Elastic Response Function Research Articles

A macro-micro elasticity-diffusion system modeling absorption-induced swelling in rubber foams: Proof of the strong solvability

In this article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid’s absorption and transport inside the material is modeled by means of a nonlinear parabolic equation derived from Darcy’s law posed in a non-cylindrical domain defined by the macroscopic deformation (which is a solution of the beam equation). Under suitable assumptions, we establish the existence and uniqueness of a suitable class of solutions to our evolution system coupling the nonlinear parabolic equation posed on the microscopic non-cylindrical domain with the beam equation posed on the macroscopic cylindrical domain. In order to guarantee the regularity of the non-cylindrical domain, we impose a singularity to the elastic response function appearing in the beam equation.

Jahn-Teller effect in the cubic fullerides A3C60

Compared to isolated C$_{60}^{3-}$ ions, characterized by a threedimensional equipotential trough at the bottom of the lowest adiabatic potential energy surface (APES), the Jahn-Teller (JT) effect in cubic fullerides is additionally influenced by the interaction of JT distortions at C$_{60}$ sites with vibrational modes of the lattice. This leads to modification of JT stabilization energy and to the warping of the trough at each fullerene site, as well as to the interaction of JT distortions at different sites. Here we investigate these effects in three fcc fullerides with A=K,Rb,Cs and in Cs$_3$C$_{60}$ with bcc (A15) structure. DFT calculations of orbital vibronic coupling constants at C$_{60}$ sites and of phonon spectra have been done for fully ordered lattices (1 C$_{60}$/u.c.). Based on them the elastic response function for local JT distortions has been evaluated and the lowest APES investigated. To this end an expression for the latter in function of trough coordinates of all sites has been derived. The results show that the JT stabilization energy slightly increases compared to an isolated C$_{60}^{3-}$ and a warping of the trough of few meV occurs. The interaction of JT distortions on nearest- and next-nearest-neighbor fullerene sites is of similar order of magnitude. These effects arise first of all due to the interaction of C$_{60}$ sites with the displacements of neighbor alkali atoms and are more pronounced in fcc fullerides than in the A15 compound. The results of this study support the picture of weakly hindered independent rotations of JT deformations at C$_{60}$ sites in cubic A$_3$C$_{60}$.

Density-functional theory prediction of the elastic constants of ice Ih.

We assess the elastic stiffness constants of hexagonal proton-disordered ice Ih as described by density-functional theory calculations. Specifically, we compare the results for a set of nine exchange-correlation functionals, including standard generalized-gradient approximations (GGAs), the strongly constrained and appropriately normed (SCAN) metaGGA functional, and a number of dispersion-corrected versions based on the van der Waals (vdW) and VV10 schemes. Compared to the experimental data, all functionals predict an excessively stiff response to tensile and compressive distortions, as well as shear deformations along the basal plane, with the SCAN metaGGA functional displaying the largest deviations as compared to the experimental values. These discrepancies are found to correlate with underestimates of inter-molecular distances, on the one hand, and overestimates of intra-molecular separations, on the other. The inclusion of non-local vdW corrections according to the vdW approach generally improves these structural parameters and softens the elastic response functions compared to their parent GGA functionals. The dispersion-corrected SCAN-rVV10 functional, however, acts in the opposite direction, further worsening the comparison to experiment. In this view, it appears useful that the database employed to gauge the quality of exchange-correlation functionals for water includes an assessment of their elastic response of ice Ih and possibly other crystalline phases.

On the formulation of anisotropic plasticity for polymeric composites‐ rate‐dependent models with non‐linear isotropic/kinematic hardening

AbstractThe use of fiber‐reinforced composites as a primary structural component in automotive and aerospace industries has significantly increased over time, due to their great weight saving potential. Consequently, the predictive modeling of the non‐linear behavior of composites has been a topic of intensive research over the last years. Experimental data shows non‐linear response that is direction dependent due to the fibrous micro‐structure. Since typical fiber materials can be considered essentially as linear elastic, the non‐linearities are attributable to the inelastic deformation in the matrix, occurring at relatively small strains.This study presents a relatively general, homogenized formulation of anisotropic plasticity for application to (mainly) polymeric composites. A constitutive model has been developed to simulate the non‐linearities exhibited by the composite, generating the anisotropic elastic and plastic response functions with the aid of representation theorems. Rate‐independent versions using isotropic hardening have previously been proposed by the authors. The main focus of this work is the extension to include rate dependency and non‐linear kinematic hardening effects. Selected numerical simulations which serve the purpose of illustrating the formulation discussed herein, are presented at the end.

Viscoelastic response of quantum Hall fluids in a tilted field

In this paper, we examine the viscoelastic properties of integer quantum Hall (IQH) states in a tilted magnetic field. In particular, we explore to what extent the tilted-field system behaves like a two-dimensional electron gas with anisotropic mass in the presence of strain deformations. We first review the Kubo formalism for viscosity in an external magnetic field, paying particular attention to the role of rotational symmetry and contact terms. Next, we compute the conductivity, stress, and viscosity tensors for IQH states in the presence of a tilted field and vertical confining potential. By comparing our results with the recently developed bimetric formalism, we show that, at the level of the contracted Hall viscosity tensor, the mapping between tilted field and effective mass anisotropy holds only if we simultaneously modify the background perpendicular magnetic field; in other words, a simultaneous measurement of the density, contracted Hall viscosity, and Hall conductivity at fixed particle number can distinguish between tilted field and effective mass anisotropy. Additionally, we show that in the presence of a tilted magnetic field, the stress tensor acquires an unusual anisotropic ground state average, leading to anomalous elastic response functions. We develop a formalism for projecting a three-dimensional Hamiltonian with confining potential and magnetic field to a two-dimensional Hamiltonian in order to further address the phenomenology of the tilted-field IQH fluid. We find that the projected fluid couples non-minimally to geometric deformations, indicating the presence of internal geometric degrees of freedom.

A one-dimensional beam-like model for double-layered pipes

A one-dimensional beam-like model is proposed to analyze the mechanical behavior of elastic double-layered pipes. Besides the classical kinematic descriptors of the Timoshenko beams, further terms are introduced to describe the local distortion of the cross-section, indeed able to warp, ovalize and perform longitudinal slipping of the two layers. A homogenization procedure is applied, making use of a corresponding three-dimensional model, in order to obtain the nonlinear, coupled, elastic response function of the beam-like model. Case studies for different load conditions are considered, and instability phenomena, which cause loss of the bearing capacity of the structure, are found by means of numerical integration of the relevant boundary value problems.

Concurrent attenuation of, and energy harvesting from, surface vibrations: experimental verification and model validation

Fundamental studies in vibrational energy harvesting consider the electromechanically coupled devices to be excited by uniform base vibration. Since many harvester devices are mass–spring systems, there is a clear opportunity to exploit the mechanical resonance in a fashion identical to tuned mass dampers to simultaneously suppress the vibration of the host structure via reactive forces while converting the ‘absorbed’ vibration into electrical power. This paper presents a general analytical model for the coupled electro-elastic dynamics of a vibrating panel to which distributed energy harvesting devices are attached. One such device is described which employs a corrugated piezoelectric spring layer. The model is validated by comparison to measured elastic and electric frequency response functions. Tests on an excited panel show that the device, contributing 1% additional mass to the structure, concurrently attenuates the lowest panel mode accelerance by >20 dB while generating 0.441 µW for a panel drive acceleration of 3.29 m s−2. Adjustment of the load resistance connected to the piezoelectric spring layer verifies the analogy between the present harvester device and an electromechanically stiffened and damped vibration absorber. The results show that maximum vibration suppression and energy harvesting objectives occur for nearly the same load resistance in the harvester circuit.

Magnetoelastic coupling within a Landau model of phase transitions: Application to the frustrated triangular antiferromagnet CsNiCl3

A model Landau free energy is proposed in order to describe elastic coupling to spin degrees of freedom in systems exhibiting phase transitions which involve long-range magnetic order. Using rigourous symmetry arguments, various interaction terms are derived for magnetic materials with a hexagonal crystal structure. The model is applied to the frustrated triangular antiferromagnet CsNiCl${}_{3}$ and used to analyze and correlate a wide variety of experimental results such as the magnetic phase diagram, magnetization, strains, and elastic constant measurements at low temperatures. Good agreement between the model and the data is obtained for the temperature and magnetic field dependence of ${C}_{33}$ and ${C}_{66}$ in the vicinity of phase transitions. In particular, the analysis shows that the anomaly observed in the field dependence of ${C}_{33}$, close to the spin-flop phase boundary (${H}_{\mathit{SF}}\ensuremath{\simeq}2$ T), is dominated by the field dependence of the magnetic susceptibility. It is also found that higher order magnetoelastic coupling terms are required to reproduce the qualitative behavior of the elastic constants in the vicinity of the phase transitions. Our results demonstrate that a straightforward mean-field model which incorporates the correct system symmetries provides a powerful tool for relating complex spin configurations to the elastic and other response functions. The present work also serves to complement and expand our earlier results [G. Quirion et al., Phys. Rev. Lett. 97, 077202 (2006)].

Interfacial Properties and Critical Avalanche Exponents of Shape Memory Alloys and Related Materials

Interfaces in martensites and ferroelastic crystals show internal structures which are not simply the interpolation of the two adjacent domains. These structures can influence solitary front propagation as observed for large depinning forces. They also contribute to local pinning of walls when the applied forces are close to the depinning threshold. Under these conditions, walls propagate in jerks and avalanches. Typical depinning is observed for very small forces in single ferroelastic needle domain. It is shown that jerks occur in elastically driven system both for planar walls (D=2) and for needle tips (which represents a line in the three dimensional crystal, D=1). The experimental power law exponents are ~ -2 for the energy exponent for collective avalanches, -1.3 for the elastic response function and -1.8 for an advancing needle domain in LaAlO3.

Brillouin Light Scattering from Thin Albumen of Chicken Egg

Chicken eggs are rich in high-quality proteins and several kinds of trace elements, and have been a very familiar nutriment for human beings since ancient times. Chicken egg white comprises two layers, namely, thin albumen and thick albumen, both surrounding the yolk within a shell. There are two types of thin albumen known as the outer and inner thin albumens. The physical properties of both types of albumen are essentially the same. Thin albumen from chicken egg, hereafter referred to as egg white (EW), gels at a gelation temperature TG of ca. 66 C. The gelation process and the rheological properties of EW have attracted the attention of biological and biochemical scientists, food scientists, and physicists. The gelation process of EW has been studied by the ultrasound pulse technique at 3MHz in the temperature range between 10 and 95 C. The velocity of a longitudinal sound wave (LSW) exhibits a broader peak with a maximum at approximately 66 C as temperature increases through the sol–gel transition. Brillouin light scattering (BLS) measurement is another powerful and convenient technique for studying thermally excited hypersonic sound waves in the GHz frequency region. By utilizing a (3+3)-pass Fabry– Perot interferometer (FPI), we expect to observe a BLS spectrum from EW even near TG. We purchased fresh Boris Brown eggs with a red shell laid within a day at a farmers market near our university just before each measurement. About 5ml of thin albumen or distilled water was transferred into a glass tube placed in a brass cell. The cell temperature was regulated up to 82 C within an accuracy of 0:1 C and monitored by an almel– chromel thermocouple. We repeated our BLS measurement six times only during a heating run. BLS spectra were probed using the 1⁄4 488 nm excitation from an Arþ laser in a single-cavity mode with an output power of 40mW. BLS spectra were obtained using a laboratory-built (3+3)-pass FPI. Since EW gradually becomes opalescent with increasing temperature above 50 C, we employed the 180 scattering geometry, which allows us to continue our BLS measurements up to 61 C. In order to determine the LSW velocity V and the refractive index n at 488 nm, we also adopted the 90A scattering geometry at 24 C. Figure 1 shows an example of the 180 and 90A BLS spectra of EW at 24 C. We performed a numerical convolution analysis of these BLS spectra by adopting the damped harmonic oscillator (DHO) function for the elastic response function Sðq; !Þ;

Jerky elasticity: Avalanches and the martensitic transition in Cu74.08Al23.13Be2.79 shape-memory alloy

Jerky elasticity was observed by dynamical mechanical analyzer measurements in a single crystal of the shape memory alloy Cu74.08Al23.13Be2.79. Jerks appear as spikes in the dissipation of the elastic response function and relate to the formation of avalanches during the transformation between the austenite and the martensite phase. The statistics of the avalanches follows the predictions of avalanche criticality P(E)∝E−ε where P(E) is the probability of finding an avalanche with the energy E. This result reproduces, within experimental uncertainties, previous findings by acoustic emission techniques.

Nonlocal elastic compliance for soft solids: Theory, simulations, and experiments

The nonlocal elastic response function is crucial for understanding many properties of soft solids. This may be obtained by measuring strain-strain autocorrelation functions. We use computer simulations as well as video microscopy data of superparamagnetic colloids to obtain these correlations for two-dimensional triangular solids. Elastic constants and elastic correlation lengths are extracted by analyzing the correlation functions. We show that to explain our observations displacement fluctuations in a soft solid need to contain affine (strain) as well as nonaffine components.

Active Elasticity of Gels with Contractile Cells

Cells play an active role in the maintenance of mechanical homeostasis within tissues and their response to elastic forces is important for tissue engineering. We predict the collective response of an ensemble of contractile cells in a three-dimensional elastic medium to externally applied strain fields. Motivated by experiment, we model the cells as polarizable force dipoles that change their orientation in response to the local elastic strain. The analogy between the mechanical response of these systems and the dielectric response of polar molecules is used to calculate the elastic response function. We use this analogy to evaluate the average cell orientation, the mean polarization stress, and the effective elastic constants of the material, as a function of the cell concentration and matrix properties.

Constitutive equations for amended non-Gaussian network models of rubber elasticity

New constitutive equations based on an amended form of the Kuhn–Grün probability distribution function due to Jernigan and Flory are derived from the standard James–Guth (JG) 3-chain and Arruda–Boyce (AB) 8-chain non-Gaussian molecular network models. The kinematics describing the stretch of a 1-chain model in an affine deformation shows that the relative stretch of a single molecular chain initially oriented along the diagonal of a cube is determined by the first principal invariant of the Cauchy–Green deformation tensor. The Kuhn–Grün probability distribution for a randomly oriented chain and its more general amended form due to Wang and Guth, are functions of only the relative chain stretch. Hence, any non-Gaussian network model for which the configurational entropy of all chains may be uniform is characterized by an elastic response function that depends on only the first principal invariant of the Cauchy–Green deformation tensor. Both the regular and amended AB 8-chain models are characterized by specific response functions in this class; the regular and amended JG 3-chain models, however, are not. An amended form of the phenomenological composite 3-chain/8-chain model suggested by Wu and van der Giessen is introduced. Analytical relations for several kinds of homogeneous deformations of the standard and amended models are compared with a variety of experimental data by others. It is found that results for the amended 3-chain and 8-chain models do not vary significantly from results for the corresponding regular models. The composite model, on the other hand, shows excellent overall agreement with the diverse data, including equibiaxial deformations for which other models show greater variance; but it offers no improvement in comparison with data for plane strain compression. Some remarks relating the chain parameters of the 3-chain and 8-chain network models, and the limiting chain and continuum stretches for these models are discussed in an appendix.

Elastic response of [111]-tunnelling impurities

We study the dynamic response of a [111] quantum impurity, suchas lithium or cyanide in alkali halides, with respect to anexternal field coupling to the elastic quadrupole moment. Becauseof the particular level structure of a eight-state system on a cubicsite, the elastic response function shows a biexponential relaxation feature and a van Vleck typecontribution with a resonance frequency that is twice the tunnelfrequency Δ/ℏ. This basically differs from the dielectricresponse that does not show relaxation. Moreover, we show thatthe elastic response of a [111]-impurity cannot be reduced to thatof a two-level system. In the experimental part, we report on recentsound velocity and internal friction measurements on KCl doped withcyanide at various concentrations. At low doping (45 ppm) we find thedynamics of a single [111]-impurity, whereas at higher concentrations(4700 ppm) the elastic response indicates rather strongly correlateddefects. Our theoretical model provides a good description of thetemperature dependence of δv/v and Q-1 at low doping, in particular the relaxation peaks, the absolute valuesof the amplitude and the resonant contributions. From our fits weobtain the value of the elastic deformation potentialγt = 0.192 eV.

Mechanics of the inelastic behavior of materials—part 1, theoretical underpinnings

Abstract This is the first of a two-part paper that is concerned with the modeling of the behavior of inelastic materials from a continuum viewpoint, taking into account changes in the elastic response and material symmetry that occur due to changes in the microstructure of the material. The first part discusses some of the fundamental issues that must be addressed when modeling the elastic response of these materials. In particular, we discuss in detail the far reaching effects of the notion of materials with families of elastic response functions with corresponding natural configurations that was introduced by Wineman and Rajagopal (1990, Archives of Mechanics, 42, 53–75) and Rajagopal and Wineman (1992, Int. J. Plasticity 8, 385–395) for the study of the inelastic behavior of polymeric materials and later generalized and extended to the study of deformation twinning of polycrystals by Rajagopal and Srinivasa (1995 Int. J. Plasticity 11(6), 653–678, 1997, 13(1/2) 1–35). For these materials, a definition of material symmetry is introduced, that makes it possible to discuss the concept of “evolving material symmetry”.

Modelling of Failure Time Distributions for Interconnects Due to Stress Voiding and Electromigration

AbstractA status report is given on a comprehensive modeling project aimed at predicting failure time distributions of interconnect lines. We discuss our novel approach to calculate the evolution of stresses in lines using elastic response functions. It is argued that this approach makes it possible to model the stress and damage evolution in a large ensemble of lines efficiently so that statistically meaningful failure time distributions can be generated.The elastic response functions enable us also to derive a generalized Korhonen equation which includes the effects of mass transport at remote locations. Basic features of this equation are demonstrated with a one-dimensional implementation and its results are compared with the classical Korhonen [8, 9, 10] model.

Simple shearing of an incompressible, viscoelastic quadratic material

Viscoelastic constitutive equations of the differential type and of the rate type are applied in the study of the mechanical response of rubberlike materials in a simple shearing deformation superimposed on a specified static longitudinal stretch. Creep, recovery and stress relaxation processes are formulated for a class of incompressible, isotropic viscoelastic quadratic materials. Equations characterizing the creep and recovery processes for materials of differential type are derived, and their exact solutions are obtained. For materials of the rate type, the stress relaxation process is independent of the elastic material response functions. A coupled system of equations for the creep and recovery processees in a material of rate type are presented ; these must be solved numerically. The finite amplitude, damped, free vibration of a rigid body supported by incompressible, viscoelastic quadratic simple shear mountings of the differential type is investigated. The motion of the body is governed by a damped Duffing equation whose solution is discussed by the averaging method.

Neutron Studies of Disordered Alloys: a Brief Review/Update

ABSTRACTBoth neutron and X-ray scattering are used to determine the pair correlation functions (short-range order), the atomic displacements arising from atom size disparity, and the (derived) effective pair interactions in binary alloy solid solutions. This enterprise has taken on a new vitality through concomitant advances in both electronic theory and computer modeling of the atomic arrangements in alloys. An overview of the diffuse scattering formalism will be presented with an emphasis on the ways in which neutron scattering is particularly effective in sorting out the separate contributions to the total scattering pattern. Aspects of contrast variation and enhancement, and the utilization of both elastic and inelastic scattering will be discussed and examples of both past and recent work will be covered, including some future directions. We will also note briefly the relation between local order studies and details of the electronic and the elastic response functions of alloys.

Determination of material response functions for prestrained rubbers

The mechanical response of two natural rubber compounds is examined in order to determine relevant material parameters by non-linear finite element analysis. The materials are subjected to (a) combined static torsion and extension, and (b) small, steady-state torsional oscillations superposed on a large static simple extension. The materials are assumed to be incompressible and isotropic in their undeformed state and a time-strain separable relaxation modulus tensor is employed in order to characterize the steady-state harmonic viscoelastic response. The combined static torsion and extension experiments are used to determine the basic delayed elastic response functions. A Rivlin-type strain energy expression of third-order accuracy is used for the purpose. The two-constant, Mooney-Rivlin form is found to be adequate for both materials in the relatively limited range of deformation magnitudes considered.