scholarly journals The Eshelby inclusion problem in ageing linear viscoelasticity

2016 ◽  
Vol 97-98 ◽  
pp. 530-542 ◽  
Author(s):  
Jean-François Barthélémy ◽  
Albert Giraud ◽  
Francis Lavergne ◽  
Julien Sanahuja
Keyword(s):  
Linear Viscoelasticity ◽  
Inclusion Problem ◽  
Eshelby Inclusion

scholarly journals The dynamic generalization of the Eshelby inclusion problem and its static limit

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160256 ◽  
Author(s):  
Luqun Ni ◽  
Xanthippi Markenscoff
Keyword(s):  
Integral Form ◽  
Transformation Strain ◽  
The Self ◽  
Eshelby Tensor ◽  
Inclusion Problem ◽  
Static Limit ◽  
Governing System ◽  
Eshelby Inclusion ◽  
Interior Domain ◽  
Eshelby Problem

The dynamic generalization of the celebrated Eshelby inclusion with transformation strain is the (subsonically) self-similarly expanding ellipsoidal inclusion starting from the zero dimension. The solution of the governing system of partial differential equations was obtained recently by Ni & Markenscoff (In press. J. Mech. Phys. Solids ( doi:10.1016/j.jmps.2016.02.025 )) on the basis of the Radon transformation, while here an alternative method is presented. In the self-similarly expanding motion, the Eshelby property of constant constrained strain is valid in the interior domain of the expanding ellipsoid where the particle velocity vanishes (lacuna). The dynamic Eshelby tensor is obtained in integral form. From it, the static Eshelby tensor is obtained by a limiting procedure, as the axes' expansion velocities tend to zero and time to infinity, while their product is equal to the length of the static axis. This makes the Eshelby problem the limit of its dynamic generalization.

Viscoelastic Eshelby Analysis of Cardiomyocyte Contraction in the Cell-in-Gel System

2018 ◽  
Author(s):  
Mohammad Kazemi-Lari ◽  
John Shaw ◽  
Alan Wineman ◽  
Rafael Shimkunas ◽  
Leighton Izu ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Mechanical Response ◽  
Time Lag ◽  
Cellular Level ◽  
Cell Stiffness ◽  
Inclusion Problem ◽  
Viscoelastic Matrix ◽  
Eshelby Inclusion ◽  
The Matrix ◽  
Transduction Mechanisms

We present a mathematical model to guide and interpret ongoing Cell-in-Gel experiments, where isolated cardiac myocytes are embedded in a constraining viscoelastic hydrogel, to study mechano-chemo-transduction mechanisms at the single cell level. A recently developed mathematical model, based on the elastic Eshelby inclusion problem, is here extended to account for viscoelasticity of the inclusion (cell) and the matrix (gel). This provides a tool to calculate time-dependent 3D stress and strain fields of a single myocyte contracting periodically inside a viscoelastic matrix, which is used to explore the sensitivity of the cell’s mechanical response to constitutive properties and geometry. A parametric study indicates that increased gel crosslink concentration significantly alters the strain and stress fields inside the cell and creates an increased time-lag in the mechanical response of the cell during contraction. It is also found that autoregulation at the cellular level in response to afterload, potentially in the form of increased cell stiffness, has a strong influence on cell contraction.

Effective Elastic Moduli of Spherical Particle Reinforced Composites Containing Imperfect Interfaces

2011 ◽  
Vol 21 (1) ◽  
pp. 97-127 ◽  
Author(s):  
K. Yanase ◽  
J. W. Ju
Keyword(s):  
Spherical Particle ◽  
Elastic Moduli ◽  
Elastic Stiffness ◽  
Inclusion Problem ◽  
Effective Elastic Moduli ◽  
Reinforced Composites ◽  
Approximate Methods ◽  
Imperfect Interfaces ◽  
Eshelby Inclusion ◽  
Particle Reinforced Composites

The effective elastic moduli of composite materials are investigated in the presence of imperfect interfaces between the inclusions and the matrix. The primary focus is on the spherical particle reinforced composites. By admitting the displacement jumps at the particle–matrix interface, the modified Eshelby inclusion problem is studied anew. To derive the modified Eshelby tensor, three approximate methods are presented and compared by emphasizing the existence of a unique solution and computational efficiency. Subsequently, the effective elastic stiffness tensor of the composite is formulated based on the proposed micromechanical framework and homogenization. Specifically, by incorporating imperfect interface, the modified versions of the Mori–Tanaka method, the self-consistent method, and the differential scheme are presented. By comparing these three methods, the effects of interfacial sliding and separation on the degradation (damage) of the effective elastic moduli of composites are analyzed and assessed. Finally, a critical aspect of the presented formulations is specifically addressed.

scholarly journals Solutions to the Eshelby conjectures

2007 ◽  
Vol 464 (2091) ◽  
pp. 573-594 ◽  
Author(s):  
L.P Liu
Keyword(s):  
Material Properties ◽  
Potential Problem ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Newtonian Potential ◽  
Inclusion Problem ◽  
The Finite Element Method ◽  
Multiply Connected ◽  
Eshelby Inclusion ◽  
Lipschitz Boundaries

We present solutions to the Eshelby conjectures based on a variational inequality. We first discuss the meanings of Eshelby's original statement. By Fourier analysis, we establish the connection between the homogeneous Eshelby inclusion problem and the classic Newtonian potential problem. We then proceed to the solutions of the Eshelby conjectures. Under some hypothesis on the material properties and restricted to connected inclusions with Lipschitz boundaries, we show that one version of the Eshelby conjectures is valid in all dimensions and the other version is valid in two dimensions. We also show the existence of multiply connected inclusions in all dimensions and the existence of non-ellipsoidal connected inclusions in three and higher dimensions such that, in physical terms and in the context of elasticity, some uniform eigenstress of the inclusion induces uniform strain on the inclusion. We numerically calculate these special inclusions based on the finite-element method.

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