Residual stresses are present in most materials as a result of thermal and mechanical loads applied during fabrication. For example metals are "forged" into a desired shape with large loads at high temperatures. But when the fabrication loads are removed and the outside surface is cooled faster at the surface, larger thermal contractions occur at the surface compared to the interior. Microstructures are also different at the surface and upon further cooling stresses are frozen even after loads are removed. Hence the name residual stress. A similar process occurs in the formation of welds. Often these stresses are unknown but sufficiently large such that cracks initiate and grow causing unexpected failure. It is common practice to remove these "aswelded" residual stresses by "post heat treatment". In some cases residual stress can be controlled for a desired beneficial result, e.g. residual stresses are intentionally created in tempered glass where compressive stresses are created near the surface that prevents crack growth and raises the fracture strength. Residual stresses are associated with space gradients, e.g. again in tempered glass compressive stresses near the surface change to tension below the surface. Residual stress gradients are important in understanding and predicting matertial response.
Below we study two different types of stress gradients in metals: 1) residual stresses near welds, Winholtz & Krawitz [1] and 2) surface residual stresses induced by plastic deformations (shot peening) in a Titanium alloy (Ti6Al4V), Harting [2]. In both cases second order tensor glyphs are used to assist in understanding the stress distribution ("gradient"). For comparison four different types of stress tensor glyphs are used: 1) Quadric, Frederick & Chang [3], 2) Reynolds, Moore & Schorn [4], 3) HWY, Hashash, Yao, and Wotring [5], 3) PNS, Yaman, Kriz, and Harting [6]. The Quadric, HWY, and PNS glyphs' shapes emphasize the shear component of the stress tensor, whereas the Reynolds glyph shapes emphasize the normal component of the stress tensor. For all glyphs a color gradient is superposed onto the glyph surface that represents pure tension with purple (0degree), pure shear with green (90degree), and pure compressive with red (180degree). Not previously known was that the Quadric glyph does not always yield an ellipsoid but when shear exists cones appear instead. As expected in all cases green appears on the conical part of the quadric glyph. Each glyph type has its advantages and disadvantages which are selected depending on the knowledge and intent of the researcher. Hence scientific data visualization is a creative process that is best realized by the applied scientist. As educators our goal is to encourage the next generation of applied scientists and engineers to become as skillful and creative with computer graphical modelling as we are currently skillful at mathematical modeling. Both will prove to be essential for problem solving in the future.
Click on images to enlarge with more detailed information All code and graphical results are archived at this link, October 2004
 
Weld Micrograph, Winholtz & Krawitz [1] Points A1 B1 E1 C1: As Welded / Post Heat Treated 
Top View Side View Residual Stress Profiles for Shot Peened Ti6AL4V, Harting [2] 

Stress Glyph Gradients Overlayed on Weld Microstructure:
VRML2 files: As Welded and Post Heat Treated 
Viewable Results:
Extract Euler Angles from the General Rotation Transformation Matrix 
References: