## Abstract

This paper analyses the effect of the form of the plastic energy potential on the (heterogeneous) distribution of the deformation field in a simple setting where the key physical aspects of the phenomenon could easily be extracted. This phenomenon is addressed through two different (rate-dependent and rate-independent) non-local plasticity models, by numerically solving two distinct one-dimensional problems, where the plastic energy potential has different non-convex contributions leading to patterning of the deformation field in a shear problem, and localization, resulting ultimately in fracture, in a tensile problem. Analytical and numerical solutions provided by the two models are analysed, and they are compared with experimental observations for certain cases.

## 1. Introduction

Recent studies have shown that strain gradient plasticity models are able to capture different localization mechanisms such as deformation patterning leading to plastic anisotropy (e.g. [1,2]) and strain localization [3] leading to failure of metallic materials through necking, as long as an appropriate non-convex energy is incorporated in a thermodynamically consistent manner. It is observed that during the localization of deformation and evolution of microstructures, the macroscopic stress–strain response shows a hardening–softening stress-plateau type of behaviour (e.g. [4,5]). The usage of standard finite-element methods for this type of problem yields post-critical results due to loss of ellipticity of the incremental boundary value problem (e.g. [6]). Several models have been proposed to remedy these issues including variational regularization methods, non-local methods, viscous regularization techniques and Cosserat theories. However, models which can fully simulate the realistic patterning of dislocation slip are not available. In order to contribute to understanding in this field, two different approaches are studied here to illustrate the ability of non-convex field models to predict the emergence and the evolution of dislocation slip microstructures and localization leading to failure in both rate-dependent (RD) and rate-independent (RI) strain gradient settings. The gradient nature of both models not only allows the numerical issues to be regularized (e.g. [7]) but it also gives the opportunity to study processes at different length scales, due to the internal length-scale parameter entering the gradient energy contribution. Even though the structure of the non-convex plastic energy functions is not physically based, in the current study and based on mathematical concerns, it gives the possibility to control the state of deformation needed for the initiation of patterning and the amount of localized strain, through the spinodal and binodal points of the non-convex plastic energy. This allows direct comparisons with experimental observations, as it is illustrated here for necking and failure of a steel bar under tensile loading.

The first framework, proposed in [1,2], parallels the formulation of strain gradient models developed in [8,9], but with additional free-energy terms. The dissipation inequality is exploited to obtain the microstress definition, and the plastic evolution equation is obtained to satisfy the reduced dissipation inequality for thermodynamic consistency. It assumes additive decomposition of the free-energy, which consists of a non-convex plastic term, and two quadratic terms with respect to the elastic deformation and the plastic deformation gradient. The resulting RD model accounts for processes where the plastic deformation is partially recoverable and partially dissipated. In the second framework, developed in [3], and extended in [10], the plastic evolution is determined by incremental minimization of a global energy functional which is equivalent to the free-energy of the previous model. In this case, the plastic term is totally dissipative, and, therefore, plastic deformation is not recoverable. The resulting framework is RI, in contrast to the rate dependency of the previous model. The energy functional considered here has many similarities with the functionals used in variational fracture models (see, for example, the seminal work [11] and the subsequent extensions [12–14]) and the damage models, such as [15,16]. However, the proposed functional is different in many respects; the elastic and inelastic terms are totally decoupled, and the elastic properties are not affected by damage parameters. The model presents similarities also with gradient plasticity theories (see [17,18] for a review). The main advantage of the proposed variational format, compared with classical plasticity theories, is that, once a proper internal energy is assigned, all the features of plasticity (yield condition, flow rule, consistency condition, etc.) are variationally deduced.

Two different problems are considered through the models above:

(i) the first problem is the evolution of the heterogeneous plastic deformation (microstructure) in metallic materials. These microstructures may macroscopically manifest themselves through softening or through plastic anisotropy in hardening under strain path changes (e.g. [19]). The RD modelling of these phenomena has been studied before [1,20] through phenomenological double-well potentials. This framework can model the formation and evolution of microstructures including the non-equilibrium stages due to its RD nature. Incorporation of more physically based relations for the plastic potential (e.g. non-convex latent hardening potential) has been addressed as well (e.g. [2,21–23]). In RI modelling, the relaxation of the associated non-convex incremental variational problem has been the main approach in the literature (e.g. [24,25]). In most of the cases, the non-convexity comes from the finite strain formulation of plasticity (e.g. [26,27]). A recent study in [28] presents an interesting comparison between the relaxation and gradient plasticity methods in finite plasticity. On the other hand, this paper addresses the RI evolution of the microstructure without a relaxation step through the incremental minimization of the total energy. The non-convex contribution is a Landau–Devonshire type of double-well functional where the second well is shifted up, similar to the form used in [1]. Using such a type of plastic potential results in a Ginzburg–Landau phase-field-like relation for the evolution of plastic slip, where the different phases are identified as regions with high plastic and low plastic strain; and

(ii) the second problem is the material localization, necking and consequent fracture problem in plasticity under uniaxial loading conditions, which is accompanied by substantial softening. The length scale of this problem, which is almost at the engineering level, differs completely from the previous one. In this case, the plastic functional has a convex–concave form, where its concavity is used to simulate the softening phase observed in tensile tests on steel bars. When the amount of plastic deformation reaches values in the concave region strain localization initiates and continues until the material ultimately ruptures. Note that the non-convexity is an intrinsic property of the current models, where the shape of the incorporated plastic potential is designed according to experimental observations. For this problem, the response is expressed in terms of nominal quantities so the equations employed here implicitly take into account the effects of both material and geometric softening. The response of steel bars to tensile loadings has recently been studied by using the RI model (e.g. [10]), and, in this study, it is solved by the RD model as well.

The purpose of this work is to demonstrate the influence of the form of the plastic energy function on the inhomogeneous distribution of the deformation field and to prove the ability of the RD and RI models to capture this phenomenon through numerical examples and by using energy functionals, which has not been considered in previous studies (except the preliminary results in recent conference proceedings [29,30]). Furthermore, the proposed study aims at providing an in-depth comparative analysis of the two models. Therefore, in order to facilitate the comparison, the two theories are formulated within a common unified framework, which allows immediately to highlight theoretical similarities and differences. In the numerical simulations, a particular attention is paid to the different capabilities of the two models in describing strain localization, considering the rate-dependency and the different nature of the plastic energy (stored or dissipative), and predictive limits and advantages of the two approaches are pointed out. To provide a thorough mechanistic understanding, the derivations and implementation are done in a one-dimensional mathematical setting. The approach can be readily extended to a multi slip strain gradient crystal plasticity setting, or to a three-dimensional isotropic softening gradient plasticity framework.

The paper is organized as follows. First, in §2, the problems are summarized in detail, and the theories of the RD and RI models are presented. In §3, the analytical solutions of the models, determined in a special case, are addressed, with a special focus on the RD case. Sections 4 and 5 study the numerical results from both models, regarding the two problems described above. Last, in §6, the concluding remarks are summarized.

## 2. Rate-dependent and rate-independent models

The basic ingredients of both RD and RI theories are presented here in a unified common framework, which facilitates comparison, highlighting the similarities and the differences of both formulations. We recall the model equations of the RD and RI models, referring to [1,3], respectively, for details.

### (a) Problem statement

Two distinct problems at different length scales described in the Introduction section, i.e. the microstructure evolution at micro–meso scales and the localization phenomena at the macro scale, are handled separately by both RD and RI frameworks, where the chosen length-scale parameter governs the nature of the problem.

One-dimensional schemes are considered in order to address the phenomena in a more simple way. Regarding the problem of microstructure evolution at micro–meso scales, an infinitely long strip subjected to a shear-strain is considered, as schematically depicted in figure 1*a*. A single slip system, with shear plane parallel to the slip boundary lines, is considered in such a way that the shear deformation depends only on the *x*-coordinate. Concerning the problem of strain localization and fracture, we consider the problem of a tensile bar described in figure 1*c* in terms of nominal quantities. It schematizes the tensile test performed on the bone-shaped sample represented in figure 1*b*, where the deformation of the enlargements at the extremities are neglected.

Given the one-dimensional domain (0,*l*) of length *l*, the displacement of a point *x*∈(0,*l*) at the time instant *t* is denoted by *u*=*u*_{t}(*x*), where the dependence on time is indicated by a subscript. The boundary conditions are assigned at the endpoints,
*ε*_{t} is an imposed positive deformation, function of time. We assume that the deformation is decomposed additively into an elastic part *ε*^{e} and a plastic part *ε*^{p}. In one-dimensional context of crystal plasticity, the plastic strain *ε*^{p} basically coincides with the plastic slip *γ*, and the strain decomposition reads
*γ* represents the accumulated plastic strain, which coincides with the plastic deformation *ε*^{p} if *ε*_{t}≥0. For an extension of this work to multi-dimension crystal plasticity, plastic strain would be calculated as a sum of plastic slips on different slip systems. In contrast, to extend the model to multi-dimension isotropic von Mises plasticity, *ε*_{p} would be defined as a purely deviatoric tensor, and the accumulated plastic strain would be *v*=*v*_{t}(*w*), which depends on a certain variable *w* and on time *t*, a prime indicates a derivative with respect to *w*, *v*′=d*v*/d*w*, and a dot means a time derivative,

### (b) Constitutive assumptions and models

#### (i) Rate-dependent model

The material is assumed to be endowed with a free energy with different contributions according to
*ψ*_{γ} is a monotonic increasing function of *γ*. The macroscopic stress and the microscopic hyperstress power-conjugated to *s* is the resistance to dislocation slip, *m* is the rate sensitivity exponent. In this paper, we assume *m*=1 in order to have a dual formulation with Ginzburg–Landau-type phase field models, which would also give a linear drag relation as used in discrete dislocation studies. If we define

#### (ii) Rate-independent model

We assume that the free-energy is
*θ*(*γ*), is not included in (2.7), as it is supposed to be totally dissipative. We assume that *θ* is a monotonic increasing function of *γ*, equal to *ψ*_{γ} of the RD model. It obeys the dissipation inequality
*θ*′(*γ*)>0, reduces to
*γ* represents the accumulated plastic strain, therefore the above condition is satisfied by definition [10], and the plastic energy has automatically a dissipative nature. This agrees with the reformulations of the Aifantis theory proposed in [32,31] in thermodynamically consistent context, according to which the local term of the Aifantis flow rule is dissipative (while the non-local term is energetic). In a multi-slip crystal plasticity setting, the relation (2.9) becomes *γ*_{i} representing the plastic slip of *i*th slip system (see [33] for details).

With these assumptions, the dissipation inequality for isothermal processes
*D* the local dissipated power and *P* the local internal power, is automatically satisfied. Indeed, the local internal powers of the RD and RI model are

and, using (2.5), (2.10) reduces to *ξ* makes the free-energy imbalance (2.10) non-local. Inequalities analogous to (2.10) were used in [31,32,34] to deduce gradient plasticity theories in thermodynamically consistent ways.

The basic difference between the two models is that the plastic energy is stored in the RD model, while it is completely dissipated in the RI model. It follows that plastic strains are partially recoverable in the RD model and totally unrecoverable in the RI models. This constitutive difference reflects on the microstress power-conjugated to *π*^{rd}=*ψ*_{γ}′+*σ*^{d} and *π*^{ri}=*θ*′ for the RD and RI model, respectively. While *π*^{rd} is sum of an energetic and a dissipative viscous term, *π*^{ri} is totally dissipative.

### (c) Equilibrium

We assume *u* and *γ* as independent kinematical descriptors. In addition to (2.1), we assign the boundary conditions on *γ*
*E*(*u*,*γ*) be the total energy of the body. A configuration (*u*,*γ*) is equilibrated if
*δu*,*δγ*), with *δE* the first variation of *E*, and *δW*_{NC} the infinitesimal work of the viscous stresses which is present only in the RD model. The total energies of the RD and RI model are

#### (i) Rate-dependent model

As admissible perturbations *δu* and *δγ* can have any sign, the inequality (2.13) reads
*δγ*=0 and *δu*′=0 (i.e. *δγ*=−*δε*^{e}), respectively. They are

### Remark (Initial elastic regime)

Let us assume that *ψ*′_{γ}(0)>0. At the initial instant *t*=0, *γ* and the dissipated force _{2} is satisfied by the elastic deformation *ε*^{e}_{0}=*ψ*_{γ}′(0)/*E*, which represents an initial elastic deformation state corresponding to the stress *σ*_{0}=*ψ*_{γ}′(0). The elastoplastic deformation evolves from such an initial elastic state, according to the evolution equation
*ψ*′_{γ}(0)>0 characterizes materials which exhibit an initial purely elastic behaviour.

#### (ii) Rate-independent model

From (2.9), *γ* can only grow, and thus an admissible perturbation is such that *δγ*≥0. The variational inequality (2.13) is rewritten as
*δγ*=0, we deduce the macroscopic stress balance (2.16)_{1}, while for arbitrary perturbations such that *δε*^{e}+*δγ*=0, we get the inequality
*σ* cannot be greater than the yield limit *θ*′−*ξ*′, and represents the yield condition of classical plasticity.

### (d) Incremental evolution problem

Let (*u*_{t},*γ*_{t}) be the solution at the instant *t*, which is supposed to be known. For a given time step *τ*, the solution at the instant *t*+*τ* is approximated by the first-order Taylor expansion
*u*_{t},*γ*_{t}).

#### (i) Rate-dependent model

By assuming

#### (ii) Rate-independent model

Assuming _{1}, and setting

In (2.25), the boundary terms are null if hard or soft boundary conditions are assigned. As *γ* evolves, and represents the flow rule of plasticity. Equations (2.23)_{1}, (2.26) and (2.24) are necessary conditions for a minimum of the total energy *E*_{t+τ}, i.e. for

### Remark (Initial elastic regime)

Initial elastic deformation regimes are found if *θ*′(0)>0, as in the RD model (see the remark in §2c). Indeed, at the initial instant *t*=0, *f*_{0}=*θ*′(0)>0, as *σ*_{0}=*ξ*_{0}=0, and, from (2.26), *σ* equals *θ*′(0). At this point, inelastic deformation can take place as *f*_{t}=0.

## 3. Analytical solutions

Analytical solutions can be determined only in some special cases. One of these is the case of evolution from homogeneous configurations. Let us assume that at the instant *t*, *γ*_{t} is homogenous. The stress is *σ*_{t}=*ψ*_{γ}′(*γ*_{t}) for the RD model, and *σ*_{t}=*θ*′_{t} for the RI model, according to (2.16)_{2} and (2.19), respectively (*ξ*′_{t}=*Aγ*′′_{t}=0 as *γ*_{t} is constant).

### (a) Rate-dependent model

Under these assumptions, the evolution equations (2.23) for the RD model reduces to equations with constant coefficients, which can be easily solved. As, from (2.23)_{1}, *l*), _{2}, we obtain an integro-differential equation for _{2}, with *δε* the deformation increment imposed in the step *τ*. If we assume hard boundary conditions, the solution of (2.23)_{2} is

In the case of soft boundary conditions, the solution is homogeneous
*t* is omitted for the sake of conciseness. Schematic of the solution as function of *ψ*_{γ}′′ are depicted in figure 2*a*.

In the case of hard boundary conditions, the solution extends in the whole domain if *full-size* solution), and it localizes if *localized* solution). In the full-size regime, we distinguish the hyperbolic solution (3.1)_{1} and trigonometric solution (3.1)_{3}. In the localized case, *γ* grows in a central zone of length 2*π*/*k*, and it reduces outside of it (plastic unloading). We note that the separation value

In the case of soft boundary conditions, *ψ*_{γ}′′<0. The numerical simulations have confirmed this result, and, furthermore, they have exhibited a correlation of direct proportionality between

### (b) Rate-independent model

When we assume *γ*_{t} homogenous and *σ*_{t}=*θ*′_{t} in the RI model, the Kuhn–Tucker problem (2.26) simplifies as follows:
_{2} of the RD model and equation (3.4)_{1} of the RI model exhibit striking similarities. Indeed, (2.23)_{2} differs from (3.4)_{1} only for terms multiplied by 1/*τ*, which disappears when *θ*′′, are schematically represented in figure 2*b*, and we refer to the above-mentioned papers for the explicit results. When hard boundary conditions are applied, *θ*′′>−4*π*^{2}*A*/*l*^{2}, and it localizes in subregions of (0,*l*) of length *θ*′′<−4*π*^{2}*A*/*l*^{2}. When soft boundary conditions are assigned, *θ*′′>−2*π*^{2}*A*/*l*^{2}, and it localizes at the boundaries, in zones of length *θ*′′<−2*π*^{2}*A*/*l*^{2}.

The analytical solutions of both models present many similarities, but some differences as well. We highlight two differences related to the two main distinguishing features of the models here: the dependence on the deformation rate of the RD model and the totally dissipative nature of the plastic energy in the RI model. The first concerns the fact that transition from full-size solutions to localized solutions depends on

## 4. Numerical results

The incremental problems (2.23), (2.24) for the RD model and (2.27) for the RI model are solved numerically, following a finite-element procedure. The domain is subdivided into finite-elements and, within each element, the displacement and plastic deformation rates are approximated by using quadratic and linear shape functions, respectively. At each time increment, the solution of the linearized problem (2.23) in the RD model is refined through a Newton–Raphson scheme, while the solution of the approximated minimum problem (2.27) in the RI model is improved by means of a Sequential Quadratic Programming algorithm. In the latter, each quadratic programming problem is solved through the projection method (e.g. [40]). Both implementation procedures are quite straightforward, and no particular convergence problems have been encountered. It is observed that viscosity in the RD model plays a regularizing role as well in addition to the non-local term. More attention must be focused on the RI model implementation: sufficiently small time steps must be considered to avoid convergence loss, when the evolution undergoes sharp deformation changes.

### (a) Numerical modelling of plastic slip patterning

As explained in §2, the first numerical example addresses the formation of microstructures in metallic materials. In this simplified one-dimensional setting, it corresponds to the analysis of the evolution of heterogeneous plastic shear deformation in a semi-infinite layer with thickness *l*, as shown in figure 1*a*. The patterning of plastic slip is investigated when soft and hard boundary conditions (2.24) are assigned in both RD and RI frameworks. According to the usual notation for shear problems, we define the macroscopic shear strain *Γ* and the shear stress *T*, which correspond to *ε* and *σ*, respectively, of the previous theoretical sections.

The layer is made of steel and its thickness is *l*=1 mm. As the shear problem is considered, Young's modulus *E* used in the above theoretical sections is replaced by the shear modulus *G*=78.9 GPa, which corresponds to Young's modulus *E*=210 GPa and Poisson's ratio *ν*=0.33. As in [1], we set *A*=*ER*^{2}/(16(1−*ν*^{2})) as, e.g., used in [41], where *R* physically represents the radius of the dislocation domain contributing to the internal stress field. In this example, *R*=0.1 mm, corresponding to *A*=147.29 N. For the plastic energy density (*ψ*_{γ} in the RD model and *θ* in the RI model), a Landau–Devonshire type of potential is assumed, which is asymmetric, where the second well is shifted up with respect to the first [1], *ψ*_{γ}=*θ*=1.525×10^{8}*γ*^{4}−5.2×10^{6}*γ*^{3}+5.25×10^{4}*γ*^{2} MPa. Graphs of the plastic energy density, and its first derivative are represented in figure 3. The binodal and spinodal points are *γ*_{b1}=0.0018, *γ*_{b2}=0.0153, *γ*_{s1}=0.0046 and *γ*_{s2}=0.0124, respectively. The stress corresponding to the Maxwell line is *T*_{M}=139.3281 MPa. which is basically *ψ*_{γ}(*γ*_{b2})−*ψ*_{γ}(*γ*_{b1})=*T*_{M}(*γ*_{b2}−*γ*_{b1}).

The stress versus strain curves obtained from the RI (solid line) and RD (dashed line) model are represented in figure 4 for the case of soft and hard boundary conditions. For the RD model, the deformation rates

Considering the deformation evolution, first we analyse the case of soft boundary conditions. In figure 5, the evolution of the plastic slip field *γ* is plotted at different values of the imposed shear strain *Γ*, for the RI model and for the RD model at low shear rate (

In the case of the RD model, for high deformation rates (

The behaviour predicted by the RI model is analogous to that of the RD model for small *γ* reaches the value *θ*′′^{−1}(−*π*^{2}*A*/*l*), which is very close to the spinodal value *γ*_{s1}.

Focusing on the softening phase, which corresponds to the microstructure formation, we observe that in the RD model, the localization front moving from right to left is 0.25 mm wide. Plastic strain has saturated at a value of *γ*=*γ*_{b2}=0.0153 to the right of the front, while at the left it rapidly reduces to a constant value *γ*=*γ*_{b1}=0.0018. The front for the RI model has a similar width, but the strain downstream of the front remains constant at *γ*≃*γ*_{s1}=0.0046, the value at which localization is initiated. In the RI model, the plastic energy is totally dissipated, and thus *γ* can only grow. On the other hand, the energy *ψ*_{γ} of the RD model is stored, and therefore *γ* can be partially recovered, which explains the reduction of *γ* outside the localization zone. Plastic energy recovery is also evident at the end of the slip patterning evolution, when the deformation becomes homogeneous. Increasing *Γ* from 0.0150 to 0.0151, results in a significant reduction in *γ* over the entire domain.

In figure 6, a thorough comparison of the plastic deformation predicted by the RI model and the RD model (with ^{−1}) is presented. Deformation profiles corresponding to eight different states, indicated by dots on the stress versus strain response curve, are analysed. In the initial hardening regime, the two models give same solutions (state 1), while differences arise in the consecutive softening phase, as pointed out above. Two further distinguishing features deserve comment. First, the transition from homogeneous to localized deformation is much sharper in the RD model than in the RI model (states 2 and 3). This difference is confirmed by the graphs of the ratio *δγ*/*δΓ* at the deformation states A and B of the sharp softening branches (figure 6), plotted in figure 7. Indeed, the values attained in the localization zone by the RD model are twice those of the RI model, although high values are obtained in both cases. In the RD model, large negative velocities are reached outside the localization zone, where the plastic deformation is recovered. The second remark concerns the delay of localization when the rate *γ* attains the values *γ*_{b1} and *γ*_{b2}, respectively, of the binodal points. As in these zones, *γ* is practically homogeneous and stationary, from (2.16)_{2}, the corresponding shear stress is *T*=*ψ*′_{γ}(*γ*_{b1})=*ψ*′_{γ}(*γ*_{b2})=*T*_{M}, which remains throughout the whole slip patterning process. A schematic sketch of the strain and stress distribution is shown in figure 8*a*. The corresponding distribution for the RI model is shown in figure 8*b*. In the high slip zone, on the right side, *γ* assumes values smaller than *γ*_{b2}, and, thus, *T*<*T*_{M}. We note that (2.19) is satisfied as equality in the zone crossed by the wave of high plastic deformation and as inequality outside of the zone. Finally, going back to figure 6, further discrepancies are evident at the end of the slip patterning (state 7). They are due to the above-discussed phenomena of strain recovery, allowed in the RD model and forbidden in the RI model, with delays in the recovery process evident at high

In the case of hard boundary conditions, the stress versus strain responses are presented in figure 4*b*. Similar to the previous soft boundary case, as the deformation rate decreases, the results of the RD model approach to the prediction of the RI model. Three different regimes are distinguished: (i) an initial hardening phase in which deformation is homogeneous, except in the boundary layer development at each end, (ii) a softening branch in which the deformation localizes in the middle of the domain, (iii) a stress-plateau characterized by the evolution of the plastic slip towards the boundaries, and (iv) a final smooth hardening branch, where the deformation field evolves in a similar way to the initial hardening phase. The corresponding evolution of the plastic slip field is presented in figure 9 for the RI and RD models with

The results show a good agreement between the two models at low deformation rates, while at high deformation rates the RD model produces an homogeneous deformation field. In the next section, localization leading to necking of a homogeneous bar is studied.

## 5. Numerical modelling of necking and fracture in tensile steel bars

We consider an homogeneous steel bar (Young's modulus *E*=210 GPa) of length *l*=140 mm, as shown in figure 1*c*, clamped at the left and subjected to the tensile displacement *εl* at the right. As we are dealing with a uniaxial tensile problem, we denote the longitudinal plastic deformation as *ε*^{p}, in place of *γ*, which usually denotes the shear strain.

We assume the plastic energy and the non-local coefficient *A* proposed in [10], which have been determined from a phenomenological fit to experimental results on smooth and notched bone-shaped samples. In this problem, we interpret *σ* and *ε* as nominal quantities, thus the phenomenological fits employed here include the effects of geometric as well as material softening.

The plastic energy is the piecewise cubic function represented in figure 10. It is a convex function in the interval 0≤*ε*^{p}<0.16, a concave function for 0.16≤*ε*^{p}<0.7, and a constant function for *ε*^{p}>0.7. The analytical expression of *ψ*_{εp} or, equivalently, *θ* is

For *ε*^{p}>0.7, the bar deforms plastically without any increase in plastic energy, and this corresponds to complete fracture. Thus, the simulations presented in the following are interrupted when *ε*^{p} reaches the fracture value of *ε*^{p}_{2}=0.7. For the non-local coefficient and the friction coefficient, we assume *A*=2 kN, and *c*=0.015 GPa s^{−1}, respectively.

Note that *A* includes a hidden phenomenological length-scale parameter; however, it could be related to the diameter of the bar, which is the dominant length scale in this problem. The viscous coefficient *c* has also been determined by a curve fitting procedure.

In figure 11, experimental and numerical *σ*−Δ*s* curves are compared. The experimental test has been performed at ‘Laboratory of Materials and Structures Testing’ of Polytechnic University of Marche (Ancona, Italy), by using a Universal Testing Machine, with deformation rate *x*-axis, Δ*s* represents the relative displacement between two points, indicated by dots in figure 1*b*, which define an initial gauge length of 80 mm and *σ* represents the nominal stress.

For the RD model, three different deformation rates ^{−2} s^{−1} are considered, which are sufficiently low to get softening and localization. The simulations predict three phases: an initial elastic phase, which is interrupted when *σ* reaches the yield value *σ*^{c}=0.375 GPa, an hardening phase, and a final softening phase. The RI and RD models give practically the same hardening branches; however, they differ in predicting the softening part. The softening response of the RI model is long and sloped, which is quite close to the experimental curve, while the response of the RD model is shorter, and it strongly depends on the deformation rate

The evolution of *ε*^{p} is described in figure 13, where the results of the RI model and RD model with *ε*^{p} in smaller and smaller portions in the middle of the bar, up to final fracture, which occurs when *ε*^{p} reaches the critical value *ε*^{p}_{2}=0.7. Thus, both the RI and RD models describe fracture as the termination of a strain localization process. But the description differ outside the localization zone: for the RI model, *ε*^{p} remains constant, and only the elastic deformation *ε*^{e} reduces, as *δε*^{e}=*δσ*/*E*<0 (perfect elastic unloading); for the RD model, *ε*^{p} reduces outside the localization zone, being partially recovered (elasto-plastic unloading). In the latter case, the strain increase in the localization zone is slightly larger than the strain reduction outside. This explains the steeper slope of the softening branches of figure 12. In figure 13*c*, *u*′ is plotted for two different states at *ε*_{a}=0.1584 and *ε*_{b}=0.1587, corresponding to the beginning and the end of the softening phase (see the enlarged zone in figure 12). As *ε*_{b}−*ε*_{a}=*l*^{−1}(*A*_{1}−2*A*_{2}−2*A*_{3})=0.0003, with *A*_{1}, *A*_{2}, *A*_{3} representing the grey areas of figure 13*c*. On the other hand, in figure 11, the displacement *s*_{b}−Δ*s*_{a}=*A*_{1}−2*A*_{2}=2.12 mm (figure 13*c*) in the softening phase. In this case, strain recovery is taken into account only partially, because the shortening occurring outside the measurement basis (*x*_{1},*x*_{2}), equal to the area 2*A*_{3} in figure 13*c*, is not taken into account. As a result, the slope of the softening branches of figure 11 is much smaller. The incremental ratio *δε*^{p}/*δε* just before fracture are reported in figure 14. Much higher local strain rates are reached in the RD model case, as the softening phase occurs much faster in the RD model than in the RI model.

High deformation rates are considered in figure 15 (^{−1}). As the deformation rate increases, the behaviour becomes stiffer, larger maximum stresses are obtained, and the softening branches extends. Figure 15*c* shows the delay in the evolution of the plastic deformation encountered when high deformation rates are considered, i.e. when the deformation is dominated by the elastic strain. Plastic deformation localization is delayed or even smoothed out at high rates, as clearly shown in figure 15*d*, where profiles of *ε*^{p} at fracture are plotted for different deformation rates. The zone where *ε*^{p} reaches the limit value of 0.7 is larger and larger as *ε*^{p}=0.7, with the exception of short boundary layers. Note that, for *ε*^{p} (local compression). The large deformation rates might not look realistic for this example as they are only considered to explore the different behaviours that the model is capable of describing.

In figures 16 and 17, the stress versus strain response during unloading is analysed. For the RI model, unloading is always elastic (figure 16*a*). For the RD model, unloading behaviour depends on the deformation rate. For large negative deformation rates, hysteretic loops are obtained (figure 16*b*), where the area of the resulting closed curves represents the dissipated plastic energy. The unloading process exhibits two phases: in the first, only elastic deformation reduces (elastic unloading), and it corresponds to the initial sloped branches of the *σ*−*ε* curves of figure 16*b*. In the second phase, plastic unloading occurs, and the corresponding *σ*−*ε* branch is parallel to the loading curve. As dissipation is related to the viscous stress (2.6), it increases with increasing ^{−1}, dissipation practically disappears, and the loading and unloading curves are coincident. Exceptions are found when low deformation rates are applied and unloading is started from points belonging to the softening branch, as shown in figure 17. In this situation, small unloading rates cannot stop and invert the process of strain localization, developed in the softening regime, and, thus, localization proceeds up to failure. Stress–strain curves for different deformation rates are plotted in figure 17*a*,*b*, and profiles of *ε*^{p} at different deformation states are presented in figure 17*c* for

Figure 18 deals with the relaxation problem solved by the RD model. Relaxation from different states of the deformation process obtained with *σ*−*ε* curve corresponding to *a*. The bar relaxes in two possible ways. (i) If relaxation is conducted from states characterized by a value of *ε* smaller than 0.156, which is the transition from a steady-states process to localization, then *ε*^{p} evolves towards the corresponding steady-state configuration. An example regarding this situation is presented in figure 18*b*. Keeping *ε*=0.15 fixed, the starting (*t*=0) and ending (*t*=10000 s) configurations of the relaxation process are demonstrated. (ii) If the bar is relaxed from states with *ε* larger than 0.156, then the process develops to fracture. This kind of process is depicted in figure 18*c*. For *ε*=0.25, the deformation localizes and very quickly terminates in final fracture at *t*=0.26 s.

## 6. Concluding remarks and perspectives

A simple demonstration of the influence of plastic energy potential on the inhomogeneous distribution of the strain field, strain localization and necking is presented through two non-local plasticity models. The first model is a RD one accounting for partially recoverable and viscously dissipated plastic deformation. The other one is a RI model, which accounts for totally dissipative plastic deformation. Both plasticity models are enhanced by a gradient energy and a non-convex plastic potential contribution. The first contribution introduces a length parameter into the model allowing studies at different length scales and the second one introduces an instability leading to decomposition or localization of the plastic strain. The instability due to the second contribution is stabilized by the gradient energy function. The numerical examples illustrate good agreement between the models where they naturally capture the evolution of deformation patterns and localization. In both the models, the non-convexity of the plastic potential drives the strain localization. At the onset of localization, the small disturbances inherent the numerical algorithm are sufficient to select a solution when multiple solutions are possible, and any external additional assumption, such as initial defects in the material or geometrical inhomogeneities, is unnecessary to trigger localization. At low deformation rates, the results of the RD model approach those of the RI model. Both models have exhibited a great versatility in modelling different plastic processes, depending on the form assigned to the plastic potential. Indeed, the convexity–concavity properties strongly influence the evolution of the plastic deformation: a convex energy leads to diffuse plastic deformations, while a concave energy produces localization. In the proposed simulations, a double-well and a convex–concave potential has been proposed to reproduce plastic processes at meso and macro scales, and their shape has been fixed by phenomenological fits to experimental data. The study of physics-based correlations between the plastic energy and material properties represents an interesting challenge for future work, requiring multidisciplinary efforts (solid-state physics, material chemistry, micro-mechanics, etc.). Differences between the two models have been highlighted. In the RD model, the viscous nature of the plastic dissipation describes the delay or, eventual, annihilation of the strain localization, at large deformation rates. It also allows the phenomena of relaxation to be described. This character is obviously absent in the RI model. The fact that the plastic energy is stored and dissipative in the RD and RI model, respectively, results in completely different predictions during unloading, as shown in §5. The RI model accounts for elastic unloading, as observed in tensile tests on steel bars, while the RD model describes plastic unloading, with the exception of a plastic deformation portion dissipated by viscosity. The current models have, however, been primarily developed to capture localization under monotonic loading. Further work is required to fully explore the most appropriate form of model to capture the full range of observed phenomena during unloading or relaxation from a pre-deformed state.

## Authors' contributions

G.L. and T.Y. conducted the research, and worked out all the derivations, implementations and examples. A.C. provided valuable discussions with the first two authors and critically revised the paper.

## Competing interests

We declare we have no competing interests.

## Funding

T.Y. greatly acknowledges financial support for this work provided by TUBITAK (The Scientific and Technological Research Council of Turkey) within project 112C023 through the 2236 Co-Circulation Scheme supported by EC-FP7 Marie Curie Actions. T.Y. and A.C. would also like to acknowledge support from the EPSRC in the UK under grant EP/K007815/1. G.L. was partially supported by the Italian Ministry of Education, Universities and Research (MIUR) by the PRIN funded Program ‘Dynamics, stability and control of flexible structures’, 2010/11N. 2010MBJK5B.

- Received April 25, 2015.
- Accepted June 26, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.