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Residual Stresses after Severe Plastic Torsion: Influence of Tension–Compression Asymmetry

This paper presents simple formulae for calculating residual stresses in solid circular cylinders after severe elastic–plastic torsion, demonstrating how tension–compression asymmetry significantly alters the stress state upon unloading and providing a tool for calibrating advanced plasticity models.

Original authors: Georgiy Sevastyanov

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Georgiy Sevastyanov

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a thick, solid rubber band or a metal rod. If you twist it really hard—so hard that it permanently bends and doesn't spring back to its original shape—you are performing what scientists call "severe plastic torsion."

Now, imagine you stop twisting it and let go. You might expect the rod to just sit there, twisted but relaxed. However, this paper reveals that the rod is actually holding its breath. It is trapped in a state of internal tension, like a spring that has been squeezed but can't fully release. These hidden forces are called residual stresses.

The author, Georgiy Sevastyanov, discovered that the way these hidden forces behave depends entirely on a quirk of the material called Tension-Compression Asymmetry (TCA).

The "Two-Faced" Material

To understand TCA, imagine a material that is "two-faced."

The Normal Material (Symmetric): Think of a standard piece of clay. It's just as easy to squish it (compression) as it is to pull it apart (tension). If you twist it and let go, the internal stresses are predictable and balanced, like a perfectly symmetrical knot.
The "Two-Faced" Material (Asymmetric): Now, imagine a material that is stubborn when you pull it but weak when you push it (or vice versa). Maybe it's like a sponge that resists being stretched but collapses easily when squeezed. This is Tension-Compression Asymmetry.

The paper argues that when you twist this "two-faced" material and let go, the internal stress doesn't just relax into a simple twist. Instead, it gets confused. The material tries to fix itself, but because it reacts differently to pulling and pushing, it ends up with a messy, uneven distribution of stress.

The "Spring" Analogy

Think of the rod as a giant, coiled spring.

Twisting: You wind the spring up tight. The outer layers get stretched and squished more than the inner core.
The Asymmetry: Because the material is "two-faced," the outer layers don't just want to unwind; they want to change length. They want to get shorter or longer depending on whether they were being pulled or pushed during the twist.
Letting Go (Unloading): When you remove the twisting force, the spring tries to unwind. But because the outer layers are fighting to change the length of the rod while the inner core is fighting to keep it the same, the rod ends up in a weird state.
- It doesn't just untwist; it might slightly lengthen or shorten.
- The internal forces aren't just a simple twist anymore; they become a complex mix of pushing, pulling, and twisting all at once.

What the Paper Actually Found

The author didn't just guess this; he used advanced math to create a "recipe" (simple formulae) to predict exactly what happens when you let go of this twisted rod. Here are the main takeaways:

The Twist Angle Changes: If the material is "two-faced," the rod won't just unwind to zero. It will stop at a slightly different angle than you might expect because the internal forces are fighting each other.
The Length Changes: The rod might get slightly longer or shorter after you let go, even though you didn't pull or push it directly. This is a side effect of the material's "two-faced" nature.
The Stress Direction Shifts: In a normal material, the strongest internal forces point at a perfect 45-degree angle. In this "two-faced" material, those forces can tilt significantly away from that angle, sometimes even pointing in the same direction (all pushing or all pulling) rather than opposing each other.
The "Lode Angle": This is a fancy way of describing the shape of the stress. The paper found that the "shape" of the stress inside the rod changes dramatically based on how "two-faced" the material is. It's like the stress pattern morphs from a perfect circle into an oval or a weird blob.
The "Reverse" Zone: Near the very center of the rod, the math shows that the material might actually try to twist backwards slightly when you let go, creating a tiny zone where the stress flips direction.

Why This Matters (According to the Paper)

The author says these findings are useful for two main reasons:

Calibration: Engineers use computer models to simulate how materials behave. These new "recipes" (formulae) allow them to quickly check if their computer models are correctly accounting for this "two-faced" behavior.
Understanding the Aftermath: If you use severe twisting to strengthen a part (like a metal shaft), you need to know what hidden stresses are left inside. If you ignore this "two-faced" effect, you might think the part is safe when it's actually sitting in a state of high, unpredictable stress.

In short, the paper tells us that if a material acts differently when you pull it than when you push it, twisting it and letting go creates a much more chaotic and complex internal stress pattern than we previously thought. The author provides the math to predict exactly how chaotic it gets.

Technical Summary: Residual Stresses after Severe Plastic Torsion: Influence of Tension–Compression Asymmetry

Problem Statement
While torsion is a standard mechanical test and a processing method for severe plastic deformation (SPD), the analytical understanding of the residual stress state following severe elastic–plastic torsion remains limited. Existing literature has largely focused on the "Swift effect" (axial force generation during fixed-end torsion) and the influence of tension–compression asymmetry (TCA) on specimen length changes. However, the specific impact of TCA on the magnitude, distribution, and type of residual stresses after unloading has not been previously investigated. This is a critical gap because TCA induces axial forces during loading, leading to non-uniform residual stress states that deviate from pure shear upon unloading. Understanding this state is essential for evaluating component service properties, predicting creep or fatigue performance, and calibrating advanced plasticity models.

Methodology
The paper presents an analytical solution for the unloaded state of a solid circular cylinder subjected to severe plastic torsion with fixed ends. The analysis is grounded in a finite elastic–plastic deformation model utilizing the multiplicative split of the deformation gradient ( $F = F_e F_p$ ).

Limit State Definition: The study builds upon a prior solution (Sevastyanov, 2026) describing the limit stress state where the entire specimen (except an infinitesimal core) has yielded. This state incorporates the Cazacu and Barlat (2004) plasticity model to account for TCA and a neo-Hookean elastic model.
Unloading Kinematics: The unloading process is modeled as the removal of torque and axial force, allowing for changes in specimen length and untwisting. The kinematics are described using velocity vectors and the Oldroyd time derivative of the elastic left Cauchy–Green deformation tensor ( $B_e$ ).
Evolution Equations: The authors derive and integrate evolution equations for the components of $B_e$ during purely elastic unloading. Initial conditions are set by the limit deformed state. The solution assumes no secondary plastic flow (purely elastic unloading), though the authors acknowledge a small zone of reversed plastic flow near the axis.
Boundary Conditions: The final unloaded state is determined by enforcing zero net torque and zero net axial force, allowing the calculation of the final relative length change and rotation angle.

Key Contributions and Results
The paper derives simple formulae for the residual stress state and demonstrates that TCA fundamentally alters the stress distribution compared to classical von Mises materials.

Deviation from Pure Shear: In materials with pronounced TCA, the residual stress state after unloading departs significantly from pure shear. The principal stress axes on the specimen surface are inclined at angles substantially different from the $45^\circ$ characteristic of symmetric materials.
Principal Stress Signs: For materials with strong TCA, the principal stresses on the specimen surface can share the same sign (both tensile or both compressive), a phenomenon not observed in symmetric materials where they are equal and opposite.
Lode Angle Distribution: The mean value of the Lode angle over the specimen volume is shown to depend strongly on the strength differential. Peak values of approximately $18^\circ$ are observed for moderate differences between compressive and tensile yield strengths ( $\sigma_{yC}/\sigma_{yT} \approx 1.1\text{--}1.2$ ).
Reversed Plastic Flow: The analysis predicts a localized zone of reversed plastic flow near the axis of symmetry during unloading, caused by TCA. For moderate asymmetry, this zone is narrow (radius $<5\%$ of the specimen radius), justifying the purely elastic unloading approximation for the bulk of the material.
Analytical Formulae: The paper provides explicit expressions for:
- The relative change in specimen length ( $\dot{L}$ ) and the angle of rotation ( $\theta$ ) upon unloading.
- The distribution of deviatoric stress components ( $s_{rr}, s_{\phi\phi}, s_{zz}, s_{z\phi}$ ) as functions of the radial coordinate and material parameters ( $a, b, c$ ).
- The inclination angle of the principal stress axes on the surface.

Significance and Claims
The authors claim that the derived formulae offer a direct method for the fast calibration of advanced plasticity models that incorporate TCA at finite strains. By providing analytical benchmarks for residual stresses, length changes, and rotation angles, the work facilitates the validation of complex constitutive models against experimental data (such as XRD measurements).

The paper modestly positions itself as a "short communication" providing a first-order analytical approximation. It acknowledges that while the assumption of purely elastic unloading introduces errors in the small central zone where reversed plastic flow occurs, the volume fraction of this zone is small enough that the presented formulae remain highly accurate for the majority of the specimen volume. The results are specifically tailored to the Cazacu and Barlat (2004) model but the solution principle is noted to be extendable to other smooth models incorporating TCA.

The "Two-Faced" Material

The "Spring" Analogy

What the Paper Actually Found

Why This Matters (According to the Paper)

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