Decomposition and Visualization of Fourth-Order Elastic-Plastic Tensors

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Date
2008
Journal Title
Journal ISSN
Volume Title
Publisher
The Eurographics Association
Abstract
Visualization of fourth-order tensors from solid mechanics has not been explored in depth previously. Challenges include the large number of components (3x3x3x3 for 3D), loss of major symmetry and loss of positive definiteness (with possibly zero or negative eigenvalues). This paper presents a decomposition of fourth-order tensors that facilitates their visualization and understanding. Fourth-order tensors are used to represent a solid's stiffness. The stiffness tensor represents the relationship between increments of stress and increments of strain. Visualizing stiffness is important to understand the changing state of solids during plastification and failure. In this work, we present a method to reduce the number of stiffness components to second-order 3x3 tensors for visualization. The reduction is based on polar decomposition, followed by eigen-decomposition on the polar "stretch". If any resulting eigenvalue is significantly lower than the others, the material has softened in that eigen-direction. The associated second-order eigentensor represents the mode of stress (such as compression, tension, shear, or some combination of these) to which the material becomes vulnerable. Thus we can visualize the physical meaning of plastification with techniques for visualizing second-order symmetric tensors.
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@inproceedings{
:10.2312/VG/VG-PBG08/121-128
, booktitle = {
IEEE/ EG Symposium on Volume and Point-Based Graphics
}, editor = {
Hans-Christian Hege and David Laidlaw and Renato Pajarola and Oliver Staadt
}, title = {{
Decomposition and Visualization of Fourth-Order Elastic-Plastic Tensors
}}, author = {
Neeman, Alisa G.
and
Brannon, Rebecca
and
Jeremic, Boris
and
Gelder, Allen Van
and
Pang, Alex
}, year = {
2008
}, publisher = {
The Eurographics Association
}, ISSN = {
1727-8376
}, ISBN = {
978-3-905674-12-5
}, DOI = {
/10.2312/VG/VG-PBG08/121-128
} }
Citation