Hyperelastic Material Model Research Articles

Roll-over shape of a prosthetic foot: a finite element evaluation and experimental validation.

Prosthetic feet have generally been designed experimentally by adopting a trial-and-error technique. The objective of this research is to introduce a novel numerical approach for the a priori evaluation of the roll-over shape (ROS) of a prosthetic foot for application in its systematic design and development. The ROS was achieved numerically by employing a non-linear finite element model incorporating the augmented Lagrangian and multi-point constraint contact formulations, a hyperelastic material model and a higher-order strain definition. The Ottobock Solid Ankle Cushion Heel (SACH) foot was chosen to experimentally validate the numerical model. The geometry of the foot was evaluated from optical scans, and the material properties were obtained from uniaxialtensile, shear and volumetric compressiontests. A new setup was designed for an improved experimental determination of the ROS, with the inclusion of an extended moment arm and variable loading. Error analysis of the radius of curvature of the ROS between the numerical and experimental results showed the percentage error to be 7.52%, thereby establishing the validity of the model. A numerical design model of this kind can be utilised to vary the input design parameters to arrive at a prosthetic foot with specified performance characteristics effectively and economically. Graphical abstract.

Architectural woven polyester fabrics: examination of possible classification of stiffness values in correlation with strength values

ABSTRACT Down to the present day, the determination of stiffness parameters for architectural fabrics, mainly used as coated woven fabrics, is a field of intense discussion. Designers and material producers are affected by an existing uncertainty. The present contribution investigates the possibility to provide tables of tensile stiffness parameters for specific materials in which tensile stiffness values are given dependent on the material’s tensile strength. The focus is on PVC-coated polyester fabrics as the most commonly used material for textile architecture. Materials of four different producers have been tested experimentally. Furthermore, from each material producer products of different strength classes have been considered in the test series. To achieve a general statement on the possibility of material classification according to their tensile stiffness, material parameters of two different material formulations have been investigated: (1) for a simplified but widely applied linear-elastic material model based on a standardized test and evaluation method and (2) for a recently developed more advanced orthotropic hyperelastic nonlinear material model. The tensile stiffness parameters are statistically evaluated and compared. In order to achieve a required basis, standardized and project-specific biaxial test procedures are presented and analysed in combination with established evaluation procedures. A tabulation of tensile stiffness parameters would be of great help for design engineers and material producers. The results of the present work will lead to a possible classified tabulation of the investigated materials in a meaningful manner.

Development of a Transient Dynamic Finite Clement Model for the Drum Testing of a Non-Pneumatic Tire

Non-pneumatic tire (NPT) or airless tire must be designed to meet the required performance such as weight, load carrying capacity and resistance to cyclic loading. The tire dynamic response is also needed to be observed since the tire are constantly subject to cyclic loading while rolling. Drum testing method is preferred over other rolling tire testing method since various factors from the vehicle suspension system and pavement structure are eliminated. This article aimed to develop the finite element (FE) model based on the transient dynamic approach for drum testing of NPT. The finite element model of NPT was created using Ogden hyperelastic material model. The FEM of NPT, which composed of Tread band and Spokes, was model using 3D hexagonal and 2D Quadrilateral elements respectively. The steel belt layers were modelled and embed using rebar elements and tying equation, respectively. The FE model of NPT was then assigned to contact with the rigid drum surface using Coulomb’s friction model. The finite element analysis of drum testing method of NPT as the drum was assigned to be rolling at speed of 11 km/hr. The NPT spoke deformations at each position and the angle was compared to the experiment to prove the validity of the model. The analysis results of spoke deformation were shown to be in a good agreement with the experiment at the average error of 3.51%. The developed FE model of NPT based on transient dynamic approach can be used to evaluate the important dynamic response of NPT during rolling and can be useful in designing the NPT.

Numerical Study about Effect of Different Boundary Conditions on Compressive Modulus Coefficient of Bonded and Non-Bonded Cylindrical Isolators

Elastomers (Rubber and rubber-like materials) are widely utilized in many engineering applications due to their specific properties like high elasticity with good static and dynamic behaviors. Rubber blocks are one of the elastomeric components and it is employed in many applications such as vibration isolators, bumpers, shock absorbers, and dampers, etc. Most of the rubber blocks are cylindrical in shape and it can undergo large deformation under different loadings and contact conditions (fixed-fixed, friction-friction and fixed-friction). In this present work, experimentally validated Ogden hyper-elastic material model is adopted in the finite element analysis (FEA). The FE model is validated with other published experimental work. In addition, the following six boundary conditions namely BC1 (Fixed – Fixed), BC2 (Fixed – Friction), BC3 (Friction – Friction), BC4 (Fixed – Cap), BC5 (Cap – Friction) and BC6 (Cap – Cap) boundary conditions are taken for comparative study. The aspect ratio (radius/height) of the rubber isolator also varied as 0.5, 0.75, and 1. From the numerical analysis carried out, it is found that the BC6 (Cap–Cap) with aspect ratio 1 showed better compressive modulus coefficient over other parameters taken for study.

Toward a Common Framework and Database of Materials for Soft Robotics.

To advance the field of soft robotics, a unified database of material constitutive models and experimental characterizations is of paramount importance. This will facilitate the use of finite element analysis to simulate their behavior and optimize the design of soft-bodied robots. Samples from seventeen elastomers, namely Body Double™ SILK, Dragon Skin™ 10 MEDIUM, Dragon Skin 20, Dragon Skin 30, Dragon Skin FX-Pro, Dragon Skin FX-Pro + Slacker, Ecoflex™ 00-10, Ecoflex 00-30, Ecoflex 00-50, Rebound™ 25, Mold Star™ 16 FAST, Mold Star 20T, SORTA-Clear™ 40, RTV615, PlatSil® Gel-10, Psycho Paint®, and SOLOPLAST 150318, were subjected to uniaxial tensile tests according to the ASTM D412 standard. Sample preparation and tensile test parameters are described in detail. The tensile test data are used to derive parameters for hyperelastic material models using nonlinear least-squares methods, which are provided to the reader. This article presents the mechanical characterization and the resulting material properties for a wide set of commercially available hyperelastic materials, many of which are recognized and commonly applied in the field of soft robotics, together with some that have never been characterized. The experimental raw data and the algorithms used to determine material parameters are shared on the Soft Robotics Materials Database GitHub repository to enable accessibility, as well as future contributions from the soft robotics community. The presented database is aimed at aiding soft roboticists in designing and modeling soft robots while providing a starting point for future material characterizations related to soft robotics research.

New numerical stress solutions to calibrate hyper-visco-pseudo-elastic material models effectively

This paper presents new numerical stress solutions for predicting the stress response of hyper-visco-pseudo-elastic solids. It is pointed out that they can be used to identify the constitutive constants of filled rubber-like materials subjected to arbitrary strain history. The constitutive constants are found by minimising the difference between the predicted and the measured stress response. The stress solutions are based on a finite time increment approach and are derived for uniaxial and equibiaxial tension/compression as well as pure and simple shear. The stress solutions make it possible to apply any hyperelastic material model and user-defined damage parameter. Furthermore, the material model parameters may be determined for each loading mode separately or any combination of the different loading modes. With the purpose of demonstrating the application of the stress solution-based parameter identification method used, as a first step, uniaxial cyclic tension tests had been performed on a carbon black filled EPDM rubber. Then, the calibration process was presented in details, applied at a strain rate of 0.01 1/s and the effectiveness of the proposed and the frequently used inverse parameter identification method was compared. Finally, the stress solution-based parameter identification was performed by considering measurements made at different strain rate simultaneously.

A stabilized mixed finite element approximation for incompressible finite strain solid dynamics using a total Lagrangian formulation

In this work a new methodology for both the nearly and fully incompressible transient finite strain solid mechanics problem is presented. To this end, the momentum equation is complemented with a constitutive law for the pressure which emerges from the deviatoric/volumetric decomposition of the strain energy function for any hyperelastic material model. The incompressible limit is attained automatically depending on the material bulk modulus. The system is stabilized by means of the Variational Multiscale-Orthogonal Subgrid Scale method based on the decomposition of the unknowns into resolvable and subgrid scales in order to prevent pressure fluctuations. Several numerical examples are presented to assess the robustness and applicability of the proposed formulation.

Electro‐thermal and ‐mechanical model of thermal breakdown in multilayered dielectric elastomers

AbstractMultiple breakdown phenomena may take place when operating dielectric elastomers. Thermal breakdown, which occurs due to Joule heating, becomes of special importance when using multilayered stacks of dielectric elastomers, due to the large volume‐to‐surface‐area‐ratio. In this article, a 2D axisymmetric finite‐element model of a multilayered stack of dielectric elastomers is set up in COMSOL Multiphysics®. Both the electro‐thermal and electro‐mechanical couplings are considered, allowing for determination of the onset of thermal breakdown. Simulation results show that an entrapped particle in the dielectric elastomer drastically reduces the possible number of layers in the stack. Furthermore, the possible number of layers is greatly affected by the ambient temperature and the applied voltage. The performance of three hyperelastic material models for modeling the elastomer deformation are compared, and it is established that the Gent model yields the most restrictive prediction of breakdown point, while the Ogden model yields the least restrictive estimation.

An embedded element based 2D finite element model for the strength prediction of mineralized collagen fibril using Monte-Carlo type of simulations

A computationally efficient statistical model for the prediction of the strength of mineralized collagen fibril (a basic building block of bone) is presented by taking into account the uncertainties associated with the geometrical and material parameters of collagen and mineral phases. The mineral plates have been considered as one-dimensional bar elements embedded in the two-dimensional plane stress collagen matrix. The mineral phase is considered as linear elastic and a hyperelastic material model is adopted for the collagen phase. Further, the crack initiation and propagation in the collagen phase have been modeled using a damage plasticity approach. Different realizations of the arrangement of mineral plates have been generated to account for the associated geometrical uncertainties using an in-house MATLAB® code. Monte-Carlo type simulations have been performed on the different realizations of mineralized collagen fibril to predict its characteristic stress-strain response under tensile load. The characteristic strength of 3.64 GPa is obtained for mineralized collagen fibril using Weibull’s analysis which is found to be in agreement with the molecular dynamics simulation data and numerical studies reported in the past. A parameter sensitivity analysis concluded that mineral modulus has a significant effect on the overall tangent modulus of mineralized collagen fibril in large strain regime.

Tests on the Mechanical Properties of Polymers in the Aspect of an Attempt to Determine the Parameters of the Mooney-Rivlin Hyperelastic Model

Abstract The article presents testing of the mechanical properties of SIKA® polymer adhesives of the type PBM, PMM, PM, and PSM in the aspect of an attempt to determine the parameters of the Mooney-Rivlin hyperelastic model. The article contains a literature review on developed models for hyperelastic materials as well as a description of the author’s own results obtained in monaxial tensile and monaxial compression tests conducted on oars and cylindrical samples, respectively. Furthermore, the results of modeling of Mooney-Rivlin hyperelastic model parameters are shown in relation to the value of average parameters for polymers after both a week and a month-and-a-half of ripening.

Hyperelastic finite deformation analysis with the unsymmetric finite element method containing homogeneous solutions of linear elasticity

SummaryA recent unsymmetric 4‐node, 8‐DOF plane finite element US‐ATFQ4 is generalized to hyperelastic finite deformation analysis. Since the trial functions of US‐ATFQ4 contain the homogenous closed analytical solutions of governing equations for linear elasticity, the key of the proposed strategy is how to deal with these linear analytical trial functions (ATFs) during the hyperelastic finite deformation analysis. Assuming that the ATFs can properly work in each increment, an algorithm for updating the deformation gradient interpolated by ATFs is designed. Furthermore, the update of the corresponding ATFs referred to current configuration is discussed with regard to the hyperelastic material model, and a specified model, neo‐Hookean model, is employed to verify the present formulation of US‐ATFQ4 for hyperelastic finite deformation analysis. Various examples show that the present formulation not only remain the high accuracy and mesh distortion tolerance in the geometrically nonlinear problems, but also possess excellent performance in the compressible or quasi‐incompressible hyperelastic finite deformation problems where the strain is large.

Understanding fracture of a carbon black filled rubber compound using material force theory

This paper is based on the determination of important fracture parameters for commercially important rubber blends through experimental and finite element modeling methods. Properties of NR/BR blends, which are mainly suitable for tire applications, of various compositions have been tested under tensile load and the result is utilized to determine the suitable hyperelastic material model. A model proposed by Yeoh has been found to be more relevant to the test materials and is used to represent them in simulation. Single Edge Notched Tensile samples (SENT) are ued for the fracture analysis. The samples are tested for uniaxial Mode I fracture (Tensile), which is the most commonly observed among the three modes of fracture, and crack growth was studied using the Virtual Crack Closure Technique (VCCT). A finite-element replica of the test specimen has been created using in-house code, which uses an implicit mesh adaptive method to study the strain potential around the crack tip. The code for each specimen is executed using the commercially available third-party software ANSYS Parametric Design Language (APDL). The stress–strain behaviors from experimental and simulation responses were compared. The energy release rate otherwise known as tearing energy has been derived for both experimental and numerical tests.

How to implement user-defined fiber-reinforced hyperelastic materials in finite element software

Finite element modeling is often used in biomechanical engineering to evaluate medical devices, treatments and diagnostic tools. Using an adequate material model that describes the mechanical behavior of biological tissues is essential for a reliable outcome of the simulation. Pre-programmed material models for biological tissues are available in many finite element software packages. However, since these pre-programmed models are presented to the user as a black box, without the possibility to modify the material description, many researchers turn to implementing their own material formulations. This is a complex undertaking, requiring extensive knowledge while documentation is limited.This paper provides a detailed description, at the level of the biomedical engineer, of the implementation of a nonlinear hyperelastic material model using user subroutines in Abaqus®, in casuUANISOHYPER_INV and UMAT. The Gasser-Ogden-Holzapfel material model is used as an example, resulting in four implementation variations: the built-in implementation, a UANISOHYPER_INV formulation, a UMAT with analytical tangent stiffness formulation and a UMAT with numerical tangent stiffness formulation. In addition, three different element formulations are used: a continuum compressible, a continuum incompressible and a plane stress incompressible. All cases are thoroughly verified by applying a series of deformations on a single cube element and by simulating an extension-inflation experiment with non-homogeneous deformations and multiple elements. In these test cases, stresses, displacements, reaction forces, the required number of iterations and the total CPU time were compared.The results show that the four implementation variations are very similar, with total relative errors between 10−3 and 10−15, number of iterations that varied by maximum one iteration, and a comparable CPU time. In addition to this detailed overview, the user subroutines are added as supplementary material to this tutorial, which can be used as the ideal starting point for biomechanical engineers to implement their own material models at different levels of complexity.

Mechanical characterization of porcine liver properties for computational simulation of indentation on cancerous tissue

An accurate characterization of soft biological tissue properties is essential for a realistic simulation of surgical procedures. Unconfined uniaxial compression tests with specimens affixed to the fixtures are often performed to characterize the stress-stretch curves of soft biological tissues, with which the material parameters can be obtained. However, the constrained boundary condition causes non-uniform deformation during the uniaxial test, posing challenges for accurate measurement of tissue deformation. In this study, we measured the deformation locally at the middle of liver specimens and obtained the corresponding stress-stretch curves. Since the effect of the constrained boundary condition on the local deformation of specimen is minimized, the stress-stretch curves are thus more realistic. Subsequently, we fitted the experimental stress-stretch curves with several constitutive models and found that the first-order Ogden hyperelastic material model was most suitable for characterizing the mechanical properties of porcine liver tissues. To further verify the characterized material properties, we carried out indentation tests on porcine liver specimens and compared the experimental data with computational results by using finite element simulations. A good agreement was achieved. Finally, we constructed computational models of liver tissue with a tumor and investigated the effect of the tumor on the mechanical response of the tissue under indentation. The computational results revealed that the liver specimen with tumor shows a stiffer response if the distance between the tumor and the indenter is small.

Design of a Biomimetic SACH Foot: An Experimentally Verified Finite Element Approach

The Solid Ankle Cushioned Heel (SACH) foot is a commonly prescribed prosthetic foot for the rehabilitation of lower limb amputees. From the viewpoint of its biomechanical performance, the foot is known to cause drop-off effect and asymmetry in amputee gait. Therefore, the objective of this work is to improvise the effective foot length ratio (EFLR) and the progression of the centre of pressure (CoP) of the SACH foot by providing a novel design approach that utilizes finite element analysis. Boundary conditions employed for evaluating the roll-over characteristics of prosthetic feet were numerically incorporated in this work. The non-linear mechanical behavior of the foot was included with the incorporation of large deformation, a hyperelastic material model and the Augmented Lagrangian contact formulation. Outcomes from the simulations were experimentally verified using an inverted pendulum-like apparatus, thereby substantiating the numerical approach. The design process of the SACH foot involved the modification of the elastic modulus of its components for enhancing the parameters of interest. Results obtained presented a 5.07% increase in the EFLR and a 9.29% increase in the anteroposterior progression of the CoP, which may improve amputee stability. The design solution presented may support the large user base of the SACH foot towards achieving enhanced gait characteristics during ambulation. Moreover, this work successfully demonstrates a novel design procedure for a prosthetic foot through an effective numerical implementation.

Calibration of hyperelastic material models for structural silicone and hybrid polymer adhesives for the application of bonded glass

To enable the introduction of new developments and concepts of bonded glass in façades, finite element modelling is a frequently used method. The implementation of the complex material behaviour of suitable adhesives into such software, however, can be labour-intensive. Usually, hyperelastic material models need to be calibrated, i.e. theoretical mathematical expressions are fitted to experimentally obtained data. In this research, three two-component structural silicones and one two-component hybrid polymer adhesive were selected. Experimental tests on H-specimens in correspondence with ETAG 002 were performed for tension, compression and simple shear at different haul-off speeds of the testing machine, more specifically 1 mm/min, 5 mm/min, 50 mm/min and 500 mm/min. The stress–strain response of the adhesives altered with different test speeds. The material response under cyclic loading regime was investigated as well. These tests confirmed the presence of strain softening, also referred to as the Mullins effect, in all selected adhesives. Next, twenty isochoric hyperelastic material models were calibrated with the experimental data of the monotonic load tests at a speed of 5 mm/min for single, dual and triple deformation states. A theoretical estimator of the relative quality of the material models, more specifically the Akaike Information Criterion, allowed for the selection of the models that fit the experimental data the best. For all investigated adhesives, the polynomial based hyperelastic material models provided the best fit. This paper presents the results of the experimental programme that enable researchers to compare numerical and analytical analyses with these results. Designers and engineers can compare the structural performance of the adhesives in correspondence with ETAG 002 and assess the importance of cyclic loading regimes in the design of bonded glass. The calibrated hyperelastic material models allow engineers to numerically implement hyperelastic material behaviour of the studied adhesives in correspondence with ETAG 002. The best-performing material model for the considered application needs to be selected and the numerical model needs to be validated using the results of additional experiments on the considered bonded glass application. Especially the calibrated material models for multiple deformation states are valuable to model complex stress distributions in bonded glass applications subjected to multiple deformation states.

Glycation alters the mechanical behavior of kidney extracellular matrix

The mechanical properties of the extracellular matrix (ECM) are important in maintaining normal physiological function, and changes in ECM mechanics drive disease. The biochemical structure of the ECM is modified with aging and in diseases such as diabetes. One mechanism of ECM modification is the non-enzymatic reaction between sugars and ECM proteins resulting in formation of advanced glycation end products (AGEs). Some AGE reactions result in formation of molecular crosslinks within or between matrix proteins, but it is not clear how sugar-mediated biochemical modification of the ECM translates to changes in kidney ECM mechanical properties. AGE-mediated changes in ECM mechanics may have pathological consequences in diabetic kidney disease. To determine how sugars alter the mechanical properties of the kidney ECM, we employ custom methodologies to evaluate the mechanical properties of isolated tubular basement membrane (TBM) and glomerular ECM. Results show that the mechanical properties of TBM and glomerular ECM stiffness were altered by incubation in glucose and ribose. Mechanical behavior of TBM and glomerular ECM were further evaluated using mechanical models for hyperelastic materials in tension and compression. Increased ECM stiffness following sugar modification corresponded to increased crosslinking as determined by ECM fluorescence and reduced pepsin extractability of sugar modified ECM. These results show that sugar-induced modifications significantly affect the mechanical properties of kidney ECM. AGE-mediated changes in ECM mechanics may be important in progression of chronic diseases including diabetic kidney disease.

Adaptation of a highly compressible elastomeric material model to simulate compressed expanded graphite and its application in the optimization of a graphite-metallic structure

The paper presents simulations of mechanical properties of compressed expanded graphite. To model the stress and strain states of compressed expanded graphite, a hyperelastic material model formulated by the Blatz–Ko was adapted. The idea of using this model resulted from the assumptions that compressed expanded graphite exhibits similar mechanical properties to polyurethane foam with high density. The material constant of the Blatz–Ko model was determined by numerical calculations where deformation energy equation was parameterized with the shear modulus. Such material model used in a compressed expanded graphite simulation was experimentally validated. Subsequently, the Blatz–Ko model was used in a numerical simulation of the graphite-metallic structure in the form of a spiral wound gasket which is composed of metallic and elastic tapes. This structure was subjected to an optimization process which included the reduction of stress in the elastic graphite tape obtaining the appropriate axial stiffness of the gasket. In the final stage, the optimal design of the gasket was experimentally tested. The experimental results were compared with the data obtained numerically, which allowed to verify the adapted material model and boundary conditions of the numerical gasket model.

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.