Discontinuities

Every correlation criterion since Correlation Criteria, and every worked example through Kernel Warping, depends on one tacit assumption: the true displacement field is smooth. A kernel window moves as a rigid or gently stretching patch. The search for its match assumes one answer exists.

Real specimens may not always have a continuous displacement field. A crack, a slip band, or a material interface can produce a genuine jump in displacement instead of a continuous displacement. Image Transformation already built exactly that jump:

(a)

(b)

A vertical crack splits the image at its vertical midline. The left half shifts down 4 pixels. The right half shifts up 4 pixels. Standard DIC can't represent that jump.

This chapter seeks to identify characteristics of the correlation map in the presence of a discontinuous displacement field. Specifically, what does a correlation surface look like when a kernel window straddles a real discontinuity instead of sitting cleanly on one side of it?

Synthetic Dislocation answers that with a known, exact ground truth: the same 4-pixel offset above, now carrying a speckle pattern a correlation can actually track. Experimental Dislocation then repeats the same straddling window on a real experimental crack image pair. There the ground truth isn't known in advance, so it checks whether the same signature shows up outside a synthetic setup.

Neither section finds the crack on its own. Detection finds where a crack runs across the whole image. Localization then measures where it crosses one chosen line.

Relation to Heaviside-DIC

Heaviside-DIC (H-DIC)¹ can represent a displacement jump like the crack dislocation above. For the continuous part of its displacement, it uses the same affine kernel warp that Kernel Warping introduced. The paper calls each kernel a subset. Its Eq. (1) writes each kernel point's deformed position as the sum of three terms: the reference position, a rigid-body translation, and a first-gradient term. The paper labels those terms "Conventional DIC method." They are the six parameters $\mathbf{p}$ of Kernel Warping's warp, $\mathbf{W}$. $(u,v)$ is the translation, and $(u_x,u_y,v_x,v_y)$ is the first gradient. The paper's Fig. 2 draws the same two steps as Kernel Warping's rigid versus warped kernel figure: a rigid-body displacement, then a first-gradient warp.

H-DIC then adds one term, a jump ${\mathbf{u}}^{\prime }$ times a Heaviside function $H$:

$$\mathbf{x}=\mathbf{X}+\mathbf{u}+\nabla \mathbf{u}(\mathbf{X}-{\mathbf{X}}_0)+{\mathbf{u}}^{\prime }H(r-r^{\ast }),r=\Delta x\cos {\theta }^{\ast }+\Delta y\sin {\theta }^{\ast }.$$

$H$ is 0 on one side of a line and 1 on the other. The line sits a distance $r^{\ast }$ from the kernel center, at angle ${\theta }^{\ast }$. Pixels on the far side of the line shift by an extra ${\mathbf{u}}^{\prime }$. When the optimized jump is zero, H-DIC reduces to the conventional (continuous) affine warp. The paper optimizes all parameters together with a Newton-based method. It doesn't say whether that method uses the inverse compositional form.

So dictk.warp covers H-DIC's continuous half. The missing half is the jump and its line: ${\mathbf{u}}^{\prime }$, $r^{\ast }$, and ${\theta }^{\ast }$. Detection finds where such a line runs. Localization finds where it crosses a chosen path.

1. Bourdin F, Stinville JC, Echlin MP, Callahan PG, Lenthe WC, Torbet CJ, Texier D, Bridier F, Cormier J, Villechaise P, Pollock TM. Measurements of plastic localization by heaviside-digital image correlation. Acta Materialia. 2018 Sep 15;157:307-25. link ↩