Mooney-Rivlin Hyperelastic Model Research Articles

New microscale constitutive model of human trabecular bone based on depth sensing indentation technique.

A new constitutive model for human trabecular bone is presented in the present study. As the model is based on indentation tests performed on single trabeculae it is formulated in a microscale. The constitutive law takes into account non-linear viscoelasticity of the tissue. The elastic response is described by the hyperelastic Mooney-Rivlin model while the viscoelastic effects are considered by means of the hereditary integral in which stress depends on both time and strain. The material constants in the constitutive equation are identified on the basis of the stress relaxation tests and the indentation tests using curve-fitting procedure. The constitutive model is implemented into finite element package Abaqus® by means of UMAT subroutine. The curve-fitting error is low and the viscoelastic behaviour of the tissue predicted by the proposed constitutive model corresponds well to the realistic response of the trabecular bone.

Influence of different cord pitch on the pullout force of steel cord conveyor belt splice

The splice is the weakest part of the entire steel cord conveyor belt. And it occurs steel cord twitch fault frequently. If this fault cannot be dealt with timely and accurately, broken belt accidents would be occurred that affecting the safety of production seriously. In this paper, we investigated the different cord pitch pullout of the steel cord conveyor belt splice by using ABAQUS. The finite element model of the ST630 type steel cord conveyor belt was established, same as to the standard of the experimental sample. The pullout force of length 50.0, 70.0, 80.0 and 100.0 mm single steel cord impacted at steel cord conveyor belt splice was numerically computed respectively. The finite element model consists of rubber, steel cord and failure unit. Failure unit is used to simulate the bonding relationship between steel cord and rubber. Mooney-Rivlin hyper-elastic model for rubber was employed in the numerical simulations. Typical results obtained have been validated by experimental result and the relative error between simulation results and experimental results is within 10%. Simulations were also done for cord pitch of 4.0, 4.5, 6.0, 7.0 and 8.0 mm to study its effect on the pullout force of steel cord conveyor belt splice. The variation of cord pitch at normal impact was found to have influence as long as the cord pitch was different. It provides us the numeric value of the pullout force with the different cord pitch, and conducts us selecting the correct steel cord configuration spacing for the steel cord conveyor belt splice, as well for the steel cord conveyor belt. This is an efficient way for avoiding the steel cord twitch fault and the broken belt accident, while ensuring their high reliability and service life.

Rigid-flexible contact analysis of an inflated membrane balloon with various contact conditions

Considering the Mooney–Rivlin hyperelastic model, a semi-analytical approach is introduced to analyze the rigid–flexible contact behaviors of an inflated membrane balloon between two plates with various interface conditions. This approach is based on the differential formulation and the coupling property of equilibrium equations are well-solved. In order to verify the reliability of the proposed theoretical model, an experimental test is designed, by which some important contact characteristics and patterns (no-slip condition) are obtained. Two special phenomena are observed for the meridian stretch ratio with different friction coefficients. One is that the intersection points of all curves fall in a small interval and the intersection of any two curves represents the same changing rate of the horizontal ordinate, resulting in the maximum difference. The other is the dividing point, where the stretch ratio decreases on the left of it and increases on the right due to the introduction of friction. Under the same contact angle, the larger displacement load should be applied to the balloon for the small friction coefficient condition, resulting in the smaller contact area and internal pressure. In addition, the vulnerable position, direction and contact condition of the balloon are found during the contact process, which happen in the center along the circumferential direction under no-slip condition. These results provide solid guidance and support for our understanding of the rigid-flexible contact behaviors of an inflated membrane balloon.

Finite element simulation and experimental verification of steel cord extraction of steel cord conveyor belt splice

The splice is the weakest part of the entire steel cord conveyor belt. And it occurs steel cord twitch fault frequently. If this fault cannot be dealt with timely and accurately, broken belt accidents would be occurred that affecting the safety of production seriously. In this paper, we investigate the steel cord pullout of the steel cord conveyor belt splice by using ABAQUS software. We selected the strength of steel cord conveyor belt ST630, the same as experiment sample in type specification. The finite element model consists of rubber, steel cord and failure unit. And the failure unit is used to simulate the bonding relationship between the steel cord and the rubber. Mooney-Rivlin hyper-elastic model for rubber was employed in the numerical simulations. The pullout force of length 50.0 mm single steel cord, on both sides of a single steel cord and on both sides of the double steel cords each impacted at steel cord conveyor belt splice were numerically computer and typical results obtained have been validated by experimental result. It shows that the relative error between simulation results and experimental results is within 10% and can be considered that the simulation model is reliable. A new method is provided for studying the steel cord twitch fault of the steel cord conveyor belt splice.

Finite element analysis of ionic liquid gel soft actuator**Project supported by the National Natural Science Foundation of China (Grant Nos. 51538009 and 51605334) and the Natural Science Foundation of Shanghai Municipality, China (Grant No. 08002360285).

A new type of soft actuator material-ionic liquid gel (ILG), which consists of HEMA, BMIMBF4, and TiO2, can be transformed into gel state under the irradiation of ultraviolet (UV) light. In this paper, Mooney–Rivlin hyperelastic model of finite element method is proposed for the first time to study the properties of the ILG. It has been proved that the content of TiO2 has a great influence on the properties of the gel, and Young’s modulus of the gel increases with the increase of its content, despite of reduced tensile deformation. The results in this work show that when the TiO2 content is 1.0 wt%, a large tensile deformation and a strong Young’s modulus can be obtained to be 325% and 7.8 kPa, respectively. The material parameters of ILG with TiO2 content values of 0.2 wt%, 0.5 wt%, 1.0 wt%, and 1.5 wt% are obtained, respectively, through uniaxial tensile tests, including C10, C01, C20, C11, C02, C30, C21, C12, and C03 elements. In this paper, the large-scaled general finite element software ANSYS is used to simulate and analyze the ILG, which is based on SOLID186 element and nonlinear hyperelastic Mooney–Rivlin model. The finite element simulation analysis based stress-strain curves are almost consistent with the experimental stress–strain curves, and hence the finite element analysis of ILG is feasible and credible. This work presents a new direction for studying the performance of soft actuator for the ILG, and also contributes to the design of soft robot actuator.

A Patient-Specific Three-Dimensional Hemodynamic Model of the Circle of Willis.

Circle of Willis (CoW) is one of the most important cerebral arteries in the human body and various attempts have been made to study the hemodynamic of blood flow in this vital part of the brain. In the present study, blood flow in a patient specific CoW is numerically modeled to predict disease-prone regions of the CoW. Medical images and computer aided design software are used to construct a realistic three-dimensional model of the CoW for this particular case. The arteries are considered as elastic conduits and the interactions between arterial walls and the blood flow are taken into account. Mooney-Rivlin hyperelastic model is used to describe the behavior of arterial walls and blood is considered as a non-Newtonian fluid obeying the Carreau model. An available experimental-based pulsatile velocity profile is used at the entrance of the CoW. The finite element-based commercial software, ADINA, is used to solve the governing equations. Blood pressure and velocity and arterial wall shear stress are calculated in different regions of the CoW. A simplified form of the model is also compared with the available published data. Results affirmed that the proposed computational model has the potential to capture the hemodynamic characteristics of the CoW. The computational results can be used to determine disease-prone locations for a given CoW.

Mechanical properties of the human spinal cord under the compressive loading

The spinal cord as the most complex and critical part of the human body is responsible for the transmission of both motor and sensory impulses between the body and the brain. Due to its pivotal role any types of physical injury in that disrupts its function following by shortfalls, including the minor motor and sensory malfunctions as well as complicate quadriplegia and lifelong ventilator dependency. In order to shed light on the injuries to the spinal cord, the application of the computational models to simulate the trauma impact loading to that are deemed required. Nonetheless, it has not been fulfilled since there is a paucity of knowledge about the mechanical properties of the spinal cord, especially the cervical one, under the compressive loading on the grounds of the difficulty in obtaining this tissue from the human body. This study was aimed at experimentally measuring the mechanical properties of the human cervical spinal cord of 24 isolated fresh samples under the unconfined compressive loading at a relatively low strain rate. The stress-strain data revealed the elastic modulus and maximum/failure stress of 40.12±6.90 and 62.26±5.02kPa, respectively. Owing to the nonlinear response of the spinal cord, the Yeoh, Ogden, and Mooney-Rivlin hyperelastic material models have also been employed. The results may have implications not only for understanding the linear elastic and nonlinear hyperelastic mechanical properties of the cervical spinal cord under the compressive loading, but also for providing a raw data for investigating the injury as a result of the trauma thru the numerical simulations.

Understanding mechanical characteristics of cellulose nanocrystals reinforced PHEMA nanocomposite hydrogel: in aqueous cyclic test

Cellulose nanocrystals (CNC) can be embedded within hydrogels to form tough and strong nanocomposite materials, which possess biomimetic properties from hydrogels including good biocompatibility, permeability and flexible mechanical characteristics. There are many potential applications for these strong nanocomposite hydrogels in medical devices, such as wound dressing or super absorbents. Whereas, the research on the mechanical properties of CNC reinforced nanocomposite remains at superficial level, and their nonlinear mechanical responses are rarely investigated in previous reports. Mechanical characteristics of CNC reinforced poly(2-hydroxyethyl methacrylate) (PHEMA) nanocomposite hydrogels, in terms of stress–strain correlations, fracture mechanism, and cyclic stretching responses, have been investigated in this work. Experimental results show that the modulus of the nanocomposite hydrogel tends to increase with increasing CNC content. Theoretical foundation for analysing the mechanical properties of hydrogels based on Mooney–Rivlin hyperelastic model, Voigt model and Reuss model has been developed and validated, which provides the prediction of the mechanical responses of CNC reinforced nanocomposite hydrogel to tension, especially the nonlinear responding behaviour.

EFFECTS OF HYPERTENSION AND PRESSURE GRADIENT IN A HUMAN CEREBRAL ANEURYSM USING FLUID STRUCTURE INTERACTION SIMULATIONS

Fluid–structure interaction (FSI) simulations were carried out in a human cerebral aneurysm model with the objective of quantifying the effects of hypertension and pressure gradient on the behavior of fluid and solid mechanics. Six FSI simulations were conducted using a hyperelastic Mooney–Rivlin model. Important differences in wall shear stress (WSS), wall displacements, and effective von Mises stress are reported. The hypertension increases wall stress and displacements in the aneurysm region; however, the effects of hypertension on the hemodynamics in the aneurysm region were small. The pressure gradient affects the WSS in the aneurysm and also the displacement and wall stress on the aneurysm. Maximum wall stress with hypertension in the range of rupture strength was found.

Finite Element Modeling with Embed Rebar Elements and Steady State Rolling Analysis for Rolling Resistance Test of Pneumatic Tire

Finite element model of tire rolling resistance test on the drum was developed using 3D steady state rolling analysis coupling with pre-inflation of 2D axisymmetric tire analysis. The complex components of the radial tires composing tread, sidewall, ply layers, steel belts, and lead wires were modeled using rebar elements which were embed into the rubber element using the tying equation. The Mooney-Rivlin hyperelastic constitutive model was employed to describe the large deformation behavior of tread and sidewall, while other components such as plies, steel belts and bead wires were assigned the linear isotropic material. The tire rolling resistance system was modeled by inflation of slick tire and compression on the drum for the footprint analysis regarding the rolling resistance test. The tire’s steady state characteristics such as footprint contact pressure, rolling resistance force, and time response characteristic of tires were predicted instead the experiment of the prototype.

Parametric Investigation of Mooney-Rivlin Material Constants on Silicone Biocomposite

Hyperelastic materials are unique materials that have high tendency to stretch and its highly non-linear behaviour is commonly investigated using hyperelastic constitutive models. The aim of this paper is to investigate the sensitivity of Mooney-Rivlin material constants; C1 and C2 values in order to observe the behavior and pattern of the stress-stretch graph for silicone-kenaf composite. There were no previous studies done in regards to assess the mechanical behaviour of the stress-stretch curve for silicone-kenaf biocomposite by varying the Mooney-Rivlin material constants. The material constant, C1 and C2 are varied into few cases and the patterns of stress-stretch curves are studied. It was found that variations of C1 and C2 material constants could contribute differently on the mechanical properties of silicone-kenaf composite. Thus, the results and findings of this study could be further enhanced by future study to gain deeper understanding on the hyperelastic materials behaviour and Mooney-Rivlin hyperelastic constitutive model.

Creeping flow of Herschel-Bulkley fluids in collapsible channels: A numerical study

In this paper, the steady flow of a viscoplastic fluid is modeled in a planar channel equipped with a deformable segment in the middle of an otherwise rigid plate. The fluid is assumed to obey the Herschel-Bulkley model which accounts for both the yield stress and the shear-thinning behavior of physiological fluids such as blood. To accommodate the large deformations of the flexible segment, it is assumed to obey the twoparameter Mooney-Rivlin hyperelastic model. The so-called fluid-structure interaction problem is then solved numerically, under creeping-flow conditions, using the finite element package, COMSOL. It is found that the yield stress leads to a larger wall deformation and a higher pressure drop as compared with Newtonian fluids. This behavior is predicted to intensify if the fluid is shear-thinning. That is, for a given yield stress, the pressure drop and the wall deformation both increase with an increase in the degree of the fluid's shear-thinning behavior.

Experimental and numerical study on vibrations and static deflection of a thin hyperelastic plate

The hyperelastic behavior of a thin square silicone rubber plate has been investigated analytically, numerically and experimentally; the case of small-amplitude vibrations has been considered, as well as the case of large static deflection under aerostatic pressure. The Mooney-Rivlin hyperelastic model has been chosen to describe the material nonlinear elasticity. The material parameters have been identified by a fitting procedure on the results of a uniaxial traction test. For the analytical model, the equations of motion have been obtained by a unified energy approach, and geometrical nonlinearities are modeled according to the Novozhilov nonlinear shell theory. A numerical model has also been developed by using a commercial Finite-Element code. In the experiments, the silicone rubber plate has been fixed to a heavy metal frame; a certain in-plane pre-load, applied by stretching the plate, has been given in order to ensure a flatness of the surface. An experimental modal analysis has been conducted; results have been used to identify the applied in-plane loads by optimization procedure with two different models: a numerical and an analytical one. The first four experimental and numerical natural modes and frequencies are in good agreement with the experiments after the pre-load identification. The static deflection has been measured experimentally for different pressures. Results have been compared to those obtained by analytical and numerical models. The static deflections are also satisfactorily compared, up to a deflection 50 times larger than the plate thickness, corresponding to a 30 percent strain.

Biomechanical characterization of the periodontal ligament: Orthodontic tooth movement

To quantify the biomechanical properties of the bovine periodontal ligament (PDL) in postmortem sections and to apply these properties to study orthodontic tooth intrusion using finite element analysis (FEA). We hypothesized that PDL's property inherited heterogeneous (anatomical dependency) and nonlinear stress-strain behavior that could aid FEA to delineate force vectors with various rectangular archwires. A dynamic mechanical analyzer was used to quantify the stress-strain behavior of bovine PDL. Uniaxial tension tests using three force levels (0.5, 1, and 3 N) and samples from two anatomical locations (circumferential and longitudinal) were performed to calculate modulus. The Mooney-Rivlin hyperelastic (MRH) model was applied to the experimental data and used in an FEA of orthodontic intrusion rebounded via a 0.45-mm step bend with three archwire configurations of two materials (stainless steel and TMA). Force levels and anatomical location were statistically significant in their effects on modulus (P < .05). The apical part had a greater stiffness than did the middle part. The MRH model was found to approximate the experimental data well (r = 0.99), and it demonstrated a reasonable stress-strain outcome within the PDL and bone for FEA intrusion simulation. The force acting on the tooth increased five times from the 0.016 × 0.022-inch TMA to the 0.019 × 0.025-inch stainless steel. The PDL is a nonhomogeneous tissue in which the modulus changed in relation to location. PDL nonlinear constitutive model estimated quantitative force vectors for the first time to compare intrusive tooth movement in 3-D space in response to various rectangular archwires.

Rate-dependent compressive behavior of EPDM insulation: Experimental and constitutive analysis

To fully characterize the compressive behavior of ethylene propylene diene monomer (EPDM), quasi-static and dynamic compression experiments were conducted for EPDM with a universal testing machine and split Hopkinson bar (SHPB), respectively. Dynamic experimental data employing the two-wave method were used to obtain the stress–strain curves, and the dynamic results were validated. The polymer was compressed under different strain rates varying from 10−4 to 103. The results indicate that the mechanical properties of EPDM insulation are obviously dependent on the strain rate. An improved Zhu–Wang–Tang (ZWT) nonlinear visco-hyperelastic constitutive model, in which the standard elastic component was replaced by the Mooney–Rivlin hyperelastic model, was developed to predict the mechanical behaviors of EPDM under quasi-static and dynamic impact loading. The model parameters were determined by fitting the model to the experimentally obtained curves by means of the least squares method. The stress–strain curves obtained from the improved ZWT model show good agreement with the experimental curves. This research will further improve the design of the internal insulation structure in solid rocket motors.

Computational local stiffness analysis of biological cell: High aspect ratio single wall carbon nanotube tip

Carbon nanotubes (CNTs) are potentially ideal tips for atomic force microscopy (AFM) due to the robust mechanical properties, nanoscale diameter and also their ability to be functionalized by chemical and biological components at the tip ends. This contribution develops the idea of using CNTs as an AFM tip in computational analysis of the biological cells. The proposed software was ABAQUS 6.13 CAE/CEL provided by Dassault Systems, which is a powerful finite element (FE) tool to perform the numerical analysis and visualize the interactions between proposed tip and membrane of the cell. Finite element analysis employed for each section and displacement of the nodes located in the contact area was monitored by using an output database (ODB). Mooney–Rivlin hyperelastic model of the cell allows the simulation to obtain a new method for estimating the stiffness and spring constant of the cell. Stress and strain curve indicates the yield stress point which defines as a vertical stress and plan stress. Spring constant of the cell and the local stiffness was measured as well as the applied force of CNT-AFM tip on the contact area of the cell. This reliable integration of CNT-AFM tip process provides a new class of high performance nanoprobes for single biological cell analysis.

MECHANICAL TEST OF HUMAN CEREBRAL ANEURYSM SPECIMENS OBTAINED FROM SURGICAL CLIPPING

The walls of the blood vessels involved with cerebral aneurysms present different mechanical properties, when compared with those of normal artery. Consequently the models which may describe then predict an aneurysm rupture need as input their mechanical properties. This paper describes the experimental determination of the mechanical properties of the tissues of the blood vessels affected with a cerebral aneurysm. In particular, we determine and discuss the critical elongation and the rupture strength. The affected tissues were obtained from surgical clipping and extraction; six specimens were tested. The mechanical tests were performed in a tensile device. The experimental data was fitted numerically with a Mooney–Rivlin hyper elastic model. The model was compared with previous published data.

Questioning numerical integration methods for microsphere (and microplane) constitutive equations

In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated.Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress–strain prediction are illustrated with a directional version of the classical Mooney–Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations.

Instability of thin circular membranes subjected to hydro-static loads

Membrane structures subjected to hydrostatic load are prone to undergo large deformations and lose stability. This paper investigates different instability phenomena for a thin, circular and initially flat and horizontal membrane. The Mooney–Rivlin hyper-elastic model is used to provide the material description. An axisymmetric and a 3D model have been set up to show the large deformations and instability behavior with different parameter settings. Numerical examples show that the methods developed are capable to describe the deformation dependent loading conditions and the instability phenomena. The numerical simulations show fundamental differences in the response and instability behavior when the horizontal membrane is loaded from above or below. The parameters of fluids and membranes and the means for introducing the pressure are of essence for interpreting the instability behavior.

Corneal hyper-viscoelastic model: derivations, experiments, and simulations.

The aim of this study is to propose a method to construct corneal biomechanical model which is the foundation for simulation of corneal microsurgery. Corneal material has two significant characteristics: hyperelastic and viscoelastic. Firstly, Mooney-Rivlin hyperelastic model of cornea obtained based on stored-energy function can be simplified as a linear equation with two unknown parameters. Then, modified Maxwell viscoelastic model of the cornea, whose analytical form is consistent with the generalized Prony-series model, is proposed from the perspective of material mechanics. Parameters of the model are determined by the uniaxial tensile tests and the stress-relaxation tests. Corneal material properties are simulated to verify the hyper-viscoelastic model and measure the effectiveness of the model in the finite element simulation. On this basis, an in vivo model of the corneal is built. And the simulation of extrusion in vivo cornea shows that the force is roughly nonlinearly increasing with displacement, and it is consistent with the results obtained by extrusion experiment of in vivo cornea. Conlusions: This paper derives a corneal hyper-viscoelastic model to describe the material properties more accurately, and explains the mathematical method for determination of the model parameters. The model is an effective biomechanical model, which can be directly used for simulation of trephine and suture in keratoplasty. Although the corneal hyper-viscoelastic model is taken as the object of study, the method has certain adaptability in biomechanical research of ophthalmology.