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Determining the three individual stress components from measured isochromatic fringes

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Abstract

This paper combines measured isochromatic information, complex stress functions and numerical concepts into a new and effective hybrid method of stress analysis. The technique simultaneously smooths the measured isochromatic data, provides accurate boundary information, and separates the isochromatic information into normal and shear stresses at nonboundary locations. No additional experimental data such as the isoclinics are needed. The technique is illustrated experimentally by application to a tensile plate containing a hole.

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Abbreviations

A kj , Aj, an, bn :

real coefficients

A 1j , A1 :

assumed initial values

C n :

complex coefficientsa n+ibn

D :

principal stress difference (p−q)

j σ :

material stress-fringe value

f, g, h :

functions

F :

isochromatic fringe order

i, j, k, l, n :

indices, as subscripts or superscripts

i :

\(\sqrt { - {\mathbf{ }}1} \)

Im :

imaginary part of a complex number

J :

number of independent coefficientsa n andb n (j≤J)

ln :

natural log

M :

number of input values or observations

N :

number of complex coefficients (n≤N)

p,q :

principal stresses

r, θ:

polar coordinates

R ζ :

region in the ζ plane as mapped fromR in the z plane

R e :

real part of a complex number

R :

region in the z physical plane of a loaded component

R′:

subregion of the regionR

S :

error sum of squares

t :

thickness

U :

Airy stress function

x, y :

Cartesian coordinates

z :

complex numberx+iy

\(\bar z\) :

complex conjugatex−iy

[z]:

derivative matrix

z kij :

derivatives

Δ:

coefficient increments or local adjustment vector

∈:

error

∇4 :

biharmonic differential operator

ξ, η:

coordinates in mapped ζ plane

ζ:

ξ+i η in mapped complex plane

Γ:

traction-free segment of boundary of regionR

Γζ :

mapped curve Γ and is a section of the real axis in the ζ plane

ϱ:

hole radius

σ, τ:

stress

σo :

far-field stress

ϕ, ψ:

complex stress function

ω:

mapping function from ζ plane to z plane

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Author information

Authors and Affiliations

  1. The Goodyear Tire and Rubber Co., P.O. Box 3531, 44309, Akron, OH

    Y. M. Huang (Senior Research Engineer)

  2. Engineering Mechanics Department, University of Wisconsin, 53706, Madison, WI

    C. H. Lin (Research Assistant)

  3. Department of Mechanical Engineering, Auburn University, 36849-3501, Auburn, AL

    J. C. Suhling (Associate Professor)

  4. Engineering Mechanics Department, University of Wisconsin, 1425 Johnson Drive, 53706, Madison, WI

    R. E. Rowlands (Professor)

Authors
  1. Y. M. Huang
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  2. C. H. Lin
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  3. J. C. Suhling
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  4. R. E. Rowlands
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Huang, Y.M., Lin, C.H., Suhling, J.C. et al. Determining the three individual stress components from measured isochromatic fringes. Experimental Mechanics 31, 310–318 (1991). https://doi.org/10.1007/BF02325987

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  • DOI: https://doi.org/10.1007/BF02325987

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