Caveats on formulating finite elasto-plasticity in curvilinear coordinates

This paper presents a practical, step-by-step methodology for formulating finite elasto-plasticity in curvilinear coordinates using explicit basis changes rather than differential geometry, clarifying the treatment of deformation gradients, Jacobians, and shifters to enable robust finite element analysis of axisymmetric problems with large deformations.

The Gist

Imagine you are an architect trying to design a massive, round silo or a deep underground tunnel. To figure out if the structure will hold up under pressure, you use a computer program (Finite Element Analysis) to simulate how the ground or metal bends and stretches.

Usually, computers are happiest when working with simple, straight grids—like graph paper with perfect squares (Cartesian coordinates). But the real world is full of curves, circles, and cylinders. When you try to force a round problem into a square grid, the computer has to do a lot of extra math to "fake" the curve, which is slow and wasteful.

It makes much more sense to use a "curved grid" (curvilinear coordinates) that naturally fits the shape, like a net thrown over a sphere. However, this paper argues that while using a curved grid is efficient, it's also dangerous if you don't know exactly how to do the math.

Here is the story of the paper, broken down with simple analogies:

1. The "Map" Problem

Think of the Cartesian grid (straight lines) as a flat map of a city. If you walk 1 mile North, you are exactly 1 mile North. Simple.

Now, think of the curvilinear grid (like latitude and longitude) as a map of the Earth. If you walk 1 mile North from the equator, you are 1 mile North. But if you walk 1 mile North from the North Pole... well, you can't really go further North! The rules change depending on where you are.

The authors found that many computer programs treat the curved world as if it were a flat world. They take the "straight line" math and just paste it onto the "curved" problem. This is like trying to measure the distance between New York and London using a ruler on a flat piece of paper without accounting for the Earth's curve. The result? The computer thinks the material is stretching or shrinking when it isn't, or vice versa.

2. The "Deformation Gradient" (The Stretchy Blanket)

In physics, the Deformation Gradient is like a magical blanket that covers a piece of dough. When you stretch the dough, the blanket stretches with it.

• In a straight world: If you pull the blanket, you can just count how many inches it stretched.
• In a curved world: The blanket is wrapped around a cylinder. If you pull it, the "stretch" isn't just about the material; it's also about the fact that the circle is getting wider.

The paper points out a sneaky trap: In a curved system, the "stretch" of the blanket (the deformation) and the "stretch" of the volume (how much space it takes up) are not the same thing. If you use the wrong formula, your computer might think a solid block of steel has suddenly turned into a sponge (or vice versa) just because you changed the coordinate system.

3. The "Shifter" (The Translator)

This is the paper's most important invention. Imagine you have two people speaking different languages. One speaks "Straight English" (Cartesian), and the other speaks "Curved French" (Curvilinear).

• The Shifter is the translator.
• In a straight world, the translator just says, "Okay, I'm moving from point A to point B."
• In a curved world, the translator has to say, "I'm moving from A to B, but because the ground is curved, I also have to rotate my orientation slightly to stay aligned."

The authors show that if you forget to use this "Shifter" (the translator), your computer program gets confused. It thinks the material is rotating or shearing when it's actually just moving along a curve. This leads to massive errors in predicting whether a tunnel will collapse or a metal part will snap.

4. The "Plastic" Problem (The Memory Foam)

The paper also deals with Elasto-plasticity.

• Elastic is like a rubber band: you stretch it, and it snaps back.
• Plastic is like chewing gum: you stretch it, and it stays stretched. It has "memory."

When you mix these two in a curved world, things get messy. The paper explains that you have to track the "volume" of the material separately for the elastic part (the rubber band) and the plastic part (the gum). If you mix them up in a curved coordinate system, the computer loses track of how much the material has permanently deformed.

5. The Solution: A Step-by-Step Recipe

The authors didn't just point out the problem; they gave a recipe to fix it.

• They showed exactly how to translate the math from the "flat world" to the "curved world" without losing information.
• They provided a specific algorithm (a set of instructions) for computer programmers to follow.
• They tested this recipe on a "cavity expansion" problem (imagine inflating a balloon inside a block of clay).

The Big Result

They compared their new "Curved Recipe" against a standard "3D Straight Recipe."

• The 3D Recipe: Took a long time to run because it had to calculate every single point in a 3D block.
• The Curved Recipe: Took a tiny fraction of the time because it only calculated a 2D slice (since the object is round, the top looks like the bottom).
• The Verdict: Both recipes gave the exact same answer, but the curved one was much faster and cheaper.

In Summary

This paper is a warning label and a user manual for engineers. It says:

"If you want to simulate round things (tunnels, pipes, tires) on a computer, don't just use the standard straight-line math. You need to add a few special 'translation' steps (the Shifter) and be very careful about how you measure volume. If you do this right, you can solve complex problems 10 times faster without losing accuracy. If you do it wrong, your simulation will be garbage."

It's about making sure our digital maps of the world are as accurate as the real world, even when the world is curved.

Technical Summary

Here is a detailed technical summary of the paper "Caveats on formulating finite elasto-plasticity in curvilinear coordinates" by Prett et al.

1. Problem Statement

The paper addresses a critical gap in the numerical implementation of finite-strain elasto-plasticity within curvilinear coordinate systems (specifically axisymmetry). While continuum mechanics is fundamentally based on tensor analysis, which is frame-invariant, Finite Element (FE) implementations rely on matrix representations in user-selected bases.

• The Core Issue: When moving from Cartesian (orthonormal) bases to curvilinear (non-orthonormal, e.g., cylindrical) bases, standard formulations often fail or produce errors. Specifically, naive extensions of Cartesian definitions for the deformation gradient, Jacobian, and stress measures lead to inaccuracies in volume transformation and kinematic descriptions.
• The Challenge: In axisymmetric problems (common in geotechnical engineering, metal forming, and biomechanics), these errors are exacerbated by large deformations and the inclusion of anelastic effects (plasticity). The paper highlights that the Cauchy stress in finite-strain plasticity depends on configuration changes beyond the current elastic state, requiring a careful distinction between elastic and plastic components of the Jacobian and deformation gradient.
• Goal: To provide a rigorous, step-by-step procedure for formulating axisymmetric finite element analysis that correctly handles these geometric complexities without resorting to abstract Riemannian manifold theory, focusing instead on practical implementation within standard Cartesian frameworks using explicit basis changes.

2. Methodology

The authors employ a cavity expansion problem (expansion of a hollow cylinder) as a benchmark model to expose and resolve kinematic inconsistencies. The methodology involves:

• Tensor Analysis in Mixed Bases: The paper works within standard Cartesian representations but explicitly derives the transformation rules to curvilinear (cylindrical) covariant and contravariant bases. It avoids differential geometry manifolds in favor of explicit change-of-basis matrices.
• Kinematic Definitions:
  • Deformation Gradient ($F$): Defined rigorously as a two-point tensor mapping material curves. The paper clarifies that $F$ is not a simple gradient of a vector field in curvilinear coordinates and requires the shifter tensor ($S$) to map between bases of different configurations.
  • Jacobian ($J$): The volume ratio is shown to depend on the determinant of the metric coefficients. The authors derive the relationship $J = \det(F) \sqrt{\det(g)}/\sqrt{\det(G)}$, proving that $\det(F_{curvilinear}) \neq J$ directly.
  • Elasto-Plastic Decomposition: The deformation gradient is multiplicatively decomposed ($F = F^e F^p$). The paper addresses the difficulty of defining the intermediate stress-free configuration's metric. It proposes a practical linearization for the elastic Jacobian ($J_e$) using the trace of the elastic logarithmic strain ($J_e = \exp(\text{tr}(\epsilon^e))$), which simplifies consistent linearization.
• Stress Measures: The paper introduces the Eshelby-zeta stress ($\zeta = J_e \sigma$) as a preferred stress measure for finite-strain plasticity with associated flow rules, ensuring thermodynamic consistency and preservation of the stress-free intermediate configuration.
• Weak Forms & Discretization: The strong forms of momentum balance are converted to weak forms for both Total Lagrangian (TL) and Updated Lagrangian (UL) formulations. These are then reduced to 2D axisymmetric forms by integrating out the circumferential coordinate, carefully handling the $2\pi r$ volume/surface terms.
• Linearization: A consistent Newton-Raphson linearization is derived, explicitly accounting for covariant derivatives and Christoffel symbols, which are non-zero in curvilinear coordinates.

3. Key Contributions

1. Clarification of the Shifter Tensor: The paper explicitly defines the role of the shifter tensor in mapping displacement components between reference and current curvilinear bases, correcting a common source of error where displacements are naively treated as simple vector differences.
2. Correct Jacobian Formulation: It demonstrates that the determinant of the deformation gradient in curvilinear components does not equal the volume ratio ($J$) unless corrected by metric determinants. It provides the correct formula for $J$ and its elastic/plastic decomposition.
3. Practical Linearization Strategy: The authors propose a specific linearization for the elastic Jacobian ($J_e = \exp(\text{tr}(\epsilon^e))$) that minimizes extra terms and integrates seamlessly with standard constitutive subroutines, avoiding the need to compute complex derivatives of the plastic deformation gradient.
4. Algorithmic Workflows: The paper provides detailed algorithms (Algorithms 1 and 2) for both UL and TL axisymmetric FE implementations, highlighting exactly where curvilinear terms (Christoffel symbols, metric updates) must be inserted compared to a standard Cartesian code.
5. Validation: It validates the formulation by comparing the proposed 2D axisymmetric code against a full 3D Cartesian implementation of the same problem.

4. Results

The authors validated their formulation using a thick-walled cylinder subjected to internal radial expansion with a linearly varying displacement along the height.

• Comparison: The results from the proposed 2D axisymmetric formulation were compared against a 3D finite element analysis using 20-noded hexahedral elements.
• Accuracy: The comparison showed no appreciable difference between the 3D and 2D axisymmetric results for key invariants:
  • Negative Cauchy pressure.
  • Deviatoric Cauchy stress.
  • Elastic and Plastic Jacobians.
• Convergence: The Newton-Raphson iterative scheme demonstrated robust convergence (typically 3–4 iterations per time step) with identical convergence rates for both the 3D and 2D formulations.
• Computational Efficiency: The axisymmetric formulation achieved 3D-like accuracy with significantly lower computational cost (2D discretization vs. 3D), confirming the viability of the proposed method for large-scale simulations.

5. Significance

This paper is significant for the computational mechanics community for several reasons:

• Error Prevention: It identifies and corrects subtle but fundamental errors that arise when implementing finite-strain plasticity in curvilinear coordinates. Incorrect definitions of kinematic quantities can lead to inconsistent tangent stiffness matrices, non-convergence, or physically incorrect stress states.
• Bridging Theory and Practice: By avoiding abstract manifold theory and focusing on explicit basis transformations, the paper provides a "cookbook" approach that allows FE developers to implement robust axisymmetric solvers without needing deep expertise in differential geometry.
• Enabling Complex Simulations: The rigorous treatment of the Jacobian and stress measures enables the reliable simulation of complex axisymmetric problems involving large plastic deformations, such as soil penetration (CPT), tunneling, and metal forming, where volume changes and anisotropic effects are critical.
• Standardization: It establishes a clear, consistent notation and procedure for handling the "shifter," metric coefficients, and covariant derivatives in axisymmetric FE codes, promoting reproducibility and accuracy in future research.

In summary, the paper provides the necessary mathematical and algorithmic corrections to ensure that finite element analyses of axisymmetric elasto-plastic problems under large strains are both mathematically rigorous and numerically robust.