Meccanica https://doi.org/10.1007/s11012-018-0873-8

Distortion-gradient plasticity theory for an isotropic body in finite deformation A. S. Borokinni

. O. O. Fadodun . A. P. Akinola

Received: 23 January 2018 / Accepted: 20 June 2018  Springer Nature B.V. 2018

Abstract This study presents a distortion-gradient model for an isotropic plastically deformed solid within the framework of finite deformation theory. The work aims to provide an alternative form of the Gurtin and Anand (Int J Plast 21:2297–2318, 2005) finite deformation strain-gradient model with a view to relaxing the constraint of irrotationality of plastic flow. The kinematic gradient of deformation is assumed to admit the Kroner–Lee decomposition into elastic and plastic parts. The obtained microforce balance, constitutive relations and plastic flow rule are similar to that obtained by Gurtin and Anand but different in that the present theory used a codirectionality hypothesis to obtain thermodynamically consistent constitutive relations for the dissipative microstresses. Keywords Distortion-gradient  Plastic spin  Isotropic body  Finite deformation  Flow rule

A. S. Borokinni (&) Distance Learning Institute, University of Lagos, Lagos, Nigeria e-mail: [email protected] O. O. Fadodun  A. P. Akinola Department of Mathematics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria e-mail: [email protected] A. P. Akinola e-mail: [email protected]

1 Introduction In the last few years, numerous experimental results have repeatedly revealed that the material behaviour at micron or sub-micron scales displays strong sizeeffects dependence. For instance, researchers like Poole et al. [1], McElhaney et al. [2], and Suresh et al. [3] reported that during micro-indentation hardness experiments, the measured hardness increases greatly as the depth of indentation decreases to microns or sub-microns scales. In the micro-torsion of thin copper wires, Fleck et al. [4] found that the scaled shear strength increases by a factor of three as the diameter of the wire decreases from 170 to 12 lm. Stolken and Evans [5] recorded similar strength increase in microbending of thin nickel foils as the foil thickness decreases from 50 to 12.5 lm. Furthermore, Lloyd [6], Nan and Clarke [7] observed substantial work hardening increase in particle-reinforced metal-matrix composites as the particle diameter is reduced from 16 to 7.5 lm at a fixed particle volume fraction. Meanwhile, it is well-known that conventional plasticity theories do not contain intrinsic material length scales in the constitutive laws; and as a result they are inappropriate for describing size-dependent phenomena in plastically deformed bodies at micron scales [8–11]. For this reason, strain gradient plasticity theories have been developed to address this shortcoming of conventional plasticity theories in applications to materials and structures whose dimension

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controlling plastic deformation is at microns or submicrons length scales [4, 12–15]. Strain gradient plasticity is a theory that involves modeling of plastically deformed bodies with focus on size range within the micron scale and consideration of the effects of plastic strain gradients. The first known phenomenological stain gradient plasticity was provided by Aifantis [16], where he introduced into the classical yield criterion terms relating to the Laplacian of accumulation of plastic strain (Cf. [17–19]). Ten years later, Fleck et al. [4] used a dislocation theory to develop rate-independent strain gradient plasticity theory. A comparison of Aifantis’ theory [16] and theory of Fleck and Hutchinson [14] is shown in the work of Gurtin and Anand [20], and reveals that Aifantis’ theory is a combination of microscopic force balance and thermodynamically consistent constitutive relations for the microscopic stresses. On the other hand, the theory of Fleck and Hutchinson ([14], 2001) is not thermodynamically consistent in general. In the context of infinitesimal deformation theory, Fleck and Hutchinson ([13, 14], 2001), Gao et al. [21], Huang et al. [22, 23], Gurtin [24], Gurtin and Anand [25, 26], Borokinni [27], and Borokinni et al. [28] have developed different strain-gradient plasticity theories. In each of these theories, strain gradients which are work conjugate to the higher-order stresses have been introduced in the constitutive models. Within the strength of each of the aforementioned strain-gradient theories, the theories have shown to be in reasonable agreement with the observation results from the micro-scale experiments. However, just as for any other approximation theory, infinitesimal deformation theory is a gross approximation of finite deformation theory. For instance, small deformation approach is inappropriate in the analysis of some micro-scale phenomena where the strain or its gradient are expected to be large, such as plastic flow localization and crack tip fields [29, 30]. Further, the obligation to adopt finite deformation approach in the problem of solid mechanics is imposed by the fact that finite deformation consideration provides the avenue to reveal some important phenomena which infinitesimal deformation theory often fails to apprehend. Now, in the framework of finite deformation theory, Hwang et al. [15] proposed a finite deformation theory of strain-gradient plasticity. In their work, the kinematics relations, equilibrium equations, and constitutive laws were expressed in the reference configuration; and in

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the special case of micro-indentation problem, the obtained result was in agreement with the experimental data. Gurtin and Anand [26] developed a finite strain-gradient plasticity for an isotropic, plastically irrotational material which generalized their earlier proposed infinitesimal deformation strain-gradient plasticity model. The theory introduced two phenomenological length parameters; and it was shown that the dependence of the microstresses on the gradient of the plastic stretch-rate resulted in the strengthening and possibly weakening of the body induced by the viscoplastic flow. The current study is motivated by the work of Gurtin and Anand [26], and to obtain flow rule of viscoplasticity that relaxes the constraint of irrotationality for finite deformation distortion gradient plasticity, and used a codirectionality hypothesis to obtain thermodynamically consistent constitutive relations for dissipative microstresses. Therefore, we develop a distortiongradient theory which accommodates defect energy due to Burgers tensor in plastically deformed bodies at micron or sub-micron scales. The present theory obtained microforce balance, plastic constitutive relations and flow rule similar to that obtained by Gurtin and Anand [26] but different in that the present theory allows for plastic spin and deduced thermodynamically consistent constitutive relations. The present theory would be appropriate for the analysis of micro-scale phenomena such as plastic flow, crack tips, micro-indentation, torsion of wire at large plastic distortion. The rest of the paper is organized as follow: Sect. 2 gives the notations and definitions of the used symbols; Sect. 3 presents the kinematic relations; Sect. 4 details the macroscopic and microscopic force balances; Sect. 5 presents the free-energy imbalance; Sect. 6 gives the constitutive relations; Sect. 7 details the plastic flow rule; Sect. 8 gives the microscopically simple boundary conditions and variational formulation of the flow rule; Sect. 9 gives the plastic freeenergy balance; Sect. 10 presents the strengthening and weakening by plastic flow; while Sect. 11 concludes the study.

2 Notations In the present work and throughout the study, we use lower-case boldface such as a; b; . . . for vectors; upper-case bold-face such as A; B; . . . for second-

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order tensors; and the hollow font such as A; B; . . . for third-order tensors. We write trA, symA, and skwA for the trace, symmetric part, and skew part of a secondorder tensor A. Similarly, we write AT , A1 , and AT for the transpose, inverse, and transpose of inverse of a second-order tensor A. Further, we denote the deviatoric part of second-order tensor A as A0 and the symmetric-deviatoric part as sym0 A. The gradient operator r, divergence operator Div, and curl operator Curl are carried out with respect to the material point X in the reference configuration and when convenient, we employ the notions of standard Cartesian tensor analysis and write ðraÞij ¼ ai;j , Diva ¼ ai;i , and ðCurlaÞi ¼ ijk ak;j for any non-zero vector function a; and ðrAÞijk ¼ Aij;k , ðDivAÞi ¼ Aij;j , and ðCurlAÞij ¼ ipq Ajq;p for non-zero second-order tensor A. We define the operator  on non-zero vector function a, non-zero second-order tensor function A, and on the pair ðA; BÞ as ðaÞij ¼ ikj ak , ðAÞrpj ¼ rij Api , and ðA  BÞrpq ¼ rij Api Bqj respectively. The respective inner products of second-order tensors and third-order tensors are denoted and defined as A : B ¼ Aij Bij and . A..B ¼ Aijk Bijk . The magnitude of non-zero secondorder tensor A is jAj ¼ ðAij Aij Þ1=2 while the magnitude pffiffiffiffiffiffiffiffiffiffiffiffiffiffi of non-zero third-order tensor A is jAj ¼ Aijk Aijk . Finally, the lightface upper case J denotes the Jacobian of transformation from reference configuration onto deformed configuration; and the operator 0 0 on the pair ðA; BÞ is defined as ðA  BÞjqp ¼ Ajk qip Bki .

3 Kinematic relations

F ¼ rx:

ð1Þ

The Kroner–Lee decomposition [31, 32] of F in Eq. (1) gives F ¼ Fe F p ;

ð2Þ

where Fe and Fp are the elastic and plastic parts of F respectively. It must be noted that Fp which characterizes the local plastic deformation at material point X maps vectors in the reference configuration onto vectors in the structural configuration while Fe which characterizes the local stretching and rotation of the coherent structure maps vectors in the structural configuration onto deformed configuration. Invoking the standard kinematic incompressibility constraint of plastic flow on Fp gives detFp ¼ 1;

and

trðF_ p Fp1 Þ ¼ 0;

ð3Þ

where F_ p is the material time derivative of Fp , detFp is the determinant of Fp , and Fp1 is the inverse of Fp . Using Eqs. (2) and (3), the Jacobian J of the transformation from the reference configuration onto deformed configuration is J ¼ detF ¼ detFe

detFp ¼ detFe :

ð4Þ

3.2 Burgers tensor The Burgers tensor, denoted by second-order tensor G measures the incompatibility or incoherency of the tensor field Fp . Basically, if e is a unit vector in the structural space for point X, the local vector GT ðXÞe describes the local Burgers vector per unit area. The Burgers tensor G is defined as

3.1 Decomposition of deformation gradient

G ¼ Fp CurlFp ;

Let an isotropic solid body B be identified with a subset of three-dimensional Euclidean space it occupies in a fixed reference configuration such that the deformation of B from the reference configuration onto deformed configuration is given by a smooth oneto-one function x ¼ xðX; tÞ, where X denotes the position vector of an arbitrary point of the body prior to the deformation and x is the position vector of the corresponding point in the deformed configuration. The gradient of deformation which measures the geometry of deformation of the body is

and its time derivative is G_ ¼ F_ p CurlFp þ Fp CurlF_ p :

ð5Þ

ð6Þ

3.3 Elastic strain tensor Applying the right polar decomposition theorem on Fe gives Fe ¼ Re Ue ;

ð7Þ

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where Re is an elastic rotation and Ue is the elastic right symmetric, positive-definite stretch tensor. The elastic strain Ee is related to tensor Ue by

in Eq. (10), the stress Se power-conjugate to F_ e transforms as Se ! Se : ð13Þ

FeT Fe ¼ Ue ReT Re Ue ¼ ðUe Þ2 ¼ I þ 2Ee ;

Then, using Eqs. (12) and (13), the power Se : F_ e transforms as _ e Þ ¼ QT Se : ðF_ e þ QT QF _ e Þ: Se : F_ e ! Se : ðQF_ e þ QF

ð8Þ

The above equation implies that 1 1 Ee ¼ ððUe Þ2  IÞ ¼ ðCe  IÞ; 2 2

ð9Þ where I is the unit tensor and Ce is the elastic Cauchy– Green tensor.

The implication of Eq. (15) is that Se ¼ QT Se ) Se ¼ QSe :

It is well-known that changes in observer (frame) are smooth time-dependent rigid transformations of the Euclidean space through which the body moves. In the present study, translations of the origin are ignored since they have no effect on the deformation gradient. Thus, the proposed theory is required to be invariant under transformation of the form ð10Þ

where QðtÞ is rotation tensor at time t. Using Eq. (10), the deformation gradient F transforms as F ! QF:

ð11Þ

The reference configuration and structural spaces are independent of the choice of frame; thus, • •

Fp is invariant under a change in frame. Furthermore, since r, Div, and Curl are gradient, divergence, and curl operators defined with respect to coordinates system in the reference configuration, then rFp , G ¼ Fp CurlFp , and DivFp are invariant under change in frame.

In view of the above observations, and using Eqs. (2) and (11), the tensors Fe and F_ e obey the transformation laws _ e: ð12Þ Fe ! QFe ; and F_ e ! QF_ e þ QF Suppose one assumes that under the change of frame

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Since the change in frame is arbitrary, one can choose the rotation Q such that Q_ ¼ 0; and knowing that power is invariant under change of frame, then Se : F_ e ! QT Se : F_ e ) ðSe  QT Se Þ : F_ e ¼ 0: ð15Þ

3.4 Material frame-indifference and its consequences

xðX; tÞ ! QðtÞxðX; tÞ;

ð14Þ

Ce ¼ FeT Fe ¼ ðUe Þ2

ð16Þ

The combination of Eqs. (13) and (16) gives the consequence of frame-indifference Se ! QSe : Now, let a second-order tensor Te be defined as ð17Þ Te ¼ Fe1 Se ; then, one finds that Se FeT ¼ Fe Te FeT :

ð18Þ

Using Eq. (18) and the symmetric property of Se FeT , the second-order tensor Te , defined in Eq. (17) is symmetric. That is, ð19Þ Te ¼ TeT : Finally, using Eq. (9), the power Se : F_ e can be expressed as Se : F_ e ¼ Fe Te : F_ e ¼ Te : ðFeT F_ e Þ ¼ Te : E_ e ; ð20Þ where the second-order tensor Te which is invariant under a change in frame is the elastic stress because its expends power via the strain-rate E_ e .

4 Macroscopic and microscopic force balances In this section, we introduce three relevant internal forces Se , Sp , K such that the elastic macrostress Se is power-conjugate to the elastic deformation rate F_ e , the plastic microstress Sp is power-conjugate to the plastic deformation-rate F_ p , the polar microstress K is power-

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conjugate to the gradient of the plastic deformationrate rF_ p . Let P be an arbitrary part (subregion) of the body B with n the outward unit vector normal on the boundary oP of P, the internal power expenditure I(P) within the part P of the body B is1  Z  . ð21Þ IðPÞ ¼ Se : F_ e þ Sp : F_ p þ K..rF_ p dV; P

Z

ðDivS þ bÞ  x~dV ¼

Z

ðSn  sðnÞÞ  x~dA: oP

P

ð24Þ Employing the fundamental lemma of calculus of variation gives the macroforce balance DivS þ b ¼ 0;

ð25Þ

and the accompanying macrotraction condition

and the power expended on part P of the body by external agencies is Z Z Z _ þ bxdV _ þ sðnÞ  xdA KðnÞ : F_ p dA; WðPÞ ¼ oP

oP

P

ð22Þ where sðnÞ, b and KðnÞ are the macrotraction, body force and microtraction respectively. Since the basic rate-like variables is a list _ F_ e ; F_ p Þ, we define a generalized virtual veloci_ F; ðx; ~ F~e ; F~p Þ subject to the constraint ties to be a list ð~ x; F; ~e p

e ~p

F~ ¼ F F þ F F

and

~p p1

trðF F

ð23Þ

Þ ¼ 0:

4.1 Macroforce balance In order to model the macroforce balance for the present theory, we employ the principle of virtual p e power and set F~ ¼ 0 in Eq. ð23a Þ so that F~ ¼ F~ Fp . Then, we have Z Z Z e Se : F~ dV ¼ sðnÞ  x~dA þ b  x~dV: oP

P

Sn ¼ sðnÞ

respectively. Moreover, using the symmetric property of Se FeT [26] and S ¼ Se FpT give the moment balance SFT ¼ FST :

ð27Þ

4.2 Microforce balance Similarly, setting x~ ¼ 0 so that F~ ¼ 0 and e p F~ ¼  Fe F~ Fp1 , the virtual power balance is  Z  .. p p p eT e pT ~ ~ ðS  F S F Þ : F þ K.rF dV P ð28Þ Z p ~ KðnÞ : F dA: ¼ oP

By divergence theorem Z Z Z .. p p ~ ~ K.rF dV ¼ Kn : F dA  DivK : F~p dV; oP

P

P

ð26Þ

P

ð29Þ

e

~ p1 in L.H.S. of Eq. (24) gives Substituting F~ ¼ FF Z Z Z e ~e e ~ p1 ~ S : F dV ¼ S : FF dV ¼ ðSe FpT Þ : FdV: P

P

P

Substituting F~ ¼ r~ x gives Z ~ ðSe FpT Þ : FdV P

¼

Z

e pT

ðS F P

Þ : r~ xdV ¼

Substituting Eqs. (29) in Eq. (28) gives Z  p  p S  FeT Se FpT  DivKp : F~ dV P Z p ¼ ½KðnÞ  Kn : F~ dA:

ð30Þ

oP

Z

sðnÞ  x~dA þ oP

Z

b  x~dV:

P

Invoking the fundamental lemma of calculus of variation yields the microscopic force balance FeT Se FpT ¼ Sp  DivK;

ð31Þ

e pT

Using divergence theorem and setting S ¼ S F give

and the associated microtraction condition KðnÞ ¼ Kn:

. 1 It should be noted that the local power expenditure K..rF_ p seems to be the most general power-expenditure associated with the gradient of plastic distortion rate.

ð32Þ

Using Eqs. (9b ) and (17) give

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FeT Se FpT ¼ FeT Fe Fe1 Se FpT ¼ Ce Te FpT ¼ Ce Pe ; Pe ¼ Te FpT : ð33Þ a relation that permits us to rewrite the microforce balance Ce Pe ¼ Sp  DivK:

ð34Þ

5 Free-energy imbalance Let w denote the free-energy in the body B per unit reference volume. The second law of thermodynamics implies that the time derivative of the total free energy of a subregion P of the body B is always less than or equal to the total power-expenditure on the subregion P. Z d wdV  WðPÞ; ð35Þ dt P where W(P) is the total power expenditure on the subregion P. Using the power balance WðPÞ ¼ IðPÞ and Eq. (21) give  Z Z  . d wdV  Se : F_ e þ Sp : F_ p þ K..rF_ p dV: dt P P ð36Þ The local form of Eq. (36) gives the free energy imbalance

6 Constitutive relation Let the free-energy w, a function of either of pair He , G or Ee , G assumes the additive form ^ e ; GÞ ¼ wðE  e ; GÞ ¼ w ^e ðHe Þ w ¼ wðH e ðEe Þ þ w p ðGÞ; ^p ðGÞ ¼ w þw

ð41Þ

e ðEe Þ ^e ðHe Þ or w where G is the Burgers tensor, w p p ^ ðGÞ is the  ðGÞ or w denotes the elastic part of w and w plastic part of w called the defect energy. Let Sp;en , Ken and Sp;dis , Kdis be the energetic and dissipative components of Sp , K such that Sp ¼ Sp;en þ Sp;dis ;

K ¼ Ken þ Kdis :

ð42Þ

From the free-energy imbalance (40) and the freeenergy function (41), we can deduce that ^e ðHe Þ e e ow _ e ðEe Þ: _ e ¼ ow ðE Þ : E_ e ¼ w : H oHe oEe

ð43Þ

Following (41) and (42), the temporal increase in the defect energy is . owp ðGÞ _ p : G ¼ R : G_ ¼ w_ ðGÞ ¼ Sp;en : F_ p þ Ken ..rF_ p ; oG owp ðGÞ ; R¼ oG

ð44Þ and the dissipative inequality . Sp;dis : F_ p þ Kdis ..rF_ p 0

ð45Þ p

. w_  S : F  S : F  K..rF_ p  0: e

_e

p

_p

ð37Þ

Let a second-order tensor He be defined through H_ e ¼ E_ e Fp ;

6.1 Energetic constitutive relations ð38Þ

then, using Eqs. (20), (33b ), and (38) give Se : F_ e ¼ Te : E_ e ¼ Te FpT : E_ e Fp ¼ Pe : H_ e : ð39Þ

e ðEe Þ of the free-energy for an Let the elastic part w isotropic body take the conventional form e ðEe Þ ¼ ljEe j2 þ j jtrEe j2 ; w o 2

ð46Þ

p ðGÞ which characterizes the and let the plastic part w defect energy assumes the form

Substituting Eq. (39) in Eq. (37) gives . w_  Pe : H_ e  Sp : F_ p  K..rF_ p  0:

respectively, where R ¼ owoGðGÞ is the thermodynamic defect stress [26].

ð40Þ

p ðGÞ ¼ 1 lL2 jGj2 ; w 2

ð47Þ

where l is the elastic shear modulus, j is the elastic bulk modulus, Ee0 is the deviatoric part of strain tensor

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Ee , trEe is the trace of Ee , and L is energetic length scales. Employing Coleman–Noll [33] procedure and Eq. (33b ) give the elastic constitutive law Te and the associated tensor Pe Te ¼

e ðEe Þ ow ¼ 2lEeo þ jðtrEe ÞI; oEe

ð48Þ

By Eqs. (55) and (56) it is clear that Ken ¼ K1 þ K2 þ K3 :

ð57Þ

Finally, using Eqs. (44a ), (55) and (56), the constitutive relations for the energetic components Sp;en , ~ v and 1 K are Sp;en ¼ lL2 GðCurlFp ÞT ;

ð58Þ

1 lL2 ~ ðIÞðFpT GÞ; v ¼ ðIÞM ¼ 2 2

ð59Þ

and Pe ¼ 2lEeo FpT þ jðtrEe ÞFpT ;

ð49Þ

respectively. Let a second-order tensor M be related to thermodynamic defect microstress R by M ¼ FpT R:

and K1 ¼

ð50Þ

 

 1 T ðM þ MÞ  I ¼ lL2 symðFpT GÞ  I : 2 ð60Þ

Using Eqs. (44b ), (47), and (50) give RðGÞ ¼

6.2 Dissipative constitutive relations

^p ðGÞ ow ¼ lL2 G; oG

ð51Þ

and M ¼ FpT R ¼ lL2 FpT G

ð52Þ

respectively. Substituting Eq. (6) in Eq. (44a ) gives w_ p ¼ R : G_ ¼ R : ðF_ p CurlFp þ Fp CurlF_ p Þ; w_ p ¼ RðCurlFp ÞT : F_ p þ M : CurlF_ p :

ð53Þ

p , we assume that Noting that M : CurlF_ p ¼ Mij irs F_ js;r en en ¼ the energetic part K has the component form Kjs;r p

p;en

Mij irs and the energetic part of S has the form S RðCurlFp ÞT so that Eq. (53) becomes . w_ p ¼ Sp;en : F_ p þ Ken ..rF_ p :

¼

. S F_ ¼ Sp;dis : F_ p þ Kdis ..rF_ p :

ð54Þ

2 Kjpq ¼ vp djq

ð62Þ

N¼

F_ ; _ jFj

_ 6¼ 0: jFj

ð63Þ

Invoking the hypothesis of codirectionality of S and N gives

and S ¼ /N; ð55Þ

where ~ v is defined through the skew part of M as ~ÞT : skwM ¼ ðv

respectively. The dissipation operator ‘ ’ is defined such that the dissipation of S through F_ is

Let N characterize the generalized flow direction and be defined by

Define the following third order tensors K1 ; K2 and K3 in component forms respectively as 1 1 ¼ ipqðMij þ Mji Þ; Kjpq 2 3 Kjpq ¼ vq djp ;

Let S ¼ ðSp;dis ; l1 Kdis Þ denote generalized plastic microstress and F_ ¼ ðF_ p ; lrF_ p Þ be its associated generalized plastic distortion rate with magnitudes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jSj ¼ jSp;dis j2 þ l2 jKdis j2 ; and ð61Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ ¼ jF_ p j2 þ l2 jrF_ p j2 ; jFj

ð56Þ

ð64Þ

where / is a scalar function. In view of Gurtin and Anand [26], the scalar function assumes the form  p m d ð65Þ / ¼ jSj ¼ S; do

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where do is the initial flow rate, m is a positive constant _ ¼ (rate sensitivity parameter) and dp ¼ jFj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jF_ p j2 þ l2 jrF_ p j2 is the effective flow rate, and S is the internal-state variable (flow resistance) with the constraint that S satisfies the evolution equation S_ ¼ HðSÞdp ;

Sðx; 0Þ ¼ SY :

ð66Þ

The parameter H(S) is the hardening function and SY is the initial flow resistance or coarse grain yield strength. The constraint of codirectionality hypothesis suggests that the constitutive relations for the dissipative components Sp;dis and Kdis are  p m  p m d S _ p dis S _p 2 d F Sp;dis ¼ ; K ¼ l rF ; p do d do d p

 SFL  lL2 GGT FpT  DivðsymðFpT GÞ  IÞ   

 T . 1 1 þ ðIÞ..rðFpT GÞ I  r ðIÞðFpT GÞ 2 2  p m  p m  d S _p d S _p 2 F  l Div r F ¼ ; do d p do d p ð71Þ e e

where SFL ¼ C P is the flow stress. It is obvious that the flow rule is a second order partial differential equation in F_ p . Remark It is observed that from the outset, the obtained flow rule in Eq. (71) • •

Accommodates plastic spin; and Ignores the constraints of symmetric and deviatoric properties on the flow stress SFL ¼ Ce Pe .

ð67Þ where S satisfies Eq. (66). Finally, using Eqs. (61b ) and (67), one writes Eq. (62) as  p m d _ ð68Þ S F¼ Sd p 0: do Eq. (68) shows that the choice of the dissipative microstresses through the codirectionality hypothesis are consistent with the laws of thermodynamics.

8 Microscopically simple boundary conditions: variational formulation of the flow rule Let vector n denote the outward unit normal on the boundary oB such that oB ¼ Chard [ Cfree , where Chard and Cfree are the microscopically hard and microscopically free portions of oB. The microscopically simple boundary conditions for the present theory are: F_ p ¼ 0 on Chard and Kn ¼ 0 on Cfree :

ð72Þ

The consequence of Eq. (72) is 7 Plastic flow rule

ðKnÞ : F_ p ¼ 0;

In general, the plastic flow rule is obtained by augmenting the microforce balance with the constitutive relations. Substituting Eq. (42) into microforce balance in Eq. (34) gives

which implies that the microscopically power expended per unit area on the boundary oB by the material in contact with body B must vanish. Now, in view of microscopically simple boundary conditions in Eq. (72), we proceed to establish a weak formulation of the plastic flow rule. Since the boundary conditions in Eq. (72) renders the power expenditure null on oB, then, using Eqs. (27), (33a ), p SFL ¼ Ce Pe ; and setting the virtual field F~ ¼ U gives  Z  .. p ð74Þ ðS  SFL Þ : U þ K.rU dV ¼ 0:

Ce Pe  Sp;en þ DivKen ¼ Sp;dis  DivKdis :

ð69Þ

This can be written as ~ÞT  ðDivv ~ÞI Ce Pe  Sp;en þ DivK1 þ ðrv ¼ Sp;dis  DivKdis :

ð70Þ

on oB;

ð73Þ

B

Substituting constitutive relations in Eqs. (58), (61), (62), and (67) in Eq. (69) gives the flow rule

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Employing the divergence theorem and the condition U ¼ 0 on Chard gives

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Z

½Sp  SFL  DivK : UdV ¼ 0:

ð75Þ

B

Equation (75) satisfies the microforce balance Eq. (34) together with assumption of the microscopically free boundary condition. The combination of Eq. (74) and the constitutive relations in Eqs. (58)–(60), and (67) yields the weak or variational form of the plastic flow rule.

9 Plastic free-energy balance In the spirit of Gurtin and Anand [26], we proceed to relate the temporal increase in the plastic free-energy to the power stress. In view of Eq. (44a ), the total defect energy in the body B is  Z Z  . d ð76Þ wp dV ¼ Sp;en : F_ p þ Ken ..rF_ p dV: dt B B Using Eq. (42) in Eq. (76) gives  Z Z  . d wp dV ¼ Sp : F_ p þ K..rF_ p dV dt B B  Z  . p;dis p S : F_ þ Kdis ..rF_ p dV 

ð77Þ

d dt

Z

Employing divergence theorem and invoking the assumption of null power expenditure on oB give Z Z d p w ðGÞdV ¼ ½Sp  DivK : F_ p dV dt B B  Z  ð78Þ . p p;dis p dis . _ _ S : F þ K .rF dV: 

Z

B

SFL : F_ p dV 

B

Z  p m d Sdp dV: B do ð80Þ

10 Effect of plastic distortion gradient on strengthening and weakening of body The rate-dependence of most metals at room temperature is very small, we therefore, pay attention to cases of rate-independent materials ðm  0Þ. In view of Gurtin and Anand [26], given any fixed time and any part P of the body: the section P of the body is strengthened by the flow if jSFL j [ SY ; while the section P of the body is weakened by the flow if jSFL j\SY . In order to examine the effect(s) of rFp , we employ the following assumptions [26]: HðSÞ ¼ 0, so that by (66) S ¼ SY ; L ¼ 0, so that wp ðGÞ ¼ 0 implies that Sp;en ¼ 0, Ken ¼ 0; The material is rate-independent i.e. m  0; At no time is the plastic distortion rate homogeneous; and The boundary conditions are microscopically simple.

1. 2. 3. 4. 5.

B

wp ðGÞdV ¼

Using the first three assumptions, Eq. (80) degenerates to Z Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jF_ p j2 þ l2 jrF_ p j2 dV: SFL : F_ p dV ¼ SY B

B

ð81Þ

B

Using the microforce balance Eq. (34) gives Z Z d wp ðGÞdV ¼ SFL : F_ p dV dt B B  Z  . p;dis p S : F_ þ Kdis ..rF_ p dV: 

For a fixed time, let maxB jSFL j denote the maximum value of jSFL j over the body B; and using the Cauchy– Schwarz inequality on the L.H.S. of Eq. (81) gives ð79Þ

B

The balance Eq. (79) which is independent of the particular constitutive relations for dissipative stresses Sp;dis , Kdis indicates that the temporal increase in the defect energy cannot exceed the plastic power. In special case, using constitutive relations in Eq. (67) gives

 Z Z Z    SFL : F_ p dV   jSFL jjF_ p jdV  maxB jSFL j jF_ p jdV:   B

B

B

ð82Þ Using the fourth assumption and Eq. (82), we conclude that Z Z p _ SFL : F dV  maxB jSFL j jF_ p jdV; and B B ð83Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 p p p jF_ j\ jF_ j þ l2 jrF_ j

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Meccanica

Finally, Eqs. (81) and (83) imply that maxB jSFL j [ SY :

ð84Þ

The inequality in Eq. (84) suggests that there is a nontrivial subregion of the body on which magnitude jSFL j of the flow stress is strictly greater than the coarsegrain yield strength SY . This subregion is strengthened by the flow stress. Next, the integration of Eq. (34) over the volume B, SFL ¼ Ce Pe ; and using divergence theorem give Z Z Z p SFL dV ¼ S dV  KndA: ð85Þ B

Chard

B

Using the second and third assumptions yield Sp ¼ SY

F_ p rF_ p ; K ¼ l2 SY p : p d d

ð86Þ

One notes that      Z  p 2   l rF_ n   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidA: KndA  SY   2 pT Chard Chard   l2 jrF_ j  Z Z   jrF_ p nj  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dA  lSY AreaðChard Þ; KndA  l2 SY  Chard Chard l2 jrF_ pT j2 Z   

In the special case, if the boundary is microscopically free, then hard ¼ 0 and Eq. (90) reduces to   Z  1    ð91Þ volðBÞ SFL dV   SY : B The above inequality shows that there exist a nontrivial region of the body weakened by the flow stress.

11 Conclusion In the context of finite deformation, we develop a distortion gradient plasticity theory in the presence of plastic spin. The obtained microforce balance, plastic flow rule, and constitutive relations are similar to that obtained by Gurtin and Anand model [26] but different in that a codirectionality hypothesis was used to obtain thermodynamically consistent constitutive relations for the dissipative microstresses, and the constraint of irrotationality is relaxed. It is observed that the presence of plastic distortion gradient in the effective flow rate improves the strengthening and weakening of the plastically deformed body. The results of this work find applications in the analysis of plastically deformed bodies at micron or sub-microns scales.

ð87Þ R

where AreaðChard Þ ¼ Chard dA is the area of the microscopically hard surface. In view of the inequality Eq. (83b ), one has    Z Z     p _  F F_ p     volðBÞ; dV  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dV    p _ j    B B jF jF_ p j2 þ l2 jrF_ pT j2   where volðBÞ is the volume of the body B. Let the quantity hard be defined as hard ¼

lAreaðChard Þ : volðBÞ

Using Eq. (87) Z  Z  Z       SFL dV    Sp dV  þ       B

B

ð88Þ

Chard

  KndA

ð89Þ

 SY volðBÞ þ lSY AreaðChard Þ: The combination of Eqs. (88) and (89) yields   Z  1    volðBÞ SFL dV   ð1 þ hard ÞSY : B

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ð90Þ

Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest.

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