Decomposition and Visualization of Fourth-Order Elastic-Plastic Tensors

Alisa NeemanSC, Rebecca BrannonU, Boris JeremicD, Allen Van GelderSC and Alex PangSC

Modeling and simulations of static and dynamic behavior of soils and structures made of various materials (soils, concrete wood, steel, etc.) is the focus of current research in civil, mechanical and other branches of engineering. We present a two-stage decomposition process to facilitate visualization of softening and failing solids and test it on soil models from Geomechanics.

ProblemVisualizing fourth-order tensor fields representing a solid’s time-varying stiffness (such as soil during an earthquake).

Basic Constitutive Equation (Hooke’s Law generalization)

σij = Eijkl εkl

σij : stress incrementEijkl : elastic-plastic stiffnessεkl : strain increment

Under stress, stiffness experiences: Eigenvalue reduction Loss of positive-definiteness Singularity

The latter two indicate localized failure.

Other Parts of a Solid Model

Yield Function F(σ) delineates stress that causes elastic (temporary) deformation versus elastic + plastic (permanent). The yield surface is a convex isosurface in stress space where F(σ)= 0.

Plastic Flow: G(σ) determines plastic flow direction. If the material is associated, the plastic flow equals the normal to the yield surface.

ApproachDecompose the stiffness to determine the mode in which the material is softening. Visualize that mode as a second-order tensor, colored by the magnitude of minimum stiffness.

Experiment

Stage 1: Self-weight compression (–Z) Stage 2: Two point loads; –Z component (0.9659 kN) and +X component (1294 kN)

Eigentensor glyphs were colored by lowest eigenvalue and scaled across all time steps. Black cubes indicate singularity with negative eigenvalues.

Results

The material experienced tension behind the point loads. The trail of points that have undergone singularity behind the point loads are green and cyan, indicating zero and slightly negative eigenvalues.

Stage 1 induced hardening at the bottom of the volume, with some singularity. Stage 2 induced softening near the point loads. It was difficult to correlate the stress to the behavior because of the changing yield surface and non-associativity.

Stage OnePolar decomposition uniquely separates a matrix into two components:

E = RS where S is a stretch (symmetric positive-semidefinite matrix), and R is a pure rotation (orthonormal).

The rotation R quantifies the misalignment between yield surface normal and the plastic flow in non-associated materials.

Stage TwoSpectral decomposition applied to the stretch, S yields second-order eigentensors and eigenvalues. A reduced eigenvalue means the solid is less stiff and the associated eigenvector is the mode of stress to which the solid is most vulnerable.

VisualizationEigentensor glyphs were drawn by stretching a unit sphere according to the formula

σN = σij n i nj

n is a unit length direction vector from the center of the sphere to a point on the surface.

Stress Modes

Triaxial Compression/Tension

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Spherical Tension/Compression Pure Shear

Deformation of volume at end of Stage 1 and Stage 2. Arrows indicate location of point loads and direction

+1

-1

Drucker-Prager: fails under tension, unchanged under compression, associated.

Dafalias and Manzari's plasticity sand model: pressure-dependent and non-associated.

2.8 x 107

-2.8 x 107

0

4.5 x105

0

0.008