Nonlinear higher-order shell theory for incompressible biological hyperelastic materials
01 Apr 2019-Computer Methods in Applied Mechanics and Engineering (North-Holland)-Vol. 346, pp 841-861
TL;DR: In this paper, a geometrically nonlinear theory for circular cylindrical shells made of incompressible hyperelastic materials is developed, which is higher-order in both shear and thickness deformations.
Abstract: In the present study, a geometrically nonlinear theory for circular cylindrical shells made of incompressible hyperelastic materials is developed. The 9-parameter theory is higher-order in both shear and thickness deformations. In particular, the four parameters describing the thickness deformation are obtained directly from the incompressibility condition. The hyperelastic law selected is a state-of-the-art material model in biomechanics of soft tissues and takes into account the dispersion of collagen fiber directions. Special cases, obtained from this hyperelastic law setting to zero one or some material coefficients, are the Neo-Hookean material and a soft biological material with two families of collagen fibers perfectly aligned. The proposed model is validated through comparison with the exact solution for axisymmetric cylindrical deformation of a thick cylinder. In particular, the shell theory developed herein is capable to describe, with extreme accuracy, even the post-stability problem of a pre-stretched and inflated Neo-Hookean cylinder until the thickness vanishes. Comparison to the solution of higher-order shear deformation theory, which neglects the thickness deformation and recovers the normal strain from the incompressibility condition, is also presented.
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TL;DR: In this article, the authors introduced a rigorous higher-order polynomial in the thickness coordinate to develop a theory with thickness and shear deformations for doubly curved, laminated composite shells.
Abstract: The present study introduces a rigorous higher-order polynomial in the thickness coordinate to develop a theory with thickness and shear deformations for doubly curved, laminated composite shells. In this theory, the nonlinear terms in all the kinematic parameters are kept. By applying the conditions of zero transverse normal and shear stresses at the top and bottom surfaces of shells, a third-order thickness and shear deformation theory with six kinematic parameters is derived. This is a particularly interesting result since the developed theory presents a single additional parameter to describe the thickness deformation with respect to the popular Reddy’s third-order shear deformation theory, which has five parameters. The accuracy of the proposed six-parameter theory is tested for static and dynamic benchmark cases. Results are also very satisfactorily compared to those obtained with a more sophisticated and computationally onerous nine-parameter theory. The considered cases are isotropic and cross-ply laminated circular cylindrical shells under radial forces and pressure, and nonlinear forced vibrations of a cross-ply laminated shell under harmonic radial excitation.
72 citations
TL;DR: A layer-specific hyperelastic and viscoelastic characterization of human descending thoracic aortas was experimentally performed and showed a positive correlation between stiffness and donor age for the three layers of the aorta in both axial and circumferential directions.
Abstract: A layer-specific hyperelastic and viscoelastic characterization of human descending thoracic aortas was experimentally performed Healthy aortas from twelve beating heart donors with an average age of 494 years, were received from Transplant Quebec Axial and circumferential strips were prepared from the specimens They were dissected into intima, media and adventitia layers Measurements of the opening angles were used to identify the circumferential residual stresses Uniaxial tensile tests on axial and circumferential strips, together with the Gasser-Ogden-Holzapfel material model, were used to characterize the hyperelastic behaviour of the three aortic layers for each donor Uniaxial harmonic excitations at different frequency, superimposed to initial stretch values, were used to characterize the viscoelastic behaviour The storage modulus and the loss tangent were obtained for each layer in both directions; comparison to intact aortic wall was also performed The generalized Maxwell model, within the framework of nonlinear viscoelasticity with internal variables, was used to obtain the constitutive material parameters Results showed a positive correlation (p
55 citations
TL;DR: The generalized fractional Maxwell model is applied to identify viscoelastic constitutive equations from layer-specific experimental data obtained by uniaxial harmonic loading of ex-vivo human descending thoracic aortas to identify constitutive parameters.
Abstract: The generalized fractional Maxwell model, formulated for hyperelastic material within the framework of the nonlinear viscoelasticity with internal variables, is applied to identify viscoelastic constitutive equations from layer-specific experimental data obtained by uniaxial harmonic loading of ex-vivo human descending thoracic aortas. The constitutive parameters are identified by using a genetic algorithm for the optimal fitting of the experimental data. The accuracy of the fitted fractional model is compared to the fitted integer order model with the same number of Maxwell elements. The formulation of an original strain energy density function for anisotropic nonlinear viscoelasticity is introduced and constitutive parameters are obtained from the experiments.
49 citations
01 Jan 2022
TL;DR: A detailed survey of the most significant literature on continuum mechanics models of micro-nano-structures can be found in this paper , which can orient researchers in their future studies in this field of research.
Abstract: Recently, the mechanical behavior of micro-/nano-structures has sparked an ongoing debate, which leads to a fundamental question: what steps can be taken to investigate the mechanical characteristics of these structures, and characterize their performance? From the standpoint of the non-classical behavior of materials, size-dependent theories of micro-/nano-structures can be considered to analyze their mechanical behavior. The application of classical theories in the investigation of small-scale structures can lead to inaccurate results. Many studies have been published in the past few years, in which continuum mechanics models have been used to investigate micro-/nano-structures with different geometry such as rods, tubes, beams, plates, and shells. The mechanical behavior of these systems under different loading – resulting in vibration, wave propagation, bending, and buckling phenomena – is the focus of the review covered in this work. The present objective is to provide a detailed survey of the most significant literature on continuum mechanics models of micro-/nano-structures, and thus orient researchers in their future studies in this field of research.
40 citations
TL;DR: A detailed survey of the most significant literature on continuum mechanics models of micro-nano-structures can be found in this article, which can orient researchers in their future studies in this field of research.
Abstract: Recently, the mechanical behavior of micro-/nano-structures has sparked an ongoing debate, which leads to a fundamental question: what steps can be taken to investigate the mechanical characteristics of these structures, and characterize their performance? From the standpoint of the non-classical behavior of materials, size-dependent theories of micro-/nano-structures can be considered to analyze their mechanical behavior. The application of classical theories in the investigation of small-scale structures can lead to inaccurate results. Many studies have been published in the past few years, in which continuum mechanics models have been used to investigate micro-/nano-structures with different geometry such as rods, tubes, beams, plates, and shells. The mechanical behavior of these systems under different loading – resulting in vibration, wave propagation, bending, and buckling phenomena – is the focus of the review covered in this work. The present objective is to provide a detailed survey of the most significant literature on continuum mechanics models of micro-/nano-structures, and thus orient researchers in their future studies in this field of research.
40 citations
References
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TL;DR: A structural continuum framework that is able to represent the dispersion of the collagen fibre orientation is developed and allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls.
Abstract: Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.
1,905 citations
Book•
01 Aug 2014
TL;DR: In this article, a comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells is presented. But the authors do not consider the effect of boundary conditions on the large-amplitude vibrations of circular cylinders.
Abstract: Introduction. 1. Nonlinear theories of elasticity of plates and shells 2. Nonlinear theories of doubly curved shells for conventional and advanced materials 3. Introduction to nonlinear dynamics 4. Vibrations of rectangular plates 5. Vibrations of empty and fluid-filled circular cylindrical 6. Reduced order models: proper orthogonal decomposition and nonlinear normal modes 7. Comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells 8. Effect of boundary conditions on a large-amplitude vibrations of circular cylindrical shells 9. Vibrations of circular cylindrical panels with different boundary conditions 10. Nonlinear vibrations and stability of doubly-curved shallow-shells: isotropic and laminated materials 11. Meshless discretization of plates and shells of complex shapes by using the R-functions 12. Vibrations of circular plates and rotating disks 13. Nonlinear stability of circular cylindrical shells under static and dynamic axial loads 14. Nonlinear stability and vibrations of circular shells conveying flow 15. Nonlinear supersonic flutter of circular cylindrical shells with imperfections.
862 citations
TL;DR: In this article, a three-dimensional constitutive model for biological soft tissues and its finite element implementation for fully incompressible material behavior is presented, along with derivations of the stress and elasticity tensors for a transversely isotropic, hyperelastic material.
Abstract: This paper describes a three-dimensional constitutive model for biological soft tissues and its finite element implementation for fully incompressible material behavior. The necessary continuum mechanics background is presented, along with derivations of the stress and elasticity tensors for a transversely isotropic, hyperelastic material. A particular form of the strain energy for biological soft tissues is motivated and a finite element implementation of this model based on a three-field variational principle (deformation, pressure and dilation) is discussed. Numerical examples are presented that demonstrate the utility and effectiveness of this approach for incompressible, transversely isotropic materials.
734 citations
01 Jan 1994
684 citations
TL;DR: In this article, the effect of thickness stretching in plate/shell structures made by materials which are functionally graded (FGM) in the thickness directions was evaluated by removing or retaining the transverse normal strain in the kinematics assumptions of various refined plate and shell theories.
Abstract: The present work evaluates the effect of thickness stretching in plate/shell structures made by materials which are functionally graded (FGM) in the thickness directions. That is done by removing or retaining the transverse normal strain in the kinematics assumptions of various refined plate/shell theories. Variable plate/shell models are implemented according to Carrera’s Unified Formulation. Plate/shell theories with constant transverse displacement are compared with the corresponding linear to fourth order of expansion in the thickness direction ones. Single-layered and multilayered FGM structures have been analyzed. A large numerical investigation, encompassing various plate/shell geometries as well as various grading rates for FGMs, has been conducted. It is mainly concluded that a refinements of classical theories that include additional in-plane variables could results meaningless unless transverse normal strain effects are taken into account.
373 citations