Adjoint-weighted variational formulation for the direct solution of plane stress inverse elasticity problems

A method of numerically approximating the solutions of plane-stress or plane-strain elasticity problems with boundary conditions consisting of concentrated forces or distributed loads is presented herein. The effect of each concentrated force (commonly termed a point load) that acts on the boundary is represented by a Flamant solution. Usually, the combined effect of these Flamant solutions indicates the presence of distributed loadings or ‘residual stresses’ on some portions of the boundary that are not consistent with the actual boundary conditions. The negatives of these ‘residual stresses’ are used as stress boundary conditions in a singular integral method of numerical analysis that is applicable to plane elasticity problems involving distributed loadings on the boundaries. Since the method presented herein involves only stress boundary conditions, the solutions are valid for both plane stress and plane strain. The accuracy of this superposition method is demonstrated by consideration of a circular disc or cylinder subjected to diametrically opposed concentrated forces for which accuracy to within 0.2 per cent of the exact solution is obtained. Parametric analyses of rectangular and elliptical compression members subjected to point loads are presented. Results determined herein are found to compare relatively well with those determined in previous numerical and experimental investigations of specific cases. These results make possible the design and analysis of compression members used to evaluate the tensile fracture strength of brittle materials.

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Hencky Bar-Grid Model for Plane Stress Elasticity Problems

Journal of Engineering Mechanics ◽

10.1061/(asce)em.1943-7889.0001931 ◽

2021 ◽

Vol 147 (5) ◽

pp. 04021021

Author(s):

Y. P. Zhang ◽

C. M. Wang ◽

D. M. Pedroso ◽

H. Zhang

Keyword(s):

Plane Stress ◽

Grid Model ◽

Elasticity Problems

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Direct error in constitutive equation formulation for plane stress inverse elasticity problem

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2016.10.026 ◽

2017 ◽

Vol 314 ◽

pp. 3-18 ◽

Cited By ~ 6

Author(s):

Olalekan A. Babaniyi ◽

Assad A. Oberai ◽

Paul E. Barbone

Keyword(s):

Constitutive Equation ◽

Plane Stress ◽

Elasticity Problem ◽

Equation Formulation ◽

Inverse Elasticity

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The use of a reflection in the direct solution of plane elasticity problems on multiply-connected domains

Quarterly of Applied Mathematics ◽

10.1090/qam/99923 ◽

1966 ◽

Vol 24 (2) ◽

pp. 127-132

Author(s):

Paul Rausch

Keyword(s):

Plane Elasticity ◽

Multiply Connected Domains ◽

Direct Solution ◽

Multiply Connected ◽

Elasticity Problems

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Computing cell tractions from particle displacements on silicone rubber substrata

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100168542 ◽

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).

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