The thermodynamics of local action is applicable to gradient models meaning that the constitutive behavior can be formulated by means of two potential functions of a set of independent state variables, namely, the free energy density function and a dissipation potential. This method is illustrated first for a simple isotropic micromorphic model and then specialized to Aifantis strain gradient plasticity, following [8].

The micromorphic theory has been proposed simultaneously by Eringen and Mindlin [9, 10]. It consists of introducing a general noncompatible field of microdeformation, in addition to the usual material deformation gradient, accounting for the deformation of a triad of microstructural directions. The micromorphic approach can, in fact, be applied to any macroscopic quantity to introduce an intrinsic length scale in the original standard continuum model in a systematic way, as done in [7].

As usual, the total strain is split into its elastic and plastic parts:

According to the micromorphic approach, a microstrain variable *p*_{χ} is associated to *p* and regarded as an additional degree of freedom. The material behavior is then assumed to depend on the micromorphic variable and its gradient. Accordingly, the sets of degrees of freedom and the state space are enhanced as follows:

The principle of virtual power is generalized to incorporate the microstructural effects. This represents a systematic use of the method of virtual power that Germain applied to Eringen’s micromorphic medium [11]. The classic power densities of internal and contact forces are extended in the following way:

in which generalized stresses *a* and

have been introduced. The application of the method of virtual power leads to the following additional local balance equation and boundary conditions, in addition to the classic local balance of momentum and traction conditions at the outer boundary:

The microstructural effects therefore arise in the balance of energy in the form:

where *ε* is the specific internal energy,

is the heat flux vector, and

*r* is the external heat sources. The free energy density function

*ψ* is assumed to be a function of the previous set

*STATE*. The entropy principle is formulated in the form of the Clausius-Duhem inequality

in the isothermal case for simplicity. In this work, the following state laws are postulated:

State equations relate the generalized stresses to the microstrain variable and its gradient, assuming, for simplicity, that no dissipation is associated with them. This restriction will be sufficient to recover the targeted class of models. The thermodynamic forces associated with internal variables are *R* and *X*. The residual dissipation therefore is

as in the classic case. The plastic behavior is characterized by the yield function

In the micromorphic model, the yield function can still be treated as the dissipation potential providing the flow and evolution rules for internal variables. This corresponds to the hypothesis of maximal dissipation:

where

is the plastic multiplier. At this stage, a coupling between the macroscopic and microscopic variables must be introduced, for instance, via the relative cumulative plastic strain

*p*-

*p*_{χ}.

A quadratic form is now proposed for the free energy density function, with respect to elastic strain, cumulative plastic strain, relative plastic strain, and micromorphic plastic strain gradient:

The corresponding classic model describes an elastoplastic material behavior with linear elasticity characterized by the tensor of elastic moduli

and the linear hardening modulus

*H*. Isotropy has been assumed for the last term for the sake of brevity. Two additional material parameters are introduced in the micromorphic extension of this classic model, namely, the coupling modulus

*H*_{χ} (MPa) and the micromorphic stiffness

*A* (MPa mm

^{2}). The thermodynamic forces associated with the state variables are given by the relations (7):

Note that when the relative plastic strain *e*=*p*-*p*_{χ} is close to zero, the linear hardening rule retrieves its classic form and the generalized stress *a* vanishes. Only the strain gradient effect **▽***p* remains in the enriched work of internal forces (3). This is the situation encountered in the strain gradient plasticity models developed in [12]. When inserted in the additional balance equation (4), the previous state laws lead to the following partial differential equation:

which is identical to the additional partial differential equation used in the so-called implicit gradient-enhanced elastoplasticity in [13]. The microstrain *p*_{χ} is called there the “nonlocal strain measure”

Note, however, that the latter model involves only one additional material parameter, namely,

instead of two in the micromorphic approach.

In the micromorphic approach, the coupling modulus *H*_{χ} plays a central role and makes it possible to have a fully consistent thermomechanical basis for the model. When its value is high enough, it acts as a penalty term forcing the micromorphic plastic strain to follow the macroscopic one as close as possible.

The necessity of an additional boundary condition associated with the nonlocal strain measure has already been recognized in [13]. The associated Neumann condition is used in the form:

It coincides with the more general boundary condition derived from the micromorphic approach:

when *a*^{c}=0 and when

depends linearly on

**▽***p*_{χ}, as it is the case for the quadratic potential (10).

The yield function is now chosen as

where σ_{eq} is an equivalent stress measure and σ_{Y} is the initial yield stress. The hardening rule then takes the following form:

After substituting the balance equation (12) into the hardening law, yielding takes place when

This expression coincides with the enhanced yield criterion originally proposed for gradient plasticity in [14–16] and used for strain localization simulations in [17–19] when the micromorphic variable remains as close as possible to the plastic strain:

In the original work, the Laplace operator is directly introduced in the yield function either as a postulate or as a consequence of dislocation flux in the elementary volume, whereas its presence is derived here from the combination of the additional balance equation and the linear generalized constitutive equations.

As a result, Aifantis’ model has been retrieved from the micromorphic approach by choosing simple linear constitutive equations and introducing the internal constraint *p*_{χ}≡*p,* stating that the micromorphic variable coincides with the plastic strain itself. The original Aifantis model can also be directly constructed through a gradient type of internal variable model, as already done by several authors [16, 20, 21]. In particular, a recent contribution [22] rests upon the energy storage due to gradient of plastic strain to derive this class of models in the same way as in [16]. More general dissipative mechanisms can also be added.

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