Mooney-Rivlin Strain Energy Function Research Articles

Micro-macro analysis of Hyperelastic auxetic lattice structures under finite-strain regime

This paper addresses the intricate analysis of lattice structures, pivotal components in engineering applications, confronted by challenges arising from their diverse unit cells (UCs) and complex behavior across micromechanical and macro-scale dimensions. The present study deals with a comprehensive analysis of lattice structure with honeycomb and re-entrant auxetic unit cells under large deformation via both a robust finite element analysis (FEA) and experimental tests. The proposed FEA is developed based on the hyperelastic Mooney–Rivlin strain energy function and the novel exact motion field that can fully describe the projection of cross-section. The results demonstrate that mechanical parameters such as Young's modulus, Poisson's ratio, and shear modulus have significant nonlinear behaviors with respect to UC geometrical parameters that are crucial for optimization across varied operational conditions. Additionally, the lattices made of TPU material and fabricated by Fused Deposition Modeling are tested under three-point bending and compression considering contact interaction. The results reveal highly nonlinear responses due to instabilities in some links and material nonlinearity. Furthermore, the behavior of lattice structures is exceedingly dependent on the orientations and types of UCs. It also can be found that both of the proposed FEA and constitutive model are in good agreement with experimental data.

A numerical modelling framework for simulating rate-dependent delamination along polyurea/steel interfaces under Mode–I loading conditions

Polyurea coated structures have drawn significant attention from researchers across the globe for various applications such as waterproof roofing, vibration control, and protection from impact and blast threats. Despite widespread studies on the rate-dependent nature of polyurea and its bulk deformation response, there are very few works which study the delamination in polyurea coated structures and the associated rate sensitivity. The present work is aimed at two objectives: (i) to develop a suitable rate-dependent traction–separation model that fits an existing experimental dataset from literature related to rate-dependent delamination of polyurea/steel interface and, (ii) to develop a simple, robust numerical modelling framework to simulate the rate-dependent delamination process in a double–cantilever beam experiment found in the same literature source. Fulfilling the first objective resulted in identifying the important parameters in a traction–separation relation and their rate-dependency. A non-zero traction at zero crack opening displacement is one of the striking features of the experimental data and the proposed model takes that into account. As part of the second objective, a numerical modelling involving separate constitutive equations for the bulk and interface regions of the polyurea adhesive has been proposed. The interface is modelled as a rate-dependent linear elastic plastic material while the bulk is considered as hyperelastic material using two parameter Mooney–Rivlin strain energy function. This model was implemented in LS-DYNA. The proposed approach could reproduce the fracture mechanisms of the dynamic delamination process reported in the literature and helped in understanding the interaction between the bulk and the interface during the delamination at different strain rates. This approach could be a potential alternative to the cohesive zone method, easing the complexity of simulations involving dynamic delamination in commercial packages.

An analytical solution to predict temperatures of dumbbell-shaped rubber specimens under cyclic deformation

Abstract The objective of this study is to propose an analytical solution that can predict temperatures of dumbbell-shaped rubber specimens under cyclic deformation. To achieve this, first, a new mathematical equation was developed based on a modified Mooney-Rivlin (MR) strain energy function, the pseudo-elasticity theory, and the inverse analysis method. This equation was used to determine the internal heat generation rates of rubber compounds. With heat generation rates, the governing equation of heat conduction and the mathematical expression of boundary conditions were further generated to describe the heat transfer in rubber compounds. Based on these equations, a novel analytical solution was developed—the RTDS solution (a solution to predict Rubber Temperatures in Dumbbell-shaped Specimens). This RTDS solution was used to predict rubber temperatures in dumbbell-shaped specimens under cyclic deformation. The results showed that the RTDS solution took 11.9 seconds to derive the rubber temperature results with an average mean absolute percent error (MAPE) of 9.2% compared with lab recordings. The RTDS solution identified a logarithmic increase in rubber temperatures at rising strain levels, and it also identified an increase in rubber temperatures with the rising strain rates. Moreover, the RTDS solution characterized an inverse proportional relationship between the rubber temperature increments and the ambient temperatures.

Experimental and numerical analysis of hyperelastic plates using Mooney-Rivlin strain energy function and meshless collocation method

In this research, the flexural behavior of a hyperelastic plate made of silicone under uniformly distributed loading was investigated. The displacements and strains were formulated using the first-order shear deformation plate theory (FSDPT). The right Cauchy-Green tensor and Mooney-Rivlin strain energy function were employed to formulate hyperelastic silicone plate governing equations. The strong-form of the governing equations of the silicone plate using Mooney-Rivlin strain energy function and Euler-Lagrange relations were derived for the first time. In the governing equations, the effects of geometrical and material nonlinearity effects were considered simultaneously. The meshless collocation method (MCM) and thin-plate spline radial basis function were utilized to discretize the governing equations. The arc-length continuation algorithm was also used for analyzing the system of non-linear equations. MCM results were compared to those of the experimental test to validate the results of the meshless method. For this purpose, a square silicone plate with simply supported boundary conditions under uniformly distributed loading was investigated via the digital image correlation (DIC) technique. According to the obtained results, the MCM based on radial basis function is an adequate method for non-linear analysis of hyperelastic plates with Mooney-Rivlin strain energy function under uniformly distributed loading.

Meshless method and experimental analysis of hyperelastic plates using Mooney-Rivlin strain energy function subjected to concentrated loading

The purpose of this research is to investigate a strong-form meshless method for experimentally and numerically analyzing hyperelastic plates subjected to concentrated loading. The silicone hyperelastic plate was formulated utilizing Mooney-Rivlin strain energy function and Lagrangian strains. The displacements and Lagrangian strains of the plate are also calculated using the first-order shear deformation plate theory (FSDPT). The hyperelastic plate’s governing equations are deduced in strong form and analyzed using the logarithmic thin-plate spline radial basis function (LTPS-RBF) and the meshless method for the first-time. The nonlinear system of governing equations is solved using the arc-length continuation algorithm. In the meshless method, the concentrated loading is modeled using the Kronecker delta function in the expression of external work. A silicone plate is employed experimentally to evaluate the accuracy of the meshless method’s results. Furthermore, the finite element method is utilized to compare the contours obtained from the meshless method. Concentrated loading is considered at various points to validate the meshless methods and results from the experimental analysis. The most significant difference between the meshless method and the experimental results is less than 15%. The results indicate that the meshless method is highly accurate when analyzing hyperelastic plates subjected to concentrated loading.

Corneal biomechanical stiffness and histopathological changes after in vivo repeated accelerated corneal cross-linking in cat eyes

Corneal cross-linking (CXL) has been proved efficiency for treating progressive keratoconus and other corneal ectasia diseases by stabilizing corneal geometry and biomechanics. However, the necessity of repeated CXL treatment in patients is unknown. This study aimed to investigate corneal biomechanical stiffness and change in corneal histopathological characteristics after repeated accelerated CXL (A-CXL) in cat eyes. A-CXL was performed with 0.1% riboflavin applied for 10 min, followed by ultraviolet A irradiation at 30 mW/cm2 for 3 min at 365 nm in 15 domestic cats. Corneas (n = 30) were divided into three groups: one-time accelerated corneal cross-linking (A-CXL*1 group), repeated accelerated corneal cross-linking (A-CXL*2 group), and an untreated control group. In A-CXL*2 group, A-CXL was repeated at 1-month intervals. In vivo ocular examinations were performed pre- and postoperatively. Biomechanical analysis was performed using a biotester biaxial testing system. We used the Mooney–Rivlin strain-energy function to describe corneal material properties. No infection in any case after A-CXL was observed. Biomechanical tests showed that the stress-strain curves of the two A-CXL groups were significantly different from those of the control group (P < 0.01), whereas stress-strain curve of the A-CXL*2 group was similar to that of the A-CXL*1 group (P > 0.05). Delayed epithelial healing and haze were observed 1 month after surgery. Stromal demarcation line depth measured with anterior spectral-domain optical coherence tomography was 187.6 ± 20.4 and 197.1 ± 11.5 μm for the A-CXL*1 and A-CXL*2 groups, respectively (P > 0.05). These results show that A-CXL can increase corneal biomechanics in cat eyes. The biomechanical enhancement of cat corneas treated with repeated A-CXL at 1-month intervals was similar to that of performing a one-time A-CXL. Repeated cross-linking procedures at short intervals may increase the risk of adverse reactions, and more caution should be taken in clinical applications.

Optimal design of phononic media through genetic algorithm-informed pre-stress for the control of antiplane wave propagation

In this paper we employ genetic algorithms in order to theoretically design a range of phononic media that can act to prevent or ensure antiplane elastic wave propagation over a specific range of low frequencies, with each case corresponding to a specific pre-stress level. The medium described consists of an array of cylindrical annuli embedded inside an elastic matrix. The annuli are considered as capable of large strain and their constitutive response is described by the popular Mooney–Rivlin strain energy function. The simple nature of the medium described is an alternative approach to topology optimization in phononic media, which although useful, often gives rise to complex phase distributions inside a composite material, leading to more complicated manufacturing requirements.

A strong-form meshfree method for stress analysis of hyperelastic materials

A strong-form based meshfree method for stress analysis of hyperelastic materials under large deformations is presented in this research. The non-linear elastic response of hyperelastic materials is modeled by the compressible Mooney–Rivlin strain energy function. Simple implementation and truly meshfree nature are some of the advantages of strong-form meshfree methods. In the presented meshfree formulation, second derivatives of the strain energy function with respect to the components of the deformation gradient tensor appear. These second derivatives are obtained analytically. Various plane stress and plane strain problems with different boundary conditions are considered. The effects of the value of the shape parameter, the number of the nodes in the support domain, and the total number of nodes on the performance of the method are investigated. Various techniques for applying boundary conditions such as the direct collocation method and the use of fictitious nodes are examined, and an alternative method is presented to apply boundary conditions in the proposed meshfree method.

Enhancement of stiffness and dynamic mechanical properties of polymers using single-walled-carbon-nanotube – a multiscale finite element formulation study

The static and dynamic mechanical properties of polymeric materials can greatly be enhanced by using carbon nanotubes as reinforcement material. However, studies still need to be carried out to characterize the dynamic mechanical properties of the Carbon-Nanotube-Reinforced-Polymer (CNRP) material. Experimental investigations for this purpose have severe limitations and, in most cases, appropriate and reliable experimental work could not be carried out. Computational modelling and simulation encompassing multiscale material behavior provides an alternate approach to study the material behavior. The objective of the present work is to study the enhancement of stiffness and dynamic mechanical properties of Carbon-Nanotube-Reinforced-Polymer (CNRP) material by using a 3D multiscale finite-element model of the representative volume element of the CNRP material. A composite material model consisting of a polymer matrix, an interface region, and a Single-Walled Carbon Nanotube (SWCNT) is constructed for this purpose. The polymer matrix is modeled with the Mooney-Rivlin strain energy function to calculate its non-linear response and the interface region is modeled via van der Waals links. The SWCNT is modeled as a space frame structure by using the Morse potential and as a thin shell model based on Donnell’s Shell Theory. The stiffness response of the CNRP is calculated and the natural frequencies of the CNRP are also determined. The viscoplastic behavior of the polymer matrix material is considered and the rate-dependent characteristics of the CNRP are studied. The damping properties of the CNRP are investigated based on its viscous and structural damping mechanisms. The effectiveness of the SWCNT reinforcement is quantified and characterized.

A Non-Linear Model of an All-Elastomer, in-Plane, Capacitive, Tactile Sensor Under the Application of Normal Forces.

In this work, a large deformation, non-linear semi-analytical model for an all-elastomer, capacitive tactile unit-sensor is developed. The model is capable of predicting the response of such sensors over their entire sensing range under the application of normal forces. In doing so the finite flat punch indentation model developed earlier is integrated with a capacitance model to predict the change-in-capacitance as a function of applied normal forces. The empirical change-in-capacitance expression, based on the parallel plate capacitance model, is developed to account for the fringe field and saturation effects. The elastomeric layer used as a substrate in these sensors is modeled as an incompressible, non-linear, hyperelastic material. More specifically, the two term Mooney-Rivlin strain energy function is used as a constitutive response to relate the stresses and strains. The developed model assumes both geometrical as well as material non-linearity. Based on the related experimental work presented elsewhere, the inverse analysis, combining finite element (FE) modeling and non-linear optimization, is used to obtain the Mooney-Rivlin material parameters. Finally, to validate the model developed herein the model predictions are compared to the experimental results obtained elsewhere for four different tactile sensors. Great agreements are found to exist between the two which shows the model capabilities in capturing the response of these sensors. The model and methodologies developed in this work, may also help advancing bio-material studies in the determination of biological tissue properties.

Visco-elasto-plastic modeling of small intestinal submucosa (SIS) for application as a vascular graft

In developing new tissue engineered for vascular grafts, the mechanical properties of the material and its evolution once implanted are of utmost importance because they determine the regeneration of the vessel and the blood flow through the conduit. In fact, compliance mismatch is considered the main determinant of graft failure. In this research, we analyze the dynamic properties of the small intestinal submucosa (SIS), and propose and validate a constitutive model to fit the material's behavior. A uniaxial creep and recovery test was performed on SIS tubes to find the constitutive parameters. The model was composed by an elastic element in series with two Kelvin-Voigt solid elements and a plastic slider. The first elastic component was defined using Mooney-Rivlin strain energy function, while the plastic component was defined using a third-degree polynomial function of the plastic stress. The viscoelastic behavior was defined using the creep compliance formulation for the Kelvin-Voigt model. The parameters for the plastic and non-linear elastic elements followed a normal distribution, while the spring and dashpot constants of the visco-elastic element had a linear dependence on the load applied. The constitutive model was then used to simulate the SIS under the geometrical and pressure conditions found in native vessels for 1000 cycles at a frequency of 60 cycles per minute. From the cases simulated, performance curve charts were obtained in terms of the compliance of the material. These curve charts can be used as a predictive tool of the graft's behavior based on its geometry.

Mechanical behaviour of short membranous liquid-filled cylinders under axial loadings

The mechanical behaviour of membranous liquid-filled cylinders is important in many industrial and biomedical applications, such as in modelling the incudostapedial joint in the middle ear. Although membranous liquid-filled containers with various geometries have been investigated under different loading conditions, no study on the mechanical behaviour of membranous circular tubes under tension-compression testing has been published. In this work, we use the theory of large deformations of elastic membranes to develop an analytical model for a liquid-filled circular tube under an axial load, neglecting the effect of gravity. We use the Mooney–Rivlin strain-energy function and numerically calculate solutions for short to moderate tube lengths. A finite-element model of the tube is also developed and the simulation results are compared with the solution from the analytical model. The finite-element model is then used for the case of a tube with an elliptical cross-section. We observe a nonlinear behaviour in the force-displacement curve of the tubes when close to the zero-force configuration. Furthermore, the tube can go through instabilities when the force changes sign. The unstable region grows if some liquid is removed from the membranous container, causing the straight walls to become convex, and also if the tube length is increased. The tube becomes stiffer and the nonlinear behaviour becomes less significant when the tube cross-section is changed from circular to elliptical. The results may provide insight into experimental observations of the mechanical behaviour of the incudostapedial joint.

Correction to ‘Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media’

[This corrects the article DOI: 10.1098/rspa.2015.0450.].

On propagation of axisymmetric waves in pressurized functionally graded elastomeric hollow cylinders

Soft materials can be designed with a functionally graded (FG) property for specific applications. Such material inhomogeneity can also be found in many soft biological tissues whose functionality is only partly understood to date. In this paper, we analyze the axisymmetric guided wave propagation in a pressurized FG elastomeric hollow cylinder. The cylinder is subjected to a combined action of axial pre-stretch and pressure difference applied to the inner and outer cylindrical surfaces. We consider both torsional waves and longitudinal waves propagating in the FG cylinder made of incompressible isotropic elastomer, which is characterized by the Mooney-Rivlin strain energy function but with the material parameters varying with the radial coordinate in an affine way. The pressure difference generates an inhomogeneous deformation field in the FG cylinder, which dramatically complicates the superimposed wave problem described by the small-on-large theory. A particularly efficient approach is hence employed which combines the state-space formalism for the incremental wave motion with the approximate laminate or multi-layer technique. Dispersion relations for the two types of axisymmetric guided waves are then derived analytically. The accuracy and convergence of the proposed approach is validated numerically. The effects of the pressure difference, material gradient, and axial pre-stretch on both the torsional and the longitudinal wave propagation characteristics are discussed in detail through numerical examples. It is found that the frequency of axisymmetric waves depends nonlinearly on the pressure difference and the material gradient, and an increase in the material gradient enhances the capability of the pressure difference to adjust the wave behavior in the FG cylinder. This work provides a theoretical guidance for characterizing FG soft materials by in-situ ultrasonic nondestructive evaluation and for designing tunable waveguides via material tailoring along with an adjustment of the pre-stretch and pressure difference.

A robust hyper-elastic beam model under bi-axial normal-shear loadings

In this paper, a simple and robust constitutive model is proposed to simulate mechanical behaviors of hyper-elastic materials under bi-axial normal-shear loadings in the finite strain regime. The Mooney–Rivlin strain energy function is adopted to develop a two-dimensional (2D) normal-shear constitutive model within the framework of continuum mechanics. A motion field is first proposed for combined normal and shear deformations. The deformation gradient of the proposed field is calculated and then substituted into right Cauchy–Green deformation tensor. Constitutive equations are then derived for normal and shear deformations. They are two explicit coupled equations with high-level polynomial non-linearity. In order to examine capabilities of the developed hyper-elastic model, uniaxial tensile responses and non-linear stability behaviors of moderately thick straight and curved beams undergoing normal axial and transverse shear deformations are simulated and compared with experiments. Fused deposition modeling technique as a 3D printing technology is implemented to fabricate hyper-elastic beam structures from soft poly-lactic acid filaments. The printed specimens are tested under tensile/compressive in-plane and compressive out-of-plane forces. A finite element formulation along with the Newton–Raphson and Riks techniques is also developed to trace non-linear equilibrium path of beam structures in large defamation regimes. It is shown that the model is capable of predicting non-linear equilibrium characteristics of hyper-elastic straight and curved beams. It is found that the modeling of shear deformation and finite strain is essential toward an accurate prediction of the non-linear equilibrium responses of moderately thick hyper-elastic beams. Due to simplicity and accuracy, the model can serve in the future studies dealing with the analysis of hyper-elastic structures in which two normal and shear stress components are dominant.

Extreme softness of brain matter in simple shear

We show that porcine brain matter can be modelled accurately as a very soft rubber-like material using the Mooney–Rivlin strain energy function, up to strains as high as 60%. This result followed from simple shear experiments performed on small rectangular fresh samples (2.5cm3 and 1.1cm3) at quasi-static strain rates. They revealed a linear shear stress–shear strain relationship (R2>0.97), characteristic of Mooney–Rivlin materials at large strains. We found that porcine brain matter is about 30 times less resistant to shear forces than a silicone gel. We also verified experimentally that brain matter exhibits the positive Poynting effect of non-linear elasticity, and numerically that the stress and strain fields remain mostly homogeneous throughout the thickness of the samples in simple shear.

DEVELOPMENT OF OPEN SOURCE FINITE ELEMENT SOLVER “LD-FEM’ FOR MODELING AND SIMULATION OF RUBBER MATERIALS

“LD-FEM” is an open source computer program working on the basis of finite element method (FEM) which is aimed to model and simulate large deformation in rubber materials. The kinematics of large deformation on the basis of the Total Lagrange framework is applied to linear 4-nodes tetrahedral element and then solved with Newton-Raphson iterative scheme. Furthermore, to obtain the material tangent stiffness directly from strain energy density functions, the Gill-Murray theory of numerical second derivative is used in LD-FEM. Finally, by using the Mooney-Rivlin strain energy function, the performance of LD-FEM is addressed for uniaxial tensile, shear and torsion loading tests. The results confirm the capability of LD-FEM to capture nonlinear behavior of the large deformation either with analytical or numerical approach on the material stiffness derivation with error less than 2%.

Modeling of human circle of Willis with and without aneurisms.

This paper includes results of the first stage of research aimed at the development of recommendations for physicians in order to help them to choose a particular type of cerebral arteries aneurysms treatment. Recent studies show that the majority of aneurysms develop as a result of hemodynamic and degenerative lesions of the vascular wall. Obviously, such wall damage can be studied using the methods of continuum mechanics and numerical simulations. Biomechanical modelling allows us to study hemodynamic parameters and stress-strain state of these arteries in health and disease, and to formulate practical recommendations for the necessity and reasonable selection of a particular type of cerebral arteries aneurysm treatment. At this stage the realistic geometric models of arterial circle of Willis were built for its normal state and in the presence of aneurysms. The ultrasound analysis of circle of Willis was conducted in order to obtain blood flow parameters and the boundary conditions for carotid and vertebral arteries. Also, the mechanical properties of these arteries were investigated and constants of the Mooney-Rivlin strain energy function were obtained. Thus, the boundary problem describing the behaviour of human Willis circle arteries was stated. Further, this problem will be solved numerically using the finite element method. The numerical results will be analyzed from the point of view of the influence of the mechanical factors on the emergence, growth and rupture of circle of Willis aneurysms.

Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder

We consider torsional wave propagation in a pre-stressed annular cylinder. Hydrostatic pressure is applied to the inner and outer surfaces of an incompressible hyperelastic annular cylinder, of circular cross-section, whose constitutive behaviour is governed by a Mooney–Rivlin strain energy function. The pressure difference creates an inhomogeneous deformation field, which modifies the inner and outer radii of the annular cylinder. We wish to determine the effect that this pre-stress, and a given axial stretch, has on the propagation of small-amplitude torsional waves through the medium. We use the theory of small-on-large to deduce the linear wave equation that governs incremental torsional waves and then determine the dispersion relation for the pre-stressed annulus by using an approximation procedure (the Liouville–Green transformation). We show that this scheme compares well to numerical solutions except in regions very close to turning points (of the transformed ordinary differential equation). In particular, we find that the inhomogeneous deformation makes the coefficients of the governing ordinary differential equation spatially dependent and affects the location of the roots of the dispersion relation. We observe that, if the pressure on the outer surface of the annular cylinder is greater (smaller) than that on the inner, then the cut-on frequencies are spaced further apart (closer) than they would be in the stress-free case. This result could potentially be used to tune the propagation characteristics of the cylinder over a range of frequencies.

Inflation of a circular elastomeric membrane into a horizontally semi-infinite liquid reservoir of finite vertical depth: Estimation of material parameters from volume–pressure data

A method is introduced for estimating values of the hyperelastic material parameters and the residual membrane tension (prestretch) that best describe a set of volume-pressure inflation response data acquired during a cell-elastomer composite diaphragm inflation (CDI) experiment. Based on a model for the quasi-static inflation of a prestretched clamped circular isotropic incompressible hyperelastic membrane into a horizontally semi-infinite incompressible liquid reservoir of finite vertical depth, the fit methodology presented here is developed explicitly for a Mooney-Rivlin (MR) strain energy function. The parameter fit is formulated as a nonlinear least-squares optimization problem (NLSOP), and two empirical constraints are postulated to restrict the domain of model fit parameters to physically relevant values. A numerical least-squares regression analysis utilizing the Nelder-Mead simplex algorithm is proposed for solving the constrained NLSOP. To demonstrate the capabilities of the method, the fit procedure is utilized to estimate the residual membrane tensions and the MR constitutive parameters C 1 and C 2 that characterize the pseudoelastic loading and unloading responses of a polydimethylsiloxane elastomer membrane as observed in the context of a CDI experiment. The material parameter estimates are verified as being both accurate and repeatable, and the MR non-uniform inflation model is shown to produce better fits (in a least-squares sense) than direct polynomial fitting of the original data set. Extension of the fit methodology to volume-pressure data measured for cell-elastomer composite diaphragm specimens, though possible, requires the development of new constitutive models that can better account for the complex rheological behavior of a living epithelial sheet.