Gent Model Research Articles

A comparison of hyperelastic constitutive models applicable to brain and fat tissues.

In some soft biological structures such as brain and fat tissues, strong experimental evidence suggests that the shear modulus increases significantly under increasing compressive strain, but not under tensile strain, whereas the apparent Young's elastic modulus increases or remains almost constant when compressive strain increases. These tissues also exhibit a predominantly isotropic, incompressible behaviour. Our aim is to capture these seemingly contradictory mechanical behaviours, both qualitatively and quantitatively, within the framework of finite elasticity, by modelling a soft tissue as a homogeneous, isotropic, incompressible, hyperelastic material and comparing our results with available experimental data. Our analysis reveals that the Fung and Gent models, which are typically used to model soft tissues, are inadequate for the modelling of brain or fat under combined stretch and shear, and so are the classical neo-Hookean and Mooney–Rivlin models used for elastomers. However, a subclass of Ogden hyperelastic models are found to be in excellent agreement with the experiments. Our findings provide explicit models suitable for integration in large-scale finite-element computations.

Limit point instability in pressurization of anisotropic finitely extensible hyperelastic thin-walled tube

Mechanical responses of materials undergoing large elastic deformations can exhibit a loss of stability in several ways. Such a situation can occur when a thin-walled cylinder is inflated by an internal pressure. The loss of stability is manifested by a non-monotonic relationship between the inflating pressure and internal volume of the tube. This is often called limit point instability. The results, known from the literature, show that isotropic hyperelastic materials with limiting chain extensibility property always exhibit a stable response if the extensibility parameter of the Gent model satisfies Jm<18.2. Our study investigates the same phenomenon but for tubes with anisotropic form of the Gent model (finite extensibility of fibers). Anisotropy, used in our study, increases the number of material parameters the consequence of which is to increase degree of freedom of the problem. It will be shown that, in stark contrast to isotropic material, the unstable response is predicted not only for large values of Jm but also for Jm≈1 and smaller, and that the existence of limit point instability significantly depends on the orientation of preferred directions and on the ratio of linear parameters in the strain energy density function (this ratio can be interpreted as the ratio of weights by which fibers and matrix contribute to the strain energy density). Especially tubes reinforced with fibers oriented closely to the longitudinal direction are susceptible to a loss of monotony during pressurization.

Influence of stiffening on elastic wave propagation in extremely deformed soft matter: from nearly incompressible to auxetic materials

We analyse the propagation of elastic waves in soft materials subjected to finite deformations. We derive explicit phase velocity relations for matter with pronounced stiffening effect, namely Gent model, and apply these results to study elastic wave propagation in (a) nearly incompressible materials such as biological tissues and polymers, (b) highly compressible and (c) negative Poisson’s ratio or auxetic materials. We find, that for nearly incompressible materials transverse wave velocities exhibit strong dependence on the direction of propagation and initial strain state, whereas the longitudinal wave velocity is not affected significantly until extreme levels of deformation are attained. For highly compressible materials, we show that both longitudinal and transversal wave velocities depend strongly on deformation and direction of propagation. Moreover, the dependence becomes stronger when stiffening effects increase. When compression is applied, the longitudinal wave velocity decreases regardless the direction of wave propagation in highly compressible materials, and increases for most of the directions in materials with negative Poisson’s ratio behaviour. We demonstrate that finite deformations can influence elastic wave propagation through combinations of induced effective compressibility and stiffness.

Crack tip fields in soft elastic solids subjected to large quasi-static deformation — A review

We review analytical solutions for the asymptotic deformation and stress fields near the tip of a crack in soft elastic solids. These solutions are based on finite strain elastostatics and hyperelastic material models, and exhibit significantly different characteristics than the classical crack tip field solutions in linear elastic fracture mechanics. Specifically, we summarize some available finite strain crack tip solutions for two dimensional cracks, namely that plane strain, plane stress, and anti-plane shear cracks in a certain class of homogeneous materials. Interface cracks between soft elastic solids and a rigid substrate are also discussed. We focus on incompressible material models with various degrees of strain stiffening effect such as generalized neo-Hookean model, exponentially hardening model and Gent model. We also explored the physical implications of the crack tip fields, and highlighted pitfalls in the applications of these solutions, particularly the J-integral and the distribution of true stress in the deformed configuration which have not been discussed in the literature.

Inflating a flat toroidal membrane

A flat toroidal geometry, composed of a stack of two flat annular membranes bonded together at the inner and outer boundaries (equators), is proposed. The finite inflation mechanics of the composite membrane under uniform internal pressure is considered. The membranes are assumed to be identical homogeneous and isotropic solids described by the Gent hyperelastic model with a relaxed strain energy density function. The variational inflation mechanics problem yields the governing equations and boundary conditions, which are recast as a nonlinear two-point boundary value problem in a suitable form for efficient computation. The effects of the inflation pressure and material properties on the state of stretch and geometry of the inflated toroidal membrane are obtained, which exhibit a number of desirable features. The phenomenon of wrinkling instability is observed near the outer equator of the torus at low pressures (in spite of a positive Gaussian curvature), which is exactly opposite to the behavior observed in a curved torus (wrinkling at the inner equator at high pressure). Semi-analytical expressions of the Cauchy stress resultants are obtained which match well with the numerical computations. Further, the limit-point instability pressure of the toroidal membrane is shown to have a remarkable power-law dependence on a geometric parameter of the uninflated flat torus and the material chain extensibility parameter involving two universal constants.

Bayesian model selection of hyperelastic models for simple and pure shear at large deformations

Hyperelastic materials are extensively employed in a wide range of applications. Although there are well-established models for describing the mechanical behavior of the hyperelastic materials, relatively few papers have attempted to rank different models. This paper aims to identify parameters of some constitutive models for pure and simple shear of an incompressible isotropic hyperelastic material under large deformations, and also aims to propose a strategy to rank different models. The constitutive models considered in the present analysis are the following: Mooney–Rivlin, Yeoh, Ogden (1 and 2 terms), Lopez-Pamies (1 and 2 terms), and Gent. In the first part of the paper, the Bayesian framework is applied for the identification of the parameters of the models, where experimental data are used to update the prior probabilistic model of the unknown parameters. The Maximum a Posteriori Estimate is obtained, and the error between the model prediction and the experimental data is computed to rank the models. In the second part of the paper, the Bayesian framework is again employed, but now as an strategy for model selection. Instead of ranking the models using the error between the model prediction and the experimental result, more ingredients, such as the Ockham factor, are taken into account for the model selection. The results indicate that all models and experimental results for pure shear are in good agreement, but Mooney–Rivlin, Gent and Yeoh models were not able to well describe the available experimental data from simple shear. The best model for pure shear is Mooney–Rivlin and the best model for simple shear is Ogden (2 terms), considering the available experimental data and the criteria proposed in the present paper.

Analysis of uniaxial Tension and circumferential Inflation on the Mechanical Property of Arterial Wall

Considering the incompressible strain hardening materials, a new strain energy density function is modified from Gent Model, which fulfills the requirement that the modified strain energy function will transform into Gent model with n=1 . When n=1 and Jm goes into infinity ,the modified strain energy density function can be transformed into neo-hooken model. Arterial wall was considered as incompressible hyper-elastic materials. Uniaxial tension and inflation of arterial wall was been researched by utilizing the modified strain energy function. Influence of constitutive parameters on the mechanical property of arterial wall has been researched. In order to verify the theoretical theory, non-linear finite element analysis of arterial wall was proposed. By utilizing the commercial finite element software MSC.Marc, numerical simulation on mechanical property of arterial wall was carried out based on the modified strain energy function, which showed the finite element results agree with the theoretical results properly. The discussions illustrate that new constitutive model describes the finite deformation property of arterial wall reasonably and the applied range has been broadened by using the new constitutive model.DOI: http://dx.doi.org/10.5755/j01.mech.21.2.8743

Material properties and constitutive modeling of infant porcine cerebellum tissue in tension at high strain rate.

BackgroundThe mechanical characterization of infant porcine cerebellum tissue in tension at high strain rate is crucial for modeling traumatic cerebellum injury, which is in turn helpful for understanding the biomechanics of such injuries suffered in traffic accidents.Material and MethodIn this study, the infant porcine cerebellum tissue was given three loading velocities, ie, 2s-1, 20s-1 and 100s-1 with up to 30% strain to investigate the tensile properties. At least six tensile tests for each strain rate were validly performed. Fung, Gent, Ogden and exponential models were applied to fit the constitutive equations, so as to obtain material parameters from the experimental data.ResultsThe Lagrange stress of infant porcine cerebellum tissue in tension appeared to be no more than 3000Pa at each loading velocity. More specifically, the Lagrange stress at 30% strain was (393.7±84.4)Pa, (928.3±56.3)Pa and (2582.4±282.2)Pa at strain rates of 2s-1, 20s-1 and 100s-1, respectively. Fung (0.833≤R2≤0.924), Gent (0.797≤R2≤0.875), Ogden (0.859≤R2≤0.944) and exponential (0.930≤R2≤0.972) models provided excellent fitting to experimental data up to 30% strain.ConclusionsThe infant cerebellum tissue shows a stiffer response with increase of the loading speed, indicating a strong strain-rate sensitivity. This study will enrich the knowledge on the material properties of infant brain tissue, which may augment the biofidelity of finite element model of human pediatric cerebellum.

A new free energy model for predicting the qual-biaxial tests of dielectric elastomers

Dielectric elastomeric actuators (DEAs) have been intensely studied in the recent decades. Their attractive features include large deformation(380%), large energy density(3.4 J/g), light weight, fast response( 1 ms), and high efficiency (80%-90%). They can be used in medical devices, space robotices and energy harvesters. The core part of DEAs is a dielectric elastomeric film with two electordes. When pre-stretched forces are exerted on the film in plane direction and voltage is applied across its thickness, the film achieves a large deformation. Usually the effect of electric field is described by Maxwell stress E2, and the effect of mechanical field is described by free energy function models (such as Neo-Hookean model, Arruda-Boyce model and Gent model). There are deviations in varying degree between every models and tests of dielectric elastomer. No model works perfectly. In the present paper, a new free energy function model is given to reduce the deviation. According to the main models above, an undetermined parameter C(1, 2) is introduced. and i (W/i)= C( 1, 2)(i2- 1-2 2-2), pi = C( 1, 2)( pi2- p1-2p2-2)(i/ pi), i = 1, 2, are assumed. The new i ( W/i) and pi are substituted into the equation of equilibrium of dielectric elastomer film pi + E2 = i ( W/i), i = 1, 2. Under equal-biaxial pre-stretched condition, P1 = P2 = P, p1 = p2 = p, C(1, 2) = C(). The parameter C()= (V2/t0)2/( 2- -4-( p- p-4)(/ p)) is obtained. Through analysing the test results of VHB4905 which contains a series of equal-biaxial pre-stretched tests, the data (, C()) are obtained from the test data (, V). C() =a + beI1-3, (I1 = 12 + 22 + 32) can be determined by data points (, C()). By computing the integral of i ( W/i)= a + beI1-3)(i2- 1-2 2-2), i = 1, 2, a new free energy function W = (a/2)(I1-3) + b[eI1-3(I1-3-1) + 1] (the new model) is achieved. The test results of VHB4905 are fitted by Neo-Hookean, Gent model and the new model. Neo-Hookean model fits well only in small deformation. Gent model fits well only in small-middle deformation, and does not work well when stretch 3.5. The new model fits well in small, middle and large deformation. It is better than Neo-Hookean and Gent model. The new model can give big support in the study of dielectric elastomer materials and structure property, and can be used in engineering practice effectively.

It Never Rains in America?

It Never Rains in America? Elya J. Zhang (bio) Sheldon Garon. Beyond Our Means: Why America Spends While the World Saves. Princeton, N.J.: Princeton University Press, 2012. 475pp. Illustrations, notes and index. $29.95. In one of Aesop’s Fables, an ant, having laid away enough grain for the bitter winter, mocked a grasshopper who asked him for food, saying: “Since you could sing all summer, you may dance all winter.” In Stuart-era England, Daniel Defoe, most famous for writing Robinson Crusoe and Moll Flanders, filled popular advice books with warnings against debt: “For a tradesman to borrow money upon interest, I take to be like a man going into a house infected with the plague.” Toward the end of the Tokugawa Shogunate, Ninomiya Kinjirō, Japan’s “Peasant Saint,” left behind his most famous phrase, “Work hard, spend little. Gather much firewood, but burn little.” In his 2012 book Beyond Our Means, Sheldon Garon connects many historical dots and presents a systematic account of household savings in thirteen countries from the eighteenth century to the present. This global account begins from a strong contemporary American agenda, as suggested by its subtitle, Why America Spends While the World Saves. Garon has spent a good part of his career studying Japan and is known for his perceptive insights into the Japanese state’s “social management” program in the modern era (Molding Japanese Minds, 1997; The State and Labor in Modern Japan, 1987). Having seen firsthand how, in East Asia, the small household savings contributed both to national and individual fortunes, Garon was intrigued by an important question even before the 2008 financial crisis exposed America’s debt problem. Why do Asians save so much and Americans so little? Don’t Americans need to save for a rainy day? It surely rains in America. To fall back on cultural explanations like Asia’s “Confucian-Buddhist heritage” has serious limitations, since leading European nations have also saved at high rates in history. Thus, in this study, Garon goes far beyond the American and Japanese borders and tries to make sense of the rival trends of saving and debt that cohabit the capitalist world. Beyond Our Means is comprised of twelve chapters, with roughly four on America, four on Japan, three on Western European nations, and one on other [End Page 756] East and Southeast Asian nations. The story spans four periods: the 1730s to the 1860s, the origin of national thrift; the 1860s to the 1900s, the rise of systems of postal savings and schools savings; the 1910s to the 1940s, campaigns for the great wars; and the 1950s to the 2000s, the balance of consumption, saving, and investment. Modern thrift promotion, in Garon’s account, originated in the wheels of Protestantism and the Industrial Revolution. Authorities in Western Europe found saving the best “vaccine” against the ailments of society—the logic being that the poor who save don’t rebel; thus German burghers established the first savings bank, Sparkassen, in 1778, and the British Parliament passed the Savings Banks Act in 1817. Both acts aimed to integrate the humbler class as self-disciplined members of the community. The British government then pioneered the combination of savings banks with its nationwide post offices. The new Post Office Savings Bank was an instant success, because its accessible, secure, and confidential accounts freed working-class savers from irritating paternalism. Meanwhile, the Belgians became the internationally recognized leader in school savings with its Gent Model, where teachers collected pupils’ coins on behalf of the state bank. By the early 1900s, the small savings deposited in post offices and savings banks had accumulated to vast pools of capital for European state agencies, especially when they needed to bypass the National Assembly or Parliament and sustained growing expenditures. On the other side of the world, Japan’s new Meiji government adopted postal savings “at the stroke of a pen” and accumulated eleven million deposit accounts by the 1910s (p. 144). Thrift promotion had evolved from a community-based charity act to a multinational trend; and Britain, Belgium, and Japan admired each other’s state mobilizing power from afar. The overwhelming costs of the two world wars further...

Large amplitude oscillatory shear studies on the strain-stiffening behavior of gelatin gels

Linear and nonlinear viscoelasticity of gelatin solutions was investigated by rheology. The dynamic mechanical properties during the sol-gel transition of gelatin followed the time-cure superposition. The fractal dimension df of the critical gel was estimated as 1.76, which indicated a loose network. A high sol fraction ws = 0.61 was evaluated from the plateau modulus by semi-empirical models. Strain-stiffening behavior was observed under large amplitude oscillatory shear (LAOS) for the gelatin gel. The strain and frequency dependence of the minimum strain modulus GM, energy dissipation Ed, and nonlinear viscoelastic parameter NE was illustrated in Pipkin diagrams and explained by the strain induced helix formation reported previously by others. The BST model described the strain-stiffening behavior of gelatin gel quite well, whereas the Gent and worm-like chain network models overestimated the strain-stiffening at large strains.

Applicability of a particularly simple model to nonlinear elasticity of slide-ring gels with movable cross-links as revealed by unequal biaxial deformation.

The strain energy density function (F) of the polyrotaxane-based slide-ring (SR) gels with movable cross-links along the network strands is characterized by unequal biaxial stretching which can achieve various types of deformation. The SR gels as prepared without any post-preparation complication exhibit considerably smaller values of the ratio of the stresses (σy/σx) in the stretched (x) and constrained (y) directions in planar extension than classical chemical gels with heterogeneous and nearly homogeneous network structures do. This feature of the SR gels leads to the peculiar characteristic that the strain energy density function (F) has no explicit cross term of strains in different directions, which is in contrast to F with explicit strain cross terms for most chemical gels and elastomers. The biaxial stress-strain data of the SR gels are successfully described by F of the Gent model with only two parameters (small-strain shear modulus and a parameter representing ultimate elongation), which introduces the finite extensibility effect into the neo-Hookean model with no explicit cross term of strain. The biaxial data of the deswollen SR gels examined in previous study, which underwent a considerable reduction in volume from the preparation state, are also well described by the Gent model, which is in contrast to the case of the classical chemical gels that the stress-strain relations before and after large deswelling are not described by a common type of F due to a significant degree of collapse of the network strands in the deswollen state. These intriguing features of nonlinear elasticity of the SR gels originate from a novel function of the slidable cross-links that can maximize the arrangement entropy of cross-linked and non-cross-linked cyclic molecules in the deformed networks.

Anisotropic hyperelastic behavior of soft biological tissues

Constitutive laws are fundamental to the studies of the mechanically dominated clinical interventions involving soft biological tissues which show a highly anisotropic hyperelastic mechanical properties. The purpose of this paper was to develop an improved constitutive law based on the Holzapfel–Gasser–Ogden's model: to replace the isotropic part with Gent constitutive law so as to model the noncollagenous matrix of the media due to its generality and capability to reproduce the Neo-Hookean model. This model is implemented into an in-house finite element program. A uniaxial tension test is considered to study the influence of material parameter in Gent model and which represents the angle between the collagen fibers and the circumferential direction. A simulation of an adventitial strip specimen under tension is performed to show the applicability of this constitutive law.

Gent models for the inflation of spherical balloons

We revisit an iconic deformation of non-linear elasticity: the inflation of a rubber spherical thin shell. We use the 3-parameter Mooney and Gent-Gent (GG) phenomenological models to explain the stretch–strain curve of a typical inflation, as these two models cover a wide spectrum of known models for rubber, including the Varga, Mooney–Rivlin, one-term Ogden, Gent-Thomas and Gent models. We find that the basic physics of inflation exclude the Varga, one-term Ogden and Gent-Thomas models. We find the link between the exact solution of non-linear elasticity and the membrane and Young–Laplace theories often used a priori in the literature. We compare the performance of both models on fitting the data for experiments on rubber balloons and animal bladder. We conclude that the GG model is the most accurate and versatile model on offer for the modelling of rubber balloon inflation.

A comparison of limited-stretch models of rubber elasticity

In this paper we describe various limited-stretch models of non-linear rubber elasticity, each dependent on only the first invariant of the left Cauchy–Green strain tensor and having only two independent material constants. The models are described as limited-stretch, or restricted elastic, because the strain energy and stress response become infinite at a finite value of the first invariant. These models describe well the limited stretch of the polymer chains of which rubber is composed. We discuss Gent׳s model which is the simplest limited-stretch model and agrees well with experiment. Various statistical models are then described: the one-chain, three-chain, four-chain and Arruda–Boyce eight-chain models, all of which involve the inverse Langevin function. A numerical comparison between the three-chain and eight-chain models is provided. Next, we compare various models which involve approximations to the inverse Langevin function with the exact inverse Langevin function of the eight-chain model. A new approximate model is proposed that is as simple as Cohen׳s original model but significantly more accurate. We show that effectively the eight-chain model may be regarded as a linear combination of the neo-Hookean and Gent models. Treloar׳s model is shown to have about half the percentage error of our new model but it is much more complicated. For completeness a modified Treloar model is introduced but this is only slightly more accurate than Treloar׳s original model. For the deformations of uniaxial tension, biaxial tension, pure shear and simple shear we compare the accuracy of these models, and that of Puso, with the eight-chain model by means of graphs and a table. Our approximations compare extremely well with models frequently used and described in the literature, having the smallest mean percentage error over most of the range of the argument.

The Gent model for rubber-like materials: An appraisal for an ingenious and simple idea

We review the main aspects of the celebrated Gent constitutive model for rubberlike materials. Emphasis is placed on the case of damageable materials describing possible damage and deformation localization.

The remarkable Gent constitutive model for hyperelastic materials

In 1996, Alan Gent published a short paper that proposed the use of a very simple two parameter phenomenological constitutive model for hyperelastic isotropic incompressible materials. The model is empirical but has the advantages of mathematical simplicity, reflects the severe strain-stiffening at large strains observed experimentally, reduces to the classic neo-Hookean model for small strains and involves just two material parameters namely the shear modulus for infinitesimal deformations and a parameter that measures a maximum allowable value of strain. The model reflects the limiting chain extensibility characteristic of non-Gaussian molecular models for rubber. Here we review some of the numerous developments, extensions and widespread applications that have resulted from that groundbreaking paper not only in rubber elasticity but also in the area of biomechanics of soft biomaterials. The Gent model is remarkably robust: its mathematical simplicity combined with physical basis has ensured that it has reached status as a fundamental canonical phenomenological constitutive model for hyperelastic materials.

Electromechanical response and failure modes of a dielectric elastomer tube actuator with boundary constraints

As a widely used configuration for dielectric elastomer (DE) actuators, DE tube actuators (or cylindrical actuators) are also found to be susceptible to electromechanical instability (EMI), which may lead to a premature electrical breakdown (EB), and inhibit the potential actuation of DE actuators. This work investigates the electromechanical response of a DE tube actuator with and without boundary constraints to demonstrate an alternative to avoid EMI while achieving large actuation. Our simulation results based on the Gent strain energy model show that the EMI of a DE tube actuator can be eliminated, and larger actuation deformation can be achieved by applying boundary constraints. As a result of these constraints, consideration is also given to the possible mechanical buckling failure that may occur. Mechanisms of possible failure modes of constrained and unconstrained DE tube actuators, such as electromechanical instability, electrical breakdown and mechanical buckling, are elucidated. This paper should provide better theoretical guidance on how to improve the actuation performance of DE actuators, thus leading to the optimal design of DE-based devices.

Phase diagrams and interface in inflating balloon

When a cylindrical balloon is inflated, inflation often takes place nonuniformly, inflated near one end first and followed by the growth of this part along the entire cylinder. This phenomenon is discussed from the phase‐transition perspective in terms of the Gent model, a free‐energy model that has the same structure as that of the gas–liquid transition in normal fluids. A phase diagram which describes the inflation behavior is constructed, showing the binodal and spinodal curves terminated at a critical point. The hysteresis effects are discussed based on the phase diagram. The interface between the coexisting inflated and weakly inflated regions along the cylindrical balloon is also examined, and the interfacial thickness and the interfacial energy are numerically calculated. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1393–1399, 2014

An invariant based damage model of stress-softening

A phenomenological model to predict the Mullins stress-softening effect in an isotropic, incompressible, hyperelastic rubber-like material is proposed which describes deformation induced microstructural damage and the same is characterised by a simple exponential softening function. The proposed isotropic damage function depends on the maximum previous value of the first invariant of the left Cauchy–Green deformation tensor. The proposed model of softening is illustrated with the theory of Gent material model and finally it is validated with experimental data provided in the literature. The model shows a simple functional form and brings out the interrelation between other models of this type.