be reflected in the cell nuclear shape and possibly the amount of mechanical stimuli applied to the cells during tendon deformation. Accordingly, restoration of a normal local mechanical environment in the healing tissue may be key to a better healing outcome of tendon injury.

Vacuum Field Photonic Trap for Excitons

AbstractThe spectral and spatial dependence of the resonant Lamb shift of an exciton inside a photonic vacuum field landscape is considered to predict an optical dipole potential generated via virtual photons quantum fluctuations. This trapping potential can be exploited for exciton self‐positioning in electric field hotspots at the nanoscale in view of quantum electrodynamics effects. By non Hermitian decomposition of the electromagnetic Green tensor in terms of quasi normal modes, the resonant part of the Lamb shift is expressed as Fano profile strictly interlinked to the Purcell enhancement of the exciton radiative lifetime. This approach explains how to tailor the depth of the vacuum photonic potential in several possible scenarios.

Small superimposed radial oscillations for a class of damaged limited elastic tubes

Small superimposed radial oscillations for a class of damaged limited elastic, incompressible, isotropic and homogeneous thick-walled circular cylindrical tube are examined. Both the damage function and the considered limited elastic class depend only on the first invariant $$ I_{1} $$ of the left Cauchy–Green deformation tensor. The present work puts some light in the context of the surgical procedure of angioplasty, used especially to treat cardiovascular diseases. Our simplified isotropic assumption allows for a much faster and analytical computation of comparably accurate results. The examples of three limited elastic material models are presented, namely the Arruda–Boyce, Gent, and Demiray. The radial oscillations are further investigated for a damaged limited elastic thin membrane, and the examples above are explored. Universal relations are found for the three material membranes. The obtained results are matched with the clinical data of the existing literature. It appears that the pressure ratio for the virgin and damaged arterial tissue is higher before angioplasty and vice versa. A similar trend for the frequency ratio also follows.

A novel phenomenological viewpoint for transversely isotropic hyperelastic materials; a new strain energy density function

In this paper, the mechanical behavior of incompressible transversely isotropic materials is modeled based on the strain energy density function proposed based on a novel framework. As far as physics is concerned, it seems plausible to assume that the strain energy density depends on the deformation. The right Cauchy–Green tensor is a fundamental tool for the deformation description of the solid elements. One of the common approaches for the deformation description of a solid element is to determining the stretches of three directions and the change in the angle of its right corners. Here, the idea of deformation description of a solid element is to determine its longitudinal stretches and the change in the area of its three faces. In this regard, the strain energy density function is proposed in terms of the longitudinal stretches and the areal stretches of a solid (continuum) element. Applying the essential requirements on the form of the strain energy density, a general framework is acquired to ensure the positive definiteness and polyconvexity conditions for modeling transversely isotropic materials. Based on this framework, the suitable mathematical forms for strain energy density function are suggested to match the requirements of the corresponding linearized theory and strain stiffening condition as well as the zero stress and strain energy density in the undeformed configuration. In the following, a polyconvex simple mathematical form for the strain energy function is proposed having few parameters. Moreover, this proposed form can establish a sensible connection between the material parameters and three independent elastic moduli of the linear theory. For evaluating the performance of the proposed strain energy density function, some test data of incompressible transversely materials with pure homogeneous deformations are employed. It is found that there exists strong compatibility between the test data and the results obtained from the proposed model.

Mechanical behavior of multilayer graphene reinforced epoxy nano-composites via a hierarchical multi-scale technique

In this paper, a multi-scale technique is presented to study the mechanical behavior of nano-composites by linking the atomistic information from lower scale to the continuum model in upper scale. Molecular dynamics simulations are employed to compute the material properties of graphene/epoxy nano-composites using the COMPASS interatomic force field. In order to obtain the atomistic stress surfaces used to evaluate the mechanical properties of material in upper scale, the biaxial loading is applied to different representative volume elements. On the continuum level, the hyperelastic strain energy functions are utilized to calculate the material parameters using the hyperelastic functions from atomistic data. The stress and elasticity tensors are obtained by computing the first and second order derivatives of hyperelastic functions with reference to the components of the right Cauchy–Green deformation tensor. The stress–strain surfaces of hyperelastic functions in lower scale are used to calculate the properties of nano-composite material in upper scale. The efficiency and applicability of the proposed technique is presented through various numerical examples. It is shown that the proposed multi-scale technique is able to solve large problems within acceptable computational time, which is not possible using conventional molecular dynamics approaches.