Engineers who design metal forming operations rely on a mathematical tool called the upper bound method to estimate how much force is needed to deform a workpiece. The approach has a long history and remains a workhorse of metal forming analysis, but most published solutions share a common limitation: they assume the von Mises or Tresca yield criteria, which describe how materials begin to flow plastically. A new study published in Discover Mechanical Engineering presents an efficient way to relax that restriction, combining a modern formulation of the upper bound theorem with a work function that can describe materials exhibiting the strength differential effect, while also handling Coulomb friction at the tool-workpiece interface.
The strength differential effect refers to a property of some plastically incompressible materials whose yielding depends on whether they are pulled in tension or pushed in compression. In other words, the yield stress measured in uniaxial tension differs from that measured in uniaxial compression, even though the classical von Mises criterion treats the two as identical. The new paper, authored by Elena Lyamina, notes that this effect is an important characteristic of certain materials, and that ignoring it can distort the predictions of forming analyses. The work was published open access on 22 October 2026 as article number 208 in volume 5 of the journal.
The central difficulty that the method addresses lies in how the plastic work rate is expressed. For the von Mises and Tresca criteria, the plastic work rate can be written simply in terms of the strain rate invariants, which makes the upper bound functional easy to evaluate. For more general yield criteria, that simplicity disappears. The study sidesteps the problem by formulating the constitutive equations through a work function instead of a yield criterion. A work function, if chosen appropriately, can closely approximate available yield criteria, and a recently proposed work function suited to materials with a strength differential effect serves as the backbone of the new approach.
A second obstacle concerns friction. In the classical version of the upper bound theorem, the load required to deform a body is evaluated using a kinematically admissible velocity field, one that satisfies all the geometric constraints of the deformation. That requirement means many boundary conditions cannot be included in the formulation, and Coulomb friction, in which the frictional traction is proportional to the normal pressure, is a prominent example. An alternative formulation of the theorem, highlighted in the paper, relaxes the admissibility requirement and works with a so-called trial velocity field that need not be kinematically admissible. This relaxation opens the door to a much larger class of boundary value problems, including those governed by Coulomb’s friction law.
To demonstrate the method, the study analyzes the upsetting of a solid cylinder compressed between two rough parallel plates, a process that is sometimes used as a friction test. The cylinder has radius R and height 2H, and the analysis exploits symmetry to consider only the upper half of the domain. The lateral surface of the cylinder is traction free, the shear stress vanishes on the symmetry plane, and Coulomb friction with a coefficient mu acts at the interface with the moving plate. The goal is to evaluate the compression force F required to drive the process.
The trial velocity field is built with care. The solution divides the material into a plastic region and a rigid region separated by a velocity discontinuity surface. A key refinement over earlier work is the treatment of the region near the symmetry axis: a previous upper bound solution that assumed Coulomb friction and the von Mises criterion did not account for a rigid region that must appear there. The present paper modifies that trial velocity field to accommodate the rigid region, introducing an arbitrary function that allows the discontinuity line to be prescribed rather than solved for, which the author notes is considerably easier and mirrors common practice in extrusion analyses. The velocity field is constructed so that the frictional boundary condition at the plate is satisfied through Coulomb’s law, fixing the angle between the resultant traction and the axis of compression.
With the velocity field in place, the compression force follows from two integrals that must be evaluated numerically: one over the volume of the cylinder involving the work function, and one over the velocity discontinuity surface accounting for the energy dissipated by the velocity jump. The work function used depends on a single material parameter q, defined as the ratio of the yield stress in tension to the yield stress in compression. When q equals one, the model reduces to the von Mises criterion, providing a direct benchmark. The parameter c that enters the work function is derived from q, and the shear yield stress entering the discontinuity contribution is likewise expressed through the same framework.
The numerical results reveal how strongly the strength differential effect shapes the predicted loads. Calculations were performed for four values of q: the von Mises benchmark at q equal to one, values of 1.28 and 0.78 drawn from polycrystalline calculations for body-centered cubic and face-centered cubic polycrystals respectively, and a value corresponding to the least value of q at which the work function remains convex. Across all investigated cases, the value of q significantly affected the compression force. When q is less than one, meaning the material is relatively stronger in compression, the predicted limit load exceeds the von Mises prediction; when q is greater than one, the load falls below it. The author concludes that this underscores the need to account for the strength differential effect in theoretical solutions.
The geometry of the deformation also matters. The dimensionless cylinder aspect ratio h, the ratio of half-height to radius, was varied together with friction coefficients of 0.2, 0.15, 0.1 and 0.05. As expected, the compression force decreases as the friction coefficient decreases and as h increases, although the force becomes nearly insensitive to h once the cylinder is sufficiently squat. In that regime the simpler solution without a rigid region applies, and the range of aspect ratios where this simpler solution holds widens as friction weakens; at a friction coefficient of 0.05 with q equal to 1.28, the simpler solution covers the entire investigated range. The two trial velocity fields, with and without the rigid region, yield the same force at a transition value of h, which varied between 0.1 and 0.55 across the parameter sets studied. That transition value matters for the flow pattern and is important for sequential limit analysis.
The practical payoff may lie in friction testing. Because cylinder compression is sometimes used to evaluate friction stress, a model that misrepresents the material’s yielding could misattribute the discrepancy to friction. The study shows that neglecting the strength differential effect in the material model can affect the interpretation of the friction test, a caution for anyone extracting friction coefficients from compression data. The author emphasizes that the method is efficient because the work function allows the upper bound functional to be computed through a unified procedure independent of the yield criterion, and the trial velocity field formulation accommodates Coulomb friction that the classical theorem cannot handle. For materials whose tension and compression yield stresses differ appreciably, the framework offers a way to keep both the material behavior and the interface physics inside the same calculation, with the strength differential effect emerging as a first-order influence on predicted forming loads rather than a refinement that can be safely ignored.
Subject of Research: An upper bound plasticity method accounting for the strength differential effect and Coulomb friction, applied to cylinder compression
Article Title: An efficient method for accounting the strength-differential effect and Coulomb friction in upper bound solutions of plasticity
Article References: Elena, L. (2026). An efficient method for accounting the strength-differential effect and Coulomb friction in upper bound solutions of plasticity. Discover Mechanical Engineering, 5(1), Article 208. https://doi.org/10.1007/s44245-026-00341-5
Image Credits: AI Generated
DOI: 10.1007/s44245-026-00341-5
Keywords: upper bound method, plasticity, strength differential effect, Coulomb friction, metal forming, cylinder compression, yield criterion, work function, friction test, limit analysis, von Mises criterion, efficient
News Source: Denise Maddox. (October 28, 2026). New Plasticity Method Captures Strength Differential and Friction in Metal Forming. Scienmag.
