Topological Derivative for Multi-Scale Linear Elasticity Models in Three Spatial Dimensions

2013 ◽  
pp. 1-9
Author(s):  
Antonio André Novotny
Keyword(s):  
Linear Elasticity ◽  
Topological Derivative ◽  
Multi Scale ◽  
Spatial Dimensions

Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures

2010 ◽  
Vol 84 (6) ◽  
pp. 733-756 ◽  
Author(s):  
S. Amstutz ◽  
S. M. Giusti ◽  
A. A. Novotny ◽  
E. A. de Souza Neto
Keyword(s):  
Linear Elasticity ◽  
Topological Derivative ◽  
Multi Scale

scholarly journals Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

2021 ◽  
Author(s):  
V. Calisti ◽  
A. Lebée ◽  
A. A. Novotny ◽  
J. Sokolowski
Keyword(s):  
Circular Inclusion ◽  
Elasticity Tensor ◽  
Second Order ◽  
Topological Derivative ◽  
Microstructural Changes ◽  
Topological Derivatives ◽  
Spatial Dimensions ◽  
Elasticity Tensors ◽  
Derivatives Of ◽  
Optimization Of Microstructures

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions

2010 ◽  
Vol 466 (2118) ◽  
pp. 1703-1723 ◽  
Author(s):  
Sebastián M. Giusti ◽  
Antonio A. Novotny ◽  
Eduardo A. de Souza Neto
Keyword(s):  
Elastic Inclusion ◽  
Contrast Ratio ◽  
Analytical Formula ◽  
Elasticity Tensor ◽  
Topological Derivative ◽  
Multi Scale ◽  
Topological Asymptotic Analysis ◽  
Potential Applicability ◽  
Circular Elastic Inclusion ◽  
Macroscopic Response

This paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions. The macroscopic response is assumed to be predicted by a well-established multi-scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element (RVE) of material. The proposed formula—a symmetric fourth-order tensor field over the RVE domain—is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the RVE. In the limits, when the inclusion/matrix phase contrast ratio tends to zero and infinity, the sensitivities to the insertion of a hole and a rigid inclusion, respectively, are rigorously obtained. The derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail. The derived fundamental formula is of interest to many areas of applied and computational mechanics. To illustrate its potential applicability, a simple finite element-based example is presented where the topological derivative information is used to automatically generate a bi-material microstructure to meet pre-specified macroscopic properties.

Computing cell tractions from particle displacements on silicone rubber substrata

1994 ◽  
Vol 52 ◽  
pp. 162-163
Author(s):  
Tim Oliver ◽  
Akira Ishihara ◽  
Ken Jacobsen ◽  
Micah Dembo
Keyword(s):  
Plane Stress ◽  
Linear Elasticity ◽  
Silicone Rubber ◽  
Mathematical Analysis ◽  
Elastic Recovery ◽  
Video Images ◽  
Cell Traction Forces ◽  
Latex Beads ◽  
Cell Traction ◽  
Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

Nanostructure of semiconductor quantum dots in a borosilicate glass matrix by complementary use of HREM and AEM

1990 ◽  
Vol 48 (4) ◽  
pp. 728-729
Author(s):  
M.J. Kim ◽  
L.C. Liu ◽  
S.H. Risbud ◽  
R.W. Carpenter
Keyword(s):  
Quantum Dots ◽  
Borosilicate Glass ◽  
Glass Matrix ◽  
Optical Switching ◽  
Response Times ◽  
Fast Response ◽  
Processing Technique ◽  
Internal Stresses ◽  
Electron Hole ◽  
Spatial Dimensions

When the size of a semiconductor is reduced by an appropriate materials processing technique to a dimension less than about twice the radius of an exciton in the bulk crystal, the band like structure of the semiconductor gives way to discrete molecular orbital electronic states. Clusters of semiconductors in a size regime lower than 2R {where R is the exciton Bohr radius; e.g. 3 nm for CdS and 7.3 nm for CdTe) are called Quantum Dots (QD) because they confine optically excited electron- hole pairs (excitons) in all three spatial dimensions. Structures based on QD are of great interest because of fast response times and non-linearity in optical switching applications.In this paper we report the first HREM analysis of the size and structure of CdTe and CdS QD formed by precipitation from a modified borosilicate glass matrix. The glass melts were quenched by pouring on brass plates, and then annealed to relieve internal stresses. QD precipitate particles were formed during subsequent "striking" heat treatments above the glass crystallization temperature, which was determined by differential thermal analysis.

The cost of mentally representing three versus two spatial dimensions

1997 ◽  
Author(s):  
Immanuel Barshi ◽  
Alice F. Healy
Keyword(s):  
Spatial Dimensions ◽  
The Cost
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