Abstract
Measuring residual stress in an orthotropic material is a difficult task due to the complex behavior of the material. Recently, two different approaches based on Smith’s simplified real value formulation and the general solution developed by Lekhnitskii have been proposed. Both solutions assume the measurement of the displacement field via interferometric optical methods and estimate stress values through solving an inverse problem. However, the high sensitivity to vibrations of interferometric techniques makes their use difficult outside optical laboratories; standard Digital Image Correlation could be used, but its low sensitivity and relatively high standard deviation of displacements severely affect the reliability of estimates. In this work we propose to integrate the residual stress displacement functions related to orthotropic materials into the shape functions of Digital Image Correlation. This makes it possible overcome most of the problems related to low sensitivity and large standard deviation because a single large patch can be used for the measurement, thus providing an accurate and reliable algoritm for the measurement of residual stress.
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Notes
Note that the stated assumption implies that intensity variation at a point is due only to motion (and that no intensity variation means no motion). This is not always true: a simple counterexample is a spinning sphere with a specular surface: if no texture is present, there will be no correlation between the motion of the surface and the image (which does not change in time). To solve this problem, a random texture must be applied on the surface of the specimen; moreover, the illumination must be isotropic and uniform.
The most commonly used functionals are the Normalized Correlation
$$ \chi^{2}_{CC}(i_{0},j_{0},u,v) \,=\, 1\,-\,\frac{\sum_{k} \sum_{l} I(k,l)\, J(k+u_{x},l+u_{y})}{\sqrt{\sum_{k}\sum_{l} I^{2}(k,l)}\,\sqrt{\sum_{k}\sum_{l} J^{2}(k+u_{x},l+u_{y})}} $$and the Least Square Difference
$$ \chi^{2}_{LSD}(i_{0},j_{0},u_{x},u_{y}) = \sum\limits_{k}\sum\limits_{l} \left[I(k,l) - J(k+u_{x},l+u_{y})\right]^{2} $$where u x and u y are the X and Y displacement components to be measured, k and l range over the rows (columns) of the block and finally I and J are the intensities of the reference and target images (i.e. the images of the specimen before and after motion).
Assuming subsets do not overlap.
It is quite easy to design a material whose behavior cannot be described using Smith’s formulation: if you consider a graphite-epoxy lamina (E 1 = 206840 MPa, E 2 = 5170 MPa, ν 12 = 0.25, G 12 = 2186 MPa) a simple computation shows that κ = 6.283; however, if you take into account the (+45,−45) s graphite-epoxy laminate, the classical lamination theory estimates a completely different set of parameters (E 1 = E 2 = 8402 MPa, ν 12 = 0.922, G 12 = 52438 MPa) and κ = − 0.8417.
Lekhnitskii [33] shows that to have real roots a material must be isotropic. Note that \(\mu _{3} = \overline {\mu _{1}}\) (the complex conjugate of μ 1) and \(\mu _{4} = \overline {\mu _{2}}\).
The general expression for the Γ i functions also includes a linear term (respectively A L z 1 and B L z 2); however, A L and B L depend on the displacement value at infinity, which is null in the residual stress case, thus both coefficients are zero and have been omitted for simplicity.
Note that equation (13) can easily be rewritten in terms of σ x , σ y and τ xy to obtain a final result which is formally identical to Kirsh’s solution for isotropic materials. However, in this case φ plays a different role: in Lekhnitskii’s solution, φ is the (unknown) orientation of principal stress, whereas in the isotropic case it is simply one of the coordinates of the inspected point in the cylindrical reference system.
Due to the orthotropy and loading configurations, the mesh cannot make use of symmetries, thus a complete circular disk has to be analyzed.
It is to be noted that the FEM and Lekhnitskii solutions are not completely equivalent even in the through-hole case: indeed, the theoretical formulas include only the first term of the solution series (equation 12), thus small numerical deviations (a few percentage points) can be observed when comparing them.
Each speckle is described by a bell-shaped polynomial function. The entire field results from the superposition of several thousand bells whose parameters (radius, location and value of maximum) are sampled from user-specified random distributions.
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Baldi, A. Residual Stress Analysis of Orthotropic Materials Using Integrated Digital Image Correlation. Exp Mech 54, 1279–1292 (2014). https://doi.org/10.1007/s11340-014-9859-1
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DOI: https://doi.org/10.1007/s11340-014-9859-1