1
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Wang X, Schiavone P. Uniformity of anti-plane stresses inside a nonlinear elastic elliptical or parabolic inhomogeneity. Math Mech Solids 2024; 29:121-128. [PMID: 38130974 PMCID: PMC10730357 DOI: 10.1177/10812865231186350] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/07/2023] [Accepted: 06/20/2023] [Indexed: 12/23/2023]
Abstract
We study the anti-plane strain problem associated with a p-Laplacian nonlinear elastic elliptical inhomogeneity embedded in an infinite linear elastic matrix subjected to uniform remote anti-plane stresses. A full-field exact solution is derived using complex variable techniques. It is proved that the stress field inside the elliptical inhomogeneity is nevertheless uniform. The uniformity of stresses is also observed inside a p-Laplacian nonlinear elastic parabolic inhomogeneity.
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Affiliation(s)
- Xu Wang
- School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai, China
| | - Peter Schiavone
- Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada
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2
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Benvenuti E, Reho GA, Palumbo S, Fraldi M. Pre-strains and buckling in mechanosensitivity of contractile cells and focal adhesions: A tensegrity model. J Mech Behav Biomed Mater 2022; 135:105413. [PMID: 36057207 DOI: 10.1016/j.jmbbm.2022.105413] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2022] [Revised: 08/04/2022] [Accepted: 08/05/2022] [Indexed: 10/31/2022]
Abstract
We demonstrate that several key aspects of the contractile activity of a cell interacting with the substrate can be captured by means of a non linear elastic tensegrity mechanical system made of a tensile element in parallel with a buckling-prone component, and exchanging forces with the surroundings through an extracellular matrix-focal adhesion complex. Mechanosensitivity of the focal adhesion plaque is triggered by pre-strain-driven buckling of the system induced either by pre-contraction or pre-polymerization of the constituents. The impact of pre-polymerization on the mechanical force and the implications of using linear and nonlinear elasticity for the focal adhesion plaque are assessed.
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Affiliation(s)
- E Benvenuti
- Engineering Department, University of Ferrara, Italy.
| | - G A Reho
- Engineering Department, University of Ferrara, Italy
| | - S Palumbo
- Department of Structures for Engineering and Architecture, University of Napoli Federico II, Italy
| | - M Fraldi
- Department of Structures for Engineering and Architecture, University of Napoli Federico II, Italy.
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3
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Lucci G, Agosti A, Ciarletta P, Giverso C. Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model. Biomech Model Mechanobiol 2022. [PMID: 35908096 DOI: 10.1007/s10237-022-01602-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2021] [Accepted: 06/17/2022] [Indexed: 11/29/2022]
Abstract
Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth.
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4
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Springhetti R, Rossetto G, Bigoni D. Buckling of Thin-Walled Cylinders from Three Dimensional Nonlinear Elasticity. J Elast 2022; 154:297-323. [PMID: 37920151 PMCID: PMC10618358 DOI: 10.1007/s10659-022-09905-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/28/2022] [Accepted: 05/24/2022] [Indexed: 11/04/2023]
Abstract
The famous bifurcation analysis performed by Flügge on compressed thin-walled cylinders is based on a series of simplifying assumptions, which allow to obtain the bifurcation landscape, together with explicit expressions for limit behaviours: surface instability, wrinkling, and Euler rod buckling. The most severe assumption introduced by Flügge is the use of an incremental constitutive equation, which does not follow from any nonlinear hyperelastic constitutive law. This is a strong limitation for the applicability of the theory, which becomes questionable when is utilized for a material characterized by a different constitutive equation, such as for instance a Mooney-Rivlin material. We re-derive the entire Flügge's formulation, thus obtaining a framework where any constitutive equation fits. The use of two different nonlinear hyperelastic constitutive equations, referred to compressible materials, leads to incremental equations, which reduce to those derived by Flügge under suitable simplifications. His results are confirmed, together with all the limit equations, now rigorously obtained, and his theory is extended. This extension of the theory of buckling of thin shells allows for computationally efficient determination of bifurcation landscapes for nonlinear constitutive laws, which may for instance be used to model biomechanics of arteries, or soft pneumatic robot arms.
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Affiliation(s)
| | | | - Davide Bigoni
- DICAM, University of Trento, via Mesiano 77, Trento, Italy
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5
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Palumbo S, Benvenuti E, Fraldi M. Actomyosin contractility and buckling of microtubules in nucleation, growth and disassembling of focal adhesions. Biomech Model Mechanobiol 2022; 21:1187-1200. [PMID: 35614374 PMCID: PMC9283365 DOI: 10.1007/s10237-022-01584-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2021] [Accepted: 04/08/2022] [Indexed: 11/24/2022]
Abstract
Building up and maintenance of cytoskeletal structure in living cells are force-dependent processes involving a dynamic chain of polymerization and depolymerization events, which are also at the basis of cells’ remodelling and locomotion. All these phenomena develop by establishing cell–matrix interfaces made of protein complexes, known as focal adhesions, which govern mechanosensing and mechanotransduction mechanisms mediated by stress transmission between cell interior and external environment. Within this framework, by starting from a work by Cao et al. (Biophys J 109:1807–1817, 2015), we here investigate the role played by actomyosin contractility of stress fibres in nucleation, growth and disassembling of focal adhesions. In particular, we propose a tensegrity model of an adherent cell incorporating nonlinear elasticity and unstable behaviours, which provides a new kinematical interpretation of cellular contractile forces and describes how stress fibres, microtubules and adhesion plaques interact mechanobiologically. The results confirm some experimental evidences and suggest how the actomyosin contraction level could be exploited by cells to actively control their adhesion, eventually triggering cytoskeleton reconfigurations and migration processes observed in both physiological conditions and diseases.
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Affiliation(s)
- S Palumbo
- Department of Structures for Engineering and Architecture, University of Napoli "Federico II", Napoli, Italy
| | - E Benvenuti
- Department of Engineering, University of Ferrara, Ferrara, Italy
| | - M Fraldi
- Department of Structures for Engineering and Architecture, University of Napoli "Federico II", Napoli, Italy.
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6
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Zhang Y, Wang X, Li K, Zhang Y, Yu X, Wang H, Wu X, Shi Z, Liu L, Zheng W, Cui Z, Xu Y, Li Q. Nanofibrous tissue engineering scaffold with nonlinear elasticity created by controlled curvature and porosity. J Mech Behav Biomed Mater 2021; 126:105039. [PMID: 34923367 DOI: 10.1016/j.jmbbm.2021.105039] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2021] [Revised: 11/29/2021] [Accepted: 12/08/2021] [Indexed: 11/27/2022]
Abstract
Micro-crimped fibers have been widely used in the field of tissue repair to mimic the natural tissue structure and mechanical properties. However, the electrospun nanofibrous membrane is a kind of dense structure, which cannot meet the requirements of mechanical properties and permeability. In this study, we prepared nanofibrous scaffold with controllable porosity and crimpness by sacrificing fiber components and releasing residual stress. The results show that the crimpness of the fiber is positively related to the porosity, and with the increase of porosity, the fiber crimpness increases greatly. Meanwhile, the scaffold modulus was reduced by 86% and the elongation at break doubled, which is similar to natural blood vessels. Moreover, it is found that the porous micro-crimped fiber scaffold promotes the adhesion and diffusion of endothelial cells, and facilitates the rapid endothelialization of the scaffold, which has a great potential for practical application.
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Affiliation(s)
- Yan Zhang
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China; National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China
| | - Xiaofeng Wang
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China; National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China.
| | - Kecheng Li
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China; National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China
| | - Yang Zhang
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China; National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China
| | - Xueke Yu
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China; National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China
| | - Haonan Wang
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China; National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China
| | - Xiaoying Wu
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China
| | - Zhijun Shi
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China
| | - Lin Liu
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China
| | - Wei Zheng
- Engineering and Technology Department, University of Wisconsin-STOUT, Menomonie, WI, USA, 54751
| | - Zhixiang Cui
- Department of Materials Science and Engineering, Fujian University of Technology, Fuzhou, 350118, China
| | - Yiyang Xu
- National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China
| | - Qian Li
- School of Mechanics and Safety Engineering, Zhengzhou University, Zhengzhou, 450001, China; National Center for International Research of Micro-nano Molding Technology, Zhengzhou University, Zhengzhou, 450001, China.
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7
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Huang R, Ogden RW, Penta R. Mathematical Modelling of Residual-Stress Based Volumetric Growth in Soft Matter. J Elast 2021; 145:223-241. [PMID: 34720362 PMCID: PMC8550432 DOI: 10.1007/s10659-021-09834-8] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2020] [Accepted: 04/08/2021] [Indexed: 05/29/2023]
Abstract
Growth in nature is associated with the development of residual stresses and is in general heterogeneous and anisotropic at all scales. Residual stress in an unloaded configuration of a growing material provides direct evidence of the mechanical regulation of heterogeneity and anisotropy of growth. The present study explores a model of stress-mediated growth based on the unloaded configuration that considers either the residual stress or the deformation gradient relative to the unloaded configuration as a growth variable. This makes it possible to analyze stress-mediated growth without the need to invoke the existence of a fictitious stress-free grown configuration. Furthermore, applications based on the proposed theoretical framework relate directly to practical experimental scenarios involving the "opening-angle" in arteries as a measure of residual stress. An initial illustration of the theory is then provided by considering the growth of a spherically symmetric thick-walled shell subjected to the incompressibility constraint.
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Affiliation(s)
- Ruoyu Huang
- Lightweight Manufacturing Centre, University of Strathclyde, Renfrew, PA4 8DJ UK
| | - Raymond W. Ogden
- School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QQ UK
| | - Raimondo Penta
- School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QQ UK
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8
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Itou H, Kovtunenko VA, Rajagopal KR. On an Implicit Model Linear in Both Stress and Strain to Describe the Response of Porous Solids. J Elast 2021; 144:107-118. [PMID: 34720361 PMCID: PMC8550286 DOI: 10.1007/s10659-021-09831-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/10/2021] [Accepted: 04/06/2021] [Indexed: 06/13/2023]
Abstract
We study some mathematical properties of a novel implicit constitutive relation wherein the stress and the linearized strain appear linearly that has been recently put into place to describe elastic response of porous metals as well as materials such as rocks and concrete. In the corresponding mixed variational formulation the displacement, the deviatoric and spherical stress are three independent fields. To treat well-posedness of the quasi-linear elliptic problem, we rely on the one-parameter dependence, regularization of the linear-fractional singularity by thresholding, and applying the Browder-Minty existence theorem for the regularized problem. An analytical solution to the nonlinear problem under constant compression/extension is presented.
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Affiliation(s)
- Hiromichi Itou
- Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku Tokyo, 162-8601 Japan
| | - Victor A. Kovtunenko
- Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstr.36, 8010 Graz, Austria
- Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk, Russia
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9
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Lucci G, Preziosi L. A nonlinear elastic description of cell preferential orientations over a stretched substrate. Biomech Model Mechanobiol 2021; 20:631-649. [PMID: 33449274 PMCID: PMC7979636 DOI: 10.1007/s10237-020-01406-4] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2020] [Accepted: 11/24/2020] [Indexed: 11/27/2022]
Abstract
The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient to reach a well-defined angle between their major axis and the main stretching direction. Recent experimental findings, also supported by a linear elastic model, indicated that the minimization of an elastic energy might drive this reorientation process. Motivated by the fact that a similar behaviour is observed even for high strains, in this paper we address the problem in the framework of finite elasticity, in order to study the presence of nonlinear effects. We find that, for a very large class of constitutive orthotropic models and with very general assumptions, there is a single linear relationship between a parameter describing the biaxial deformation and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\cos ^2\theta _{\mathrm{eq}}$$\end{document}cos2θeq, where \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\theta _{\mathrm{eq}}$$\end{document}θeq is the orientation angle of the cell, with the slope of the line depending on a specific combination of four parameters that characterize the nonlinear constitutive equation. We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy–Green strain tensor. This leads to departures from the linear relationship mentioned above, that are again critically compared with experimental data.
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Affiliation(s)
- Giulio Lucci
- Department of Mathematical Sciences “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
- Department of Mathematics “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Turin, Italy
| | - Luigi Preziosi
- Department of Mathematical Sciences “G.L. Lagrange” Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
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10
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Clayton JD. Modeling lung tissue dynamics and injury under pressure and impact loading. Biomech Model Mechanobiol 2020; 19:2603-2626. [PMID: 32594333 DOI: 10.1007/s10237-020-01358-9] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/24/2020] [Accepted: 06/10/2020] [Indexed: 11/29/2022]
Abstract
A nonlinear viscoelastic model for the lung is implemented and evaluated for high-rate loading. Principal features of the model include a closed-cell approximation of the bulk compressibility accounting for air inside the lung and a damage-injury component by which local trauma is induced by cumulative normalized internal energy and amplified by gradients of energy density. The latter feature is adapted for use in standard numerical (i.e., explicit finite element) simulations in terms of the local rate of strain energy density and the longitudinal wave speed. Injury predictions for direct loading of a block of extracted lung material, rather than the entire thorax, via pressure pulses are in reasonably close agreement with experimental observations for an extracted rabbit lung: a threshold applied pressure exists above which edema is observed experimentally, correlating with low but non-negligible damage in the numerical results. Responses to impact by cylindrical and spherical projectiles are also interrogated. Penetration depths are comparable to those observed experimentally, as is drastically increasing damage with increasing impact velocity. Damage initiates and propagates from the impact surface, with local severity of injury decreasing with distance from the impact zone, in agreement with some empirical evidence. The model predicts more severe local injury, relative to the aforementioned surface pressure loading, than what is observed experimentally. Possible reasons for the discrepancy are analyzed, and adjustments to the model, with caveats, are suggested accordingly.
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Affiliation(s)
- J D Clayton
- Impact Physics, CCDC ARL, Aberdeen, MD, 21005, USA. .,University of Maryland, College Park, MD, 20742, USA.
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11
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Chang B, Reighard C, Flanagan C, Hollister S, Zopf D. Evaluation of human nasal cartilage nonlinear and rate dependent mechanical properties. J Biomech 2019; 100:109549. [PMID: 31926590 DOI: 10.1016/j.jbiomech.2019.109549] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2019] [Revised: 11/25/2019] [Accepted: 11/25/2019] [Indexed: 11/25/2022]
Abstract
Nasal reconstruction frequently requires donor cartilage and tissue, and ideally, donor tissue will closely emulate native nasal cartilage mechanics. Tissue engineering scaffolds, especially 3D printed scaffolds, have been proposed for nasal reconstruction, and the success of these constructs may depend on how well scaffolds reflect native nasal cartilage mechanical properties. Therefore, consistent and comprehensive characterization of native nasal cartilage mechanical properties is a foundation for nasal cartilage tissue engineering and reconstruction in general by providing design targets for reconstructive materials. Our group has previously shown the feasibility of producing scaffolds with porous architecture permitting chondrocyte growth and cartilage production. In this study, we determined the nonlinear and stress relaxation behavior of human nasal cartilage under unconfined compression. We then fit this experimental data to nonlinear elastic, nonlinear viscoelastic and nonlinear biphasic constitutive models. The resulting coefficients will provide design targets for nasal reconstruction and scaffold design as well as outcome measures for assessment of tissue engineered nasal cartilage.
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Affiliation(s)
- Brian Chang
- University of Michigan Medical School, 1500 East Hospital Drive, Ann Arbor, MI 48109, USA
| | - Chelsea Reighard
- University of Michigan Kellogg Eye Center, Department of Ophthalmology and Visual Sciences, 1000 Wall Street, Ann Arbor, MI 48105, USA
| | - Colleen Flanagan
- Department of Biomedical Engineering, University of Michigan, 2200 Bonisteel Blvd., Ann Arbor, MI 48109, USA
| | - Scott Hollister
- Coulter Department of Biomedical Engineering, Georgia Institute of Technology, 313 Ferst Drive, Atlanta, GA 30332, USA.
| | - David Zopf
- Department of Biomedical Engineering, University of Michigan, 2200 Bonisteel Blvd., Ann Arbor, MI 48109, USA; Department of Otolaryngology - Head and Neck Surgery, CS Mott Children's Hospital, 1540 East Hospital Drive, Ann Arbor, MI 48109, USA.
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12
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Gao K, Rougier E, Guyer RA, Lei Z, Johnson PA. Simulation of crack induced nonlinear elasticity using the combined finite-discrete element method. Ultrasonics 2019; 98:51-61. [PMID: 31200274 DOI: 10.1016/j.ultras.2019.06.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2018] [Revised: 03/11/2019] [Accepted: 06/05/2019] [Indexed: 06/09/2023]
Abstract
Numerical simulation of nonlinear elastic wave propagation in solids with cracks is indispensable for decoding the complicated mechanisms associated with the nonlinear ultrasonic techniques in Non-Destructive Testing (NDT). Here, we introduce a two-dimensional implementation of the combined finite-discrete element method (FDEM), which merges the finite element method (FEM) and the discrete element method (DEM), to explicitly simulate the crack induced nonlinear elasticity in solids with both horizontal and inclined cracks. In the FDEM model, the solid is discretized into finite elements to capture the wave propagation in the bulk material, and the finite elements along the two sides of the crack also behave as discrete elements to track the normal and tangential interactions between crack surfaces. The simulation results show that for cracked models, nonlinear elasticity is generated only when the excitation amplitude is large enough to trigger the contact between crack surfaces, and the nonlinear behavior is very sensitive to the crack surface contact. The simulations reveal the influence of normal and tangential contact on the nonlinear elasticity generation. Moreover, the results demonstrate the capabilities of FDEM for decoding the causality of nonlinear elasticity in cracked solid and its potential to assist in Non-Destructive Testing (NDT).
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Affiliation(s)
- Ke Gao
- Geophysics, Los Alamos National Laboratory, NM, USA.
| | | | - Robert A Guyer
- Geophysics, Los Alamos National Laboratory, NM, USA; Department of Physics, University of Nevada at Reno, NV, USA
| | - Zhou Lei
- Geophysics, Los Alamos National Laboratory, NM, USA
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13
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Melchor J, Parnell WJ, Bochud N, Peralta L, Rus G. Damage prediction via nonlinear ultrasound: A micro-mechanical approach. Ultrasonics 2019; 93:145-155. [PMID: 30529738 DOI: 10.1016/j.ultras.2018.10.009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2018] [Revised: 10/02/2018] [Accepted: 10/20/2018] [Indexed: 06/09/2023]
Abstract
Nonlinear constitutive mechanical parameters, predominantly governed by micro-damage, interact with ultrasound to generate harmonics that are not present in the excitation. In principle, this phenomenon therefore permits early stage damage identification if these higher harmonics can be measured. To understand the underlying mechanism of harmonic generation, a nonlinear micro-mechanical approach is proposed here, that relates a distribution of clapping micro-cracks to the measurable macroscopic acoustic nonlinearity by representing the crack as an effective inclusion with Landau type nonlinearity at small strain. The clapping mechanism inside each micro-crack is represented by a Taylor expansion of the stress-strain constitutive law, whereby nonlinear terms arise. The micro-cracks are considered distributed in a macroscopic medium and the effective nonlinearity parameter associated with compression is determined via a nonlinear Mori-Tanaka homogenization theory. Relationships are thus obtained between the measurable acoustic nonlinearity and the Landau-type nonlinearity. The framework developed therefore yields links with nonlinear ultrasound, where the dependency of measurable acoustic nonlinearity is, under certain hypotheses, formally related to the density of micro-cracks and the bulk material properties.
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Affiliation(s)
- J Melchor
- Department of Structural Mechanics, University of Granada, Granada, Spain; IBS Biosanitary Research Institute, Granada, Spain; MNat Scientific Unit of Excellence, University of Granada, Spain.
| | - W J Parnell
- School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
| | - N Bochud
- Institut Langevin, ESPCI Paris, CNRS (UMR 7587), PSL Research University, 75005 Paris, France
| | - L Peralta
- Biomedical Engineering Department, School of Biomedical Engineering and Imaging Sciences, King's College London, London, UK
| | - G Rus
- Department of Structural Mechanics, University of Granada, Granada, Spain; IBS Biosanitary Research Institute, Granada, Spain; MNat Scientific Unit of Excellence, University of Granada, Spain
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14
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Erlich A, Moulton DE, Goriely A. Are Homeostatic States Stable? Dynamical Stability in Morphoelasticity. Bull Math Biol 2019; 81:3219-44. [PMID: 30242633 DOI: 10.1007/s11538-018-0502-7] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2018] [Accepted: 09/03/2018] [Indexed: 01/20/2023]
Abstract
Biological growth is often driven by mechanical cues, such as changes in external pressure or tensile loading. Moreover, it is well known that many living tissues actively maintain a preferred level of mechanical internal stress, called the mechanical homeostasis. The tissue-level feedback mechanism by which changes in the local mechanical stresses affect growth is called a growth law within the theory of morphoelasticity, a theory for understanding the coupling between mechanics and geometry in growing and evolving biological materials. This coupling between growth and mechanics occurs naturally in macroscopic tubular structures, which are common in biology (e.g., arteries, plant stems, airways). We study a continuous tubular system with spatially heterogeneous residual stress via a novel discretization approach which allows us to obtain precise results about the stability of equilibrium states of the homeostasis-driven growing dynamical system. This method allows us to show explicitly that the stability of the homeostatic state depends nontrivially on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasis-driven growth laws.
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15
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Abstract
Thin samples adherent to a rigid substrate are considerably less compliant to indentation when compared to specimens that are not geometrically confined. Analytical corrections to this so-called substrate effect exist for various types of indenters but are not applicable when large deformations are possible, as is the case in biological materials. To overcome this limitation, we construct a nonlinear scaling model characterized by one single exponent, which we explore employing a parametric finite element analysis. The model is based on asymptotes of two length scales in relation to the sample thickness, i.e., indentation depth and radius of the contact area. For small indentation depth, we require agreement with analytical, linear models, whereas for large indentation depth and extensive contact area, we recognize similarity to uniaxial deformation, indicating a divergent force required to indent nonlinear materials. In contrast, we find linear materials not to be influenced by the substrate effect beyond first order, implying that nonlinear effects originating from either the material or geometric confinement are clearly separated only in thin samples. Furthermore, in this regime the scaling model can be derived by following a heuristic argument extending a linear model to large indentation depths. Lastly, in a large indentation setting where the contact is small in comparison with sample thickness, we observe nonlinear effects independent of material type that we attribute to a higher-order influence of geometrical confinement. In this regime, we define a scalar as the ratio of strains along principal axes as obtained by comparison with the case of a point force on a half-space. We find this scalar to be in quantitative agreement with the scaling exponent, indicating an approach to distinguish between nonlinear effects in the scaling model. While we conjecture our findings to be applicable to other flat-ended indenters, we focus on the case of a flat-ended cylinder in normal contact with a thin layer. The analytical solution for small indentation associated with this problem has been given by Hayes et al. (J Biomech 5:541-551, 1972), for which we provide a convenient numerical implementation.
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Affiliation(s)
- Adrian Fessel
- Institut für Biophysik, Universität Bremen, Bremen, Germany.
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16
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Hasan A, Kolahdouz EM, Enquobahrie A, Caranasos TG, Vavalle JP, Griffith BE. Image-based immersed boundary model of the aortic root. Med Eng Phys 2017; 47:72-84. [PMID: 28778565 PMCID: PMC5599309 DOI: 10.1016/j.medengphy.2017.05.007] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2017] [Revised: 05/04/2017] [Accepted: 05/24/2017] [Indexed: 10/19/2022]
Abstract
Each year, approximately 300,000 heart valve repair or replacement procedures are performed worldwide, including approximately 70,000 aortic valve replacement surgeries in the United States alone. Computational platforms for simulating cardiovascular devices such as prosthetic heart valves promise to improve device design and assist in treatment planning, including patient-specific device selection. This paper describes progress in constructing anatomically and physiologically realistic immersed boundary (IB) models of the dynamics of the aortic root and ascending aorta. This work builds on earlier IB models of fluid-structure interaction (FSI) in the aortic root, which previously achieved realistic hemodynamics over multiple cardiac cycles, but which also were limited to simplified aortic geometries and idealized descriptions of the biomechanics of the aortic valve cusps. By contrast, the model described herein uses an anatomical geometry reconstructed from patient-specific computed tomography angiography (CTA) data, and employs a description of the elasticity of the aortic valve leaflets based on a fiber-reinforced constitutive model fit to experimental tensile test data. The resulting model generates physiological pressures in both systole and diastole, and yields realistic cardiac output and stroke volume at physiological Reynolds numbers. Contact between the valve leaflets during diastole is handled automatically by the IB method, yielding a fully competent valve model that supports a physiological diastolic pressure load without regurgitation. Numerical tests show that the model is able to resolve the leaflet biomechanics in diastole and early systole at practical grid spacings. The model is also used to examine differences in the mechanics and fluid dynamics yielded by fresh valve leaflets and glutaraldehyde-fixed leaflets similar to those used in bioprosthetic heart valves. Although there are large differences in the leaflet deformations during diastole, the differences in the open configurations of the valve models are relatively small, and nearly identical hemodynamics are obtained in all cases considered.
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Affiliation(s)
- Ali Hasan
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
| | - Ebrahim M Kolahdouz
- Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA
| | | | - Thomas G Caranasos
- Division of Cardiothoracic Surgery, Department of Surgery, University of North Carolina School of Medicine, Chapel Hill, NC, USA
| | - John P Vavalle
- Division of Cardiology, Department of Medicine, University of North Carolina School of Medicine, Chapel Hill, NC, USA
| | - Boyce E Griffith
- Department of Mathematics and McAllister Heart Institute, University of North Carolina, Chapel Hill, NC, USA.
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17
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Itou H, Kovtunenko VA, Rajagopal KR. Nonlinear elasticity with limiting small strain for cracks subject to non-penetration. Math Mech Solids 2017; 22:1334-1346. [PMID: 29750007 PMCID: PMC5935035 DOI: 10.1177/1081286516632380] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/04/2015] [Accepted: 01/22/2016] [Indexed: 06/08/2023]
Abstract
A major drawback of the study of cracks within the context of the linearized theory of elasticity is the inconsistency that one obtains with regard to the strain at a crack tip, namely it becoming infinite. In this paper we consider the problem within the context of an elastic body that exhibits limiting small strain wherein we are not faced with such an inconsistency. We introduce the concept of a non-smooth viscosity solution which is described by generalized variational inequalities and coincides with the weak solution in the smooth case. The well-posedness is proved by the construction of an approximation problem using elliptic regularization and penalization techniques.
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Affiliation(s)
- Hiromichi Itou
- Department of Mathematics, Tokyo University of Science, Tokyo, Japan
| | - Victor A Kovtunenko
- Victor A Kovtunenko, Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstr.36, 8010 Graz, Austria.
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18
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Aggarwal A. An improved parameter estimation and comparison for soft tissue constitutive models containing an exponential function. Biomech Model Mechanobiol 2017; 16:1309-27. [PMID: 28251368 DOI: 10.1007/s10237-017-0889-3] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2016] [Accepted: 02/10/2017] [Indexed: 01/05/2023]
Abstract
Motivated by the well-known result that stiffness of soft tissue is proportional to the stress, many of the constitutive laws for soft tissues contain an exponential function. In this work, we analyze properties of the exponential function and how it affects the estimation and comparison of elastic parameters for soft tissues. In particular, we find that as a consequence of the exponential function there are lines of high covariance in the elastic parameter space. As a result, one can have widely varying mechanical parameters defining the tissue stiffness but similar effective stress–strain responses. Drawing from elementary algebra, we propose simple changes in the norm and the parameter space, which significantly improve the convergence of parameter estimation and robustness in the presence of noise. More importantly, we demonstrate that these changes improve the conditioning of the problem and provide a more robust solution in the case of heterogeneous material by reducing the chances of getting trapped in a local minima. Based upon the new insight, we also propose a transformed parameter space which will allow for rational parameter comparison and avoid misleading conclusions regarding soft tissue mechanics.
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19
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Lin T, Guyader CL, Dinov I, Thompson P, Toga A, Vese L. Gene Expression Data to Mouse Atlas Registration Using a Nonlinear Elasticity Smoother and Landmark Points Constraints. J Sci Comput 2012; 50:10.1007/s10915-011-9563-6. [PMID: 24273381 PMCID: PMC3838306 DOI: 10.1007/s10915-011-9563-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
This paper proposes a numerical algorithm for image registration using energy minimization and nonlinear elasticity regularization. Application to the registration of gene expression data to a neuroanatomical mouse atlas in two dimensions is shown. We apply a nonlinear elasticity regularization to allow larger and smoother deformations, and further enforce optimality constraints on the landmark points distance for better feature matching. To overcome the difficulty of minimizing the nonlinear elasticity functional due to the nonlinearity in the derivatives of the displacement vector field, we introduce a matrix variable to approximate the Jacobian matrix and solve for the simplified Euler-Lagrange equations. By comparison with image registration using linear regularization, experimental results show that the proposed nonlinear elasticity model also needs fewer numerical corrections such as regridding steps for binary image registration, it renders better ground truth, and produces larger mutual information; most importantly, the landmark points distance and L2 dissimilarity measure between the gene expression data and corresponding mouse atlas are smaller compared with the registration model with biharmonic regularization.
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Affiliation(s)
- Tungyou Lin
- Department of Mathematics, UCLA, Los Angeles, CA, USA
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