Green Tensor Research Articles

A trust region algorithm for finite-strain plasticity with strongly coupled hardening

As extensions to our Lagrangian finite-strain plasticity framework based on the approximate exponential integrator, we introduce two new algorithms: (i) a fixed-radius trust region root finder for the nonlinear system involving the elastic right Cauchy Green tensor and plastic multiplier (ii) a partitioned approach for the hardening variables, which are determined in a staggered form using the strongly-coupled concept typically adopted for multiphysics problems. This allows the use of intricate hyperelastic laws combined with recent yield functions, which otherwise would involve a laborious treatment, and the use of corresponding work-hardening. Work-hardening would introduce significant nonlinearities in the constitutive system if used in a fully-coupled form. For the partitioned approach, a dynamic relaxation algorithm is adopted. This allows the efficient solution of the two nonlinear equations without significant drifting. Results show that robustness and efficiency are significantly improved. Herein, algorithms are described in detail. Significant testing is performed with imposed strains up to 100 for a carbon steel. This contributes to the robustness of equilibrium iterations. Drifting is also assessed as a function of number of steps in the dynamic relaxation algorithm. Numerical experimentation is performed in the 3D tension test for an anisotropic yield function.

Fourier Transform Approach to Boundary Domain Integral Equations for Elastic Composites With Domain Decomposition and Multi Reference Parameters

ABSTRACTIn this article, displacement and strain periodic boundary domain integral equations for homogenization problems of elastic composites are derived in the context of FFT homogenization methods. The resolution methods based on regular grids and discrete Green's tensors are presented. The displacement based equations can be used to solve problems in arbitrary domains under periodic and non‐periodic boundary conditions. The strain based integral equation is obtained from the combination of the displacement based equations for different domains, each one having its own reference elasticity tensor. In the latter, the strain values inside every phases are connected to material mismatch parameters on the phase boundary. It was shown that by decomposing suitably domains by stiffness and using adapted reference parameters, the iteration schemes converge faster.

Nonlinear free vibrations of a structurally tailored anisotropic pre-twisted thin-walled beam subjected to large deformations

It is essential to understand the nonlinear free vibration behaviour of a thin-walled composite structure as they find their applications in harsher environments prone to large deformations. Further, these structures made of fibrous materials, have directional properties for different fibre orientations resulting in a diversified and a complex nonlinear behaviour. In this regard, present work provides a comprehensive mathematical formulation on nonlinear vibration of pre-twisted thin-walled anisotropic box beam. The mathematical model is derived as coupled (flap-lag-extension-torsion) model considering shear deformation and green's strain tensor for large displacements in order to study the influence of nonlinear couplings on the dynamic behaviour. The derived energy equations are presented depending on degree of nonlinearity. These nonlinear equations those derived are nondimensionalized and solved with in the frame work of energy method using classical Ritz approximation. An iterative method is used to solve the nonlinear equations pertaining displacement terms. This nonlinear behaviour is evaluated for two important types of structural tailoring techniques available for thin-walled composite beams namely Circumferentially Asymmetric Stiffness (CAS) and Circumferentially Uniform Stiffness (CUS). It is found that CAS layup experiences significantly severe nonlinear effects compared to CUS due to the presence of 3rd degree nonlinear coupling terms. The variation of nonlinear frequency ratios over different ply angles showing the influence of nonlinearity on fibres orientation is presented for the first time for the two ply lay ups. The influence of degree of nonlinearity on the accuracy of the nonlinear frequencies is evaluated and presented. Moreover, the impact of nonlinearity on the pre-twisted beam is presented for different pre-twist angles.

Three-Dimensional and Two-Dimensional Green Tensors of Piezoelectric Quasicrystals

Three-Dimensional and Two-Dimensional Green Tensors of Piezoelectric Quasicrystals

Assembling algorithm for Green's tensors and absorbing boundary conditions for Galbrun's equation in radial symmetry

Solar oscillations can be modeled by Galbrun's equation which describes Lagrangian wave displacement in a self-gravitating stratified medium. For spherically symmetric backgrounds, we construct an algorithm to compute efficiently and accurately the coefficients of the Green's tensor of the time-harmonic equation in vector spherical harmonic basis. With only two resolutions, our algorithm provides values of the kernels for all heights of source and receiver, and prescribes analytically the singularities of the kernels. We also derive absorbing boundary conditions (ABC) to model wave propagation in the atmosphere above the cut-off frequency. The construction of ABC, which contains varying gravity terms, is rendered difficult by the complex behavior of the solar potential in low atmosphere and for frequencies below the Lamb frequency. We carry out extensive numerical investigations to compare and evaluate the efficiency of the ABCs in capturing outgoing solutions. Finally, as an application towards helioseismology, we compute synthetic solar power spectra that contain pressure modes as well as internal-gravity (g-) and surface-gravity (f-) ridges which are missing in simpler approximations of the wave equation. For purpose of validation, the locations of the ridges in the synthetic power spectra are compared with observed solar modes.

Full-waveform automatic location of small seismic events in an underground mine using synthetic strain Greens tensors

A reliable method to locate microseismic evets is an important objective for routine mine seismic monitoring. We developed an algorithm that can automatically and reliably locate an event from a single uniaxial trace. Signals created by microseismic events are recorded as seismograms that consist of direct P- and S-waves as well as the waves reflected off the underground excavations. We take advantage of the extra complexity these reflections introduce as they make the signal more unique. In our investigation we cross-correlated unaltered synthetic seismograms against trial waveforms created from a linear combination of appropriate waveforms from a library of numerically calculated strain Green's tensors. We use a single uniaxial synthetic seismogram to invert for the source location, six moment tensor components, and frequency. The resulting cost functions are constructed as maximal correlations at each node in the mine grid through the true source x-planes. This function is maximized by differential evolution to simultaneously yield the source parameters. Sensitivity tests such as shifting the mine plans and altering the background velocities of the media are investigated to test the robustness of the method. We contaminate the data with 40% white noise to test the resolution against simulated real recorded seismograms. The method is demonstrated using synthetic data calculated with a realistic underground 3D velocity model.

A Constitutive Framework for Modeling of the Visco-Hyperelastic Materials: Application to the Mechanical Behavior of Cylinders Made of the Rate-Dependent Elastomers

In this paper, the formulation of the mechanical behavior of visco-hyperelastic materials is developed in a thermo-dynamically compatible framework. In this viewpoint, the stress power is assumed to be equal to the sum of the reversible (elastic) energy rate stored and the irreversible energy rate dissipated. Based on this assumption and using the second law of thermodynamics, the constitutive modeling framework is obtained for the rate-dependent materials. Inspired by the constitutive law of a Newtonian fluid and also, models of density strain energy of an elastic material, a framework for the dissipative energy rate related to the irreversible part is proposed for viscoelastic materials. In this framework, the reversible energy part is considered a function of the right Cauchy–Green deformation tensor, and the irreversible energy part is considered a function of the same tensor’s rate. To evaluate the proposed model for nonlinear viscoelastic behavior modeling, the results of tests performed on non-compressible materials at different rates are used, and the proposed model provided acceptable results. Finally, the proposed model is implemented to find a closed-form analytical solution for mechanical behavior modeling of the thick-walled cylindrical shells made of visco-hyperelastic materials, also, stress and stability analyses are assessed for these types of cylinders. To achieve this goal, the effect of pre-stretch on the stability of cylinders has been substantially investigated. Furthermore, based on the proposed model, the effect of the parameters such as the wall thickness and behavior of the material used in the cylinder are examined on the stability of open-ended and closed-ended cylinders.

Azimuthal crustal variations and their implications on the seismic impulse response in the Valley of Mexico

AbstractPrevious studies have suggested prominent variations in the seismic wave behavior at the 5 s period when traveling across the Valley of Mexico, associating them with the crustal structure and contributing to the anomalous seismic wave patterns observed each time an earthquake hits Mexico City. This article confirms the variations observed at 0.2 Hz by analyzing the Green tensor diagonal retrieved from empirical Green functions (EGF) calculated using seismic noise data cross-correlations of the vertical and horizontal components. We observe time and phase shifts between the east and north EGF components and show that they can be explained by the crustal structure from the surface up to 20 km depth; we also observe that the time and phase shifts are more significant if the distance between the source and the station increases. Additionally, the article presents an updated version of the velocity model from receiver functions and dispersion curves (VMRFDC v2.0) for the crustal structure under the Valley of Mexico. To validate this model, we compare the EGFs with synthetic Green functions determined numerically. To do so, we adaptatively meshed this model using an iterative algorithm to numerically simulate the impulse response up to 0.5 Hz. Finally, the comparisons between noise and synthetic EGF showed that the VMRFDC v2.0 model explains the time shifts and phase differences at 0.2 Hz, previously observed by independent studies, suggesting it correctly characterizes the crustal structure anomalies beneath the Valley of Mexico.

Quasidiscrete spectrum Cherenkov radiation by a charge moving inside a dielectric waveguide

We investigate the Cherenkov radiation by a charge uniformly moving inside a dielectric cylindrical channel in a homogeneous medium. The expressions for the Fourier components of the electric and magnetic fields are derived by using the electromagnetic field Green tensor. The spectral distribution of the Cherenkov radiation intensity in the exterior medium is studied for the general case of frequency dispersion of the interior and exterior dielectric functions. It is shown that, under certain conditions on the dielectric permittivities, strong narrow peaks appear in the spectral distribution. The spectral locations of those peaks are specified and their heights and widths are estimated analytically on the base of the dispersion equation for the electromagnetic eigenmodes of the cylinder.

Energy variation and stress fields of spherical inclusions with eigenstrain in three-dimensional anisotropic bi-materials

In this study, expressions for the interior stress fields of a spherical inclusion with uniform eigenstrain embedded in an anisotropic bi-material featuring a planar interface are derived. These expressions involve surface integrals of the imaginary term of the first derivative of the Green tensor in an anisotropic bi-material which are well-suited for standard numerical integrations. Specific formulations are provided for cases where the inclusion belongs to either the same material or both materials. Additionally, expressions are presented for the equivalent Eshelby tensor. The interior stress fields and variations in elastic strain energy are computed using cubic elastic constants of Cu. Various inclusion positions relative to the interface, eigenstrain forms, and crystallographic orientations are considered. For instances involving dilatational eigenstrain, the elastic strain energy variation with the inclusion’s position may exhibit multiple extrema. The global minimum and maximum consistently occur when the inclusion spans both materials. In the context of symmetrical tilt boundaries, energy variations are perfectly symmetric, with a global minimum on the interface that decreases with the tilt angle. The significance of the observed energy variations for defects segregation is quantitatively assessed by comparing them with the interaction energy between an eigenstrain and a grain boundary stress field.

Relativistic constitutive modeling of inelastic deformation of continua moving in space-time

Since Einstein introduced the theory of relativity, several scientific observations have proven that it thoroughly approximates the motion of materials in space-time. Thus, efforts have been made to expand classical continuum mechanics to relativistic continuum models to consider relativistic effects. However, most models consider only the elastic range without accounting for inelastic deformation. A relativistic inelasticity model should provide objective measures of the inelastic deformation for different observers with nonzero relative velocities over a wide speed range, even when the moving speed is close to the speed of light. This study presents a mathematical modeling structure of the relativistic constitutive equations of the inelastic deformation of a material moving over a wide speed range to provide objective measures for observers. In this model, the deformation tensors of classical mechanics are expanded to four-dimensional tensors (referred to relativistic Cauchy–Green deformation tensors) in space-time, based on which constitutive equations of inelastic deformation are introduced. The four-dimensional tensors are objective about the homogeneous Lorentz transformation in the Minkowski space-time, and the material dissipation, determined employing the proposed modeling of inelastic deformation, satisfies the second law of thermodynamics. In addition, an example illustrates the application of this theory by explaining the physical meaning of modeling. Finally, it is also demonstrated that the proposed relativistic inelasticity model collapses into a classical inelasticity model when the speed of motion is much slower than the speed of light.

Deformation induced evolution of plasmonic responses in polymer grafted nanoparticle thin films.

Multi-functional nanoparticle thin films are being used in various applications ranging from biosensing to photo-voltaics. In this study, we integrate two different numerical approaches to understand the interplay between the mechanical deformation and optical response of polymer grafted plasmonic nanoparticle (PGPN) arrays. Using numerical simulations we examine the deformation of thin films formed by end-functionalised polymer grafted nanoparticles subject to uniaxial elongation. The induced deformation causes the particles in the thin film network to rearrange their positions by two different mechanisms viz. sliding and packing. In sliding, the particles move in the direction of induced deformation. On the other hand, in packing, the particles move in a direction normal to that of the induced deformation. By employing a Green's tensor formulation in polarizable backgrounds for evaluating the optical response of the nanoparticle network, we calculate the evolution of the plasmonic response of the structure as a function of strain. The results indicate that the evolution of plasmonic response closely follows the deformation. In particular, we show that the onset of relative electric field enhancement of the optical response occurs when there is significant rearrangement of the constituent PGPNs in the array. Furthermore, we show that depending on the local packing/sliding and the polarization of the incident light there can be both enhancement and suppression of the SERS response.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Mathematical micro–macro modeling of fully coupled nonlinear magneto-elastic reinforced composites

In this paper, integral equation formulations and field-dependent micromechanical modeling of global nonlinear magneto-mechanical behavior of magneto-elastic composites under high strain and magnetic field are elaborated. The modeling is obtained based on an extension of micro–macro transition inclusion problem to nonlinear behavior using field-dependent and highly nonlinear localization tensors. A methodological procedure is elaborated based on newly introduced strain and magnetic field-dependent Green tensors, strain and magnetic field-dependent integral equations linked to tensors of Eshelby and micromechanical approaches. The field-dependent global moduli are predicted based on the Mori–Tanaka approach and self-consistent method. Iterative incremental schemes based on the Newton–Raphson and fixed-point algorithms are examined and established precise semi-analytic equations of the effective magneto-elastic properties of composites that are dependent on the magnetic-strain field for different types of inclusions. A numerical code is elaborated for numerical predictions and the obtained field-dependent effective magneto-elastic coefficients are obtained and presented for various volume fractions of inclusion, types and shapes of the reinforced nonlinear composites.

Parameter determination and verification of ZWT viscoelastic dynamic constitutive model of Al/Ep/W material considering strain rate effect

As a new type of energetic structural material, Al/Ep/W(Aluminum/epoxy/tungsten), whether as a fragment or projectile material, should comprehensively consider the impact energy release and structural strength, especially the structural strength is the premise to ensure the material integrity. The dynamic constitutive model of materials is the theoretical basis for describing the strength and dynamic behavior of materials. Therefore, it is great significance to determine the dynamic constitutive model parameters of Al/Ep/W materials considering strain rate effect, and to verify the reliability of the model. In this paper, Al/Ep/W material is prepared, and the stress/strain relationship curve of the material under high strain rate is obtained through Split Hopkinson Pressure Bar (SHPB) test. The ZWT viscoelastic constitutive model is selected based on the composition of the material and the distribution of the internal components, and the simplified ZWT constitutive model with damage under constant strain rate is derived, and the parameters of the constitutive model are fitted by the least square method. By using Kirchhoff stress and Green strain tensor theory, the incremental form of the constitutive model is derived, and the Abaqus user material subroutine is written and embedded in the finite element software. The accuracy of material VUMAT subroutine is verified by comparing the results of SHPB test and numerical simulation under high strain rate. Combined with the verified material model subroutine, the experiment and numerical simulation of Al/Ep/W cylindrical projectile impacting aluminum plate are conducted. The results show that the experimental and numerical simulation results are in good agreement. The proposed model can be used to evaluate the dynamic response characteristics of the new energetic material projectile to the lightweight aluminum armor protective structure.

A note on the response of elastic bodies whose material moduli depend on the density and the mechanical pressure

In this note we study the response of an inhomogeneous body, whose material moduli depend on the density and the mechanical pressure, that is described by an implicit constitutive relation between the Cauchy stress and the right Cauchy–Green tensor. Such a constitutive relation is relevant to the study of the deformation of porous elastic solids as the porous structure varies and the local density changes. The boundary value problems of a plate of such a material subject to uniaxial and bi-axial extension and compression are considered. We find that regions that are more porous, and thus of lower macroscopic density, have the propensity to fail than regions of higher density, in the sense the load carrying capacity decreases, when other measures such as the stress are comparable.

Consistent C-bar radial-basis Galerkin method for finite strain analysis

For finite-strain problems of quasi-incompressible materials, with hyperelastic and hyperelasto-plastic behavior, we develop a mixed polynomial-enriched radial-basis function (RBF) interpolation. It is characterized by decoupling quadrature and polynomials for the deviatoric and volumetric terms of the right Cauchy–Green tensor. With an enriched interpolation, first and second derivatives of the deformation gradient are continuous. A finite-element mesh is employed for quadrature purposes with the corresponding distribution of Gauss points. In terms of discretization, linear, quadratic and cubic polynomials are combined and function support is established from the number of pre-assigned nodes. Due to the adoption of a Lagrangian kernel, finite strain elasto-plastic constitutive developments are based on the Mandel stress. Combined with the C¯ technique, these developments are found to be convenient from an implementation perspective, as RBF formulations for finite strain plasticity have been limited in terms of generality and amplitude of deformations in the quasi-incompressibility case. Three benchmark tests are successfully solved, and it was found that a combination of reduced integration with a deviatoric cubic basis and volumetric quadratic basis provides superior results in the quasi-incompressible case. Numerical testing shows very good convergence behavior of the results. Besides absence of locking with selective interpolation, outstanding Newton convergence was observed regardless of strain levels.

Strong Coupling Dynamics of a Quantum Emitter near a Topological Insulator Nanoparticle.

We study the spontaneous emission dynamics of a quantum emitter near a topological insulator Bi2Se3 spherical nanoparticle. Using the electromagnetic Green's tensor method, we find exceptional Purcell factors of the quantum emitter up to 1010 at distances between the emitter and the nanoparticle as large as half the nanoparticle's radius in the terahertz regime. We study the spontaneous emission evolution of a quantum emitter for various transition frequencies in the terahertz and various vacuum decay rates. For short vacuum decay times, we observe non-Markovian spontaneous emission dynamics, which correspond perfectly to values of well-established measures of non-Markovianity and possibly indicate considerable dynamical quantum speedup. The dynamics turn progressively Markovian as the vacuum decay times increase, while in this regime, the non-Markovianity measures are nullified, and the quantum speedup vanishes. For the shortest vacuum decay times, we find that the population remains trapped in the emitter, which indicates that a hybrid bound state between the quantum emitter and the continuum of electromagnetic modes as affected by the nanoparticle has been formed. This work demonstrates that a Bi2Se3 spherical nanoparticle can be a nanoscale platform for strong light-matter coupling.

Bounding nonlinear stretching about spacecraft trajectories using tensor eigenpairs

Local linearization is often used to enable analytical methods in spacecraft dynamics, navigation, and control. However, the linear approximation becomes inaccurate at large distances from the linearization point and/or when the underlying dynamical system is strongly nonlinear. As a result, linearization-based methods like the finite time Lyapunov exponent (FTLE) can underestimate stretching in phase space. Alternatively, this paper presents a semianalytical method to quantify nonlinear stretching: the eigenpairs of higher-order tensors are used to bound the maximum stretching of a state deviation over time. The tenors in question are derived from the Taylor series expansion of a nonlinear solution flow about a reference trajectory, i.e., the state transition tensor (STT) expansion. It is well known that the eigenvalues of the Cauchy–Green tensor (CGT) bound the stretching of state deviations over time for a linear system, and the largest eigenvalue can be related to the FTLE. This paper will present higher-order, nonlinear analogs to the CGT and the FTLE to more accurately bound nonlinear stretching for applications in guidance, navigation, and control.

Bespoke two-dimensional elasticity and the nonlinear analogue of Cauchy’s relations

Is it possible to design an architectured material or structure whose elastic energy is arbitrarily close to a specified continuous function? This is known to be possible in one dimension, up to an additive constant (Dixon et al., Bespoke extensional elasticity through helical lattice systems, Proc. R. Soc. A. (2019)). Here, we explore the situation in two dimensions. Given (1) a continuous energy function E ( C ) , defined for two-dimensional right Cauchy–Green deformation tensors C contained in some compact set and (2) a tolerance ϵ > 0 , can we construct a spring-node unit cell (of a lattice) whose energy is approximately E , up to an additive constant, with L ∞ -error no more than ϵ ? We show that the answer is yes for affine E s (i.e., for energies E that are quadratic in the deformation gradient), but that the general situation is more subtle and is related to the generalisation of Cauchy’s relations to nonlinear elasticity.