Guidelines for Constructing Strain Gradient Plasticity Theories

Transcription

1 . A. Fleck Deartment of Engineering, Cambridge University, Trumington Street, Cambridge CB2 1Z, UK J. W. Hutchinson School of Engineering and Alied Sciences, Harvard University, Cambridge, MA 2138 J. R. Willis Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 WA, UK Guidelines for Constructing Strain Gradient lasticity Theories Issues related to the construction of continuum theories of strain gradient lasticity which have emerged in recent years are reviewed and brought to bear on the formulation of the most basic theories. Elastic loading gas which can arise at initial yield or under imosition of nonroortional incremental boundary conditions are documented and analytical methods for dealing with them are illustrated. The distinction between unrecoverable (dissiative) and recoverable (energetic) stress quantities is highlighted with resect to elastic loading gas, and guidelines for eliminating the gas are resented. An attractive ga-free formulation that generalizes the classical J 2 flow theory is identified and illustrated. [DOI: / ] Keywords: strain gradient lasticity, roortional loading, nonroortional loading, elastic loading gas 1 Introduction This aer builds on a recent aer by the authors [1] which investigated two classes of rate-indeendent continuum strain gradient lasticity theories, dubbed incremental, and nonincremental. In articular, the earlier aer illustrated markedly different redictions of the two classes of theories for two roblems involving nonroortional loading. The first roblem is a layer of material stretched in lane strain tension into the lastic range which, then, undergoes surface assivation that blocks further lastic straining at its surfaces as additional stretch is imosed. The incremental theory redicts continued lastic flow following assivation, although reduced by the constraint imosed by surface assivation. The nonincremental theory redicts that lastic flow is interruted after assivation and does not resume until the layer exeriences additional tensile stress which can be substantial. In other words, according to the nonincremental theory, assivation gives rise to a delay in lastic flow which will be referred to here as an elastic loading ga, or more briefly as a ga. Similar behavior has been revealed in Ref. [2] for the nonincremental theory for a cylindrical wire that is twisted into the lastic range, assivated, and then subject to further twist. The second roblem considered in Ref. [1] is an unassivated layer in lane strain that is first stretched into the lastic range in tension and then is subject to bending with no further overall stretch. In this case, the incremental theory redicts continued lastic flow over the half of the layer exeriencing increasing tensile strain as soon as bending commences, just as in conventional lasticity, but with the lastic flow constrained by gradient effects. By contrast, the nonincremental theory redicts an initial elastic resonse at the onset of bending followed by slowly develoing lastic flow. The two classes of rate-indeendent theories are distinguished from one another by the fact that the constitutive law for the nonincremental theory has certain stress variables exressed in terms of strain increments, whereas the other class emloys incremental relations between all the stress and the strain variables. The nonincremental stress quantities arise due to a constitutive construction roosed in Refs. [3 5] to ensure that stresses associated with dissiative lastic straining (unrecoverable lastic straining in the terminology of this aer) roduce non-negative lastic work. This same construction has been emloyed in the formulation of Contributed by the Alied Mechanics Division of ASME for ublication in the JOURAL OF ALIED MECHAICS. Manuscrit received December 21, 214; final manuscrit received March 12, 215; ublished online June 3, 215. Assoc. Editor: Erik van der Giessen. nonincremental strain gradient lasticity theories for single crystals and similar consequences for roblems involving nonroortional loading conditions can be anticiated. In this aer, conditions under which theories are exected to redict elastic loading gas will be further exlored, including conditions where a ga occurs at initial yield. It will be seen that conditions must be imosed on both incremental and nonincremental theories if a ga at initial yield is to be avoided. The attitude taken in this aer is agnostic as to whether elastic loading gas should or should not occur. ew exeriments will be required to establish the validity or invalidity of such behavior. Instead, the aroach here is to rovide guidance to what asects of the theories give rise to the gas and to how they can be excluded in the constitutive formulation if so desired. The discussion is within the context of small strain, rate-indeendent strain gradient lasticity. The underlying ideas can be extended to a broader class of theories, including those for single crystals. The starting oint in Sec. 2 is a discussion of a deformation theory of strain gradient lasticity which can generally be invoked to model history-deendent lasticity, at least as an aroximation, for alications where straining is roortional or nearly so. This is a good lace to start because the issue of a ga at initial yield arises here in erhas the simlest context where the formulation is straightforward. The issue is whether lastic flow starts at the conventional initial yield stress or whether there is a delay beyond this stress. The two classes of lasticity theories, incremental and nonincremental, are introduced in Sec. 3 and discussed as to whether gas are exected to occur both at initial yield and also subsequently after lastic straining when nonroortional loading occurs due to abrut changes in the incremental boundary conditions. Section 4 resents a detailed analysis of the onset of lastic flow at initial yield for a layer that is assivated from the start and then stretched into the lastic range. This analysis comlements the analysis in Ref. [1] for the case where an unassivated layer is first stretched into the lastic range and then assivated before more stretch occurs. The analysis in Sec. 4 illustrates the comlexity of the solutions in the early stages of yield whether a ga occurs or not. An incremental version of strain gradient lasticity generalizing classical J 2 flow theory constructed such that elastic loading gas do not occur is resented and discussed in Sec otation and General Framework for the Gradient lasticity. There is an imortant distinction in this aer between recoverable and unrecoverable lastic strain quantities reflected Journal of Alied Mechanics Coyright C 215 by ASME JULY 215, ol. 82 / 712-1

2 by the following notation used throughout the aer. Small strain, rate-indeendent lasticity is considered throughout. With _e as the lastic strain increment, or rate, and e Ð _e as the lastic strain, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi define a recoverable effective lastic strain as e 2e e =3 which can increase or decrease. Define the accumulated effective lastic strain used in classical J 2 flow theory as e Ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _e, where _e 2_e _e =3 which is monotonically increasing. In this aer, e will be referred to as the unrecoverable lastic strain. Under monotonic roortional straining, e and e coincide. Two analogous measures qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the lastic strain gradients used in this aer are e 2e ;k e ;k =3 and e Ð _e with _e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2_e ;k _e ;k =3. More general isotroic measures of the strain gradients have been identified in Ref. [6], but e and e adequately exose the issues relevant to the resent investigation. Two generalized effective lastic strain quantities will also aear in the sequel which bring in a material length arameter,. The recoverable measure is E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 þ 2 R e2, and the accumulated, or unrecoverable measure, is E Ð _E with _E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _e 2 þ 2 UR _e2. The two sets of measures coincide when the straining is monotonic and roortional if UR R, i.e., _e ; _e ;k Þ ke _ ; e ;k Þ with e ; e ;kþ indeendent of k, and k increasing from zero. The small strain framework for strain gradient lasticity will be adoted [3 8]. The rincile of virtual work is n o r de e þ q de þ s kde ;k d S T i du i þ t de ds (1.1) with volume of the solid, surface S, dislacements u i, total strains e u i;j þ u j;i Þ=2, lastic strains e (e kk ), and elastic strains e e e e. The symmetric Cauchy stress is r, and the stress quantities work conjugate to increments of e and e ;k are q (q q ji, q kk ) and s k (s k s jik, s jjk ). The surface tractions are T i r n j and t s k n k with n i as the outward unit normal to S. The equilibrium equations are r ;j ; s þ q s k;k (1.2) with s r r kk d =3. The effective Cauchy stress is r e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3s s =2. Elasticity is isotroic with Young s modulus E and oisson s ratio. The initial tensile yield stress is r Y with the associated yield strain e Y r Y =E. umerical results will be resented for incomressible materials with a uniaxial tensile stress strain curve ) e r=e & e ; r r Y (1.3) e r=e þr r Y Þ=kÞ 1= ; r > r Y with < < 1 such that beyond yield r r Y 1 þ ke (1.4) We have deliberately chosen for the inut uniaxial stress strain behavior a curve with continuous sloe at yield rather than a curve with a discontinuous sloe such as a bilinear relation. Had an inut curve been adoted with a shar break in sloe at yield, gas at initial yield would be more clearly delineated, but, as will be seen, gas are also quite evident with the smooth curve. A continuous sloe is more reresentative of the initial yielding behavior of annealed metals than a curve with a shar discontinuity. Moreover, as will be seen in the sequel, this choice will enable us to illustrate an imortant oint concerning recoverable contributions of the gradients of lastic strain to the free energy: namely that these contributions are not necessarily quadratic in the gradients quantities, as is usually assumed. 2 Deformation Theories and the Onset of lastic Flow The deformation theories under consideration characterize small strain, nonlinear elastic solids with a strain energy density of the form w 1 2 L kle e ee kl þ w e ; e Þ (2.1) with isotroic moduli, L kl, and w as the lastic contribution. The associated stresses are e L kl e e kl @w 2e 2e (2.2) 3e >; ;k The otential energy of a body is regarded as a functional of u i and e Fu i ; e Þ wd T i u i þ t e ds (2.3) S T with rescribed T i and t on ortions of the surface, S T, and with u i and e rescribed on the remaining surface S U. The solution to the boundary value roblem minimizes the otential energy among admissible u i and e. Continuity of the stress variables (q ; s k ) under continuing overall deformation lies at the heart of the issues being addressed in this aer. If the strain variables vary continuously and =@e =@e are continuous functions of e and e, then the stresses given by Eq. (2.2) will vary continuously excet ossibly when e and/or e vanish. Within the linear elastic range, (e, e ) vanish and (q ; s k ) are not defined by Eq. (2.2) for the deformation theory. The onset of yield is where the ossible existence of a delay in yielding deends in a critical way on the behavior of w for small e and e. Two distinct behaviors will be illustrated with the following choices for w, each of which reduces to Eq. (1.4) in uniaxial tension: w r Y E þ k= þ 1Þ w r Y e þ k= þ 1Þ ÞE þ1 ÞE þ1 (2.4) (2.5) The first choice Eq. (2.4) follows the roosal in Ref. [6] by relacing e everywhere in the energy density of the classical theory by E ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2 þ 2 R e2, while the second choice Eq. (2.5) retains e in the lowest order contribution. The overall stress strain curve for the tensile stretching in lane strain of a layer of thickness 2h whose surfaces are assivated from the start is lotted in Fig. 1 for the two choices, Eqs. (2.4) and (2.5), for :2, ke Y :5, and R=h 1. The classical limit with no gradient effect corresonding to R =h is also shown. A assivated surface is assumed to block dislocations requiring zero lastic strain to be imosed at the surfaces of the layer in the continuum model. From an analytical ersective, deformation theory roblems are attractive because solutions can be roduced at any load without recourse to rior history. For the second choice Eq. (2.5) there is no elastic loading ga at the onset of yield and lastic flow initiates when r e r Y (at r 11 ffiffi 3 ry =2 in lane strain tension). By contrast, there is a substantial ga for choice Eq. (2.4) and the lastic flow delayed to / ol. 82, JULY 215 Transactions of the ASME

3 Fig. 1 Comarison overall stress strain resonse based of deformation theory for a layer of thickness 2h with surfaces assivated from the start and stretched in lane strain. The material is incomressible. The lower curve alies to an unassivated layer or, equivalently, a layer with R =h. The uer two curves have R =h 1. The to curve is based on formulation (2.4), and it has an elastic loading ga on the vertical axis from 1 to The middle curve is based on Eq. (2.5) and it has no elastic loading ga. r e 1:825r Y. For Eq. (2.4), the ga deends on R =h; it is lotted in Fig. 2. This is recisely the same elastic loading ga identified in Ref. [1] for a articular family of nonincremental theories for the case when assivation is imosed after the layer has been stretched into the lastic range. The question as to why one form of the deformation theory roduces a ga and the other does not is now addressed for the case of initial yield. In the current state with no rior lastic straining, assume r is an equilibrium state of stress (r ;j ) such that on the boundary with outward normal n i, T i r n j. Let e be an admissible trial field associated with the onset of yield and assume that the boundary conditions are such that either t ore such that t e on the boundary. Minimization of F in Eq. (2.3) requires df. At the onset of yield, for arbitrary small variations with de e and dee, df can be obtained df e r e d where the derivatives of w are evaluated at e e. The boundary conditions are homogeneous with either unconstrained e or e. This is an eigenvalue roblem for the stress r at the onset of yield and the nonzero associated eigenfield e. First consider the case where r is uniform. If the lowest order contribution to w is r Y e, as in Eqs. (2.5), then (2.6) becomes Ð r Ye r e d. This has no deendence on the gradients of lastic strain and no enalty for satisfying e on the boundary. For either set of boundary conditions the eigen solution is r e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3s s =2 r Y with e =e 3s =2r Y such that, by Eq. (2.2), q s and s k. There is no ga at initial yield in this case. On the other hand, for the choice Eqs. (2.4) and (2.6) becomes Ð r YE r e d which does bring in a deendence on the lastic strain gradients. If zero lastic strain on the boundary is required, there must be nonzero gradients for any nonzero solution and, thus, E > e over some ortion of the body. It follows that any eigen stress associated with the onset of yield must satisfy r e > r Y. The eigenvalue functional governing the delay in yielding for Eq. (2.4) also arises for roblems based on the nonincremental theories, as first noted in Ref. [1], as will be discussed further in Sec. 4. Fig. 2 Elastic loading ga at the onset of yield for a assivated layer in lane strain for the deformation theory based on formulation (2.4) with R. This same ga arose for the nonincremental theory considered in Ref. [1] for a layer stretched into the lastic range and then assivated followed by further stretch with UR. For the deformation theory (2.1), initial yield will occur in a uniformly stressed body when r e r Y if, at e and =@e r Y, =@e. This is tantamount to the requirement q! s and s k! ase and e aroach zero. While according to Eq. (2.2), q and s k are indeterminate when e and e are zero, the assignment q s and s k within the linear elastic range ensures that all the stress variables will vary continuously at yield for materials meeting the above conditions on the first artial derivatives. This assignment is consistent with the second of equilibrium Eq. (1.2). ow consider situations where the body, or a subregion of the body, has not yet yielded and r in Eq. (2.6) is not uniform. Assume w =@e r Y =@e when e e. lastic yield must begin locally at any location where r e r Y. This can be seenfrom the fact that at this location the integrand of Eq. (2.6) is r Y e r e, which is non-negative for all e if r e r Y and is negative for e =e 3s =2r Y if r e > r Y. Thus, because there is no local deendence on the gradient and no restriction on continuity of e at the onset of yield, lastic flow in the form e =e 3s =2r Y at any location where r e > r Y will lead to smaller values of F than if no flow occurred. In summary, for deformation theory materials =@e r Y =@e ate and e, the onset of lastic flow is a local condition met where r e r Y. For materials not satisfying this condition, the onset of lastic flow is generally governed by a nonlocal condition and an elastic loading ga beyond r e r Y should be exected. The material secified by Eq. (2.4) =@e r Y e =E =@e r Y 2 R e =E which do not satisfy the requirement for no ga at initial yield. 3 Theories of Strain Gradient lasticity With Guidance as to Whether They Generate Elastic Loading Gas A fairly general set of theories will be considered, but secial cases that have aeared in the literature will be discussed. The theory laid out is nonincremental, but it will be secialized to a class of incremental theories. The general thermodynamic framework is consistent with that develoed in Refs. [3 5], but here secifically for rate-indeendent lasticity. The free energy of the solid w has the form given by Eq. (2.1) with recoverable stresses (energetic stresses in the terminology of Refs. [3 5]) Journal of Alied Mechanics JULY 215, ol. 82 / 712-3

4 r L kl e e kl 2e 3e ; s 2e 3e (3.1) A non-negative dissiation function ue ; e ; _e ; _e Þ is assumed that is homogeneous of degree one in _e and _e. Two examles which reduce to Eq. (1.4) in uniaxial tension are h i u r Y 1 þ ke _e þ k UR e Þ UR _e (3.2) and the couled form using E adoted in Ref. [8] factors if there are multile disconnected regions of ongoing lastic straining. A second minimum rincile [8] closely resembles the classical rincile for an incremental roblem, and it rovides the amlitudes of the eigenfields _e and the dislacement rate field. rincile II minimizes where U II 1 2 _r _e e þ _q _e þ _s k _e ;k d S T T_ i _u i ds (3.7) u r Y 1 þ ke _ E (3.3) The dissiation otential u generates the unrecoverable stresses 2_e ; s UR _e 2_e ;k _e 3 _e (3.4) _r _e e þ _q _e þ _s k _e ;k L kl_e _e Þ_e kl _e kl _e w _e 2 e þ w _e _e 2 e þ _e 2 1 _e2 þ _e 2 _e 2 (3.8) Homogeneity of u gives _e þ sur k _e _e Þ _e þ@u=@ _e Þ _e u (3.5) ensuring that the work rate of the unrecoverable stresses is nonnegative if u is non-negative. It follows _e Þ _e Þ must also be non-negative. The general form Eq. (3.4) derives from the constitutive construction roosed in Refs. [3 5] to ensure ositive lastic dissiation of the unrecoverable stresses. The stresses are the sum of the recoverable and unrecoverable contributions, i.e., r, q q R þ qur and s k s R k þ sur k.an imortant distinction between the recoverable and unrecoverable stresses, which has imlications related to the elastic loading gas, is that the recoverable stresses (3.1) are known and fixed in the current state while generally the unrecoverable stresses are not. The unrecoverable stresses in Eq. (3.4) deend on the lastic strain rate and its gradient and thus are not known in the current state they deend on the boundary conditions imosed for the incremental roblem. The unrecoverable stresses can change discontinuously [8,9] from one increment of loading to another if boundary conditions for the incremental roblem change abrutly. It is this feature that motivated the designation nonincremental for theories with such stresses in Ref. [1]. Alternative formulations which introduce extra gradientlike variables to meet the requirement of ositive lastic dissiation have been considered in a broad overview of strain gradient lasticity in Ref. [1], but they will not be considered here. When unrecoverable stresses are resent, the second equilibrium equation in Eq. (1.2) becomes an equation for the lastic strain rates, and the following minimum rincile I was devised in Ref. [8] to satisfy this equation. In the current state with known distributions of r, e, e, and e, a functional homogenous of degree one in _e is defined as U I u þ w _ s _e d (3.6) noting that u _e þ sur k _e ;k and _w q R _e þ sr k _e ;k. In arriving at Eq. (3.6), for all cases considered in this aer, it has been assumed that the boundary conditions on the surface and on any internal elastic lastic boundary are either t or_e. Among all nonzero admissible fields _e, the field that minimizes U I satisfies the second equilibrium equation in Eq. (1.2). Due to the homogeneous nature of U I and the boundary conditions under consideration, the minimum has U I and _e is determined only to within an amlitude factor, or to within multile amlitude Traction rates T_ i are rescribed on S T while on the remainder of the surface _u i are rescribed, and attention here is restricted to either _t or_e ons. For the issues at hand it should be noted that, if u has no deendence on the strain gradients, i.e., if u ge Þ _e, minimum rincile I based on Eq. (3.6) is identically satisfied because all the stress quantities are known and fixed in the current equilibrium state, i.e., s UR k and, from Eq. (1.2), s q R þ sr k;k. Thus, when the unrecoverable contributions do not involve the lastic strain gradients, is known in the current state and the entire incremental field is delivered by minimum rincile II. This is an imortant class of incremental theories discussed later. Minimum rincile I based on Eq. (3.6) and the associated homogeneous boundary conditions can be thought of as an eigenvalue roblem for s, similar to that discussed in Sec. 2. The solution _e is always available, although it may not rovide the minimum to rincile II. If a body is deformed lastically under a sequence of boundary loads which change smoothly, then one can anticiate that at each incremental ste the stresses and the associated strain rates will vary continuously. In other words, under a sufficiently smooth loading history, when lastic straining starts, a nonzero solution to minimum rincile I is exected to exist at each ste with the stresses and strain rates varying continuously. What will haen, however, if there is an abrut change in the incremental boundary conditions? As such an examle, consider the stretch assivation roblem in Ref. [1], where a layer is stretched into the lastic range with lasticity unconstrained on its surfaces (t ) and then assivated such that for subsequent increments _e on the surfaces. The abrut imosition of the constraint on lastic flow at the surfaces results in the fact that the only solution to the minimum roblem for Eq. (3.6) for the case considered in Ref. [1] is_e for a finite range of stress above the stress at assivation. In this case, the abrut change in the boundary condition is the origin of the elastic loading ga. 3.1 Conditions for Eliminating an Elastic Ga at Initial Yield. Conditions on u and w to eliminate a ga at initial yield for the theory in this section are first derived, after which conditions at every stage of loading will be addressed. The condition at initial yield to ensure that a nonzero solution exists in minimizing U I for any s satisfying r e r Y is derived in a manner similar to that for the deformation theory. Let the current deviator stress distribution be s with all the lastic strain quantities zero. In any region where the first increment of lastic strain occurs, _e _e and _e _e, such that Eq. (3.6) becomes / ol. 82, JULY 215 Transactions of the ASME

5 U @ _e s _e d (3.9) with the artial derivatives evaluated at zero lastic strain. Suose these derivatives _e ar =@e 1aÞr Y with a _e =@e. Then, Eq. (3.9) reduces to U I Ð r Y _e s _e d, which is the same eigenvalue functional discussed in Sec. 2. For such materials, initial yield occurs at r e r Y with q s and s k. These conditions on u and w eliminate the roles of _e and the material length arameters at the onset of yield. Conversely, if the artial derivatives of u and w with resect to _e and _e in Eq. (3.9) are not zero, a delay in initial yielding beyond r e r Y must be anticiated. These guidelines are consistent with the numerical examles generated in Ref. [11] for a variety of theories, some of which have gas at initial yield and others which do not. 6, will necessarily have such gas for some roblems. Consider the two nonincremental versions with dissiation otential secified by Eqs. (3.2) and (3.3) and take w, which is not essential to the discussion. For the roblem considered in Ref. [1], where a layer is first stretched into the lastic range and then undergoes assivation followed by 3.2 Conditions for Eliminating an Elastic Loading Ga After lastic Deformation has Occurred. ow suose the body has been deformed into the lastic range and inquire whether an abrut change in boundary conditions for the incremental roblem is likely to roduce an elastic loading ga where a lastic resonse would otherwise be redicted by conventional theory. We begin by illustrating with a secific examle the assertion that any nonincremental version which has unrecoverable stresses generated by Eq. (3.4) with s UR k further stretch, Eq. (3.3) was emloyed, i.e., u r Y 1 þ ke E _. This choice gave rise to the elastic ga alluded toh earlier. Had the choice Eq. (3.2) been made, i.e., u r Y 1 þ ke _e i þk UR e Þ UR _e, no ga would have occurred, as will be discussed further in Sec. 4. The difference between the two choices for this roblem is that Eq. (3.3) has a nonzero contribution of order _e at the onset of the ga while the corresonding contribution from Eq. (3.2) is zero because the current lastic strain is uniform with e. Suose, however, if instead of stretching the roblem is ure bending into the lastic range with no surface constraint followed by surface assivation and continued bending. Then, because of the existence of a gradient of lastic strain at assivation, there will be a nonzero contribution of order _e from both Eqs. (3.2) and (3.3), and, indeed, from any dissiation otential u with a deendence on the strain gradients. Figure 3(a) resents the moment-curvature relation for ure bending in lane strain for a secific examle comuted using Eq. (3.2) in the same manner as in Ref. [1]. A distinct elastic loading ga is evident. The ga, as measured by the curvature change Dj after assivation without any lastic deformation, has been comuted based on a numerical imlementation of minimum rincile I in Eq. (3.9) and lotted in Fig. 3(b). As in the stretch-assivation examles, the ga can be large corresonding to elastic strain increases on the order of 5% of the yield strain or more. The torsion roblem in Ref. [2] is another examle which will generate a ga following assivation for any nonincremental formulation with dissiation deendent on the gradients of lastic strain. In conclusion, these examles illustrate the fact that nonincremental theories with unrecoverable stress quantities s UR k will always generate elastic loading gas for some roblems. In the remainder of this section, we resent what we believe to be an attractive incremental secialization of the theories considered above with no deendence of e and no elastic loading gas either at initial yield or under continued lastic straining. 3.3 A Basic Incremental Theory Extension of J 2 Flow Theory With no Elastic Loading Gas. For this theory, Eqs. (3.1), (3.4), and (3.5) defining the constitutive relation continue to aly, but the work-rate of the lastic strain rate is artitioned between nonrecoverable and recoverable contributions using a factor a in the range a 1. The dissiation otential is taken as u ar e Þ _e, where r e Þ is the relation of stress to effective lastic strain in uniaxial tension with r Þ r Y. The free energy is taken to be w 1 e 2 L kle e ee kl þ1aþ r e Þe þ f e Þ (3.1) As in classical J 2 flow theory, the conventional accumulated effective lastic strain e is unrecoverable. The limit a isa deformation theory, but the concern here is with < a 1, including the limit a 1 for which q R. As the guidelines in Sec. 3.1 indicate, gas at initial yield will be eliminated if f df =de ate. As noted earlier, this theory is incremental with s q R þ sr k;k known in the current state. Rather than an equation for in terms of the lastic strain rate, the first equation in Eq. (3.4) now becomes a constraint on the lastic strain rate. The lastic strain rate must satisfy the normality condition _e 3 2 a _e r e Þ ; _e (3.11) With e qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 =2, is on the surface e ar e Þ and _e is normal to this surface. For elastic resonses ( _e ) with e take < ar e Þ, we define changes in by s and rior to any lastic deformation. With this extended definition of, the second equilibrium equation in Eq. (1.2) is always satisfied. lastic reloading occurs when e returns to the yield surface. Minimum rincile I has no role in this theory. The distributions of _u i and _e are given by minimizing U II in Eq. (3.7) whose integrand (3.8) becomes _r _e e þ _q _e þ _s k _e ;k L kl_e _e Þ_e kl _e kl Þþa dr e Þ _e 2 de þ1aþ dr e Þ _e 2 de þ 1 r e Þ _e 2 e _e2 þ d2 f e Þ d 2 e _e 2 þ 1 df e Þ e de _e 2 _e2 (3.12) By Eq. (3.11), _e a _e e =r e Þe Þ and _e a _e = r e ÞÞ ;k e ;k =e Þ. Further discussion of this theory and illustrative solutions are resented in Sec Analysis of the First Increment of lastic Strain for a assivated Layer in lane-strain Stretch 4.1 Basics. For the urose of this section, define _e to be any ositive, ositively homogeneous function of degree 1 of _e, and _E to be any ositive, ositively homogeneous function of degree 1in_e and UR _e ;k. The free energy w has the general form we e ; e ; re Þ 1 2 ee L kle e þ U e ; e ;k Þ (4.1) Journal of Alied Mechanics JULY 215, ol. 82 / 712-5

6 and the dissiation otential u is taken to have the form The constitutive relations are u_e ; r_e Þr 1 e Þ _e þ r 2 E Þ _E e s _e ; ; s ; ;k r 1 e _e þ r 2 E _e r 2 E _E _e : (4.3) ;k (The formulae for the latter two aly when _e and _E are ositive; when either one is zero, the derivatives must be relaced by subgradients.) For later use, introduce the otentials 1 e Þ and 2 E Þ such that r 1 e Þ1 e Þ; r 2 E Þ2 E Þ (4.4) 4.2 ariational Formulation for an Increment. An incremental formulation will be adoted, for which the solution is sought at discrete times t k t þ kdt. Corresondingly, the value of any function f tþ at time t k is denoted f t k Þf k. The finite difference f kþ1 f k gives Dt f_ t c Þ at some time t c t k þ cdt with < c < 1, at which time f t c Þ itself equals f k f k þ kf kþ1 f k Þ with < k < 1. The values of the arameters c and k are generally not known but still it will rove convenient to resent the formulation as though they were. 1 To see what haens next, note Þ c, with _e t cþ given by its finite difference and e at time t c exressed by linear interolation like that emloyed for e, can be exressed as 1e Þ k Þþ 2 E Þ k Þg k@e Þ kþ1 (4.5) where, for examle, e Þ k e Þ k þ kdt _e t c Þ. The higher-order traction s UR k may be exressed similarly. ow consider, for a body occuying a domain, the variational statement d we c e k ; e k ; re k Þþ 1e Þ k Þþ 2 E Þ k Þ o r e Þ c s k e Þ k (4.6) the variation being taken with resect to e kþ1 and e kþ1. Assuming that c >, the variation with resect to e kþ1 rovides the equation of equilibrium for the Cauchy stress over the domain, and any associated traction boundary conditions on the boundary S, at time t c. Assuming that k >, the variation with resect to e kþ1 yields the second equation of equilibrium in Eq. (1.2) and any higherorder traction condition at time t c. The fields r, q and s k are required to satisfy the equations of equilibrium and any given traction boundary conditions but are otherwise arbitrary. (A similar incremental variational formulation can be develoed for ratedeendent material resonse but this is not required in the resent work.) 4.3 lane-strain Tension of a assivated Stri. The domain is now the stri defined by 1 < x 1 < 1; h < x 2 < h. The material is assumed to be isotroic and incomressible. The only 1 The forward difference aroximation would take c k. For the backward difference aroximation, c k 1. Both have an error of order Dt. The central difference aroximation is defined by c k 1=2 and has an error of order DtÞ 2. nonzero comonents of total strain are e 11 and e 22 e 11 and similarly for the lastic strains: e 22 e 11. These quantities are functions only of x 2 and the timelike variable t. It will be convenient to write e for e 11 and e for e 11. The strain e can be rescribed to be uniform, and is henceforth identified as the timelike variable. The ste size Dt becomes De and e k e þ kde. Since all boundary conditions (aart from those that define e) are homogeneous, r and s k can be chosen to be zero. With these secializations, the variation in the rincile (4.6) is taken only with resect to e and the integration is only over h < x 2 < h. Attention will be focused on the first increment, k 1, and the notation y e 1 e will be emloyed.2 For this first increment, the variational functional has no exlicit deendence on x 2, and therefore, writing the integrand in the variational functional as f y; y Þ, the associated Euler Lagrange equation has first integral f y; y =@y constant. To make rogress, some further secialization is necessary. The free energy w is taken as we e ; e ; re Þ E 3 ee ee þ1 aþw e ; e Þ (4.7) with w given by Eq. (2.5), the form (2.4) already having been exosed as unsatisfactory in the sense of giving an elastic ga, even for deformation theory; and for the rate theory, in which w is identified hysically as the free energy, it is not accetable that q R and s R k are not uniquely defined when E. ote that, for the resent roblem, E 2= ffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Þ e Þ 2 þ 2 R e Þ 2.Thevariable _e is taken as equivalent lastic strain-rate, and this becomes, for the resent roblem, _e 2= ffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 Þj_e j.thevariable _E, in the first instance, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi will be taken as _e 2 þ 2 UR _e Þ2,with _e 2_e ;k _e ;k =3 which becomes, in the resent case, _e 2= ffiffi 3 Þj_e j. The otentials 1 and 2 are taken as 1 e Þ1aþabÞr Y e and 2 E Þ ar Y 1 bþe þ k þ 1 Eþ1 (4.8) with a; b 2½; 1Š. The forms Eqs. (4.7) and (4.8) deliver the basic ower-law (1.4) in uniaxial tension, and they generalize the law (3.3). To make the first integral of the Euler-Lagrange equation for the first increment comletely exlicit and of manageable length e will be set to zero, and the definitions Y R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 2 þ 2 R y2 ; Y UR ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 2 þ 2 UR y2 will be emloyed. The required first integral is 2E 3 e c kyþ 2 þ1aþr Y ½ 2ky ffiffiffi k 2k þ1 ffiffiffi Y R 3 þ 1 3 2k þ1 þ k ffiffiffi y 2 Y 1 2ky R Šþ1aþabÞr Y ffiffiffi 3 3 þ ar Y ½1 bþ 2ky2 ffiffiffi k 2k þ1 ffiffi Y UR 3 YUR þ 1 3 2k þ1 þ k ffiffiffi y 2 YUR 1 Šc (4.9) 3 The constant c is fixed from the symmetry requirement that y Þ which imlies that Y R Þ Y UR Þ yþ y where y has yet to be determined. Thus, 2 For the first increment, e. Retention of e allows the general formula to aly, with re-numbering, to any increment and also to the case of uniform straining with surfaces unassivated u to a uniform lastic strain e, as considered in Ref. [1] / ol. 82, JULY 215 Transactions of the ASME

7 Fig. 3 ure bending in lane strain with no assivation followed by continued bending with assivation. The constitutive law is secified by a dissiation otential u given by Eq. (3.2) with no recoverable contributions. The material is taken to be incomressible and the comutation in (a) is carried out using the ratedeendent version with a strain-rate exonent m :1, as in Ref. [1]. The elastic loading ga as secified by the curvature increase Dj after assivation without lastic flow is lotted as a function of the curvature j at assivation in (b). The redictions in (b) are based on the rate-indeendent formulation and minimum rincile I (3.9). " c 2E 3 e c ky Þ 2 2ky þ r Y ffiffiffi þ k # 2ky þ1 ffiffi 3 þ 1 3 (4.1) Before continuing, normalized variables are introduced by defining e Y r Y =E, and then, z 2ky ffiffiffi ; z 2ky ffiffiffi ; 3 ey 3 ey Z R Z R ; Z UR 2kY UR ffiffi ; z 3 ey z z z ; Z R 2kYR ffiffi ; 3 Z UR Z UR z (4.11) Equations (4.9) and (4.1) now give 1 aþ z ke Y þ 1 z ZR þ1 þ ke Y z z2 Z R 1 þ a 1 bþ z2 ke Y Z UR þ 1 z ZUR þ1 þ ke Y z z2 Z UR 1 where þ abz 1 R c 1 zþþ 1 2 z 1 z 2 Þþ ke Y z þ 1 ; (4.12) R c 2e c ffiffiffi (4.13) 3 ey It is exedient now to consider secial cases, as follows The Case a. Equation (4.12) becomes ke Y z Z þ 1 R þ1 z 2 Z R 1 R c 1Þ1 zþ 1 2 z 1 z 2 Þ ke Y z þ 1 (4.14) and finally for consistency, the requirement that z when x 2 h, h R 1 dz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4.16) z2 fixes z. For the urose of asymtotic analysis, the term of order z in Eq. (4.14) can be neglected to leave the equation Z þ 1 R þ1 z 2 Z R 1 Z 2 R R c 1Þ1 zþ ke 1 Y z þ 1 (4.17) This equation cannot be solved in closed form, but substitution of its solution into Eq. (4.15) would yield an equation for the arameter R c 1 = ke Y z, requiring z to be of order R c 1Þ 1=. Thus, R c should be close to 1, imlying that e e Y. Thus, as exected, there is no ga and z /DeÞ 1=. ote that the form of deendence of z on De is redicted consistently, for any choice of c and k. The constant of roortionality is given correctly by taking c =1Þ and k 1=1Þ. ote also that, if the boundary of the stri were not assivated, the increment in lastic strain would have the same deendence on De, though with different amlitude The Case a 1. Equation (4.12) becomes where ke Y z Z þ 1 UR þ2 z2 1 b þ ke Y z ZUR azþ Z UR (4.18) Once this equation is solved for Z R, the solution of the differential equation to which it is equivalent follows as: azþ 1 bz R c 1 zþþ 1 2 z 1 z 2 Þþ ke Y z þ 1 (4.19) x 2 R 1 z dz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (4.15) z2 Z 2 R The lowest-order asymtotic solution to Eq. (4.18) as z! isas follows: Journal of Alied Mechanics JULY 215, ol. 82 / 712-7

8 þ1 1 Z UR R c 1R c bþz ke Y z þ1 if z < z and Z UR z 2 1 R c b 1 1 zþ 1 b if z < z 1 (4.2) where z R c 1 R c b (4.21) Substituting the asymtotic forms Eq. (4.2) into Eq. (4.16) requires the calculation of two integrals z dz z dz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ke z2 Z þ 1Þz =þ1þ Y Þ1=þ1Þ UR Z 2 UR Z 2 UR R c 1Þ =þ1þ R c b and 3 " 1 1=2 # dz 2 ^R ^R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi tan 1 þ 1 z z2 ^R 2 1 ^R 1 2 where ^R R c b 1 b (4.22) (4.23) (4.24) The integral (4.23) decreases monotonically from þ1 to as ^R increases from 1 to 1. It is therefore imossible to satisfy Eq. (4.16) unless ^R is at least ^R c, the value of ^R for which that integral equals h= UR. Thus, an elastic ga is redicted. 4 lastic flow does not commence until e reaches a value e > e Y, corresonding to the attainment of ^R c. ow when e is increased to e þ De, the associated value of z is obtained when the integral (4.22) exactly comensates for the shortfall of Eq. (4.23) below h= UR. To first order 1 z dz 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h= Z UR 2 UR z2 ^R c ^R 2 c 1Þ h= UR þ =2 þ ^R c ^R ^R c Þ Comleting the algebra gives the result 2ky þ1 ffiffi 3 ey k 1= e Y 1bÞ ^R c 1Þ " #þ1 h= UR þ=2þ ^R c ^R 2 c 1 2cDe ffiffi 3 ey (4.25) þ1 (4.26) This result is asymtotically exact if c = þ 1ÞÞ and k = þ 1ÞÞ. Remarkably, this exact result for y is also roduced by the choices c k 1, corresonding to the use of the (inexact) backward Euler aroximation. It should be noted that the derivation given is far from rigorous: the asymtotic aroximations (4.2) break down near z z, and 3 The integral to follow is obtained via the variable transformation transformation cos h z=½1 ^R1 zþš, as in Ref. [1]. 4 Strictly, it is necessary to demonstrate that there exist fields and s UR that do not exceed the yield criterion, when ^R < ^R c. This demonstration was made in a slightly different context in Ref. [1]; it is omitted here. there is also a serious roblem in obtaining a good aroximation near z 1 when z >. We have, however, erformed analysis that shows that terms neglected are of lower order than those retained; these details are omitted here, for the sake of brevity The Case R UR. Equation (4.12) becomes ke Y z Z þ 1 R þ2 z 2 a1 bþþke Y z ZR 1 þ1aþabþz R c 1 zþ þ z2 Þþ ke Y z Z R (4.27) þ 1 This has exactly the same form as Eq. (4.18) and gives the same tye of delay, basically induced by the term a1 bþz 2 = Z UR in Eq. (4.12) The Case a 1, b 1. The analysis of Sec becomes nonuniform as b aroaches 1. The value R c of R c that corresonds to ^R c (which is fixed) tends to 1 as b! 1, imlying that the elastic ga reduces to zero. Corresondingly, the size of De for which the asymtotic formula has validity tends to zero. Furthermore, when b 1, Eq. (4.18) reduces exactly to Eq. (4.17) to leading order, excet that R is relaced by UR,soz becomes roortional to DeÞ 1= A Class of onrecoverable Laws That Dislay no Ga Under Stretch-assivation. There does, however, remain a difference between the cases for which the gradient term is recoverable or nonrecoverable. The resent roblem does not show it, but if the stri were subjected to lane-strain tension with unassivated boundaries, and then strain increased following assivation, as discussed in Ref. [1], a ga would still be dislayed with the resent constitutive law. 5 ow here is a nonrecoverable law that will dislay no ga under stretch-assivation; it is a slight generalization of the law (3.2). The free energy is unchanged, but E is chosen to be UR e and 1 e Þr Y e þ ak þ 1 eþ1 ; k 2 UR e Þar Y þ 1 URe Þþ1 (4.28) This leads to the equation 1 aþ Z þ 1 R þ1 z 2 Z R 1 a þ 1 zþ1 þ ak =kþ þ 1 URjz jþ þ1 R c 1Þ1 zþ ke Y z 1 þ 1 (4.29) This is similar in character to Eq. (4.17) and dislays no ga. In the case a 1, the arameter R c 1 = ke Y z is fixed by the requirement k 1 k þ1 1 þ 1Þ R c 1 1 ke 1 zþ1z þ1 Þ Y z þ1 dz h UR (4.3) This is, of course, only one reresentative of a class of laws. The essential feature is that 2 should be a function of any ositively homogeneous function UR e of degree 1, of URe ;k only, with the additional roerty that 2 Þ. However, as argued in 5 We refrain from recording the analysis, in the interest of conciseness / ol. 82, JULY 215 Transactions of the ASME

9 5 A Basic Ga-Free Incremental Theory The incremental formulation introduced in Sec. 3.3 is a gafree incremental strain gradient lasticity which reduces to classical J 2 flow when the gradients are sufficiently small. It will be imlemented to illustrate several asects of behavior of a stretched layer under assivation. In the examles, the inut tensile relation (1.4), r e Þr Y 1 þ ke, is again used and we take a 1 such that the dissiation function is u r e Þ _e. The contribution of the lastic strain gradients to the free energy in Eq. (3.1) is taken to be f e Þ r Yk þ 1 Re Þþ1 with s R k 2r Yk R R e Þ e ;k 3 e (5.1) Fig. 4 Average stress versus stretching strain for a assivated layer secified by the ga-free incremental theory defined in Sec. 5. The material is incomressible and the deformation is lane strain. Sec. 3.2, such theories will still dislay gas for roblems in which nonuniform lastic strain is develoed rior to assivation. Still continuing with the case a 1, if there is already a lastic strain e, the equation governing the increment is 2 ½e c e 3e þ kyþš2 þ 2e þ ffiffiffi kyþ þ k 2e þ kyþ þ1 ffiffi Y 3 þ 1 3 þ k 2 þ1 UR ffiffi je þ 1 3 þ ky j e y Þsgne þ ky Þc (4.31) If the stri is stretched uniformly rior to assivation at lastic strain e, then y Þ and this equation imlies " k e Y þ 1 URjz jþ þ1 2ec ffiffi 1 þ ke 2ke!# Y ffiffi z zþ 3 3 ey (4.32) which delivers no ga. If, however, e deends on x 2, the resence of e alters this conclusion. The derivation of Eq. (4.32) made use of a Taylor exansion, valid for De e, which is why it does not reduce exactly to Eq. (4.29) (with a 1) when e!. In making the above choice for f e Þ, we have followed [12,13] by assuming that geometrically necessary dislocations associated with e contribute to the hardening with a functional deendence that is similar to that of the statistically stored dislocations generated by e. In this articular case, the stress increase due to e is r Y ke while the corresonding stress generated by f e Þ is r Y k R e Þ. We will return to the issue of identifying f e Þ shortly. The average stress as a function of strain for a layer of thickness 2h which is assivated from the start and stretched in lane strain tension has been comuted for the theory defined above. For this one-dimensional roblem, because _e _e and _e _e, it is readily shown that the solution is identical to that of the corresonding deformation theory with a in Sec This corresondence has been exloited in generating the numerical results. The stress strain behavior is lotted in Fig. 4 with associated results in Fig. 5. Figure 5(a) shows the emergence of the lastic strain at the center of the layer after yield, comaring the exact numerical result with an analytical asymtotic result. Figure 5(b) lots the distribution of the normalized lastic strain across one half of the layer at a articular imosed strain. The stress strain curves in Fig. 4 have no gas at initial yield yet they reveal substantial increases in flow strength in the early stages of lastic deformation due to strain gradient effects. Thus, the functional form for f e Þ adoted in Eq. (5.1) gives rise to both early flow strength elevation and subsequent hardening elevation, even though the gradient contributions are entirely recoverable. There are more than a few theories in the literature with asects in common with the basic theory in this section with Fig. 5 (a) The lastic strain at the center of the assivated layer as a function of the strain imosed on the layer a comarison betweenasymtotic and exact results. (b) The distribution of the normalized lastic strain across the layer at 2e 11 = ffiffiffi 3 3eY. In both arts, for the incomressible, incremental material in Sec. 5 with :2 and :5. Journal of Alied Mechanics JULY 215, ol. 82 / 712-9

10 Fig. 6 An unassivated layer of thickness 2h stretched into the lastic range and then assivated followed by additional stretch, as redicted by the incremental theory in Sec. 5 for an incomressible layer in lane strain unrecoverable contributions to q and only recoverable contributions to s k. Among them are aers on isotroic theories [4,7,11,14] and on single crystal theories [15,16]. To our knowledge, all excet [16] have taken f e Þ, or its equivalent, to have a quadratic deendence on the gradients of lastic strain. A quadratic deendence on gradients does not dislay the strength elevation seen in Fig. 4, but, instead, it only reveals an increase in linear hardening behavior. Aarently, for this reason, a mistaken notion has taken hold in the literature that recoverable gradient effects contribute to hardening but not to strengthening. The argument for taking f e Þ Re Þþ1 with as the tensile hardening exonent is henomenological but with some hysical basis for metals with well-develoed microstructures such as reciitates or dislocation cell structures [12,13]. The single crystal formulation alied to the grain size effect on strength in Ref. [16] estimated the free energy of geometrically necessary dislocations using the self-energy of a dilute distribution of dislocations giving a strictly linear deendence of the free energy on the lastic strain gradients. This choice gives a ga at initial yield noted by the authors. Evaluation of f e Þ using fundamental dislocation comutations is likely to be a fruitful oint of contact between continuum theory and discrete dislocation theory. Some reliminary results [17] along these lines for an elementary, nondilute distribution of geometrically necessary dislocations suggest that f e Þ is not quadratic but nearly linear in e. The theory in this section has also been alied to the stretchassivation roblem considered in Ref. [1] where an unassivated layer is first stretched into the lastic range and then assivated followed by further stretch. rior to assivation the stress and strain distributions are uniform. After assivation the distributions become nonuniform and the roblem requires an incremental ste-by-ste solution rocedure. A numerical examle is shown in Fig. 6 comuted using minimum rincile II in Eq. (3.7). There is no elastic loading ga after assivation, but there is a short raid rise in the average stress analogous to that at initial yield. This is due to the fact that the stress contribution of the gradients is roortional to R e Þ. 6 Conclusions This aer has focused on identifying, analyzing, and ossibly eliminating elastic loading gas which arise in some formulations of strain gradient lasticity at initial yield and under nonroortional loading histories. While hysical arguments against elastic loading gas can be ut forward, the view taken in this aer is that it is remature to rejudge the outcome on this matter until exeriments and more fundamental dislocation studies concerning the existence of gas become available. Discrete dislocation models of boundary value roblems of the tye analyzed in this aer, if roerly formulated and interreted, should be caable of roviding qualitative insight into the existence, or lack thereof, of elastic loading gas. Also, the insights so gained might assist the design of exeriments to test the existence or otherwise of gas. The aroach here has been to identify the features of the continuum constitutive laws which give rise to the gas and to resent a selection of examles which illustrate how to analyze the gas and the early stage when lastic flow resumes. These roblems can be fairly comlex with unusual boundary layer behavior. While not exhaustive, the analysis in Sec. 4 illustrates a variety of behaviors that can arise. Relatively simle guidelines emerge for ensuring that there are no gas at initial yield. A general finding is that all nonincremental formulations which contain unrecoverable (dissiative) contributions deendent on the gradients of lastic strain will necessarily roduce elastic loading gas for some roblems. To date, it aears no thermodynamically accetable recies exist for an incremental formulation with dissiative contributions deendent on the gradients of lastic strain. An attractive ga-free, generalization of J 2 flow theory incororating strain gradients has been identified. The theory is incremental with recoverable and unrecoverable contributions and a welldefined yield surface. The contributions from the gradients of lastic strain are entirely recoverable. The examles considered in this work offer some guidance for the interretation of exeriments on assivated layers. Acknowledgment The authors are indebted to rofessor L. Bardella for a careful reading of the original manuscrit resulting in helful suggestions. References [1] Fleck,. A., Hutchinson, J. W., and Willis, J. R., 214, Strain Gradient lasticity Under on-roortional Loading, roc. R. Soc., A, 47(217), [2] Bardella, L., and anteghini, A., 215, Modelling the Torsion of Thin Metal Wires by Distortion Gradient lasticity, J. Mech. hys. Solids, 78, [3] Gurtin, M. E., 23, On a Framework for Small-Deformation iscolasticity: Free Energy, Microforces, Strain Gradients, Int. J. last., 19(1), [4] Gudmundson,. A., 24, A Unified Treatment of Strain Gradient lasticity, J. Mech. hys. Solids, 52(6), [5] Gurtin, M. E., and Anand, L., 25, A Theory of Strain-Gradient lasticity for Isotroic, lastically Irrotational Materials. art I: Small Deformations, J. Mech. hys. Solids, 53(7), [6] Fleck,. A., and Hutchinson, J. W., 1997, Strain Gradient lasticity, Adv. Al. Mech., 33, [7] Muhlhaus, H. B., and Aifantis, E. C., 1991, A ariational rincile for Gradient lasticity, Int. J. Solids Struct., 28(7), [8] Fleck,. A., and Willis, J. R., 29, A Mathematical Basis for Strain Gradient lasticity Theory. art I: Scalar lastic Multilier. art II: Tensorial lastic Multilier, J. Mech. hys. Solids, 57(1), [9] Hutchinson, J. W., 212, Generalizing J 2 Flow Theory: Fundamental Issues in Strain gradient lasticity, Acta Mech. Sin., 28(4), [1] Forest, S., and Sievert, R., 23, Elastoviscolastic Constitutive Frameworks for Generalized Continua, Acta Mech., 16(12), [11] Danas, K., Deshande,. S., and Fleck,. A., 21, Comliant Interfaces: A Mechanism for Relaxation of Dislocation ile-us in a Sheared Single Crystal, Int. J. last., 26(12), [12] Fleck,. A., Muller, G. M., Ashby, M. F., and Hutchinson, J. W., 1994, Strain Gradient lasticity: Theory and Exeriment, Acta Metall. Mater., 42(2), [13] ix, W. D., and Gao, H., 1998, Indentation Size Effects in Crystalline Materials: A Law for Strain Gradient lasticity, J. Mech. hys. Solids, 46(3), [14] iordson, C.., and Legarth, B.., 21, Strain Gradient Effects in Cyclic lasticity, J. Mech. hys. Solids, 58(4), [15] Bittencourt, E., eedleman, A., Gurtin, M. E., and an der Giessen, E., 23, A Comarison of onlocal Continuum and Discrete Dislocation lasticity redictions, J. Mech. hys. Solids, 51(2), [16] Ohno,., and Okumura, D., 27, Higher-Order Stress and Grain Size Effects Due to Self-Energy of Geometrically ecessary Dislocations, J. Mech. hys. Solids, 55(9), [17] Evans, A. G., and Hutchinson, J. W., 29, A Critical Assessment of Theories of Strain Gradient lasticity, Acta Mater., 57(5), / ol. 82, JULY 215 Transactions of the ASME