Mooney-Rivlin Material Model Research Articles

THE POTENTIAL FIELD-BASED TRAJECTORY PLANNING FOR NEEDLE INSERTION IN A SOFT-TISSUE MODEL

This paper presents a 3D dynamic trajectory planning method for the insertion of a rigid needle into soft tissue. The optimal needle tip orientation was calculated in which applying an artificial potential field method to determine the 3D distribution of repulsive and attractive forces surrounding the target object and adjacent obstacles, e.g. bones, nerves, or arteries. Soft-tissue deformation occurs dynamically and continuously during the needle insertion. The trajectory planning was therefore temporally discretized, and the compartment searching method used in each time step. This trajectory planning method was then validated by a dynamic finite element method (FEM) simulation. The dynamic finite element model is built for the important displacement parameters of deformation node in needle insertion process. The Mooney–Rivlin material model combined with solid cubic element and an explicit center differencing scheme was used to compute the soft-tissue deformation at each time step and dynamically identify the target and obstacle positions. The proposed trajectory planning method can optimize the insertion path to achieve the target position while avoiding obstacles.

Deformation of polyurea-coated steel plates under localised blast loading

Experimental and numerical studies were conducted to investigate the effect of polyurea coatings on the blast resistance of mild steel plates. Square steel plates with and without polyurea coatings applied to the back surface were subjected to localised blast loading, both experimentally and numerically using ANSYS® AUTODYN®. Two coating thicknesses were considered which were chosen such that each of the plates had the same areal density of 4.7 g/cm2 over the test area. The residual deformations of the plates were measured after each event and found to increase with coating thickness. The transient deformations of the plates were captured using high speed video. The video revealed that the polyurea coatings de-bonded over a circular area, resulting in a hyperelastic extension of the polyurea and a maximum transient deformation approximately twice that of the bare steel plates. Close agreement was found between the experimental and numerical results for the residual deformations; however, the peak transient deformations were under-predicted in the numerical models. This lead to the development of new Mooney–Rivlin material model constants for the polyurea which gave improved results. It was found that a numerical bond strength of 80 MPa between the polyurea and the steel gave the closest match with the experimental results. Further numerical modelling which varied coating thickness and location found no coating solutions which deformed less than a bare steel plate of equivalent areal density.

Effect of ambient temperature on stress, deformation and temperature of dump truck tire

The influence of the ambient temperature on the stress, deformation and temperature of a dump truck tire has been predicted with the virtual tire model established in this work by employing the finite element method (FEM) and Algor software. The nonlinear stress–strain relationship of tire rubber material has been given by Mooney–Rivlin material model. The hysteresis and loss strain energy density have been related to give the heat generation rate. A two-dimensional (2D) FE model of rolling radial tire has been built in Algor environment according to the actual construction of bridgestone 24.00R35 tire adopted for Caterpillar Earth Mover 775E Model. The model consists of the tread, belt, carcass, air filled volume and tire–rim interface. The material properties of the components and boundary conditions from tire–rim contact and tire–road contact are considered. The mechanical event simulation (MES) with nonlinear material model and the steady-state heat transfer analysis have been performed. The analysis results of the tire stress, deformation and temperature are presented under the ambient temperatures ranging from −40 to 40°C conditions at the Syncrude mine in Canada. The results show that the effect of the ambient temperature on the tire deformation is larger than on the tire stress. The increase of ambient temperature will cause the tire temperature rise. This study provides a comprehensive modeling and analysis that provide key to avoid tire temperature related failure.

Digital Image Correlation in the Classroom: Determining Stress Concentration Factors with Webcams

In this study, we introduce graduate students to an experiment with digital image correlation (DIC). From an educational point of view, the aim is to get students familiar with basic aspects of stereovision and DIC, for example, speckle pattern, subset size, focus, aperture, etc. First, a homogeneous uniaxial tensile test is conducted on a rubber dogbone, allowing the determination of the hyperelastic stress—strain relationship. Next, a nondestructive 2D deformation test on a rubber specimen with a specific geometry is performed on a small tensile setup. At various load steps, images are captured with a low-cost webcam and are processed with our in-house DIC software which is available for each individual student. Assuming a Mooney—Rivlin material model based on the determined stress—strain data, one can derive stress concentration factors for the specific holes and notches present in the specimen. A comparison is made to analytical calculations of the stress concentration factors and eventually a finite element analysis of the experiment can be performed. In this way, a synergy between experiment, simulation, and theoretical calculations is achieved. Moreover, we have accomplished all stages in the engineering process of determination of material properties, simulation, and validation. The particular experimental setup is chosen to avoid the use of special equipment, for example, sophisticated tensile devices and expensive high-tech cameras, without losing the focus of the objective.

Finite Element Analysis of Rubber Bumper Used in Air-springs

Abstract In this paper a solution of a contact problem for large displacements and deformations are analyzed where a rubber bumper applied in air–spring. The nonlinear load-displacement curve is determined. A FEM code written in FORTRAN has been developed for the analysis of nearly incompressible axially symmetric rubber parts. The Hu Washizu type variational principle is formulated for the Mooney-Rivlin material model. Stability and sensitivity analyses are also investigated.

Estimation of the macroscopic stress-strain curve of a particulate composite with a crosslinked polymer matrix

The main focus of the present paper is the estimation of the macroscopic stress–strain behavior of a particulate composite. A composite with a cross-linked polymer matrix in a rubbery state filled with an alumina-based mineral filler is investigated by means of the finite-element method. The hyperelastic material behavior of the matrix is described by the Mooney–Rivlin material model. Numerical models on the basis of unit cells are developed. The existence of a discontinuity (breaking) in the matrix at higher loading levels is taken into account to obtain a more accurate estimate for the stress–strain behavior of the particulate composite investigated. The numerical results obtained are compared with an experimental stress–strain curve, and a good agreement is found to exist. The paper can contribute to a better understanding of the behavior and failure of particulate composites with a polymer matrix.

Design optimization of backup seal for sodium cooled fast breeder reactor

Design optimization of static, fluoroelastomer backup seals for the 500MWe, Prototype Fast Breeder Reactor (PFBR) is depicted. 14 geometric variations of a solid trapezoidal cross-section were studied by finite element analysis (FEA) to arrive at a design with hollowness and double o-ring contours on the sealing face. The seal design with squeeze of 5mm assures failsafe operation for at least 10years under a differential pressure of 25kPa and ageing influences of fluid (air), temperature (110°C) and γ radiation (23mGy/h) in reactor. Hybrid elements of 1mm length, regular integration, Mooney–Rivlin material model and Poisson’s ratio of 0.493 were used in axisymmetric analysis scheme. Possible effects of reduced fluoroelastomer strength at 110°C, ageing, friction, tolerances in reactor scale, testing conditions during FEA data generation and batch-to-batch/production variations in seal material were considered to ensure adequate safety margin at the end of design life. The safety margin and numerical prediction accuracy could be improved further by using properties of specimens extracted from seal. The approach is applicable to other low pressure, moderate temperature elastomeric sealing applications of PFBR, mostly operating under maximum strain of 50%.

A note on the Mooney–Rivlin material model

In finite elasticity, the Mooney–Rivlin material model for the Cauchy stress tensor T in terms of the left Cauchy–Green strain tensor B is given by $$T = -pI + s_1 B + s_2 B^{-1},$$ where p is the pressure and s1, s2 are two material constants. It is usually assumed that s1 > 0 and s2 ≤ 0, known as E-inequalities, based on the assumption that the free energy function be positive definite for any deformation. In this note, we shall relax this assumption and with a thermodynamic stability analysis, prove that s2 need not be negative so that some typical behavior of materials under contraction can also be modeled.

A nonlinear viscoelastic finite element model of polyethylene.

A nonlinear viscoelastic finite element model of ultra-high molecular weight polyethylene (UHMWPE) was developed in this study. Eight cylindrical specimens were machined from ram extruded UHMWPE bar stock (GUR 1020) and tested under constant compression at 7% strain for 100 sec. The stress strain data during the initial ramp up to 7% strain was utilized to model the "instantaneous" stress-strain response using a Mooney-Rivlin material model. The viscoelastic behavior was modeled using the time-dependent relaxation in stress seen after the initial maximum stress was achieved using a stored energy formulation. A cylindrical model of similar dimensions was created using a finite element analysis software program. The cylinder was made up of hexahedral elements, which were given the material properties utilizing the "instantaneous" stress-strain curve and the energy-relaxation curve obtained from the experimental data. The cylinder was compressed between two flat rigid bodies that simulated the fixtures of the testing machine. Experimental stress-relaxation, creep and dynamic testing data were then used to validate the model. The mean error for predicted versus experimental data for stress relaxation at different strain levels was 4.2%. The mean error for the creep test was 7% and for dynamic test was 5.4%. Finally, dynamic loading in a hip arthroplasty was modeled and validated experimentally with an error of 8%. This study establishes a working finite element material model of UHMWPE that can be utilized to simulate a variety of postoperative arthroplasty conditions.

Finite Element Simulation of Permanent Magnetoelastic Thin Films

We are interested in the deformation of a thin magnetoelastic film. This has many possible applications, such as in microfluidic actuators and sensors. In previous articles a finite element method has been used for transient simulation of the deformation of a film, for the special case of soft magnetic material only. Here the approach is extended to the case of permanent magnetoelastic materials. The key enhancements required for simulation of magnetoelastic materials are a nonlinear hyperelastic Mooney-Rivlin material model, addition of an implicit system of equations for the scalar magnetic potential, and a more complex asymmetric Maxwell stress tensor. A simple test problem was constructed, and simulations for a soft magnetic film and a permanent magnetic film are compared.

Numerical Modeling of Macroscopic Behavior of Particulate Composite with Crosslinked Polymer Matrix

Particulate composites with crosslinked polymer matrix and solid fillers are one of important classes of materials such as construction materials, high-performance engineering materials, sealants, protective organic coatings, dental materials, or solid explosives. The main focus of a present paper is an estimation of the macroscopic Young’s modulus and stress-strain behavior of a particulate composite with polymer matrix. The particulate composite with a crosslinked polymer matrix in a rubbery state filled by an alumina-based mineral filler is investigated by means of the finite element method. A hyperelastic material behavior of the matrix was modeled by the Mooney-Rivlin material model. Numerical models on the base of unit cell were developed. The numerical results obtained were compared with experimental stress-strain curve and value of initial Young’s modulus. The paper can contribute to a better understanding of the behavior and failure of particulate composites with a crosslinked polymer matrix.

Hyperelastic properties of human meniscal attachments

Meniscal attachments are ligamentous tissues anchoring the menisci to the underlying subchondral bone. Currently little is known about the behavior of meniscal attachments, with only a few studies quantitatively documenting their properties. The objective of this study was to quantify and compare the tensile mechanical properties of human meniscal attachments in the transverse direction, curve fit experimental Cauchy stress–stretch data to evaluate the hyperelastic behavior, and couple these results with previously obtained longitudinal data to generate a more complete constitutive model. Meniscal attachment specimens were tested using a uniaxial tension test with the collagen fibers oriented perpendicular to the loading axis. Tests were run until failure and load-optical displacement data was recorded for each test. The medial posterior attachment was shown to have a significantly greater elastic modulus (6.42±0.78 MPa) and ultimate stress (1.73±0.32 MPa) when compared to the other three attachments. The Mooney–Rivlin material model was selected as the best fit for the transverse data and used in conjunction with the longitudinal data. A novel computational approach to determining the transition point between the toe and linear regions is presented for the hyperelastic stress–stretch curves. Results from piece-wise non-linear longitudinal curve fitting correlate well with previous linear elastic and SEM findings. These data can be used to advance the design of meniscal replacements and improve knee joint finite element models.

Effects of Scleral Stiffness Properties on Optic Nerve Head Biomechanics

The biomechanical environment within the optic nerve head, important in glaucoma, depends strongly on scleral biomechanical properties. Here we use a range of measured nonlinear scleral stress-strain relationships in a finite element (FE) model of the eye to compute the biomechanical environment in the optic nerve head at three levels of intraocular pressure (IOP). Three stress-strain relationships consistent with the 5th, 50th and 95th percentiles of measured human scleral stiffness were selected from a pool of 30 scleral samples taken from 10 eyes and implemented in a generic FE model of the eye using a hyperelastic five-parameter Mooney-Rivlin material model. Computed strains within optic nerve head tissues depended strongly on scleral properties, with most of this difference occurring between the compliant and median scenarios. Also, the magnitudes of strains were found to be substantial even at normal IOP (up to 5.25% in the lamina cribrosa at 15 mmHg), being larger than previously reported values even at normal levels of IOP. We conclude that scleras that are "weak", but still within the physiologic range, will result in appreciably increased optic nerve head strains and could represent a risk factor for glaucomatous optic neuropathy. Estimations of the deformation at the optic nerve head region, particularly at elevated IOP, should take into account the nonlinear nature of scleral stiffness.

A mathematical model for cylindrical, fiber reinforced electro-pneumatic actuators

In recent years, dielectric elastomers have received increasing attention due to their unparalleled large strain actuation response (>100%). The force output, however, has remained a major limiting factor for many applications. To address this limitation, a model for a fiber reinforced dielectric elastomer actuator based on the deformation mechanism of McKibben actuators is presented. In this novel configuration, the outer cylindrical surface of a dielectric elastomer is enclosed by a network of helical fibers that are thin, flexible and inextensible. This configuration yields an axially contractile actuator, in contrast to unreinforced actuators which extend. The role of the fiber network is twofold: (i) to serve as reinforcement to improve the load-bearing capability of dielectric elastomers, and (ii) to render the actuator inextensible in the axial direction such that the only free deformation path is simultaneous radial expansion and axial contraction. In this paper, a mathematical model of the electromechanical response of fiber reinforced dielectric elastomers is derived. The model is developed within a continuum mechanics framework for large deformations. The cylindrical electro-pneumatic actuator is modeled by adapting Green and Adkins’ theory of reinforced cylinders to account for the applied electric field. Using this approach, numerical solutions are obtained assuming a Mooney–Rivlin material model. The results indicate that the relationship between the contractile force and axial shortening is bilinear within the voltage range considered. The characteristic response as a function of various system parameters such as the fiber angle, inflation pressure, and the applied voltage are reported. In this paper, the elastic portion of the modeling approach is validated using experimental data for McKibben actuators.

Finite-element analysis of middle-ear pressure effects on static and dynamic behavior of human ear

A finite-element analysis for static behavior of middle ear under variation of the middle-ear pressure was conducted in a 3D model of human ear by combining the hyperelastic Mooney-Rivlin material model and geometry nonlinearity. An empirical formula was then developed to calculate material parameters of the middle-ear soft tissues as the stress-dependent elastic modulus relative to the middle-ear pressure. Dynamic behavior of the middle ear in response to sound pressure in the ear canal was predicted under various positive and negative middle-ear pressures. The results from static analysis indicate that a positive middle ear pressure produces the static displacements of the tympanic membrane (TM) and footplate more than a negative pressure. The dynamic analysis shows that the reductions of the TM and footplate vibration magnitudes under positive middle-ear pressure are mainly determined by stress dependence of elastic modulus. The reduction of the TM and footplate vibrations under negative pressure was caused by both the geometry changes of middle-ear structures and the stress dependence of elastic modulus.

Nonlinear finite element modelling and incremental analysis of a truck tyre

A non-linear finite element (FE) model of a radial truck tyre is developed to analyse the tensile stress distributions, deformation fields and inter-ply shear stresses as a function of the normal load. The tyre model is established based on consideration of the geometry and orientation of the cords in belts and carcass layers, the stacking sequence of different layers and the large deformation of the multiple layers in order to predict the interactions between the plies. The study incorporates the nearly incompressible property of the tread rubber block and anisotropic material properties of the layers. A non-linear incremental analysis of the rubber element is performed using the modified Mooney-Rivlin material model. The non-linear FE tyre model is analysed using commercial software, the ANSYS® program in order to determine the deformation fields, contact patch geometry, contact pressure distribution, principal stress fields, distributions of fibre stresses and inter-ply shear stresses. The influence of normal load and thus the tyre deflection on the above response parameters are discussed. The computed footprint geometry is qualitatively compared in a limited sense with the measured data to examine the validity of the model. It is concluded that the proposed model can provide reliable predictions about the three-dimensional stress, strain and deformation fields in an inflated radial truck tyre based on the known tyre material properties and structural features for given internal air pressures and normal loading conditions.

The Mullins effect in equibiaxial deformation

Common experience teaches that the pressure required to inflate a balloon is noticeably reduced by prestretching it several times prior to its primary inflation. This preconditioning phenomenon is known as stress-softening and often is called the Mullins effect. A theory of stress-softening in incompressible isotropic materials is applied to study this effect in an equibiaxial extension for which some general results are presented. It is shown, for example, that effects of stress-softening in a simple uniaxial compression can be adduced from those demonstrated for equibiaxial extension under plane stress. The general equibiaxial results are applied to study the Mullins effect in the inflation of a spherical membrane. The stress-softening phenomenon in cyclic inflation and deflation of the balloon is investigated for Mooney-Rivlin and biotissue parent material models. It is shown analytically that the effect of preconditioning in uniaxial extension is to significantly reduce the pressure required to first inflate a balloon to an equivalent strain intensity. This result characterizes the familiar softening phenomenon associated with balloon inflation.

Plane deformations in incompressible nonlinear elasticity

The substantially general class of plane deformation fields, whose only restriction requires that the angular deformation not vary radially, is considered in the context of isotropic incompressible nonlinear elasticity. Analysis to determine the types of deformations possible, that is, solutions of the governing systems of nonlinear partial differential equations and constraint of incompressibility, is developed in general. The Mooney-Rivlin material model is then considered as an example and all possible solutions to the equations of equilibrium are determined. One of these is interpreted in the context of nonradially symmetric cavitation, i.e., deformation of an intact cylinder to one with a double-cylindrical cavity. Results for general incompressible hyperelastic materials are then discussed. The novel approach taken here requires the derivation and use of a material formulation of the governing equations; the traditional approach employing a spatial formulation in which the governing equations hold on an unknown region of space is not conducive to the study of deformation fields containing more than one independent variable. The derivation of the cylindrical polar coordinate form of the equilibrium equations for the nominal stress tensor (material formulation) for a general hyperelastic solid and a fully arbitrary cylindrical deformation field is also given.

Buckling and Postbuckling of Elastomeric Components: Tactile Feel in Electric Switches

Abstract Within the range of the parameters of the switch designs considered, only very slight cocking of the switch structure occurred. Thus, with minimal off-axis cocking, the axisymmetric assumption remained valid, resulting in a considerable finite-element economization when compared to a full three dimensional model. Some switch designs encounter localization problems, such as surface folding and stiffness mismatches. Mesh refinement is not a general panacea for these localization problems. Regardless, a fold of this type or stiffness mismatch is generally indicative of a potentially poor switch design, with the possibility of associated fatigue problems. For the switch designs considered, the peak strains were found to be less than 70 percent; thus the Mooney-Rivlin and Ogden material models yielded essentially similar results. Furthermore, a Hookean material model may be adequate for some purposes, because the tactile-feel response was found to be dominated more by switch geometry than anything else. Overall, we have seen that combining FE analysis together with the full continuum-constitutive-boundary formulation can successfully address the problem of buckling and postbuckling of elastomeric structures. In the case of switches, such behavior is an intrinsic property. This is an interesting contrast with traditional structures.

Stability of a body supported by a simple vehicular shear suspension system

Stability of the oscillatory motion of a body supported by simple shear springs is examined relative to a vehicle that executes a steady circular turn about an arbitrary axis fixed in the vertical plane. The problem is formulated for a general isotropic elastic material whose shear response function is a positive, even function of the amount of shear. A general sufficient condition bounding the normalized vehicle angular speed Ω by a certain critical angular speed Ω c , which depends on the vehicle and load suspension orientation design parameters alone, is obtained for periodic, and hence stable, free relative oscillations of the load for arbitrary initial data. Stability in the forced vibration problem also is discussed. For the special class of materials whose shear response function is a constant, a familiar class that includes the Mooney-Rivlin, Blatz-Ko, and Hadamard material models, for example, the free oscillatory motion of the load is stable if and only if Ω < Ω c . It is shown also for this class that in the forced vibration problem for which the oscillations generally are aperiodic, resonance will occur at an angular speed Ω r smaller than Ω c .