A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling

Madeo, Angela; George, Daniel; Lekszycki, Tomasz; Nierenberger, Mathieu; Rémond, Yves

doi:10.1016/j.crme.2012.05.003

Cited by 96 publications

(69 citation statements)

References 73 publications

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“…[8,24,25]) and they often use rather subtle material models, including second gradient approaches (see e.g. [26,27] and [28] for a recent overview). To simplify our model, however, we simplify the material behaviour to that of an ideal fluid (i.e.…”

Section: Introductionmentioning

confidence: 99%

Tissue growth controlled by geometric boundary conditions: a simple model recapitulating aspects of callus formation and bone healing

Fischer

Zickler

Dunlop

et al. 2015

J. R. Soc. Interface.

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The shape of tissues arises from a subtle interplay between biochemical driving forces, leading to cell growth, division and extracellular matrix formation, and the physical constraints of the surrounding environment, giving rise to mechanical signals for the cells. Despite the inherent complexity of such systems, much can still be learnt by treating tissues that constantly remodel as simple fluids. In this approach, remodelling relaxes all internal stresses except for the pressure which is counterbalanced by the surface stress. Our model is used to investigate how wettable substrates influence the stability of tissue nodules. It turns out for a growing tissue nodule in free space, the model predicts only two states: either the tissue shrinks and disappears, or it keeps growing indefinitely. However, as soon as the tissue wets a substrate, stable equilibrium configurations become possible. Furthermore, by investigating more complex substrate geometries, such as tissue growing at the end of a hollow cylinder, we see features reminiscent of healing processes in long bones, such as the existence of a critical gap size above which healing does not occur. Despite its simplicity, the model may be useful in describing various aspects related to tissue growth, including biofilm formation and cancer metastases.

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Section: Introductionmentioning

confidence: 99%

Tissue growth controlled by geometric boundary conditions: a simple model recapitulating aspects of callus formation and bone healing

Fischer

Zickler

Dunlop

et al. 2015

J. R. Soc. Interface.

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“…Then a quick list of possible extensions and applications are considered. For an exhaustive treatment of presented results the interested reader is referred to [33][34][35][40][41][42].…”

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confidence: 99%

Second Gradient and Generalized Continua. A workshop held on 12–16 March 2012 in Cisterna di Latina

Dell’Isola

Forest

2014

Z Angew Math Mech

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This editorial intends to complement the scientific content of the papers which follow in this special issue. The workshop "Second Gradient and Generalized Continua" held on 12-16 March 2012 at Palazzo Caetani in Cisterna di Latina, Italy was organized by Francesco dell'Isola and Samuel Forest. Very interesting, exciting and creative discussions started nearly immediately during the seminars and the round tables: the spirit of all participants was very productive and open minded and nobody felt necessary to hide his views, also when they were in contrast with those of other participants, so that a truly frank and direct scientific discussion was possible. The video-recorded seminars are available at the website http://memocs.univaq.it/?page id=2858, http://memocs.univaq.it/?page id=3071.In the following paragraphs we will reproduce, following the seminar order, only the abstracts of those presentations which did not produce a paper published in the present issue of ZAMM.Francesco dell'Isola: "Contact interactions in N-th gradient continua". Cauchy format for continuum mechanics is not general enough to include the description of some important phenomena occurring in a large class of "microscopically" inhomogeneous bodies. This circumstance has been recognized already by E. and F. Cosserat and later by Mindlin, Rivlin, Toupin, and Green. Particularly detailed is the development of second gradient model due to Paul Germain, which is the simplest continuum model which cannot be included in the Cauchy format. Actually Cauchy Continuum model has a wide but limited scope of predictivity: therefore the logical possibility of generalizing it truly corresponds to the need of describing some new and interesting phenomena, which include, but are not limited to, the formation of many different kinds of boundary layers. The class of generalized (N -th gradient) continua which is presented here is constituted by those models in which deformation energy may depend on N + 1 gradients of displacement field. The precise definition of N -th gradient continua and the nature of contact interactions which may arise in them can be mathematically formulated and characterized only by using suitable concepts from the theory of distribution and differential geometry, but it is definitively not so difficult to generalize -in the considered context -the Cauchy concept of stress state. The mathematical abstraction required does not seem inappropriate as it actually supplies a useful description of some phenomena of relevance in the mechanics of growing tissues and opens interesting perspectives in the design of artificial materials showing "exotic" mechanical behavior. More details about the presented ideas can be found in [17,18].Pierre Seppecher: "Linear elastic trusses leading to continua with exotic mechanical interactions". When studying the statics of a truss made of a discrete network of nodes joined by linear elastic springs, the kernel of the quadratic potential energy (space of floppy modes) plays an essential role. In particular w...

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“…The link between the atomistic and continuum worlds is of technological interest to new, rapidly growing, fields of engineering such as porous materials, nano-composites and microstructured continuous media [13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Reviewing the roots of continuum formulations in molecular systems. Part III: Stresses, couple stresses, heat fluxes

Davydov

Steinmann

2013

Mathematics and Mechanics of Solids

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This contribution is the third part in a series devoted to the fundamental link between discrete particle systems and continuum descriptions. The basis for such a link is the postulation of the primary continuum fields such as density and kinetic energy in terms of atomistic quantities using space and probability averaging.In this part, solutions to the flux quantities (stress, couple stress, and heat flux), which arise in the balance laws of linear and angular momentum, and energy are discussed based on the Noll's lemma. We show especially that the expression for the stress is not unique. Integrals of all the fluxes over space are derived. It is shown that the integral of both the microscopic Noll-Murdoch and Hardy couple stresses (more precisely their potential part) equates to zero. Space integrals of the Hardy and the Noll-Murdoch Cauchy stress are equal and symmetric even though the local NollMurdoch Cauchy stress is not symmetric. Integral expression for the linear momentum flux and the explicit heat flux are compared to the virial pressure and the Green-Kubo expression for the heat flux, respectively.It is proven that in the case when the Dirac delta distribution is used as kernel for spatial averaging, the Hardy and the Noll-Murdoch solution for all fluxes coincide.The heat fluxes resulting from both the so-called explicit and implicit approaches are obtained and compared for the localized case. We demonstrate that the spatial averaging of the localized heat flux obtained from the implicit approach does not equate to the expression obtained using a general averaging kernel. In contrast this happens to be true for the linear momentum flux, i.e. the Cauchy stress.

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A second gradient continuum model accounting for some effects of micro-structure on reconstructed bone remodelling

Cited by 96 publications

References 73 publications

Tissue growth controlled by geometric boundary conditions: a simple model recapitulating aspects of callus formation and bone healing

Tissue growth controlled by geometric boundary conditions: a simple model recapitulating aspects of callus formation and bone healing

Second Gradient and Generalized Continua. A workshop held on 12–16 March 2012 in Cisterna di Latina

Reviewing the roots of continuum formulations in molecular systems. Part III: Stresses, couple stresses, heat fluxes

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