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Chelstrom BP, Chawla D, Henak CR. Failure in articular cartilage: Finite element predictions of stress, strain, and pressure under micro-indentation induced fracture. J Mech Behav Biomed Mater 2024; 150:106300. [PMID: 38104488 DOI: 10.1016/j.jmbbm.2023.106300] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2023] [Revised: 10/31/2023] [Accepted: 12/02/2023] [Indexed: 12/19/2023]
Abstract
Articular cartilage is found at the distal end of long bones and is responsible for assisting in joint articulation. While articular cartilage has remarkable resistance to failure, once initially damaged, degeneration is nearly irreversible. Thus, understanding damage initiation is important. There are a few proposed mechanisms for articular cartilage failure initiation: (A) a single collagen fibril stress-based regime; (B) a rate-dependent regime captured by brittle failure at slow displacement rates (SDR) and ductile failure at fast displacement rates (FDR); and (C) a rate-dependent regime where failure is governed by pressurization fragmentation at SDR and governed by strain at FDR. The objective of this study was to use finite element (FE) models to provide evidence to support or refute these proposed failure mechanisms. Models were developed of microfracture experiments that investigated osmolarity (hypo-osmolar, normal osmolarity, and hyper-osmolar) and displacement rate (FDR and SDR) effects. Cartilage was modeled with a neo-Hookean ground matrix, strain-dependent permeability, nonlinear fibril reinforcement with viscoelastic fibril terms, and Donnan equilibrium swelling. Total stress, solid matrix stress, Lagrange strain, and fluid pressure were determined under the indenter tip at the moment of microfracture. Results indicated significant rate dependence across multiple outputs, which does not support (A) a single failure regime. Larger solid and fluid pressures at FDR than SDR did not support (C) a rate-dependent regime split by pressurization at SDR and strain at FDR. Consistent solid shear stresses at SDR and consistent third principal solid stresses at FDR support (B) the ductile-brittle failure regime. These findings help to shed light on the underlying mechanisms of articular cartilage failure, which have implications for the development of osteoarthritis.
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Affiliation(s)
- Brandon P Chelstrom
- Department of Biomedical Engineering, University of Wisconsin-Madison, Madison, WI, USA
| | - Dipul Chawla
- Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI, USA
| | - Corinne R Henak
- Department of Biomedical Engineering, University of Wisconsin-Madison, Madison, WI, USA; Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI, USA; Department of Orthopedics and Rehabilitation, University of Wisconsin-Madison, Madison, WI, USA.
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Chakraborti S, Chakraborty T, Das A, Dandekar R, Pradhan P. Transport and fluctuations in mass aggregation processes: Mobility-driven clustering. Phys Rev E 2021; 103:042133. [PMID: 34005942 DOI: 10.1103/physreve.103.042133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2020] [Accepted: 03/31/2021] [Indexed: 11/07/2022]
Abstract
We calculate the bulk-diffusion coefficient and the conductivity in nonequilibrium conserved-mass aggregation processes on a ring. These processes involve chipping and fragmentation of masses, which diffuse on a lattice and aggregate with their neighboring masses on contact, and, under certain conditions, they exhibit a condensation transition. We find that, even in the absence of microscopic time reversibility, the systems satisfy an Einstein relation, which connects the ratio of the conductivity and the bulk-diffusion coefficient to mass fluctuation. Interestingly, when aggregation dominates over chipping, the conductivity or, equivalently, the mobility of masses, is greatly enhanced. The enhancement in the conductivity, in accordance with the Einstein relation, results in large mass fluctuations and can induce a mobility-driven clustering in the systems. Indeed, in a certain parameter regime, we show that the conductivity, along with the mass fluctuation, diverges beyond a critical density, thus characterizing the previously observed nonequilibrium condensation transition [Phys. Rev. Lett. 81, 3691 (1998)10.1103/PhysRevLett.81.3691] in terms of an instability in the conductivity. Notably, the bulk-diffusion coefficient remains finite in all cases. We find our analytic results in quite good agreement with simulations.
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Affiliation(s)
- Subhadip Chakraborti
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India.,International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Tanmoy Chakraborty
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India
| | - Arghya Das
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Rahul Dandekar
- The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600113, India.,Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Punyabrata Pradhan
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India
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Saha J, Kumar J, Heinrich S. On the approximate solutions of fragmentation equations. Proc Math Phys Eng Sci 2018. [DOI: 10.1098/rspa.2017.0541] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A numerical model based on the finite volume scheme is proposed to approximate the binary breakage problems. Initially, it is considered that the particle fragments are characterized by a single property, i.e. particle’s volume. We then investigate the extension of the proposed model for solving breakage problems considering two properties of particles. The efficiency to estimate the different moments with good accuracy and simple extension for multi-variable problems are the key features of the proposed method. Moreover, the mathematical convergence analysis is performed for one-dimensional problems. All mathematical findings and numerical results are validated over several test problems. For numerical validation, we propose the extension of Bourgade & Filbet (2008
Math. Comput.
77
, 851–882. (
doi:10.1090/S0025-5718-07-02054-6
)) model for solving two-dimensional pure breakage problems. In this aspect, numerical treatment of the two-dimensional binary breakage models using finite volume methods can be treated to be the first instance in the literature.
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Affiliation(s)
- Jitraj Saha
- Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
| | - Jitendra Kumar
- Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
| | - Stefan Heinrich
- Institute of Solids Process Engineering and Particle Technology, Hamburg University of Technology, Hamburg 21073, Germany
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Das A, Kundu A, Pradhan P. Einstein relation and hydrodynamics of nonequilibrium mass transport processes. Phys Rev E 2017; 95:062128. [PMID: 28709216 DOI: 10.1103/physreve.95.062128] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2017] [Indexed: 06/07/2023]
Abstract
We derive hydrodynamics of paradigmatic conserved-mass transport processes on a ring. The systems, governed by chipping, diffusion, and coalescence of masses, eventually reach a nonequilibrium steady state, having nontrivial correlations, with steady-state measures in most cases not known. In these processes, we analytically calculate two transport coefficients, bulk-diffusion coefficient and conductivity. Remarkably, the two transport coefficients obey an equilibrium-like Einstein relation even when the microscopic dynamics violates detailed balance and systems are far from equilibrium. Moreover, we show, using a macroscopic fluctuation theory, that the probability of large deviation in density, obtained from the above hydrodynamics, is in complete agreement with the same derived earlier by Das et al. [Phys. Rev. E 93, 062135 (2016)2470-004510.1103/PhysRevE.93.062135] using an additivity property.
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Affiliation(s)
- Arghya Das
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, TIFR, Bangalore 560012, India
| | - Punyabrata Pradhan
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India
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Inoue C, Izato YI, Miyake A, Villermaux E. Direct Self-Sustained Fragmentation Cascade of Reactive Droplets. PHYSICAL REVIEW LETTERS 2017; 118:074502. [PMID: 28256875 DOI: 10.1103/physrevlett.118.074502] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2016] [Indexed: 06/06/2023]
Abstract
A traditional hand-held firework generates light streaks similar to branched pine needles, with ever smaller ramifications. These streaks are the trajectories of incandescent reactive liquid droplets bursting from a melted powder. We have uncovered the detailed sequence of events, which involve a chemical reaction with the oxygen of air, thermal decomposition of metastable compounds in the melt, gas bubble nucleation and bursting, liquid ligaments and droplets formation, all occurring in a sequential fashion. We have also evidenced a rare instance in nature of a spontaneous fragmentation process involving a direct cascade from big to smaller droplets. Here, the self-sustained direct cascade is shown to proceed over up to eight generations, with well-defined time and length scales, thus answering a century old question, and enriching, with a new example, the phenomenology of comminution.
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Affiliation(s)
- Chihiro Inoue
- The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
| | - Yu-Ichiro Izato
- Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
| | - Atsumi Miyake
- Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
| | - Emmanuel Villermaux
- Aix-Marseille University, CNRS, Centrale Marseille, IRPHE, 13013 Marseille, France
- Institut Universitaire de France, 75005 Paris, France
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Das A, Chatterjee S, Pradhan P. Spatial correlations, additivity, and fluctuations in conserved-mass transport processes. Phys Rev E 2016; 93:062135. [PMID: 27415236 DOI: 10.1103/physreve.93.062135] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2016] [Indexed: 11/07/2022]
Abstract
We exactly calculate two-point spatial correlation functions in steady state in a broad class of conserved-mass transport processes, which are governed by chipping, diffusion, and coalescence of masses. We find that the spatial correlations are in general short-ranged and, consequently, on a large scale, these transport processes possess a remarkable thermodynamic structure in the steady state. That is, the processes have an equilibrium-like additivity property and, consequently, a fluctuation-response relation, which help us to obtain subsystem mass distributions in the limit of subsystem size large.
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Affiliation(s)
- Arghya Das
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India
| | - Sayani Chatterjee
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India
| | - Punyabrata Pradhan
- Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India
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Vledouts A, Vandenberghe N, Villermaux E. Fragmentation as an aggregation process: the role of defects. Proc Math Phys Eng Sci 2016; 472:20150679. [PMID: 26997900 PMCID: PMC4786045 DOI: 10.1098/rspa.2015.0679] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2015] [Accepted: 12/09/2015] [Indexed: 11/12/2022] Open
Abstract
A cohesive object will eventually break into fragment when experiencing a strong deformation, during an impact for instance. Using necklaces of cohesive magnetized spheres suddenly expanded, we have shown that the fragmentation of this one-dimensional material results from an inverse aggregation cascade (Vledouts et al. 2015 Proc. R. Soc. A 471, 20150678. (doi:10.1098/rspa.2015.0678)). Here, we explore a variant of this process by changing the force law between the attracting spheres, using hydrogel beads linked by capillary bridges. We also investigate the role of (weak) defects in the cohesion strength and the consequences of a distribution of forces between the beads. It is found that fragment do form by a cascade of aggregations, which is interrupted earlier when the force disorder is stronger.
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Affiliation(s)
- A. Vledouts
- Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille 13384, France
| | - N. Vandenberghe
- Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille 13384, France
| | - E. Villermaux
- Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille 13384, France
- Institut Universitaire de France, Paris 75005, France
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