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Reformulating plastic instabilities within a damage-like variational framework
Baptiste Reyne
TL;DR
Plastic instabilities share softening and localization across distinct materials, while existing descriptions face experimental and computational challenges. This paper introduces an explicit damage-like variable to separate hardening from softening and builds a variational, regularization-compatible model. Simulations show localization spreading as the softening variable’s Lipschitz constraint increases, and the framework is compatible with regularization approaches from continuum damage mechanics.
Problem
Plastic instabilities share softening and localization across materials, but linking local events to global measurements and obtaining convergent, non-spurious, mesh-insensitive computations remain challenging.
Source: §1 · p. 2, §3 · p. 14Method
The framework represents softening with an explicit damage-like variable, separates it from hardening, and formulates a variational model compatible with regularization methods.
Source: §2.1 · p. 5, §4 · p. 16Results
As the softening variable’s Lipschitz constraint increases, strain distributions become less localized and localization-band width increases.
Source: §3 · pp. 12–13Takeaways & Limitations
The framework is compatible with several regularization approaches inherited from continuum damage mechanics while describing plastic instabilities.
Source: §3 · p. 16, §4 · p. 16The implementation’s solutions are non-unique, sensitive to initial plastic strain, and often take strain steps larger than δ, likely through snap-through.
Source: §3 · p. 14
Abstract
from arXiv · showhide
Yield point phenomena in metals, anticracks in soils and snows, or planewise collapses in lattice metamaterials, are different manifestations of plastic instabilities characterized by stress softening and strain localization. Several independent constitutive descriptions exist and suffer from similar challenges: experimental difficulties to link local events to global measurements, and computational lack of convergence, spurious localization and mesh sensitivity. This work proposes a minimal field-agnostic constitutive framework where plastic softening is represented by a damage-like internal variable which explicitly separates stable hardening behavior from unstable softening. The resulting formulation admits a variational structure and can leverage regularization tools originally developed by the ductile fracture community. The framework aims at emphasizing similarities within plastic instability phenomena and establishing a bridge with continuum damage mechanics, to open a path towards regularized variational formulations of localized plastic flows.
1. Introduction
Plastic instabilities occur when uniform loads produce inhomogeneous deformation, appearing across metals, soils, snows, and architected materials. The paper proposes separating stable hardening from unstable softening with dedicated variables in a variational, regularization-compatible framework.
1. Introduction
- Plastic instabilities include localization bands in metals, compaction bands or anticracks in soils and snows, and planewise collapse or yielding in architected materials.The examples span materials with near-constant-volume plasticity, volume-changing plasticity, and local buckling.Source: §1 · p. 2
- Across these phenomena, loading produces peak resistance followed by softening and localized deformation, while global measurements typically show a plateau.The plateau is characteristic of a phase change in many contexts.Source: §1 · p. 2
- Existing macroscopic descriptions often do not separate stable hardening from unstable softening because both respond to a positive strain rate.Approaches include negative hardening, negative strain-rate sensitivity, and internal-variable models such as KEMC for the PLC effect.Source: §1 · p. 3
- Damage and ductile-fracture models use an independent softening variable and derive constitutive equations from a scalar potential, enabling variational solution methods.Phase-field regularizes with gradient penalization, whereas Lip-field constrains the softening variable to be Lipschitz-continuous [14, 6].Source: §1 · p. 3
- The proposed minimal proof of concept must capture softening followed by possible re-hardening while distinguishing hardening and softening with dedicated independent variables.It should also admit a variational structure or remain compatible with gradient approaches.Source: §1 · pp. 3–4
- The paper adapts a Lip-field model–solver couple to softening plasticity and aims to promote a shared computational toolbox for softening mechanics.It formulates and regularizes a minimal model, then illustrates and analyzes a Lip-field-compatible solver.Source: §1 · p. 4
2. Model–solver formulation
The model separates material hardening from softening in a variational formulation, then extends it to a one-dimensional bar. Without regularization, the discretized problem is ill-posed; Lip-field constraints and a staggered solver address localization and non-convex minimization.
2.1. Minimal recurrent softening model
- The constitutive variables derive from a state potential, and their evolution is imposed as equality or inequality constraints in its minimization.The softening variable's evolution is bound to accumulated plastic strain through the condition μ = 0.Source: §2.1 · pp. 4, 6
- The local model separates hardening and softening variables; its potential is independently convex in each group but non-convex overall, providing a source of localization.The resulting constrained problem has stable solutions, although its Hessian is only conditionally positive definite.Source: §2.1 · pp. 5, 7
- A plastic increment p step of δ represents a localization event as the bar operator progresses from 0 to 100%.The event is accompanied by a controlled stress drop bounded by the plastic increment, even as the material globally hardens.Source: §2.1 · pp. 4, 5, 7
2.2. Extension to one dimension
- The non-regularized bar problem is ill-posed: it admits mesh-sensitive localized strain bursts or homogeneous internal-variable evolution with successive load drops.Some satisfying solutions cannot be reached with standard minimization.Source: §2.2 · p. 9
2.3. One-dimensional solver with spatial regularization
- Lip-field regularization constrains the softening variable using a distance metric, imposing regularity without requiring gradient computation or differentiability.The internal length sets the spatial scale of the constraint, which is implemented as a linear constraint.Source: §2.3 · pp. 10–11
- A staggered scheme splits the remaining non-convex minimization into hardening and softening steps, fixing one group of variables while solving for the other.The iterations repeat until the converged solutions stagnate.Source: §2.3 · p. 11
- With increasing regularization, strain distributions become less localized and localization bands widen; the initial localization occurs at the bar's central defect.In the reported computations, localization propagates rightward before leftward.Source: Figure 2 · p. 12
3. Results and discussion
The simulations show that regularization controls the width and homogeneity of localized deformation, while the proposed framework connects plastic-softening work to measurable parameters and established regularization approaches. The discussion also identifies solver sensitivities and implementation trade-offs.
3. Results and discussion
- In the 15-node tensile test, internal lengths above an element effectively regularize localization; constrained softening widens bands and makes strains more homogeneous.Localization starts at the seeded center defect, then propagates right before left.Source: §3 · pp. 12–13
- The implementation produces nonunique solutions, is sensitive to defect magnitude, and can jump from softening to re-hardening as localization starts.Arc-length solvers or altered staggered algorithms may alleviate the jump; the general implementation requires specialization to improve efficiency.Source: §3 · p. 14
- Softening-related plastic work can be calibrated from regularization and localization profiles, while δ and σa can be inferred from monotonic tests.For Lip-field with piecewise-linear softening, the localization profile is triangular, with base 2ℓ, width w, and thickness b.Source: §3 · p. 15
- The field-agnostic softening variable d can represent continuous phase transitions and connect macroscopic behavior to structural or chemical processes.Examples include solute densities during dislocation (un)pinning and density changes in lattices or anticracks.Source: §3 · p. 15
- Lip-field regularization acts only after softening begins, whereas Phase-field requires domain-wide gradient computations at every time step.Phase-field needs less solver intervention, while Lip-field requires dedicated implementations; the framework can also add a gradient contribution to the softening potential.Source: §3 · pp. 15–16
- The framework is compatible with regularization approaches from continuum damage, and its internal length corresponds to localization-band thickness.The authors expect the observations to hold when the computational strategy is scaled to higher dimensions.Source: §3 · pp. 14, 16
4. Conclusions
The paper reformulates plastic instabilities as variational softening through an explicit damage-like variable, while identifying proof-of-concept scope and future development needs.
4. Conclusions
- The formulation represents plastic instabilities as a variational softening problem with an explicit damage-like variable, enabling use of ductile-fracture regularization and solution strategies.Source: §4 · p. 16
- The proof of concept uses linear elasticity, linear work-hardening, linear softening, and one localization event, while allowing arbitrary hardening and an extension to recurring softening.Source: §4 · p. 16
- Future work includes solver optimization, higher-dimensional models, and applications to sheet-metal forming and compressive responses of architected and granular media.Source: §4 · p. 17
Appendix A.2. Bar operator derivative
The bar operator connects plastic evolution to the softening variable; its derivative is undefined at p = δ, and the text notes a pragmatic treatment.
Appendix A.2. Bar operator derivative
- The bar operator links plastic evolution to the softening variable, and its derivative is undefined at p = δ; the indicator function is also used.Source: Appendix A.2 · p. 18
Appendix A.3. Alternative bar operator definition for recurring softening
The passage states that the alternative bar operator’s derivative is defined almost everywhere.
Appendix A.3. Alternative bar operator definition for recurring softening
- The alternative bar operator’s derivative is defined almost everywhere.Source: Appendix A.3 · p. 18