# Plastic Hardening

Plastic hardening refers to increases in the flow stress, \(\bar{\sigma}\), with plastic deformation. As such, hardening is described via a functional relationship between the flow stress and isotropic hardening variable (effective plastic strain), \(\bar{\sigma}\left(\bar{\varepsilon}^p\right)\). Over the course of nearly a century of work in metal plasticity, a variety of relationships have been proposed to describe the interactions associated with different physical interpretations, deformation mechanisms, and materials. To enable the utilization of the same plasticity models for different material systems, a modular implementation of plastic hardening has been adopted such that the analyst may select different hardening models from the input deck thereby avoiding any code changes or user subroutines. In this section, additional details are given for the different models to enable the user to select the appropriate choice of model. Note, the models being discussed here are only for isotropic hardening in which the yield surface expands. Kinematic hardening in which the yield surface translates in stress-space with deformation and distortional hardening where the shape of the yield surface changes shape with deformation are not treated. For a larger discussion of the phenomenology and history of different hardening types, the reader is referred to [[1], [2], [3]].

Given the ubiquitous nature of these hardening laws in computational plasticity, some (if not most) of this material may be found elsewhere in this manual. Nonetheless, the discussion is repeated here for the convenience of the reader.

## Linear

Linear hardening is conceptually the simplest model available in LAMÉ. As the name implies, a linear relationship is assumed between the hardening variable, \(\bar{\varepsilon}^p\), and flow stress. The hardening modulus, \(H^{\prime}\), is a constant giving the rate of change of flow stress with plastic flow. The flow stress expression may therefore be written,

\[\bar{\sigma}=\sigma_y + H^{\prime}\bar{\varepsilon}^p.\]

The simplicity of the model is its main feature as the constant slope,

\[\frac{d\bar{\sigma}}{d\bar{\varepsilon}^p}=H^{\prime},\]

makes the model attractive for analytical models and cheap for computational implementations (e.g. radial return algorithms require only a single correction step). Unfortunately, the simplicity of the representation also means that it has limited predictive capabilities and can lead to overly stiff responses.

## Power Law

Another common expression for isotropic hardening is the power-law hardening model. Due to its prevalence, a dedicated `ELASTIC-PLASTIC POWER LAW HARDENING` model may be found in LAMÉ (see Section 4.8.1). This expression is given as,

\[\bar{\sigma}=\sigma_y + A<\bar{\varepsilon}^p-\varepsilon_L>^n,\]

in which \(<\cdot>\) are Macaulay brackets, \(\varepsilon_L\) is the Luders strain, \(A\) is a fitting constant, and \(n\) is an exponent typically taken such that \(0<n\leq1\). The Luders strain is a positive, constant strain value (defaulted to zero) giving an initially perfectly plastic response in the plastic deformation domain (see Fig. 4.20). The derivative is then simply,

\[\frac{d\bar{\sigma}}{d\bar{\varepsilon}^p}=nA<\bar{\varepsilon}^p-\varepsilon_L>^{\left(n-1\right)}.\]

Note, one difficulty in such an implementation is that when the effective equivalent plastic strain is zero, numerical difficulties may arise in evaluating the derivative and necessitate special treatment of the case.

## Voce

The Voce hardening model (sometimes referred to as a saturation model) uses a decaying exponential function of the equivalent plastic strain such that the hardening eventually saturates to a specified value (thus the name). Such a relationship has been observed in some structural metals giving rise to the popularity of the model. The hardening response is given as,

\[\bar{\sigma}=\sigma_y + A\left(1-\exp\left(-n\bar{\varepsilon}^p\right)\right),\]

in which \(A\) is a fitting constant and \(n\) is a fitting exponent controlling how quickly the hardening saturates. Importantly, the derivative is written as,

\[\frac{d\bar{\sigma}}{d\bar{\varepsilon}^p}=nA\exp\left(-n\bar{\varepsilon}^p\right),\]

and is well defined everywhere giving the selected form an advantage over the aforementioned power law model.

## Johnson-Cook

The Johnson-Cook hardening model is a variant of the classical Johnson-Cook [[4], [5]] expression. In this instance, the temperature-dependence is neglected to focus on the rate-dependent capabilities while allowing for arbitrary isotropic hardening forms via the use of a user-defined hardening function. With these assumptions, the flow stress may be written as,

\[\bar{\sigma}=\tilde{\sigma}_y\left(\bar{\varepsilon}^p\right)\left[1+C\left<\ln\left(\frac{\dot{\bar{\varepsilon}}^p}{\dot{\varepsilon}_0}\right)\right>\right],\]

in which \(\tilde{\sigma}_y\left(\bar{\varepsilon}^p\right)\) is the user-specified rate-independent hardening function, \(C\) is a fitting constant and \(\dot{\varepsilon}_0\) is a reference strain rate. The Macaulay brackets ensure the material behaves in a rate independent fashion when \(\dot{\bar{\varepsilon}}^p < \dot{\varepsilon}_0\).

## Power Law Breakdown

Like the Johnson-Cook formulation, the power-law breakdown model is also rate-dependent. Again, a multiplicative decomposition is assumed between isotropic hardening and the corresponding rate-dependence dependent. In this case, however, the functional form is derived from the analysis of Frost and Ashby [[6]] in which power-law relationships like those of the Johnson-Cook model cease to appropriately capture the physical response. The form used here is similar to the expression used by Brown and Bammann [[7]] and is written as,

\[\bar{\sigma} = \tilde{\sigma}_y\left(\bar{\varepsilon}^p\right)\left[1+\text{asinh}\left(\left(\frac{\dot{\bar{\varepsilon}}^p}{g}\right)^{\left(1/m\right)}\right)\right],\]

with \(\tilde{\sigma}_y\left(\bar{\varepsilon}^p\right)\) being the user supplied rate independent expression, \(g\) is a model parameter related to the activation energy required to transition from climb to glide-controlled deformation, and \(m\) dictates the strength of the dependence.