Preliminary Concepts

At its basic level, a DIC system comprises a camera, light(s), and an image processing unit. However, a single camera setup is only able to capture displacements and translations fixed in a two-dimensional plane. Upgrading to a two camera setup enables monitoring of displacements and translations in a three-dimensional sense in much the same way as the human eye. Some DIC systems allow for a multitude of cameras, and the setup used in this study implemented a four camera configuration.

5.2.1.1 Solid Mechanics Deformation

The fundamentals of DIC begin with the basic deformation theory of solid mechanics [Bow09;Mal69]. The following reflects the formulation described in “Applications of digital-image-correlation techniques to experimental mechanics” by Chu et al. [CRS85]. While DIC system manufactures might employ a slightly different formulation,

Fig. 5.9: Deforming line segment PQ on body B.

Consider a body B, which deforms in a Euclidean space E (Figure 5.9).

Now, consider the line segment PQ, which deforms to the line segment P⁰Q⁰, lying on the newly deformed body B⁰. If (u, v, w) denote the components of displacement of an arbitrary point in the x, y, and z directions, respectively, the points P and Q are located at (x, y, z) and (x + dx, y + dy, z + dz) prior to deformation, respectively. Deformed points P and Q are represented by P⁰ and Q⁰defined by

P⁰ x⁰,y⁰,z⁰
_ 

x + u(P), y + v(P), z + w(P) _(5.1)

Q⁰ x⁰+ dx⁰, y⁰+ dy⁰,z⁰+ dz⁰

[x + u(P) + u(Q) − u(P) + dx, y + v(P) + v(Q) − v(P) + dy, z + w(P) + w(Q) − w(P) + dz]

(5.2)

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The vector lengths between line segments PQ and P⁰Q⁰are defined by

|PQ|  ( ds)²  dx²+ dy²+ dz²

|P⁰Q⁰|  ( ds⁰)² ( dx⁰)²+ dy⁰
2+( dz⁰)²

(5.3)

Using the relationships dx⁰  u(Q) − u(P) + dx, dy⁰  v(Q) − v(P) + dy, and dz⁰ w(Q) − w(P) + dz derived from Equation (5.2), Equation (5.3) can be rewritten as

|P⁰Q⁰|  [u(Q) − u(P) + dx]²+. . . . . . + 

v(Q) − v(P) + dy2+[w(Q) − w(P) + dz]²

(5.4)

A linear Taylor’s expansion of the displacement functions about point P gives

u(Q) − u(P)u ^∂u_∂x dx + ∂u

∂ydy + ∂u

∂z dz v(Q) − v(P)u ^∂v_∂x dx + ∂v

∂ydy + ∂v

∂z dz w(Q) − w(P)u ^∂w_∂x ^{dx + ∂w}_∂y ^{dy + ∂w}_∂z ^dz

(5.5)

From Equation (5.5), projections of the deformed lengths dx⁰, dy⁰, and dz⁰ are

dx⁰u

Equation (5.6) are the basis for deriving the finite-strain-tensor equations. If P and Q are initially oriented along the x-axis, dz  dy  0 and Equation (5.6) generate the final values of ( dx⁰, dy⁰, dz⁰). Thus, the strain tensor is equal to the engineering strain shown by

xx  |P⁰Q⁰| − |PQ|

For DIC analysis, the observed pattern is a two-dimensional projection of an object onto a plane; therefore, the finite-strain equation used to compute strain is given by

Equation (5.8) is relatively general, but follows along the assumptions inherent in digital image processing. First, the image processing assumes that

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plane deformations and displacements are not affected by out-of-plane displacements. Second, out-of-plane displacements (i.e. ∂w/∂x) are much less than terms like ∂u/∂x so that the effect is excluded from Equation (5.8).

5.2.1.2 Image Correlation

Performing a DIC analysis starts with applying a speckle pattern to the object in question. Applying the speckled pattern can somewhat be an art in itself, but the general concept is to apply a stochastic-like pattern composed of two colors in large contrast to one another. The most common colors are black and white. There are several application methods and the choice is largely dependent on the individual test. The most simple method is to use canned aerosol paint, painting the background either black or white with the speckles being the opposing color. The size and distribution of the speckles is largely dependent on the camera’s proximity to the object.

Figure 5.10shows a schematic of a two-camera DIC configuration. On the object’s surface is a speckle pattern indicative of the pattern generally used for analysis. Consider the measured light-intensity pattern reflected from the object surface. The intensity pattern in its initial state is represented by f (x, y) and after deformation is represented by f⁰(x⁰,y⁰). Between these states an unique, one-to-one correspondence is assumed. Thus, discretizing the surface into a number of subset images allows for tracking the strains and displacements as the object deforms and/or translates.

An example of a measured light-intensity pattern f (x, y) is shown in Fig-ure5.11. As the image is scanned over the x-y plane, the reflected light is

Fig. 5.10: Schematic of two-camera DIC configuration.

measured by the camera’s sensor. In regions where a black dot exists, most of the light is absorbed which shows as a dip in f (x, y). Having a sufficiently random speckle pattern is important because repeating patterns can lead to misregistration issues [SOS09]. Correlating for each individual speckle is unfeasible, since the speckles and the surrounding area are continously deforming. Instead, the domain is discretized, forming a subset of gray value patterns, which are used to track relative displacements.

Consider the initially undeformed subset B in Figure 5.12. Subset B lies on a scanning area (light-intensity pattern) discretized by sampling grids.

At the center of B exists point P, which after deformation translates to P⁰ on deformed subset B⁰. Using the theory of deformation discussed in the

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x

y f (x, y)

Fig. 5.11: Light-intensity pattern of speckled image.

previous section, the light-intensity values at points P and P⁰can be written as

f (P)  f (x, y) f⁰(P⁰)  f⁰

x + u(P), y + v(P)

(5.9)

Similarly, for a point Q at position (x + dx, y + dy) on subset B, the light-intensity values for points Q and Q⁰can be written as

f (Q)  f x + dx, y + dy
f⁰(Q⁰)  f⁰

x + u(Q) + dx, y + v(Q) + dy

(5.10)

Assuming the local intensity value does not change due to deformation, f (Q)  f⁰(Q⁰) so that

Fig. 5.12: Deformation analysis using DIC.

f (P)  f 

x + u(P), y + v(P) f (Q)  f 

x + u(Q) + dx, y + v(Q) + dy

(5.11)

Referencing Figure5.12, if subsets B and B⁰are small enough that strain lines remain straight after deformation, Equations (5.5) and (5.6) can be used to describe the position of point Q⁰as

x⁰⁰, y⁰⁰
_{ x0+ dx0}

, y⁰+ dy⁰


x + u(P) + dx⁰,y + v(P) + dy⁰





x + u(P) + ∂u

∂x dx + ∂u

∂ydy + dx , . . . . . . ,y + v(P) +∂v

∂xdx + ∂v

∂y dy + dy

(5.12)

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Using Equation (5.5), Equation (5.11) can be rewritten as

f⁰(Q⁰)  f

Therefore, if the displacement of point P and its the derivative terms ∂u/∂x(P) and ∂u/∂y(P) are known, the position of any nearby point Q⁰ can be de-termined. Similarly, if values for u(P), v(P), ∂u/∂x(P), ∂u/∂y(P), ∂v/∂x(P), and ∂v/∂y(P) are assumed, estimates for point P⁰and all points Q⁰can be obtained. This statement forms the foundation for the numerical computation of local deformation.

DIC systems are able to correct for camera lens distortions, lighting conditions, and interpolation/noise biases. Depending on the individual DIC system, the user may be able to apply various image matching and/or interpolation methods.