Plastic Flow Rules and Hardening Models

In the study of solid mechanics, the ability to accurately predict how a material behaves after it surpasses its elastic limit is fundamental to engineering design and failure analysis. This transition into permanent, or plastic deformation, is captured through constitutive modeling. A robust plasticity model is built upon three essential pillars: the Yield Criterion, which defines the boundary of elastic behavior; the Plastic Flow Rule, which dictates the direction of deformation; and the Hardening Model, which describes how that boundary evolves as deformation progresses.

This article delves into the theoretical frameworks of flow rules and hardening models, which are critical for simulating complex material behaviors.
Once the stress state reaches or exceeds the yield surface, the material begins to undergo plastic flow. The primary objective of a flow rule is to establish a mathematical relationship between the plastic strain increment $\dot{\varepsilon}{ij}^p$ and the current stress state $\sigma{ij}$.

1. The Normality Rule and Associated Flow

In classical plasticity theory, the most widely adopted assumption is the Normality Rule (also known as the principle of orthogonality). This principle posits that the plastic strain increment vector $\Delta \varepsilon_{ij}^p$ must be perpendicular to the yield surface at the current stress point.

If we define the yield function as $f(\sigma_{ij}, \kappa) = 0$ (where $\kappa$ represents an internal hardening variable), the normality rule expresses the plastic strain increment as:
$$\dot{\varepsilon}{ij}^p = \dot{\lambda} \frac{\partial f}{\partial \sigma{ij}}$$
In this equation, $\dot{\lambda} \ge 0$ is known as the plastic multiplier. The magnitude of $\dot{\lambda}$ is determined by the loading rate and the material's specific hardening characteristics.

When the yield function $f$ is identical to the plastic potential function $g$, the rule is referred to as an Associated Flow Rule. For many ductile metals, such as those modeled using the Von Mises criterion, the associated flow rule provides highly accurate predictions of plastic deformation.

2. Non-associated Flow Rules

While associated flow rules work well for metals, they often fail when applied to granular materials like sand, soil, or rock. In these materials, an associated flow rule tends to significantly overpredict the dilatancy (the tendency of the material to increase in volume during shear).

To rectify this discrepancy, researchers employ Non-associated Flow Rules. In this framework, the plastic potential function $g$ is treated as a distinct entity from the yield function $f$:
$$\dot{\varepsilon}{ij}^p = \dot{\lambda} \frac{\partial g}{\partial \sigma{ij}}, \quad f \neq g$$
By independently defining $g$, engineers can calibrate the direction of plastic strain increments to better match experimental observations, allowing for a more realistic simulation of volumetric changes during shear.

Hardening Models

As plastic deformation accumulates, the material's internal structure changes, causing the yield surface to evolve. This evolution is known as hardening. Based on the geometric transformation of the yield surface in stress space, hardening models are categorized into three main types.

1. Isotropic Hardening

Isotropic hardening assumes that the yield surface expands or contracts uniformly in all directions within the stress space, while its center remains fixed at the origin.

• Characteristics: The size of the yield surface increases as a function of the equivalent plastic strain $\bar{\varepsilon}^p$.
• Mathematical Representation: If the initial yield surface is $f(\sigma_{ij}) = 0$, the hardened surface is expressed as $f(\sigma_{ij}, \bar{\varepsilon}^p) = 0$.
• Application: This model is highly effective for describing materials undergoing monotonic loading (e.g., a simple tension test), where the material's strength increases uniformly.

2. Kinematic Hardening

Standard isotropic models cannot account for the Bauschinger Effect, a phenomenon observed during cyclic loading. The Bauschinger effect describes how plastic deformation in one direction (e.g., tension) reduces the yield strength in the opposite direction (e.g., compression).

To capture this, Kinematic hardening assumes that the yield surface undergoes a translation in stress space without changing its shape or size.

• Characteristics: This model introduces a tensor known as the Back Stress ($\alpha_{ij}$), which represents the internal stress field caused by microstructural changes like dislocation pile-ups.
• Mathematical Representation: The yield function is modified to account for the shift in the center:
  $$f(\sigma_{ij} - \alpha_{ij}) = 0$$
• Physical Significance: The back stress $\alpha_{ij}$ effectively "moves" the yield surface, allowing the model to predict the reduced yield point during reverse loading.

3. Combined Hardening

In most industrial applications, particularly in fatigue analysis and metal forming, materials exhibit both isotropic and kinematic characteristics. Consequently, a Combined Hardening model is often employed.

• Isotropic Component: Responsible for the expansion of the yield surface (representing overall strength increase).
• Kinematic Component: Responsible for the translation of the yield surface (capturing the Bauschinger effect).

By integrating both, the model can accurately simulate the complex hysteresis loops observed during cyclic loading.

Summary and Practical Illustration

To synthesize these concepts, consider the cyclic loading of a metal rod:

1. Initial State: The rod begins with an initial Von Mises yield circle centered at the origin.
2. Unidirectional Tension: As stress exceeds the yield point, the material deforms plastically. Isotropic hardening causes the circle to expand, and the flow rule determines the direction of the strain.
3. Reverse Compression: When the load is reversed, an isotropic-only model would predict a very high compressive yield strength. However, due to kinematic hardening, the yield circle has shifted toward the tensile direction. This means the material will yield in compression much earlier than expected, perfectly mirroring the experimental Bauschinger Effect.

In modern Finite Element Analysis (FEA) software like Abaqus or ANSYS, selecting the appropriate hardening model and flow rule is a critical step. Whether simulating the deep drawing of a car panel or the fatigue life of an aircraft component, the accuracy of the numerical prediction depends entirely on how well these theoretical frameworks represent the underlying physics of the material.