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Determining reliable edge isopachic data from interior thermoelastic measurements

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Abstract

The maximum stresses in a plane-stressed component typically occur on the boundary. However, it is generally difficult to obtain reliable experimental data at an edge and thermoelastic stress analysis is no exception. The inability to measure reliable edge isopachic stresses has caused many previous thermoelastic stress analyses to be more qualitative than quantitative. This paper develops and implements an effective iterative least-squares method for calculating reliable edge isopachic stresses from measured interior values. The method is based upon the plane-stress isotropic compatibility equation. A regularization scheme is employed to minimize the sensitivity to measurement error and to improve the stability of the algorithm by controlling the rate of convergence. An illustrative example with actual measured thermoelastic data is included. The processes thermoelastically determined results compare well with those obtained using strain gages.

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Abbreviations

l, m :

direction cosines on Γ

n :

number of nodes per element

nel :

number of elements

N i :

shape functions

S :

isopachic stress,S = σx + σy = σ1 + σ2

\(\bar S\) :

approximating field for isopachics

[A], [B]:

coefficient matrices composed of pseudo-stiffness terms

{F}:

load vector resulting from imposing boundary conditions

[k]:

element pseudo-stiffness matrix

[K]:

global pseudo-stiffness matrix

[K′]:

global pseudo-stiffness matrix with applied boundary conditions

[N]:

shape function row vector

[R]:

regularization matrix

{S}:

global vector of nodal isopachic values

{S e}:

vector of element nodal isopachics

{S i}:

vector of interior nodal isopachic values

{Ŝ i}:

vector of experimentally measured interior isopachic values

{S u}, {S k}:

vector of unknown and known isopachic edge values, respectively

[X]:

sensitivity matrix

\(\varepsilon _1 ,\varepsilon _2 \) :

convergence criteria parameters

σx, σy :

principal normal stresses

σx, σy :

normal stresses in coordinate directions

Γ:

boundary of a domain

Ω:

interior of a domain

\(\nabla ^2 \) :

Laplacian operator,\(\nabla ^2 = \partial ^2 0/\partial x^2 0/\partial y^2 \)

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Rauch, B.J., Rowlands, R.E. Determining reliable edge isopachic data from interior thermoelastic measurements. Experimental Mechanics 35, 174–181 (1995). https://doi.org/10.1007/BF02326477

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