## Abstract

A critical examination is made of two classes of strain gradient plasticity theories currently available for studying micrometre-scale plasticity. One class is characterized by certain stress quantities expressed in terms of increments of strains and their gradients, whereas the other class employs incremental relationships between all stress quantities and the increments of strains and their gradients. The specific versions of the theories examined coincide for proportional straining. Implications stemming from the differences in formulation of the two classes of theories are explored for two basic examples having non-proportional loading: (i) a layer deformed into the plastic range by tensile stretch with no constraint on plastic flow at the surfaces followed by further stretch with plastic flow constrained at the surfaces and (ii) a layer deformed into the plastic range by tensile stretch followed by bending. The marked difference in predictions by the two theories suggests that critical experiments will be able to distinguish between them.

## 1. Introduction

The first strain gradient theories of plasticity were proposed over two decades ago [1,2]. An early objective was to extend the classical isotropic hardening theory of plasticity, *J*_{2} flow theory, by incorporating a dependence on gradients of plastic strain. This has turned out to be more difficult than was first anticipated. An otherwise attractive formulation by Fleck & Hutchinson [3] was found, under some non-proportional straining histories, to violate the thermodynamic requirement that plastic dissipation must be positive. Gudmundson [4] and Gurtin & Anand [5], who noted this violation, proposed alternative formulations which ensured that the thermodynamic dissipation requirement was always met. The manner in which these authors circumvented the problem was unusual for a rate-independent solid—they proposed a constitutive relation in which certain stress quantities are expressed in terms of increments of strain. This class of formulations admits the possibility of finite stress changes due to infinitesimal changes in strain under non-proportional straining. By contrast, the constitutive relation proposed by Fleck & Hutchinson [3] was incremental in nature with increments of all stress quantities expressed in terms of increments of strain. This constitutive relation has been modified so that it now satisfies the thermodynamic requirements [6]. The consequences of the two classes of formulations for problems involving distinctly non-proportional loading histories will be investigated in this paper. Here, in the interest of brevity and for lack of a better terminology, a constitutive construction in the class proposed by Gudmundson [4] and Gurtin & Anand [5] will be referred to as non-incremental, whereas that proposed by Fleck and Hutchinson will be termed incremental.

To bring out the differences in predictions for the two classes of theories, it is essential to consider problems with non-proportional loading, yet to the best of our knowledge no such studies have been made. Non-proportional loading has played a central role in the history of plasticity not only because it arises in applications, but also by serving to clarify critical aspects of constitutive behaviour. Under nearly proportional histories, the predictions of the two theories differ only slightly. Indeed, the two formulations employed in this paper coincide with the prediction of a deformation theory of strain gradient plasticity under strictly proportional straining histories. The deformation theory is a nonlinear elasticity theory devised to mimic elastic–plastic behaviour under monotonic loading for problems with little or no departure from proportional straining. Almost all investigations in the literature employing strain gradient plasticity, whether based on the incremental or the non-incremental formulation, have focused on problems with loads applied proportionally. Here, two basic non-proportional loading problems are studied. The first is a layer of material stretched uniformly in plane strain tension into the plastic range with no constraint on plastic flow at its surfaces. Then, at a prescribed stretch, plastic flow is constrained such that no further plastic strain occurs at the surfaces as the layer undergoes further stretch. The constraint models passivation of the surfaces at the prescribed stretch whereby a very thin layer is deposited on the surface blocking dislocations from passing out of the surface. The second problem is again a layer stretched uniformly in plane strain tension into the plastic range to a prescribed stretch at which point bending is imposed on the layer with no additional average stretch. In the first problem, non-proportionality arises due to the abrupt change in the distribution of the strain rate caused by passivation, whereas in the second problem by the switch from stretching to bending.

For each example, the most important aspects of the predictions of the two theories are illustrated and contrasted. The calculations involved in these examples expose some interesting and unusual mathematical aspects of the non-incremental theories; these are identified and analysed. The paper is organized as follows. Section 2 introduces specific versions of the two classes of theories together with the deformation theory with which they coincide for proportional straining. Section 3 deals with the two plane strain problems: stretch passivation and stretch–bend. Section 4 presents a detailed analysis of mathematical aspects of the non-incremental theory for the stretch-passivation problem with further details given in appendix A. Finally, in §5, an overview summary is presented for both the mathematical and physical findings from this study. Differences in the predictions of the two classes of theories that have significant physical implications are highlighted.

## 2. The two classes of strain gradient plasticity

The established small strain framework for strain gradient plasticity will be adopted [7–9]. Equality of the internal and external virtual work is
*V* , surface *S*, displacements *u*_{i}, total strains *ε*_{ij}=(*u*_{i,j}+*u*_{j,i})/2, plastic strains *σ*_{ij}, and the stress quantities work conjugate to increments of *q*_{ij} (*q*_{ij}=*q*_{ji} and *q*_{kk}=0) and *τ*_{ijk} (*τ*_{ijk}=*τ*_{jik} and *τ*_{jjk}=0). The surface tractions are *T*_{i}=*σ*_{ij}*n*_{j} and *t*_{ij}=*τ*_{ijk}*n*_{k} with *n*_{i} as the outward unit normal to *S*. The equilibrium equations are
*s*_{ij}=*σ*_{ij}−*σ*_{kk}*δ*_{ij}/3.

Isotropic elastic behaviour will be assumed with elastic moduli *σ*_{0}(*ε*_{P}), which is assumed to be monotonically increasing with *σ*_{Y}=*σ*_{0}(0) as the initial tensile yield stress.

We begin by defining the *deformation theory version of strain gradient plasticity*. The deformation theory will be used as the template for the two theories used in this paper by defining them such that they each coincide with the deformation theory for proportional plastic straining. The deformation theory is a version of small strain, nonlinear elasticity, with energy density dependent on *ε*_{ij} and *σ*_{0}(*ε*_{P}), is reproduced when (2.5) is specialized to uniaxial tension. The potential energy of a body is
*T*_{i} and *t*_{ij} on portions of the surface, *S*_{T}, and with *u*_{i} and *S*_{U}. The solution to the boundary value problem minimizes the potential energy among all admissible *u*_{i} and

The notion of *proportional plastic straining* will be important in the sequel. Within the context of strain gradient plasticity, proportional straining histories are the limited set for which the plastic strains and their gradients increase in proportion according to

### (a) Non-incremental theories with certain stresses expressed in terms of strain increments

The Cauchy stress continues to be given by *q*_{ij} and *τ*_{ijk}, follows the idea proposed by Gudmundson [4] and Gurtin & Anand [5], who were motivated to ensure that the dissipative plastic work rate is never negative. Gudmundson considered both rate-dependent and -independent materials, whereas Gurtin and Anand worked within the framework of rate-dependent materials with well-defined rate-independent limits. For the purposes of this paper, it will suffice to construct a version of this class of theories with unrecoverable plastic work—the notation *t* is time, and note that this monotonically increasing measure of the effective plastic strain is defined differently from *ϵ*_{P} in (2.3). The latter is not monotonic and is zero when the plastic strain and its gradient vanish. The two measures coincide for proportional plastic straining. In the absence of plastic strain gradients, or if ℓ=0, *E*_{P} reduces to the effective plastic strain used in conventional *J*_{2} flow theory, *e*_{P}, which is non-decreasing, and *ε*_{P} defined in (2.3), which can increase or decrease, is important and analogous to the distinction between *E*_{P} and *ϵ*_{P}.

The construction of Gudmundson [4] and Gurtin & Anand [9,5] specifies ** Σ** to be co-directional to

*J*

_{2}flow theory when ℓ=0. A change in the direction of loading can lead to a finite change in the distribution

**can undergo finite changes. In other words, an infinitesimal change in loads on the boundary of the solid can produce finite changes in**

*Σ*Using the definitions in (2.8) and (2.10), one finds ** Σ**=

*σ*

_{0}(

*E*

_{P}). In the class of theories introduced above, normality exists in the sense that

**=**

*Σ**σ*

_{0}(

*E*

_{P}) [10,11]. However, the correct interpretation is that

**locates itself on this surface depending on**

*Σ***is defined in terms of**

*Σ***are not fixed in the current state. They depend on the current strain rates, which in turn depend on the prescribed incremental boundary conditions. This is analogous to conventional stresses in the theory of a rigid-plastic solid for which**

*Σ***remains on the yield surface but its components undergo finite changes when directional changes in**

*Σ*Fleck & Willis [10,11] derived two coupled minimum principles governing the incremental boundary value problem for this class of theories. In the current state, *E*_{P} and *σ*_{ij} are known but ** Σ** is not known. Minimum principle I is used to determine the spatial distribution of

**), whereas principle II determines**

*Σ**t*

_{ij}, when prescribed on a surface, are taken to be zero and

*t*

_{ij}=0. Under these conditions, the amplitude of the distribution is undetermined and a normalizing constraint on the distribution of

Rate-dependent versions of this class of theories have proved to be relatively straightforward to implement in numerical codes and widely adopted. To illustrate the influence of the rate dependence on the issue of non-proportional loading, we will present some results based on the following standard incorporation of time dependence following Fleck & Willis [10,11]. Let *m* is the strain-rate exponent which delivers the rate-independent limit when *m*→0. The associated stress quantities are
*Φ*_{I}. The rate-dependent form of principle II will not be needed in the examples considered in this paper.

### (b) Incremental theories with stress increments expressed in terms of strain increments

In this class of theories, the Cauchy stress is known in the current state and continues to be given by the isotropic relation, *J*_{2} yield surface is retained with *m*_{ij}=3*s*_{ij}/2*σ*_{e}, *δe*_{P} give a single constrained equilibrium equation *m*_{ij} of the three incremental equilibrium equations from (2.2) of the unconstrained theory.

The specification adopted is a modification of the Fleck–Hutchinson [3] theory outlined in Hutchinson [6], such that the dissipative contribution is always non-negative. In this paper, the measure of the plastic strain gradients in (2.3) is *ϵ*_{P} and *ε*_{P} defined in (2.3) and *U*_{P} in (2.4). The contribution of the plastic strains and their gradients to the free energy, *ψ*_{P}=*U*_{P}(*ϵ*_{P})−*U*_{P}(*ε*_{P}), vanishes when the gradients vanish and is otherwise non-negative. The recoverable stresses generated from (2.7) are
*J*_{2} flow theory
*U*_{P}(*e*_{P}) is non-decreasing. The unrecoverable stress components are taken to be *σ*_{ij}, *q*_{ij} and *τ*_{ijk} coincide with those in (2.5) for deformation theory. In addition, the theory reduces to conventional *J*_{2} flow theory in the limit ℓ→0. Thus, both classes of theories introduced and used in this paper coincide with the deformation theory for proportional plastic straining and both reduce to *J*_{2} flow theory when ℓ→0. If gradient effects are important, significant differences between the two theories arise under distinctly non-proportional straining, as illustrated in this paper.

The minimum principle for the incremental boundary value problem for this theory is similar in structure to that for conventional *J*_{2} flow theory except that it brings in gradients of the plastic strain rate. The principle requires the quadratic functional *F* to be minimized with respect to *S*_{T}, and (*S*_{U}. A direct calculation gives
*S*(*ε*)≡*dσ*_{0}(*ε*)/*dε*−*σ*_{0}(*ε*)/*ε*. Because *C*'s depend on the current distribution of plastic strain, *m*_{ij} and ℓ^{2}. A rate-dependent version of this theory can also be introduced, but it is not needed in this paper.

The yield condition for this theory [6] is based on the Cauchy stress: *Y* =*σ*_{0}(0). During plastic straining, *Y* is updated by *J*_{2} flow theory yielding condition, but it differs in that

An alternative, but completely equivalent, statement of the yield condition is as follows. When loading occurs, the unrecoverable stress quantities have been defined as *s*_{ij} returns to *e*_{P} in the same manner as the yield surface expressed in terms of *s*_{ij} in conventional *J*_{2} flow theory.

The examples in this paper take the elastic response to be isotropic and incompressible with Young's modulus *E*. For both theories, the input tensile curve is *σ*_{0}(*ε*_{P})=*σ*_{Y}(1+*kε*^{N}_{P}), with initial yield stress *σ*_{Y}=*σ*_{0}(0) and yield strain *ε*_{Y}=*σ*_{Y}/*E*. In dimensionless form

## 3. Two plane strain problems for an infinite layer

Non-proportional conditions in this section are created for an initially uniform layer of thickness 2 *h* undergoing plane strain tension by abruptly changing the constraint on plastic flow at the top and bottom surfaces of the layer or by abruptly switching from stretching to bending. By constraining the plastic strain rate to vanish at the surfaces, one can model the effect of surface passivation which blocks dislocation motion across the surfaces. In the first example, it is imagined that surface passivation is done under load following unconstrained plastic straining. Passivation blocks additional plastic flow at the surfaces. This relatively simple example provides insights into basic aspects of the behaviour predicted by the two classes of models. Even though plane strain conditions prevail throughout, non-proportionality arises due to the abrupt change in plastic strain-rate distribution across the layer, altering the ratio of the gradient of plastic strain rate to the plastic strain rate itself. In the second example, the layer is stretched uniformly into the plastic range and then, with no further overall stretch, is subject to pure bending. The surfaces are unconstrained throughout the entire history such that gradients of plastic flow and non-proportionality arise owing to the switch from stretch to bending.

The layer occupies −*h*≤*x*_{2}≤*h* and is stretched along the *x*_{1}-direction and is subject to *u*_{3}=0. Under these conditions, the total strains are uniform if there is no bending or vary linearly if bending occurs, with only two non-zero components: *ε*_{22}=−*ε*_{11}. The non-zero plastic strain components are *ε**_{P}=|*dε*_{P}/d*x*_{2}| and *σ*_{33}=*σ*_{11}/2, *s*_{22}=−*s*_{11}=−*σ*_{11}/2, *q*_{22}=−*q*_{11} and *τ*_{222}=−*τ*_{112}. The stresses are functions only of *x*_{2} and the equilibrium equations in (2.2) are satisfied except for −*s*_{11}+*q*_{11}−*τ*_{112,2}=0. In addition, *σ*_{11}>0 and *m*_{11,2}=0.

The boundary conditions on the top and bottom surfaces will have *Ti*=0 in all cases and either *constrained plastic flow*, *unconstrained plastic flow*, * κ*, such that the strain in the layer is

*ε*

_{11}will be prescribed, only the distribution of

The minimum principle (2.19) for the incremental theory becomes, for a unit length of layer,
*C*'s are obtained using expression (2.20).

### (a) The stretch-passivation problem

The first example considers stretch of the layer into the plastic range with no constraint on plastic flow at the surfaces until *ε*_{11}=*ε*_{T} when constraint at the surfaces is switched on (for example, by passivating the surfaces under load) for the subsequent increments of stretch. The boundary conditions in this problem are ones which a strain gradient plasticity theory must be able to handle. The problem has the additional advantage that its mathematical formulation is relatively simple. With no constraint at the surfaces, *x*_{2}=±*h* if constraint is active. Uniform plane strain tension holds for both theories for *ε*_{11}≤*ε*_{T}. With *ε*_{Y}=*σ*_{Y}/*E* plastic yield occurs at

#### (i) Consider first the non-incremental theory

For the first increment after passivation at *ε*_{11}=*ε*_{T}, *Φ*_{I}, among all *x*_{2}=±*h*, is *ε*_{T}. As long as no additional plastic strain occurs, principle I minimizes
*σ*_{11} is uniform and *σ*_{11} at *ε*_{11}=*ε*_{T}. Plastic straining resumes when the stress *σ*_{11} becomes large enough such that a non-zero solution *Φ*_{I}=0. This is an eigenvalue problem for *x*_{2} to obtain
*R*=*R*_{C}>1, plotted in figure 1*a*. The associated solution *y*(*x*_{2}) (with *y*(0)=1) is plotted in figure 1. It has the undesirable property that *y*(±*h*)≠0. Thus, strictly, the only acceptable solution is *y*(*x*_{2})=0. Computations with admission of small rate dependence (figure 2) nevertheless strongly suggest that plastic flow resumes at *R*=*R*_{C}. The eigenvalue problem will be discussed fully in §4.

The implication of the results in figure 1*a* is that the class of theories with non-incremental stresses predicts a significant delay in the resumption of plastic flow following passivation. This delay is also evident in the predictions from the rate-dependent version of the theory, as seen in the example in figure 2. For the lowest strain-rate sensitivity (*m*=0.01) and ℓ/*h*=0.2, approximately a 10% increase of stress above the stress at passivation is predicted to occur with essentially no plastic straining. This elastic gap is similar to that predicted by the eigenvalue problem for the rate-independent limit for ℓ/*h*=0.2. Care has been taken to establish that the results presented in figure 2 are insensitive to the increment in the time step.

The *incremental theory* predicts no elastic gap in plastic straining following passivation, only reduced plastic straining. Specifically, for the first increment following passivation, the solution to minimum principle (3.4) can be obtained analytically with the result

### (b) Stretch–bend with no constraint of plastic flow at the surfaces

The problem considered has no constraint on plastic flow at the surfaces at any stage of the history. Uniform stretch in plane strain tension to a strain, *ε*_{11}=*ε*_{T}, is followed by plane strain bending with no further overall stretch. That is, for *κ*=0 and

For the rate-independent *non-incremental theory*, the first increment following the onset of bending, minimum principle I is still given by (3.6) and (3.7), except that there is no constraint on the plastic strain rate at the surfaces. Principle I says that the plastic strain-rate distribution must be uniform. Application of principle II then says that the amplitude of this uniform plastic strain-rate distribution must be zero. Thus, according to this theory, *R*=*R*_{C}.) Predictions based on the rate-dependent version of the theory in figure 5*a* are consistent with the behaviour described above. In the example shown, the layer is stretched well into the plastic range (*σ*_{11}/*σ*_{Y}=2, *ε*_{T}/*ε*_{Y}=7.32) and then subject to bending. The slope of the moment–curvature relation governing elastic incremental behaviour, *a*. The early stage of the bending response is nearly elastic and relatively insensitive to the values of the strain-rate sensitivity exponent chosen. After the onset of bending, there is no elastic gap but additional plasticity develops slowly.

For the *incremental theory*, the boundary value problem (3.4) for the first increment following the imposition of bending can be solved analytically with the result
*K* and *β* are given in (3.8). The limit ℓ→0, *b*. The full response is generated by solving the minimum principle (3.4) sequentially, increment by increment. No reversed plastic straining occurs on the compressive side of the layer over the range of curvature imposed in figure 5*b*, 0≤*κh*/*ε*_{Y}≤1. The bending moment increases almost linearly over this range and is in close agreement with (3.10).

The incremental theory predicts that the moment–curvature relation following initial uniform stretch is increased above the classical plasticity prediction (ℓ=0), depending on ℓ/*h*. The response is significantly reduced below the initial elastic response predicted by the non-incremental theory. From a physical standpoint, there is a significant difference between the predictions from the two types of theories for both this stretch–bend problem and the earlier passivation problem.

## 4. Detailed analysis of the re-emergence of plastic strain following passivation for the stretch problem for the formulation based on non-incremental stresses

As revealed in §3.1, the non-incremental theory suggests that plastic flow is interrupted when a layer which has been stretched uniformly into the plastic range to a stress *a* presents the dependence of *h* based on the solution to the problem posed by (3.7). The resumption of plastic flow after the gap of elastic deformation gives rise to some challenging and interesting mathematical issues which will be addressed in this section.

The starting point is the solution to the eigenvalue problem (3.7), which is valid in this form as long as *σ*_{11} remains uniform. Denote the integrand of (3.7) by *f*(*y*′,*y*) with dependence on ℓ/h and *R* implicit. Because the integrand has no explicit *x*_{2}-dependence, a first integral of the Euler–Lagrange equation is *f*−*y*′∂*f*/∂*y*′=*c*. By symmetry, *y*′(0)=0, and because the equation is homogeneous, one can require *y*(0)=1, such that the first integral is
*x*_{2} is achieved when *θ*(*x*_{2})=*π*/2 for which the corresponding value of *y*(*x*_{2}) is *y**=(*R*−1)/*R*. Thus,
*R*, *R*=*R*_{C}, for which this is true satisfies the equation

To facilitate discussion of the problem after the resumption of plastic flow, it is convenient to define *σ*_{11} and *σ*_{33}=*σ*_{11}/2, and the non-zero components of plastic strain are *x*_{2} only, whereas *ε*_{11} is uniform and prescribed. Then,
*Σ*^{UR}<*σ*_{0}(*E*_{P}). Since *suggests* that plastic flow will resume as *R*_{C}, it only permits the firm conclusion that *R*_{C}. The remainder of this section is devoted to a resolution of this dilemma.

It is assumed that *ε*_{11} is prescribed as a monotone increasing function of time. Since rate-independent behaviour is considered, *ε*_{11} itself can be taken as the time-like variable; passivation commenced at *Henceforth, the suffixes 11 and 112 will be dropped*.

### (a) Direct derivation of *R*_{C}

Consider first the range *ε*^{T}<*ε*<*ε*^{C} (the latter to be determined). By hypothesis, no plastic deformation has occurred since passivation so *ε*^{P} remains at the value *ε*^{PT}, and the stress *σ*^{T} corresponding to strain *ε*^{T} has the value *σ* exceeds *σ*^{T} but still the yield criterion is not met. Thus, it must be possible to construct (*q*^{UR}, *τ*^{UR}) satisfying equation (4.10), for which *ρ* is a constant and *θ* depends on *x*_{2}. The yield criterion will not be violated so long as
*θ* is an odd function of *x*_{2}). Note that this integral is identical to the one developed in (4.4) with *R* replaced by *x*_{2} provided *R*<*R*_{C}, as defined in (4.5b). This, together with inequality (4.12), implies
*ρ* exist so long as *R*<*R*_{C}.

### (b) Solution beyond *R*_{C}

The system of equations comprising (4.5) and (4.8), together with the boundary conditions *ε*^{P}=*ε*^{PT} can be approached by discretizing the time-like variable *ε* into finite steps of magnitude Δ*ε*. A scheme for doing this is outlined in appendix A. The main point of this section is to investigate the first development of the plastic deformation close to the resumption of yield. This requires study of the first increment, *k*=0, as defined in appendix A, where a variational principle for individual time steps is derived. This can be treated analytically because *x*_{2} at *ε*=*ε*^{T}. The variational principle (A.8) with *k*=0 implies
*ε*_{1}=*ε*^{C}+Δ*ε*. The constant *c* is obtained below. It will be convenient to drop the reference to *k*=0 and to write
*ε*→0, (*σ*_{0})_{1/2} will be approximated and replaced by
*α* denotes the rate of hardening d*σ*_{0}/d*E*_{P} evaluated at

Equation (4.16) can then be written in the form
*y*′(0)=0,
*Y* related to *y* via (4.20) and (4.21), and the boundary condition *y*(*h*)=0 now satisfied, can be expressed as
*y*_{0}≡*y*(0) follows from the requirement for consistency that
*X*=*F*(*a*) be the (unique) positive real solution of (4.26) such that

The numerical solution of the identity (4.24) for *y*_{0} at selected values of Δ*ε* can be obtained by straightforward numerical iteration. At each value of *F*(*a*) obtained numerically. A convenient normalization uses *y*_{0}/*ε*_{Y}, Δ*ε*/*ε*_{Y}, *R*_{C}=*σ*^{C}/*σ*^{T} and *σ*^{T}/*σ*_{Y} such that *y*_{0}*α*/*σ*^{T}=(*y*_{0}/*ε*_{Y})(*α*/*E*)/(*σ*^{T}/*σ*_{Y}) and, by (2.22), *α*/*E*=*pN*(*ε*^{PT}/*ε*_{Y})^{N−1}. The other terms in (4.27) and (4.28) can be expressed similarly such that the problem is completely specified by the set of parameters: *N*, *p*, ℓ/*h* and *σ*^{T}/*σ*_{Y}. Results for a specific example are plotted in figure 6. It is seen that *y*_{0}/*ε*_{Y} varies quadratically for small Δ*ε*/*ε*_{Y} and then linearly at larger values. Plots of the distribution of the plastic strain increment, *y*(*x*), normalized by *y*_{0}=*y*(0) are presented in figure 7. The asymptotic results in these figures are derived below.

Since plastic flow re-initiates at *ε*=*ε*^{C}, it follows that *y*_{0}→0 as Δ*ε*→0. This motivates consideration of equation (4.25) as *y*_{0}→0. With a slight departure from earlier notation, let
*r*=(4*E*/(3*σ*^{T}))Δ*ε*. Also, define *y** so that *a*=0 when *y*_{0}→0, and for any *y*_{0}→0 for any fixed *δ*>0. Note that
*R*→*R*_{C} as Δ*ε*→0.

Since
*y*_{0}→0,
*y*_{0}→0,
*δ*>0 (but *δ*<1−*y**) and Δ*ε* sufficiently small. This inequality remains true when *δ*=0. Note that (4.2) contains the same integral, evaluated in (4.4). Thus, *R*≥*R*_{C}.

### (c) Asymptotic solution for small Δ*ε*

The asymptotic relationship between *y*_{0} and Δ*ε* as Δ*ε*→0 can be obtained from the asymptotic approximations (4.32). By direct integration,
*R*=*R*_{C}+*r*/2,

Figure 6*b* shows an example plot of *y*_{0}/*ε*_{Y} against Δ*ε*/*ε*_{Y} computed from the exact form of *a*, the relationship between these two quantities is essentially linear for Δ*ε*/*ε*_{Y}>0.001. However, for Δ*ε*/*ε*_{Y}<0.0001 the relationship approaches the quadratic dependence on Δ*ε* implied by (4.37). Note the fact that the asymptotic result gives *y*_{0}/Δ*ε*→0 as Δ*ε*→0 provides the conclusion asserted earlier that *ε*=*ε*_{C}. The remarkably small range of validity of the asymptotic result reflects the highly singular nature of the problem and the unusual character of the boundary layer discussed next.

An asymptotic relation is also obtained for *y*(*x*_{2}) in the boundary layer near the surface. For *x*_{2}/*h*→1, *y*_{0} neglected, *x*_{2}=*h*,
*y*/*y*_{0} in figure 7*b* in the boundary layer have been computed with the above equation using the values of *y*_{0} from the exact numerical scheme and thus they are not restricted to the small range of validity noted in connection with (4.37). The width of the boundary layer scales with *ε*=0 and increases as Δ*ε* increases, as seen in figure 7*b*. The strain-dependent width of the boundary gives rise to the singular behaviour of the solution associated with resumption of plastic flow.

This problem also illustrates limitations of the non-incremental formulation with regard to determination of the stress quantities *q*=*q*_{11} and *τ*=*τ*_{112}. During uniform plastic stretch prior to passivation, *q*=*s*_{11} and *τ*=0. In the elastic gap period following passivation, *q* and *τ* cannot be determined by the theory. However, immediately after the resumption of plastic flow the distributions of *q* and *τ* are determined. In the boundary layer in the first increment of resumed plastic flow, |ℓ*y*′|≫*y* and, by (4.7), *q*≅0 and

### (d) An improved estimate for *y*_{0}

The reasoning as presented above provides convincing evidence that the plastic strain increment *ε*^{P}−*ε*^{PT} is of order (Δ*ε*)^{2} as Δ*ε*→0. If the variation were exactly quadratic, the central difference approximation that has been employed would be exact, and hence ensuring satisfaction of the governing equations at *ε*_{1/2}=*ε*^{C}+(1/2)Δ*ε* is appropriate. This requires, however, an expression for *E*_{P}∝Δ*ε* so that
*α* by *α**=1/2*α*. The stress *s*_{1/2}=*σ*_{1/2}/2 now becomes
*y*_{0}, the only effect on the asymptotic result (4.37) is to replace *α* by *α**=*α*/2, thus doubling the coefficient of *r*^{2}.

## 5. Summary: implications of the examples of non-proportional loading

As noted in the Introduction, applications of strain gradient plasticity to problems with proportional, or nearly proportional, loading are not problematic. For such applications even a deformation theory will generally give predictions that are similar to those of a genuine plasticity theory. The class of constitutive laws with non-incremental stresses proposed by Gudmundson [4] and Gurtin & Anand [9,5] was specifically constructed to be applicable to non-proportional loading problems, because it is under these conditions that violations of the constraint on plastic dissipation will generally arise. The examples in this paper reveal that this construction gives rise to unanticipated mathematical and physical consequences. By contrast, the incremental theory generates mathematical problems and predictions which are less exceptional, mathematically and physically, and the predictions do not diverge in an unexpected manner from those widely explored for non-proportional loading problems with the context of conventional theories.

For the stretch-passivation problem, the non-incremental theory predicts a substantial ‘elastic gap’ following passivation with no plastic straining. The extent of the gap depends on the material length parameter. Within the elastic gap, the stress quantities *q*_{11} and *τ*_{112} are undetermined. As laid out in §4, the problem for the additional plastic strain following the resumption of plastic flow is a non-standard incremental problem that is inherently nonlinear. No elastic gap is predicted for the incremental theory, and the incremental relationship between the average stress and stretch increment after passivation deviates from conventional elastic–plastic behaviour in a continuous manner that depends on the amplitude of the material length parameter. Unlike the non-incremental theory, the stress quantities *q*_{11} and *τ*_{112} are well defined throughout the history and vary continuously with stretch.

The moment–curvature behaviours predicted by the two theories following the onset of bending in the stretch–bend problem are also markedly different. For the non-incremental theory, there is a substantial range of curvature in which the moment–curvature response is nearly elastic. Within this same curvature range, the prediction based on the incremental theory indicates that the moment–curvature behaviour is significantly less stiff and approaches that from conventional plasticity theory as the material length parameter becomes small.

Several interesting and unusual mathematical problems based on the non-incremental theory for resumption of plastic flow following passivation have been analysed. Minimum principle I of Fleck & Willis [10,11] leads to a nonlinear eigenvalue problem with no acceptable solution. Problematically, it is not possible to impose the desired boundary condition that the plastic strain increment vanishes at the surfaces. This is traced to the feature that minimum principle I has the character of a forward Euler scheme, which is adequate for the analysis of continued plastic flow. To deliver the correct asymptotic behaviour, it was essential to employ an incremental scheme that samples at the end of the load step. Following resumption of plastic flow, the solution has a steep boundary layer adjacent to the passivated surfaces, and the boundary layer width increases from zero.

For the non-incremental theory, the problem for the initial plastic yield stress, *σ*_{C}, of a layer passivated from the start and subject to a plane strain stretch displays the same behaviour, with yield initiating at *R*_{C}=*σ*_{C}/*σ*_{Y}. The same feature arises for shearing of a layer with constrained plastic flow at its surfaces. Yield initiates at a stress level *τ*_{C}>*τ*_{Y}, where *τ*_{Y} is the initial yield stress in the absence of gradients. The delay in yielding in the shear problem was computed as a function of the dimensionless material length parameter by Niordson & Legarth [13] using a rate-dependent version of the non-incremental theory. The computed ratio, *τ*_{C}/*τ*_{Y}, in fig. 3*a* of Niordson & Legarth [13] agrees with the results for *R*_{C} in figure 1 to within several per cent when account is taken for different definitions of the material length parameter. Nielsen & Niordson [14] have presented further results, including for rate-independent behaviour. Their finite-element discretization employed minimum principles I and II of Fleck & Willis [10,11]. They could not capture details of the very steep boundary layer in the early stages of plastic flow as their algorithm was initiated by a small elastic step, but otherwise their numerical method produces satisfactory predictions of overall shear stress–strain behaviour and of shear strain distributions beyond the early stage of plastic flow.

A third class of strain gradient plasticity theories has been proposed in the literature (e.g. [15,16]) which has not been considered in this paper. In this third class of theories, the governing equations are postulated in weak form. It is not necessary to define additional stress quantities, such as **q** and *τ*, although, in principle, they could be identified. This class of theories is intrinsically incremental. Thus, we conjecture that their application to the stretch-passivation and stretch–bend problems would generate results similar to those predicted by the incremental theory, but we have not carried out the requisite calculations.

From a physical standpoint, there are significant differences between the predictions of the two classes of theories considered here for the stretch-passivation and the stretch–bend problems. For both problems, the non-incremental theory predicts an initial response that is either elastic or nearly elastic, whereas the incremental theory predicts an initial response that is much less stiff due to continued plastic flow. The difference is most marked for the stretch-passivation problem, where for the non-incremental theory it can be noted from figure 1 that a moderate value of the material length parameter, ℓ/*h*=0.5, predicts an elastic gap having almost a 40% increase in stress before resumption of plastic flow following passivation. The incremental theory predicts that plastic flow is not interrupted by passivation, only constrained, giving rise to an increase in effective incremental stiffness. This clear difference in predictions suggests critical experiments to clarify the physical relevance of the two theories.

## Author profile

**John W. Hutchinson** is the James and Abbott Lawrence Research Professor of Engineering in the School of Engineering and Applied Sciences at Harvard University. He obtained his undergraduate degree in engineering mechanics at Lehigh University and his PhD in Mechanical Engineering at Harvard. After a post-doctoral period at the Technical University of Denmark he joined the faculty at Harvard, where he has spent his career. Hutchinson works broadly in the areas of solid mechanics and structures with current focus on films, coatings and layered materials, on fracture and on plasticity at the micrometre scale. He was elected as a Foreign Member of the Royal Society in 2013.

## Appendix A. Discretization in the time-like variable

Define *ε*_{k}=*ε*^{C}+*k*Δ*ε*, and let *ε*^{P} at load level *εk*. Assuming that *ε* between *ε*_{k} and *ε*_{k+1}. However, the form of the resulting differential equation for *ε*_{k}. This, however, is of no use at the first step, *k*=0, because it will give the result already found from minimum principle I, i.e. *ε*, whereas the central difference approximation
*ε*)^{2}. The use of this approximation is now pursued. It implies that (4.7) and (4.10) are satisfied at *ε*_{k+1/2}. Equation (4.10) thus requires an expression for *E*_{P})_{k+1/2}. The only simple choice is to assume that *E*_{P} varies linearly on the interval (*ε*_{k},*ε*_{k+1}), which gives

With these approximations (now treated as though they are exact), the system that defines *ε*.

## Footnotes

An invited Perspective to mark the election of John Woodside Hutchinson to the fellowship of the Royal Society in 2013.

- Received April 2, 2014.
- Accepted July 16, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.