## Abstract

The construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the Helmholtz free energy function of the constitutive material model. The combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator. Within the small strain framework, the choice of a quadratic potential w.r.t. the gradient term provides the widely used Helmholtz operator whose regularization properties are well known: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and finite width localization bands in softening materials. The objective is to review and propose nonlinear extensions of micromorphic and strain/damage gradient models along two lines: the first one introducing nonlinear relations between generalized stresses and strains; the second one envisaging several classes of finite deformation model formulations. The generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling. Two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach: additive split of some Lagrangian strain measure or choice of a local objective rotating frame. Three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient. A new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable.

## 1. Introduction

Regularization differential or integral operators are widely used in mechanics in order to smooth discontinuities or restore the well-posedness of boundary-value problems. They are phenomenologically introduced in convection–diffusion problems of fluid mechanics and heat transfer [1] and in the damage of solids [2], with close links to filtering techniques in image analysis [3]. Discontinuous variables like plastic strain in conventional plasticity can be smoothed at boundaries and interfaces to better reproduce physical deformation mechanisms, or in strain localization bands in the case of softening mechanical behaviour. Sharp interfaces are replaced by smooth interfaces in phase field models to simulate moving boundaries and thus avoid complex front-tracking methods [4–6]. The regularization is used in the two latter cases to obtain discretization-objective simulations results, i.e. fields that do not depend on the finite element, finite difference or Fourier grid size.

Various types of regularization methods are available relying on non-local integral operators [7], gradient formulations [8] or extra-degrees of freedom for smoothing strain or damage fields [9]. The last two techniques very often involve so-called Helmholtz-type partial differential equations (PDEs) including a Laplace operator [9] or even bi-Helmholtz operators [10], see [11] for an anisotropic Helmholtz operator. Diffusion-like operators are used in phase field models in the form of Ginzburg–Landau or Allen–Cahn equations. Close relations exist between regularization operators used in continuum damage mechanics and in phase field theory, as recognized recently [12–15].

The regularization is often seen as a mathematical tool to be introduced into the physical model in an heuristic or *ad hoc* manner. It is then presented in the form of computational recipes to enhance existing algorithms. This is for instance the case of the regularization strategy proposed for softening plasticity and damage in [9,16–19] where the additional field equations and their coupling to the physical equations are postulated as PDEs independent of the specific form of the mechanical constitutive equations. By contrast, thermodynamically based formulations have been proposed where regularization differential operators are derived from new balance equations of generalized forces [20–23]. The form of the regularization operator then follows from the choice of constitutive equations linking generalized stresses and generalized strains and is not given *a priori*. In particular, the theories resorting to extra-degrees of freedom and of their gradients can be formulated within the micromorphic approach to gradient elastoviscoplasticity, as proposed in [21], the name micromorphic denoting additional strain-like degrees of freedom describing microstructure evolution according to Mindlin [24] and Eringen & Suhubi [25]. The proposed procedure is especially useful to derive the form of the regularization operators under anisothermal or multiphysics conditions [21]. The thermodynamic foundations of phase field models are well established [26] and a unified thermomechanical formulation is possible to reconcile the gradient, micromorphic and phase-field model classes [21,27–29].

The enhancement of the free energy density function by gradient terms of strain, damage or phase-field variables is usually limited to a quadratic contribution leading to a linear relationship between the generalized stresses and the gradient terms. Most applications are also limited to the small strain framework. They deal not only with problems of strain and damage localization in the mechanics of materials and structures, but also more recently with biomechanics and energy storage [30,31]. The linearized structure of these theories ensures in most cases good regularization properties of Helmholtz or diffusion operators. The formulation and performance of such operators in nonlinear cases like non-quadratic dependence or large deformations remain largely open.

First, extensions of regularization approaches to finite deformations dealt with the simulation of strain localization in gradient materials [32,33] or non-local media [34] where Lagrangian and Eulerian formulations were presented and compared. The approach based on extra-degrees of freedom was first extended using an Eulerian formulation, a logarithmic elastic strain and an expression of the Helmholtz equation involving the Laplace operator w.r.t. to the current configuration [35]. Generalized Helmholtz equations governing micromorphic-like variables were proposed within the Lagrangian framework in [36,37] with quadratic potentials w.r.t. the Lagrangian gradient of the extra-degrees of freedom. A constitutive formulation using local rotating frames, i.e. hypoelastic laws, was developed in [17,38]. The authors in [39] made used of a multiplicative decomposition of the deformation gradient.

Guidelines for the design of regularization operators at large deformations can be deduced from a hierarchy of micromorphic and strain gradient constitutive models as proposed in [40–45]. In particular, decompositions of higher order deformation measures into elastic and plastic parts were proposed. These constitutive settings are more general than the ones targeted in this work. They aim at the formulation of physically based higher order constitutive equations and not only at regularization purposes. The aim of the present class of models is different, it concentrates on regularization operators and therefore focuses on extensions of standard elastoviscoplasticity models involving strain-like or hardening/softening variable-like additional degrees of freedom. The enhanced constitutive equations should be kept as simple as possible. In particular, dissipative contributions of higher order variables are not considered as a first step as such simplified models already provide powerful regularization properties. More specific dissipative higher order constitutive equations can be found in [40,41,46] and in the references quoted therein.

Motivations for nonlinear potentials with respect to gradient of the extra-degrees of freedom stem from recent strain gradient crystal plasticity models making use of rank one, power law or logarithmic functions of the gradient terms for better description of dislocation behaviour [47–50]. Non-quadratic potentials are seldom in the phase field community. Examples can be found for the modelling of grain boundary migration and recrystallization [51,52]. Such nonlinear relations between generalized stresses and gradient terms lead to fundamentally new kinds of differential operators whose properties are essentially unknown.

The objective of this article is to review and propose nonlinear extensions of previous micromorphic and strain/damage gradient models along two distinct lines: the first one introducing nonlinear material relations between higher order stresses and strains; the second one envisaging different classes of finite deformation formulations of the models. Following both directions, nonlinear regularization operators, with up to now mostly unknown mathematical properties, will be exhibited. We insist here on the generic character of the proposed approach, applicable in a systematic way to a large class of elastoviscoplastic and damage models allowing for anisothermal and multiphysics coupling.

The original micromorphic approach used to generate Helmholtz-like regularization operators is first recalled in §2 in the case of scalar and tensor microstrain degrees of freedom. Nonlinear extensions of the relation between generalized stresses and strains are proposed in §3, still within the small strain hypothesis. Three methods of extension to finite deformations are presented then: a fully Lagrangian method in §§4 and 5, the use of local objective rotating frames in §6 and finally in combination with the multiplicative decomposition of the deformation gradient (§7) which represents the best-suited method for anisotropic materials. In the final conclusions, the new contributions of this article are pointed out, together with the limitations and need for future developments of the approach.

An intrinsic notation is used throughout where zeroth, first, second, third and fourth order tensors are denoted by **∇**. For example, the component *ijk* of *a*_{ij,k}, the comma denoting partial differentiation with respect to the *i*th coordinate. In particular, Δ=**∇**⋅**∇** is the Laplace operator.

## 2. The original approach based on quadratic potentials

The micromorphic approach delivering linear Helmholtz-type regularization operators is illustrated for three kinds of micromorphic degrees of freedom, namely a microstrain tensor in the spirit of Eringen's theory, and scalar variables associated with equivalent total or plastic strain measures, respectively.

### (a) Microstrain tensor model

According to Eringen's and Mindlin's original approach [24,25], the material point is endowed with the usual translational degrees of freedom (d.f.), the displacement vector, *microstrain tensor* *χ*_{ij} are generally distinct from those of the classical small strain tensor *ε*_{ij}=(*u*_{i,j}+*u*_{j,i})/2. The method of virtual power is used to derive the balance equations of the theory, following [54]. The power density of internal forces is assumed to be the following linear form:

The Helmholtz free energy volume density is a function *α*, the microdeformation tensor and its gradient.

The entropy principle of thermodynamics is adopted in its local form:

Substituting the dependency of the free energy function on the chosen state variables leads to the Clausius–Duhem inequality:
*ψ*_{ref} refers to any classical reference mechanical model and *ψ*_{χ} to the generalized contributions including the microdeformation gradient and a constitutive variable *ψ*_{χ}:
*H*_{χ} and *A*. Then, the PDE (2.8) reduces to
*H*_{χ}>0, *A*>0. This is an essential property of the model ensuring regularization properties.

In the simplified version, the coupling between strain and microstrain is apparent in the generalized Hooke law:
*H*_{χ} enforce the internal constraint *gradelas* model [55,56].

### (b) Scalar microstrain

A variant of the previous model consists in reducing the number of extra-degrees of freedom from 6 to 1:
*χ* is a scalar microstrain variable. In the microdilatation model described in [41,57,58], this degree of freedom is related to the trace of Eringen's microdeformation tensor accounting for instance for microvoid volume changes. The generalized stresses of the theory then reduce to the scalar *a* and the vector *χ* at the boundary.

The two last state laws in (2.4) are replaced by the lower order ones:
*e*:=*ε*_{eq}−*χ* involves the equivalent strain measure, *ε*_{eq}, function of the three invariants of the strain tensor *ψ*_{α} related to hardening is left unspecified and can be non-quadratic. Accordingly,

### (c) Scalar plastic microstrain

As an alternative, the following relative strain measure is adopted in equation (2.15): *e*:=*p*−*χ*, where *p* is the cumulative plastic strain arising in the plasticity theory in such a way that the plastic power reads: *σ*_{eq} is the stress measure involved in the yield function:
*p* is used here as an internal variable, together with *R*_{0}, from (2.22).

The choice of the following partly quadratic potential
*R*_{ref}(*p*)=*ρ*∂*ψ*_{α}/∂*p* is the standard hardening law. It is apparent that the classical Hooke law is unaffected, in contrast with (2.20), whereas the hardening law is modified by the coupling between plastic macro- and microstrain.

It was noted in [21] that the regularized hardening law (2.24) differs from the function *R*(*χ*) which is obtained by substituting the variable *p* by the micromorphic/regularized variable *χ* in the hardening function, as initially proposed in [59]. This inconsistency leads to difficulties in combining hardening and softening material behaviour, as recognized in [20,60,61], difficulties that are settled by adopting the previous thermodynamic framework.

The hardening rule can be related to higher order space derivatives of the plastic microstrain by combining the state laws (2.24) with the balance equation (2.13):
*χ* or Neumann conditions giving the flux *R*_{ref}/∂*p*. When the internal constraint *e*≡0⇔*p*=*χ* is enforced, or, equivalently when the penalty modulus *H*_{χ} is high enough, the Laplacian of the cumulative plastic strain itself appears in the hardening rule, which corresponds to Aifantis celebrated strain gradient plasticity model [62,56]. Strain gradient plasticity there arises as a limit case of the micromorphic model. Note that the sign of the material parameter *A* in (2.25) was heavily debated in heuristically introduced higher order terms in Aifantis-like models. The present thermodynamic approach requires *A*>0 to ensure the convexity of the gradient term in the free energy, and therefore, the model regularizing power.

When the material point is under plastic loading conditions, the combination of the yield function (2.22) and of the hardening law (2.25) provides the current value of the equivalent stress measure
*σ*_{eq} is homogeneous. Considering the limit case *e*≡0 and a simple linear hardening rule, *R*_{ref}(*p*)=*Hp*, *H* being the plastic modulus, the cumulative plastic strain is solution of the differential equation

—

*Hardening materials,*: the plastic field is of hyperbolic/exponential type plus a possible parabolic contribution coming from the constant term. This model describes size dependent boundary layer effects related to size effects in the behaviour of metals for instance. The found size effects are discussed in [63].*H*≥0—

*Softening materials,*: the plastic field is of harmonic type in addition to the possible parabolic contribution. This corresponds to the localization of plastic strain into a band of finite width in the form of a sinus arch. This regularization property has been widely used, for instance, in the simulation of strain localization phenomena like shear banding [64,39].*H*<0

Both types of solutions arise in the description of propagating localization bands as encountered in the Lüders phenomenon in steels [65,66]. They illustrate the regularizing character of the original model: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and description of finite width localization bands in softening materials. The proposed micromorphic approach in the present isotropic version introduces only two additional parameters. Parameter *A* can be identified from the scaling laws for size effects in hardening plasticity or from the characteristic width of strain localization bands for softening behaviour. The parameter *H*_{χ} can be seen as a penalty term high values of which lead to the underlying gradient model, or as a true micromorphic constitutive parameter that can be identified again to better describe the size-dependent structural responses, as done in [67] for micromorphic crystal plasticity. The anisotropic case involving a large number of additional parameters is much more challenging from the identification perspective. Extended homogenization procedures can be used to identify the whole set of parameters from the consideration of an underlying periodic microstructure, as reviewed in [68–70].

## 3. Nonlinear strain gradient potentials

Recent results in the plasticity of metals have revealed the limitations of a quadratic potential *ψ*_{∇} in equation (2.15), especially regarding the scaling of size effects in dislocation plasticity [47–50,63,71,72]. Two classes of nonlinear potentials are explored below motivated by the latter references. The motivations come from crystal plasticity theory mainly but the models are developed here in the context of phenomenological laws for polycrystalline materials. In this section, the specific form of the studied free energy potential is
*χ* is a plastic microstrain degree of freedom. A normalized anisotropic norm of its gradient is introduced in Appendix A:
_{c} and a definite positive symmetric second-order tensor

### (a) Power law potential

The following power law gradient potential is investigated:
*m*≥1 for reasons of convexity. The elastic shear modulus *μ* sets the physical dimension of this energy contribution and ℓ_{c} is the actual constitutive length of the model, although other lengths could be defined from the ratio between *ψ*_{∇}, *m*=2, as in §2c. The original regularization operator (2.19) is retrieved in the isotropic case *m*=2:
*m*=1 is of particular importance as it has been considered at various places in the literature [47,48,50]:
*g*=0 due to the non-differentiability of the potential *ψ*_{∇}, and additional regularization techniques must be used in order to perform finite-element simulations in this context [50,73]. The coupling between plasticity and the micromorphic variables takes place at the level of the hardening law, with the hardening variable *α*:=*p*:
*m*=2. The special case *m*=1 for a rank one potential *ψ*_{∇} leads to the following hardening law deduced from (3.12):
*g* can be regarded as the phenomenological counterpart of the norm of the dislocation density tensor in the physical crystal plasticity theory. The quadratic case *m*=2 was used in strain gradient plasticity as a first constitutive proposal [74,75]. It turns out that the size effects predicted based on a quadratic potential are not consistent with results from physical metallurgy [67,63]. The singular case *m*=1 provides consistent scaling laws for the yield stress as a function of channel width in laminate microstructures [47,50]. In particular, the singular character of the model results in a size-dependent abrupt increase of the apparent yield stress. By contrast, regular potentials lead to a size-dependent apparent hardening modulus. The reader is referred to [73] for applications of the power law model to size effects in hardening crystal plasticity.

### (b) Logarithmic potential

Motivated by energy considerations in dislocation theory, several authors have considered logarithmic functions of scalar dislocation densities or of the norm of the dislocation density tensor [49,50,71,72]. In the present context of phenomenological metal plasticity, a logarithmic function of the norm of the gradient of the scalar plastic microstrain is proposed:
*g* still given by equation (3.2) and *g*_{0} is a constant. This function is convex with respect to *g* for *g*≥0, with *ψ*_{∇}(0)=0. It is not differentiable at *g*=0. A possible regularization is to consider that the initial value of *g* is non-zero, for instance equal to *g*_{0}. Other regularizing choices are possible like the use of a quadratic potential in the interval 0≤*g*≤*g*_{0} [50]. The generalized stress vector
*g* and vanishes at *g*=*g*_{0}. The divergence of the generalized stress vector follows:

### (c) One-dimensional example

In the one-dimensional case with a single non-vanishing stress component *σ*(*x*) (e.g. simple tension or shear) and with all variables depending on *x* only, the plastic microstrain variable is solution of the following differential equation derived from equation (3.7) for a power-law potential:
*m*=2, the linear regularization Laplace operator (2.19) and Aifantis-like model (2.25) are retrieved.

The case *m*=1 leads to the condition *p*=*χ* and no extra-hardening. This leaves the possibility of localization of plastic strain and plastic strain gradient in the form of interface dislocations as discussed in [49,50]. The corresponding singular distribution of plastic microstrain in conjunction with the relation (3.15) was shown in the latter reference to lead to a size-dependent overall increase of the apparent yield stress and to no extra-hardening.

The logarithmic potential (3.14), the associated regularization operator (3.18) and the enhanced hardening law (3.19) are now specialized to the one-dimensional case:
*R*_{ref}(*p*)=*R*_{0}, a constant initial threshold, the differential equation, *χ*′′=*μ*ℓ_{c}|*χ*′|, admits exponential solutions with cusps as illustrated in the example given in gradient plasticity in reference [49]. The logarithmic potential is inspired from strain gradient crystal plasticity models emerging from the statistical theory of dislocations [49]. It has the advantage with respect to the *m*=1 case that it can account for both enhanced strength and hardening at small sizes, as demonstrated in [50]. It is expected that similar behaviour can be obtained from the present isotropic polycrystal model, as suggested by Ohno [48].

## 4. Micromorphic and gradient hyperelasticity

Nonlinearity arises not only from nonlinear material response but also from the consideration of finite deformations. The impact of finite strains on regularization operators is largely unexplored. It is first illustrated in the pure hyperelastic case, leaving aside for a moment the inclusion of plastic effects. The Lagrangian coordinates of the material points are denoted by *Ω*_{0}, whereas their positions in the current configuration *Ω* are called **∇**^{0} and **∇**, respectively. The deformation gradient is

### (a) Finite microstrain tensor model

The linear microstrain model discussed in §2a is now extended to the finite deformation case by applying the general approach presented in [41]. The additional degrees of freedom are the six components of a microstrain tensor *p*^{(i)} being the Eulerian internal power density and the Jacobian

As an example and straightforward generalization of equation (2.6), the following potential is proposed:
*H*_{χ} was introduced and where *ψ*_{ref} is a standard hyperelastic strain energy potential (neo-Hookean, etc.). The higher order term involves a sixth-rank tensor of elasticity moduli which is symmetric and assumed definite positive [70]. The stress–strain relations (4.6) become
*A*:
^{0}(•)=(•)_{,KK} in a Cartesian frame where capital indices refer to Lagrange coordinates and the comma to partial derivation. It is linear w.r.t. Lagrangian coordinates but the full problem is of course highly nonlinear. The associated Eulerian partial differential operator is nonlinear in the form: *χ*_{IJ,KK}=*χ*_{IJ,kl}*F*_{kK}*F*_{lK}, where small index letters refer to the current Cartesian coordinate.

An Eulerian formulation of the proposed constitutive equations is possible. It will be illustrated in the next section in the case of a scalar microstrain variable for the sake of brevity.

### (b) Equivalent microstrain model at finite deformation

The set of degrees of freedom of the proposed model is given by equation (2.12) and contains a scalar microstrain variable *C*_{eq}, function of the invariants of *χ* is
^{0} is the Laplace operator with respect to Lagrangian coordinates.

An example of equivalent strain measure which the microstrain is compared with is

The formulation of a constitutive law based on Eulerian strain measures, **∇***χ*, is now envisaged. It relies on the choice of a free energy potential *ψ*. Representation theorems are available for such functions, *B*_{i} are the eigenvalues of

The hyperelastic state laws then take the form
*ψ*_{ref} refers to a standard isotropic elasticity potential in classical mechanics. Note that *B*_{eq}=*C*_{eq} as

It is essential to note that the isotropic regularization operators (4.15) and (4.21) are distinct. For, if the Lagrangian higher order elastic law is linear with respect to the constitutive quantities involved, the deduced Eulerian law is NOT linear:

## 5. Finite deformation micromorphic elastoviscoplasticity using an additive decomposition of a Lagrangian strain

The most straightforward extension of the previous framework to viscoplasticity is to introduce a finite plastic strain measure in the decomposition of a Lagrangian total strain tensor. Such Lagrangian formulations of elastoviscoplasticity involve the additive decomposition of some Lagrangian total strain measure into elastic and viscoplastic parts:
*h* with restrictions ensuring that *m*=2 corresponds to the Green–Lagrange strain measure for which this finite deformation theory was first formulated by Green and Naghdi, see [79,80] for the pros and the cons of various such formulations. This Lagrangian formulation is preferable to Eulerian ones based on corresponding Eulerian strain measures in order not to limit the approach to isotropic material behaviour [81]. The additive decomposition of the Lagrangian logarithmic strain is put forward in the computational plasticity strategies developed in [82,83]. However, there is generally no physical motivation for the selection of one or another Lagrangian strain measure within this framework. This approach favours one particular reference configuration for which the corresponding strain is decomposed into elastic and plastic parts, again without clear physical argument. Changes of reference configuration lead to complex hardly interpretable transformation rules for the plastic strain variables, see [84] for a comparison of finite deformation constitutive laws with respect to this issue. Note also that limitations arise from using a symmetric plastic strain variable

A Lagrangian conjugate stress tensor

Two straightforward extensions of the micromorphic approach to finite strain viscoplasticity based on an additive decomposition of a Lagrangian strain measure are proposed
*E*_{heq} is an equivalent total strain measure, or, alternatively,
^{0} in the same way as in equation (4.10).

## 6. Finite deformation micromorphic viscoplasticity using local objective frames

An alternative and frequently used method to formulate anisotropic elastoviscoplastic constitutive equations at finite deformations that identically fulfil the condition of Euclidean form invariance (also called material frame indifference [78]), is to resort to local objective rotating frames, as initially proposed by Dogui & Sidoroff [89,90]. A local objective rotating frame is defined by the rotation field

This method is now applied to a micromorphic model including a scalar additional d.f. *χ* as in equation (2.12). Extension to higher order tensor-valued additional degrees of freedom is straightforward. The field equations are still given by (7.6). The stresses w.r.t. the local objective frames are

The yield function and the flow rule are formulated within the rotated frame
*H*. The yield radius is chosen as the following expression inspired by the previous thermodynamically based models:

This extension of the micromorphic approach to finite deformations using rotating frames has been proposed first by [38] and used by these authors for metal forming simulations involving regularized damage laws. As a result, the regularization operator can be written as

Among all choices of rotating frames, the one associated with the logarithmic spin rate tensor [92] was claimed to be the only one such that, when

Alternative constitutive choices are possible for the higher order stresses if Laplacian operators are preferred. They amount to restricting the use of the rotating frame only for the classical elastoviscoplasticity equations and to writing independently, ^{0}, see (7.17) in the next section.

Note that limitations in the formulation of anisotropic plasticity arise from using symmetric plastic strain variable

## 7. Finite deformation micromorphic elastoviscoplasticity based on the multiplicative decomposition

The most appropriate thermodynamically based framework for the formulation of finite deformation anisotropic elastoviscoplasticity relies on the multiplicative decomposition of the deformation gradient, as settled by Mandel [94]. This method is applied here to a generalized continuum model again limited to one scalar degree of freedom, *χ*, in addition to the displacement vector,

A multiplicative decomposition is envisaged in this section in the form

In the present work, the microdeformation gradient

The power density of internal and contact forces are
*p*^{(i)} with respect to any change of observer requires the Cauchy stress

### (a) Lagrangian formulation

The Lagrangian free energy density is a function *α* a set of internal variables accounting for material hardening. Note that the usual elastic strain tensor *a*_{0}=*Ja* and

As a result the dissipation rate becomes
*R*, the positivity of dissipation is ensured. Specific expressions for *Ω* within the context of viscoplasticity can be found in [78].

As an example, the following free energy potential is proposed:
*ψ*_{α} is the appropriate free energy contribution associated with usual work-hardening (not specified here) and *p* defined as
^{0} in the isotropic case, i.e. *α*=*p*, according to (7.13), as an internal variable in (7.12). The dissipation potential *Ω* can be chosen in such a way that the residual dissipation takes the form
*f* is the yield function. As a result, the yield stress *R* is given by the following enhanced hardening law:

### (b) Eulerian formulation

In the Eulerian formulation, the free energy density is taken as a function of *ψ* to be isotropic with respect to

The constitutive choice (7.12) is now replaced by
*ψ*_{∇} is necessarily of the form

### (c) Formulation using the local intermediate configuration only

In the two previous formulations, Lagrangian or Eulerian generalized strain variables were mixed with the elastic strain variable

The yield criterion is taken as a function *Π*^{M}_{eq} is an equivalent stress measure based on the generalized Mandel stress tensor. Choosing *α*=*p*, where *p* is still given by equation (7.13), the residual dissipation takes the form
*R*. This is a specific feature of the model formulation w.r.t. the intermediate configuration.

As an example, a typical form of the free energy density function based on constitutive variables defined on the intermediate configuration, and hyperelastic laws are

## 8. Conclusion

The construction of regularization operators presented in this work is based on the introduction of strain or damage-like micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the Helmholtz free energy density function of the constitutive model. The combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the wanted regularization operator. Within the small strain framework, the choice of a quadratic potential w.r.t. the gradient term provides the widely used Helmholtz operator whose regularization properties are well demonstrated in the literature: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and description of finite width localization bands in softening materials. The thermodynamic theory also predicts the form of the coupling between the standard and micromorphic variables. When the micromorphic degrees of freedom are related to total strain tensor or equivalent strain measures, the coupling arises in an extended elasticity law. For microplastic variables, the elasticity law is unaffected whereas the hardening law is modified.

Non-quadratic potentials w.r.t. the gradient term were proposed including power and logarithmic laws motivated by recent advances in strain gradient plasticity. They provide new strongly nonlinear differential operators whose mathematical properties were not explored and are largely unknown. They were formulated considering anisotropy, as required for applications to crystalline metals and damaging composites for instance.

The presented extensions to finite deformations show that the regularization operator cannot be postulated in an intuitive way. It is rather the result of a constitutive choice regarding the dependence of the free energy function on the gradient term. Purely Lagrangian and Eulerian formulations are straightforward and lead to Helmholtz-like operators w.r.t. Lagrangian of Eulerian coordinates. Two alternative standard procedures of extension of classical constitutive laws to large strains, widely used in commercial finite-element codes, have been combined with the micromorphic approach. In particular, the choice of local objective rotating frames leads to new nonlinear regularization operators that are not of the Helmholtz type. Three distinct operators were proposed within the context of the multiplicative decomposition of the deformation gradient. A new feature is that a free energy density function depending on variables solely defined with respect to the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic degree of freedom.

Note that the results obtained for the micromorphic theory with additional degrees of freedom are also valid for gradient theories (gradient plasticity or gradient damage) once an internal constraint is imposed linking the additional degrees of freedom to strain or internal variables. This amounts to selecting high values of parameter *H*_{χ} or introducing corresponding Lagrange multipliers. The analysis was essentially limited to scalar micromorphic degrees of freedom for the sake of simplicity, even though tensorial examples were also given. Scalar plastic microstrain approaches suffer from limitations like indeterminacy of flow direction at cusps of the cumulative plastic strain in bending, for instance, see [60,97,98]. Those limitations can be removed by the use of tensorial micromorphic variable (microstrain or microdeformation tensors). The micromorphic approach is not limited to the gradient of strain-like, damage or phase field variables. It can also be applied to other internal variables, as demonstrated for hardening variables in [38,99].

It remains that the regularization properties of the derived nonlinear operators are essentially unknown, except through examples existing in the mentioned literature. For instance, the Eulerian and Lagrangian variants of the Helmholtz-type equation for scalar micromorphic strain variables have been assessed recently in [100] giving the advantage to the latter, based on finite element simulations of specific situations. The regularizing properties of more general operators should be explored in the future from the mathematical and computational perspectives in order to select the most relevant constitutive choices that may depend on the type of material classes.

It may be surprising that the constitutive theory underlying the construction of regularization operators for plasticity and damage, mainly relies on the enhancement of the free energy density function instead of the dissipative laws. It is in fact widely recognized that plastic strain gradients, e.g. associated with the multiplication of geometrically necessary dislocations, lead to energy storage that can be released by further deformation or heat treatments. However, the limitation to the enhancement of free energy potential is mainly due to the simplicity of the theoretical treatment and to the computational efficiency of the operators derived in that way. Dissipative higher order contributions remain to be explored from the viewpoint of regularization, as started in [21]. Constitutive models of that kind are already available for plasticity, damage and fracture [101,102].

## Competing interests

The author has no competing interests.

## Funding

Part of this study was carried out in the framework of project MICROMORFING (ANR-14-CE07-0035-03) funded by the Agence Nationale de la Recherche (ANR, France). This support is gratefully acknowledged.

## Acknowledgements

The author thanks his colleagues and PhD students at Centre des Matériaux Mines ParisTech for many fruitful discussions.

## Appendix A. Notations for higher order gradients of a scalar field

Notations are introduced for the normalized gradient of a scalar field *ϕ* corresponding to a micromorphic variable, and also to a phase field: _{c} is a characteristic length introduced for normalization purposes. The direction of the gradient is given by the vector

The gradient of the anisotropic norm is calculated in a similar way:

- Received October 30, 2015.
- Accepted March 29, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.