Diffraction Assisted Image Correlation: A Novel Method for Measuring Three-Dimensional Deformation using Two-Dimensional Digital Image Correlation

Abstract

Digital Image Correlation (DIC) provides a full-field non-contact optical method for accurate deformation measurement of materials, devices and structures. The measurement of three-dimensional (3D) deformation using DIC in general requires imaging with two cameras and a 3D-DIC code. In the present work, a new experimental technique, namely, Diffraction Assisted Image Correlation (DAIC) for 3D displacement measurement using a single camera and 2D-DIC algorithm is presented. A transmission diffraction grating is placed between the specimen and the camera, resulting in multiple images which are then used to obtain apparent in-plane displacements using 2D-DIC. The true in-plane and out-of-plane displacements of the specimen are obtained from the apparent in-plane displacements and the diffraction angle of the grating. The validity and accuracy of the DAIC method are demonstrated through 3D displacement measurement of a small thin membrane. This technique provides new avenues for performing 3D deformation measurements at small length scales and/or dynamic loading conditions.

Appendix: Location of a virtual image formed by diffraction of a transmission grating

The ray diagram in Fig. 9(a) shows two selected rays from a monochromatic point source of light, P, which is placed at a distance, d, from a transmission line grating. The definitions of the three unit vectors (e _⊥, e _// , e _n) are the same as those in Fig. 1. The first-order diffraction angle of the grating is $\theta ={\sin^{-1 }}\left( {{\lambda \left/ {p} \right.}} \right)$ where λ is the wavelength of the light and p is the pitch of the grating. The first ray (PB) impinges the grating at an angle of θ, and bends toward the normal to the grating surface after passing through the grating. The second incident ray (PC) deviates from the first by an angle of Δθ, resulting in a different angle of diffraction, Δα. When one traces back the diffracted rays, a virtual image is formed at P ⁺, where the two back-projected rays intersect. According to the geometric relationship depicted in Fig. 9(a), the distance between P ⁺ and the grating is

$$ {d^{+}}\left( {\varDelta \theta } \right)=\frac{{\left| {BC} \right|}}{{\tan \varDelta \alpha }}=d\frac{{\tan \left( {\theta +\varDelta \theta } \right)-\tan \theta }}{{\tan \varDelta \alpha }}. $$

(A1)

Fig. 9

(a) Ray diagram illustrating formation of a virtual image by first-order diffraction of a transmission grating; (b) plots of d ⁺ as a function of Δθ for different values of first-order diffraction angle. d ⁺ is normalized by the limiting value of d ⁺ as Δθ approaches zero

Then, the grating equation $\sin \left( {\theta +\varDelta \theta } \right)-\sin \varDelta \alpha ={\lambda \left/ {p} \right.}=\sin \theta$ is invoked to obtain an expression for Δα. By substituting this expression into equation (A1) one gets,

$$ {d^{+}}\left( {\varDelta \theta } \right)=d\frac{{\tan \left( {\theta +\varDelta \theta } \right)-\tan \theta }}{{\tan \left\{ {{\sin^{-1 }}\left[ {\sin \left( {\theta +\varDelta \theta } \right)-\sin \theta } \right]} \right\}}}. $$

(A2)

The limiting value of d ⁺ as Δθ approaches zero can be obtained analytically. Denote the numerator in equation (A2) by f (Δθ), and the denominator by g(Δθ). Both f(Δθ) and g(Δθ) vanish as Δθ approaches zero. Using L’Hospital’s rule, in the limit as Δθ approaches zero,

$$ {{\left. {{d^{+}}} \right|}_{{\varDelta \theta \to 0}}}=d\frac{{{{{\left. {f\prime } \right|}}_{{\varDelta \theta \to 0}}}}}{{{{{\left. {g\prime } \right|}}_{{\varDelta \theta \to 0}}}}}=\frac{d}{{\cos^3 \theta }}. $$

(A3)

Figure 9(b) plots the curves of d ⁺ (normalized by ${d \left/ {{\cos^3 \theta }} \right.}$) versus Δθ for various values of θ. It is seen that d ⁺ varies around ${d \left/ {{\cos^3 \theta }} \right.}$ with change in Δθ, implying a non-zero thickness of the virtual image. d ⁺ is observed to be less sensitive to change in Δθ when θ is small.