Flow rule

In metal plasticity, the assumption that the plastic strain increment and deviatoric stress tensor have the same principal directions is encapsulated in a relation called the flow rule. Rock plasticity theories also use a similar concept except that the requirement of pressure-dependence of the yield surface requires a relaxation of the above assumption. Instead, it is typically assumed that the plastic strain increment and the normal to the pressure-dependent yield surface have the same direction, i.e.,

${\displaystyle d{\boldsymbol {\varepsilon }}_{p}=d\lambda \,{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}}$

where ${\displaystyle d\lambda >0}$ is a hardening parameter. This form of the flow rule is called an associated flow rule and the assumption of co-directionality is called the normality condition. The function ${\displaystyle f}$ is also called a plastic potential.

The above flow rule is easily justified for perfectly plastic deformations for which ${\displaystyle d{\boldsymbol {\sigma }}=0}$ when ${\displaystyle d{\boldsymbol {\varepsilon }}_{p}>0}$, i.e., the yield surface remains constant under increasing plastic deformation. This implies that the increment of elastic strain is also zero, ${\displaystyle d{\boldsymbol {\varepsilon }}_{e}=0}$, because of Hooke's law. Therefore,

${\displaystyle d{\boldsymbol {\sigma }}:{\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}=0\quad {\text{and}}\quad d{\boldsymbol {\sigma }}:d{\boldsymbol {\varepsilon }}_{p}=0\,.}$

Hence, both the normal to the yield surface and the plastic strain tensor are perpendicular to the stress tensor and must have the same direction.

For a work hardening material, the yield surface can expand with increasing stress. We assume Drucker's second stability postulate which states that for an infinitesimal stress cycle this plastic work is positive, i.e.,

${\displaystyle d{\boldsymbol {\sigma }}:d{\boldsymbol {\varepsilon }}_{p}\geq 0\,.}$

The above quantity is equal to zero for purely elastic cycles. Examination of the work done over a cycle of plastic loading-unloading can be used to justify the validity of the associated flow rule.^[2]

Consistency condition

The Prager consistency condition is needed to close the set of constitutive equations and to eliminate the unknown parameter ${\displaystyle d\lambda }$ from the system of equations. The consistency condition states that ${\displaystyle df=0}$ at yield because ${\displaystyle f({\boldsymbol {\sigma }},{\boldsymbol {\varepsilon }}_{p})=0}$, and hence

${\displaystyle df={\frac {\partial f}{\partial {\boldsymbol {\sigma }}}}:d{\boldsymbol {\sigma }}+{\frac {\partial f}{\partial {\boldsymbol {\varepsilon }}_{p}}}:d{\boldsymbol {\varepsilon }}_{p}=0\,.}$

Kinematics of multiplicative plasticity

The concept of multiplicative decomposition of the deformation gradient into elastic and plastic parts was first proposed independently by B. A. Bilby,^[3] E. Kröner,^[4] in the context of crystal plasticity and extended to continuum plasticity by Erasmus Lee.^[5] The decomposition assumes that the total deformation gradient (F) can be decomposed as:

${\displaystyle {\boldsymbol {F}}={\boldsymbol {F}}^{e}\cdot {\boldsymbol {F}}^{p}}$

where F^e is the elastic (recoverable) part and F^p is the plastic (unrecoverable) part of the deformation. The spatial velocity gradient is given by

${\displaystyle {\begin{aligned}{\boldsymbol {l}}&={\dot {\boldsymbol {F}}}\cdot {\boldsymbol {F}}^{-1}=\left({\dot {\boldsymbol {F}}}^{e}\cdot {\boldsymbol {F}}^{p}+{\boldsymbol {F}}^{e}\cdot {\dot {\boldsymbol {F}}}^{p}\right)\cdot \left[({\boldsymbol {F}}^{p})^{-1}\cdot ({\boldsymbol {F}}^{e})^{-1}\right]\\&={\dot {\boldsymbol {F}}}^{e}\cdot ({\boldsymbol {F}}^{e})^{-1}+{\boldsymbol {F}}^{e}\cdot [{\dot {\boldsymbol {F}}}^{p}\cdot ({\boldsymbol {F}}^{p})^{-1}]\cdot ({\boldsymbol {F}}^{e})^{-1}\,.\end{aligned}}}$

where a superposed dot indicates a time derivative. We can write the above as

${\displaystyle {\boldsymbol {l}}={\boldsymbol {l}}^{e}+{\boldsymbol {F}}^{e}\cdot {\boldsymbol {L}}^{p}\cdot ({\boldsymbol {F}}^{e})^{-1}\,.}$

The quantity

${\displaystyle {\boldsymbol {L}}^{p}:={\dot {\boldsymbol {F}}}^{p}\cdot ({\boldsymbol {F}}^{p})^{-1}}$

is called a plastic velocity gradient and is defined in an intermediate (incompatible) stress-free configuration. The symmetric part (D^p) of L^p is called the plastic rate of deformation while the skew-symmetric part (W^p) is called the plastic spin:

${\displaystyle {\boldsymbol {D}}^{p}={\tfrac {1}{2}}[{\boldsymbol {L}}^{p}+({\boldsymbol {L}}^{p})^{T}]~,~~{\boldsymbol {W}}^{p}={\tfrac {1}{2}}[{\boldsymbol {L}}^{p}-({\boldsymbol {L}}^{p})^{T}]\,.}$

Typically, the plastic spin is ignored in most descriptions of finite plasticity.

Elastic regime

The elastic behavior in the finite strain regime is typically described by a hyperelastic material model. The elastic strain can be measured using an elastic right Cauchy-Green deformation tensor defined as:

${\displaystyle {\boldsymbol {C}}^{e}:=({\boldsymbol {F}}^{e})^{T}\cdot {\boldsymbol {F}}^{e}\,.}$

The logarithmic or Hencky strain tensor may then be defined as

${\displaystyle {\boldsymbol {E}}^{e}:={\tfrac {1}{2}}\ln {\boldsymbol {C}}^{e}\,.}$

The symmetrized Mandel stress tensor is a convenient stress measure for finite plasticity and is defined as

${\displaystyle {\boldsymbol {M}}:={\tfrac {1}{2}}({\boldsymbol {C}}^{e}\cdot {\boldsymbol {S}}+{\boldsymbol {S}}\cdot {\boldsymbol {C}}^{e})}$

where S is the second Piola-Kirchhoff stress. A possible hyperelastic model in terms of the logarithmic strain is ^[6]

${\displaystyle {\boldsymbol {M}}={\frac {\partial W}{\partial {\boldsymbol {E}}^{e}}}=J\,{\frac {dU}{dJ}}+2\mu \,{\text{dev}}({\boldsymbol {E}}^{e})}$

where W is a strain energy density function, J = det(F), μ is a modulus, and "dev" indicates the deviatoric part of a tensor.

Flow rule

Application of the Clausius-Duhem inequality leads, in the absence of a plastic spin, to the finite strain flow rule

${\displaystyle {\boldsymbol {D}}^{p}={\dot {\lambda }}\,{\frac {\partial f}{\partial {\boldsymbol {M}}}}\,.}$

Loading-unloading conditions

The loading-unloading conditions can be shown to be equivalent to the Karush-Kuhn-Tucker conditions

${\displaystyle {\dot {\lambda }}\geq 0~,~~f\leq 0~,~~{\dot {\lambda }}\,f=0\,.}$

Consistency condition

The consistency condition is identical to that for the small strain case,

${\displaystyle {\dot {\lambda }}\,{\dot {f}}=0\,.}$