Flow plasticity theory

Flow plasticity is a solid mechanics theory that is used to describe the plastic behavior of materials. Flow plasticity theories are characterized by the assumption that a flow rule exists that can be used to determine the amount of plastic deformation in the material.
In flow plasticity theories it is assumed that the total strain in a body can be decomposed additively (or multiplicatively) into an elastic part and a plastic part. The elastic part of the strain can be computed from a linear elastic or hyperelastic constitutive model. However, determination of the plastic part of the strain requires a flow rule and a hardening model.
Small deformation theory.
Typical flow plasticity theories for unidirectional loading (for small deformation perfect plasticity or hardening plasticity) are developed on the basis of the following requirements:
The above requirements can be expressed in three dimensional states of stress and multidirectional loading as follows.
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.,
where formula_18 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 formula_19 is also called a plastic potential.
The above flow rule is easily justified for perfectly plastic deformations for which formula_20 when formula_21, i.e., the yield surface remains constant under increasing plastic deformation. This implies that the increment of elastic strain is also zero, formula_22, because of Hooke's law. Therefore,
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., 
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.
Consistency condition.
The Prager consistency condition is needed to close the set of constitutive equations and to eliminate the unknown parameter formula_25 from the system of equations. The consistency condition states that formula_26 at yield because formula_27, and hence
Large deformation theory.
Large deformation flow theories of plasticity typically start with one of the following assumptions:
The first assumption was widely used for numerical simulations of metals but has gradually been superseded by the multiplicative theory.
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, E. Kröner, in the context of crystal plasticity and extended to continuum plasticity by Erasmus Lee. The decomposition assumes that the total deformation gradient (F) can be decomposed as:
where Fe is the elastic (recoverable) part and Fp is the plastic (unrecoverable) part of the deformation. The spatial velocity gradient is given by 
where a superposed dot indicates a time derivative. We can write the above as
The quantity
is called a plastic velocity gradient and is defined in an intermediate (incompatible) stress-free configuration. The symmetric part (Dp) of Lp is called the plastic rate of deformation while the skew-symmetric part (Wp) is called the plastic spin:
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:
The logarithmic or Hencky strain tensor may then be defined as
The symmetrized Mandel stress tensor is a convenient stress measure for finite plasticity and is defined as
where S is the second Piola-Kirchhoff stress. A possible hyperelastic model in terms of the logarithmic strain is 
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
Loading-unloading conditions.
The loading-unloading conditions can be shown to be equivalent to the Karush-Kuhn-Tucker conditions
Consistency condition.
The consistency condition is identical to that for the small strain case, 