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Continuum mechanics

About: Continuum mechanics is a research topic. Over the lifetime, 5042 publications have been published within this topic receiving 181027 citations.


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TL;DR: In this paper, the authors have studied the growth of thin hyperelastic samples and showed that the corresponding equilibrium equations are of the Foppl-von Karman type where growth acts as a source of mean and Gaussian curvatures.
Abstract: The shape of plants and other living organisms is a crucial element of their biological functioning. Morphogenesis is the result of complex growth processes involving biological, chemical and physical factors at different temporal and spatial scales. This study aims at describing stresses and strains induced by the production and reorganization of the material. The mechanical properties of soft tissues are modeled within the framework of continuum mechanics in finite elasticity. The kinematical description is based on the multiplicative decomposition of the deformation gradient tensor into an elastic and a growth term. Using this formalism, the authors have studied the growth of thin hyperelastic samples. Under appropriate assumptions, the dimensionality of the problem can be reduced, and the behavior of the plate is described by a two-dimensional surface. The results of this theory demonstrate that the corresponding equilibrium equations are of the Foppl–von Karman type where growth acts as a source of mean and Gaussian curvatures. Finally, the cockling of paper and the rippling of a grass blade are considered as two examples of growth-induced pattern formation.

147 citations

BookDOI
01 Jan 2013

146 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a general framework for fractional calculus in continuum mechanics by defining the laws of motion and the stresses using fractional derivatives, and apply this framework to two one-dimensional model problems: the deformation of an infinite bar subjected to a self-equilibrated load distribution, and the propagation of longitudinal waves in a thin finite bar.
Abstract: Although there has been renewed interest in the use of fractional models in many application areas, in reality fractional analysis has a long and distinguished history and can be traced back to the likes of Leibniz (Letter to L’Hospital, 1695), Liouville (J Ec Polytech 13:71, 1832), and Riemann (Gesammelte Werke, p 62, 1876) Recent publications (Podlubny in Math Sci Eng 198, 1999; Sabatier et al in Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer, Berlin, 2007; Das in Functional fractional calculus for system identification and controls, Springer, Berlin, 2007) demonstrate that fractional derivative models have found widespread applications in science and engineering Late fundamental considerations have led to the introduction of fractional calculus in continuum mechanics in an attempt to develop non-local constitutive relations (Lazopoulos in Mech Res Commun 33:753–757, 2006) Attempts have also been made to model microscopic forces using fractional derivatives (Vazquez in Nonlinear waves: classical and quantum aspects, pp 129–133, 2004) Our approach in this paper differs from previous theoretical work, in that we develop a general framework directly from the classical continuum mechanics, by defining the laws of motion and the stresses using fractional derivatives The timeliness and relevance of this work is justified by the surge in interest in applications of fractional order models to biological, physical and economic systems The aim of the present paper is to lay the foundations for a new non-local model of continuum mechanics based on fractional order derivatives which we will refer to as the fractional model of continuum mechanics Following the theoretical development, we apply this framework to two one-dimensional model problems: the deformation of an infinite bar subjected to a self-equilibrated load distribution, and the propagation of longitudinal waves in a thin finite bar

146 citations

Journal ArticleDOI
TL;DR: The crenated, echinocytic shapes of human red blood cells are model and it is shown how they may arise from a competition between the bending energy of the plasma membrane and the stretching/shear elastic energies of the membrane skeleton.
Abstract: We study the shapes of human red blood cells using continuum mechanics. In particular, we model the crenated, echinocytic shapes and show how they may arise from a competition between the bending energy of the plasma membrane and the stretching/shear elastic energies of the membrane skeleton. In contrast to earlier work, we calculate spicule shapes exactly by solving the equations of continuum mechanics subject to appropriate boundary conditions. A simple scaling analysis of this competition reveals an elastic length which sets the length scale for the spicules and is, thus, related to the number of spicules experimentally observed on the fully developed echinocyte.

146 citations

Journal ArticleDOI
TL;DR: In this article, a unified framework of balance laws and thermodynamically-consistent constitutive equations is proposed for Cahn-Hilliard-type species diffusion with large elastic deformations of a body.
Abstract: We formulate a unified framework of balance laws and thermodynamically-consistent constitutive equations which couple Cahn–Hilliard-type species diffusion with large elastic deformations of a body. The traditional Cahn–Hilliard theory, which is based on the species concentration c and its spatial gradient ∇ c , leads to a partial differential equation for the concentration which involves fourth-order spatial derivatives in c; this necessitates use of basis functions in finite-element solution procedures that are piecewise smooth and globally C 1 - continuous . In order to use standard C 0 - continuous finite-elements to implement our phase-field model, we use a split-method to reduce the fourth-order equation into two second-order partial differential equations (pdes). These two pdes, when taken together with the pde representing the balance of forces, represent the three governing pdes for chemo-mechanically coupled problems. These are amenable to finite-element solution methods which employ standard C 0 - continuous finite-element basis functions. We have numerically implemented our theory by writing a user-element subroutine for the widely used finite-element program Abaqus/Standard. We use this numerically implemented theory to first study the diffusion-only problem of spinodal decomposition in the absence of any mechanical deformation. Next, we use our fully coupled theory and numerical-implementation to study the combined effects of diffusion and stress on the lithiation of a representative spheroidal-shaped particle of a phase-separating electrode material.

146 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202363
2022136
2021150
2020176
2019181
2018185