Three-dimensional continuum dislocation theory

International Journal of Plasticity 76 (2016) 213e230

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Three-dimensional continuum dislocation theory K.C. Le* €t Bochum, D-44780 Bochum, Germany Lehrstuhl für Mechanik e Materialtheorie, Ruhr-Universita

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 May 2015 Received in revised form 16 July 2015 Available online 25 August 2015

A three-dimensional continuum dislocation theory for single crystals containing curved dislocations is proposed. A set of governing equations and boundary conditions is derived for the true placement, plastic slips, and loop functions in equilibrium that minimize the free energy of crystal among all admissible functions, provided the resistance to dislocation motion is negligible. For the non-vanishing resistance to dislocation motion the governing equations are derived from the variational equation that includes the dissipation function. A simplified theory for small strains is also provided. An asymptotic solution is found for the two-dimensional problem of a single crystal beam deforming in single slip and simple shear. © 2015 Published by Elsevier Ltd.

The paper is dedicated to the 70th birthday of my teacher V. Berdichevsky. Keywords: A. Dislocations B. Crystal plasticity B. Finite strain C. Variational calculus

1. Introduction Dislocations appear in ductile crystals to reduce their energy. In view of a huge number of dislocations appearing in plastically deformed crystals (which typically lies in the range 108 ÷ 1015 dislocations per square meter) the necessity of developing a physically meaningful continuum dislocation theory (CDT) to describe the evolution of dislocation network and predict the formation of microstructure in terms of mechanical and thermal loading conditions becomes clear to all researchers in crystal plasticity. One of the main guiding principles in seeking such a continuum dislocation theory has first been proposed by Hansen and Kuhlmann-Wilsdorf (1986) in form of the so-called LEDS-hypothesis: the true dislocation structure in the final state of deformation minimizes the energy of crystal among all admissible dislocation configurations. This turns out to be the consequence of Gibbs variational principle applied to crystals with dislocations in case of vanishingly small Peierls stress (see Berdichevsky, 2009). Because of numerous experimental evidences supporting this hypothesis (see, i.e., Hughes and Hansen, 1997; Kuhlmann-Wilsdorf, 1989, 2001; Laird et al., 1986), the latters use in constructing the continuum dislocation theory seems to be quite reasonable and appealing. For the practical realization one needs to (i) specify the whole set of unknown functions and state variables of the continuum dislocation theory, and (ii) lay down the free energy of crystals as their functional to be minimized. Berdichevsky and Sedov (1967) were the firsts to have proposed the variational formulation of CDT that extended Gibbs variational principle to crystals with continuously distributed dislocations (see also its further development in Berdichevsky (2006a)). The developed CDT has been successfully applied to various twodimensional problems of dislocation pileups, bending, torsion, as well as formation of dislocation patterns in single crystals with straight dislocation lines (see Baitsch et al., 2015; Berdichevsky and Le, 2007; Kaluza and Le, 2011; Kochmann and Le, 2008a,b, 2009; Koster et al., 2015; Koster and Le, 2015; Le and Günther, 2015; Le and Nguyen, 2012, 2013; Le and Sembiring,

* Tel.: þ49 234 32 26033. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ijplas.2015.07.008 0749-6419/© 2015 Published by Elsevier Ltd.

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2008a,b, 2009). Let us mention here the similar approaches suggested in Acharya and Bassani (2000), Acharya (2001), Engels € et al. (2012), Gurtin (2002), Gurtin et al. (2007), Mayeur and McDowell (2014) and Oztop et al. (2013) which do not use the LEDS-hypothesis explicitly but employ instead the extended principle of virtual work for the gradient plasticity. However, as experiences and experiments show, dislocation lines are in general loops that, as a rule, can change their directions and curvatures depending on the material properties, loading condition, and crystal's geometry. Therefore the extension of CDT to networks of dislocations whose lines are curves in the slip planes is inevitable. To the best of author's knowledge, such threedimensional continuum dislocation theory based on the LEDS-hypothesis for curved dislocations has not been developed until now. It became also clear to him that the latter's absence was due to the missing scalar dislocation densities for the network of curved dislocations. The first attempt at constructing a continuum theory that can predict in principle not only the dislocation densities but also the direction and curvature of the dislocation lines has been made by Hochrainer et al. (2007) in form of the so-called continuum dislocation dynamics. Their theory starts with the definition of the dislocation density that contains also the information about the orientation and curvature of the dislocation lines. Then the set of kinematic equations is derived for the dislocation density and curvature that requires the knowledge about the dislocation velocity. The relation between the dislocation density and the macroscopic plastic slip rate via the dislocation velocity is postulated in form of Orowan's equation. The couple system of crystal plasticity and continuum dislocation dynamics becomes closed by the constitutive € hlke, 2015). In addition equation of a flow rule type (see Hochrainer et al., 2014; Sandfeld et al., 2011, 2015; Wulfinghoff and Bo to the heavy computational cost of such theory, the relation to thermodynamics of crystal plasticity and to the LEDShypothesis is completely lost: the equilibrium solution found in this theory may not minimize the energy of crystal among all admissible dislocation configurations. Let us mention also a continuum approach proposed by Xiang (2009), Zhu and Xiang (2010), Zhu et al. (2014) and Zhu and Xiang (2015) in which the three-dimensional dislocation network is characterized by two families of disregistry functions that may take only integer values. As will be seen later, a loop function introduced in the present paper to describe the kinematics of curved dislocations is quite similar to one of the disregistry functions employed in the above cited references. However, this similarity between two approaches is restricted only to the kinematics. As the governing equations for the displacements, plastic slip, and loop function and the basic principle to derive them are concerned, our approach differs strongly from that proposed in Zhu et al. (2014) and Zhu and Xiang (2015). The coupled system of equations in Zhu and Xiang (2015) is derived from the underlying discrete dislocation dynamics for the displacement and disregistry functions. This approach is subject to the same critics as that developed in Hochrainer et al. (2007). The aim of this paper is to extend the nonlinear continuum dislocation theory (CDT) developed recently by Le and Günther (2014) to the case of crystals containing curved dislocations. Provided the dislocation network is regular in the sense that nearby dislocations have nearly the same direction and orientation, we introduce a loop function whose level curves coincide with the dislocation lines. Taking an infinitesimal area perpendicular to the dislocation line at some point of the crystal, we express the densities of edge and screw dislocations at that point through the resultant Burgers vectors of dislocations. Such scalar densities, found in this paper for the first time, contain not only the information about the number of dislocations, but also the information about the orientation and curvature of the dislocation lines. In case of dislocation motion we introduce the vector of normal velocity of dislocation line through the time derivative of the loop function. € ner (1992) and Berdichevsky (2006b) we require that the free energy density of crystal depends only on the Following Kro elastic deformation tensor and on the above scalar densities of dislocations. Then we formulate a new variational principle of CDT according to which the placement, the plastic slip, and the loop function in the final state of equilibrium minimize the free energy functional among all admissible functions. We derive from this variational principle a new set of equilibrium equations, boundary conditions, and constitutive equations for these unknown functions. In case the resistance to dislocation motion is significant, the variational principle must be replaced by the variational equation that takes the dissipation into account. This enables crystals with dislocations to stay in equilibrium not at the minimum of the free energy. Note, however, that the incremental minimization of the “relaxed” energy can be applied instead (Ortiz and Repetto, 1999; Carstensen et al., 2002). The constructed theory is generalized for single crystals having a finite number of active slip systems. We provide also the simplifications of the theory for small strains. As compared to the continuum dislocation dynamics proposed in Hochrainer et al. (2007) and Zhu and Xiang (2015) our theory is advantageous not only in the computational cost due to its simplicity, but also in its full consistency with the LEDS-hypothesis. In the problem of single crystal beam having only one active slip system and deforming in simple shear, the energy minimization problem reduces to the two-dimensional variational problem. We solve this problem analytically for the circular cross section and asymptotically for the rectangular cross section. We will show that this solution reduces to that found in Berdichevsky and Le (2007) for the beams with thin and long cross-sections. The paper is organized as follows. After this introduction we present in Section 2 the three-dimensional kinematics for single crystals deforming in single slip. Section 3 formulates the variational principles of the three-dimensional CDT and derives its governing equations. Section 4 extends this nonlinear theory to the case of single crystals with n active slip system. Section 5 studies the three-dimensional small strain CDT. Section 6 is devoted to the analytical and asymptotic solutions of the two-dimensional energy minimization problem of a single crystal beam deforming in simple shear. Section 7 concludes the paper and discusses several open issues. Finally, in Appendix we give a detailed asymptotic analysis of the energy minimization problem posed in Section 6.

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2. 3-D kinematics for single crystals deforming in single slip Nonlinear CDT starts from the basic kinematic resolution of the deformation gradient F ¼ vy/vx into elastic and plastic parts (Bilby et al., 1957)

F ¼ Fe $Fp :

(1)

We attribute an active role to the plastic deformation: Fp is the deformation creating dislocations (either inside or at the boundary of the volume element) or changing their positions in the crystal without distorting the lattice parallelism (see Fig. 1). On the contrary, the elastic deformation Fe deforms the crystal lattice having frozen dislocations (Le and Günther, 2014). Note that the lattice vectors remain unchanged when the plastic deformation is applied, while they change together with the shape vectors by the elastic deformation. We consider first a single crystal deforming in single slip. In this case let us denote the right-handed triad of unit lattice vectors of the active slip system by s, p, and m, where s points to the slip direction, p lies in the slip plane and is perpendicular to s, and m is normal to the slip plane. Without restricting generality we may choose the rectangular cartesian coordinate system (x1,x2,x3) in the reference configuration such that its basis vectors coincide with these lattice vectors (see Fig. 2)

e1 ¼ s;

e2 ¼ p;

e3 ¼ m:

The plastic deformation is then given by

Fp ¼ I þ bðxÞs5m ¼ I þ bðxÞe1 5e3 ;

(2)

with b being the plastic slip. We assume that all dislocations causing this plastic deformation lie completely in the slip planes and the dislocation network is regular in the sense that nearby dislocations have nearly the same direction and orientation. This enables one to introduce a scalar function l(x1,x2,x3) (called a loop function) such that its level curves

lðx1 ; x2 ; c3 Þ ¼ c;

(3)

with c3 and c being constants, coincide with the dislocation lines (cf. with the disregistry function introduced by Xiang (2009) and Zhu and Xiang (2010)). Thus, in this three-dimensional kinematics we admit, according to equation (3), only the conservative motion of dislocations and exclude from consideration the dislocation climb which is an important mechanism of temperature-dependent creep. We denote by n and t the plane unit vectors normal and tangential to the dislocation line. From equation (3) follow

  1 n ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l;1 e1 þ l;2 e2 ; 2 2 l;1 þ l;2

  1 t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  l;2 e1 þ l;1 e2 ; 2 2 l;1 þ l;2

where the comma before an index denotes the partial derivative with respect to the corresponding coordinate. Note that n, t, m form a right-handed basis vectors of the three-dimensional space (see Fig. 2).

Fe

F

Fp Fig. 1. Multiplicative decomposition.

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Fig. 2. A dislocation loop in the chosen coordinate system.

Ortiz and Repetto (1999) (see alsoOrtiz et al., 2000) introduced the resultant Burgers vector of excess dislocations, whose lines cross the area A in the reference configuration, in the following way

I br ¼

Fp $dx;

(4)

C

where C is the close contour surrounding A . Le and Günther (2014) have shown that, in the continuum limit, when the atomic distance goes to zero at the fixed sizes of the representative volume element and the fixed density of dislocations per area of unit cell, integral (4) gives the total closure failure induced by Fp which must be equal to the resultant Burgers vector. It is natural to assume Fp continuously differentiable in this continuum limit, so, applying Stoke's theorem we get from (4)

Z br ¼ 

ðFp  VÞ$nda;

A

where  denotes the vector product, da the surface element, and n the unit vector normal to A . This legitimates the introduction of the dislocation density tensor

T ¼ Fp  V: For the plastic deformation taken from (2)

T ¼ Fp  V ¼ s5ðVb  mÞ: If we choose now an infinitesimal area da with the unit normal vector t, then the resultant Burgers vector of all excess dislocations, whose dislocation lines cross this area at right angle is given by

br ¼ T$t da ¼ sðVb$nÞda ¼ s vn b da: This resultant Burgers vector can be decomposed into the sum of two vectors

br ¼ br⊥ þ brk ¼ ðnsn þ tst Þvn bda; where sn ¼ s$n ¼ n1 and st ¼ s$t ¼ t1 are the projections of the slip vector onto the normal and tangential direction to the dislocation line, respectively. This allows us to define two scalar densities (or the numbers of excess dislocations per unit area) of edge and screw dislocations

   1 l;1 b;1 l;1 þ b;2 l;2  jbr⊥ j 1 ¼ jsn vn bj ¼  r⊥ ¼ ;  b b b l2;1 þ l2;2        brk  1 1 l;2 b;1 l;1 þ b;2 l;2  rk ¼ ¼ jst vn bj ¼  ;  b b b l2 þ l2 ;1

;2

(5)

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with b the magnitude of Burgers vector. We see that the three-dimensional dislocation densities r⊥ and rk depend on both the gradient of the plastic slip and the gradient of the loop function l(x) through the vectors n and t. Note also that, in case of mixed dislocations, the scalar quantities (5) tell us about the resultant screw and edge components of excess dislocations per unit area. Consider now the case of motion of dislocation loops in the slip plane. In this case we allow the loop function to depend explicitly on time t such that equation

lðx1 ; x2 ; c3 ; tÞ ¼ c

(6)

with fixed constants c3 and c describes one and the same dislocation line during its motion in the slip plane. Letting z be the variable along the dislocation line, we may represent the level curve defined by (6) in the parametric form

x1 ¼ x1 ðz; tÞ;

x2 ¼ x2 ðz; tÞ:

Fixing z and taking the differential of (6) we obtain

l;1 dx1 þ l;2 dx2 þ l;t dt ¼ 0 that yields

l;1

dx1 dx þ l;2 2 ¼ l;t : dt dt

1 2 Since v ¼ dx e þ dx e is the velocity of the fixed point on the dislocation line with coordinate z, we define the normal dt 1 dt 2 velocity of the dislocation line as follows

  l_ 1 dx dx vn ¼ v$n ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l;1 1 þ l;2 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; dt dt 2 2 2 l;1 þ l2;2 l;1 þ l;2

(7)

with l_ ¼ l;t (cf. with formula (10) in Zhu and Xiang (2010)). This kinematic quantity will be used in the case of non-vanishing resistance to dislocation motion. 3. Governing equations for single crystals deforming in single slip € ner (1992), the elastic deformation Fe and the dislocation densities r⊥ and rk characterize the current state According to Kro of the crystal, so these quantities are the state variables of the continuum dislocation theory. The reason why the plastic deformation Fp cannot be qualified for the state variable is that it depends on the cut surfaces and consequently on the whole history of creating dislocations. Likewise, the gradient of plastic deformation tensor Cp ¼ FpTFp cannot be used as the state variable by the same reason. In contrary, the dislocation densities depend only on the characteristics of dislocations in the current state (Burgers vector and positions of dislocation lines) and not on how they are created, so r⊥ and rk , in addition to Fe, are the proper state variables. Thus, if we consider isothermal processes of deformation, then the free energy per unit volume of crystal (assumed as macroscopically homogeneous) must be a function of Fe, r⊥, and rk

 j ¼ j Fe ; r⊥ ; rk : Now, if we superimpose an elastic rotation R onto the actual deformation of the body, then the total and elastic deformation change according to

F ¼ R$F;

Fe* ¼ R$Fe :

At the same time, the dislocation densities r⊥ and rk remain unchanged. As such superimposed elastic rotation does not change the elastic strain and the dislocation densities, we expect that the energy remains unchanged. The standard argument (see, e.g., Gurtin, 1981) leads then to

 j ¼ j Ce ; r⊥ ; rk ; where Ce is the elastic deformation tensor defined by

Ce ¼ FeT $Fe :

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Let the undeformed single crystal occupy some region V of the three-dimensional space. The boundary of this region, vV , is assumed to be the closure of union of two non-intersecting surfaces, vk and vs. Let the placement be a given smooth function of coordinates

yðxÞ ¼ x þ u0 ðxÞ at vk ;

(8)

where u0(x) ¼ y(x)x is the given displacement vector. Under this condition it is natural to assume that dislocations cannot reach this part vk of the boundary, because otherwise they would form small legs (or steps) at the boundary conflicting the continuity of the displacements. Therefore we set

bðxÞ ¼ 0;

lðxÞ ¼ 0

at vk :

(9)

Note that such Dirichlet's boundary conditions for b and l resemble the micro-clamped boundary conditions as classified in Gurtin (2002). At the remaining part vs the “dead” load (traction) t is specified. As the free energy density is invariant with respect to the transformation (scaling) of the loop function l(x) / cl(x), with c being an arbitrary constant factor, we can impose on this scalar function the following constraint

Z

  lðxÞdx ¼ V ;

(10)

V

where dx ¼ dx1dx2dx3 denotes the volume element and jV j is the volume of the region V .1 If no body force acts on this crystal, then its energy functional is defined as

Z I½yðxÞ; bðxÞ; lðxÞ ¼

Z wðF; b; Vb; VlÞdx 

t$y da;

(11)

vs

V

where

 wðF; b; Vb; VlÞ ¼ j Ce ; r⊥ ; rk :

(12)

Provided the resistance to the dislocation motion is negligibly small and no surfaces of discontinuity occur inside crystals, then the following variational principle is valid for single crystals with one active slip system: the true placement vector yðxÞ, the true plastic slip bðxÞ, and the true loop function lðxÞ in the final equilibrium state of deformation minimize energy functional (11) among all continuously differentiable fields y(x), b(x), and l(x) satisfying constraints (8)e(10). Let us derive the equilibrium equations from this variational principle. Introducing Lagrange's multiplier ll into the functional (11) to get rid of the constraint (10), we compute its first variation

Z  P : dyV þ

dI ¼

 Z vw vw vw db þ $Vdb þ $Vdl þ ll dl dx  t$dy da; vb vVb Vl vs

V

where P ¼ vw/vF. Integrating the first, third, and fourth term by parts with the help of Gauss' theorem and taking the conditions (8) and (9) into account, we obtain

Z dI ¼



    dy$ðP$VÞ þ wb  V$wVb db  ðV$wVl  ll Þdl dx þ

Z



 ðP$n  tÞ$dy þ wVb $n db da þ

vs

V

Z wVl $n dl da ¼ 0: (13) vV

Equation (13) implies that the minimizer must satisfy in V the equilibrium equations

P$V ¼ 0;

wb þ V$wVb ¼ 0;

V$wVl  ll ¼ 0;

(14)

subjected to the kinematic boundary conditions (8) and (9) at vk, and the following natural boundary conditions

P$n ¼ t;

1

wVb $n ¼ 0;

wVl $n ¼ 0 at vs :

Alternatively, the following constraint

R V

ðl2;1 þ l2;2 Þdx ¼ const can be imposed instead that might be advantageous in numerical simulations.

(15)

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219

We call P the first Piola-Kirchhoff stress tensor, tr ¼ wb the resolved shear stress (or Schmid stress), and 2 ¼ V$wVb the back stress. The first equation of (14) is nothing else but the equilibrium of macro-forces acting on the crystal, the second equation represents the equilibrium of micro-forces acting on dislocations, while the last one expresses the equilibrium condition for the curved dislocation lines. The constitutive equations for P ¼ wF, wb, wVb, and wVl can easily be obtained from the free energy density (12). First, we express Fe in terms of F and b with the use of (1) and (2)

Fe ¼ F$Fp1 ¼ F$ðI  bs5mÞ: Now, the standard differentiation using the chain rule and the above relation yields the first Piola-Kirchhoff stress tensor

P ¼ wF ¼ 2Fe $jCe $FpT :

(16)

For the resolved shear stress (Schmid stress) we get

tr ¼ wb ¼ 2s$FpT $Ce $jCe $m:

(17)

Likewise, from (5) follows

wVb ¼

i 1h jr⊥ signðsn vn bÞsn þ jrk signðst vn bÞst n: b

(18)

Thus, the vector wVb is two-dimensional. Finally, we compute wVl directly in components using formulas (5). Since r⊥ and rk do not depend on l;3 , so wl;3 ¼ 0, and the vector wVl is also two-dimensional. For its first two components we have

0 1 13     2 2l l þb l b 2b l þb l 2l l l þb l b l b 16 ;2 ;2 B B ;1 ;1 ;1 ;1 ;1 ;2 ;2 C ;1 ;2 ;1 ;1 ;2 ;2 ;1 ;2 C7 wl;1 ¼ 4jr⊥ signðsn vn bÞ@  þ þ 2 2 A5; A þjrk signðst vn bÞ@   2 2  2 2 b l þl l þl;2 2 2 2 2 ;1 ;2 ;1 l;1 þl;2 l;1 þl;2 0 0 2 1 13     2 b;2 l;1 C 16 B 2l;1 l;2 b;1 l;1 þb;2 l;2 B 2l;2 b;1 l;1 þb;2 l;2 b;1 l;1 þ2b;2 l;2 C7 wl;2 ¼ 4jr⊥ signðsn vn bÞ@  þ 2 2 A þjrk signðst vn bÞ@  þ A5:  2 2  b l;1 þl;2 l2;1 þl2;2 l2;1 þl2;2 l2;1 þl2;2 0

2

(19) Substituting the constitutive equations (16)e(19) into (14)e(15) we get the completely new system of equations and boundary conditions which, together with (8)e(10), enable one to determine yðxÞ, bðxÞ, and lðxÞ. Note that equation (14)1 and (14)2 are coupled via the first Piola-Kirchhoff stress tensor and the Schmid stress containing both F and b, while equation (14)2 and (14)3 are coupled because both contain the gradients of b and l. All equations are strongly nonlinear partial differential equations. The above theory has been developed for the case of negligibly small resistance to dislocation motion and plastic slip. In real crystals there is however always the resistance to the dislocation motion and plastic slip causing the energy dissipation that changes the above variational principle as well as the equilibrium conditions. We assume that the dissipation function _ Thus, depends on the plastic slip rate b_ and on the normal velocity of the dislocation loop vn given by (7) (or, equivalently, on l).

 _ l_ : D ¼ D b; When the dissipation is taken into account, the above formulated variational principle must be modified. Following (Sedov, 1965; Berdichevsky and Sedov, 1967) we require that the true placement yðx; tÞ, the true plastic slips bðx; tÞ, and the true loop function lðx; tÞ obey the variational equation

Z  dI þ V

 vD vD db þ dl dx ¼ 0 vb_ vl_

(20)

for all variations of admissible fields y(x,t), b(x,t), and l(x,t) satisfying the constraints (8)e(10). Together with the above formula for dI and the arbitrariness of dy, db, and dl in V as well as at vs, equation (20) yields

P$V ¼ 0;

wb þ V$wVb ¼

vD ; vb_

V$wVl  ll ¼

vD ; vl_

(21)

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K.C. Le / International Journal of Plasticity 76 (2016) 213e230

which are subjected to the kinematic boundary conditions (8) and (9), the natural boundary conditions (15), and the constraint (10). The constitutive equations remain exactly the same as (16)e(19). For the rate-independent theory the dissipation function can be assumed in a simple form

    D ¼ K1 b_  þ K2 l_; with K1 and K2 being positive constants. We call K1 the critical resolved shear stress and K2 the Peierls threshold. In this case equations (21)2,3 become

V$wVl  ll ¼ K2 signl_

_ wb þ V$wVb ¼ K1 signb;

for non-vanishing b_ and l._ These are the yield conditions for b and l: b_ and l_ are non-zero if and only if

    wb þ V$wVb  ¼ K1 ;

jV$wVl  ll j ¼ K2 :

On the contrary, if the expressions on the left-hand sides are less than K1 and K2, the plastic slip cannot evolve and the dislocation lines cannot move: b_ ¼ 0 and l_ ¼ 0. Thus, they are frozen in the crystal. 4. Extension to multiple slip The extension to the case of single crystals having n active slip systems can be done straightforwardly under the assumption2

Fp ¼ I þ

n X

ba ðxÞsa 5ma ;

(22)

a¼1

with ba being the plastic slip, where the pair of constant and mutually orthogonal unit vectors sa and ma is used to denote the slip direction and the normal to the slip planes of the corresponding a-th slip system, respectively. Here and later, the Gothic upper index a running from 1 to n numerates the slip systems, so one could clearly distinguish ba from the power function. We denote by pa the unit vector lying in the slip plane such that sa , pa , and ma form a right-handed basis vectors. For each slip system we can introduce the coordinates associated with these basis vectors

xa1 ¼ sa $x;

xa2 ¼ pa $x;

xa3 ¼ ma $x:

(23)

Equations (23) can be regarded as the one-to-one linear transformation relating xa and x according to

xa ¼ Ma $x;

x ¼ Ma1 xa ;

where Ma is the 3  3 matrix whose rows are basis vectors sa , pa , and ma . Thus, any function of x can be expressed as function of xa and vice versa. For the plastic slip ba caused by dislocations of the slip system a we assume that their lines lie completely in the slip planes parallel to the ðxa1 ; xa2 Þ-plane. To describe the latter we introduce the loop function la ðxa1 ; xa2 ; xa3 Þ such that its level curves

  la xa1 ; xa2 ; c3 ¼ c;

(24)

where c3 and c are constants, coincide with the dislocation lines. We denote by na and ta the plane unit vectors normal and tangential to the dislocation line. From equation (24) follow

 1 a a a a na ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi l;1 s þ l;2 p ; la;1

2

þ la;2

2

 1 a a a a ta ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi  l;2 s þ l;1 p ; la;1

2

þ la;2

2

where the semicolon in indices denotes the partial derivatives of the loop function with respect to xa1 , xa2 , so these vectors lie in the slip planes parallel to the (x1,x2)-plane as expected. For the plastic deformation (22) the dislocation density tensor becomes

2 In conventional crystal plasticity the kinematic equation for Fp is usually formulated in rate form that does not always reduces to (22) (Ortiz and Repetto, 1999; Ortiz et al., 2000).

K.C. Le / International Journal of Plasticity 76 (2016) 213e230

T ¼ Fp  V ¼

n X

sa 5ðVba  ma Þ:

221

(25)

a¼1

To characterize the geometrically necessary dislocations belonging to one slip system we consider one term Ta ¼ sa 5ðVba  ma Þ in the sum (25). Let us choose an infinitesimal area da with the unit normal vector ta and compute the resultant Burgers vector of all excess dislocations of the system a, whose dislocation lines cross this area at right angle

bar ¼ Ta $ta da ¼ sa ðVba $na Þda ¼ sa vna ba da: This resultant Burgers vector can be decomposed into the sum of two vectors

  bar ¼ bar⊥ þ bark ¼  na san þ ta sat vna ba da; where san ¼ sa $na and sat ¼ sa $ta are the projections of the slip vector onto the normal and tangential direction to the dislocation line, respectively. This allows us to define two scalar densities of edge and screw dislocations of the corresponding slip system

   a a a l;1 b;1 l;1 þ ba;2 la;2    1 1   ra⊥ ¼ r⊥ ¼ san vna ba  ¼   2  2 ;  b b b a a  l;1 þ l;2       a a a  a a a  brk  1  1l;2 b;1 l;1 þ b;2 l;2  a a a rk ¼ ¼ st vna b  ¼   2  2 :  b b b  la;1 þ la;2   a  b 

(26)

We see that the dislocation densities ra⊥ and rak depend only on the partial derivatives ba;a and la;a , a ¼ 1,2. For the moving dislocations we allow the loop functions to depend on time t such that the level curves

  la xa1 ; xa2 ; c3 ; t ¼ c; with c3 and c being constants, coincide with the dislocation lines during their motion. Similar to the single slip we introduce the normal velocities of dislocation lines as follows a l_ van ¼ va $na ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi;

la;1

2

þ la;2

2

with l_ ¼ l;t . These kinematic quantities will be used in the model with dissipation. From the above discussion of kinematics we see that a single crystal with n active slip systems is a generalized continuum with 3 þ 2n degrees of freedom at each point: y(x), ba ðxÞ, and la ðxa ðxÞÞ, a ¼ 1; …; n. We require that the free energy per unit volume of crystal (assumed as macroscopically homogeneous) must be a function of Ce ¼ FeT,Fe (where Fe ¼ F,Fp1), ra⊥ , and rak

 j ¼ j Ce ; ra⊥ ; rak :

Under the same loading condition as for the crystal with single slip we write down the energy functional

I½yðxÞ; ba ðxÞ; la ðxa ðxÞÞ ¼

Z

wðF; ba ; Vba ; Vla Þdx 

V

Z t$y da;

(27)

vs

where

 wðF; ba ; Vba ; Vla Þ ¼ j Ce ; ra⊥ ; rak : Provided the resistance to the dislocation motion is negligibly small, we formulate the following variational principle for a single crystals with n active slip systems: the true placement vector yðxÞ, the true plastic slips b ðxÞ, and the true loop a a  functions l ðx ðxÞÞ in the final equilibrium state of deformation minimize energy functional (27) among all continuously differentiable fields yðxÞ, ba ðxÞ, and la ðxa ðxÞÞ satisfying the constraints

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yðxÞ ¼ x þ u0 ðxÞ;

ba ðxÞ ¼ 0;

la ðxÞ ¼ 0

at vk ;

(28)

and

Z

  la ðxÞdx ¼ V :

(29)

V

Applying the same calculus of variation and taking into account the arbitrariness of the variations of y(x), b(x), and lðxa ðxÞÞ in V as well as at vs, one can show that the minimizer must satisfy in V the equilibrium equations

P$V ¼ 0;

wab þ V$wVba ¼ 0;

V$wVla  lal ¼ 0;

(30)

subjected to the kinematic boundary conditions (28) at vk and the following natural boundary conditions

P$n ¼ t;

wVba $n ¼ 0;

wVla $n ¼ 0

at vs :

(31)

The constitutive equations for P ¼ wF, wba , wVba , and wVla can easily be obtained from the above free energy density by standard differentiation. For the first Piola-Kirchhoff stress tensor and the Schmid stresses we have

P ¼ wF ¼ 2Fe $jCe $FpT ;

(32)

tar ¼ wba ¼ 2sa $FpT $Ce $jCe $ma :

(33)

Likewise, from (26) follows

wVba ¼

    i 1h jra⊥ sign san vna ba san þ jra sign sat vna ba sat na : k b

(34)

Thus, the vectors wVba are two-dimensional. Finally, for wVla we have

wVla ¼

jra⊥

! vrak vra⊥ a a þ j rk a s þ vla;1 vl;1

jra⊥

! vrak vra⊥ a a þ j rk a p : vla;2 vl;2

(35)

Differentiating formulas (26) for the dislocation densities ra⊥ and rak with respect to la;1 and la;2 , we get

1  2  a a a b;1 la;1 þ b;2 la;2 2ba;1 la;1 þ ba;2 la;2 C  B vra⊥ 1 C B 2 l;1 ¼ sign san vna ba B   þ  2  2 C;  a 2  A @ vl;1 b 2 2 a a l;1 þ l;2 la;1 þ la;2 0

1  a a a a C b;1 l;1 þ b;2 l;2 vrak 1 ba;1 la;2  B C B ¼ sign sat vna ba B   þ  2  2 C;  a 2  A @ vl;1 b 2 2 a a l;1 þ l;2 la;1 þ la;2 0

2la;1 la;2

0 2la;1 la;2



ba;1 la;1

þ ba;2 la;2



 B vra⊥ 1 B ¼ sign san vna ba B   a  2 2 þ  a @ vl;2 b 2 a l;1 l;1 þ la;2

1 ba;2 la;1 2 

þ la;2

C C ; 2 C A

1  2  a a a a a a a a a b;1 l;1 þ b;2 l;2 b;1 l;1 þ 2b;2 l;2 C  B 1 C B 2 l;2 : ¼ sign sat vna ba B   2 þ  2  2 C  A @ b 2 2 a a l;1 þ l;2 la;1 þ la;2 0

vrak vla;1

Substituting the constitutive equations 32e35 into (30)e(31) we get the completely new system of equations and a a boundary conditions which, together with (28)e(29), enables one to determine yðxÞ, b ðxÞ, and l ðxÞ. Note that the

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interaction between different slip systems can be taken into account by incorporating the cross-terms between different dislocation densities in the free energy density. For the case of non-zero resistance to dislocation motion leading to the energy dissipation we take the dissipation function in the form

 a a D ¼ D b_ ; l_ : a

a

We require that the true placement yðx; tÞ, the true plastic slips b ðx; tÞ, and the true loop functions l ðx; tÞ obey the variational equation

Z dI þ V

n  X vD a¼1

 vD a a dx ¼ 0 db þ dl a a vb_ vl_

(36)

for all variations of admissible fields y(x,t), ba ðx; tÞ, and la ðx; tÞ satisfying the constraints (28). It is then easy to show by exactly the same arguments like those used at the end of the previous Section that equation (36) yields

P$V ¼ 0;

wba þ V$wVba ¼

vD vb_

a;

V$wVla  lal ¼

vD a

vl_

;

which are subjected to the boundary conditions (28) and (31). The constitutive equations remain exactly the same as (32)e(35). Note that the latent hardening of one slip system due to another can be taken into account by incorporating the a a cross-terms between different b_ and l_ in the energy dissipation. 5. Small strain theory Let us simplify the above theory for small strains. In this case, instead of the placement y(x) we regards the displacement u(x) that is related to the former by

uðxÞ ¼ yðxÞ  x as the unknown function. Thus, the total compatible deformation is

F¼

vy ¼ I þ uV: vx

We assume that the displacement gradient uV (called distortion) is small compared with I. Concerning the plastic deformation given by (22) we also assume that the plastic slips ba are much smaller than 1. Using the multiplicative resolution (1) to express Fe through F and Fp1 and neglecting the small nonlinear terms in it, we get

Fe ¼ F$Fp1 ¼ I þ be ; where

be ¼ uV 

n X

ba sa 5ma :

a¼1

The last equation can be interpreted as the additive resolution of the total distortion into the plastic and elastic parts. The total compatible strain tensor field can be obtained from the displacement field according to

ε¼

1 ðuV þ VuÞ: 2

The incompatible plastic strain tensor field is the symmetric part of the plastic distortion field

εp ¼

n 1X 1 b þ bT ¼ ba ðsa 5ma þ ma 5sa Þ: 2 2 a¼1

Accordingly, the elastic strain tensor field is equal to

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εe ¼ ε  εp : The dislocation densities remain exactly the same as in the finite strain theory. They are given by the formulas (26). Concerning the free energy density we will assume that it depends on the elastic strain εe and on the dislocation densities ra⊥ and rak

 j ¼ j εe ; ra⊥ ; rak :

The energy of crystal containing dislocations reads

Z I½uðxÞ; bðxÞ; lðxÞ ¼

wðuV; ba ; Vba ; Vla Þdx 

Z t,u da;

(37)

vs

V

where wðuV; ba ; Vba ; Vla Þ ¼ jðεe ; r⊥ ; rk Þ. Provided the resistance to the dislocation motion can be neglected, then the a following variational principle is valid for single crystals: the true displacement field uðxÞ, the true plastic slips b ðxÞ, and the a true loop functions l ðxÞ in the final state of deformation in equilibrium minimize energy functional (37) among all admissible fields satisfying the constraints

uðxÞ ¼ u0 ðxÞ;

ba ðxÞ ¼ 0;

la ðxÞ ¼ 0 at vk ;

(38)

and

Z

  la ðxÞdx ¼ V :

(39)

V

The standard calculus of variation similar to the previous case leads to the equilibrium equations

s$V ¼ 0;

wab  V$wVba ¼ 0;

V$wVla  lal ¼ 0;

(40)

subjected to the kinematic boundary conditions (38) at vk and the following natural boundary conditions

s$n ¼ t;

wVba $n ¼ 0;

wVla $n ¼ 0

at vs :

(41)

The constitutive equation for the Cauchy stress tensor becomes

s¼

vj : vεe

(42)

For the Schmid stress we obtain

tar ¼ wba ¼ sa $s$ma :

(43)

The constitutive equations for wVba and wVla remain unchanged as compared with (34) and (35). The new set of governing equations and boundary conditions are obtained by substituting (42), (43), (34), and (35) into the equilibrium equation (40) and boundary conditions (41). Note that even for small strain the system of governing equations remain as a whole nonlinear. It is a simple matter to modify the theory for the case of non-zero resistance to dislocation motion and plastic slip leading to the energy dissipation. 6. Simple shear deformation of a single crystal beam Let us consider now the simple shear deformation of a single crystal beam having only one active slip system. The crystal occupies in its initial configuration a long cylinder of an arbitrary cross section such that ðx1 ; x2 Þ2A and 0x3L (see Fig. 3 for the beam of rectangular cross section). As before, the slip system is chosen such that the vectors s, p, and m coincide with e1, e2, and e3, respectively. We realize the simple shear deformation by placing this crystal beam in a “hard” device with the prescribed displacements at the boundary of the crystal such that

y1 ¼ x1 þ gx3 ;

y2 ¼ x2 ;

y3 ¼ x3 :

(44)

We assume that the length of the crystal L is large enough compared with the sizes of the cross section so that the uniform simple shear deformation state holds true in the first approximation. If the overall shear g is sufficiently small, then it is

K.C. Le / International Journal of Plasticity 76 (2016) 213e230

225

Fig. 3. Simple shear deformation of a beam of rectangular cross section.

natural to expect that the crystal deforms elastically and the plastic slip as well as the loop function vanish. If this parameter exceeds some critical threshold, then dislocation loops may appear (see one dislocation loop in Fig. 3). Due to the almost translational invariance in x3-direction we may assume that b(x) and l(x) depend only on x1 and x2 (except perhaps the neighborhoods of x3 ¼ 0 and x3 ¼ L). For simplicity let us consider the small strain theory. Then the only non-zero components of the total strain tensor, under the condition that (44) is valid everywhere, are

ε13 ¼ ε31 ¼

1 g: 2

(45)

Since the plastic distortion tensor b has only one non-zero component b13 ¼ b(x1,x2), the non-zero components of the elastic strain tensor read

εe13 ¼ εe31 ¼

1 ðg  bðx1 ; x2 ÞÞ: 2

(46)

Note that, strictly speaking, the stress obtained from this formula does not satisfy the macro-equilibrium equation exactly. However, it can be shown by the variational-asymptotic analysis of the exact variational problem that the correction terms to the displacement field u1 ¼ gx3, u2 ¼ u3 ¼ 0 enable one to satisfy the macro-equilibrium within the first approximation (see Appendix). With the loop function being l(x1,x2) the dislocation densities are given by

   1l;1 b;1 l;1 þ b;2 l;2  r⊥ ¼  ;  b l2;1 þ l2;2     1l;2 b;1 l;1 þ b;2 l;2  rk ¼  :  b l2 þ l2 ;1

(47)

;2

For the small strain theory we propose the free energy per unit volume of the undeformed crystal in the most simple form

1  r2k 1 r2 1 2 j εe ; r⊥ ; rk ¼ lðtrεe Þ þ mtrðεe $εe Þ þ mk1 ⊥2 þ mk2 2 : 2 2 rs 2 rs

(48)

 constants, k1 and k2 are material constants, while rs can be Here εe is the elastic strain tensor, l and m are the Lame interpreted as the saturated dislocation density. The first two terms in (48) represent the free energy of the crystal due to the macroscopic elastic strain, where we assume that the crystal is elastically isotropic. Elastic anisotropy can also be incorporated by modifying these two terms to 12 Cijkl εeij εekl , where Cijkl is the fourth-rank tensor of elastic moduli. The last two terms in (48) correspond to the energy of the dislocation network for moderate dislocation densities (Gurtin, 2002; Gurtin et al., 2007). Note that, for the small or extremely large dislocation densities close to the saturated value, the logarithmic energy proposed by Berdichevsky (2006a, b) turns out to be more appropriate. Substituting formulas (46) and (47) into (48) we obtain

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2 3  2  2 2 b l þb l 2 b l þb l l l 1 1 m ;1 ;1 ;2 ;2 ;1 ;1 ;2 ;2 6 7 ;1 ;2 wðb; Vb; VlÞ ¼ mðg  bÞ2 þ þ k2 4 k1 5:  2  2 2 2 b2 r2s l2;1 þ l2;2 l2;1 þ l2;2 Since in this case the side boundary does not allow dislocations to reach it, we pose the Dirichlet boundary conditions to both functions b(x1,x2) and l(x1,x2)

bðx1 ; x2 Þ ¼ 0;

lðx1 ; x2 Þ ¼ 0 for ðx1 ; x2 Þ2vA ;

(49)

The variational problem reduces to minimizing the two-dimensional functional

Z I½bðx1 ; x2 Þ; lðx1 ; x2 Þ ¼ L

wðb; Vb; VlÞdx1 dx2

(50)

A

among all admissible functions b(x1,x2) and l(x1,x2) satisfying the boundary conditions (49) and the constraint

Z

  lðx1 ; x2 Þdx1 dx2 ¼ A :

(51)

A

It is convenient to simplify the functional and minimize it in the dimensionless form. Introducing the dimensionless variables and quantity

x1 ¼ brs x1 ;

x2 ¼ brs x2 ;

ðx1 ; x2 Þ2A ;

I¼

Ib2 r2s ; mL

we write functional (50) in the form

I¼

1 2

Z A

2

3  2  2 2 b l þb l 2 b l þb l l l ;2 ;2 ;2 ;2 7 6 ;1 ;1 ;1 ;2 ;1 ;1 2 þ k2 4ðg  bÞ þ k1 5dx1 dx2 ;  2  2 2 2 2 2 l;1 þ l;2 l;1 þ l;2

(52)

where the bar over the quantities are dropped for short. The problem is to minimize functional (52) among all admissible functions b(x1,x2) and l(x1,x2) satisfying Dirichlet boundary conditions (49) and the constraint (51). Since the integrand of (52) is positive definite, the existence of the minimizer in this variational problem is guaranteed. In one special case the problem degenerates and admits an analytical solution. Indeed, if we choose k1 ¼ k2 ¼ k and take A to be a circular cross section whose boundary q isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given ffiin the polar coordinates by r ¼ R, then due to the symmetry we may assume that both b and l are functions of r ¼ x21 þ x22 only. It is now a simple matter to show that

 2  2 l2;1 b;1 l;1 þ b;2 l;2 l2;2 b;1 l;1 þ b;2 l;2 þ ¼ b2;r :  2  2 2 2 2 2 l;1 þ l;2 l;1 þ l;2 Thus, functional (52) (normalized by 2p) does not depend on l(r) and takes the form

ZR I¼

1 1 ðg  bÞ2 þ kb2;r rdr; 2 2

0

which leads to the Euler equation

ðg  bÞ þ

 k b r ¼ 0: r ;r ;r

This is nothing else but the inhomogeneous modified Bessel equation that yields the following solution (regular at r ¼ 0)

 .pffiffiffi k ; bðrÞ ¼ g þ CI0 r with I0(x) being the modified Bessel function of the first kind. The coefficient C must be found from the boundary condition b(R) ¼ 0 giving

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227

 .pffiffiffi k : C ¼ g=I0 R Thus,

 .pffiffiffi .  .pffiffiffi i h bðrÞ ¼ g 1  I0 r k I0 R k :

(53)

The plot of solution (53) for different values of g is shown in Fig. 4, where we choose the radius such that R ¼ brs R ¼ 1 and the parameter k ¼ 104. As g increases, the plastic slip increases too. It is seen from this Figure that the plastic slip is nearly constant in the middle of the cross section and changes strongly only in the thin layer in form of ring near the boundary. Since l is a function of r, dislocations in form of circles pile up against the boundary of the cross section, leaving the middle of the cross section almost dislocation-free. Since the dislocation loops are circles, they have the purely edge character at 4 ¼ 0 and 4 ¼ p and the purely screw character at 4 ¼ ±p/2. For all other angles the dislocation loops have the mixed character. Note that if k1 s k2, the strictly axi-symmetric solution is no longer valid because the contributions of the edge and screw components to the energy of dislocation network are not equal. Note also that, if the logarithmic energy is used instead of the quadratic energy, b becomes non-zero only if g > gc, so there is a threshold stress for the dislocation nucleation (see Berdichevsky and Le, 2007). Besides, the existence of the dislocation-free zone in the middle of the cross-section can be established. For an arbitrary cross section and for k1 s k2 the problem does not admit exact analytical solution. However, based on the character of solution that changes strongly only in the normal direction to the boundary observed in the previous case, we may use the asymptotic method to find the solution in the thin boundary layer. Take for example the rectangular cross-section (x1,x2)2(0,W)  (0,H), with W and H being its width and height. In this case let us assume that there are two boundary layers near the left and right boundaries x1 ¼ 0 and x1 ¼ W, and two other boundary layers near the bottom and top boundaries x2 ¼ 0 and x2 ¼ H. Near the boundaries parallel to the x2 axis the derivative with respect to x2 can be neglect as compared to the derivative with respect to x1, while the opposite is true near the boundaries parallel to x1-axis. In the middle of the cross section the plastic slip remains constant. Functional (52) reduces to the sum of four integrals, and all of them do not depend on l. Consider for instance the integral along the left boundary layer

Zl I1 ½bðx1 Þ ¼

1 1 ðg  bÞ2 þ k1 b2;1 dx1 ; 2 2

0

with l being still an unknown length. We omit here the integration over x2 (because in this boundary layer it plays just the role of a parameter) and try to find the minimum among b. The standard variational calculus leads to the Euler equation

g  b þ k1 b;11 ¼ 0

(54)

which must be subjected to the boundary conditions

bð0Þ ¼ 0:

(55)

The solution of (54) and (55) that does not grow exponentially as x1 / ∞ reads

Fig. 4. The plastic slip b(r): i) g ¼ 0.001, ii) g ¼ 0.005, iii) g ¼ 0.01.

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K.C. Le / International Journal of Plasticity 76 (2016) 213e230

pffiffiffiffi  bðx1 Þ ¼ g 1  ex1 k1 :

Due to the mirror symmetry of the problem, the plastic slip in the boundary layer near x1 ¼ W must be

pffiffiffiffi  bðxÞ ¼ g 1  eðWx1 Þ k1 :

Similarly, the solution in the boundary layers parallel to the x1 axis equals

pffiffiffiffi  g 1  ex2 k2 pffiffiffiffi  bðx2 Þ ¼ : g 1  eðHx2 Þ k2 8 <

near x2 ¼ 0; near x2 ¼ H:

(56)

pffiffiffiffiffi Thus, the width of the boundary layers parallelp toffiffiffiffiffi the x1 axis must be of the order k2 =ðbrs Þ while that of the boundary layers parallel to the x2-axis must be of the order k1. Since l depends only on the normal coordinate to the boundary, the dislocation lines must be parallel to the boundary of the cross section except at four corners where the asymptotic solution becomes no longer valid (the numerical solution should lead to a smoothing of the corners of dislocation loops). We see that near the vertical boundaries the dislocation loops have the edge character, while near the horizontal boundaries they have the screw character. This agrees well with the results obtained by Zhu et al. (2014) for the pile-up of dislocation loops against the boundary of a rectangle due to the Frank-Read sources. As the consequence, the widths of the boundary layers are determined by the corresponding contributions of the edge and screw components to the energy of the dislocation network. For very thin rectangular cross section with H ≪ W we may neglect the influence of the edges near x1 ¼ 0 and x1 ¼ W by considering the dislocation network in the central part of the beam. In this case we will have only screw dislocations which pile up against two obstacle at x2 ¼ 0 and x2 ¼ H. The solution (56) reduces to that found in Berdichevsky and Le (2007). 7. Conclusions and discussions In this paper we have developed the nonlinear CDT for crystals containing curved dislocations based on the LEDShypothesis. The completely new set of equilibrium equations, boundary conditions and constitutive equations have been derived from the principle of minimum free energy. As the outcome, we have obtained the system of strongly nonlinear partial differential equations for the placement, the plastic slip, and the loop function. In the case of non-vanishing resistance to dislocation motion we have derived the governing equations from the variational equation that takes the dissipation into account. We have extended the theory to the case of multiple slip and simplified it for small strains. The application of the theory has been illustrated on the problem of single crystal beam having one slip system and deforming in simple shear. Under the simplified assumption k1 ¼ k2 the analytical solution of this problem has been found for the circular cross section. For arbitrary cross sections the problem has been solved by the asymptotic method. We have shown that the asymptotic solution found for the rectangular cross section reduces to the well-known solution in Berdichevsky and Le (2007) if it is thin and long. Let us point out also some limitations of the proposed 3-D CDT. First, the regularity assumption of the dislocation network exclude from the consideration the so-called statistically stored dislocations, so the full Taylor hardening cannot be captured. Thus, the typical length scale of the problems that can be solved by this theory must be much larger that the mean distance between two small dislocation loops of opposite sign in the dipoles of dislocations. Second, the dynamic version of the present CDT neglects the inertial effect that could be essential for the case of dislocations moving with the velocity comparable with the sound speed. Third, the present theory neglects also the thermal fluctuation and its influence on the nucleation of statistically stored dislocations. Fourth, the problem of experimental determination of the material parameters in the above models is also not addressed. Finally, we mention a large number of open problems which can be solved by this 3-D CDT like 3-D dislocation pile-ups, size-effects in bending, torsion, indentation, formation of dislocation patterns in crystals during severe plastic deformations et cetera. Some of the above mentioned issues will be addressed in forthcoming papers. Acknowledgments The financial support by the German Science Foundation (DFG) through the research projects LE 1216/4-2 and GP01-G within the Collaborative Research Center 692 (SFB692) is gratefully acknowledged. The author would like to thank all reviewers for their constructive criticism and suggestions.

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Appendix. Variational-asymptotic analysis This Appendix shows the detailed asymptotic analysis of the exact variational problem of the single crystal beam deforming in single slip under a simple shear posed in Section 6. In this variational problem we need to minimize the threedimensional functional

Z I½uðxÞ; bðxÞ; lðxÞ ¼

  w ui;j ; b; b;a ; l;a dx

(57)

V

under the constraints

Z u1 ¼ gx3 ;

u2 ¼ u3 ¼ 0;

for x2vV ;

lðxÞdx ¼ jV j;

(58)

V

where V ¼ A  ð0; LÞ is the three-dimensional region occupied by the undeformed beam, and where, according to the energy density (48),

w¼

2 1  2 2 1  2 1  1 l u1;1 þ u2;2 þ u3;3 þ mu21;1 þ mu22;2 þ u1;2 þ u2;1 þ m u1;3 þ u3;1  b þ m u2;3 þ u3;2 þ mu23;3 2 2 2 2 2 3  2  2 2 b l þb l 2 b l þb l l l 1 m 6 ;1 ;1 ;1 ;2 ;2 ;2 ;2 7 ;2 ;1 ;1 þ þ k2 4k1 5:  2  2 2 b2 r2s 2 2 2 2 l;1 þ l;2 l;1 þ l;2

(59)

This variational problem possesses a small parameter H/L, with H being the characteristic length of the cross-section (say, its height). To recognize this and to make the small parameter entering the energy functional (57) explicitly we rescale the coordinates as follow:

z1 ¼ x1 =H;

z2 ¼ x2 =H;

z3 ¼ x3 =L:

Then

ui;1 ¼

1 u ; H ij1

ui;2 ¼

1 u ; H ij2

ui;3 ¼

1 u ; L ij3

where the vertical bar before an index denotes the derivatives with respect to the corresponding z. Similar formulas hold for the derivatives of b and l. Thus, we see that ui,3 are small compared with ui,1 and ui,2 if uijj are of the same order of smallness. Because of the presence of this small parameter in the energy functional the variational-asymptotic analysis can be applied. At the first step of the variational-asymptotic procedure (see, e.g., Le, 1999) we maintain only the formally principal asymptotic terms in the functional (57). This leads to uncoupled problems for ui and b, l. At this step the displacements ui are found from minimizing the two-dimensional functional

I0 ¼ L

Z  2 2 1 1 1 1 l u1j1 þ u2j2 þ mu21j1 þ mu22j2 þ u1j2 þ u2j1 þ mu23j1 þ mu23j2 dz1 dz2 2 2 2 2

A~

under the constraints (58)1 at each point z3, where A~ is the rescaled region of z1 and z2. It is easy to see that the minimizer is

u1 ¼ gLz3 ;

u2 ¼ u3 ¼ 0:

Fixing now this displacement field, we seek at the second step the plastic slip and the loop function. This leads to the variational problem of minimizing functional (50) under the constraints (49) and (51) that has been solved in Section 6. Then at the next step we fix these b and l and seek for the correction to displacement field in the form

u1 ¼ gLz3 þ v1 ;

u2 ¼ v2 ;

u 3 ¼ v3 :

(60)

Substituting formulas (60) into the functional (57) and keeping there only the asymptotically principal terms containing vi, we obtain the functional

I1 ¼ L

Z  2 2 1  2 1 1 1 l v1j1 þ v2j2 þ mv21j1 þ mv22j2 þ v1j2 þ v2j1 þ m Hg  Hb þ v3j1 þ mv23j2 dz1 dz2 2 2 2 2

A~

that should be minimized under the constraints

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v1 ¼ v2 ¼ v3 ¼ 0 at vA~ : Thus, it is easy to see that v1 ¼ v2 ¼ 0, while v3 is the solution of the Poisson's equation

v3j11 þ v3j22 ¼ Hbj1 ;

v3 ¼ 0

at vA~ ;

yielding the correction of the order H/L of smallness as compared with the asymptotically principal term u1 ¼ gLz3. This correction term v3 enables one to satisfy the macro-equilibrium in the asymptotic sense far away from edges the of the beam. References Acharya, A., 2001. A model of crystal plasticity based on the theory of continuously distributed dislocations. J. Mech. Phys. Solids 49, 761e784. Acharya, A., Bassani, J.L., 2000. Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys. Solids 48, 1565e1595. Baitsch, M., Le, K.C., Tran, T.M., 2015. Dislocation structure during indentation. Int. J. Eng. Sci. 94, 195e211. Berdichevsky, V.L., 2006a. Continuum theory of dislocations revisited. Contin. Mech. Therm. 18, 195e222. Berdichevsky, V.L., 2006b. On thermodynamics of crystal plasticity. Scr. Mater 54, 711e716. Berdichevsky, V.L., 2009. Variational Principles of Continuum Mechanics: I. Fundamentals. Springer, Berlin. Berdichevsky, V.L., Le, K.C., 2007. 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