Hyperelastic Material Model Research Articles

Equilibrium for Multiphase Solids with Eulerian Interfaces

We describe a general phase-field model for hyperelastic multiphase materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic deformation, as it measures interfaces in the deformed configuration. We prove existence of energy minimizing equilibrium states and $\Gamma $ -convergence of diffuse-interface approximations to the sharp-interface limit.

Abstract 16886: Wall Stress Profiles in Tricuspid Aortic Valve Associated Ascending Thoracic Aortic Aneurysms: The Effect of Surgical Threshold Diameter ≥5.5cm

Introduction: Ascending thoracic aortic aneurysms (aTAAs) carry a risk of dissection. Elective repair guidelines are designed around size thresholds, but the one-dimensional parameter is insufficient to predict acute events in small aneurysms. Biomechanically, aortic events can occur when wall stress exceeds wall strength. Patient-specific aTAA wall stresses may be a better predictor of complications. Our aim was to compare wall stresses in aTAAs associated with a tricuspid aortic valve (TAV) based on diameter. Methods: Patients with TAV-aTAA and diameter >4.0cm (n=448) were divided into groups by 0.5 cm diameter increments. Pre-stress three-dimensional aneurysm geometries were reconstructed from ECG-gated computer tomography images. A fiber-embedded hyperelastic material model was applied to obtain longitudinal and circumferential wall stress distributions under systolic pressure. Medians with interquartile ranges are reported. The Kruskal-Wallis test is used for comparisons between size groups. Results: Peak longitudinal wall stresses for TAV-aTAA were 308[282-338] kPa for size 4.0-4.4cm vs 341[309-362] kPa for 4.5-4.9cm vs 339[289-370] kPa for 5.0-5.4cm vs 319[297-355] kPa for 5.5-5.9cm vs 373[364-449] for 6.0cm (p=0.003). Peak circumferential wall stresses were 487[448-579] kPa for size 4.0-4.4cm vs 516[473-619] kPa for 4.5-4.9cm vs 506[422-580] kPa for 5.0-5.4cm vs 540[468-591] kPa for 5.5-5.9cm vs 565[506-634] for >6.0cm (p=0.19) (figure). 95th-percentile longitudinal peak stress for TAV-aTAA <5.5cm vs ≥5.5cm is 408 vs 465 kPa. Conclusions: Longitudinal wall stresses are higher as diameter increases. The 95% percentile longitudinal peak stress for diameter ≥5.5cm is ~450 kPa, which correlates with established ~5% dissection risk for size ≥5.5cm. Wall stress thresholds may be a better predictor of patient-specific risk of dissection than diameter and require testing in clinical trials.

Transformation of Test Data for the Specification of a Viscoelastic Marlow Model

The combination of hyperelastic material models with viscoelasticity allows researchers to model the strain-rate-dependent large-strain response of elastomers. Model parameters can be identified using a uniaxial tensile test at a single strain rate and a relaxation test. They enable the prediction of the stress–strain behavior at different strain rates and other loadings like compression or shear. The Marlow model differs from most hyperelastic models by the concept not to use a small number of model parameters but a scalar function to define the mechanical properties. It can be defined conveniently by providing the stress–strain curve of a tensile test without need for parameter optimization. The uniaxial response of the model reproduces this curve exactly. The coupling of the Marlow model and viscoelasticity is an approach to create a strain-rate-dependent hyperelastic model which has good accuracy and is convenient to use. Unfortunately, in this combination, the Marlow model requires to specify the stress–strain curve for the instantaneous material response, while experimental data can be obtained only at finite strain rates. In this paper, a transformation of the finite strain rate data to the instantaneous material response is derived and numerically verified. Its implementation enables us to specify hyperelastic materials considering strain-rate dependence easily.

Multiscale modelling of the mechanical response of 3D multi-axial knitted 3D spacer composites

Multiscale modelling of 3D multi-axial knitted 3D spacer composites is proposed. The developed modeling approach is based on a full-field finite element homogenization and conducted at each microstructural level. The relevant scales and their related material features have been identified by means of micro-tomography imaging. The homogenized mechanical response at each level has been identified using appropriate (isotropic or anisotropic) hyperelastic material model and used in the upper level. Thus, the multiscale modeling is carried out through the transfer of information between different length scales rather than by coupling different simulation techniques. The multiscale modeling presented in this work sheds light on the complex mechanics of the textile-rubber microstructure. Moreover, our study may support the design of this class of 3D-fabric-rubber materials by enabling the optimization of the different constituents to obtain specific mechanical properties. The correlation between models and experiments is good, both in terms of details in the architecture and mechanical properties. There are however some deviations that could be explained by the models being more regular than the real material microstructure.

{text {FE}}^r method with surrogate localization model for hyperelastic composite materials

This study presents a method for constructing a surrogate localization model for a periodic microstructure, or equivalently, a unit cell, to efficiently perform micro-macro coupled analyses of hyperelastic composite materials. The offline process in this approach is to make a response data matrix that stores the microscopic stress distributions in response to various patterns of macroscopic deformation gradients, which is followed by the proper orthogonal decomposition (POD) of the matrix to construct a reduced order model (ROM) of the microscopic analysis (localization) with properly extracted POD bases. Then, response surfaces of the POD coefficients are constructed so that the ROM can be continuous with respect to the input datum, namely, the macroscopic deformation gradient. The novel contributions of this study are the application of the L2 regularization to the interpolation approximations of the POD coefficients by use of radial basis functions (RBFs) to make the response surfaces continuous and the combined use of the cross-validation and the Bayesian optimization to search for the optimal set of parameters in both the RBFs and L2regularization formula. The resulting model can be an alternative to microscopic finite element (FE) analyses in the conventional {text {FE}}^2 method and realizes {text {FE}}^r with 1<r<<2 accordingly. Representative numerical examples of micro-macro coupled analysis with the {text {FE}}^r are presented to demonstrate the capability and promise of the surrogate localization model constructed with the proposed approach in comparison with the results with high-fidelity direct {text {FE}}^2.

A general higher‐order shell theory for compressible isotropic hyperelastic materials using orthonormal moving frame

SummaryThe primary objective of this study is threefold: (1) to present a general higher‐order shell theory to analyze large deformations of thin or thick shell structures made of general compressible hyperelastic materials; (2) to formulate an efficient shell theory using the orthonormal moving frame, and (3) to develop and apply the nonlinear weak‐form Galerkin finite element model for the proposed shell theory. The displacement field of the line normal to the shell reference surface is approximated by the Taylor series/Legendre polynomials in the thickness coordinate of the shell. The use of an orthonormal moving frame makes it possible to represent kinematic quantities (e.g., the determinant of the deformation gradient) in a far more efficient manner compared with the nonorthogonal covariant bases. Kinematic quantities for the shell deformation are obtained in a novel way in the surface coordinate described in the appendix of this study with the help of exterior calculus. Furthermore, the governing equation of the shell deformation has been derived in the general surface coordinates. To obtain the nonlinear solution in the quasi‐static cases, we develop the weak‐form finite element model in which the reference surface of the shell is modeled exactly. The general invariant based compressible hyperelastic material model is considered. The formulation presented herein can be specialized for various other nonlinear compressible hyperelastic constitutive models, for example, in biomechanics and other soft‐material problems (e.g., compressible neo‐Hookean material, compressible Mooney–Rivlin material, Saint Venant–Kirchhoff model, and others). A number of numerical examples are presented to verify and validate the formulation presented in this study. The scope of potential extensions are outlined in the final section of this study.

Key considerations for finite element modelling of the residuum-prosthetic socket interface.

Finite element modelling has long been proposed to support prosthetic socket design. However, there is minimal detail in the literature to inform practice in developing and interpreting these complex, highly nonlinear models. To identify best practice recommendations for finite element modelling of lower limb prosthetics, considering key modelling approaches and inputs. Computational modelling. This study developed a parametric finite element model using magnetic resonance imaging data from a person with transtibial amputation. Comparative analyses were performed considering socket loading methods, socket-residuum interface parameters and soft tissue material models from the literature, to quantify their effect on the residuum's biomechanical response to a range of parameterised socket designs. These variables had a marked impact on the finite element model's predictions for limb-socket interface pressure and soft tissue shear distribution. All modelling decisions should be justified biomechanically and clinically. In order to represent the prosthetic loading scenario in silico, researchers should (1) consider the effects of donning and interface friction to capture the generated soft tissue shear stresses, (2) use representative stiffness hyperelastic material models for soft tissues when using strain to predict injury and (3) interrogate models comparatively, against a clinically-used control.

Lifetime prediction of carbon black filled elastomers based on the probability distribution of particle using an inelastic and hyperelastic material model

In this work, lifetime of carbon black filled elastomers is predicted based on a micro-flaw statistics lifetime concept. Within this concept two material models, the hyperelastic Yeoh model and the inelastic material model of rubber phenomenology (MORPH), are compared. The Yeoh model is chosen as a demonstrator for any hyperelastic material model. This model is not able to reproduce the characteristic material behaviour such as permanent set, softening effects and hysteresis and can only reproduce the nonlinearity of the stress-strain response. To overcome these deficiencies the hyperelastic models are typically fitted to the softening state of the material at a maximal (cyclic) load level. However, the results show that it is not foreseeable, which load level should be used for fitting the model to get a reasonable lifetime prediction. This is even more influenced by the restriction that only one parameter set has to be chosen to represent the inhomogeneous softening state of the material when arbitrary loads are applied. The above-mentioned characteristic properties of filled elastomers can be reproduced by the MORPH model. The lifetime prediction using this inelastic material model shows very good agreement with the measured values. Furthermore, the variable softening within the model adumbrate a good lifetime prediction quality of components with complex geometries.

Transversely isotropic magnetoactive elastomers: theory and experiments

The current contribution is concerned with the development of novel constitutive equations for anisotropic magnetoactive elastomers (MAEs). A hyperelastic material model for an incompressible magnetoelastic medium representing transversely isotropic MAEs has been developed to investigate their response behavior in the presence of the applied magnetic field while undergoing finite deformation. Transversely isotropic MAE samples in circular cylindrical geometry with 15% iron particle volume fraction are then fabricated and experimentally tested to measure their permeability and torque–twist response. The experimental results have then been effectively utilized to identify the constant material parameters in the proposed material model. Finally, the accuracy and validity of the proposed constitutive equations are demonstrated through the comparison of the simulation and experimental results.

Finite element simulation of friction and adhesion attributed contact of bio-inspired gecko-mimetic PDMS micro-flaps with SiO2 spherical surface

The remarkable tribological attributes of the gecko feet have grown much interest in the field of biomimetic tribology over the past two decades. It has been shown that the complexity of friction and adhesion phenomena made it difficult to transfer these exceptional properties into fully functional smart, dry, micro patterned adhesives. The latter, combined with the relative lack of literature on computational oriented studies on these phenomena, is the motive of the current work. Here, a 2D time-dependent finite element model of friction and adhesion attributed contact of polydimethysiloxane (PDMS) micro flaps with a smooth SiO2 spherical surface is presented. The model is tested through simulations concerning changes in the disc curvature, the flap density, as well as different disc mounting heights, representing the effect of preload. Furthermore, the effect of tribological parameters of adhesion and friction coefficient is discussed. Finally, the effect of the use of two hyperelastic material models was examined.

The Influence of Anterior Cruciate Ligament Matrix Mechanical Properties on Simulated Whole-Knee Biomechanics.

Knee finite element (FE) models are used to study tissue deformation in response to complex loads. Typically, ligaments are modeled using transversely isotropic, hyperelastic material models fitted to tension data along the predominant fiber direction (longitudinal) and, less commonly, to tension data orthogonal to the fiber direction (transverse). Currently, the shear and bulk responses of the anterior cruciate ligament (ACL) are not fitted to experimental data. In this study, a newly proposed material model was fitted to longitudinal tension, transverse tension, and shear experimental data. The matrix transverse tensile, shear, and bulk stiffnesses were then varied independently to determine the impact of each property on knee kinematics and tissue deformation in a whole-knee FE model. The range of values for each parameter was chosen based on published FE studies of the knee. For a knee at full extension under 134 N anterior tibial force (ATF), increasing matrix transverse tensile stiffness, shear stiffness, or bulk stiffness decreased anterior tibial translation (ATT), ACL longitudinal strain, and ACL shear strain. For a knee under 134 N ATF and 1600 N compression, changing the ACL matrix mechanical properties caused variations in ATT and thus changed cartilage deformation contours by changing the point of contact between the femoral and the tibial cartilage. These findings indicate that material models for the ACL must describe matrix material properties to best predict the in vivo response to applied loads.

Design, Modeling, and Control of a 3D Printed Monolithic Soft Robotic Finger With Embedded Pneumatic Sensing Chambers

This article presents a directly 3-D printed soft monolithic robotic finger with embedded soft pneumatic sensing chambers (PSC) as position and touch sensors. The monolithic finger was fabricated using a low-cost and open-source fused deposition modeling 3-D printer that employs an off-the-shelf soft and flexible commercially available thermoplastic polyurethane. A single soft hinge with an embedded PSC was optimized using finite element modeling and a hyperelastic material model to obtain a linear relationship between the internal change in the volume of its PSC and the corresponding input mechanical modality, to minimize its bending stiffness and to maximize its internal volume. The soft hinges with embedded PSCs have several advantages, such as fast response to very small changes in their internal volume (~0.0026ml/°), linearity, negligible hysteresis, repeatability, reliability, long lifetime, and low power consumption. Also, the flexion of the soft robotic finger was predicted using a geometric model for use in real-time control. The real-time position and pressure/force control of the soft robotic finger were achieved using feedback signals from the soft hinges and the touch PSC embedded in the tip of the finger. This article contributes to the development of seamlessly embedding optimized sensing elements in the monolithic topology of a soft robotic system and controlling the robotic system using the feedback data provided by the sensing elements to validate their performance.

Mechanical Characterization and Modeling of the Porcine Cerebral Meninges.

The cerebral meninges, made up of the dura, arachnoid, and pia mater, is a tri-layer membrane that surrounds the brain and the spinal cord and has an important function in protecting the brain from injury. Understanding its mechanical behavior is important to ensure the accuracy of finite element (FE) head model simulations which are commonly used in the study of traumatic brain injury (TBI). Mechanical characterization of freshly excised porcine dura-arachnoid mater (DAM) was achieved using uniaxial tensile testing and bulge inflation testing, highlighting the dependency of the identified parameters on the testing method. Experimental data was fit to the Ogden hyperelastic material model with best fit material parameters of μ = 450 ± 190 kPa and α = 16.55 ± 3.16 for uniaxial testing, and μ = 234 ± 193 kPa and α = 8.19 ± 3.29 for bulge inflation testing. The average ultimate tensile strength of the DAM was 6.91 ± 2.00 MPa (uniaxial), and the rupture stress at burst was 2.08 ± 0.41 MPa (inflation). A structural analysis using small angle light scattering (SALS) revealed that while local regions of highly aligned fibers exist, globally, there is no preferred orientation of fibers and the cerebral DAM can be considered to be structurally isotropic. This confirms the results of the uniaxial mechanical testing which found that there was no statistical difference between samples tested in the longitudinal and transversal direction (p = 0.13 for μ, p = 0.87 for α). A finite element simulation of a craniotomy procedure following brain swelling revealed that the mechanical properties of the meninges are important for predicting accurate stress and strain fields in the brain and meninges. Indeed, a simulation using a common linear elastic representation of the meninges was compared to the present material properties (Ogden model) and the intracranial pressure was found to differ by a factor of 3. The current study has provided researchers with primary experimental data on the mechanical behavior of the meninges which will further improve the accuracy of FE head models used in TBI.

Laplace stretch: Eulerian and Lagrangian formulations

Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations) and work-conjugate stress measures are analyzed and compared for each formulation. While the Lagrangian formulation has been used in prior work for constitutive modeling of anisotropic and hyperelastic materials, the Eulerian formulation, which may be advantageous for modeling isotropic solids and fluids with no physically identifiable reference configuration, does not seem to have been used elsewhere in a continuum mechanical setting for the purpose of constitutive development, though it has been introduced before in a purely kinematic setting.

Design and implementation of variable inclined air pillow soft pneumatic actuator suitable for bioimpedance applications

The technological revolution has caused the modernization of human–machine relationship changing our approach in problem solving our society issues and deviated the science of robotic all together. An example for one of the most important pawn in this revolution is soft robotics, the soft robots are robots that are made of deformable materials that provide an alternative approach to rigid robots. The soft pneumatic actuator (SPA) is one of the most widely used and studied form of this type of robotics. In this study, a new geometrical parameter of the SPAs is introduced by studying the effect of the change of the air pillow Inclination angle on the SPA work envelope and tip reaction force against wide range of positive and vacuum pressure applied on SPAs inner surfaces. We are elaborating this effect by using nonlinear static Finite Element Analysis. (FEA) basing it on the hyper-elastic material model identified by testing a specimen from the SPA fabricated material. Three SPAs are fabricated using Fused Deposition Modeling (FDM) 3D printer for the experimental validation of two aspects the work envelope and the tipping force. The test was performed by the image processing of the work envelope comparing the FEA and the experimental work envelope results and the tipping force was validated by using the load force scales. Based on the FEA and the experimental results, two different grippers with two work envelopes and tips was created to observe the change in the effect of the gripper on the weight and surfaces of the grasped object. By the integration of the SPA gripper with bio-impedance measurements, the SPA gripper was enabled to be translated to an agricultural application for fruit picking. A device that is monitored by having soft gripper to have no detrimental effect on the collected fruit's tissues, hybrid with Electrochemical Impedance Spectroscopy (EIS) test for quality assurance. Finally, this work can be interrogated in the future to a gripper that have modifiable synchronized work envelope and tip force for each object.

Numerical simulation of deformed red blood cell by utilizing neural network approach and finite element analysis

In order to have research on the deformation characteristics and mechanical properties of human red blood cells (RBCs), finite element models of RBC optical tweezers stretching and atomic force microscope (AFM) indentation were established. Non-linear elasticity of cell membrane was determined by using the neo-Hookean hyperelastic material model, and the deformation of RBC during stretching and indentation had been researched in ABAQUS, respectively. Considering the application of machine learning (ML) in material parameters identification, ML algorithm was combined with finite element (FE) method to identify the constitutive parameters. The material parameters were estimated according to the deformation characteristics of RBC obtained from the change of cell diameter with stretching force when RBC was stretched. The non-linear relationship between material parameter and RBC deformation was established by building a FE-model. The FE simulation of RBC stretching was used to construct the training set and the neural network trained by a large number of samples was used to predict the material parameter. With the predicted parameter, FE simulation of RBC under AFM indentation to explore the local deformation mechanism was completed.

Phase field simulation for fracture behavior of hyperelastic material at large deformation based on edge-based smoothed finite element method

Phase Field Method (PFM) has remarkable performance in simulating crack patterns with complex topology and the edge-based smoothed finite element method (ES-FEM) has intrinsic advantages in modeling the fracture behavior at large deformation. A phase field fracture model for modeling hyperelastic material at large deformation is proposed based on ES-FEM. The weak forms and numerical framework of proposed model are thoroughly derived and constructed. The Neo-Hooken and Mooney-Rivlin constitutive models are adopted to describe the large deformation response of rubber-like materials under displacement load. Fracture behaviors of the penny shaped pre-cracked specimen and the double edge notched specimen in tension are simulated. The load-displacement curves of the double edge notched specimen agree well with that of the experiments. The influence of material constitutive model on Mode II crack propagation is investigated. It is shown that the developed model can capture the Mode I and the mode II cracks of rubber-like material and is insensitive to the distorted mesh at large deformation.

Modeling of the elastic properties of compressed expanded graphite - A material used in spiral wound gaskets

The paper presents the method of modeling the elastic deformations of expanded graphite, a material used as flexible filling in spiral wound gaskets. Five methods (from A to E) were proposed, by means of which the mean values of the effective elasticmodulus were determined on the basis of the experimentally determined stress-strain characteristics of expanded graphite. The values determined in this way constituted the data set into the isotropic material model, reflecting the elastic properties of the expanded graphite tape in the numerical spiral wound gasket model. A different way of modeling the graphite's elastic properties using the numerical model was method F, in which a nonlinear hyperelastic material model proposed by Blatz-Ko was adapted. The basic purpose of the numerical calculations was to determine both the compression curve of the gasket and the stress distribution in the gasket's windings. The obtained numerical results of the gasket's compression were compared with the experimental test, which allowed the assumptions of the numerical models to be validated. The best way to reflect the elastic behavior of expanded graphite in the gasket's numerical model turned out to be the nonlinear material model presented in method F.

A membrane theory for circular graphene sheets, based on a hyperelastic material model for large deformations

Large deflections relevant for suspended circular graphene sheets with simply supported boundaries are computed by a theory for 2D membranes subjected to several types of vertical axisymmetric forces, based on the principle of virtual power (PVP). Corresponding stress–strain relations are provided in the form of a nonlinear hyperelastic material model for graphene. When approximating the deflections through Fourier series, the PVP yields a nonlinear algebraic system of equations, which is solved by the iterative Newton–Raphson procedure. The new computational efficient method is validated through comparison of the numerical results it provides, with predictions obtained from experimental nanoindentaion measurements.

Topology optimization of hyperelastic structures using a modified evolutionary topology optimization method

Soft materials are finding widespread implementation in a variety of applications, and it is necessary for the structural design of such soft materials to consider the large nonlinear deformations and hyperelastic material models to accurately predict their mechanical behavior. In this paper, we present an effective modified evolutionary topology optimization (M-ETO) method for the design of hyperelastic structures that undergo large deformations. The proposed M-ETO method is implemented by introducing the projection scheme into the evolutionary topology optimization (ETO) method. This improvement allows nonlinear topology optimization problems to be solved with a relatively big evolution rate, which significantly enhances the robustness. The minimal length scale is achieved as well. Numerical examples show that the proposed M-ETO method can stably obtain a series of optimized structures under different volume fractions with smooth boundaries. Moreover, compared with other smooth boundary methods, another merit of M-ETO is that the problem of the dependency on initial layout can be eliminated naturally due to the inherent characteristic of ETO.