Limit Of Small Deformations Research Articles

Features of deformation resistance of polypropylene polymer films

The article studies the physical and mechanical properties of experimental polymer materials, which are determined by parameters such as Tensile strength s, tensile model E, relative elongation at break ε, which affect the quality of printing under other equal conditions. Experimental studies on the elongation of biaxially oriented polypropylene and polyethylene polymer films were carried out in accordance with the requirements of GOST 14236-81. The tensile modulus E was determined according to Hooke’s law when observing a rectilinear area in the limit of small deformations on the stretching diagram of polymer films.

Virtual Elements for computational anisotropic crystal plasticity

In this contribution, the Virtual Element Method (VEM) with a linear ansatz is applied to a computational crystal plasticity framework in a micro-structural environment. Furthermore, a simple anisotropic energetic contribution, based on invariant-formulations of tensorial deformation measures and structural tensors, is presented for the cubic elastic anisotropy of the underlying crystal structure. The anisotropic elastic formulation recovers the elasticity tensor structure of a cubic material in the limit of small deformations. The authors propose a new stabilization degradation formulation which is purely based on the dissipative response of the problem. Representative examples illustrate the robustness and performance of VEM with regard to locking phenomena in the crystal plasticity framework, when bench-marked against the solutions of classical finite element approaches. Further examples investigate the performance and current limitations of VEM within a crystal plasticity framework, when being applied to heterogeneous microstructures for both, structured element topology as well as flexible element topology.

Soft-lubrication interactions between a rigid sphere and an elastic wall

The motion of an object within a viscous fluid and in the vicinity of a soft surface induces a hydrodynamic stress field that deforms the latter, thus modifying the boundary conditions of the flow. This results in elastohydrodynamic interactions experienced by the particle. Here, we derive a soft-lubrication model, in order to compute all the forces and torque applied on a rigid sphere that is free to translate and rotate near an elastic wall. We focus on the limit of small deformations of the surface with respect to the fluid-gap thickness, and perform a perturbation analysis in dimensionless compliance. The response is computed in the framework of linear elasticity, for planar elastic substrates in the limiting cases of thick and thin layers. The EHD forces are also obtained analytically using the Lorentz reciprocal theorem.

Modeling of many-body interactions between elastic spheres through symmetry functions.

Simple models for spherical particles with a soft shell have been shown to self-assemble into numerous crystal phases and even quasicrystals. However, most of these models rely on a simple pairwise interaction, which is usually a valid approximation only in the limit of small deformations, i.e., low densities. In this work, we consider a many-body yet simple model for the evaluation of the elastic energy associated with the deformation of a spherical shell. The resulting energy evaluation, however, is relatively expensive for direct use in simulations. We significantly reduce the associated numerical cost by fitting the potential using a set of symmetry functions. We propose a method for selecting a suitable set of symmetry functions that capture the most relevant features of the particle's environment in a systematic manner. The fitted interaction potential is then used in Monte Carlo simulations to draw the phase diagram of the system in two dimensions. The system is found to form both a fluid and a hexagonal crystal phase.

On the two-potential constitutive modeling of dielectric elastomers

This work lays out the two-potential framework for the constitutive modeling of dielectric elastomers. After its general presentation, where the constraints imposed by even electromechanical coupling, material frame indifference, material symmetry, and entropy imbalance are all spelled out, the framework is utilized to put forth a specific constitutive model for the prominent class of isotropic incompressible dielectric elastomers. The model accounts for the non-Gaussian elasticity and electrostriction typical of such materials, as well as for their deformation-enhanced shear thinning due to viscous dissipation and their time-dependent polarization due to electric dissipation. The key theoretical and practical features of the model are discussed, with special emphasis on its specialization in the limit of small deformations and moderate electric fields. The last part of this paper is devoted to the deployment of the model to fully describe the electromechanical behavior of a commercially significant dielectric elastomer, namely, the acrylate elastomer VHB 4910 from 3M.

Prediction and measurement of the fastest-growing mode in two-liquid systems

The paper is concerned with the capillary instability of a liquid thread surrounded by another immiscible liquid. We study the occurrence of the dominant mode of perturbation for different combinations of Newtonian and viscoelastic fluids. When a viscoelastic fluid is present, in the limit of small deformations, the dispersion relation is obtained via a Maxwell type approximation of the viscosity curve which can be directly incorporated into an existing model of thread instability. Measurements of the fastest-growing mode are then performed and compared with theoretical predictions. A satisfactory agreement between the theory and experiment is found in terms of the wavenumber of the fastest-growing mode.

Electro-osmotic oscillatory flow of viscoelastic fluids in a microchannel

This work presents an analytical solution for electro-osmotic flow (EOF) in small amplitude oscillatory shear (SAOS) as a measuring tool suitable to characterize the linear viscoelastic properties of non-Newtonian fluids in microchannel flow. The flow in the straight microchannel is driven by applying oscillating sinusoidal electric potentials. Fourier series are used to derive an expression for the velocity field, under an externally imposed generic potential field aimed at the practical application of SAOS in characterizing the rheological properties of viscoelastic fluids. This extensive investigation covers a wide range of parameters and considers the multi-mode upper-convected Maxwell (UCM) model, which represents the rheology of viscoelastic fluids in the limit of small and slow deformations. Particular focus is given to two cases of practical interest: equal wall zeta potentials at both channel walls and negligible zeta potential at one of the walls. The results show that in flows with thin electric double layers (EDL), Reynolds numbers below 0.001, and Deborah numbers below 100, corresponding to a viscoelastic Mach number of 0.32, the velocity field outside the EDLs is linear and has a large enough amplitude of oscillation, which may allow the quantification of the storage and loss moduli in the linear regime. This technique requires the use of significantly smaller sample sizes than the traditional SAOS in a rotational rheometer.

ESTIMATING DIFFEOMORPHIC MAPPINGS BETWEEN TEMPLATES AND NOISY DATA: VARIANCE BOUNDS ON THE ESTIMATED CANONICAL VOLUME FORM.

Anatomy is undergoing a renaissance driven by the availability of large digital data sets generated by light microscopy. A central computational task is to map individual data volumes to standardized templates. This is accomplished by regularized estimation of a diffeomorphic transformation between the coordinate systems of the individual data and the template, building the transformation incrementally by integrating a smooth flow field. The canonical volume form of this transformation is used to quantify local growth, atrophy, or cell density. While multiple implementations exist for this estimation, less attention has been paid to the variance of the estimated diffeomorphism for noisy data. Notably, there is an infinite dimensional unobservable space defined by those diffeomorphisms which leave the template invariant. These form the stabilizer subgroup of the diffeomorphic group acting on the template. The corresponding flat directions in the energy landscape are expected to lead to increased estimation variance. Here we show that a least-action principle used to generate geodesics in the space of diffeomor-phisms connecting the subject brain to the template removes the stabilizer. This provides reduced-variance estimates of the volume form. Using simulations we demonstrate that the asymmetric large deformation diffeomorphic mapping methods (LDDMM), which explicitly incorporate the asymmetry between idealized template images and noisy empirical images, provide lower variance estimators than their symmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for the variances in the limit of small deformations. Analytical results are shown for the Jacobian in terms of perturbations of the vector fields and divergence of the vector field.

Modified Unruh effect from generalized uncertainty principle

We consider a generalized uncertainty principle (GUP) corresponding to a deformation of the fundamental commutator obtained by adding a term quadratic in the momentum. From this GUP, we compute corrections to the Unruh effect and related Unruh temperature, by first following a heuristic derivation, and then a more standard field theoretic calculation. In the limit of small deformations, we recover the thermal character of the Unruh radiation. Corrections to the temperature at first order in the deforming parameter are compared for the two approaches, and found to be in agreement as for the dependence on the cubic power of the acceleration of the reference frame. The dependence of the shifted temperature on the frequency is also pointed out and discussed.

A general result for the magnetoelastic response of isotropic suspensions of iron and ferrofluid particles in rubber, with applications to spherical and cylindrical specimens

This paper puts forth an approximate solution for the effective free-energy function describing the homogenized (or macroscopic) magnetoelastic response of magnetorheological elastomers comprised of non-Gaussian rubbers filled with isotropic suspensions of either iron or ferrofluid particles. The solution is general in that it applies to N=2 and 3 space dimensions and any arbitrary (non-percolative) isotropic suspension of filler particles. By construction, it is exact in the limit of small deformations and moderate magnetic fields. For finite deformations and finite magnetic fields, its accuracy is demonstrated by means of direct comparisons with full-field simulations for two prominent cases: (i) isotropic suspensions of circular particles and (ii) isotropic suspensions of spherical particles.With the combined objectives of demonstrating the possible benefits of using ferrofluid particles in lieu of the more conventional iron particles as fillers and gaining insight into recent experimental results, the proposed homogenization-based constitutive model is deployed to generate numerical solutions for boundary-value problems of both fundamental and practical significance: those consisting of magnetorheological elastomer specimens of spherical and cylindrical shape that are immersed in air and subjected to a remotely applied uniform magnetic field. It is found that magnetorheological elastomers filled with ferrofluid particles can exhibit magnetostrictive capabilities far superior to those of magnetorheological elastomers filled with iron particles. The results also reveal that the deformation and magnetic fields are highly heterogenous within the specimens and strongly dependent on the shape of these, specially for magnetorheological elastomers filled with iron particles. From an applications perspective, this evidence makes it plain that attempts at designing magnetrostrictive devices based on magnetorheological elastomers need to be approached, in general, as structural problems, and not simply as materials design problems.

Effect of body deformability on microswimming.

In this work we consider the following question: given a mechanical microswimming mechanism, does increased deformability of the swimmer body hinder or promote the motility of the swimmer? To answer this we run immersed-boundary-lattice-Boltzmann simulations of a microswimmer composed of deformable beads connected with springs. We find that the same deformations in the beads can result in different effects on the swimming velocity, namely an enhancement or a reduction, depending on the other parameters. To understand this we determine analytically the velocity of the swimmer, starting from the forces driving the motion and assuming that the deformations in the beads are known as functions of time and are much smaller than the beads themselves. We find that to the lowest order, only the driving frequency mode of the surface deformations contributes to the swimming velocity, and comparison to the simulations shows that both the velocity-promoting and velocity-hindering effects of bead deformability are reproduced correctly by the theory in the limit of small bead deformations. For the case of active deformations we show that there are critical values of the spring constant - which for a general swimmer corresponds to its main elastic degree of freedom - which decide whether the body deformability is beneficial for motion or not.

Nonlinear electroelastic deformations of dielectric elastomer composites: II — Non-Gaussian elastic dielectrics

This paper presents an analytical framework to construct approximate homogenization solutions for the macroscopic elastic dielectric response — under finite deformations and finite electric fields — of dielectric elastomer composites with two-phase isotropic particulate microstructures. The central idea consists in employing the homogenization solution derived in Part I of this work for ideal elastic dielectric composites within the context of a nonlinear comparison medium method — this is derived as an extension of the comparison medium method of Lopez-Pamies et al. (2013) in nonlinear elastostatics to the coupled realm of nonlinear electroelastostatics — to generate in turn a corresponding solution for composite materials with non-ideal elastic dielectric constituents. Complementary to this analytical framework, a hybrid finite-element formulation to construct homogenization solutions numerically (in three dimensions) is also presented.The proposed analytical framework is utilized to work out a general approximate homogenization solution for non-Gaussian dielectric elastomers filled with nonlinear elastic dielectric particles that may exhibit polarization saturation. The solution applies to arbitrary (non-percolative) isotropic distributions of filler particles. By construction, it is exact in the limit of small deformations and moderate electric fields. For finite deformations and finite electric fields, its accuracy is demonstrated by means of direct comparisons with finite-element solutions. Aimed at gaining physical insight into the extreme enhancement in electrostriction properties displayed by emerging dielectric elastomer composites, various cases wherein the filler particles are of poly- and mono-disperse sizes and exhibit different types of elastic dielectric behavior are discussed in detail. Contrary to an initial conjecture in the literature, it is found (inter alia) that the isotropic addition of a small volume fraction of stiff (semi-)conducting/high-permittivity particles to dielectric elastomers does not lead to the extreme electrostriction enhancements observed in experiments. It is posited that such extreme enhancements are the manifestation of interphasial phenomena.

On the accuracy and fitting of transversely isotropic material models

Fiber reinforced structures are central to the form and function of biological tissues. Hyperelastic, transversely isotropic material models are used widely in the modeling and simulation of such tissues. Many of the most widely used models involve strain energy functions that include one or both pseudo-invariants (I4 or I5) to incorporate energy stored in the fibers. In a previous study we showed that both of these invariants must be included in the strain energy function if the material model is to reduce correctly to the well-known framework of transversely isotropic linear elasticity in the limit of small deformations. Even with such a model, fitting of parameters is a challenge. Here, by evaluating the relative roles of I4 and I5 in the responses to simple loadings, we identify loading scenarios in which previous models accounting for only one of these invariants can be expected to provide accurate estimation of material response, and identify mechanical tests that have special utility for fitting of transversely isotropic constitutive models. Results provide guidance for fitting of transversely isotropic constitutive models and for interpretation of the predictions of these models.

How bio-filaments twist membranes

We study the deformations of a fluid membrane imposed by adhering stiff bio-filaments due to the torques they apply. In the limit of small deformations, we derive a general expression for the energy and the deformation field of the membrane. This expression is specialised to different important cases including closed and helical bio-filaments. In particular, we analyse interface-mediated interactions and membrane wrapping when the filaments apply a local torque distribution on a tubular membrane.

The Overall Elastic Dielectric Properties of Fiber-Strengthened/Weakened Elastomers

By employing recent results (Lopez-Pamies, O., 2014, “Elastic Dielectric Composites: Theory and Application to Particle-Filled Ideal Dielectrics,” J. Mech. Phys. Solids, 64, p. 6182 and Spinelli, S. A., Lefèvre, V., and Lopez-Pamies, O., “Dielectric Elastomer Composites: A General Closed-Form Solution in the Small-Deformation Limit,” J. Mech. Phys. Solids, 83, pp. 263–284.) on the homogenization problem of dielectric elastomer composites, an approximate solution is generated for the overall elastic dielectric response of elastomers filled with a transversely isotropic distribution of aligned spheroidal particles in the classical limit of small deformations and moderate electric fields. The solution for such a type of dielectric elastomer composites is characterized by 13 (five elastic, two dielectric, and six electrostrictive) effective constants. Explicit formulae are worked out for these constants directly in terms of the elastic dielectric properties of the underlying elastomer and the filler particles, as well as the volume fraction, orientation, and aspect ratio of the particles. As a first application of the solution, with the objective of gaining insight into the effect that the addition of anisotropic fillers can have on the electromechanical properties of elastomers, sample results are presented for the case of elastomers filled with aligned cylindrical fibers. These results are confronted to a separate exact analytical solution for an assemblage of differential coated cylinders (DCC), wherein the fibers are polydisperse in size, and to full-field simulations of dielectric elastomer composites with cylindrical fibers of monodisperse size. These results serve to shed light on the recent experimental findings concerning the dielectric elastomers filled with mechanically stiff fibers. Moreover, they serve to reveal that high-permittivity liquid-like or vacuous fibers—two classes of filler materials yet to be explored experimentally—have the potential to significantly enhance the electrostriction capabilities of dielectric elastomers.

Dielectric elastomer composites: A general closed-form solution in the small-deformation limit

A solution for the overall electromechanical response of two-phase dielectric elastomer composites with (random or periodic) particulate microstructures is derived in the classical limit of small deformations and moderate electric fields. In this limit, the overall electromechanical response is characterized by three effective tensors: a fourth-order tensor describing the elasticity of the material, a second-order tensor describing its permittivity, and a fourth-order tensor describing its electrostrictive response. Closed-form formulas are derived for these effective tensors directly in terms of the corresponding tensors describing the electromechanical response of the underlying matrix and the particles, and the one- and two-point correlation functions describing the microstructure. This is accomplished by specializing a new iterative homogenization theory in finite electroelastostatics (Lopez-Pamies, 2014) to the case of elastic dielectrics with even coupling between the mechanical and electric fields and, subsequently, carrying out the pertinent asymptotic analysis.Additionally, with the aim of gaining physical insight into the proposed solution and shedding light on recently reported experiments, specific results are examined and compared with an available analytical solution and with new full-field simulations for the special case of dielectric elastomers filled with isotropic distributions of spherical particles with various elastic dielectric properties, including stiff high-permittivity particles, liquid-like high-permittivity particles, and vacuous pores.

Incipient mixing by Marangoni effects in slow viscous flow of two immiscible fluid layers

Ignoring inertia, a deformable interface separating two fluid films is considered, subject to non-uniform tension driven by the solutal Marangoni effect in the presence of a scalar concentration field. Detailed description of adsorption kinetics is abrogated by a simple ansatz directly relating interfacial tension and bulk solute concentration. Consequently, the formal mathematical treatment and some of the results share features in common with the Rayleigh–Benard–Marangoni thermocapillary problem. Normal mode perturbation analysis in the limit of small interface deformations establishes the existence of an unstable response for low wavenumber excitation. In the classification of Cross & Hohenberg (1993, Pattern formation outside of equilibrium. Rev. Mod. Phys., 65, 851–1112), both type I and type II behaviour are observed. By considering the zero wavenumber situation exactly, it is proved that all eigenvalues are purely imaginary with non-positive imaginary part; hence, a type III instability is not possible. For characteristic timescales of mass diffusion much shorter than the relaxation time of interfacial fluctuations (infinite crispation number): the response growth rate is obtained explicitly; only a single excitation mode is available, and a complete stability diagram is constructed in terms of the relevant control parameters. Otherwise, from a quiescent base state, an infinite discrete spectrum of modes is observed that exhibit avoided crossing and switching phenomena, as well as exceptional points where stationary state pairs coalesce into a single oscillatory standing wave pattern. A base state plane Poiseuille flow, driven by an external pressure gradient, generally exaggerates the response: growth rates of instabilities are enhanced, and stable decay is further suppressed with increasing base flow speed, but the inherent symmetry breaking destroys stationary and standing wave response. Results are obtained in this most general situation by implementing a numerical Chebyshev collocation scheme. The model was motivated by hydrodynamic processes supposed to be involved in gastric digestion of humans.

Statistical-mechanics based modeling of anisotropic viscoplastic deformation

Statistical mechanics-based coarse graining is used for constitutive modeling of finite elasto-viscoplastic deformation behavior of transversely isotropic materials, without relying on the associated flow rule or classical yield criteria. It was shown previously (Hütter and Tervoort, 2008c) that a detailed expression for the plastic velocity gradient can be obtained in terms of correlations of the fluctuations of the elastic deformation gradient. In this paper, we demonstrate that it is crucial to include cross-correlations between fluctuations of different components of the elastic deformation gradient in order to describe materials with strong plastic anisotropy. Subsequently, the expression for the equivalent stress is obtained from a steady-state analysis during plastic deformation, and is shown to coincide with the Hill equivalent stress in the limit of small elastic deformations. In this case, only two material parameters describe in a self-consistent manner the anisotropic structure of both the plastic velocity gradient and the equivalent stress. Finally, the new constitutive viscoplastic description is successfully applied to describe experimental yield data of oriented isotactic polypropylene, specifically the influence of anisotropy and deformation rate on yielding. Particularly, a step-by-step procedure is given to identify all model parameters.

The overall elastic dielectric properties of a suspension of spherical particles in rubber: An exact explicit solution in the small-deformation limit

A solution is constructed for the homogenization problem of the elastic dielectric response of rubber filled with a random isotropic distribution of polydisperse spherical particles in the classical limit of small deformations and moderate electric fields. In this limit, the overall elastic dielectric response is characterized by five (two elastic, one dielectric, and two electrostrictive) effective constants. Explicit formulas are derived for these constants directly in terms of the corresponding constants describing the elastic dielectric response of the underlying rubber and the filler particles, as well as the concentration of particles. By means of comparisons with finite-element simulations, these formulas are shown to also be applicable to isotropic suspensions of monodisperse spherical particles, provided that the particle concentration is sufficiently away from percolation. With the aim of gaining physical insight into the extreme enhancement in electrostrictive properties displayed by emerging dielectric elastomer composites, specific results are examined for the case of suspensions wherein the rubber is incompressible and the particles are mechanically rigid and of infinite permittivity.

The nonlinear elastic response of suspensions of rigid inclusions in rubber: I—An exact result for dilute suspensions

A solution is constructed for the problem of the overall elastic response of ideal (Gaussian or, equivalently, Neo-Hookean) rubber reinforced by a dilute isotropic distribution of rigid particles under arbitrarily large deformations. The derivation makes use of a novel iterative homogenization technique in finite elasticity that allows to construct exact solutions for the homogenization problem of two-phase nonlinear elastic composites with particulate microstructures. The solution is fully explicit for axisymmetric loading, but is otherwise given in terms of an Eikonal partial differential equation in two variables for general loading conditions. In the limit of small deformations, it reduces to the classical Einstein–Smallwood result for dilute suspensions of rigid spherical particles. The solution is further confronted to 3D finite-element simulations for the large-deformation response of a rubber block containing a single rigid spherical inclusion of infinitesimal size. The two results are found to be in good agreement for all loading conditions. We conclude this work by devising a closed-form approximation to the constructed solution which is remarkably accurate and — as elaborated in Part II — proves particularly amenable as a fundamental building block to generate approximate solutions for suspensions with finite concentration of particles.