Streamline visualization of tensor fields in solid mechanics

Stress net view of maximum shear lines inferred from molecular dynamics simulation of crack growth. Image from

Brazilian stress net before and after material failure. Colors indicate maximum principal stress (showing tension in the center of this axially compressed disk). Lines show directions of max principal stress.

A stress net is simply a graphical depiction of principal stress directions (or other directions derived from them, such as rotating them by 45 degrees to get the maximum shear lines.) 

The Brazilian plot in this post confirms that, for the classic Brazilian indirect tension test, vertically compressing the top and bottom edges of a disk induces a horizontal tension in the interior of the disk (red region).  Just after failure, the material suffers a vertical tensile crack.  In the Brazilian plot, the white lines (i.e., the stress net) are aligned with the directions of maximum principal stress, which would indicate good placement of composite reinforcing fibers if you were to try to strengthen a disk against Brazilian tensile failure.

In the publication entitled “Exploring 2D Tensor Fields Using Stress Nets” (See for citation data), an interesting result emerged from a detailed visualization of the stress field inferred from molecular dynamics simulation of dynamic Mode I crack growth. Rather than merely looking at a color plot of stress invariants, the plot of crack growth overlays the color plot of equivalent shear stress with a grid of white lines that are aligned with the planes of maximum shear (exactly 45deg off of principal directions). Note that this plot reveals something that would have been missed with only an invariant color plot: there is a “wake discontinuity” in the principal stress directions without a jump in the principal stresses themselves!

To download a detailed description of the method use to generate these stress net plots, see the above cited publication.

The stress net concept applies to any 2D tensor field.  Its extension to 3D to plot 3D streamlines should be self-evident if not straightforward.  As the requisite functionality for 2D nets was developed for the open-source VTK graphics suite, the ability to make plots like these has arrived in most mathematics software packages such as Mathematica (where the command is ListStreamDensityPlot).

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