Mooney-Rivlin Hyperelastic Model Research Articles

Nonlinear dynamics of cantilevered hyperelastic pipes conveying fluid: Comparative study of linearelasticity and hyperelasticity

In light of the softness and lightweight characteristics, hyperelastic pipes are prone to large deformation. Intending to provide accurate predictions of their large deformation vibrations, the present study aims to establish a rotation-angle-based geometrically exact model for cantilevered pipes conveying fluid composed of hyperelastic materials. Additionally, given the presence of nonlinear geometric relations, it is imperative to assess the role of hyperelasticity in the nonlinear dynamics of fluid-conveying pipes in comparison to linearelastic pipes. The modeling process integrates the Mooney-Rivlin hyperelastic model with a geometrically exact description of the pipe. And the final governing equation is developed through energy analysis and the extended Hamilton's principle. Subsequently, Galerkin's method and the finite difference method are employed to address the governing equation. Following the verification of the proposed model, parametric studies are conducted in the form of planar oscillation diagrams, time histories, phase trajectories, and bifurcation diagrams. The findings of the present study highlight three critical insights. Firstly, the hyperelastic model introduces numerous complicated nonlinear terms as function of the rotation angle with high-order nonlinearity. In the present study, at least fifth-order nonlinear terms should be preserved to adequately predict large deformation vibrations. Secondly, the Mooney-Rivlin hyperelastic model imparts a hardening effect in comparison to the linearelastic model, significantly reducing vibration amplitudes. Lastly, the third-order approximate model is demonstrated to be inadequate in predicting large deformation vibrations at high flow velocity, as it contains the approximation on not only the geometric description but the hyperelasticity-related terms.

Novel hyper-viscoelastic approach to modelling elastomer mechanic behaviour with relaxation spectrum

Elastomers, such as polyurethane, exhibit hyperelastic behaviour and distinct rate dependence under a wide range of strain rates. Herein, a methodology for developing a hyper-viscoelastic constitutive model for succinctly characterising the mechanical behaviour of polyurethane under large deformations and a wide range of strain rates is proposed. The model is constructed using a long-term hyperelastic formulation combined with a rate-dependent viscoelastic model. This model is based on internal state variables and a finite strain formulation. A polyurethane elastomer was used as a sample to develop this model, by way of testing wide strain-rate mechanical behaviour (0.001–5000s−1) and viscoelastic behaviour. The viscoelastic behaviour was obtained using the equivalent time–temperature superposition relation. A parameter-fitting algorithm was adopted to separate the long-term behaviour from the viscoelastic behaviour. The long-term behaviour was modelled using a Mooney–Rivlin hyperelastic model, and its parameters were identified using the least-squares method. The time-dependent viscous effect calculated using the Prony series-based model was approximately determined based on the continuous relaxation spectrum. The characterisation capacity was confirmed through a thorough comparison with experimental curves. This model can be extended to other elastomers, even under tension, without any specific limitations of the internal components.

Stability study of the compressible Mooney-Rivlin hyperelastic model

The unstable behavior of the isotropic, compressible Mooney-Rivlin hyperelastic model is investigated and described. The constitutive equation is parameterized with the help of the ground-state Poisson’s ratio and the dimensionless ratio of the material parameters [Formula: see text] and [Formula: see text]. Transverse stretch solutions are obtained for standard homogeneous loading modes, and the stress solutions are computed numerically for the physically permitted range of the ground-state Poisson’s ratio. We introduce a numerical technique to isolate subdomains with non-unique transverse stretch responses. Our analysis revealed some limitations of the model and allowed us to make critical observations about the strengths and weaknesses of the model. The analyses we present are essential to understand the characteristics of this widely used hyperelastic model. The novel results have important implications for the application of the isotropic, compressible Mooney-Rivlin hyperelastic material model.

Ogden and Mooney-Rivlin hyperelastic models comparison in porcine lobular tissue

Ogden and Mooney-Rivlin hyperelastic models comparison in porcine lobular tissue

Quantification and comparison of the nonlinear mechanical properties of transcutaneous and human aortic valve leaflets: Experimental and numerical studies

BackgroundExtracting the mechanical behaviors of tissue leaflets in prosthetic aortic valves is necessary for designing and manufacturing appropriate prostheses for percutaneous deployment. The goal of this study was to opt a proper tissue for bioprostheses. MethodsIn this study, donkey and bovine pericardia and human aortic valve leaflets are compared mechanically. The uniaxial and biaxial tensile tests were performed on the tissue specimens. The measured uniaxial data were then fitted into Yeoh, and Mooney-Rivlin hyperelastic models, and the elastic modulus, failure strain, and ultimate tensile strength (UTS) for each sample were calculated. Furthermore, to determine the degree of anisotropy of the specimens, the biaxial hyperelastic properties of the samples were extracted using the Gasser-Ogden-Holzapfel (GOH) and Mooney-Rivlin models. In addition, the extracted anisotropic parameters were exerted to the finite element modeling of the bovine and donkey pericardia. ResultsDonkey pericardium exhibited a low stiffer behavior, based on its lowest strain energy magnitude and the average slope of stress-stretch curves. This tissue was also high distensible than the bovine pericardium, due to its highest areal strain. Furthermore, the donkey finite element model induced low stress regions during the systole and diastole phases. On the other hand, decreased mechanical stress on the bioprosthetic leaflets may reduce tissue dehiscence and increase the long-term durability of the valve. ConclusionThe nonlinear behavior of the pericardial tissues can be well-characterized by the constitutive functions. The mechanical properties of the donkey pericardium are even closer to the native leaflets. The donkey pericardium might be a good candidate valve leaflet material for bioprostheses.

Temperature dependent tensile fracture strength model of rubber materials based on Mooney-Rivlin model

With the increasingly widespread application of rubber in many fields, the demand for quantitative characterization of temperature dependent mechanical properties in service environments over a wide temperature range is increasing. Tensile fracture strength is one of the most important mechanical properties of rubber materials, but its temperature dependent model without fitting parameters is still rarely reported. In this work, based on Mooney-Rivlin hyperelastic constitutive model, a strain energy expression that explicitly includes tensile fracture strength is derived, and a temperature dependent tensile fracture strength model for rubber materials without fitting parameters is further developed in combination with the Force-Heat Equivalence Energy Density Principle. The prediction results of the model in a wide temperature range are in good agreement with the experimental results. Furthermore, a quantitative analysis of the effect of elastic modulus on tensile fracture strength at different temperatures is conducted. Finally, based on the above conclusions, some useful suggestions on the selection of rubber materials for service in a wide temperature range are proposed.

Reducing the urban environment impact of joint leakage of underground prefabricated structures: Waterproof performance improvement of WSR-EPDM gasket

Composite cross-section gaskets with ethylene propylene diene monomer (EPDM) and water swelling rubber (WSR) have been adopted for specific underground prefabricated structures to prevent water leakage. Research on the sealant behavior and waterproof failure mechanism of WSR-EPDM gaskets needs to be strengthened, and this paper presents a combined experimental and computational study to investigate the leakage behavior of WSR-EPDM gaskets. The mechanical performance of the WSR-EPDM gasket based on the 3-stages load-deformation relationship was detected using the electronic universal testing machine. And the waterproof performance of the WSR-EPDM gasket was tested based on the developed waterproof capacity test setup to monitor the water leakage pressure of joints under various combinations of joint openings and offsets. Three groups of mechanical tests and 12 groups of waterproof tests were carried out. Then, a series of finite element models were developed using the coupled Eulerian-Lagrangian method to reveal the failure mechanism of the WSR-EPDM gasket. The hyperelastic behavior of the WSR-EPDM gaskets was described using the two-coefficient Mooney-Rivlin hyperelastic model and verified against the gasket-in-groove mechanical test results. A supplementary parametric study investigated the effect of joint opening and offset. The correlation between gasket contact stress and supplied water pressure was analyzed. Finally, an improved gasket was developed, and the mechanical and waterproofing behavior was evaluated. The results indicated that the joint opening and offset negatively influenced the gasket waterproof behavior. The water-leakage path of the WSR-EPDM gasket was more likely to happen via the gasket-groove interface when the joint offset was larger. Furthermore, the mean contact pressure ratio to the water leakage pressure of 1:1 can be used as the criteria for a preliminary estimate of the potential water leakage path. The improved WSR-EPDM gasket showed excellent waterproof performance, with a maximum increasing ratio of 127.6%. The developed test setup and numerical methodology provide novel insights into the waterproof performance improvement of composite cross-section gaskets in complex situations.

Modeling of hyperelastic polymer gels under blunt ballistic impact with three-dimensional flexibilities

Polymer gels are promising surrogates for human tissues, whereas simulating the behavior of these materials in 3D scenarios, particularly when they are subjected to dynamic loads, can be a complex and computationally demanding task. This work presents a solution to this challenge through developing a three-dimensional meshfree framework based on bond-based peridynamics. The novelties of this method are threefold: (1) The peridynamic bond force–stretch​ relationship of the Mooney–Rivlin hyperelastic model is original formulated and implemented based on thermomechanical theories and energy equivalence principles; (2) A novel contact algorithm is proposed to reduce the complexity of realistic loading and boundary conditions in blunt ballistic impact simulations; (3) This computational framework enables 3D dynamic simulations of hyperelastic polymer gels with complex geometric configurations. Tensile and compressive tests with varying loading rates are studied to validate the capability and accuracy of the proposed method. The proposed computational framework outperforms its finite element counterparts in modeling blunt impacts on polymer gels. It can be readily extended to penetration impact problems since accounting for failures is straightforward in this peridynamics framework.

Modeling of hyperelasticity in polyamide 12 produced by selective laser sintering

Polyamide 12 (PA12) is a core material in many 3D-printing techniques, including selective laser sintering (SLS), and its mechanical characterization helps to better understand behaviors of additively manufactured parts made from this polymer. In this paper, the elastic response of SLS-produced PA12 is shown to be nonlinear. Standard test samples with different orientations with regard to the scanning direction are 3D-printed with the use of PA2200 powder, and their elastic response is investigated under uniaxial tension at different strain rates. Mooney–Rivlin hyperelastic models are proposed to address the observed nonlinear elasticity of the samples. Cyclic response of the specimens is shown to be stabilized after a few transient cycles so the material parameters are determined for trained samples after shakedown in their response. The obtained parameters are found to depend on the loading speed; thus, a rate-dependent hyperelastic constitutive model is presented for PA12 produced by selective laser sintering. This model is validated by comparing its numerical prediction with empirical responses under simple tension tests.

3D printed flexible wearable sensors based on triply periodic minimal surface structures for biomonitoring applications

Soft piezoresistive wearable conductors have led to a paradigm shift in the monitoring of human bodily motions. Cellular additively manufactured conductors are promising piezoresistive components as they offer mechanical tunability and provide controllable percolation pathways. In the present study, we engineer high surface-area cellular structures with the triply periodic minimal surface (TPMS)-based architectures to tailor their piezoresistive response for use in wearable devices. A simple and economical fabrication process is proposed, wherein a fused deposition modeling 3D printing technique is utilized to fabricate flexible thermoplastic polyurethane (TPU) cellular structures. Interconnectivity of TPMS designs enables the coating of a continuous graphene layer over the TPU internal surfaces via a facile dip-coating process. The effects of pore shape on piezoresistivity are studied in four different TPMS structures (i.e. Primitive, Diamond, Gyroid, and I-WP). Mechanical properties of sensors are evaluated through experimental procedures and computation methods using finite element analysis of the Mooney–Rivlin hyperelastic model. The piezoresistive performance of sensors exhibits durability under cyclic compression loading. Finally, we conclude that the Primitive structure offers suitable piezoresistive characteristics for sensing of walking, whereas the Diamond structure presents favorable results for respiration monitoring.

Multi-scale constitutive model of human trabecular bone

The present study aims to formulate a new multiscale constitutive model of human trabecular bone. The trabecular bone was modelled as a nonlinear viscoelastic material. The viscoelastic effects of single trabeculae were considered by means of a hereditary integral in which stress depends on time and strain, while the elastic response was described by the hyperelastic Mooney–Rivlin model. The cuboid bone sample was extracted from the femoral head during the hip replacement surgery. The material constants in the constitutive equation were identified based on the stress relaxation test performed on the cuboid sample and the microindentation tests performed on trabeculae using the curve-fitting procedure. The microindentation tests were performed using a spherical tip instead of Vickers or Berkovich tip to minimize plastic effects during trabecular deformation. In order to validate formulated constitutive model, results from a FE simulation of stress relaxation test and uniaxial compression test were compared to the results of the corresponding experiments conducted on a macroscopic bone sample. Good agreement was observed between numerical and experimental results. The viscoelastic behaviour predicted by the proposed constitutive equation corresponds well to the response of human trabecular bone under various types of load conditions. This demonstrates the high ability of our constitutive model to simulate the behaviour of trabecular bone on a micro- and macroscopic scale. Thus, we conclude that the model, which was formulated for a single trabecula, can be successfully applied to simulate mechanical behaviour of the tissue in a macroscale.

Theory and experiment for dynamics of hyperelastic plates with modal interactions

This paper presents a joint experimental and theoretical approach to the dynamics and mass sensitivity of hyperelastic plates including cases with modal interactions. For the theoretical approach, the plate structure is assumed to undergo large strains and deformations using the Mooney-Rivlin hyperelastic strain energy density model and the von-Kármán geometric nonlinearity, respectively. The plate is modelled using the continuum mechanics definitions and the Kirchhoff–Love plate theory. The coupled in-plane and out-of-plane equations of motion are obtained using Hamilton's equation and solved afterwards using a combination of Galerkin's procedure together with a dynamic equilibrium technique. For the experimental analysis, the properties of the material are first obtained by performing a set of stress-strain tests on the samples following the ASTM D-412 standard, and then from the same rubber material, a plate structure is fabricated, and an externally actuated vibration test is performed. The nonlinear frequency response of the structure both with and without a concentrated mass is investigated and the capability of the structure for mass sensing is discussed. By obtaining the amplitude-frequency response of the structure due to the applied external periodic load, using both theoretical and experimental approaches, it is shown that the given model has good accuracy in simulating the nonlinear dynamics of the hyperelastic plate structure under different conditions. Furthermore, a set of analyses on the nonlinear forced vibration behaviour of hyperelastic plates at different internal resonances, using concentrated mass and the length-to-width ratio is presented. The results of this study are useful in designing systems involving hyperelastic plates, such as soft robots and soft functional devices.

Analytical pressure–deflection curves for the inflation of pre-stretched circular membranes

The inflation of circular membranes is a benchmark problem in finite elasticity. Due to its mathematical complexity, so far most efforts have been focused on numerical solutions whereas analytical solutions are still lacking. The present work proposes an analytical formula to compute the pressure–deflection curves of pre-stretched circular membranes. The Mooney–Rivlin hyperelastic model is assumed as constitutive law. Following the semi-inverse method, an analytical model assuming spherical deformed configurations is developed and a pressure–deflection relation is derived. Due to the simplifying hypothesis on the kinematics, the analytical expression for the pressure is not always accurate and requires an adjustment. Two corrective polynomial functions are implemented in the formula to capture the effect of the pre-stretch, which affects the pressure–deflection response depending on the magnitude of deformation and material parameters. The polynomial coefficients are calibrated by fitting numerical solutions of the differential equilibrium equations of the exact theory. The calibration is performed over a wide range of material parameters and pre-stretch, covering all the values of practical interest. As a result, an accurate and ready-to-use pressure–deflection relation for practical applications is obtained. The proposed formula is validated by finite element simulations. Differently from other solutions in literature, our formula holds for both compressible and incompressible materials. Experimental bulge tests on pre-stretched rubber membranes are carried out and the proposed formula is used for the calibration of material parameters. A code developed in the software Mathematica is made available upon request to directly compute the proposed formula.

Effects of Hysteresis on the Dynamic Deformation of Artificial Polymeric Heart Valve

The deformation behavior of an artificial heart valve was analyzed using the explicit dynamic finite element method. Time variations of the left ventricle and the aortic pressure were considered as the mechanical boundary conditions in order to reproduce the opening and closing movements of the valve under the full cardiac cycle. The valve was assumed to be made from a medical polymer and hence, a hyperelastic Mooney–Rivlin model was assigned as the material model. A simple formula of the damage mechanics was also introduced into the theoretical material model to express the hysteresis response under the unloading state. Effects of the hysteresis on the valve deformation were characterized by the delay of response and the enlargement of displacement. Most importantly, the elastic vibration observed in the pure elastic response under the full close state was dramatically reduced by the conversion of a part of elastic energy to the dissipated energy due to hysteresis.

Highly stretchable, shape memory and antioxidant ionic conductive degradable elastomers for strain sensing with high sensitivity and stability

Ionic conductive elastomers with high sensitivity, good mechanical property, shape memory and antioxidant capacity are of great significance in flexible smart wearable devices, but remains a challenge. Here we designed a series of degradable elastomers based on hexamethylene diisocyanate crosslinked poly(polycaprolactone citric acid)–co-dopamine (PCD) and 4, 4′-diaminodiphenyldisulfide with excellent mechanical property, shape memory property and antioxidation, and the ionic liquid is further introduced into elastomer leading to ionic conductive elastomer. The elastomer with 10 % 4, 4′-diaminodiphenyldisulfide shows tensile strain of 829 % and tensile stress of 5.48 MPa. After introducing 10 % ionic liquid, the ionic conductive elastomer’s conductivity increases to 1.18 × 10-6 S/cm while still maintaining 580 % break elongation. The elastomers can maintain good stability after cyclic stress–strain test and show high strain sensitivity to small deformation in joint motion signal monitoring. In addition, the elastomers have good shape memory property, cytocompatibility and in vivo biocompatibility. The mechanical response of elastomers was studied by employing the Mooney-Rivlin hyperelastic model. Further comparative analysis shows that the local stress concentration is the main factor leading to the failure of the elastomer. The ionic conductive elastomers with good conductivity, sensing sensitivity, mechanical strength and antioxidation are promising in the field of flexible wearable devices.

Synthesizing of Metallized Acrylic Containing Both Gadolinium and Lead as a Transparent Radiation Shielding Material and Its Physical Properties

In this study, a series of optically transparent metallized acrylics containing Gd and Pb were synthesized by the bulk polymerization of Gd(MAA)3, Pb(MAA)2 and AM according to different polymerization procedures. The variation of their optic transmittance and mechanical performance with Gd contents was investigated. Then, quasi-static uniaxial tensile tests under different strain rates and temperatures were performed to study the influence of strain rate and temperature on the mechanical properties of radiation-shielding metallized acrylic containing both Gd and Pb. The tensile responses of this material distinctly exhibit nonlinear characteristics and strongly depend on both temperature and strain rate. Based on the experimental results, a modified Zhu–Wang–Tang (ZWT) constitutive model, in which the standard elastic component was replaced by the Mooney–Rivlin hyperelastic model, was implemented to characterize the observed both hyperelastic and viscoelastic behaviors. The constitutive parameters were expressed as functions of temperature and determined by experimental data. The model fitting results indicate that the selected constitutive model can accurately describe the nonlinear tensile stress–strain responses of metallized acrylic containing Gd and Pb. Furthermore, the great difference in constitutive parameters implies that the viscoelastic behavior of the as-prepared metallized acrylic affects the response to quasi-static tensile loading the most.

A method for determining elastic constants and boundary conditions of three-dimensional hyperelastic materials

Although there have been many studies on the identification of elastic constants and boundary conditions, it is still challenging to simultaneously estimate the unknown elastic constants and boundary conditions of a three-dimensional hyperelastic material under locally observed boundary conditions with noise. In this paper, a novel inverse method based on two different objective functions is proposed. The first objective function is constructed by a “dual fields” construction method, which utilizes redundant Cauchy boundary conditions (displacement boundary conditions and force boundary conditions) on the observable boundary. The second objective function is based on the gap between measured and calculated displacements of the observable boundary. The inverse analysis is conducted by an adjoint method and a linear approximation method based on the theories of continuum mechanics and finite element method to obtain the Jacobian matrices of the objective functions with respect to different variables. An alternating iterative algorithm based on Gauss–Newton method is proposed to minimize the objective functions. The proposed inverse algorithm is tested using nearly incompressible Neo-Hookean and nearly incompressible Mooney–Rivlin hyperelastic model in numerical experiments and physical experiments with a silicone-based tissue-mimicking material. The results of both numerical and physical experiments show the feasibility and accuracy of the proposed method.

Novel curve fitting method based on constrained optimization for the modelling of human brain aneurysms using Mooney–Rivlin hyperelastic materials in the entire range of deformations til rupture

Purpose: Brain aneurysms often prove fatal if ruptured, therefore, understanding their mechanical behaviour in the coupled system of vessels and blood flow can significantly help preventive surgical treatment. The purpose of this work was to analyse measurement data and to determine material parameters for the hyperelastic Mooney–Rivlin model for model building and numerical simulations of aneurysms. Methods: A total of 88 human brain aneurysm specimens of 41 patients obtained from surgery were processed in this work based on the tests performed by the authors in a previous project. A novel algorithm was proposed and applied in this work to fit stress–stretch ratio curves for multiple measurement data using constrained optimization with hard conditions to comply with known mechanical behaviour. Results: The method produced parameters of stretch ratio–stress curves for a number of groups of the specimens representing the average as well as the extreme stresses, separately for male and female subsamples. Stretch range both in compression and in tension up to rupture was covered and material stability for the entire range was also verified. Conclusions: The fitted curves with recommended range of validity are directly applicable to numerical finite element or coupled simulations of aneurysms supporting preventive medical treatment or decision making

Some aspects related to the indentation-based viscoelastic modelling of trabecular bone tissue

Purpose The research aims to verify to what extent the shape of an indenter tip influences the final form of the constitutive equation for the trabecular bone. Methods Trabecular bone was formulated as a non-linear viscoelastic material with Mooney-Rivlin hyperelastic model to describe the purely elastic response of the bone tissue. Tests of the mechanical properties of the trabecular bone, resected from the femoral head of a 56-year-old patient, were carried out with two types of indenter: the spherical tip of a diameter of 200 µm and pyramid Vickers tip with 136º between plane faces. Tests with both indenters included loading and unloading phases with no hold at peak force and with hold time t=20 s and were conducted with a maximum load P_max = 500 mN and loading/unloading rate V = 500 mN/min. Results The formulated constitutive model describes the trabecula behaviour very well. The model curves match the experimental results in the loading phase, holding period and most of the unloading ramp. The purely viscoelastic material constants are very close in value for both considered tips, but purely elastic constants differ. Conclusions The results indicate that the constitutive model based on the indentation with the Vickers tip does not cover the plastic residual deformation. When a viscoelastic response of bone is expected, a model with constants calibrated for the spherical tip should be used, and the other set of parameters values (Vickers tip) when trabeculae may undergo plastic deformation.

The Effect of Catalysts Addition to Silicone Rubber as Pressure Transmitting Medium in the Pebble Fuel Production Process

Pebble Fuel is a spherical fuel for high temperature gas-cooled reactors (HTGR). This fuel must have a homogeneous density distribution. Hyperelastic material is used as a pressure transmitting medium (PTM) material in making Pebble fuel using the cold quasi-isostatic pressing method. PTM material properties and characteristics were predicted using the finite element analysis method. The problem is the type of material used and its suitable composition to make a pressure-transmitting medium that has the properties and characteristics of the material as predicted. This research discusses the manufacture of tensile specimens for pressure-emitting media using RTV-586 silicone rubber. The composition comprises three different variants with two major ingredients, namely RTV-586 silicone rubber and catalyst. The test results are then analyzed using the finite element method to determine the material composition that is appropriate or close to the predicted properties and characteristics of the PTM material. This initial study used the Mooney-Rivlin hyperelastic model. The Mooney-Rivlin model shows good similarity to the test result data. In future studies, it will make comparisons with other hyperelastic models to get a suitable PTM material constant.