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Residual-stress determination by single-axis holographic interferometry and hole drilling—Part I: Theory

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Abstract

A method is described for the rapid, accurate determination of residual stresses from a holographic interference fringe pattern. The pattern is generated by the displacement field caused by localized relief of residual stresses via the introduction of a small, shallow hole into the surface of a component or test specimen. The theoretical development of the holographic method is summarized. An example is given showing how the method can be applied to a typical experimentally observed fringe pattern to determine principal residual stresses and directions.

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Abbreviations

\(\tilde a,\tilde b,\tilde c\) :

nondimensional coefficients derived from\(\tilde A,\tilde B,\tilde C\)

h :

depth of a blind hole

n :

fringe order

n 0 :

fringe order in a fringe pattern that does not include out-of-plane displacements

n (1),n (1′) :

fringe orders at diametrically opposite points around a hole

n (i),n (i′) :

fringe orders at diametrically opposite points around a hole

r :

radial coordinate

r 0 :

radius of a blind hole

t :

specimen thickness

u x ,u y ,u z :

displacement components in a Cartesian coordinate system

u r ,u θ ,u z :

displacement components in a cylindrical coordinate system

z vp :

z-coordinate of the observer's position

\(\tilde A,\tilde B,\tilde C\) :

coefficients in the expressions for the displacement field [eq (2)]

C ij :

elements of the matrix of coefficients in eqs (19) and (21)

D :

diameter of a blind hole

E :

Young's modulus

K x ,K y ,K z :

Cartesian components of the sensitivity vector

K 0 x ,K 0 y ,K 0 z :

Cartesian components of the sensitivity vector when the observer is at infinity

\(\bar e_x ,\bar e_y ,\bar e_z \) :

unit vectors in a Cartesian coordinate system

\(\bar k_1 \) :

propagation vector in the illumination direction

\(\bar k_2 \) :

propagation vector in the viewing direction

ū:

displacement vector

\(\bar K\) :

sensitivity vector

\(\bar K^0 \) :

sensitivity vector when the observer is at infinity

α:

angle defined by tan−1(r/z vp )

β:

orientation of the principal axes

\(\gamma _1 \) :

grazing angle of illumination

ζ:

inclination of the illumination direction

θ:

circumferential coordinate in a cylindrical coordinate system

λ:

wavelength of the illumination source

ν:

Poisson's ratio

ρ:

ratio of hole radius to radial coordinate (r 0 /r)

\(\sigma _{xx} ,\sigma _{yy} ,\tau _{xy} \) :

components of the stress tensor (Cartesian coordinate system)

\(\sigma _1 ,\sigma _2 \) :

principal stresses

ϕ:

phase shift

ϕ(x, y), ϕ(r, θ):

fringe function

\(\Phi _0 \) :

net phase shift of the object beam

\(\Phi _R \) :

net phase shift of the reference beam

ΔΦ:

net phase shift change

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Authors and Affiliations

  1. Mechanical Engineering Department, Stanford University, MIC 4021, 94305-4021, Stanford, CA

    A. Makino (Research Assistant) & D. Nelson (Associate Professor)

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  1. A. Makino
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  2. D. Nelson
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Makino, A., Nelson, D. Residual-stress determination by single-axis holographic interferometry and hole drilling—Part I: Theory. Experimental Mechanics 34, 66–78 (1994). https://doi.org/10.1007/BF02328443

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