Ogden-type Model Research Articles

A visco-hyperelastic model of brain tissue incorporating both tension/compression asymmetry and volume compressibility

The understanding of the mechanical behavior of the brain tissue is essential to prevent the occurrence of potential brain damage, such as traumatic brain injury. Recent experimental results showed that brain tissue exhibits significant tension/compression asymmetry. Due to the migration and diffusion of interstitial fluid, brain tissue also shows a moderate volume compressibility during loading. These mechanical characteristics have a strong impact on the deformation response of brain tissue. In this paper, a visco-hyperelastic constitutive model incorporating both tension/compression asymmetry and volume compressibility is proposed to describe the mechanical behavior of brain tissue under various loading modes. An Ogden-type model with the addition of a viscoelastic part is used to characterize the tension/compression asymmetry as well as the viscoelastic properties. Poisson’s ratio was introduced as a phenomenological index to represent the total volume change as well as the compressibility/recoverability. The mechanical responses of brain tissue under uniaxial tension, unconfined compression, stress relaxation, and cyclic compression were reproduced with a good capture of the tension/compression asymmetry, volume compressibility, significant viscoelastic properties, and cyclic hysteresis behaviors. The good agreement with the experimental data implies that the proposed model has a strong capability to describe the complex mechanical performance of brain tissue under a variety of loading conditions.

Rheological characterization of human brain tissue.

While recent experiments have shaped our understanding of the time-independent, hyperelastic response of human brain tissue, its time-dependent behavior at finite strains and under various loading conditions remains insufficiently understood. In this manuscript, we characterize the rheology of human brain tissue through a family of finite viscoelastic Ogdentype models and identify their parameters for multiple loading modes in four different regions of the brain. We show that even the simplest model of this family, with only one viscoelastic mode and five material parameters, naturally captures the essential features of brain tissue: its characteristic nonlinearity, pre-conditioning, hysteresis, and tension-compression asymmetry. For the first time, we simultaneously identify a single parameter set for shear, compression, tension, shear relaxation, and compression relaxation loading. This parameter set is significant for computational simulations under physiological conditions, where loading is naturally of mixed mode nature. Understanding the rheology of the human brain will help us predict neurosurgical procedures, inform brain injury criteria, and improve the design of protective devices.

An Invariant-Based Ogden-Type Model for Incompressible Isotropic Hyperelastic Materials

The Ogden model for an incompressible isotropic hyperelastic material is versatile enough to match complicated data for rubber-like materials at large deformations. However, the tensorial expression for the Cauchy stress in the Ogden model requires determination of the eigenvalues and eigenvectors of the left Cauchy-Green deformation tensor \(\mathbf{B}\). The objective of this paper is to propose an invariant-based Ogden-type model for isotropic incompressible hyperelastic materials. The strain energy function in this new model depends on classical invariants of \(\mathbf{B}\) and the Cauchy stress tensor can be expressed directly in terms of the tensor \(\mathbf{B}\) without need for its spectral form. Examples show that this new Ogden-type model retains the versatility of the original Ogden model in characterizing material response.

On the behavior of rotating thick-walled cylinders made of hyperelastic materials

This paper focuses on the behavior of rotating thick-walled cylinders made of the rubber-like materials. These materials are characterized by high deformability which their stress-stretch curves are arranged in the range of S-shaped to J-shaped forms. Here, it is considered an Ogden-type model with only integer powers to give a satisfactory correspondence with experimental results over wide ranges of different types of deformation. This type of constitutive model is mathematically simple and can provide the foundation of existence of a closed-form solution for the boundary value problems in the finite deformation elasticity. In this study, as an application, the Ogden-type strain energy density with only integer powers is applied to a rotating thick-walled cylinder made of the hyperelastic materials. It is shown that this constitutive model gives a closed-form solution, not involving the integral form, for the stress distribution through the wall thickness. Also, the results predicted from classic strain energy density models (Mooney-Rivlin, Neo-Hookean and Varga) and the Ogden-type model are compared for (i) a solid circular cylinder under rotation, (ii) a rotating circular cylinder with a concentric circular rigid inclusion, (iii) a rotating cylinder shrink-fitted to a rigid spindle and (iv) an unconstrained cylinder under rotation. These comparisons are done in order to investigate the accuracy of the simplification carried out by applying the well-known classic models instead of the Ogden-type model. It is concluded that this simplification is allowable if the classic models can follow the test data to some acceptable extent. While, the classic models are not capable to analyze an unconstrained rotating cylinder when experiences large deformations. Finally, influence of different parameters such as wall thickness, presence of the axial pre-stretch and form of mechanical behavior or strain-stiffening behavior of the rubber utilized in cylinders examined on stability of an unconstrained rotating cylinder.

On the Hyperelastic Pressurized Thick-Walled Spherical Shells and Cylindrical Tubes Using the Analytical Closed-Form Solutions

This paper focuses on thick-walled spherical shells and cylindrical tubes made of the soft tissues and the rubber-like materials. These materials are characterized by high deformability in which their stress–stretch curves are arranged in the range of S-shaped to J-shaped forms. From the continuum viewpoint, a strain energy density function is postulated for modeling the behavior of these materials. In order to fulfill the main aims of this paper, among all existing energy functions including polynomial, power law, logarithmic and exponential functions, or a linear combination of them, we deduced to evaluate the performance of an Ogden-type model with only integer powers for the mechanical behavior modeling of the S-shaped to J-shaped materials. Most of all, this strain energy function because of its mathematical form can play a constructive role in presentation of the analytical closed-form solutions for the boundary value problems in the field of the finite deformation elasticity. This constitutive model due to the high performance in constitutive modeling and the simplicity of its mathematical form is applied to pressurized thick-walled spherical shells and cylindrical tubes in order to find a closed-form analytical solution for their analysis. Using these analytical solutions, a comprehensive study is done on vanishing circumstance of the snap-through instability that occurs in the inflation of internally pressurized spherical shells and cylindrical tubes. It was observed that the parameters such as shell thickness, the elastic material properties specially the materials with J-shaped mechanical behaviors and the absence and presence of axial forces in cylindrical tubes have significant influence on vanishing of the snap-through instability in the thick-walled pressurized spherical shells and cylindrical tubes.

Implementation of Material Stiffness Coefficients in Finite Element Applications to Rubber

Abstract Two classes of constitutive models are widely used in the current literature to describe the general strain state of rubber under large deformations: the Invariant-based (Rivlin type) and the Principal-value based (Ogden type). However, their predictive capabilities, coupled with the finite element method, will greatly depend on the material coefficients determined from experimental stress-strain curves by nonlinear curve fitting. In addition, special care must be exercised in their numerical implementations, which is particularly true for the Ogden type models, when evaluating the material tangent stiffness coefficients. Several issues in connection with these are discussed and a number of simulations and test comparisons are presented.