## Abstract

This work outlines a novel variational-based theory for the phase-field modelling of ductile fracture in elastic–plastic solids undergoing large strains. The phase-field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modelling. It is linked to a formulation of gradient plasticity at finite strains. The framework includes two independent length scales which regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones, and guarantees on the computational side a mesh objectivity in post-critical ranges.

## 1. Introduction

Ductile fracture is a phenomenon that couples at the macroscopic level *plastic deformations* with the *accumulation of damage* and *crack propagation*. The process of damage that follows extensive plastic deformations covers the macroscopic effects of degrading stiffness, strength and ductility up to a critical state where rupture occurs. Damage is caused by deformation mechanisms at the microscopic level, such as void nucleation, growth and coalescence, the formation of micro-shear bands and micro-cracks. The prediction of these failure mechanisms plays an extremely important role in various engineering applications. A large number of phenomenological and micro-mechanical approaches exist for the continuum modelling of ductile fracture; see Lemaitre & Chaboche [1], Besson [2] and Li *et al.* [3] for overviews.

However, the demanding *tracking of sharp crack surfaces* causes substantial difficulties in numerical implementations and is often restricted to simple crack topologies. This difficulty can be overcome by recently developed *phase-field approaches* to fracture, which regularize sharp crack discontinuities within a pure continuum formulation. This *diffusive crack modelling* allows the resolution of complex failure patterns, such as crack branching phenomena in dynamic fracture of brittle solids. In contrast with computational models which model sharp cracks, the phase-field approach is a spatially smooth continuum formulation that can be implemented in a straightforward manner by coupled multi-field finite-element solvers.

Three basic approaches to the *regularized modelling* of Griffith-type brittle fracture mechanics may be distinguished. (i) The phase-field approach by Karma *et al.* [4] and Hakim & Karma [5] applies a Ginzburg–Landau-type evolution of an unconstrained crack phase field, using a non-convex degradation function that separates unbroken and broken states. It *lacks an explicit definition of irreversibility constraints* for the crack evolution. (ii) The approach of Francfort & Marigo [6] and Bourdin *et al.* [7] adopts the variational structure and *Γ*-convergent regularization of image segmentation developed by Mumford & Shah [8] and Ambrosio & Tortorelli [9] for the analysis of finite increments in quasi-static crack evolution. The irreversibility of the fracture process is modelled by *evolving Dirichlet-type boundary conditions*, while the scalar auxiliary field used for the regularization is not constrained. This needs the implementation of non-standard code structures in typical finite-element solvers. (iii) The phase-field approach by Miehe *et al.* [10] is a gradient damage theory with a *local irreversibility constraint* on the crack phase field, but is equipped with constitutive structures rooted in fracture mechanics. It incorporates regularized crack surface density functions as central constitutive objects, which are motivated in a descriptive format based on geometric considerations. Such a formulation can easily be implemented by a multi-field finite-element solver with monolithic or staggered solution of the coupled problem. Recent works on brittle fracture along this third line are Pham *et al.* [11], Borden *et al.* [12] and Verhoosel & de Borst [13]. Extensions to the phase-field modelling of ductile fracture are exclusively related to the third line, representing conceptually a coupling of gradient damage mechanics with models of elasto-plasticity. Variational-based approaches to combined brittle–ductile fracture are outlined in Ulmer *et al.* [14] and Alessi *et al.* [15]. The model suggested in Ambati *et al.* [16] uses a characteristic degradation function that couples damage to plasticity in a multiplicative format. However, these settings combine models of *local* plasticity *without inherent length scales* to the gradient-damage-type phase-field modelling of fracture. This does not meet the demand of related plastic and damage length scales, keeping regularized fracture zones inside of plastic zones, and does not guarantee on the computational side a mesh objectivity in post-critical ranges. This is achieved in the recent works of Miehe *et al.* [17]; 18 that couples gradient plasticity to gradient damage.

This paper presents a rigorous variational-based framework for the phase-field modelling of ductile fracture in elastic–plastic solids undergoing large strains. It links a formulation of *variational gradient plasticity*, as recently outlined in Miehe [19,20], to a specific setting of *variational gradient damage*, rooted in the phase-field approach of fracture suggested in Miehe *et al.* [10]. The basic ingredients of the formulation proposed here are as follows:

— a phase-field model for ductile fracture that combines ingredients of gradient plasticity and gradient damage, offering a

*scaling of plastic to damage length scales*;— a thermodynamic framework that is fully variational in nature, based on a split of a

*work density function*into energetic and dissipative parts, and a*dissipation function*with separate thresholds for plasticity and damage; and— the

*micromorphic regularization*of the variational principle of gradient-enhanced plasticity damage, allowing a convenient and robust numerical implementation.

The approach is embedded in the theory of *gradient-extended continuum modelling* as outlined by Maugin [21], and in a more general context by Capriz [22], Mariano [23] and Frémond [24]. For simplicity of the representation, only two scalar field variables are considered to describe length scales of the dissipative response: the *equivalent plastic strain* and the *fracture phase field*. These variables are postulated to be *passive* in the sense that external driving via boundary conditions is not allowed. Section 3 outlines a *minimization principle* in terms of these global fields, which fully governs the quasi-static evolution problem of the coupled gradient plastic-damage response. A further important aspect is the *micromorphic regularization* of the variational principle that is performed in §4. Following conceptually Forest [25], this is achieved by considering an extended set of plastic and damage variables linked by penalty terms in a *modified work density function*. The advantage of such a formulation lies on the computational side, in particular, of the side of gradient plasticity. It allows a straightforward finite-element formulation of gradient plasticity that does not need to account for sharp plastic boundaries.

## 2. Introduction of primary field variables

### (a) Finite gradient plasticity in the logarithmic strain space

#### (i) Finite deformations of a solid

Let ** φ**(

**,**

*X**t*) with initial condition

**(**

*φ***,**

*X**t*

_{0})=

**be the deformation map at time**

*X**t*that maps material positions of the reference configuration onto points of the current configuration as visualized in figure 1. The material deformation gradient

**:=∇**

*F*

*φ*_{t}(

**) satisfies det[**

*X***]>0. The solid is loaded by prescribed deformations and tractions on the boundary, defined by time-dependent (‘active’) Dirichlet and Neumann conditions 2.1 on the surface of the undeformed configuration. The first Piola stress tensor**

*F***is the thermodynamic dual to**

*P***.**

*F*#### (ii) Definition of an elastic strain measure

Following conceptually Miehe *et al.* [26,27], we focus on a phenomenological setting of finite plasticity based on an additive decomposition of a *Lagrangian Hencky strain* ** ε**. This allows us to define a stress-producing elastic strain measure in the

*additive format*2.2 where

**with coordinates**

*C**C*

_{AB}=

*g*

_{ab}

*F*

^{a}

_{A}

*F*

^{b}

_{B}is the right Cauchy–Green tensor, i.e. the pull-back of the spatial metric

**to the reference configuration. The**

*g**logarithmic plastic strain*

*ε*^{p}(

**,**

*X**t*) with coordinates

*ε*

^{p}

_{AB}is a

*local*variable, related by to a plastic metric

*G*^{p}∈Sym

_{+}(3). It starts to evolve from the initial condition

*ε*^{p}(

**,**

*X**t*

_{0})=

**0**. The additive decomposition (2.2) in the logarithmic strain space allows for a simple extension of constitutive structures for the geometrically linear setting to the nonlinear case. Finally,

*ε*^{e}is

*a priori*an objective variable due to its Lagrangian nature.

#### (iii) Isotropic strain gradient plasticity

We consider a setting of isotropic finite gradient plasticity at fracture. To this end, a scalar *isotropic hardening variable* *α*(** X**,

*t*) is introduced, that defines an equivalent plastic strain in the logarithmic strain space by the evolution equation 2.3 consistent with von Mises-type isochoric plasticity. It starts to evolve from the initial condition

*α*(

**,**

*X**t*

_{0})=0. In the subsequent treatment, we introduce the

*plastic length scale*

*l*

_{p}that accounts for size effects to overcome the non-physical mesh sensitivity in ductile fracture. To this end, we focus on a first-order setting of gradient plasticity where the gradient ∇

*α*(

**,**

*X**t*) enters the constitutive functions. The generalized internal variable field

*α*is considered as passive in the sense that an external driving is not allowed. This is consistent with the time-independent (‘passive’) Dirichlet and Neumann conditions 2.4 on the surface of the undeformed configuration, defining ‘micro-clamped’ and ‘free’ constraints for the evolution of the plastic deformation.

### (b) The phase-field approximation of sharp cracks

Following the previous treatments of Miehe *et al.* [10], we consider the phase-field approach to fracture as a specific formulation of gradient damage mechanics. It is based on a geometric regularization of sharp crack discontinuities that is governed by a *crack phase field*
2.5
with *irreversible growth*. It characterizes locally for the initial condition *d*(** X**,

*t*)=0 the unbroken and for

*d*(

**,**

*X**t*)=1 the fully broken state of the material. In contrast with traditional approaches to gradient damage mechanics, the crack phase field

*d*is considered to have a geometric meaning. It governs the

*regularized crack surface*2.6 that is formulated in terms of an isotropic

*crack surface density function*per unit volume of the solid. The regularization is governed by a

*fracture length scale*2.7

such that the regularized crack zone lies inside of the plastic zone as sketched in figure 1. The function in (2.6) already appears in the approximation by Ambrosio & Tortorelli [9] of the Mumford & Shah [8] functional of image segmentation. The functional *Γ*_{l}(*d*) converges to sharp-crack topology for vanishing fracture length scale as schematically visualized in figure 2, which depicts in addition a possible higher-order approximation suggested by Borden *et al.* [28]. When assuming a given sharp crack surface topology by prescribing the Dirichlet condition *d*=1 on , the regularized crack phase field *d* in the full domain is obtained by a *minimization principle of diffusive crack topology* with the limit ; see Miehe *et al.* [10] for more details. The crack phase field *d* is passive in the sense that an external driving is not allowed. Only time-independent (‘passive’) Dirichlet and Neumann conditions
2.8
are allowed, defining a sharp ‘initial crack’ and ‘free’ evolution of the crack phase field on the full boundary.

### (c) Global primary fields and constitutive state variables

The above-introduced variables characterize a multi-field setting of gradient plasticity at fracture based on three *global primary fields*
2.9
the finite deformation map ** φ**, the strain-hardening variable

*α*and the crack phase field

*d*. In addition, the plastic strain field

*ε*^{p}serves as an additional

*local primary field*. The subsequent approach focuses on the set of constitutive state variables 2.10 reflecting a combination of first-order gradient plasticity with a first-order gradient damage modelling. The state is automatically objective, due to the dependence on the deformation gradient ∇

**through the Lagrangian Hencky strain**

*φ***defined in (2.2).**

*ε*## 3. Variational phase-field approach to ductile fracture

This section develops a theory for the coupling of gradient plasticity to a phase-field modelling of fracture that is fully variational in nature. It is based on the definition of *six constitutive functions* with a clear physical meaning, which allow the construction of a *minimization principle* for the coupled evolution system.

### (a) Coupling gradient plasticity to gradient damage mechanics

Consider the stress power acting on a local material element that undergoes elastic–plastic deformation and damage. It is the inner product of stress and rate of strain, the thermodynamic external variables acting on the material element. We use the Lagrangian logarithmic Hencky tensor ** ε** defined in (2.2) and its dual stress tensor

**. Let**

*σ**W*denote the time-accumulated work per unit volume and its accumulation in space 3.1 i.e. the total work needed to deform and crack the solid within the process time [0,

*T*]. We base the subsequent development on a constitutive representation of this work 3.2 It is governed by a

*constitutive work density function*that describes the

*rate-independent part*of the global work . The

*a priori*dissipative

*rate-dependent part*

*D*

_{vis}due to viscous resistance forces vanishes in the rate-independent limit. Equation (3.2) holds for particular boundary conditions of the ‘non-local’ generalized internal variable fields

*α*and

*d*. These must be ‘passive’ in the sense that an external driving of these fields is not allowed, which is consistent with (i)

*constant Dirichlet data*and (ii)

*zero Neumann data*of

*α*and

*d*on the surface of the solid, as defined in (2.4) and (2.8) above. The rate-independent part is assumed to depend on the array of constitutive state variables introduced in (2.10). We focus on the particular structure 3.3 already suggested in Miehe

*et al.*[17], which provides a particular coupling of gradient plasticity with gradient damage mechanics. The function splits up into elastic and plastic contributions according to 3.4 The derivatives of the potential density determine the rate-independent parts of stresses, the driving forces and the thresholds for the evolution of the plastic strains and the fracture phase field. It is based on four constitutive functions with a clear physical meaning:

F1. The

*effective elastic work density function*models the stress response and the plastic driving force of the undamaged material.F2. The

*effective plastic work density function*models the local and strain gradient plastic hardening response of the undamaged material.F3. The

*degradation function*describes the transition of the elastic–plastic work density towards the constant crack threshold parameter*w*_{c}.F4. The

*crack surface density function*provides the geometric regularization of a sharp crack topology as already introduced in (2.6).

The work density function models for *d*∈[0,1] by the first two terms a phase transition of the effective elastic–plastic work density towards the constant threshold value *w*_{c}, and by the third term the accumulated fracture work density. Here, *w*_{c}>0 is a *specific critical fracture energy* per unit volume, that enters the formulation as the key material parameter on the side of fracture mechanics. The second material parameter *ζ* *controls the post-critical range* after crack initialization by scaling the work needed for the generation of the regularized crack surface. Figure 3 gives a visual interpretation of the parameters *w*_{c} and *ζ* for a local homogeneous response, where *D*^{pf}:=(1+1/*ζ*)*w*_{c} is the maximum dissipated work density at fracture *d*=1. The constitutive representation for in (3.3) provides the basis for the coupling of a model of gradient plasticity (governed by ) with a gradient damage formulation (governed by ), realized by the degradation function .

### (b) Effective elastic–plastic work and degradation functions

#### (i) Effective elastic work density

The effective elastic work density function in (3.4) models the stored elastic energy of the unbroken material, depending on the elastic strain measure *ε*^{e}. For the subsequent model problems, the elastic work density is assumed to have the simple quadratic form
3.5
characterizing an isotropic, linear stress response in the logarithmic strain space. Here *κ*>0 and *μ*>0 are the elastic bulk and shear moduli, respectively. The function provides a structure identical to the geometrical linear theory of elasticity at small strains. Note that is *convex* with respect to *ε*^{e}, but, due to the nonlinear relationship (2.1), *not poly-convex* with respect to ** F**. This restricts the model of elasto-plasticity under consideration to a range of small elastic strains ∥

*ε*^{e}∥<

*ϵ*, however, accompanied by large plastic strains. This is a typical scenario applicable to the plasticity of metals and glassy polymers.

#### (ii) Effective plastic work density

The effective plastic work density function in (3.4) models the dissipated plastic work of the unbroken material per unit volume, in terms of variables that describe the strain gradient hardening effect. For the modelling of length scale effects in isotropic gradient plasticity, we focus on the equivalent plastic strain *α* and its gradient. It is assumed to have the form
3.6
where *l*_{p}≥0 is a plastic length scale related to a strain gradient hardening effect. Here is an isotropic local hardening function obtained from homogeneous experiments. A typical example is the saturation-type function in terms of the four material parameters *y*_{0}>0, , *η*>0 and *h*≥0, where the initial yield stress *y*_{0} determines the threshold of the effective elastic response.

#### (iii) Fracture degradation function

The degradation function in (3.3) models the degradation of the elastic–plastic work density due to fracture. It interpolates between the unbroken response for *d*=0 and the fully broken state at *d*=1 by satisfying the constraints , , and . In particular, the last constraint ensures that the local driving force dual to *d* causes an upper bound of the phase field *d*∈[0,1]. A *convex* function that satisfies this constraint is
3.7
The quadratic nature of this function is an important ingredient for the construction of a *linear equation* for the evolution of the phase field *d*. Note that the total work density introduced in (3.3) applies the same degradation function on the effective elastic and plastic work densities and , respectively. This is an important assumption with regard to the subsequent construction of a *gradient plasticity model related to the effective quantities* of the undamaged material, where the effective plastic work density serves as a ductile contribution to the crack driving force.

### (c) Stored energy, dissipation and thermodynamic consistency

#### (i) Energetic–dissipative split of work density

In order to quantify both the energy stored in the material and the dissipation, a further assumption is needed that postulates a split of the work density function into energetic and dissipative parts. To this end, (3.3) is decomposed
3.8
into a *stored energy density* and the *accumulated dissipative part* due to plasticity and fracture. This split assumes that the elastic strain energy is the only part of the total work density that is stored in the material. The constitutive expression for this part obtained from (3.3) is
3.9
governed by the degrading function and the elastic strain energy function of the unbroken material. Consequently, the remaining part of the work density function (3.3)
3.10
models the accumulated dissipation in terms of the plastic work density function of the unbroken material and the crack surface density function .

#### (ii) Plasticity–fracture split of dissipated work density

Note that provides a coupled constitutive expression for the accumulated dissipation due to plasticity *and* fracture. It does not allow one to separate the two contributions. In order to investigate this, define the dissipation *locally* as the difference of the external stress power and the evolution of the energy storage, by the standard Clausius–Duhem–Planck inequality
3.11
that should hold for any rates involved. Applying a standard argument, a reduced expression for this dissipation splits up into plastic and fracture parts
3.12
in terms of the energetic plastic and fracture *driving forces*
3.13
obtained from the energy storage function in (3.9). When introducing the time- and space-accumulated dissipative work
3.14
in analogy to (3.1), insertion of (3.12) allows a *separate identification* of the contributions due to plasticity and fracture. In particular, we have
3.15
with the definitions
3.16
These expressions can be evaluated numerically and provide for a rate-independent model with *D*_{vis}=0 in (3.2) under homogeneous conditions with ∇*α*=∇*d*=**0** the closed form in (3.10). The split (3.15) is visualized in figure 3 for a one-dimensional model problem of non-hardening ideal plasticity.

### (d) Driving, resistance and thresholds for plasticity and fracture

The evolution of the plastic strains and fracture phase field is constructed in a normal-dissipative format related to threshold functions. These functions are formulated in terms of energetic driving forces and dissipative resistance forces, related to the split (3.8) of the work density function into the energetic and dissipative parts and .

#### (i) Threshold function for plasticity

The energetic driving force dual to the plastic strain *ε*^{p} and the rate-independent part of the dissipative resistance dual to the hardening variable *α* are defined by
3.17
where denotes the variational derivative of with respect to *α*, reflecting characteristics of the gradient-extended plasticity model under consideration. Clearly, for the kinematic assumption (2.2), the energetic driving force *f*^{p} is the stress tensor ** σ** dual to the Hencky strain in the logarithmic strain space. An

*elastic domain*associated with the plastic deformation in the space of the plastic driving force is defined by 3.18 in terms of the

*plastic yield function*. A classic example is the von Mises function 3.19 in the logarithmic stress space.

#### (ii) Threshold function for fracture

The energetic driving force and the rate-independent part of the dissipative resistance dual to the fracture phase field *d* are defined by
3.20
The variational derivative characterizes the phase-field model of fracture as a gradient-extended damage formulation. An *elastic domain* associated with the crack propagation in the space of the crack driving force is defined by
3.21
in terms of the *crack threshold function* . We focus on the constitutive representation
3.22
where the energetic driving force *f*^{f} is bounded by the crack resistance *r*^{f}.

### (e) Evolution equations for the generalized internal variables

With the above introduced threshold and resistance functions at hand, a *dissipation potential function* is constructed based on the standard concept of maximum dissipation. A rate-dependent definition in a non-constrained manner is
3.23
in terms of the *dual dissipation potential function*
3.24
where 〈*x*〉:=(*x*+|*x*|)/2 is the Macaulay bracket, and *η*_{p} and *η*_{f} are additional material parameters which characterize viscosity of the plastic deformation and the crack propagation. Note that the dual dissipation potential can mathematically be interpreted as a quadratic penalty term that enforces approximately the threshold conditions (3.18) and (3.21). Hence, the phase-field model of ductile fracture is completed by the following two functions which govern the evolution of the internal variables:

F5. The

*plastic yield function*determines the elastic domain in terms of the plastic driving force*f*^{p}.F6. The

*fracture threshold function*determines the initiation of fracture in terms of the fracture driving force*f*^{d}.

The necessary conditions of the local optimization problem (3.23) yield the *plastic flow rules*
3.25
and the normal-dissipative *evolution equation for the crack phase field*
3.26
along with the two loading–unloading conditions
3.27
Note that the positiveness of the parameters λ^{p} and λ^{f} imply via (3.25) and (3.26) the *monotonic growth*
3.28
of the equivalent plastic strain and the fracture phase field.

### (f) Proof of thermodynamic consistency and its consequences

The above evolution equations (3.25) and (3.26) satisfy the thermodynamic constraints (3.12). In particular, we have
3.29
This is obvious due to the *a priori* positive parameters λ^{p} and λ^{f} and the positive driving terms, caused by the *convexity of the threshold functions* and in (3.19) and (3.22), respectively.

For the case of both plastic as well as fracture loading with and , the driving forces can be expressed in terms of the rate-independent and rate-dependent resistances
3.30
yielding the representation of the dissipation
3.31
Hence, the total dissipation splits into rate-independent and rate-dependent parts
3.32
The viscous part is positive for positive material parameters *η*_{p}>0 and *η*_{f}>0. Using the definitions of *r*^{p} and *r*^{f} in (3.17) and (3.20) in terms of variational derivatives of the function , integration over the volume of the solid gives
3.33
Here, the surface term vanishes as a consequence of the restriction to ‘passive’ boundary conditions (2.4) and (2.8), representing (i) *constant Dirichlet data* and (ii) *zero Neumann data* of *α* and *d* on the surface of the solid. When integrating over the process time [0,*T*], we end up with the representation of the space- and time-accumulated *total dissipative work* needed for the generation of plastic deformation and fracture
3.34
with the definition
3.35
This identifies the dissipative part of the work density function introduced in (3.3) with decomposition (3.8) as the time- and space-accumulated dissipative work done to the solid.

### (g) Minimization principle for the multi-field evolution problem

With the above introduced functions at hand, the boundary-value problem is fully governed by a rate-type minimization principle for the quasi-static case, where inertia effects are neglected. In line with recent treatments on variational principles of gradient-extended materials outlined in Miehe [19], consider the constitutive rate potential density
3.36
in terms of the two basic constitutive functions and defined in (3.3) and (3.23), respectively. With this potential density at hand, the evolution of the boundary-value problem of gradient plasticity coupled with gradient damage mechanics is governed by the *global rate potential*
3.37
where is an external load functional, is a given body force per unit volume of the reference configuration and is a given traction field on the surface of the reference configuration. The evolution of all primary fields introduced in §2 at a given state is determined by the *minimization principle*
3.38
where the evolutions of the global fields are constrained by Dirichlet-type boundary conditions defined in (2.1), (2.4) and (2.8) above. Note that the minimization structure of this variational principle is governed by the *convexity of the dissipation potential function* in (3.23), which states the thermodynamical consistency with the second axiom of thermodynamics. The combination of the global minimization principle (3.38) with the local maximum problem (3.23) for the definition of the dissipation potential provides a *mixed variational principle*, that defines all equations of the problem of gradient plasticity at fracture. When introducing the *mixed potential density*
3.39
the Euler equations of the variational principle (3.38) appear in the following form:
3.40
along with Neumann-type boundary conditions of the form defined in (2.1), (2.4) and (2.8) above. The loading parameters λ^{p} and λ^{f} are defined in (3.27). Note that the above Euler equations are exclusively related to variational derivatives of the potential density *π** defined in (3.39).

## 4. Micromorphic regularized minimization principle

The implementation of the above outlined formulation of coupled gradient plasticity damage by finite-element methods is demanding. In particular, problems arise on the side of plasticity, where the gradient part is restricted to the plastic zone. Here, the realization of the sharp plastic boundary in finite-element implementations induces substantial difficulties as demonstrated in figure 4. It needs additional concepts such as those outlined in a series of recent papers by Miehe [20] and Miehe *et al.* [29,30]. An alternative is provided by the *micromorphic approach to gradient-extended models* outlined in Forest [25]. Here, the key concept is the introduction of *dual local–global field variables* via a penalty method, where only the *global fields are restricted by boundary conditions*. Hence, the above-mentioned problem of restricting the gradient variable to the plastic domain is relaxed, which makes the formulation very attractive for finite-element implementation. This section develops a *minimization principle* for this micromorphic regularization of (3.38).

### (a) Micromorphic modification of constitutive functions

#### (i) Micromorphic work density function

The micromorphic setting, applied here in the generalized sense of Forest [25] to the scalar hardening and fracture phase-field variables, is based on an *extension* of the set (2.10) of state variables
4.1
by the *local* equivalent plastic strain and the *local* crack phase-field variable . The dual *global* fields *α* and *d* are then denoted as the micromorphic variables. The micromorphic regularization of the work density function (3.3) reads
4.2
Note first that the local slot *d* in (3.3) is replaced by the additional local variable introduced above. This local variable is then linked to the global micromorphic field variable *d* by the quadratic penalty term, where *ϵ*_{f} is an additional material parameter. Next, the additive split of according to (3.4) into elastic and plastic parts now reads
4.3
where the plastic work density function (3.6) is represented in the micromorphic regularization
4.4
where the local slot *α* in (3.6) is replaced by the additional local variable introduced above. This variable is then linked to the global micromorphic field variable *α* by the quadratic penalty term, where *ϵ*_{p} is an additional material parameter. Note that, for and the above micromorphic extensions (4.2) and (4.4) recover the original settings (3.3) and (3.6) of the gradient-extended theory in §3.

#### (ii) Micromorphic energetic–dissipative split

The energetic–dissipative split of the work density function is performed in full analogy to (3.8)
4.5
with the energetic contribution (3.9) now formulated as
4.6
in terms of the *local* crack phase field . As a consequence, the micromorphic formulation of the dissipative part (3.10) reads
4.7
Recall at this point that governs the thresholds against plasticity and fracture. As a consequence, the global micromorphic fields *α* and *d* are considered as *passive* in the sense that an external driving is not allowed, i.e. determined by the boundary conditions (2.4) and (2.8).

#### (iii) Plastic and fracture driving forces

As a consequence of the energetic–dissipative split (4.5), the dissipation is fully local in nature, where (3.12) now reads
4.8
with plastic and fracture driving forces *f*^{p} and *f*^{f} fully determined by the *local* variables *ε*^{p} and . On the side of *plasticity*, the driving force and resistance are determined by the micromorphic extension of (3.17)
4.9
Note that the plastic resistance *r*^{p} is formally local in nature, i.e. governed by the variables and *α*. However, is linked to *α* by the *additional condition* (4.9)_{3}
4.10
This equation can be recast into the standard form of a modified Helmholtz equation for the link of the local variable to the micromorphic variable *α*
4.11
conceptually similar to Engelen *et al.* [31], where is the plastic length scale of the micromorphic theory. Note carefully that the variable *α* is now defined in the full domain, and not restricted to the plastic zone. This provides a substantial simplification with regard to the finite-element implementation. On the side of regularized *fracture*, the driving force and resistance are determined by the micromorphic extension of (3.20)
4.12
The fracture resistance *r*^{f} is formally local in nature, i.e. governed by the variables and *d*. However, is linked to *d* by the *additional condition* (4.12)_{3}
4.13
This equation can be recast into the standard form of a modified Helmholtz equation for the link of the local variable to the micromorphic variable *d*
4.14
conceptually similar to the gradient damage formulation by Peerlings *et al.* [32], where is the fracture length scale of the micromorphic theory. The above two equations (4.11) and (4.14) exclusively govern the gradient extension of a purely local formulation of plasticity coupled to damage. In particular, the representation (4.11) on the side of plasticity provides a substantial simplification with regard to the numerical implementation.

#### (iv) Evolution equations of local variables

The dissipation potential (3.23) is in the micromorphic theory exclusively formulated in terms of the additional *local* variables and
4.15
where the dual dissipation potential function attains the identical constitutive structure to (3.24), but now formulated in terms of the local driving forces and resistances defined in (4.11) and (4.14). The necessary conditions of the local optimization problem (4.15) give evolution equations for all *local* variables, i.e. the plastic flow rules
4.16
and the evolution equation for the crack phase field
4.17
along with the two loading–unloading conditions (3.27) which determine λ^{p} and λ^{f}, respectively.

### (b) Micromorphic modification of the minimization principle

The micromorphic modification of the minimization principle (3.38) for the evolution problem associated with the extended set of state variables (4.1) reads
4.18
It now covers the evolution of the additional local fields and by the *modified global rate potential*
4.19
in terms of the constitutive potential density function *π* defined in (3.36) in terms of the modified work density function in (4.2) and the dissipation function in (4.15). When introducing the *mixed potential density*
4.20
in analogy to (3.39), the Euler equations of the variational principle (4.18) are as follows:
4.21
along with Neumann-type boundary conditions of the form defined in (2.1), (2.4) and (2.8) above. The loading parameters λ^{p} and λ^{f} are defined in (3.27). Note that the above Euler equations are exclusively related to variational derivatives of the potential density *π** defined in (4.20). Figure 5 depicts a typical numerical example that demonstrates the effect of related plastic and damage length scales *l*_{p} and *l*_{f}≤*l*_{p}.

## 5. Conclusion

A phase-field model of brittle and ductile fracture in elastic–plastic solids at large strains was proposed. It couples a gradient damage formulation with geometric terms rooted in fracture mechanics to finite gradient plasticity. The theory includes two independent length scales which regularize both the plastic response as well as the crack discontinuities, and ensures that the damage zones of ductile fracture are inside of plastic zones. The formulation is conceptually identical to the recent work of Miehe *et al.* [17]; however, it is represented here in a rigorous format based on a minimization principle for the evolution problem. The proposed micromorphic regularization of this framework, in particular, on the side of gradient plasticity, offers a setting that is highly attractive and robust for numerical implementation.

## Authors' contributions

C.M. developed the variational framework of gradient damage coupled with gradient plasticity and drafted the manuscript. S.T. and F.A. were involved in the construction of the variational-based formulation of the coupled problem and did the numerical examples.

## Competing interests

We declare we have no competing interests.

## Funding

Funding was provided within the Cluster of Excellence Exc 310 *Simulation Technology* at the University of Stuttgart

## Acknowledgements

We thank Arun Raina for his support.

## Footnotes

One contribution of 11 to a theme issue ‘Trends and challenges in the mechanics of complex materials’.

- Accepted January 21, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.