Hyperelastic Material Model Research Articles

Biaxial test and hyperelastic material models of silicone elastomer fabricated by extrusion-based additive manufacturing for wearable biomedical devices

Biaxial tensile testing of the silicone elastomer fabricated by additive manufacturing (AM) for wearable biomedical devices is presented. Silicone elastomer has unique mechanical properties ideal for stretchable parts in wearable biomedical devices. Biaxial tensile testing, in comparison to the uniaxial tensile and compression tests, is more representative for materials with large, multidirectional stretch. This study investigates the experimental setup and methods to measure and characterize mechanical properties of AM solid and porous silicone elastomer sheets on equi-biaxial stretch (with the stretch ratio up to 2). Four hyperelastic material models were applied to characterize the AM silicone sheet and evaluated through the finite element analysis (FEA) to quantify the fit to biaxial tensile test results. The Yeoh model with C10 = 80.7 kPa, C20 = –2.11 kPa, and C30 = 0.22 kPa has the best match to the experimentally measured stress-strain curve as well as the force and strain in FEA. This study demonstrates that the effects of raster angle and thickness are insignificant in the stress-strain curve of an AM silicone elastomer with solid cross-section. This study also demonstrates that AM can fabricate anisotropic porous silicone elastomer parts with different stiffness in x- and y-directions.

Different types of solitary waves in a thermo-hyperelastic neo-Hookean cylindrical shell

Based on the hyperelastic material models, the propagation of nonlinear travelling waves in cylindrical rods has been studied. Here, different types of nonlinear travelling waves in a thermo-hyperelastic neo-Hookean cylindrical shell are determined. By using the variational principle, a complex differential dynamical system describing the axisymmetric motion of the shell is derived. The effect of the temperature fields is considered. Moreover, the structure is extended from a one-dimensional rod to a three-dimensional cylindrical shell. Then in terms of the bifurcation theory, a detailed qualitative analysis of the system is carried out, and for the bifurcation parameters, their corresponding critical bifurcation values are determined. Combining with the orbits in phase diagrams under different parameters, solitary waves and periodic waves are found in the shell. It is worth pointing out that the solitary waves with the valley form may appear in the radial direction of the shell. The implicit analytical solutions and profiles of these travelling waves are given. There is a promising prospect for the propagation of strongly nonlinear travelling waves to detect structural defects and determine material parameters.

Prediction of mechanical behavior of 3D bioprinted tissue-engineered scaffolds using finite element method (FEM) analysis

Three-dimensional (3D) printing can be a promising tool in tissue engineering applications for generating tissue-specific 3D architecture. The 3D printing process, including computer-aided design (CAD), can be combined with the finite element method (FEM) to design and fabricate 3D tissue architecture with designated mechanical properties. In this study, we generated four types of 3D CAD models to print tissue-engineered scaffolds with different inner geometries (lattice, wavy, hexagonal, and shifted microstructures) and analyzed them by FEM to predict their mechanical behaviors. For the validity of computational simulations by FEM, we measured the mechanical properties of the 3D printed scaffolds. Results showed that the theoretical compressive elastic moduli of the designed constructs were 23.3, 56.5, 67.5, and 1.8 MPa, and the experimental compressive elastic moduli were 23.6 ± 0.6, 45.1 ± 1.4, 56.7 ± 1.7, and 1.6 ± 0.2 MPa for lattice, wavy, hexagonal, and shifted microstructures, respectively, while maintaining the same construct dimension and porosity. In addition, van der Waals hyperelastic material model was successfully utilized to predict the nonlinear mechanical behavior of the printed scaffolds with different inner geometries. These findings indicated that the CAD-based FEM prediction could be used for designing tissue-specific constructs to mimic the mechanical properties of targeted tissues or organs.

Wall Stress Distribution in Bicuspid Aortic Valve–Associated Ascending Thoracic Aortic Aneurysms

Bicuspid aortic valve-associated ascending thoracic aortic aneurysms (BAV-aTAAs) carry a risk of acute type A dissection. Biomechanically, dissection may occur when wall stress exceeds wall strength. Our aim was to develop patient-specific computational models of BAV-aTAAs to determine magnitudes of wall stress by anatomic regions. Patients with BAV-aTAA diameter greater than 4.5 cm (n= 41) underwent electrocardiogram-gated computed tomography angiography. Three-dimensional aneurysm geometries were reconstructed after accounting for prestress and loaded to systemic pressure. Finite element analyses were performed with fiber-embedded hyperelastic material model using LS-DYNA software (LSTC Inc, Livermore, CA) to obtain wall stress distributions. The 99th percentile longitudinal and circumferential stresses were determined at systole. The 99th percentile longitudinal wall stresses for BAV-aTAAs at sinuses of Valsalva, sinotubular junction (STJ), and ascending aorta were 361 ± 59.8 kPa, 295 ± 67.2 kPa, and 224 ± 37.6 kPa, respectively, with significant differences in ascending aorta vs sinuses (P< 1× 10-13) and STJ (P < 1× 10-6). The 99th percentile circumferential wall stresses were 474 ± 88.2 kPa, 634 ± 181.9 kPa, and 381 ± 54.0 kPa for sinuses, the STJ, and the ascending aorta, respectively, with significant differences in the ascending aorta vs sinuses (P= .002) and STJ (P < 1× 10-13). Wall stresses, both circumferential and longitudinal, were greater in the aortic root, sinuses, and STJ than in the ascending aorta on BAV-aTAAs. These results fill a fundamental knowledge gap regarding biomechanical stress distribution in BAV-aTAA patients, which when related to wall strength may provide prognostication of aTAA dissection risk by patient-specific modeling.

A model for hyperelastic materials reinforced with fibers resistance to extension and flexure

A model for the mechanics of a soft hyperelastic material reinforced with long fibers is presented in finite plane elastostatics. The strain energy potential of the composite is refined by the Mooney Rivlin model to accommodate the hyperelastic behaviors of the matrix material. Within the framework of the strain gradient theory, the kinematics of the fibers is formulated and subsequently integrated into the model of continuum deformation. A rigorous derivation of the Euler equation and the associated boundary conditions are presented by virtue of variational principles and a virtual work statement. In particular, the obtained model successfully predicts the ‘J-shaped’ stress-strain responses, deformation profiles and shear strain angles of the elastomer - polyester fiber composites subjected to uniaxial extension. The proposed model may also serve as an alternative of the Holzapfel model in the design and analysis of biomimetic composites with rapid strain-stiffening behaviors by providing direct estimations of the stress-strain responses of intended composites.

An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials

Computational biomechanics plays an important role in biomedical engineering: using modeling to understand pathophysiology, treatment and device design. While experimental evidence indicates that the mechanical response of most tissues is viscoelastic, current biomechanical models in the computational community often assume hyperelastic material models. Fractional viscoelastic constitutive models have been successfully used in literature to capture viscoelastic material response; however, the translation of these models into computational platforms remains limited. Many experimentally derived viscoelastic constitutive models are not suitable for three-dimensional simulations. Furthermore, the use of fractional derivatives can be computationally prohibitive, with a number of current numerical approximations having a computational cost that is O(NT2) and a storage cost that is O(NT) (NT denotes the number of time steps). In this paper, we present a novel numerical approximation to the Caputo derivative which exploits a recurrence relation similar to those used to discretize classic temporal derivatives, giving a computational cost that is O(NT) and a storage cost that is fixed over time. The approximation is optimized for numerical applications, and an error estimate is presented to demonstrate the efficacy of the method. The method, integrated into a finite element solid mechanics framework, is shown to be unconditionally stable in the linear viscoelastic case. It was then integrated into a computational biomechanical framework, with several numerical examples verifying the accuracy and computational efficiency of the method, including in an analytic test, in an analytic fractional differential equation, as well as in a computational biomechanical model problem.

Theoretical Modeling and Exact Solution for Extreme Bending Deformation of Hard-Magnetic Soft Beams

Abstract Hard-magnetic soft materials (HMSMs) manufactured by embedding hard-magnetic particles in soft materials belong to a new type of soft active materials. The abilities of fast and complicated transformations of hard-magnetic soft structures provide a promising technology for soft robotics, flexible electronics, and biomedical devices. It is significant to investigate the mechanical behaviors of hard-magnetic soft structures for their better applications. In this work, a hard-magnetic soft beam under an external magnetic field is theoretically modeled and the exact solutions for its mechanical responses are presented. First, the governing equations and boundary conditions are derived based on the principle of minimum potential energy. To solve the derived governing equations analytically, a new polynomial fitting model for hyperelastic materials is proposed for the hard-magnetic soft beam. Then, the exact solutions of a cantilevered hard-magnetic soft beam actuated by a uniform magnetic field in any direction are obtained. The newly derived exact solutions are further verified by comparing current results with those from recent simulations and experiments. For large bending angles up to 90 deg and extreme bending angle up to 180 deg, quite consistent agreement among exact solutions, numerical simulations, and experimental observations can be achieved. Finally, using our theoretical model, the deformation of the hard-magnetic soft beam actuated by magnetic fields in an arbitrary direction with non-zero magnetic declination is explored. When the magnetic actuation is increased from a small level gradually, the hard-magnetic soft beam deflects and it would undergo small, large, and extreme bending deformations in sequence. It is very interesting that, when the magnetic actuation is sufficiently large, the hard-magnetic soft beam is stretched and its centerline tends to align with the external magnetic field direction, implying that the hard-magnetic soft beam undergoes a uniaxial tension. The theoretical modeling and exact solutions for hard-magnetic soft beams are expected to be useful in the analysis and design of soft materials and structures.

Computational characterization of the porous-fibrous behavior of the soft tissues in the temporomandibular joint.

The prevalence and severity of temporomandibular joint (TMJ) disorders have led to growing research interest in the development of new biomaterials and medical devices for TMJ implant designs. In computational designs, however, the time and stretch direction dependences of the TMJ soft tissues behavior are not considered and they are frequently based on measurements taken from non‐human species or from joints that differ markedly from the human TMJ. The aim of this study was to accurately characterize the porous‐fibrous properties of the TMJ soft tissues by simulating previously published experimental tests, to assist professionals in the design of new TMJ implants. To that end, material parameters were determined assuming a uniform fiber orientation throughout the entire sample. This assumption was then tested by comparing these results with those of considering multiple regions and distinct fiber orientations in each sample. Our findings validated the use of a transversely isotropic hyperelastic material model to characterize the direction dependent behavior of TMJ soft tissues and its combination with porous hyperfoam material models to mimic the compressive response of the TMJ disc. In conclusion, constitutive model proposed accurately reproduce the mechanical response of the TMJ soft tissues at different strain rates and stretch directions.

How does stent expansion alter drug transport properties of the arterial wall?

Stents have become the most successful device to treat advanced atherosclerotic lesions. However, one of the main issues with these interventions is the development of restenosis. The coating of stents with antiproliferative substances to reduce this effect is now standard, although such drugs can also delay re-endothelialization of the intima. The drug release strategy is therefore a key determinant of drug-eluting stent efficacy. Many mathematical models describing drug transport in arteries have been developed and, usually separately, models describing the mechanics of arterial tissue have been devised. However, the literature is lacking a comprehensive model that adequately takes into account both the mechanical deformation of the porous arterial wall and the resulting impact on drug transport properties. In this paper, we provide the most comprehensive study to date of the effect of stent mechanical expansion on the drug transport properties of a three-layer arterial wall. Our model incorporates the state-of-the art description of the mechanical properties of arterial tissue though an anisotropic, hyperelastic material model and includes a nonlinear saturable binding model to describe drug transport in the arterial wall. We establish relationships between mechanical force generated through device expansion and alteration in diffusion within the arterial wall and perform simulations to elucidate the impact of such alterations in spatio-temporal drug release and tissue uptake. Mechanical deformation of the arterial wall results in modified drug transport properties and tissue drug concentrations, highlighting the importance of coupling solid mechanics with drug transport.

Optimization of graphene in carbon black‐filled nitrile butadiene rubber: Constitutive modeling and verification using finite element analysis

AbstractGraphene nanoplatelet loading is varied in carbon black‐filled nitrile butadiene rubber (NBR) to form a hybrid system. The mechanical properties of the hybrid system are determined by performing tension and compression tests. It is found that graphene nanosheets are well dispersed possessing strong interfacial interaction with NBR and carbon black. There is an increment of 9% in tensile strength of the carbon black‐filled hybrid system with a graphene loading of 4 phr without hampering ultimate strain. Further addition of graphene resulted in degraded properties. Hyperelastic material models describing nonlinear mechanical behavior are fitted with experimental data to acquire material constants. Young's modulus is determined using various models, and a synergistic effect is observed with graphene loading of 4 phr. A comparison between experimental result and finite element analysis (FEA) result is presented for uniaxial tension and compression. Maximum principal stresses at break are also visualized with the help of FEA.

Characterization of Hyperelastic Material Using Experimental Data and Finite Element Simulation

Structural integrity of propellant grain is investigated by carrying out numerical simulations of rocket motor for various loading conditions like pressure, temperature etc to carry out these numerical simulations, one must have material properties of each component of rocket motor. Propellant material behaves like a rubber material. It may be characterized using hyperelasticity. It can be used to analyze rubber like materials that undergo large strains and displacements with small volume change nearly incompressible materials. Large strain theory is required. The hyperelastic material models are isotropic and constant with respect to temperature. The hyperelastic materials are also assumed nearly or purely incompressible. The strain energy function will be used to describe the material model. Experimental testing data can be input for a material model like mooney-rivlin material model. The allowed experimental testing data are uniaxial test data, biaxial test data, and pure shear test data. It is planned to design and realize the test specimen, fixtures that are needed to carry out the above tests. Mooney Rivlin material model will be used and material coefficients are estimated using the test results by curve fitting. Later, these coefficients will be used to study the structural integrity of propellant grain for various loading conditions using any numerical simulation software like ANSYS.

Identification of Damping Characteristics of EPDM-RUBBER with applications to sandwiched beams and considerations to Engine Mounts for Performance Evaluation

Vibrations are undesirable in many fields owing to its adverse effects therefore as a control procedure, methods have been developed to prevent these vibrations. One of the methods is using dampers. Rubbers are widely used for damping the vibrations of various structures because of its low elastic modulus and high bulk modulus. Behavior of rubber can be classified as Hyper-elastic and Visco-elastic. In FEA rubbers or more precisely elastomers are modeled as Hyper-elastic. Hyper-elastic is a type of constitutive model for rubber like materials and provides a means of specifying their behavior which cannot be exactly specified by linear elastic models. In view of this an attempt is made to identify the damping characteristics of EPDM-RUBBER-rubber, which were identified by using sandwiched structures. Then the engine mount was modeled using this EPDM-RUBBER-rubber as a Hyper-elastic rubber, using the best fit Hyper-elastic material model, and then the performance of selected engine mounts with EPDM-RUBBER as their core rubber was studied for static, harmonic and time varying loads.

PERFORMANCE EVALUATION OF HYPERELASTIC MODELS FOR CARBON-BLACK–FILLED SBR VULCANIZATES

ABSTRACTThe performance of four recently developed hyperelastic material models (HM models) was evaluated by using stress–stretch relationships of filled styrene butadiene rubber vulcanizates with different contents of carbon black (CB) tested under uniaxial tension (UT), pure shear (PS), and equi-biaxial tensile (BT) conditions. The stress–stretch relationships for the PS and BT tests were obtained with a specially designed specimen having higher accuracy than those obtained with an ordinary specimen. The performances of the HM models were compared via the following three evaluations: (1) ability to reproduce the UT, PS, and BT tests; (2) ability to predict the PS and BT test results with material constants identified from the UT test; and (3) correlation between the identified material constants and CB contents. The evaluations based on a combination of the stress–stretch relationships with high accuracy and four recently developed HM models revealed that the HM model activates only the I1 term, which is the first invariant of the right Cauchy–Green tensor, exhibiting good reproducibility and predictability.

Numerical analysis of an elastomeric bearing pad by hyperelastic models

Elastomeric bearing pads are responsible for transferring loads at the junction between beams and columns of bridges and viaducts, providing restrict freedom of movement in the superstructure. The elastomeric material of the bearing pad is a synthetic rubber reinforced with carbon black particles and subjected to a process of vulcanization, also represented by hyperelastic material models based on strain energy density functions. The objective of the present paper is to use the finite element analysis software Abaqus® to select the most appropriate hyperelastic model, as well as its constants, applying them in a bearing pad installed in an existing viaduct, evaluating its behavior and displacements resulting from the application of usual loads. The proposed methodology presents results coherent with technical specifications limits for available bearing pads products.

Nonlinear modeling and dynamic analysis of bioengineering hyper-elastic tubes based on different material models.

In this research, nonlinear vibrations of a hyper-elastic tube accounting for large deflection and moderate rotation have been examined. The hyper-elastic tube is assumed to be surrounded by a nonlinear hardening elastic medium. Different types of hyper-elastic material models are presented and discussed including neo-Hookean, Mooney-Rivlin, Ishihara and Yeoh models. The efficacy of these models in nonlinear vibration modeling and analysis of hyper-elastic tubes has been examined. Modified von-Karman strain is used to consider both large deflection and moderate rotation. The governing equations are obtained based on strain energy function of above-mentioned hyper-elastic material models. The nonlinear governing equation of the tube contains cubic and quantic terms which is solved via extended Hamiltonian method leading to a closed form of nonlinear vibration frequency. The effect of hyper-elastic models and their material parameters on nonlinear vibrational frequency of tubes has been studied.

Comparing quantum, molecular and continuum models for graphene at large deformations

In this paper, the validity and accuracy of three interatomic potentials and the continuum shell model of Ghaffari and Sauer [1] are investigated. The mechanical behavior of single-layered graphene sheets (SLGSs) under uniaxial stretching, biaxial stretching and pure bending is studied for this comparison. The validity of the molecular and continuum models is assessed by direct comparison with density functional theory (DFT) data available in the literature. The molecular simulations are carried out employing the MM3, Tersoff and REBO + LJ potentials. The continuum formulation uses an anisotropic hyperelastic material model in the framework of the geometrically exact Kirchhoff-Love shell theory and isogeometric finite elements. Results from the continuum model are in good agreement with those from DFT. The results from the MM3 potential agree well up to the point of material instability, whereas those from the REBO + LJ and Tersoff potentials agree only for small deformations. Only the Tersoff potential is found to yield auxetic response in SLGSs under uniaxial stretch. Additionally, the transverse vibration frequencies of a pre-stretched graphene sheet and a carbon nanocone are obtained using the continuum model and molecular simulations with the MM3 potential. The variations of the frequencies from these approaches agree within an error of ≈5%.

Prediction of hyperelastic material sealing pressure using experimental and numerical analyses

In commercial vehicle air braking system, leakage is one of the major problems and will affect the performance of the vehicle braking in terms of brake pedal travel and stopping distance. If there is any leakage in the brake system, the vehicle stopping distance will not meet the safety regulations. One of the main reasons for braking system leakage is ineffective sealing mechanism. The majority of sealing elements used in the air brake system are O-rings, lip seals, and gaskets. This article presents an experimental procedure for measuring the sealing pressure between an O-Ring and its mating parts. The contact pressure measurement was performed in a static condition by means of an experimental test rig using Fuji film. For the sealing pressure study, a test rig was properly designed to replicate the actual operating conditions. Contact pressure was evaluated by means of Fuji film interposed between the O-ring and its mating parts. The sealing pressure tests were carried out for different clamping load conditions. The experimental results were compared with the numerical result using the finite-element analysis (FEA). A good correlation was found between the experimental and the numerical results. The outcome from the experimental results will be useful for finalizing the hyperelastic material models, which are input to the FEA for future reference.

Constitutive Properties Determination of Human Cranium by an Experimental–Computational Modal Analysis

Abstract In this paper, we identified the material constitutive parameters of the human skull from reported tensile test results. Initially, we applied both linear-elastic and Mooney–Rivlin nonlinear hyperelastic constitutive models to the available tensile test data at different strain rates of 0.005, 0.10, 10, and 150 1/sec. It was shown that the suggested hyperelastic model fitted the test results with higher accuracy in comparison with the linear-elastic model. In the next step, the experimental modal analysis was carried out through roving hammer-impact tests on a dried human skull. The first four natural frequencies of the skull were measured to be 496, 543, 1250, and 1287 Hz, and these values were verified by the modal assurance criterion. Then, a 3D finite element (FE) model of that human skull was created by a 3D scanner and discretized to carry out a computational modal analysis. The performance of the determined material properties for the human skull from both linear and hyperelastic material models was evaluated using FE modal analysis. The calculated modal frequencies were then compared to the experimentally measured frequencies. It was shown that the material parameters from both the linear and hyperelastic constitutive models obtained at a strain rate of 0.10 1/sec, provided the best performance in computational modal analysis with minimum deviations relative to the experimental results. These results confer a better understanding of the human skull behavior among different strain rates, which could increase the accuracy of nonlinearity dynamic simulations on the skull.

Finite element modeling of the periodontal ligament under a realistic kinetic loading of the jaw system

PurposeThe stresses and deformations in the periodontal ligament (PDL) under the realistic kinetic loading of the jaw system, i.e., chewing, are difficult to be determined numerically as the mechanical properties of the PDL is variably present in different finite element (FE) models. This study was aimed to conduct a dynamic finite element (FE) simulation to investigate the role of the PDL (PDL) material models in the induced stresses and deformations using a simplified patient-specific FE model of a human jaw system. MethodsTo do that, a realistic kinetic loading of chewing was applied to the incisor point, contralateral, and ipsilateral condyles, through the experimentally proven trajectory approach. Three different material models, including the elasto-plastic, hyperelastic, and viscoelastic, were assigned to the PDL, and the resulted stresses of the tooth FE model were computed and compared. ResultsThe results revealed the highest von Mises stress of 620.14 kPa and the lowest deformation of 0.16 mm in the PDL when using the hyperelastic model. The concentration of the stress in the elastoplastic and viscoelastic models was in the mid-root and apex of the PDL, while for the hyperelastic model, it was concentrated in the cervical margin. The highest deformation in the PDL regardless of the employed material model was located in the caudal direction of the tooth. The viscoelastic PDL absorbed the transmitted energy from the dentine and led to lower stress in the cancellous bone compared to the elastoplastic and hyperelastic material models. ConclusionThese results have implications not only for understanding the stresses and deformations in the PDL under chewing but also for providing comprehensive information for the medical and biomechanical experts in regard of the role of the material models being used to address the mechanical behavior of the PDL in other components of the tooth.

Nitsche’s method for non-conforming multipatch coupling in hyperelastic isogeometric analysis

Complex geometric models are usually built with multiple NURBS patches with non-conforming interfaces, which bring difficulties within the isogeometric analysis. In this paper, Nitsche’s method is employed to glue different patches for nonlinear isogeometric analysis of hyperelastic material models which are widely used to describe the material behavior of rubbers, foams, biological tissues, etc. Nitsche based weakly governing equations and discretized stiffness matrices are detailedly developed in total Lagrangian form for isogeometric implementation. Different popular hyperelastic materials including Neo–Hookean, Mooney–Rivlin and Yeoh materials are employed for derivation. The Legendre–Gauss quadrature rule is used for numerical calculation. Several numerical examples in two dimensions are performed and compared with the results from commercial software to verify the validity of the proposed method and show the prospect in solving engineering problems.