neo-Hookean Behavior Research Articles

A complete direct approach to nonlinear modeling of dielectric elastomer plates

In this paper, we present a complete direct approach to nonlinear modeling of thin plates, which are made of incompressible dielectric elastomers. In particular, the dielectric elastomers are assumed to exhibit a neo-Hookean elastic behavior, and the effect of electrostatic forces is incorporated by the purely electrical contribution to the augmented Helmholtz free energy. Our approach does not involve any extraction-type procedure from the three-dimensional energy to derive the plate augmented free energy, but directly postulates the form of this energy for the structural plate problem treated in this paper. Results computed within the framework of this novel approach are compared to results available in the literature as well as to our own three-dimensional finite element solutions. A very good agreement is found.

Swelling-induced twisting and shearing in fiber composites: the effect of the base matrix mechanical response

If a helical network of fibers is embedded in a swellable matrix, and if the fibers themselves resist swelling, then a change in the amount of swelling agent will cause a corresponding twisting motion in the material. This effect has recently been analyzed in highly deformable soft material tubes using the theory of hyperelasticity, suitably modified to incorporate the swelling effect. Those studies examined the effect of spiral angle and fiber-to-matrix inherent stiffness in the context of a ground state matrix material that exhibited classical neo-Hookean behavior in the absence of swelling. While such a ground state material is nonlinear in general, its shear response is linear. As we describe here, it is this shear response that governs the matrix contribution to the twist-swelling interaction. Because gels, elastomers, and even biological tissue can exhibit complex ground state behavior in shear—behavior that may depart significantly from a linear response—we then examine the effect of alternative ground state behaviors on the twist-swelling interaction. The range of behaviors considered includes materials that harden in shear, materials that soften in shear, materials that have an ultimate shear stress bound, and materials that collapse in shear. Matrix materials that either soften or collapse in shear are found to amplify the twisting effect.

Deformation Field in Diametrically Loaded Soft Cylinders

Deformation fields at the surface of diametrically squeezed shallow cylinders in the large deformation regime are measured experimentally and numerically for different material behaviour in the large deformation regime. By means of a digital image correlation method optimized for large displacements, strain fields are measured and compared with finite element simulations. Assuming a neo-Hookean behaviour for cylinders made of rubber silicone, the strain field is found to be in quantitative agreement with non-linear finite element simulations up to the highest deformations reached in our experiments (15%). For materials that follow an elastoplastic constitutive law, agreement is lost after few percents of deformation and location of the strain field differences are identified up to strains as high as 30%. Strain field evolution is also measured for solid foam cylinders up to 60% of global deformation strain. This method that can be applied to a broad variety of materials, even in the occurrence of large deformations, provides a way to study quantitatively local features of the mechanical contact.

Strain smoothing for compressible and nearly-incompressible finite elasticity

We present a robust and efficient form of the smoothed finite element method (S-FEM) to simulate hyperelastic bodies with compressible and nearly-incompressible neo-Hookean behaviour. The resulting method is stable, free from volumetric locking and robust on highly distorted meshes. To ensure inf-sup stability of our method we add a cubic bubble function to each element. The weak form for the smoothed hyperelastic problem is derived analogously to that of smoothed linear elastic problem. Smoothed strains and smoothed deformation gradients are evaluated on sub-domains selected by either edge information (edge-based S-FEM, ES-FEM) or nodal information (node-based S-FEM, NS-FEM). Numerical examples are shown that demonstrate the efficiency and reliability of the proposed approach in the nearly-incompressible limit and on highly distorted meshes. We conclude that, strain smoothing is at least as accurate and stable, as the MINI element, for an equivalent problem size.

Finite Stretching and Shearing of an Internally Balanced Elastic Solid

When seeking to separately describe elastic and inelastic effects in solids undergoing finite strain, it is typical to factor the deformation gradient \(\bf F\) into parts, say \(\mathbf{F}={\hat {\mathbf{F}}}{\mathbf{F}}^{*}\). Such a factorization can also be used to describe a notion of internal elastic balance. For equilibrium deformations this balance emerges naturally from an appropriately formulated energy minimization by considering the variation with respect to the multiplicative decomposition itself. The internal balance requirement is then a tensor relation between F, \({\hat {\mathbf{F}}}\) and F∗. Such theories holds promise for describing various substructural reconfigurations in solids. This includes the development of surfaces that localize slip once a threshold value of load is obtained. We consider an incompressible internally balanced elastic material with a constitutive law that is motivated by neo-Hookean behavior in the standard hyperelastic theory. The role of the internal balance relation is examined in uniaxial loading and also in simple shearing. In uniaxial loading the principle directions remain fixed. In simple shearing the principle directions change as the deformation proceeds. To address the latter, we establish a specific sense in which the internal balance relation can be solved analytically for general isochoric \(\bf F\).

The rheology of non-dilute dispersions of highly deformable viscoelastic particles in Newtonian fluids

AbstractWe present a model for the rheological behaviour of non-dilute suspensions of initially spherical viscoelastic particles in viscous fluids under uniform Stokes flow conditions. The particles are assumed to be neutrally buoyant Kelvin–Voigt solids undergoing time-dependent finite deformations and exhibiting generalized neo-Hookean behaviour in their purely elastic limit. We investigate the effects of the shape dynamics and constitutive properties of the viscoelastic particles on the macroscopic rheological behaviour of the suspensions. The proposed model makes use of known homogenization estimates for composite material systems consisting of random distributions of aligned ellipsoidal particles with prescribed two-point correlation functions to generate corresponding estimates for the instantaneous (incremental) response of the suspensions, together with appropriate evolution laws for the relevant microstructural variables. To illustrate the essential features of the model, we consider two special cases: (i) extensional flow and (ii) simple shear flow. For each case, we provide the time-dependent response and, when available, the steady-state solution for the average particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspensions. The results exhibit shear thickening for extensional flows and shear thinning for simple shear flows, and it is found that the volume fraction and constitutive properties of the particles significantly influence the rheology of the suspensions under both types of flows. In particular, for extensional flows, suspensions of particles with finite extensibility constraints are always found to reach a steady state, while this is only the case at sufficiently low strain rates for suspensions of (less realistic) neo-Hookean particles, as originally reported by Roscoe (J. Fluid Mech., vol. 28, 1967, pp. 273–293) and Gao et al. (J. Fluid Mech., vol. 687, 2011, pp. 209–237). For shear flows, viscoelastic particles with high viscosities can experience a damped oscillatory motion of decreasing amplitude before reaching the steady state.

Development of an inverse approach for the characterization of in vivo mechanical properties of the lower limb muscles.

The purpose of this study was to develop an inverse method, coupling imaging techniques with numerical methods, to identify the muscle mechanical behavior. A finite element model updating (FEMU) was developed in three main interdependent steps. First, a 2D FE modeling, parameterized by a Neo-Hookean behavior (C10 and D), was developed from a segmented thigh muscle 1.5T MRI (magnetic resonance imaging). Thus, a displacement field was simulated for different static loadings (contention, compression, and indentation). Subsequently, the optimal mechanical test was determined from a sensitivity analysis. Second, ultrasound parameters (gain, dynamic, and frequency) were optimized on the thigh muscles in order to apply the digital image correlation (DIC), allowing the measurement of an experimental displacement field. Third, an inverse method was developed to identify the Neo-Hookean parameters (C10 and D) by performing a minimization of the distance between the simulated and measured displacement fields. To replace the experimental data and to quantify the identification error, a numerical example was developed. The result of the sensitivity analysis showed that the compression test was more adapted to identify the Neo-Hookean parameters. Ultrasound images were recorded with a frequency, gain, and dynamic of 9 MHz, 34 dB, 42 dB, respectively. In addition, the experimental noise on displacement field measurement was estimated to be 0.2 mm. The identification performed on the numerical example revealed a low error for the C10 (<3%) and D (<7%) parameters with the experimental noise. This methodology could have an impact in the scientific and medical fields. A better knowledge of the muscle behavior will help to follow treatment and to ensure accurate medical procedures during the use of robotic devices.

On compressible versions of the incompressible neo-Hookean material

We consider three different compressible versions of the conventional incompressible neo-Hookean material model. The different versions are not new and have been used in various model studies. They each give neo-Hookean behavior in an appropriate incompressible limit. The three versions each show some basic differences with respect to each other as regards the qualitative nature of the approach to the neo-Hookean limit. The purpose of this study is to exhibit these differences.

On the macroscopic response, microstructure evolution, and macroscopic stability of short-fibre-reinforced elastomers at finite strains: I – Analytical results

This paper presents a homogenization-based constitutive model for the mechanical behaviour of particle-reinforced elastomers with random microstructures subjected to finite deformations. The model is based on a recently improved version of the tangent second-order (TSO) method (Avazmohammadi and Ponte Castañeda, J. Elasticity 112 (2013) p.139–183) for two-phase, hyperelastic composites and is able to directly account for the shape, orientation, and concentration of the particles. After a brief summary of the TSO homogenization method, we describe its application to composites consisting of an incompressible rubber reinforced by aligned, spheroidal, rigid particles, undergoing generally non-aligned, three-dimensional loadings. While the results are valid for finite particle concentrations, in the dilute limit they can be viewed as providing a generalization of Eshelby’s results in linear elasticity. In particular, we provide analytical estimates for the overall response and microstructure evolution of the particle-reinforced composites with generalized neo-Hookean matrix phases under non-aligned loadings. For the special case of aligned pure shear and axisymmetric shear loadings, we give closed-form expressions for the effective stored-energy function of the composites with neo-Hookean matrix behaviour. Moreover, we investigate the possible development of “macroscopic” (shear band-type) instabilities in the homogenized behaviour of the composite at sufficiently large deformations. These instabilities whose wavelengths are much larger than the typical size of the microstructure are detected by making use of the loss of strong ellipticity condition for the effective stored-energy function of the composites. The analytical results presented in this paper will be complemented in Part II (Avazmohammadi and Ponte Castaneda, Phil. Mag. (2014)) of this work by specific applications for several representative microstructures and loading configurations.

Local axial compressive mechanical properties of human carotid atherosclerotic plaques—characterisation by indentation test and inverse finite element analysis

The fibrous cap of an atherosclerotic plaque may be prone to rupture if the occurring stresses exceed the strength of the cap. Rupture can cause acute thrombosis and subsequent ischaemic stroke or myocardial infarction. A reliable prediction of the rupture probability is essential for the appropriate treatment of atherosclerosis. Biomechanical models, which compute stresses and strain, are promising to provide a more reliable rupture risk prediction. However, these models require knowledge of the local biomechanical properties of atherosclerotic plaque tissue. For this purpose, we examined human carotid plaques using indentation experiments. The test set-up was mounted on an inverted confocal microscope to visualise the collagen fibre structure during the tests. By using an inverse finite element (FE) approach, and assuming isotropic neo-Hookean behaviour, the corresponding Young's moduli were found in the range from 6 to 891kPa (median 30kPa). The results correspond to the values obtained by other research groups who analysed the compressive Young's modulus of atherosclerotic plaques. Collagen rich locations showed to be stiffer than collagen poor locations. No significant differences were found between the Young's moduli of structured and unstructured collagen architectures as specified from confocal collagen data. Insignificant differences between the middle of the fibrous cap, the shoulder regions, and remaining plaque tissue locations indicate that axial, compressive mechanical properties of atherosclerotic plaques are independent of location within the plaque.

Folding wrinkles of a thin film stiff layer on a soft substrate

[This corrects the article DOI: 10.1098/rspa.2011.0567.].

Deformation and flow of matter: Interrogating the physics of materials using rheological methods

Rheological measurements offer a unique means of interrogating the physics of amorphous solids, including crosslinked rubbers and polymeric glasses. Here, we present several vignettes to demonstrate the ability of both classical and novel rheological experiments to resolve important questions in condensed matter physics. First, results from torque and normal force measurements aimed at understanding the thermodynamics and mechanics of polymer networks in both dry and swollen states are presented. In particular, we examine the validity of the Frenkel–Flory–Rehner theory of rubber network swelling. Torsion and normal force measurements are also described for a series of polymeric glasses that exhibit similar shear moduli but, surprisingly, very different normal force responses, with one set of materials showing extreme deviations from neo-Hookean behavior and the other being close to neo-Hookean. We then describe the use of a novel torsional dilatometer, which allows simultaneous measurement of mechanical properties and volume recovery, to investigate the aging and rejuvenation behaviors of glassy polymers. Also, the temperature dependence of dynamics is probed in glassy polymers that have been aged into equilibrium below the nominal glass transition temperature and evidence is presented that time-scale divergence may not be a true signature of the glass transition itself. Finally, we describe a reduction in scale of the classical membrane inflation test to allow measurement of the biaxial creep compliance of nanometer thick polymeric films using an atomic force microscope. In each instance, emphasis is placed on how the measurements are designed to interrogate the physics of interest in the materials investigatedx.

Finite Element Analysis Animated Simulation of Velopharyngeal Closure

To use finite element analysis animated simulations to investigate factors affecting velopharyngeal closure. A coronal section multicomponent finite element analysis model of a human soft palate was created in Simulia Abaqus 6.5-1 from high resolution MRI images of a single adult female subject, interpreted by reference to published anatomic dissections. Tissues were assigned hyperelastic property coefficients for neo-Hookean behavior, with gravity at 9.8 ms(-2) in the y-axis. Vector forces based on estimations in previous publications were applied throughout levator veli palatini and palatopharyngeus muscles, using a nonlinear analysis algorithm, to produce animated simulations of velopharyngeal space closure. Variation of levator veli palatini angle from 60° to 49°, the contribution of palatopharyngeus muscle, and the effect of submucous cleft were investigated for their effects on velopharyngeal closure. The animated simulations showed anthropomorphic behavior and supported the previously suggested effects of the levator veli palatini angle, with reduced effectiveness of velopharyngeal closure as levator veli palatini angle decreases. Palatopharyngeus action reduced the efficiency of closure for a levator veli palatini angle of 60°, and a submucous cleft reduced this for both our normal subject and for a levator veli palatini angle of 60°, but both palatopharyngeus action and a submucous cleft enhanced closure for a levator veli palatini angle of 49°. This study advances soft palate finite element analysis to a real-subject-based multicomponent hyperelastic model that demonstrates anthropomorphic behavior. Animated simulations using the model demonstrate the possible effects of levator veli palatini angle, a submucous cleft, and the contribution of the palatopharyngeus.

Folding wrinkles of a thin stiff layer on a soft substrate

We present the mechanics of folding surface-layer wrinkles on a soft substrate, i.e. inter-touching of neighbouring wrinkle surfaces without forming a cusp. Upon laterally compressing a stiff layer attached on a finite-elastic substrate, certain material nonlinearities trigger a number of bifurcation processes to form multi-mode wrinkle clusters. Some of these clusters eventually develop into folded wrinkles. The first bifurcation of the multi-mode wrinkles is investigated by a perturbation analysis of the surface-layer buckling on a pre-stretched neo-Hookean substrate. The post-buckling equilibrium configurations of the wrinkles are then trailed experimentally and computationally until the wrinkles are folded. The folding process is observed at various stages of wrinkling, by sectioning 20–80 nm thick gold films deposited on a polydimethylsiloxane substrate at a stretch ratio of 2.1. Comparison between the experimental observation and the finite-element analysis shows that the Ogden model deformation of the substrate coupled with asymmetric bending of the film predicts the folding process closely. In contrast, if the bending stiffness of the film is symmetric or the substrate follows the neo-Hookean behaviour, then the wrinkles are hardly folded. The wrinkle folding is applicable to construction of long parallel nano/micro-channels and control of exposing functional surface areas.

On the consequences of non linear constitutive modelling of brain tissue for injury prediction with numerical head models

The objective of this work was to investigate the influences of constitutive non linearities of brain tissue in numerical head model simulations by comparing the performance of a recently developed non linear constitutive model [10, 11] with a simplified version, based on neo-Hookean elastic behaviour, and with a previously developed constitutive model [6]. Numerical simulation results from an existing 3D head model in the explicit Finite Element code MADYMO were compared. A head model containing a sliding interface between the brain and the skull was used and results were compared with the results obtained with a previously validated version possessing a tied skull-brain interface. For these head models, the effects of different constitutive models were systematically investigated for different loading directions and varying loading amplitudes in both translation and rotation. In the case of the simplified and fully non linear version of the model of Hrapko et al. [10, 11], the response predicted with a head model for varying conditions (i.e. severity and type of loading) varies consistently with the constitutive behaviour. Consequently, when used in a finite element head model, the response can be scaled according to the constitutive model used. However, the differences found when using the non linear model of Brands et al. [5] were dependent on the loading conditions. Hence this model is less suitable for use in a numerical head model.

The stress field in a pulled cork and some subtle points in the semi-inverse method of nonlinear elasticity

In an attempt to describe cork-pulling, we model a cork as an incompressible rubber-like material and consider that it is subject to a helical shear deformation superimposed onto a shrink fit and a simple torsion. It turns out that this deformation field provides an insight into the possible appearance of secondary deformation fields for special classes of materials. We also find that these latent deformation fields are woken up by normal stress differences. We present some explicit examples based on the neo-Hookean, the generalized neo-Hookean and the Mooney–Rivlin forms of the strain-energy density. Using the simple exact solution found in the neo-Hookean case, we conjecture that it is advantageous to accompany the usual vertical axial force by a twisting moment, in order to extrude a cork from the neck of a bottle efficiently. Then we analyse departures from the neo-Hookean behaviour by exact and asymptotic analyses. In that process, we are able to give an elegant and analytic example of secondary (or latent) deformations in the framework of nonlinear elasticity.

Non-linear, rate-dependent strain-hardening behavior of polymer glasses

This study is concerned with the finite, large strain deformation behavior of polymeric glasses. True stress–strain curves in uniaxial compression obtained for five different polymeric glasses: polycarbonate, polystyrene, poly(2,6-dimethyl-1,4-phenylene oxide), and linear and cross-linked poly(methylmethacrylate), revealed a strain-hardening response during plastic deformation that is strain-rate dependent and deviates from neo-Hookean behavior. An empirical modification of the so-called compressible Leonov model by a strain dependent activation volume is suggested, which describes the strain-rate dependent large strain behavior of these glassy polymers in good agreement with experimental data.

Instability analysis in pressurized transversely isotropic Saint-Venant–Kirchhoff and neo-Hookean cylindrical thick shells

Under the effect of an inner pressure, a thick hyperelastic shell cylinder is susceptible to developing mechanical instabilities, leading to a bifurcated shape that no longer has the initial cylindrical symmetry. The considered problem has a strong resonance in a biomedical context, considering pathologies encountered for arteries, such as aneurysm. In this contribution, the mechanical behavior of thick elastic shells has been analyzed, considering a thick-walled cylindrical hyperelastic model material obeying a transversely isotropic behavior, first in a large-displacement situation and then in a large-deformation case. The response of the material is assumed to be instantaneous, so that time-dependent effects shall not be considered in this paper. The case of a Saint-Venant–Kirchhoff material is considered with special emphasis to exemplify a large-displacement small-strain situation; the neo-Hookean behavior is next considered to enlarge the constitutive law toward consideration of large strains. The stability conditions of the shell are studied and bifurcation conditions formulated in terms of the applied pressure and of the geometrical and mechanical parameters that characterize the shell. Analytical solutions of some bifurcation points are evidenced and calculated when the direction of the fibers coincide with the cylinder axis.

Chemical structure—normal force relationships in polymer glasses

In previous work, we found that the normal force responses in poly(methyl methacrylate) (PMMA) and poly(ethyl methacrylate) (PEMA) subjected to torsional deformations were strongly influenced by their prominent β-relaxation mechanisms which are related to the n-alkyl methacrylate side group motions. Using the concept of an elastic stress relaxing material we were able to extract the derivatives of the strain energy density function with respect to the first and second invariants of the deformation tensor ( W 1 and W 2). In the present work, we report results from two materials, polysulfone (PSF) and polycarbonate (PC), with weak β-relaxations that are due to main chain motions. We find that for the two materials, the difference from neo-Hookean behavior is real though very small, contrary to the large deviations from neo-Hookean behavior observed for PMMA and PEMA. The results support the contention that the large normal force responses and the consequent strongly non neo-Hookean behavior of the glassy n-alkyl methacrylates is due to the side chain.

Analysis of pulled axisymmetric membranes with wrinkling

The nonlinear behavior of an axisymmetric hyperelastic membrane subjected to pulling forces is analyzed. The membrane is considered to be ideal in the sense that it cannot carry compressive stress resultants. If the membrane has a positive initial Gaussian curvature, the pulling gives rise to wrinkles which form over parts of the surface. The full nonlinear equations governing the membrane behavior in the doubly tense and in the wrinkled regions are formulated, and then solved using a numerical integration procedure. Solutions for various examples are presented, with Hookean and neo-Hookean constitutive behavior. These include a few examples of wrinkled membranes with positive initial Gaussian curvatures, and one example of a membrane with a negative initial Gaussian curvature, where no wrinkles are formed.