Ogden Model Research Articles

Hyper-viscoelastic modelling of brain tissues based on the Ogden model

Hyper-viscoelastic modelling of brain tissues based on the Ogden model

Nonlinear geometrically exact dynamics of hyperelastic pipes conveying fluid: Comparative study of different hyperelastic models

This work presents a comparative analysis of the nonlinear dynamic responses of fluid-conveying pipes modelled by four different phenomenological hyperelastic constitutive models, namely the Mooney-Rivlin model, the Neo-Hookean model, the Yeoh model, and the Ogden model. Based on the strain energy density functions and the exact geometric relation, the geometrically exact dynamic models corresponding to different hyperelastic contitutive models are established. Then the Galerkin's truncation is adopted to transfer the obtained nonlinear integral-partial differential equations into the nonlinear ordinary differential equations. Subsequently, linear dynamic study and finite difference method are taken to explore the natural frequencies and stabilities and nonlinear dynamic responses of the pipe, respectively. And the fitting coefficients for various models obtained from Treloar's experiment data are utilized in the numerical calculation. It is found the Hopf bifurcation and the limit cycle oscillations in the post-flutter status are influenced by the choice of the hyperelastic models, particularly under large flow velocity. However, hyperelastic constitutive models do not exert a significantly extreme influence on the results, which can be attributed to the relatively small strains even though large deformations occur in the pipe. Mathematically, it is proven that the symmetry observed in the pipe's displacement response stems from the symmetry of its cross-section and the inextensibility assumption of the centerline, with hyperelasticity having no effect on this symmetry.

Characterization of pentenoate-functionalized hyaluronic acid and pentenoate-functionalized gelatin hydrogels for printing and future surgical placement in regenerative medicine applications

Injectable hydrogels with in situ crosslinking may be more suitable than pre-fabricated scaffolds for surgical delivery to clinical injuries. However, low viscosity hydrogel precursors may be challenging to surgically place into an injury if the precursor leaks or is washed out. The biomaterials field for extrusion bioprinting is a fertile ground for discovering biomaterials with injectable and paste-like precursor rheology with in situ gelation capabilities, which may promote better material retention in clinical injuries. We previously developed and evaluated one formulation of a pentenoate-functionalized hyaluronic acid (PHA) / pentenoate-functionalized gelatin (PGel) hydrogel with a paste-like, printable precursor and rapid photocrosslinking in a spinal cord injury application. Further characterization of the material and cell response to PHA/PGel hydrogel formulations is needed to expand the bioprinting and other regenerative medicine opportunities for PHA/PGel hydrogels. In the current study, we utilized 2D NMR methods (i.e., 1H–1H TOCSY) to confirm and quantify a high degree of pentenoate functionalization of PGel and PHA. We characterized the stiffness, swelling, and cell viability using varying formulations of PGel or PHA/PGel hydrogels. For compression testing, a straightforward application of the Ogden model enabled evaluation of the full stress-strain range for improved moduli comparisons. We identified two formulations that best supported cell viability (i.e., 3%/10% and 4%/5% PHA/PGel). Furthermore, one of the identified formulations (4%/5% PHA/PGel) had superior printability compared to the other. With better printability and potentially better clinical surgical placement, the new PHA/PGel hydrogel formulations may be more widely applied in the bioprinting and regenerative medicine fields.

Constitutive Model for Thermal-Oxygen-Aged EPDM Rubber Based on the Arrhenius Law.

Ethylene-propylene-diene monomer (EPDM) is a key engineering material; its mechanical characterization is important for the safe use of the material. In this paper, the coupled effects of thermal degradation temperature and time on the tensile mechanical behavior of EPDM rubber were investigated. The tensile stress-strain curves of the aged and unaged EPDM rubber show strong nonlinearity, demonstrating especially rapid stiffening as the strain increases under small deformation. The popular Mooney-Rivlin and Ogden (N = 3) models were chosen to fit the test data, and the results indicate that neither of the classical models can accurately describe the tensile mechanical behavior of this rubber. Six hyperelastic constitutive models, which are excellent for rubber with highly nonlinearity, were employed, and their abilities to reproduce the stress-strain curve of the unaged EPDM were assessed. Finally, the Davis-De-Thomas model was found to be an appropriate hyperelastic model for EPDM rubber. A Dakin-type kinetic relationship was employed to describe the relationships between the model parameters and aging temperature and time, and, combined with the Arrhenius law, a thermal aging constitutive model for EPDM rubber was established. The ability of the proposed model was checked by independent testing data. In the moderate strain range of 200%, the errors remained below 10%. The maximum errors of the prediction results at 85 °C for 4 days and 100 °C for 2 and 4 days were computed to be 17.06%, 17.51% and 19.77%, respectively. This work develops a theoretical approach to predicting the mechanical behavior of rubber material that has suffered thermal aging; this approach is helpful in determining the safe long-term use of the material.

Fluid-structure interaction analysis of a healthy aortic valve and its surrounding haemodynamics.

The opening and closing dynamics of the aortic valve (AV) has a strong influence on haemodynamics in the aortic root, and both play a pivotal role in maintaining normal physiological functions of the valve. The aim of this study was to establish a subject-specific fluid-structure interaction (FSI) workflow capable of simulating the motion of a tricuspid healthy valve and the surrounding haemodynamics under physiologically realistic conditions. A subject-specific aortic root was reconstructed from magnetic resonance (MR) images acquired from a healthy volunteer, whilst the valve leaflets were built using a parametric model fitted to the subject-specific aortic root geometry. The material behaviour of the leaflets was described using the isotropic hyperelastic Ogden model, and subject-specific boundary conditions were derived from 4D-flow MR imaging (4D-MRI). Strongly coupled FSI simulations were performed using a finite volume-based boundary conforming method implemented in FlowVision. Our FSI model was able to simulate the opening and closing of the AV throughout the entire cardiac cycle. Comparisons of simulation results with 4D-MRI showed a good agreement in key haemodynamic parameters, with stroke volume differing by 7.5% and the maximum jet velocity differing by less than 1%. Detailed analysis of wall shear stress (WSS) on the leaflets revealed much higher WSS on the ventricular side than the aortic side and different spatial patterns amongst the three leaflets.

Effect of temperature and humidity on mechanical properties and constitutive modeling of pressure-sensitive adhesives

Pressure-sensitive adhesives (PSAs) are crucial for the structural and functional integrity of flexible displays. Investigating the intricate mechanical properties of PSAs can help enhance product quality and performance. This study conducts systematic mechanical tests, including uniaxial tensile, compression, planar shear, and stress relaxation, on PSAs at temperatures ranging from – 25 to 85 ℃ and relative humidity levels from 0 to 90%. Our findings reveal that the Anssari-Benam model accurately describes the hyperelastic behavior of PSA materials under large deformation, outperforming the Ogden model by requiring fewer parameters and better preserving convexity. Moreover the results show that temperature markedly affects PSA properties, particularly near the glass transition temperature (Tg), with lower temperatures leading to decreased elasticity and higher temperatures aiding in stress relaxation. Similarly, humidity impacts PSA behavior, increasing elasticity and decreasing stiffness, especially noticeable in stress relaxation tests. These findings highlight the substantial influence of environmental conditions on the material properties of PSAs and underscore the necessity of understanding both hyperelastic and viscoelastic responses for their application in flexible technologies. This research provides critical insights for the optimal utilization of PSAs in the rapidly evolving field of flexible electronics, including OLED displays.

Soft tissue material properties based on human abdominal in vivo macro-indenter measurements

Simulations of human-technology interaction in the context of product development require comprehensive knowledge of biomechanical in vivo behavior. To obtain this knowledge for the abdomen, we measured the continuous mechanical responses of the abdominal soft tissue of ten healthy participants in different lying positions anteriorly, laterally, and posteriorly under local compression depths of up to 30 mm. An experimental setup consisting of a mechatronic indenter with hemispherical tip and two time-of-flight (ToF) sensors for optical 3D displacement measurement of the surface was developed for this purpose. To account for the impact of muscle tone, experiments were conducted with both controlled activation and relaxation of the trunk muscles. Surface electromyography (sEMG) was used to monitor muscle activation levels. The obtained data sets comprise the continuous force-displacement data of six abdominal measurement regions, each synchronized with the local surface displacements resulting from the macro-indentation, and the bipolar sEMG signals at three key trunk muscles. We used inverse finite element analysis (FEA), to derive sets of nonlinear material parameters that numerically approximate the experimentally determined soft tissue behaviors. The physiological standard values obtained for all participants after data processing served as reference data. The mean stiffness of the abdomen was significantly different when the trunk muscles were activated or relaxed. No significant differences were found between the anterior-lateral measurement regions, with exception of those centered on the linea alba and centered on the muscle belly of the rectus abdominis below the intertubercular plane. The shapes and areas of deformation of the skin depended on the region and muscle activity. Using the hyperelastic Ogden model, we identified unique material parameter sets for all regions. Our findings confirmed that, in addition to the indenter force-displacement data, knowledge about tissue deformation is necessary to reliably determine unique material parameter sets using inverse FEA. The presented results can be used for finite element (FE) models of the abdomen, for example, in the context of orthopedic or biomedical product developments.

Approach for contact medical device development via integrated testing, skeletal muscle modeling, and finite element analysis

Development of novel medical devices for the treatment of musculoskeletal pain associated with neuro-muscular trigger points requires a model for relating the mechanical responses of in vivo biological tissues to applied palliative physical pressures and a method to design treatments for optimal effects. It is reasonable to hypothesize that the efficacy of therapeutic treatment is proportional to the maximum tensile strain at trigger point locations. This work presents modeling of the mechanical behavior of biological tissue structures and treatment simulations, supported by indentation experiments and finite element (FE) modeling. The steady-state indentation responses of the tissue structure of the posterior neck were measured with a testing device, and an FE model was constructed using a first-order Ogden hyperelastic material model and calibrated with the experimental data. The error between experimental and FE-generated displacement-load curves was minimized via a two-stage optimization process comprised of an Optimal Latin Hypercube design-of-experiments analysis and a Bayesian optimization loop. The optimized Ogden model had an initial shear modulus (μ) of 5.16 kPa and a deviatoric exponent (α) of 11.90. Another FE model was developed to simulate the deformation of the tissue structures in the posterior neck adjacent to the C3 vertebrae in response to indentation loading, in order to determine the optimal location and angle to apply an indentation force for maximum therapeutic benefit. The optimal location of indentation was determined to be 28° lateral from the sagittal plane along the surface of the skin, measured from the centerline of the spine, at an angle of 8° counterclockwise from the surface normal vector. The optimized spatial orientation of the indentation corresponded to the average of the maximum principal strain across the deep muscle region of the model.

Mechanical Insights into Biomaterial Wound Healing Patches and Polyvinyl Acetate Elasticity

This study investigates the development of biomaterial healing patches, specifically focusing on the elastomer characteristics and hyperelastic parameters of polyvinyl acetate in lab-made skin. Three sets were evaluated: plain PVAc (Set A), a mix of PVAc with Polysiloxanes (Set B), and a combination of PVAc with PVM-MA &CMC (Set C). Mechanical properties and hyperelastic behavior were assessed using ASTM D412 type-C standard and a 40 mm/min tensile test. Set A shows higher tensile stress (0.072 MPa) and higher tensile strain (1.1324) than Sets B and C. The Mooney-Rivlin and Ogden models were identified as suitable for elastomer characteristics among various hyperelastic models; specific constants for Set A were validated through these models: μ = -0.0211, α = -2.2919 for the Ogden model, C1 = -0.0036, C2 = 0.0300 for the Mooney-Rivlin model. These findings reveal similarities in mechanical characteristics between studied materials previously developed to mimic skin seen in the pattern of the hyperelastic graph, which has significant implications in developing skin-mimicking compounds for medical applications. The paper establishes potential advances in surgical practices through medicinally treated artificial skin but underscores further research into behavior when used within artificial skins in this technology sector.

Mechanical analysis of PDMS films based on different hyperelastic numerical constitutive models

PDMS(polydimethylsiloxane) is widely employed as a substrate material in flexible electronic devices, necessitating its ability to undergo camber deformation while maintaining excellent ductility and flexibility. Understanding the mechanical behavior of PDMS is imperative for its practical applications. Consequently, to numerically investigate the tensile behavior of PDMS, two commonly used hyperelastic constitutive models, the Mooney-Rivlin model and the 3-term Ogden model, have been employed to describe its mechanical characteristics. The material coefficients were determined by fitting the uniaxial tensile experimental data. Subsequently, separate finite element models were developed for the PDMS membrane and the Cu-PDMS composite layer structure. The numerical findings demonstrate that both the Mooney-Rivlin model and the 3-term Ogden model adequately fit the experimental data. Nevertheless, in comparison to the 3-term Ogden model, the PDMS stress distribution exhibits higher values at the same tensile rate, consequently resulting in the fracture of the Cu layer first.

Comparative analysis of nonlinear dynamic behaviors of hyperelastic curved structure modelled by different constitutive laws

This work focuses on conducting numerical studies and comparisons on the nonlinear dynamic behaviors of hyperelastic curved structures subjected to external dynamic loadings. A numerical model of the hyperelastic structure is developed using the nonlinear finite element method, which considers both the geometric nonlinearity resulting from large deformations and the material nonlinearity due to hyperelasticity. Four commonly used hyperelastic constitutive laws, namely the Mooney-Rivlin model, the neo-Hookean model, the Yeoh model, and the Ogden model, are employed to describe the nonlinear stress-strain relationship of the hyperelastic material. The convergence performance of the numerical model with respect to grid element and time step is thoroughly examined through several numerical examples. Subsequently, based on the developed numerical model, the nonlinear dynamic behaviors of the hyperelastic curved structures excited by uniform pressure loadings with single and dual frequencies are studied and compared using various constitutive laws. The material constants used in these constitutive laws are estimated based on the well-known Treloar's test data on vulcanized rubber material. Additionally, comprehensive parametric analyses are conducted on the nonlinear dynamic behaviors of the hyperelastic structure simulated by different constitutive laws. The deformation evolution process, resonance curves, time-frequency analysis, bifurcation diagrams, and phase portraits are employed to demonstrate the similarities and differences in the nonlinear dynamic behaviors (i.e., super- and sub-harmonic resonance, internal resonance, chaotic motion and bifurcations) of the hyperelastic structure modeled by different constitutive laws.

Ogden and Mooney-Rivlin hyperelastic models comparison in porcine lobular tissue

Ogden and Mooney-Rivlin hyperelastic models comparison in porcine lobular tissue

Wave propagation characteristics in a rotating soft cylinder

Wave propagation characteristics in rotating cylindrical structures, which reflect the dynamic properties of the structures, play a crucial role in various applications. This paper systematically investigates the wave propagation characteristics in a rotating soft cylinder, based on the Neo-Hookean and Ogden soft material models. The finite deformation theory and the linear incremental theory are employed to solve the large deformation induced by rotation and the small perturbation caused by wave propagation. It is found that rotation changes the geometry size, induces the asymmetry of the forward and backward waves, and leads to the discontinuous frequency change. At large wave numbers, the effect of asymmetry can be neglected for a thin cylinder. However, for small wavenumbers, the effect of asymmetry is significant for both thin and thick cylinders. As the speed increases, the frequency will undergo a sudden and discontinuous change in both Neo-Hookean and Ogden models. In the Ogden model, the snap-through in frequency is observed, during which the dispersion curves will change dramatically. The results of this work will benefit the application of the rotating systems with soft rollers and wheels, and the frequency characteristics can be used in structural detection and signal generation.

Identifying composition-mechanics relations in human brain tissue based on neural-network-enhanced inverse parameter identification

The mechanical properties of human brain tissue remain far from being fully understood. One aspect that has gained more attention recently is their regional dependency, as the brain’s microstructure varies significantly from one region to another. Understanding the correlation between tissue components and the mechanical behavior is an important step toward better understanding how human brain tissue properties change in space and time and to develop highly spatially resolved constitutive models for large-scale brain simulations. Here, we analyze the correlation between human brain tissue components quantified through enzyme-linked immunosorbent assays (ELISA) and material parameters obtained through an inverse parameter identification scheme based on a hyperelastic Ogden model and multimodal mechanical testing data for eight regions of the brain. We use neural networks as a metamodel to save computational costs. The networks are trained on finite element simulation outputs and are able to replace the simulations in the initial optimization step. We identified strong dependencies between mechanical properties and Iba1 associated with microglia cells, collagen VI, GFAP associated with astrocytes, and collagen IV. These results advance our understanding of microstructure-mechanics relations in human brain tissue and will contribute to the development of highly spatially resolved microstructure-informed constitutive models.

Surface–dislocation interaction by various models of surface elasticity

The effect of applying different surface elasticity models related to the Gurtin–Murdoch and Steigmann–Ogden theories to the problem on an interaction of a dislocation row with a flat surface of a semi-infinite three-dimensional body is analyzed in the paper. The boundary conditions in the case of an arbitrary shape of a cylindrical surface under the plane strain are derived within the framework of the Steigmann–Ogden model involving all simplified versions of the Gurtin–Murdoch model and the Gurtin–Murdoch model itself. The boundary condition used in the paper for the flat surface is a particular case of the general one. The analytical solution of the corresponding boundary value problem is obtained in the paper in a closed form for the elastic field. Based on this solution, some numerical examples of the stress distribution at the surface and the image force acting on each dislocation are presented. It is shown that incorporating the bending resistance of the surface in the Steigmann–Ogden model leads to the decrease of some stresses and increase the other ones at the surface and to the decrease of the image force, as compared with those obtained by the Gurtin–Murdoch membrane theory of the surface.

Torsional waves in hyperelastic shells: Appearing shock waves and energy dissipation

Nonlinear torsional waves propagating in cylindrical shells made of hyperelastic material obeying Ogden model, are studied by a combined approach comprising theoretical and finite element analysis. It was found, apparently for the first time, that delta-like pulses of torsional waves attenuate with distance due to the appearance of shock wave fronts. Moreover, both strain and kinetic mechanical energy dissipate due to the release of thermal energy.

Circular inclusion with a refined linearized version of Steigmann–Ogden model

The plane deformation of an infinite elastic matrix enclosing a single circular inclusion incorporating stretching and bending resistance for the inclusion–matrix interface is revisited using a refined linearized version of the Steigmann–Ogden model. This refined version of the Steigmann–Ogden model differs from other linearized counterparts in the literature mainly in that the tangential force of the interface defined in this version depends not only on the stretch of the interface but also on the bending moment and initial curvature of the interface (the corresponding bending moment relies on the change in the real curvature of the interface during deformation). Closed-form results are derived for the full elastic field in inclusion–matrix structure induced by an arbitrary uniform in-plane far-field loading. It is identified that with this refined version of the Steigmann–Ogden model a uniform stress distribution could be achieved inside the inclusion for any non-hydrostatic far-field loading when R = 3 χ int / λ int (where R is the radius of the inclusion, while λ int and χ int are the stretching and bending stiffness of the interface). Explicit expressions are also obtained for the effective transverse properties of composite materials containing a large number of unidirectional circular cylindrical inclusions using, respectively, the dilute and Mori–Tanaka homogenization methods. Numerical examples are presented to illustrate the differences between the refined version and two typical counterparts of the Steigmann–Ogden model in evaluating the stress field around a circular nanosized inclusion and the effective properties of the corresponding homogenized composites.

Mathematical Analysis of Aortic Deformation in Aneurysm and Wall Dissection

Aortic dissection is an extremely severe pathology. From the mechanics point of view, the aorta is a multilayered anisotropic reinforced shell, which is subjected to periodic loading under the action of pulse blood pressure. The questions of mathematical modeling of aorta and large arteries dissection are considered. A review of modern mathematical models of aortic and arterial wall structure obtained on the basis of experimental data processing on biaxial stretching of samples is carried out. Mathematical models can be conditionally divided into two classes: 1) effective models, when the internal structure of the wall structure is ignored, but the mechanical parameters of the material "averaged" over the wall thickness are introduced; 2) structured models, when the multilayer (up to three layers) structure of the artery is taken into account with the addition of one to four families of reinforcing fibers. One of the most widely used artery models, the Holzapfel – Hasser – Ogden model, is considered in detail. This model describes a two or three-layered artery with two families of reinforcing fibers. For this model tables of design parameters are given, numerical calculations of arterial rupture and dissection are carried out. The blood vessel is subjected to pulse pressure of blood flowing through it. It is shown that rupture of the inner layer of the vessel leads to an increase in the stress on the outer wall of the vessel. Increasing the thickness and length of the rupture increases the stresses on the outer wall of the vessel. The presence of an aneurysm of the vessel increases stresses twice as much as a vessel without an aneurysm. Splitting of the inner wall of the vessel leads to an increase in stresses on the wall – stresses fall with increasing rupture width for a straight vessel and rise for a vessel with an aneurysm. Stress calculations at the “tip” of delamination showed that the maximum stress is reached at the outer wall of the rupture.

Accuracy meets simplicity: A constitutive model for heterogenous brain tissue

We present a general, hyperelastic, stretch-based potential that shows promise for modeling the mechanics of brain tissue. A specific four-parameter model derived from this general potential outperforms alternative models, such as the modified Ogden model, the Gent model, Demiray model, and machine-learning models, in capturing brain tissue elasticity. Specifically, the stretch-based model achieved R2 values of 0.997, 0.992, and 0.993 (tension, compression, and shear) for the cortex, 0.995, 0.983, and 0.983 for the basal ganglia, 0.994, 0.929, and 0.970 for the corona radiata, and 0.990, 0.896, and 0.969 for the corpus callosum. This work has the potential to advance our understanding of brain tissue mechanics and provides a valuable tool to improve finite element models for the investigation of brain development, injuries, and disease.

Mechanical differences of anterior and posterior spinal nerve roots revealed by tensile testing

Cervical spondylotic myelopathy can damage the nerves and structures of the subarachnoidal canal. The spinal cord is attached to the subarachnoidal canal via structures such as nerve roots (NR), which play an important role in its positioning and can be affected by cervical spondylotic myelopathy Understanding the tensile mechanical properties of nerve roots is therefore crucial.A total of 37 swine nerve samples (15 bundles, 12 posterior roots, and 10 anterior roots) were mechanically tested within 12 h of sacrifice by a tensile test on a Mach 1 system (Biomomentum, Montreal, Canada) equipped with a 17 N load cell. Bi-linear piecewise fitting was performed to determine the elastic modulus, maximal strain and stress at failure, as well as the coefficients of the 1st- and 3rd-order Ogden models. Additionally, a sensitivity analysis on the Ogden coefficients was performed.The elastic modulus for the bundles, posterior roots, and anterior roots were 3.7 ± 3.1 MPa, 0.23 ± 0.18 MPa, and 0.31 ± 0.27 MPa, respectively. Significant differences (P < 0.05) were found between the bundles and the anterior/posterior roots. Coefficients for the 1st-, 2nd- and 3rd-order Ogden models were provided for each type of sample.Nerve roots have different properties depending on their types. The 1st-order Ogden model may be used to model the mechanical properties of NR using constitutive laws.