Plasticity overview

Let $\sigma$$,$ $\varepsilon$$, {\rm and}\$    $\varepsilon$$^p$ be the stress, total strain, and plastic strain vectors, respectively. It is assumed that the total strain is decomposed into reversible elastic and irreversible plastic parts

$$\mbox{\boldmath$\varepsilon $} = \mbox{\boldmath$\varepsilon $} ^e + \mbox{\boldmath$\varepsilon $} ^p$$ (4.1)

The elastic response is characterized in terms of elastic constitutive matrix $D$ as

$$\mbox{\boldmath$\sigma$} = \mbox{\boldmath$D$} \mbox{\boldmath$\varepsilon $} ^e = \mbox{\boldmath$D$} ( \mbox{\boldmath$\varepsilon $} - \mbox{\boldmath$\varepsilon $} ^p)$$ (4.2)

As long as the stress remains inside the elastic domain, the deformation process is purely elastic and the plastic strain does not change. It is assumed that the elastic domain, denoted as $IE$ is bounded by a composite yield surface. It is defined as

$$IE=\{( \mbox{\boldmath$\sigma$} , \mbox{\boldmath$\kappa$} )\vert f_i( \mbox{\boldmath$\sigma$} , \mbox{\boldmath$\kappa$} )<0, \rm {for\ all\ }i\in\{1,\cdots,m\}\}$$ (4.3)

where $f_i($$\sigma$$,$$\kappa$$)$ are $m\ge1$ yield functions intersecting in a possibly non-smooth fashion. The vector $\kappa$ contains internal variables controlling the evolution of yield surfaces (amount of hardening or softening). The evolution of plastic strain $\varepsilon$$^p$ is expressed in Koiter's form. Assuming the non-associated plasticity, this reads

$$\dot{\mbox{\boldmath$\varepsilon$}}^p=\sum^{m}_{i=1} \lambda^i \p... ...\mbox{\boldmath$\sigma$}}g_i(\mbox{\boldmath$\sigma$},\mbox{\boldmath$\kappa$})$$ (4.4)

where $g_i$ are plastic potential functions. The $\lambda^i$ are referred as plastic consistency parameters, which satisfy the following Kuhn-Tucker conditions

$$\lambda^i\ge0,\;f_i\le0,\;{\rm and}\ \lambda^i f_i=0$$ (4.5)

These conditions imply that in the elastic regime the yield function must remain negative and the rate of the plastic multiplier is zero (plastic strain remains constant) while in the plastic regime the yield function must be equal to zero (stress remains on the surface) and the rate of the plastic multiplier is positive. The evolution of vector of internal hardening/softening variables $\kappa$ is expressed in terms of a general hardening/softening law of the form

$$\dot{\mbox{\boldmath$\kappa$}} = \dot{\mbox{\boldmath$\kappa$}}(\mbox{\boldmath$\sigma$}, \mbox{\boldmath$\lambda$})$$ (4.6)

where $\lambda$ is the vector of plastic consistency parameters $\lambda_i$.