Ogden Material Model Research Articles

Driving torque model of the bionic soft arm’s hyperelastic bellows

ABSTRACT The technology of soft continuum robots represents an advance in the field of robotics to benefit a wide range of industries such as healthcare, manufacturing or environmental exploration. Soft continuum robots can be driven pneumatically by bellows as the soft continuum manipulator presented in this article. A driving torque model of the bellows considering hyperelastic material properties, friction and restoring torques is derived, whose purpose is suitability for model-based control design and subsequent trajectory generation. Hence, the driving torque model must be real-time capable. This is realized by an iterative algorithm calculating the bellows’ torque transmission by assuming a two-dimensional no-slip membrane contact. Nonlinear strain behaviours and hysteresis effects of the bellows are considered by the Ogden material model. The driving torque model’s performance is validated experimentally by measuring the external torques of the bellows.

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.

Ogden material calibration via magnetic resonance cartography, parameter sensitivity and variational system identification.

Contemporary material characterization techniques that leverage deformation fields and the weak form of the equilibrium equations face challenges in the numerical solution procedure of the inverse characterization problem. As material models and descriptions differ, so too must the approaches for identifying parameters and their corresponding mechanisms. The widely used Ogden material model can be comprised of a chosen number of terms of the same mathematical form, which presents challenges of parsimonious representation, interpretability and stability. Robust techniques for system identification of any material model are important to assess and improve experimental design, in addition to their centrality to forward computations. Using fully three-dimensional displacement fields acquired in silicone elastomers with our recently developed magnetic resonance cartography (MR-u) technique on the order of greater than [Formula: see text], we leverage partial differential equation-constrained optimization as the basis of variational system identification of our material parameters. We incorporate the statistical F-test to maintain parsimony of representation. Using a new, local deformation decomposition locally into mixtures of biaxial and uniaxial tensile states, we evaluate experiments based on an analytical sensitivity metric and discuss the implications for experimental design. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Hyper-dual number-based numerical differentiation of eigensystems

Considering application to nonlinear material models, this study proposes a novel numerical method for computing arbitrary-order derivatives of eigenvalues and eigenvectors (eigensystems) for second-order tensors (matrices) based on hyper-dual numbers (HDNs). This study introduces two numerical differentiation approaches for eigensystems: one obtains derivatives of both eigenvalues and eigenvectors by inductively solving linear equations derived from an HDN formulation of eigensystem, and the other specializes in solving derivatives of a function represented by eigenvalues, such as the Helmholtz free energy function in the Ogden model, a representative example of the eigenvalue-based hyperelastic material models. Because these approaches differ considerably in applicability and computational cost, this study proposes a numerical differentiation method that uses both complementarily. The proposed method can accurately compute higher-order derivatives even if the second-order tensors have multiple solutions for eigenvalues. Moreover, applying the proposed method to an incremental variational formulation (IVF) leads to robust and accurate computations for the Ogden material model. These results imply that IVF facilitates the implementation of several constitutive models in a unified manner, and the proposed method broadens the scope of these models, including eigensystem expressions.

Application of a deepest‐path algorithm to study the objective function landscape during fitting for the Yeoh and Ogden model

AbstractIn this paper we demonstrate the application of the deepest‐path algorithm for better understanding of objective function landscapes resulting from the optimization process. For this purpose, the sensitivities of the Yeoh and Ogden material model parameters are compared for different load cases. This analysis shows a much higher variation of the material parameters for the Ogden model than for the Yeoh model at approximately constant objective function values. The reasons for this may be local minima or shallow gradients in the objective function landscape. Afterwards, the deepest‐path algorithm is performed between selected designs from the sensitivity analysis. It can be seen that the deepest‐path algorithm provides further information about local minima in the objective function landscape, which are not clear identifiable from a sensitivity analysis.

Investigation of design and performance improvements on solid resilient tires through numerical simulation

Solid tires are often utilized to bear excessive loads. Therefore, sidewall cracks and internal heat build-up affect their durability more significantly. It is a challenge to minimize these factors and improve tire performance while maintaining the functionality of the solid tire. Moreover, there are no specific standard guidelines for modifying the solid tire design to achieve this objective. This study proposes several design and performance improvements to the solid resilient tire and investigates the performance of these modified designs using the Finite Element (FE) method under static and dynamic conditions. For these FE simulations, suitable hyperelastic models are obtained using curve fitting combined with three standard error measures. The results show that the Mooney-Rivlin, Ogden, and Yeoh material models show good agreement with experimental data for modelling the base, cushion, and tread layers of the tire, respectively. The developed FE models are validated using experimental data obtained from a leading tire manufacturing company in Sri Lanka. The validated model is used to develop and analyse two distinct tire models which have two different cavity geometries (circular and elliptical) in their cushion layers to minimize heat build-up and sidewall cracks. The tire reinforcements are also rearranged to further improve the performance of the models. The introduction of cavities helps to reduce strain energy dissipation and to increase the dissipation of internal heat by convection. Furthermore, the stress intensity on the side walls is also reduced, thereby minimizing sidewall cracks. Numerical experiments show that the modified designs have a lower strain energy density, lower stress concentration, and less material utilization when compared to the basic tire design.

A novel semi-implicit scheme for elastodynamics and wave propagation in nearly and truly incompressible solids

This paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

The effects of body position on the material properties of soft tissue in the human thigh

Forty percent of patients with a spinal cord injury acquire a pressure ulcer during rehabilitation, and sixty percent of individuals in elderly care facilities have at least one pressure ulcer upon admittance. A commonality between those populations is the increased amount of time they spend in the seated position. The loading on the buttocks and thighs while in the seated position has been cited as a risk factor for pressure ulcer formation, especially for wheelchair users. Finite element models provide a tool with which to evaluate the internal tissue stresses, but they are reliant upon accurate material properties for the soft tissue. Thus the goals of this research were to determine and compare the material properties of the soft tissue in the thigh and buttock regions in the seated, quadruped (a universally accessible position with the knee and hip articulations similar to the seated position), and prone positions. A custom indenter was designed to collect force and deflection data for the buttocks/proximal thigh, middle thigh, and distal thigh regions of twenty able-bodied individuals. The force and deflection data were converted into stress and stretch data, which were used to obtain parameters from an Ogden material model. Our results indicated that the prone position yielded significantly stiffer tissue properties than in the seated and quadruped positions for both males and females, meaning that position should be taken into account when obtaining material properties that are input into finite element models. Realistic material properties of the soft tissue will lead to better understanding of tissue injury risk.

A virtual element method for isotropic hyperelasticity

This work considers the application of a displacement-based virtual element method to plane hyperelasticity problems with a novel approach to the selection of stabilization parameters. The method is applied to a range of numerical examples and well known strain energy functions, including neo-Hookean, Mooney-Rivlin and Ogden material models. For each of the strain energy functions the performance of the method under varying degrees of compressibility, including near-incompressibility, is investigated. Through these examples the convergence behaviour of the virtual element method is demonstrated. Furthermore, the method is found to be robust and locking free for a variety of element geometries, including elements with a high degree of concavity.

Inverse finite element characterization of the human thigh soft tissue in the seated position.

Pressure ulcers are localized damage to the skin and underlying tissues caused by sitting or lying in one position for a long time. Stresses within the soft tissue of the thigh and buttocks area play a crucial role in the initiating mechanism of these wounds. Therefore, it is crucial to develop reliable finite element models to evaluate the stresses caused by physiological loadings. In this study, we compared how the choice of material model and modeling area dimension affect prediction accuracy of a model of the thigh. We showed that the first-order Ogden and Fung orthotropic material models could approximate the mechanical behavior of soft tissue significantly better than neo-Hookean and Mooney-Rivlin. We also showed that, significant error results from using a semi-3D model versus a 3D model. We then developed full 3D models for 20 participants employing Ogden and Fung material models and compared the estimated material parameters between different sexes and locations along the thigh. We showed that males tissues are less deformable overall when compared to females and the material parameters are highly dependent on location, with tissues getting softer moving distally for both men and women.

Biomechanical Restoration Potential of Pentagalloyl Glucose after Arterial Extracellular Matrix Degeneration.

The objective of this study was to quantify pentagalloyl glucose (PGG) mediated biomechanical restoration of degenerated extracellular matrix (ECM). Planar biaxial tensile testing was performed for native (N), enzyme-treated (collagenase and elastase) (E), and PGG (P) treated porcine abdominal aorta specimens (n = 6 per group). An Ogden material model was fitted to the stress–strain data and finite element computational analyses of simulated native aorta and aneurysmal abdominal aorta were performed. The maximum tensile stress of the N group was higher than that in both E and P groups for both circumferential (43.78 ± 14.18 kPa vs. 10.03 ± 2.68 kPa vs. 13.85 ± 3.02 kPa; p = 0.0226) and longitudinal directions (33.89 ± 8.98 kPa vs. 9.04 ± 2.68 kPa vs. 14.69 ± 5.88 kPa; p = 0.0441). Tensile moduli in the circumferential direction was found to be in descending order as N > P > E (195.6 ± 58.72 kPa > 81.8 ± 22.76 kPa > 46.51 ± 15.04 kPa; p = 0.0314), whereas no significant differences were found in the longitudinal direction (p = 0.1607). PGG binds to the hydrophobic core of arterial tissues and the crosslinking of ECM fibers is one of the possible explanations for the recovery of biomechanical properties observed in this study. PGG is a beneficial polyphenol that can be potentially translated to clinical practice for preventing rupture of the aneurysmal arterial wall.

Persistent occiput posterior position and stress distribution in levator ani muscle during vaginal delivery computed by a finite element model

Introduction and hypothesisObjective of this study was to develop an MRI-based finite element model and simulate a childbirth considering the fetal head position in a persistent occiput posterior position.MethodsThe model involves the pelvis, fetal head and soft tissues including the levator ani and obturator muscles simulated by the hyperelastic nonlinear Ogden material model. The uniaxial test was measured using pig samples of the levator to determine the material constants. Vaginal deliveries considering two positions of the fetal head were simulated: persistent occiput posterior position and uncomplicated occiput anterior position. The von Mises stress distribution was analyzed.ResultsThe material constants of the hyperelastic Ogden model were measured for the samples of pig levator ani. The mean values of Ogden parameters were calculated as: μ1 = 8.2 ± 8.9 GPa; μ2 = 21.6 ± 17.3 GPa; α1 = 0.1803 ± 0.1299; α2 = 15.112 ± 3.1704. The results show the significant increase of the von Mises stress in the levator muscle for the case of a persistent occiput posterior position. For the optimal head position, the maximum stress was found in the anteromedial levator portion at station +8 (mean: 44.53 MPa). For the persistent occiput posterior position, the maximum was detected in the distal posteromedial levator portion at station +6 (mean: 120.28 MPa).ConclusionsThe fetal head position during vaginal delivery significantly affects the stress distribution in the levator muscle. Considering the persistent occiput posterior position, the stress increases evenly 3.6 times compared with the optimal head position.

The Effect of Pentagalloyl Glucose on the Wall Mechanics and Inflammatory Activity of Rat Abdominal Aortic Aneurysms.

An abdominal aortic aneurysm (AAA) is a permanent localized expansion of the abdominal aorta with mortality rate of up to 90% after rupture. AAA growth is a process of vascular degeneration accompanied by a reduction in wall strength and an increase in inflammatory activity. It is unclear whether this process can be intervened to attenuate AAA growth, and hence, it is of great clinical interest to develop a technique that can stabilize the AAA. The objective of this work is to develop a protocol for future studies to evaluate the effects of drug-based therapies on the mechanics and inflammation in rodent models of AAA. The scope of the study is limited to the use of pentagalloyl glucose (PGG) for aneurysm treatment in the calcium chloride rat AAA model. Peak wall stress (PWS) and matrix metalloproteinase (MMP) activity, which are the biomechanical and biological markers of AAA growth and rupture, were evaluated over 4 weeks in untreated and treated (with PGG) groups. The AAA specimens were mechanically characterized by planar biaxial tensile testing and the data fitted to a five-parameter nonlinear, hyperelastic, anisotropic Holzapfel–Gasser–Ogden (HGO) material model, which was used to perform finite element analysis (FEA) to evaluate PWS. Our results demonstrated that there was a reduction in PWS between pre- and post-AAA induction FEA models in the treatment group compared to the untreated group using either animal-specific or average material properties. However, this reduction was not statistically significant. Conversely, there was a statistically significant reduction in MMP-activated fluorescent signal between pre- and post-AAA induction models in the treated group compared to the untreated group. Therefore, the primary contribution of this work is the quantification of the stabilizing effects of PGG using biomechanical and biological markers of AAA, thus indicating that PGG could be part of a new clinical treatment strategy that will require further investigation.

Optimal tread design for agricultural lug tires determined through failure analysis

Agricultural lug tires, commonly used in tractors, must provide safe and stable support for the body of the vehicle and bear any additional load while effectively traversing rough, poor-quality ground surfaces. Many agricultural lug tires fail unexpectedly. In this study, we optimised and validated a tread design for agricultural lug tires intended to increase their durability using failure analysis. Specifically, we identified tire failure modes using indoor driving tests and failure mode effects analysis. Next, we developed a threedimensional tire model using the Ogden material model and finite element method. Using sensitivity analysis and response surface methodology, we optimised the tread design. Finally, we evaluated the durability of the new design using a tire prototype and drum test equipment. Results indicated that the optimised tread design decreased the tire tread stress by 16% and increased its time until cracking by 38% compared to conventional agricultural lug tires.

Modelling peeling- and pressure-driven propagation of arterial dissection

An arterial dissection is a longitudinal tear in the vessel wall, which can create a false lumen for blood flow and may propagate quickly, leading to death. We employ a computational model for a dissection using the extended finite element method with a cohesive traction-separation law for the tear faces. The arterial wall is described by the anisotropic hyperelastic Holzapfel–Gasser–Ogden material model that accounts for collagen fibres and ground matrix, while the evolution of damage is governed by a linear cohesive traction-separation law. We simulate propagation in both peeling and pressure-loading tests. For peeling tests, we consider strips and discs cut from the arterial wall. Propagation is found to occur preferentially along the material axes with the greatest stiffness, which are determined by the fibre orientation. In the case of pressure-driven propagation, we examine a cylindrical model, with an initial tear in the shape of an arc. Long and shallow dissections lead to buckling of the inner wall between the true lumen and the dissection. The various buckling configurations closely match those seen in clinical CT scans. Our results also indicate that a deeper tear is more likely to propagate.

A new pseudo-energy function to simulate the Mullins effect

Several authors have proposed different parameters to include the softening effect in hyperelastic models; however, for a number of materials, softening parameters could be further improved. This article proposes a new softening parameter to include Mullins effect in hyperelastic material models. The methodology employed can be also used in cases with hysteresis or damage in a hyperelastic material, however this methodology modifies the behavior of the material differently from damage theories. Common hyperelastic constitutive models do not include dissipation effects and so the present work intends to fill this gap. Experimental data for silicone in uniaxial tensile test, equibiaxial, and pure shear tests were modeled in order to calibrate the models. The softening parameters essentially changes the constitutive law from the loading to the unloading path. Therefore, it is still necessary to use a hyperelastic model, and here Ogden and Hoss-Marczak material models were used. The obtained results show good agreement with experimental data even when simulating with a compressible finite element code and it can model isotropic Mullins effect.

A multivariate stochastic material model with correlated material parameters

AbstractThis paper concentrates on the aleatoric uncertainty of hyperelastic material models caused by material parameters which are characterized by random variables. To this end, the Polynomial Chaos Expansion (PCE) is applied to represent the random variables by a series of deterministic coefficients and random polynomials. To determine the corresponding PC coefficients and the polynomials of correlated stochastic material parameters we use the Cholesky decomposition and the Gram‐Schmidt algorithm. Based on experimental results and artificial data parameter identification of an Ogden material model for rubber is applied. As a numerical example we consider a static problem for uniaxial tension of the rectangular plate. This structure is investigated in order to represent the conditions for the experimental setup. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Non-linear finite element model to assess the effect of tendon forces on the foot-ankle complex

A three-dimensional foot finite element model with actual geometry and non-linear behavior of tendons is presented. The model is intended for analysis of the lower limb tendon forces effect in the inner foot structure. The geometry of the model was obtained from computational tomographies and magnetic resonance images. Tendon tissue was characterized with the first order Ogden material model based on experimental data from human foot tendons. Kinetic data was employed to set the load conditions. After model validation, a force sensitivity study of the five major foot extrinsic tendons was conducted to evaluate the function of each tendon. A synergic work of the inversion-eversion tendons was predicted. Pulling from a peroneus or tibialis tendon stressed the antagonist tendons while reducing the stress in the agonist. Similar paired action was predicted for the Achilles tendon with the tibialis anterior. This behavior explains the complex control motion performed by the foot. Furthermore, the stress state at the plantar fascia, the talocrural joint cartilage, the plantar soft tissue and the tendons were estimated in the early and late midstance phase of walking. These estimations will help in the understanding of the functional role of the extrinsic muscle-tendon-units in foot pronation-supination.

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.

Realistic kinetic loading of the jaw system during single chewing cycles: a finite element study.

Although knowledge of short-range kinetic interactions between antagonistic teeth during mastication is of essential importance for ensuring interference-free fixed dental reconstructions, little information is available. In this study, the forces on and displacements of the teeth during kinetic molar biting simulating the power stroke of a chewing cycle were investigated by use of a finite-element model that included all the essential components of the human masticatory system, including an elastic food bolus. We hypothesised that the model can approximate the loading characteristics of the dentition found in previous experimental studies. The simulation was a transient analysis, that is, it considered the dynamic behaviour of the jaw. In particular, the reaction forces on the teeth and joints arose from contact, rather than nodal forces or constraints. To compute displacements of the teeth, the periodontal ligament (PDL) was modelled by use of an Ogden material model calibrated on the basis of results obtained in previous experiments. During the initial holding phase of the power stroke, bite forces were aligned with the roots of the molars until substantial deformation of the bolus occurred. The forces tilted the molars in the bucco-lingual and mesio-distal directions, but as the intrusive force increased the teeth returned to their initial configuration. The Ogden material model used for the PDL enabled accurate prediction of the displacements observed in experimental tests. In conclusion, the comprehensive kinetic finite element model reproduced the kinematic and loading characteristics of previous experimental investigations.