Hyperelastic Material Model Research Articles

Dynamic instability of dielectric elastomer actuators subjected to unequal biaxial prestress

The paper presents a Hamiltonian approach for extracting the dynamic instability parameters of homogeneously deforming dielectric elastomer actuators subjected to an unequal biaxial prestress, and driven by a suddenly applied electric load. The approach relies on setting up the balance between the kinetic, strain, and electrostatic energy at the point of maximum overshoot in an oscillation cycle. The equation of the stagnation curve, obtained by invoking aforestated statement of energy-balance, is operated upon by the condition of instability to determine the instability parameters. The underlying principles of the approach are elucidated by considering the Ogden family of hyperelastic material models. The approach is however portrayed generically, and hence, can be extended to the other hyperelastic material models of interest. The estimates of the dynamic instability parameters are corroborated by examining the saddle-node bifurcation points in the time-history response obtained by integrating the equation of motion. A parametric study is conducted to bring out the effect of unequal biaxial prestress, and the trends of variation of the critical electric field and the thickness-stretch on the onset of dynamic instability are presented. A quantitative comparison with the static instability parameters reveals that the dynamic instability gets triggered for electric fields that are lower than those corresponding to the static instability. In contrast, the maximum stretch experienced by the actuator at the dynamic instability is significantly higher than that at the static instability. The crucial inferences can find their potential use in the design of DEAs subjected to a transient motion.

Adhesive connections in glass structures—part II: material parameter identification on thin structural silicone

The present paper proposes two methodologies of identifying hyperelastic material parameters of thin structural silicones based on so-called direct and inverse methods. Based on part I of this paper, analytical investigations were performed to conduct homogeneous experiments with structural silicones. To obtain more insight wether or not an experiment provides a homogeneous stress state, the so-called triaxiality was introduced, which allows one to illustrate differences between homogeneous and inhomogeneous experiments. With the help of this scalar, it was possible to design experimental test setups, which ensure a homogeneous stress and strain distribution within the tested rubber-like material. Furthermore an engineering approach to determine the testing speed of arbitrary experiments dependent on one reference testing speed and experiment was presented. This approach ensured equivalent strain energies between arbitrary and reference test specimens during testing, by which expensive strain rate controlled experiments can be relinquished. Based on these analytical studies, experimental data could be provided for the material parameter identification, which exhibits firstly a nearly homogeneous stress state in accordance to the desired stress and strain field of the applied mathematical model and secondly providing nearly equivalent strain energies within different experimental test set-ups and geometries of test specimens. Returning to the present paper, the first methodology identifies simultaneously hyperelastic material parameters based on a set of conventional and homogeneous experimental tests, like uniaxial tension and uniaxial compression, biaxial tension as well as shear-pancake tests. The second methodology determines inversely hyperelastic material parameters utilizing the inverse Finite Element Method based on one single unconventional and inhomogeneous experimental test, here a microindentation test. The main idea is to obtain reliable hyperelastic material parameters based on a single, inhomogeneous experiment to avoid many, time-consuming homogeneous experiments. To validate the inversely determined hyperelastic material parameters, simultaneous multi-experiment data fits are performed to relate the obtained material parameters to those of the microindentation tests. Considering the set of homogeneous experiments, two classical hyperelastic constitutive equations (Neo-Hooke and Mooney–Rivlin) were utilized to determine constitutive parameters. Due to the simplicity of the classical material laws, a more sophisticated, novel phenomenological hyperelastic material law will be proposed and compared with the results of the classical models respectively the results obtained by a modern hyperelastic material model after Kaliske & Heinrich, which generally delivers outstanding results for the material parameter identification.

A coupled hyperelastic-plastic-continuum damage model for studying cyclic behavior of unfilled engineering polymers

History of accumulated fatigue damage and induced plastic strain are salient design criteria in many structural elements in aerospace, automotive and healthcare industries. A successful fatigue engineering tool should be able to predict the cyclic permanent set as well as the fatigue damage, and lifetime. When an engineering polymer is subjected to cyclic loads, the unloading curves may deviate from the so-called linear unloading response. It is argued that previously developed fatigue computational tools, e.g. Shojaei and Wedgewood (2017) and Shojaei and Volgers (2017), might overpredict the residual plastic strain in real applications. This work aims at accommodating the observed nonlinear unloading curves into the previously developed fatigue modeling framework that significantly improves the predicted permanent deformation. To accomplish this task a hyperelastic material model is coupled with plasticity, and Continuum Damage Mechanics (CDM). The developed hyperelastic-plastic-CDM framework is implemented in Finite Element Analysis (FEA) software, viz. Abaqus, through user-defined material laws for structural analysis. The hyperelastic material model within the small strain framework can effectively capture elastic loading and unloading responses. The enforced coupling between hyperelastic, plastic, and CDM models results in more accurate prediction of the progressive fatigue damage and permanent set in unfilled polymers. The performance of the developed framework is compared with the experimental data and it is shown that the developed framework performs well in capturing progressive permanent set, and fatigue damage in unfilled polymers. The proposed framework may improve the precision of cyclic analysis in unfilled polymers and may help engineers in their design optimization cycles.

Topology optimization of hyperelastic structures using a level set method

Soft rubberlike materials, due to their inherent compliance, are finding widespread implementation in a variety of applications ranging from assistive wearable technologies to soft material robots. Structural design of such soft and rubbery materials necessitates the consideration of large nonlinear deformations and hyperelastic material models to accurately predict their mechanical behaviour. In this paper, we present an effective level set-based topology optimization method for the design of hyperelastic structures that undergo large deformations. The method incorporates both geometric and material nonlinearities where the strain and stress measures are defined within the total Lagrange framework and the hyperelasticity is characterized by the widely-adopted Mooney–Rivlin material model. A shape sensitivity analysis is carried out, in the strict sense of the material derivative, where the high-order terms involving the displacement gradient are retained to ensure the descent direction. As the design velocity enters into the shape derivative in terms of its gradient and divergence terms, we develop a discrete velocity selection strategy. The whole optimization implementation undergoes a two-step process, where the linear optimization is first performed and its optimized solution serves as the initial design for the subsequent nonlinear optimization. It turns out that this operation could efficiently alleviate the numerical instability and facilitate the optimization process. To demonstrate the validity and effectiveness of the proposed method, three compliance minimization problems are studied and their optimized solutions present significant mechanical benefits of incorporating the nonlinearities, in terms of remarkable enhancement in not only the structural stiffness but also the critical buckling load.

Fully-coupled fluid-structure interaction simulation of the aortic and mitral valves in a realistic 3D left ventricle model.

In this study, we present a fully-coupled fluid-structure interaction (FSI) framework that combines smoothed particle hydrodynamics (SPH) and nonlinear finite element (FE) method to investigate the coupled aortic and mitral valves structural response and the bulk intraventricular hemodynamics in a realistic left ventricle (LV) model during the entire cardiac cycle. The FSI model incorporates valve structures that consider native asymmetric leaflet geometries, anisotropic hyperelastic material models and human material properties. Comparison of FSI results with subject-specific echocardiography data demonstrates that the SPH-FE approach is able to quantitatively predict the opening and closing times of the valves, the mitral leaflet opening and closing angles, and the large-scale intraventricular flow phenomena with a reasonable agreement. Moreover, comparison of FSI results with a LV model without valves reveals substantial differences in the flow field. Peak systolic velocities obtained from the FSI model and the LV model without valves are 2.56 m/s and 1.16 m/s, respectively, compared to the Doppler echo data of 2.17 m/s. The proposed SPH-FE FSI framework represents a further step towards modeling patient-specific coupled LV-valve dynamics, and has the potential to improve our understanding of cardiovascular physiology and to support professionals in clinical decision-making.

Application of Hyperelastic Models in Mechanical Properties prediction of Mouse Oocyte and Embryo Cells at Large Deformations

Biological cell studies have many applications in biology, cell manipulation and diagnosis of diseases such as cancer and malaria. In this study, inverse finite element method (IFEM) combined with Levenberg-Marquardt optimization algorithm has been used to extract and characterize material properties of mouse oocyte and embryo cells at large deformations. Then, the simulation results have been validated using data from experimental works. In this study, it is assumed cell material is hyperelastic, isotropic, homogenous and axisymmetric. For inverse analysis, FEM model of cell injection experiment which implemented in Abaqus software has been coupled with Levenberg-Marquardt optimization algorithm written in Matlab; based on this coupling the optimum hyperelastic coefficients which give the best match between experimental and simulated forces are extracted. Results show that among different hyperelastic material models, Ogden material is well suitable for characterization of mouse oocyte cell and Mooney-Rivlin or polynomial are suitable for characterization of mouse embryo cell. Moreover the evaluated Poisson ratio of the cell is obtained to be equal to 0.5, which indicates the structural material of mouse oocyte and embryo, are compressible.

Dynamic characterization and modeling of rubber shock absorbers: A comprehensive case study

Rubber or elastomeric materials are widely used for shock absorbers having elastic and viscous properties such as high inherent damping, deflection capacity, and energy storage. The dynamic properties of these components are of primary concern in designing rubber absorbers to reduce the shock loading given as well as the structure-borne noise transmissibility. Besides, the dynamic response of the mechanical systems, at where the rubber shock absorbers are used, is directly associated with the properties of the shock absorbers. In order to determine these properties of the rubber, mathematical models are created in terms of hyperelasticity and viscoelasticity. The hyperelastic and viscoelastic material models represent the nonlinear elastic and strain rate dependencies of the overall rubber behavior, respectively. Hyperelastic material model captures the material’s nonlinear elasticity with no-time dependence whereas viscoelastic model describes the material response which contains an elastic and viscous part depending on time, frequency, and temperature. This paper presents the dynamic characterization of rubber shock absorbers, having different shore hardness values, in terms of hyperelastic and viscoelastic constitutive models. The parameters of the constitutive models are determined from the uniaxial tensile and relaxation tests. These parameters are used for the numerical model of the rubber components and the accuracy of the characterization is presented by means of a numerical case study.

Non-contact experimental methods to characterise the response of a hyper-elastic membrane

BackgroundMembranes often feature in dynamic structures. The design of such structures generally includes the evaluation of their dynamic characteristics, such as natural frequecies and mode shapes.MethodsThe quasi-statics ad dyamic responses of thin rubber sheeting were investigated through non-contact experimental techniques. The rubber sheeting was modelled as a membrane structure and the material was assumed to be hyper-elastic, isotopic and incompressible. Two hyper-elastic material models were considered, namely the Mooney-Rivlin model and the Neo-Hookean model. The natural frequencies and mode shapes of the hyprt-elastic membrane were anatically and numerically calculated by assuming small linear vibrations and an equi-bi-axial stress state in the membrane. To validate the mathematical analyses, experimental modal analysis was performed where the vibration response was measured with a laser Doppler vibrometer.Results and conclusionsThe analytical model, shows that the natural frequencies of the membrane depend on the initial stretch. Mathematical and experimental results agree well at the lower modes. However, measurement resolution is found to be a vital factor which limits the extraction of closely spaced modes due to difficulties with the accurate identification of nodal line in a purely experimental approach.

Mechanical properties of the human spinal cord under the compressive loading

The spinal cord as the most complex and critical part of the human body is responsible for the transmission of both motor and sensory impulses between the body and the brain. Due to its pivotal role any types of physical injury in that disrupts its function following by shortfalls, including the minor motor and sensory malfunctions as well as complicate quadriplegia and lifelong ventilator dependency. In order to shed light on the injuries to the spinal cord, the application of the computational models to simulate the trauma impact loading to that are deemed required. Nonetheless, it has not been fulfilled since there is a paucity of knowledge about the mechanical properties of the spinal cord, especially the cervical one, under the compressive loading on the grounds of the difficulty in obtaining this tissue from the human body. This study was aimed at experimentally measuring the mechanical properties of the human cervical spinal cord of 24 isolated fresh samples under the unconfined compressive loading at a relatively low strain rate. The stress-strain data revealed the elastic modulus and maximum/failure stress of 40.12±6.90 and 62.26±5.02kPa, respectively. Owing to the nonlinear response of the spinal cord, the Yeoh, Ogden, and Mooney-Rivlin hyperelastic material models have also been employed. The results may have implications not only for understanding the linear elastic and nonlinear hyperelastic mechanical properties of the cervical spinal cord under the compressive loading, but also for providing a raw data for investigating the injury as a result of the trauma thru the numerical simulations.

Computationally Informed Design of a Multi-Axial Actuated Microfluidic Chip Device

This paper describes the computationally informed design and experimental validation of a microfluidic chip device with multi-axial stretching capabilities. The device, based on PDMS soft-lithography, consisted of a thin porous membrane, mounted between two fluidic compartments, and tensioned via a set of vacuum-driven actuators. A finite element analysis solver implementing a set of different nonlinear elastic and hyperelastic material models was used to drive the design and optimization of chip geometry and to investigate the resulting deformation patterns under multi-axial loading. Computational results were cross-validated by experimental testing of prototypal devices featuring the in silico optimized geometry. The proposed methodology represents a suite of computationally handy simulation tools that might find application in the design and in silico mechanical characterization of a wide range of stretchable microfluidic devices.

A plate model for multilayer graphene sheets and its finite element implementation via corotational formulation

An equivalent continuum model for multilayer graphene sheets (MLGSs) and its plate model are developed to analyze the deformation behavior of MLGSs. Hyperelastic material models are introduced for the MLGS continuum model by examining the atomistic structures of MLGSs and obtaining their mechanical properties by means of molecular statics simulations. The MLGS plate model, a structural model for MLGSs, is developed by applying kinematics assumptions to the MLGS continuum model subjected to infinitesimal deformation. Finite element methods (FEM) with the corotational formulation are adopted to analyze the mechanical behavior of MLGSs under small-strain deformation and large rotation conditions. The MLGS plate element passes several basic numerical tests, including patch tests, eigenvalue analyses, and geometrically nonlinear benchmark problems. Finally, the deflections of a plane-strain cantilever and spherical indentations are analyzed by the proposed MLGS plate element and molecular dynamics (MD) simulations. These results show that the MLGS plate element properly represents the deformation behaviors of MLGSs from the atomic scale to the macroscopic continuum scale.

Parameter identification methods for visco- and hyperelastic material models

Many materials in modern civil engineering applications, such as interlayers for laminated safety glass, are polymer-based. These materials are showing distinct viscoelastic (strain-rate) and temperature dependent behaviour. In literature, different mathematical representations of these phenomena exist. A common one is the ‘Prony-series’ representation, which is implemented in many state-of-the-art Finite-Element-Analysis-Software to incorporate linear viscoelastic material behaviour. The Prony-parameters at a certain reference temperature can either be determined by relaxation or retardation experiments in the time domain or with a steady state oscillation in the frequency domain in the so called ‘Dynamic Mechanical Thermal Analysis’ followed by a ‘Time–Temperature-Superposition-Principle’. However, present research shows that polymeric materials also may need to have constitutive equations which include hyperelasticity (nonlinear stress–strain behaviour in a quasi-static condition) when undergoing large deformations, so that the material model should be expanded for a more realistic representation in numerical simulations. A novel method for the whole identification process for a numerical material model in terms of a linear Generalized Maxwell Model (Prony-series) based on experimental data will be presented. Furthermore, material parameters for different hyperelastic material models based on experimental investigations will be shown and compared. Future research activities as well as extensions of the presented novel method are also highlighted within this paper.

Optimal design of compliant mechanisms using functionally graded materials

This research applies topology optimization to create feasible functionally graded compliant mechanism designs with the aim of improving structural performance compared to traditional homogeneous compliant mechanism designs. Converged functionally graded designs will also be compared with two-material compliant mechanism designs. Structural performance is assessed with respect to mechanical/geometric advantage and stress distributions. Two design problems are presented – a gripper and a mechanical inverter. A novel modified solid isotropic material with penalization (SIMP) method is introduced for representing local element material properties in functionally graded structures. The method of moving asymptotes (MMA) is used in conjunction with adjoint sensitivity analysis to find the optimal distribution of material properties. Geometric non-linear analysis is used to solve the mechanics problem based on the Neo-Hookean model for hyperelastic materials. Functionally graded materials (FGMs) have material properties that vary based on spatial position. Here, FGMs are implemented using two different resource constraints – one on the mechanism’s volume and the other on the integral of the Young’s modulus distribution throughout the design domain. Tensile tests are performed to obtain the material properties used in the analysis. Results suggest that FGMs can achieve the desired improvements in mechanical/geometric advantage when compared to both homogeneous and two-material mechanisms.

A hyperelastic fractional damage material model with memory

In this paper a scalar damage model for hyperelastic materials is considered. The novelty of the proposed approach lies in the evolution law for the damage variable that is formulated with the application of fractional calculus. In this way damage evolution includes the memory, or in other words the current intensity of damage evolution, which is based on information from the past - whose length is included in the fractional operator. Based on illustrative examples, the flexibility of the model to mimic experimentally observed material behaviour is presented.

Mechanical characterization and finite element implementation of the soft materials used in a novel anthropometric test device for simulating underbody blast loading

Soft materials (e.g. polymers) are widely used in biomechanical devices to represent the nonlinear viscoelastic properties inherent in biological soft tissues. Knowledge of their mechanical properties is used to inform design choices and develop accurate finite element (FE) models of human surrogates. The goal of this study was to characterize the behavior of eight polymeric materials used in the design of a novel anthropomorphic test device (ATD) and implement these materials in an FE model of the ATD.Tensile and compressive tests at strain rates ranging from 0.01s−1 to 1000s−1 were conducted on specimens from each material. Stress-strain relationships at discrete strain rates were used to define strain rate-dependent hyper-elastic material models in a commercial finite element solver. Then, the material models were implemented into an FE model of the ATD. The performance of the material models in the FE model was evaluated by simulating experiments that were conducted on the ATD lower limb.The material characterization tests revealed viscoelastic strain rate-dependent properties in the flesh and compliant elements of the ATD. Higher modulus polymers exhibited rate-dependent, strain-hardening properties. A strong agreement was seen between the material model simulations and corresponding experiments. In component simulations, the materials performed well and the model reasonably predicted the forces observed in experiments.

Numerical Approximation of Elasticity Tensor Associated With Green-Naghdi Rate.

Objective stress rates are often used in commercial finite element (FE) programs. However, deriving a consistent tangent modulus tensor (also known as elasticity tensor or material Jacobian) associated with the objective stress rates is challenging when complex material models are utilized. In this paper, an approximation method for the tangent modulus tensor associated with the Green-Naghdi rate of the Kirchhoff stress is employed to simplify the evaluation process. The effectiveness of the approach is demonstrated through the implementation of two user-defined fiber-reinforced hyperelastic material models. Comparisons between the approximation method and the closed-form analytical method demonstrate that the former can simplify the material Jacobian evaluation with satisfactory accuracy while retaining its computational efficiency. Moreover, since the approximation method is independent of material models, it can facilitate the implementation of complex material models in FE analysis using shell/membrane elements in abaqus.

Closed‐form stress solutions for incompressible visco‐hyperelastic solids in uniaxial extension

AbstractThis paper is concerned with the theory of incompressible visco‐hyperelastic solids. Closed‐form stress solutions are derived for five different visco‐hyperelastic material models in uniaxial extension. Two loading scenarios are considered: uniaxial extension with constant true strain‐rate; uniaxial extension with constant engineering strain‐rate. The obtained stress solutions are expressed using only elementary functions, when the applied stretch is linear in the true strain. However, for the constant engineering strain‐rate case, all the solutions are expressed using special functions, even for the very simple Hencky's hyperelastic material model. The novel solutions can be used to improve the performance and accuracy of the parameter‐fitting procedure of a particular visco‐hyperelastic solid. In addition, the closed‐form expressions for the stresses may serve reference solution for benchmark problems.

Sealing capability and hyperelastic behaviour of silicone biocomposites via compression test

AbstractDue to the excellent chemical flexibility and elastic properties of silicone rubber, Arenga pinnata‐silicone rubber biocomposites were introduced as a candidate for sealing applications. Its sealing properties were investigated via compressive properties between oil soaked and unsoaked specimens. Compression test based on ASTM D395 was employed to obtain the compression sets and its material constants are acquired using hyperelastic material models i. e. Neo‐Hookean and Mooney‐Rivlin models. Unsoaked specimens consist of 0 wt %, 4 wt %, 8 wt % and 16 wt % fibre content while for soaked specimens, only those with 0 wt % (lowest) and 16 wt % (highest) were prepared for comparison purposes. The results indicated that 16 wt % soaked specimens exhibit the highest compressive stress compared to both soaked and unsoaked 0 wt % specimens, while the compression sets of both 0 wt % and 16 wt % soaked specimens showed lower values compared to all unsoaked specimens. Using hyperelastic material models, the 16 wt % soaked specimens indicated higher material constants than unsoaked specimens while pure silicone showed the opposite. Thus, this study has found that the silicone biocomposites have the capability in sealing applications as its compressive stress shows superior properties especially for soaked specimens compared to pure silicone rubber.

Assessment of hyperelastic material models for the application of adhesive point-fixings between glass and metal

For the investigation of adhesive point-fixings a computationally demanding finite element model is required. The accuracy of the numerical results depends highly on the validity of the used material models, which describe the deformation behaviour of the adhesive. The material models are derived from curve-fitting the mathematical expressions to experimental data mostly derived from uniaxial and equibiaxial experiments. In literature the suitability of the used material models is determined by comparing the numerical results from the same uniaxial and equibiaxial experiments to the experimental results. In contrast, in this contribution, the material models are validated by two additional validation experiments, i.e. an adhesive point-fixing loaded in uniaxial tension and an adhesive point-fixing loaded in a combination of tension and shear.After comparison of the numerical and experimental displacements, it appears that the material models that are calibrated by shear tests or by a combination of shear tests yield the best results. In addition, most numerical load-displacement curves have an almost linear gradient at small strains. Such behaviour is also demonstrated in the experimental measurements of the deformation.

The counterintuitive out-of-plane strength of some incompressible orthotropic hyperelastic materials

Constitutive models for incompressible orthotropic hyperelastic materials are of interest here. We first consider the corresponding linearised theory which has six independent elastic moduli and examine its response in simple tension. Expressions for the Poisson’s ratios and the ratios of the three Young’s moduli are obtained in terms of derivatives of the strain-energy density in the undeformed configuration. The positive semi-definiteness of the linear strain-energy imposes restrictions on these derivatives, which are advocated here as being necessary conditions for physically realistic response of the non-linear model. Additionally, it is proposed that two further restrictions are desirable, restrictions that ensure that the out-of-plane Young’s modulus is always smaller than the Young’s moduli in the plane of the fibres. Since the constitutive laws for orthotropic materials are complex, simplifications in the general form of the strain-energy function are often made for mathematical tractability, the most common being strain energies that depend only on the I1, I4, I6 invariants. It is shown that these reduced models do not satisfy the relative stiffness inequalities, even though they can satisfy the positive semi-definiteness conditions. This gives rise to rather counterintuitive behaviour. Firstly, the out-of-plane Young’s modulus for infinitesimal deformations is always larger than at least one of the in-plane moduli and is greater than both for fibre angles that are close to orthonormal symmetry. For these fibre angles, it is also shown that the out-of-plane Young’s modulus can be orders of magnitude larger than the in-plane moduli and that as the fibres are stiffened, the out-of-plane modulus is increased proportionally more than the in-plane moduli. Since the defining characteristic of fibre-reinforced composite materials is that they should be much stiffer in the plane of the fibres than out of the plane, it is argued that it is desirable to include both the I5, I7 invariants to obtain more realistic constitutive models. It is shown by means of an example that the addition of a simple linear term in these invariants can have a significant effect on the mechanical response.