Residual Stress Gradient Visualization
All code, images, and results are copyright protected
Ronald D. Kriz, Assoicate Professor
Department of Engineering Science and Mechanics, and
Director of the University Visualization and Animation Group
Mecit Yaman, Ph.D Graduate Candidate
(Advisor: Professor M. Harting)
Department of Physics
University of Cape Town, South Africa

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. Micro-structures 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 "as-welded" 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 (Ti-6Al-4V), 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 (0-degree), pure shear with green (90-degree), and pure compressive with red (180-degree). 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 Micro-graph, Winholtz & Krawitz [1]
Points A1 B1 E1 C1: As Welded / Post Heat Treated

Top View              Side View
Residual Stress Profiles for Shot Peened Ti-6AL-4V, Harting [2]

Stress Glyph Gradients Overlayed on Weld Microstructure:
VRML-2 files: As Welded and Post Heat Treated

Viewable Results: Creating Rotational Gradients:
Extract Euler Angles from the General Rotation Transformation Matrix


  1. R.A. Winholtz and A.D. Krawitz, Methods for depth profiling complete stress tensors using neutron diffraction, Advances in X-Ray analysis, Vol. 37, Plenum Press, New York, 1994.
  2. M. Harting, "A seminumerical method to determine the depth profile of the three dimensional residual stress state with x-ray diffraction", Acta Mater, Elsevier Science, Vol. 46, no. 4, pp. 1427-1436,1998.
  3. D. Frederick and T.S. Chang, Continuum Mechanics, Scientific Publishers, Inc. Boston, pp. 38-40, 1972.
  4. Y.M.A. Hashash, J.I. Yao, and D.C. Worting, "Glyph and hyperstreamline representation of stress and starin tensors and material constitutive response," Int. J. Numer. and Anal. Meth. in Geomech., Vol. 27, pp. 603-626, 2003.
  5. J.G. Moore, S.A. Schorn, and J. Moore, "Methods of classical mechanics applied to Turbulence Stresses in a Tip Leakage Vortex. Conference Proceedings of the ASME Gas Turbine, Houston, Texas October 1994, (also Turbomachinery Research Group Report No. JM/94-90).
  6. M. Yaman, R.D. Kriz, and M. Harting, "Stress Tensor Visualization With Applications To Materials Research," submitted for review, 2005.