Isotropic Hyperelastic Material Research Articles

Extending Marlow’s general first-invariant constitutive model to compressible, isotropic hyperelastic materials

PurposeA general first-invariant constitutive model has been derived in literature for incompressible, isotropic hyperelastic materials, known as Marlow model, which reproduces test data exactly without the need of curve-fitting procedures. This paper aims to describe how to extend Marlow’s constitutive model to the more general case of compressible hyperelastic materials.Design/methodology/approachThe isotropic constitutive model is based on a strain energy function, whose isochoric part is solely dependent on the first modified strain invariant. Based on Marlow’s idea, a principle of energetically equivalent deformation states is derived for the compressible case, which is used to determine the underlying strain energy function directly from measured test data. No particular functional of the strain energy function is assumed. It is shown how to calibrate the volumetric and isochoric strain energy functions uniquely with uniaxial or biaxial test data only. The constitutive model is implemented into a finite element program to demonstrate its applicability.FindingsThe model is well suited for use in finite element analysis. Only one set of test data is required for calibration without any need for curve-fitting procedures. These test data are reproduced exactly, and the model prediction is reasonable for other deformation modes.Originality/valueMarlow’s basic concept is extended to the compressible case and applied to both the volumetric and isochoric part of the compressible strain energy function. Moreover, a novel approach is described on how both compressive and tensile test data can be used simultaneously to calibrate the model.

A note on a class of generalized neo-Hookean models for isotropic incompressible hyperelastic materials

In a recent paper in this journal by Anssari-Benam and Bucchi (2021), the authors have proposed a new two-parameter constitutive model for isotropic incompressible hyperelastic generalized neo-Hookean materials. The model reflects the limiting chain extensibility characteristic of non-Gaussian molecular models for rubber. A major contribution of Anssari-Benam and Bucchi (2021) is in showing that the model proposed there is superior to the well-known two-parameter Gent model when fitting with a large variety of experimental data for rubber for the homogeneous deformations of uniaxial, equi-biaxial and pure shear. Moreover, for all of these deformations, a fitting with data is achieved with a single set of values for the parameters with a narrow range of variation. In the present note, we establish a simple direct relation between the new model and the classical Gent model. The large body of research results on the mechanical behavior of rubber-like materials based on the latter model are thus readily applicable to the new model. Some other directions for widening the applicability of the new model are also suggested.

Can biomechanical analysis shed some light on aneurysmal pathophysiology? Preliminary study on ex vivo cerebral arterial walls

BackgroundThe pathophysiology of cerebral aneurysm is complex and poorly understood, and it can have the most catastrophic clinical presentation. Flow dynamics is a key player in the initiation and progression of aneurysm. Better understanding the interaction between hemodynamic loading and biomechanical wall responses can help to add the missing piece on aneurysmal pathophysiology. In this laboratory study we aimed to analyze the effect of the application of a mechanical force to cerebral arterial walls. MethodsDisplacement control tests were performed on five porcine cerebral arteries. The test machine was the T150 Nanotensile. The stiffness variation with the increment of the strain level is modeled as the outcome of an isotropic hyperelastic material model. FindingsThrough the application of an axial force we obtained Stress/Strain curves that showed a marked isotropic hyperelastic behavior, characterized by an increasing of stiffness with the level of strain. This behavior of the cerebral arterial wall is different from the well-established behavior of other arterial vessel (as the aortic vessel) characterized by a marked anisotropic behavior. Additionally, the data scattering observed for higher values of the applied stress are related to different individual packing of collagen fibers that represent the load-bearing mechanics at higher level of the strain. InterpretationThe data obtained by test in this paper represent a first step in our ongoing research about the mechanics of multi-axial loads on cerebral arterial walls, and in producing more comprehensive patient-specific calculations for potential applications on cerebral aneurysm management.

Effective time step restrictions for explicit MPM simulation

AbstractTime steps for explicit MPM simulation in computer graphics are often selected by trial and error due to the challenges in automatically selecting stable time step sizes. Our time integration scheme uses time step restrictions that take into account forces, collisions, and even grid‐to‐particle transfers calculated near the end of the time step. We propose a novel set of time step restrictions that allow a time step to be selected that is stable, efficient to compute, and not too far from optimal. We derive the general solution for the sound speed in nonlinear isotropic hyperelastic materials, which we use to enforce the classical CFL time step restriction. We identify a single‐particle instability in explicit MPM integration and propose a corresponding time step restriction in the fluid case. We also propose a reflection‐based boundary condition for domain walls that supports separation and accurate Coulomb friction while preventing particles from penetrating the domain walls.

Mechanical effects of distributed fibre orientation in the periodontal ligament of an idealised geometry

In this study, we computationally assess the effects of the distributed fibre orientation in the periodontal ligament (PDL) on mechanical responses of the tooth-PDL complex. An idealised axial-symmetric geometry of a tooth-PDL complex was constructed. The fibre orientation in the PDL was modelled as a trigonometric function based on anatomical knowledge, and the PDL was modelled as a transversely isotropic hyperelastic material dependent on fibre orientations. Parametric studies of the fibre orientation on the mechanical responses of the tooth-PDL complex were conducted. Obtained results showed that the anatomically consistent fibre orientation functions as a supporting structure against not only vertical but also horizontal loads.

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.

The Anelastic Ericksen Problem: Universal Deformations and Universal Eigenstrains in Incompressible Nonlinear Anelasticity

Ericksen’s problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold. Physically, this situation corresponds to the case where nontrivial finite eigenstrains are present. We characterize explicitly the universal eigenstrains that share the symmetries present in the classical problem, and show that in the presence of eigenstrains, the six known classical families of universal solutions merge into three distinct anelastic families, distinguished by their particular symmetry group. Some generic solutions of these families correspond to well-known cases of anelastic eigenstrains. Additionally, we show that some of these families possess a branch of anomalous solutions, and demonstrate the unique features of these solutions and the equilibrium stress they generate.

Zona pellucida shear modulus, a possible novel non-invasive method to assist in embryo selection during in-vitro fertilization treatment

The present study investigated the association between oocyte zona pellucida shear modulus (ZPSM) and implantation rate (IR). Ninety-three oocytes collected from 38 in-vitro fertilization patients who underwent intracytoplasmic sperm injection were included in this case–control study. The ZP was modeled as an isotropic compressible hyperelastic material with parameter C_{10}, which represents the ZPSM. Computational methodology was used to calculate the mechanical parameters that govern ZP deformation. Fifty-one developed embryos were transferred and divided into two groups—implanted and not implanted. Multivariate logistic regression analysis was performed to identify the association between ZPSM and IR while controlling for confounders. Maternal age and number of embryos per transfer were significantly associated with implantation. The IR of embryos characterized by C_{10} values in the range of 0.20–0.40 kPa was 66.75%, while outside this range it was 6.70%. This range was significantly associated with implantation (p < 0.001). Geometric properties were not associated with implantation. Multivariate logistic regression analysis that controlled for relevant confounders indicated that this range was independently associated with implantation (adjusted OR 38.03, 95% confidence interval 4.67–309.36, p = 0.001). The present study suggests that ZPSM may improve the classic embryo selection process with the aim of increasing IR.

Formation of finite-time singularities for nonlinear hyperbolic systems with small initial disturbances

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough.

Kearsley-type instabilities in finite deformations of transversely isotropic and incompressible hyperelastic materials

We assume that the strain energy density, W, for a transversely isotropic and incompressible hyperelastic solid is a complete quadratic function of components of the right Cauchy-Green strain tensor, C. We first discuss restrictions on the seven material parameters appearing in the expression for W. It is shown that for the reference configuration to be stress-free, both terms linear in (I4 − 1) and (I5 − 1) must simultaneously appear in the expression for W where I4 = A • CA, I5 = A • C2A, and A is a unit vector along the axis of transverse isotropy. Subsequently, we show that for a prismatic body comprised of this material deformed either in dead-load or displacement-controlled uniaxial tension/compression along A, depending upon the sign of the material parameter h14 multiplying the term (I1 − 3)(I4 − 1) in the expression for W, the two lateral stretches may bifurcate from being equal to being unequal for a stable solution defined as the one that has lower free energy. Here I1 = tr(C) where tr is the trace operator. This is similar to Kearsley's instability in an isotropic square Mooney-Rivlin membrane deformed with equal biaxial dead loads on the edges for which the stable deformed configuration shifts from having equal to unequal biaxial stretches.

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.

Transient swelling response of pH-sensitive hydrogels: A monophasic constitutive model and numerical implementation.

Due to their ability to swell/deswell under alternating pH conditions, pH-sensitive hydrogels present themselves as a potential candidate for controlled drug delivery. In this paper, a coupled electro-chemo-mechanical transient large deformation homogeneous and inhomogeneous swelling theory is developed for pH-sensitive hydrogels. The hydrogel is treated as a single-phase compressible isotropic hyperelastic material. The Nernst-Planck equation is used for the flux of the ionic species inside the hydrogel through its boundaries. The main reason which causes the hydrogel to swell is the osmotic pressure. The osmotic pressure is treated as external stress to the hydrogel. As soon as the concentration of the ions inside the hydrogel is known one may compute the osmotic pressure. Finally, the mechanical equilibrium equations give the deformation of the hydrogel. After presenting the swelling theory, the theory is implemented into the finite element method, and the experimental data is used in order to validate the theory. The simulation matches closely the experimental data which shows the accuracy of the proposed theory. The free swelling and constrained swelling of the pH-sensitive hydrogel are also simulated to show the capabilities of the proposed formulation and well as its numerical counterpart.

Nonlinear analytical solution of nearly incompressible hyperelastic cylinder with variable thickness under non-uniform pressure by perturbation technique

In this paper, nonlinear analytical solution of pressurized thick cylindrical shells with variable thickness made of hyperelastic materials is presented. The governing equilibrium equations for the cylindrical shell with variable thickness under non-uniform internal pressure are derived based on first-order shear deformation theory (FSDT). The shell is assumed to be made of isotropic and homogenous hyperelastic material in nearly incompressible condition. Two-term Mooney-Rivlin type material is considered which is a suitable hyperelastic model for rubbers. Boundary Layer Method of the perturbation theory which is known as Match Asymptotic Expansion (MAE) is used for solving the governing equations. In order to validate the results of the current analytical solution in analyzing pressurized hyperelastic thick cylinder with variable thickness, a numerical solution based on Finite Element Method (FEM) have been investigated. Afterwards, for a rubber case study, displacements, stresses and hydrostatic pressure distribution resulting from MAE and FEM solution have been presented. Furthermore, the effects of geometry, loading, material properties and incompressibility parameter have been studied. Considering the applicability of the rubber elasticity theory to aortic soft tissues such as elastin, the behaviour of blood vessels under non-uniform pressure distribution has been investigated. The results prove the effectiveness of FSDT and MAE combination to derive and solve the governing equations of nonlinear problems such as nearly incompressible hyperelastic shells.

Equilibrium paths of a three-bar truss in finite elasticity with an application to graphene

This paper presents the formulation of the equilibrium problem of a three-bar truss in the nonlinear context of finite elasticity. The bars are composed of a homogeneous, isotropic, and compressible hyperelastic material. The equilibrium equations in the deformed configuration are derived under the assumption of homogeneous deformations and the stability of the solutions is assessed through the energy criterion. The general formulation is then specialized for a compressible Mooney–Rivlin material. The results for both vertical and horizontal load cases show unexpected post-critical behaviors involving several branches, stable asymmetrical configurations, bifurcation, and snap-through. The three-bar truss studied here is not only a benchmark test for the numerical analysis of nonlinear truss structures, but also a representative system for the unit cell of the graphene hexagonal lattice. Therefore, an application to graphene is performed by simulating the covalent bonds between carbon atoms as the bars of the truss, characterized by the modified Morse potential. The results provide insights on the internal mechanisms that take place when graphene undergoes large in-plane deformations, whose influence should be considered when developing molecular mechanics and continuum models in nonlinear elasticity.

A hyper-elastic thermal aging constitutive model for rubber-like materials

Different phenomenological, empirical, and micromechanical constitutive models have been proposed to describe the behavior of incompressible isotropic hyper-elastic materials. Among these models, very few have accounted for the thermal aging effect on the model constants and parameters. This article introduces a new empirical constitutive hyper-elastic model for thermally aged hyper-elastic materials. The model named “the weight function based (WFB) model” considers the effect of aging temperature and time on its parameters. The WFB model formulation can facilitate fatigue analysis and lifetime prediction of rubber-like materials under aging conditions. The WFB model in this article defines all rubber-like material properties, such as fracture stretch, strength, and stiffness, by predicting the full stress–strain curve at any aging time and temperature. The WFB model was tested on natural rubber for uniaxial and biaxial loading conditions. More than 100 specimens were aged and tested uniaxially under various temperatures and aging times to extract the stress–strain behavior. The temperatures used in the test ranged from 76.7°C to 115.5°C, and the aging time ranged from 0 to 600 hours (hrs). A classical bulge test experiment was generated to extract the biaxial natural rubber material behavior. An ABAQUS finite element analysis model was created to simulate and verify the generated biaxial stress–strain curve. The proposed model represents the aging effect on the tested natural rubber under uniaxial and biaxial loading conditions with an acceptable error margin of less than 10% compared to experimental data.

Analysis of Hooke-like isotropic hypoelasticity models in view of applications in FE formulations

This paper presents an analysis of the constitutive relations of Hooke-like isotropic hypoelastic material models in Lagrangian and Eulerian forms generated using corotational stress rates with associated spin tensors from the family of material spin tensors. Explicit expressions were obtained for the Lagrangian and Eulerian tangent stiffness tensors for the hypoelastic materials considered. The main result of this study is a proof that these fourth-order tensors have full symmetry only for material models generated using two corotational stress rates: the Zaremba–Jaumann and the logarithmic ones. In the latter case, the Hooke-like isotropic hypoelastic material is simultaneously the Hencky isotropic hyperelastic material. For the material models considered, basis-free expressions for the material and spatial tangent stiffness tensors are obtained that can be implemented in FE codes. In particular, new basis-free expressions are derived for the tangent stiffness (elasticity) tensors for the Hencky isotropic hyperelastic material model.

Fully-Coupled Transient Fluid–Solid Interaction Simulation of the pH-Sensitive Hydrogel-Based Microvalve

The pH-sensitive hydrogels are attractive candidates to act like a microvalve in microfluidic devices. In this study, we build a theory for the transient simulation of a pH-sensitive hydrogel-based microvalve. Three fields are involved in the theory, namely the electrochemical, mechanical and fluid fields. We utilize the Nernst–Planck equation to describe the ionic flux into the hydrogel through diffusion, electrical migration and convection. We model the hydrogel as a compressible isotropic hyperelastic material with the Gent model. Then we implement the theory in a nonlinear finite element framework to simulate the time-dependent fluid–solid interaction (FSI) behavior of the pH-sensitive microvalve. Our focus is on exploring the physics and phenomena involving in the simulation rather than simulating a complex geometry or presenting a new design. We manifest the significance of the FSI by comparing the transient FSI and non-FSI simulation of the microvalve. The most highlighted novelty of our study is accounting for time-dependent effects. The results demonstrate that the microvalve perfectly closes the channel much before it reaches its stationary state and the closing state, which is of high interest in the microvalve study is different from the stationary state.

An energy-stable mixed formulation for isogeometric analysis of incompressible hyper-elastodynamics.

We develop a mixed formulation for incompressible hyper-elastodynamics based on a continuum modeling framework recently developed in [41] and smooth generalizations of the Taylor-Hood element based on non-uniform rational B-splines (NURBS). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf-sup condition is utilized to provide a bound for the pressure field. The generalized-α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure [42]. The inf-sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.

Contributions of elastic fibers, collagen, and extracellular matrix to the multiaxial mechanics of ligament

Elastin is a biopolymer known to provide resilience to extensible biologic tissues through elastic recoil of its highly crosslinked molecular network. Recent studies have demonstrated that elastic fibers in ligament provide significant resistance to tensile and especially shear stress. We hypothesized that the biomechanics of elastic fibers in ligament could be described as transversely isotropic with both fiber and matrix components in a multi-material mixture. Similarly, we hypothesized that material coefficients derived using the experimental tensile response could be used to predict the experimental shear response. Experimental data for uniaxial and transverse tensile testing of control tissues, and those enzymatically digested to disrupt elastin, were used as inputs to a material coefficient optimization algorithm. An additive decomposition of the strain energy was used to model the total stress as the sum of contributions from collagen fibers, elastic fibers, elastic matrix, and ground substance matrix. Matrices were modeled as isotropic Veronda-Westmann hyperelastic materials, whereas fiber families were modeled as piecewise exponential-linear hyperelastic materials. Optimizations provided excellent fits to the tensile experimental data for each treatment case and material model. Given the disparity in magnitude of stresses between longitudinal and transverse/shear tests and agreement between models and experiments, the hypothesized transversely isotropic material of elastin symmetry was supported. In addition, the coefficients derived from uniaxial and transverse tensile experiments provided reasonable predictions of the experimental behavior during shear deformation. The magnitudes of coefficients representing stress, nonlinearity, and stiffness supported the experimental evidence that elastic fibers dominate the low strain tensile and shear response of ligament. These findings demonstrate that the additive decomposition modeling strategy can represent each discrete fiber and matrix constituent and their relative contribution to the material response of the tissue. These experimental data and the validated constitutive model provide essential inputs and a framework to refine existing computational models of ligament and tendon mechanics by explicitly representing the mechanical contributions of elastic fibers.

A plane strain analysis of the elastostatic fields near the notch-tip of a Blatz-Ko material

An asymptotic plane strain analysis is carried out in order to study the notch problem in a homogeneous isotropic compressible hyperelastic material whose behavior is governed by the Blatz-Ko constitutive law. The work is realized within the framework of the fully nonlinear fracture mechanics at large strain. Close to the notch tip, a stress-free state is assumed in the notch edges, while at infinity (sufficiently far to the notch tip) fracture modes I and II are considered. The results emphasizes many discrepancies with linear theory predictions like the emergence of a logarithmic singularity and show that solution depends on the notch initial opening angle.