Yielding, Plasticity & Non-Linear Material Behaviour

Yield and the Onset of Plastic Behaviour

Yielding is the transition from reversible elastic deformation to permanent plastic deformation. Below the yield stress, the material deforms elastically and returns to its original shape on unloading. Above the yield stress, a portion of the deformation is permanent — the material retains a residual strain when the load is removed. The yield stress is therefore a critical response property: it defines the boundary between the elastic regime (where the linear elastic constants govern) and the plastic regime (where a plasticity model must govern). For most structural analyses, the design intent is that the structure remains below yield under the limit loads; but for analyses that consider ultimate loads, crash, impact, forming, or post-buckling behaviour, the structure may go significantly beyond first yield, and the plasticity model becomes the primary determinant of the predicted response. The yield stress is a response property that enters the plasticity model; a yield allowable, derived statistically, is a separate quantity that enters the margin calculation.

Why Yield Stress Alone Does Not Define Post-Yield Behaviour

A single yield-strength number tells the engineer when plastic deformation begins, but it tells nothing about how the material behaves after yield. Does the material harden as it deforms, and if so, how rapidly? Does it harden isotropically (the yield surface expands uniformly) or kinematically (the yield surface translates)? Does it reach a plateau and behave as a perfect plastic? Does the hardening saturate? The post-yield behaviour is defined by the hardening law and the hardening curve, not by the yield point alone. A structure that is loaded just past yield behaves very differently under a perfect-plastic model (no hardening, collapse at the plastic moment) than under a model with strong isotropic hardening (the material strengthens as it deforms, carrying more load). If the structure is expected to go significantly beyond first yield, the hardening behaviour must be characterised and entered into the model — a single yield-strength number is not a material model.

IF THE STRUCTURE IS EXPECTED TO GO SIGNIFICANTLY BEYOND FIRST YIELD, A SINGLE YIELD-STRENGTH NUMBER IS NOT A MATERIAL MODEL. The post-yield behaviour — how the material hardens, whether it hardens isotropically or kinematically, whether it reaches a plateau — is defined by the hardening law and the hardening curve, not by the yield point. A yield stress without a hardening definition is insufficient for plastic analysis.

Plastic Strain and the Decomposition of Strain

In plasticity theory, the total strain is decomposed into an elastic part and a plastic part. The elastic part is the reversible deformation that is recovered on unloading, and it is related to the stress through the elastic constants. The plastic part is the permanent deformation that remains after unloading. The plasticity model tracks the evolution of the plastic strain as the material yields and hardens. The hardening curve — true stress versus plastic strain — defines how the yield stress increases as the plastic strain accumulates. This is why the conversion to plastic strain (covered in the previous article) is essential: the plasticity model works in plastic strain, not total strain, and the hardening curve must be provided in the true-stress-versus-plastic-strain form that the solver expects. The decomposition of strain is a response-property concept — it defines how the model computes the permanent deformation, which is part of the predicted structural response.

Isotropic Hardening

Isotropic hardening is the simplest hardening rule. As the material deforms plastically, the yield surface expands uniformly in all directions in stress space — the material gets stronger in every direction, including the direction opposite to the current loading. This means that if the material is loaded in tension past yield, then unloaded and reloaded in compression, it yields at a higher stress in compression than the original yield (because the yield surface has expanded). Isotropic hardening is appropriate for monotonic loading — loading in one direction without reversal — and for many forming and crash analyses where the loading is predominantly in one direction. It is not appropriate for cyclic loading, where the material is loaded alternately in tension and compression, because it does not capture the Bauschinger effect: the real material yields at a lower stress in the reverse direction than in the forward direction, due to the residual micro-stress field developed during the forward loading. Isotropic hardening is a response-property choice that determines how the plasticity model behaves under the loading history.

Kinematic Hardening and the Bauschinger Effect

Kinematic hardening addresses the Bauschinger effect. Instead of expanding the yield surface, kinematic hardening translates the yield surface in stress space — the yield stress in the reverse direction decreases while the yield stress in the forward direction increases. This captures the real behaviour of metals under cyclic loading: after yielding in tension, the material yields in compression at a stress that is lower in magnitude than the tensile yield, because the residual micro-stresses assist the reverse yielding. Kinematic hardening is essential for cyclic loading analysis — low-cycle fatigue, seismic loading, vibration with reversed plasticity — where the loading reverses direction and the Bauschinger effect governs the response. A combined (mixed) hardening model, which both translates and expands the yield surface, can capture both the Bauschinger effect and the long-term cyclic hardening that some materials exhibit. The choice of hardening rule is a response-property decision that must match the loading character: monotonic loading needs isotropic hardening; cyclic loading needs kinematic or combined hardening.

Perfect Plasticity and Multilinear Curves

Perfect plasticity is the limiting case where the material yields but does not harden — the stress remains constant at the yield stress as the plastic strain increases. This is an idealisation, but it is useful for limit analysis and for estimating the plastic collapse load of a structure: the structure collapses when enough of it has yielded to form a mechanism, and the perfect-plastic assumption gives a lower-bound estimate of the collapse load. Multilinear hardening curves approximate the true stress-strain curve with a series of linear segments — a bilinear curve has an elastic slope and a plastic slope (a tangent modulus), a multilinear curve has several segments that approximate the curve more closely. The choice between a smooth curve and a multilinear approximation is a trade-off between accuracy and simplicity: a smooth curve (many points) is more accurate but requires more data; a bilinear curve is simple but may misrepresent the early hardening. The analyst must ensure that the approximation is adequate for the strain range the structure will experience.

von Mises Plasticity for Ductile Metals

For ductile metals, the most common yield criterion in FE analysis is the von Mises (or J2) yield criterion. The von Mises criterion is based on the deviatoric stress — the distortional component of the stress state, which drives plastic flow in metals. It is pressure-independent: a hydrostatic stress (uniform pressure in all directions) does not cause yielding, regardless of its magnitude. This is a good approximation for most ductile metals, where plastic flow is driven by shear and the hydrostatic component has little effect on yield. The von Mises yield surface is a cylinder in principal stress space, centred on the hydrostatic axis, and the material yields when the deviatoric stress reaches a critical value (related to the yield stress in uniaxial tension). The von Mises criterion is a response-property choice that defines when the plasticity model predicts the onset of yielding for a multiaxial stress state.

The Hydrostatic Stress Limitation of von Mises

The pressure-independence of the von Mises criterion is a limitation, not a feature, for materials whose yield is pressure-dependent. Some materials — porous metals, foams, soils, polymers under certain conditions, and materials with significant void growth or damage — yield at different stresses under tension and compression, or under different levels of hydrostatic pressure. For these materials, the von Mises criterion misrepresents the yield behaviour, and a pressure-dependent yield criterion (such as Drucker-Prager, Mohr-Coulomb, or Gurson-Tvergaard-Needleman for porous metals) is needed. The analyst must check whether the material's yield is pressure-dependent before applying von Mises. For most sound wrought metals under typical structural loading, von Mises is appropriate; for foams, cast metals with porosity, or polymers, it may not be. This is a response-property consideration: the yield criterion determines when and how the model predicts yielding, which is part of the predicted structural response.

VON MISES PLASTICITY IS PRESSURE-INDEPENDENT — IT DOES NOT CAPTURE PRESSURE-DEPENDENT YIELD. For ductile metals under typical structural loading this is appropriate. For foams, porous metals, soils, and some polymers where tension and compression yield differ, a pressure-dependent yield criterion is required. Do not apply von Mises without checking whether the material's yield is pressure-dependent.

Monotonic vs Cyclic Material Models

The choice between a monotonic and a cyclic material model is determined by the loading character. A monotonic model (isotropic hardening with a monotonic stress-strain curve) is appropriate for analyses where the load is applied in one direction and does not reverse — a static ultimate load analysis, a forming simulation, a crash analysis with predominantly one-directional loading. A cyclic model (kinematic or combined hardening) is required for analyses where the load reverses — low-cycle fatigue, seismic loading, vibration with reversed plasticity, ratcheting analysis. Using a monotonic model for cyclic loading over-predicts the reverse yield stress (because isotropic hardening expands the yield surface in all directions) and misses the Bauschinger effect, producing an unconservative estimate of the plastic strain accumulated under cyclic loading. The analyst must identify the loading character and select the hardening rule that matches it.

Elastic vs Plastic Material Model Comparison

The following table summarises the distinction between a linear elastic material model and a plasticity model, including when each is appropriate and what data is needed.

Cross-Link to Non-Linear Structural Analysis

A plasticity model is a non-linear material model, and it is typically used within a non-linear structural analysis — one that accounts for both material non-linearity (the plasticity) and possibly geometric non-linearity (large deformation, large rotation, contact). The interaction between the material non-linearity and the geometric non-linearity can be significant: a structure that yields may undergo large deformation that changes the load path, which in turn changes the stress distribution and the extent of yielding. The solver must iterate to find the equilibrium configuration at each load step. The analyst should cross-reference the non-linear structural analysis articles for the solution procedures, the convergence controls and the interpretation of non-linear results. The material model is one part of the non-linear analysis; the solution procedure and the convergence behaviour are the other part, and both must be correct for a credible result.

Key Takeaways

• Yield stress defines the onset of plasticity; it does not define the post-yield behaviour
• Plastic strain is the permanent component of total strain; the plasticity model tracks its evolution
• Isotropic hardening expands the yield surface — appropriate for monotonic loading
• Kinematic hardening translates the yield surface — captures the Bauschinger effect, required for cyclic loading
• Perfect plasticity is an idealisation useful for limit analysis and collapse-load estimation
• von Mises plasticity is pressure-independent — appropriate for ductile metals, not for pressure-dependent materials
• A single yield-strength number is not a material model if the structure goes beyond first yield