Thermodynamic dislocation theory: Finite deformations
TThermodynamic dislocation theory: Finite deformations
K.C. Le a,b a Materials Mechanics Research Group, Ton Duc Thang University, Ho Chi Minh City,Vietnam b Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Abstract
The present paper extends the thermodynamic dislocation theory initiatedby Langer, Bouchbinder and Lookman [2010] to non-uniform finite plastic de-formations. The equations of motion are derived from the variational equa-tion involving the free energy density and the positive definite dissipationfunction. We also consider the simplified theory by neglecting the excessdislocations. For illustration, the problem of finite strain constrained shearof single crystals with one active slip system is solved within the proposedtheory.
Keywords: dislocations, thermodynamics, configurational temperature,plastic yielding, strain rate.
1. Introduction
The novel approach initiated by Langer et al. [27], called LBL-theoryfor short (see also [28, 31]), has opened new perspectives in the dislocationmediated plasticity. Its main idea is to decouple the system of crystals con-taining dislocations into configurational and kinetic-vibrational subsystems.The configurational degrees of freedom describe the relatively slow, i.e. in-frequent, atomic rearrangements associated with the irreversible motion ofdislocations; the kinetic-vibrational degrees of freedom correspond to the fastoscillations of atoms about their equilibrium positions in the lattice. Due tothe two different time scales characterizing these subsystems, two well-definedentropies (or temperatures) can be introduced. The governing equations of E-mail: [email protected]
Preprint submitted to International Journal of Plasticity February 20, 2019 a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b BL-theory are based on the kinetics of thermally activated dislocation de-pinning and the irreversible thermodynamics of driven systems involving theconfigurational temperature. This LBL-theory has been successfully usedto simulate the stress-strain curve for copper over fifteen decades in strainrates and for temperatures between room temperature and about one thirdof the melting temperature. Only a single fitting parameter is required toachieve full agreement with the experiments conducted by Follansbee andKocks [11] over a wide range of temperatures and strain rates. The theorywas extended to include the interaction between two subsystems by Langer[29] and used to simulate the stress-strain curves for aluminum [43, 45] andsteel [45], which capture the thermal softening in full agreement with theexperiments performed in [57, 1]. The extension of the LBL-theory to non-uniform plastic deformations without including the excess dislocations wasproposed by Langer [30], Le et al. [39], Le and Piao [38] and was appliedto the adiabatic shear band for steel and torsion of aluminum bars. Thetheory has again shown excellent agreement with the experiments conductedby Marchand and Duffy [51], Horstemeyer [16], and Zhou and Clode [63].The LBL-theory and the above extensions can be used to describe the de-formations of crystals whose dislocations are neutral in that their resultantBurgers vector vanishes. Ashby [2] called this kind of dislocations statisti-cally stored, but we prefer the shorter and more precise name of redundantdislocations given earlier by Cottrell [9]. With non-uniform plastic deforma-tions, like e.g. the torsion of a bar, the bending of a beam, or the deformationof polycrystals containing obstacles such as grain boundaries, solute atoms,or precipitates, redundant dislocations are accompanied by another type ofdislocation to accommodate the plastic deformation gradient and ensure com-patibility of the total deformation [53, 8, 24, 2]. Most of contemporary ex-perts in dislocation theory accept the proposal of Ashby [2] to call thesedislocations “geometrically necessary”. From the point of view of statisticalmechanics of dislocations though (see e.g. [6, 49, 55, 61]) the name excessdislocations seems more pregnant. Note that in recent years the density of ex-cess dislocations can be measured indirectly by the high-resolution electronbackscattering technique (EBSD) [26]. Although the percentage of excessdislocations in severly and plastically deformed crystals is low, they play aprominent role in the formation of the microstructure [54, 17, 19, 34, 22, 23]and the Bauschinger and size effects [10, 52, 14, 7, 20, 18, 35, 3]. Thecontinuum dislocation theory (CDT), which includes the density of excessdislocations as developed by Berdichevsky and Sedov [4], Le and Stumpf241, 42], Gurtin [12, 13], Berdichevsky [5], Le and G¨unther [33], can actuallycapture microstructural formation and Bauschinger and size effects. Let usmention also the alternative approaches based on the discrete dislocation dy-namics [58, 62], the continuum dislocation dynamics [15, 47], and the phase-field modeling [59, 48]. However, the main drawback of these approaches tothe dislocation mediated plasticity is the absence of redundant dislocationsand configurational entropy (or its dual, configurational temperature). Sincethese two quantities are decisive for isotropic strain hardening [27], these ap-proaches need a substantial revision. The thermodynamic dislocation theory,which accounts for redundant dislocations and configurational entropy, wasfirst developed by Le [32] for small plastic deformations (see also [37, 44]).It captures all experimentally observed features of the plastic flow: both theisotropic and kinematic hardening, the sensitivity of the stress-strain curvesto temperature and strain rate, as well as the Bauschinger and size effects.Its application to the torsion of microwires [36] has shown an excellent agree-ment with the experiments of Liu et al. [50] in the wide range of twist rates.The above TDT and its extensions have been proposed for small defor-mations. However, most of the experimental results were performed at finiteplastic deformations. For logical consistency and comparison with experi-ments, the finite deformation TDT has to be developed. The aim of thispaper is to provide the TDT for finite and non-uniform plastic deformations.By including its two missing quantities, configurational entropy (or its dual,configurational temperature) and density of redundant dislocations, as statevariables in the constitutive equations, we obtain the synthesis of CDT andLBL-theory, which is a truly dynamic theory that applies to any non-uniformfinite plastic deformations. We call it finite deformation thermodynamic dis-location theory. The latter is reduced to the previously proposed version ofTDT if the total and plastic deformations are small. It is also consistent withthe second law of thermodynamics. As an illustrative example, the problemof finite strain constrained shear of single crystal deforming in single slip isconsidered within the proposed theory. It will be shown that, for the speci-mens of macroscopic sizes, this problem can approximately be solved if theexcess dislocations are neglected. The simulated stress-strain curves showtemperature, velocity and orientation sensitivity.The paper is organized as follows. In Section 2 the finite deformationkinematics of TDT is laid down. Section 3 proposes the thermodynamicframework for this nonlinear TDT. In Section 4 the problem of finite strainconstrained shear is analyzed. Section 5 presents the numerical solution of3his problem and discusses the stress-strain curves and their sensitivity ontemperature, strain rate, and orientation of slip system. Finally, Section 6concludes the paper.
2. Finite deformation kinematics
For simplicity, we restrict ourselves to the plane strain deformations ofa single crystal slab of thickness L having only one active slip system. Wechoose a fixed rectangular cartesian coordinate system associated with thisslab such that the displacements u and u depend only on x and x , butnot on x , while the displacement u = 0. The position vector of the samematerial point ( x , x , x ) in the deformed state is y α ( x β , t ) = x α + u α ( x β , t ) , y = x , α, β = 1 , . Thus, the deformation gradient is given by F ( x α , t ) = I + u ∇ = u , u , u , u ,
00 0 1 , (1)where the comma in indices denotes partial differentiation. Kinematicalquantities characterizing the observable deformation of this single crystalare the displacement field u ( x α , t ) and the plastic deformation field F p ( x α , t )that is in general incompatible. For single crystals having one active slipsystem, the plastic deformation is given by F p ( x α , t ) = I + β ( x α , t ) s ⊗ m , (2)with β ( x α , t ) being the plastic slip, where the pair of constant and mutuallyorthogonal unit vectors s = ( s , s ,
0) and m = ( m , m ,
0) indicates the slipdirection and the normal to the slip planes. The edge dislocations causingthis plastic slip have dislocation lines parallel to the x -axis and the Burgersvector parallel to s . Thus, there are altogether three degrees of freedom ateach point of this generalized continuum. Later on, two additional internalvariables will be introduced. In this paper, Greek indices run from 1 to 2,while Latin indices from 1 to 3. Summation over repeated indices withintheir range is understood. We see immediately from (2) that det F p = 1, sothe plastic deformation is volume preserving.4e assume that the deformation gradient F is decomposed into elasticand plastic part according to F = F e · F p . Thus, the elastic deformation field equals F e = F · F p − = F · ( I − β ( x α , t ) s ⊗ m ) . (3)The relationship between the three deformation fields is illustrated schemat-ically in Fig. 1. Looking at this Figure we see that the non-uniform plasticdeformation F p is that creating dislocations (either inside or at the boundaryof the volume element) or changing their positions in the crystal withoutdeforming the crystal lattice. In contrary, the elastic deformation F e distortsthe crystal lattice having frozen dislocations [33]. F F p F e Figure 1: Multiplicative decomposition F = F e · F p With these deformation fields various measures of strain can be intro-duced. The relevant measures that will be used in TDT are the right Cauchy-Green deformation tensor C = F T · F and the similar elastic deformation tensor C e = F eT · F e . a perpendicular to the x -axis through which a total number N of dislocation lines cross. Assume that,among them, there are N + dislocations with the Burgers vector b s and N − dislocations with the Burgers vector − b s , with b being the magnitude ofthe Burgers vector. We define the number of excess dislocations as N g = | N + − N − | and the number of redundant dislocations as N r = N − N g .Correspondingly, the density of excess and redundant dislocations can bedefined as ρ g = N g d a , ρ r = N r d a . It turns out that, under the plane strain deformation, the density of excessdislocations can be related to the plastic slip β by the kinematic equation ρ g = | d B | b d a = 1 b | ∂ s β | . (4)This is based on the fact that the incompatibility tensor, introduced by Nye[53], Bilby [8], Kr¨oner [24], α = − F p × ∇ , has only two non-zero components α i = s i ( ∂ s β ), where ∂ s = s i ∂ i denotes the derivative in the direction s .Thus, the resultant Burgers vector of all dislocations, whose lines cross theinfinitesimal area d a perpendicular to the x -axis isd B i = α i d a = s i ( ∂ s β )d a. Since | d B | = b | N + − N − | , the scalar density of excess dislocations is given by(4). This density can be indirectly measured by the high resolution electronbackscatter diffraction technique (EBSD) [26].In contrast to the excess dislocations, the other family of dislocations,called by Cottrell [9] and Weertman [60] redundant, cannot be expressedthrough the plastic distortion but nevertheless may have significant influ-ences on the nucleation of excess dislocations and the work hardening ofcrystals. For any closed circuit surrounding an infinitesimal area (in thesense of continuum mechanics) the resultant Burgers vector of these dislo-cations always vanishes, so the closure failure caused by the incompatibleplastic deformation is not affected by them. As a rule, the redundant dislo-cations in unloaded crystals at low temperatures exist in form of dislocationdipoles. The simple reason for this is that the energy of a dislocation dipoleis much smaller than that of dislocations apart, so this bounded state of6islocations of opposite sign renders low energy to the whole crystal. Fromthe other side, due to their low energy, the dislocation dipoles can be createdby the mutual trapping of dislocations of different signs in a random wayor, eventually, by thermal fluctuations in the presence of a stress field. Letus denote the density of redundant dislocations by ρ r . The total dislocationdensity is thus ρ = ρ g + ρ r .
3. Thermodynamic dislocation theory
To set up phenomenological models of crystals with continuously dis-tributed dislocations using the methods of non-equilibrium thermodynamicsof driven system let us begin with the free energy density. As a function ofthe state, the free energy density may depend only on the state variables.Following Kr¨oner [25] and Langer [29] we will assume that the elastic de-formation tensor C e = F eT · F e , the dislocation densities ρ r and ρ g , thekinetic-vibrational temperature T , and the configurational temperature χ characterize the current state of the crystal, so these quantities are the statevariables of the thermodynamic dislocation theory. The reason why the plas-tic deformation F p cannot be qualified for the state variable is that it dependson the cut surfaces and consequently on the whole history of creating disloca-tions (for instance, climb or glide dislocations are created quite differently).Likewise, the gradient of plastic deformation tensor C p = F pT · F p cannot beused as the state variable by the same reason. In contrary, the dislocationdensities ρ r and ρ g depend only on the characteristics of dislocations in thecurrent state (Burgers vector and positions of dislocation lines) and not onthe history of their creation, so they are the proper state variable. We restrictourself to the isothermal processes, so the kinetic-vibrational temperature T is assumed to be constant and can be dropped in the list of arguments of thefree energy density. Two state variables, C e and ρ g are dependent variablesas they are expressible through the degrees of freedom u and β , while twoothers, ρ r and χ , can be regarded as independent internal variables. Ourmain assumption for the free energy density is ψ ( C e , ρ r , ρ g , χ ) = w ( C e ) + γ D ρ r + ψ m ( ρ g ) − χ ( − ρ ln (cid:0) a ρ (cid:1) + ρ ) /L. (5)The first term in (5), w ( C e ), describes the stored energy density of crystaldue to the elastic deformation tensor C e . The second term is the self-energy7ensity of redundant dislocations, with γ D being the energy of one dislocationper unit length. The third term is the energy density of excess dislocations.The last term has been introduced by Langer [29], with S C = A ( − ρ ln( a ρ ) + ρ ) having the meaning of the configurational entropy of dislocations, A beingthe area occupied by the crystal slab.We propose the following stored energy density of neo-Hookean compress-ible material w ( C e ) = 12 µ ( I −
3) + 12 λ (1 − J ) − µ ln J, (6)with I = tr C e , J = det F e = det F , and with λ and µ being the Lameconstants (cf. [21]). It is easy to check that this formula reduces to theclassical quadratic energy density of isotropic elastoplastic materials for smallstrain. Indeed, using Eqs. (1) and (3) we find that, up to the cubic terms in u ∇ and β , I = 3 + 2 tr ε e + ε e : ε e + ω e : ω e − β s · u ∇ · m , and J = 1 + tr ε e + 12 (tr ε e ) − ε e : ε e + 12 ω e : ω e − β s · u ∇ · m , where ε e = ( u ∇ + ∇ u − β s ⊗ m − β m ⊗ s ) is the small elastic strain tensorand ω e = ( u ∇ − ∇ u − β s ⊗ m + β m ⊗ s ) the small elastic rotation tensor.Expanding function f ( J ) = λ (1 − J ) − µ ln J into the Taylor series about J = 1 and keeping in Eq. (6) only the quadratic terms, we reduce it to theclassical formula w ( ε e ) = 12 λ (tr ε e ) + µ ε e : ε e . Furthermore, we employ the following formula for the energy of excess dis-locations ψ m ( ρ g ) ψ m ( ρ g ) = γ D a ln 11 − a ρ g . This logarithmic energy term stems from two facts: (i) energy of excess dis-locations for small dislocation densities must be γ D ρ g like that of redundantdislocations, and (ii) there exists a saturate state of maximum disorder andinfinite configurational temperature characterized by the admissibly closestdistance between excess dislocations, a . The logarithmic term [6] ensures a8inear increase of the energy for small dislocation density ρ g and tends to in-finity as ρ g approaches the saturated dislocation density 1 /a hence providingan energetic barrier against over-saturation.With this free energy density we can now write down the energy functionalof the dislocated crystal. Let the cross section of the undeformed singlecrystal occupy the region A of the ( x , x )-plane. The boundary of thisregion, ∂ A , is assumed to be the closure of union of two non-intersectingcurves, ∂ k and ∂ s . Let the displacement vector u ( x , t ) be a given smoothfunction of coordinates (clamped boundary), and, consequently, the plasticslip β ( x , t ) vanishes u ( x , t ) = ˜ u ( x , t ) , β ( x , t ) = 0 for x ∈ ∂ k . (7)The remaining part ∂ s of the boundary is assumed to be traction-free. If nobody force acts on this crystal, then its energy functional per unit depth isdefined as I [ u ( x , t ) , β ( x , t ) , ρ r ( x , t ) , χ ( x , t ))] = (cid:90) A ψ ( C e , ρ r , ρ g , χ ) d a , with d a = d x d x denoting the area element.Under the increasing load the resolved shear stress also increases, andwhen it reaches the Taylor stress, dislocations dipoles dissolve and begin tomove until they are trapped again by dislocations of opposite sign. Duringthis motion dislocations always experience the resistance causing the energydissipation. The increase of dislocation density as well as the increase ofconfigurational temperature also lead to the energy dissipation. Neglectingthe dissipation due to internal viscosity associated with the strain rate, wepropose the dissipation potential in the form D ( ˙ β, ˙ ρ, ˙ χ ) = τ Y ˙ β + 12 d ρ ˙ ρ + 12 d χ ˙ χ , (8)where τ Y is the flow stress during the plastic yielding, d ρ and d χ are stillunknown functions, to be determined later. The first term in (8) is theplastic power which is assumed to be homogeneous function of first orderwith respect to the plastic slip rate [56]. The other two terms describe thedissipation caused by the multiplication of dislocations and the increase ofconfigurational temperature [27].Since the dislocation mediated plastic flow is the irreversible process, wederive the governing equations from the following variational principle: the9rue displacement field ˇ u ( x , t ), the true plastic slips ˇ β ( x , t ), the true densityof redundant dislocations ˇ ρ r ( x , t ), and the true configurational temperatureˇ χ ( x , t ) obey the variational equation δI + (cid:90) A (cid:18) ∂D∂ ˙ β δβ + ∂D∂ ˙ ρ δρ + ∂D∂ ˙ χ δχ (cid:19) d a = 0 (9)for all variations of admissible fields u ( x , t ), β ( x , t ), ρ r ( x , t ), and χ ( x , t ) sat-isfying the constraints (7).Varying the energy functional with respect to u we obtain the quasi-staticequations of equilibrium of macro-forces T · ∇ = 0 , T = ∂ψ∂ F = µ F e · F p − T − [ λ (1 − J ) J + µ ] F − T , (10)which are subjected to the boundary conditions (7) and T · n = 0 on ∂ s . Taking the variation of I with respect to three other quantities β , ρ r , and χ and requiring that Eq. (9) is satisfied for their admissible variations, we getthree equations τ − τ B − τ Y = 0 , ( e D + χ ln (cid:0) a ρ (cid:1) ) /L + d ρ ˙ ρ = 0 , ( ρ ln (cid:0) a ρ (cid:1) − ρ ) /L + d χ ˙ χ = 0 , (11)with e D = γ D L . Here, τ = − ∂ψ∂β = µ s · ( C · F p − ) · m is the resolved shear stress (Schmid stress), while τ B = − ∂ ψ m ∂ ( ρ g ) β ,ss is the back stress. The first equation of (11) can be interpreted as the bal-ance of microforces acting on dislocations. This equation is subjected to theDirichlet boundary condition (7) on ∂ k and ∂ψ m ∂ρ g = γ D on ∂ s .
10e [32], Le and Piao [36] have shown that Eqs. (11) , reduces to thecorresponding equations for χ and ρ of LBL-theory describing the motion ofthe system driven by the constant shear rate ˙ γ = q /t [27]˙ χ = K χ τ q ( τ, ρ ) µt (cid:20) − χχ ss ( q ) (cid:21) , ˙ ρ = K ρ τa µν ( θ, ρ, q ) q ( τ, ρ ) t (cid:20) − ρρ ss ( χ ) (cid:21) , if we choose d χ = ρ − ρ ln( a ρ ) LK χ τ Y q ( τ Y ,ρ ) µt (cid:104) − χχ (cid:105) ,d ρ = − e D − χ ln( a ρ ) LK ρ τ Y a µν ( θ,ρ,q ) q ( τ Y ,ρ ) t (cid:104) − ρρ ss ( χ ) (cid:105) , (12)where t is the characteristic microscopic time scale. In these equations thesteady-state configurational temperature is denote by χ , while the steady-state dislocation density equals ρ ss ( χ ) = 1 a e − e D /χ . Finally, ν ( θ, ρ, q ) is defined as follows ν ( θ, ρ, q ) = ln ( θ ) − ln (cid:20)
12 ln (cid:18) b ρq (cid:19)(cid:21) . Note that, for ρ changing between 0 and ρ ss < /a , both numerators onthe right-hand sides of (12) are positive, and the dissipative potential (8) ispositive definite as required by the second law of thermodynamics.To close this system an evolution equation for τ Y is required. We considerfirst the uniform deformation and take the time derivative of the equationfor the resolved shear stress˙ τ = µ s · ( ˙ C · F p − + C · ˙ F p − ) · m = µ [ s · ( ˙ C · F p − ) · m − ˙ β s · C · s ] . (13)Based on the Orowan’s equation ˙ β = bρv and the assumption that disloca-tions spend most of time in the pinned state, the plastic slip rate must be11etermined by the kinetics of thermally activated dislocation depinning [27].This yields ˙ β = q ( τ, ρ ) t , q ( τ, ρ ) = b √ ρ [ f P ( τ, ρ ) − f P ( − τ, ρ )] , (14)where f P ( τ, ρ ) = exp (cid:104) − θ e − τ/τ T ( ρ ) (cid:105) . (15)In Eq. (15) the dimensionless temperature is introduced as θ = T /T P , with T P being the pinning energy barrier, while τ T = µ T b √ ρ is the Taylor stress.Substituting (14) into (13), we obtain the evolution of the resolved shearstress for the uniform plastic deformation. However, for non-uniform plasticdeformations producing excess dislocations q ( γ ) /t does not equal ˙ β , andwe associate q ( γ ) /t to the plastic shear rate caused by the depinning ofredundant dislocations only. In this case the equation for the flow stress inrate form is proposed as follows˙ τ Y = µ (cid:104) s · ( ˙ C · F p − ) · m − q ( τ Y , ρ ) t s · C · s (cid:105) . (16)Eqs. (10) and (11), combined with (14) and (16), yield the equations ofmotion of dislocated crystal. Note that, for the fast loading when the inertiaterm becomes essential, Eq. (10) should be modified to (cid:37) ¨ u = T · ∇ , with (cid:37) being the initial mass density.For uniform deformations the theory is considerably simplified. Indeed,since ρ g = 0, the back stress τ B vanishes, and Eq. (11) implies that τ = τ Y .Besides, as the equilibrium of macro-forces (10) is satisfied automatically, thewhole system reduces to ˙ β = q ( τ, ρ ) t , ˙ τ = µ (cid:104) s · ( ˙ C · F p − ) · m − q ( τ, ρ ) t s · C · s (cid:105) , ˙ χ = K χ τ q ( τ, ρ ) µt (cid:20) − χχ (cid:21) , ˙ ρ = K ρ τa µν ( θ, ρ, q ) q ( τ, ρ ) t (cid:20) − ρρ ss ( χ ) (cid:21) . (17)12 . Finite strain constrained shear h γ L x m s w ϕ x x Figure 2: Finite strain constrained shear
As an application of the proposed theory let us consider the single crystallayer having a rectangular cross-section of width w and height h , 0 ≤ x ≤ w ,0 ≤ y ≤ h and undergoing a finite plane strain constrained shear deformation(see Fig. 2). The single crystal is placed in a hard device with prescribeddisplacements at its upper and lower sides as u (0 , t ) = 0 , u (0 , t ) = 0 , u ( h, t ) = γ ( t ) h, u ( h, t ) = 0 , (18)where u ( y, t ) and u ( y, t ) are the longitudinal and transverse displacements,respectively, with γ ( t ) being the overall shear regarded as a control param-eter. We assume the shear rate ˙ γ = q /t to be constant. If γ is small,the layer deforms elastically. However, when γ becomes sufficiently large,dislocations will be depinned initiating the plastic flow. We admit only oneactive slip system, with the slip directions inclined at an angle ϕ to the x -axis and the edge dislocations whose lines are parallel to the x -axis. Underthese conditions the deformation can be assumed uniform for the specimensof macroscopic sizes. Strictly speaking, the conditions (18) do not allowdislocations to reach the upper and lower boundaries, so these boundariesact as obstacles. This leads to an accumulation of excess dislocations and13o nonuniformity of plastic deformation near these boundaries. For macro-scopic specimens, however, the percentage of excess dislocations is negligiblylow and the assumption of uniformity of plastic deformation is a good ap-proximation everywhere except for the very thin layers near the upper andlower boundaries. We aim at determining the displacements, the disloca-tion densities, the configuration temperature and the stress-strain curve as afunction of γ within the finite deformation thermodynamic dislocation theorysummarized in (17).Under these conditions the deformation gradient and the right Cauchy-Green deformation tensor are given by F = γ
00 1 00 0 1 , C = F T · F = γ γ γ
00 0 1 . (19)The active slip system inclined at the angle ϕ to the x -axis has the vec-tors s = (cos ϕ, sin ϕ,
0) and m = ( − sin ϕ, cos ϕ, F p = − β sin ϕ cos ϕ β cos ϕ − β sin ϕ β sin ϕ cos ϕ
00 0 1 , F p − = β sin ϕ cos ϕ − β cos ϕ β sin ϕ − β sin ϕ cos ϕ
00 0 1 . (20)Because the system is undergoing steady-state plane constrained shearwith the constant shear rate ˙ γ = q /t , we can replace the time t by the totalamount of shear γ so that t d/d t → q d/d γ . The equations of motion forthis system become d β d γ = q ( τ, ρ ) q , d τ d γ = µ (cid:104) s · ( d C d γ · F p − ) · m − q ( τ, ρ ) q s · C · s (cid:105) , d χ d γ = K χ τ q ( τ, ρ ) µq (cid:20) − χχ (cid:21) , d ρ d γ = K ρ τa µν ( θ, ρ, q ) q ( τ, ρ ) q (cid:20) − ρρ ss ( χ ) (cid:21) . (21)14ith C from (19) and F p − from (20) we find that f ( β, γ, ϕ ) ≡ s · ( d C d γ · F p − ) · m = − βγ + (1 + βγ ) cos 2 ϕ + ( γ − β ) sin 2 ϕ,f ( γ, ϕ ) ≡ s · C · s = 1 + γ sin ϕ + γ sin 2 ϕ. In terms of the introduced function Eq. (21) can be written as followsd τ d γ = µ (cid:104) f ( β, γ, ϕ ) − f ( γ, ϕ ) q ( τ, ρ ) q (cid:105) . It is convenient to rewrite Eqs. (21) in terms of the following dimensionlessvariables ˜ ρ = a ρ, ˜ χ = χe D . Then we present Eq. (14) in the form q ( τ, ρ ) = ba ˜ q ( τ, ˜ ρ ) , where ˜ q ( τ, ˜ ρ ) = (cid:112) ˜ ρ [ ˜ f P ( τ, ˜ ρ ) − ˜ f P ( − τ, ˜ ρ )] . We set ˜ µ T = ( b/a ) µ T = µr and assume that r is independent of temperatureand strain rate. Then ˜ f P ( τ, ˜ ρ ) = exp (cid:104) − θ e − τ/ ( µr √ ˜ ρ ) (cid:105) . We define ˜ q = ( a/b ) q so that q/ ( q ) = ˜ q/ ˜ q . Function ν becomes˜ ν ( θ, ˜ ρ, ˜ q ) ≡ ln (cid:16) θ (cid:17) − ln (cid:104) ln (cid:16) √ ˜ ρ ˜ q (cid:17)(cid:105) . The dimensionless steady-state quantities are˜ ρ ss ( ˜ χ ) = e − / ˜ χ , ˜ χ = χ /e D . Using ˜ q instead of q as the normalized plastic strain rate means that we areeffectively rescaling t by a factor b/a . Since t − is a microscopic attemptfrequency, of the order 10 s − , we take ( a/b ) t = 10 − s.15n terms of the introduced quantities the governing equations readd β d γ = q ( τ, ρ ) q , d τ d γ = µ (cid:104) f ( β, γ, ϕ ) − f ( γ, ϕ ) ˜ q ( τ, ˜ ρ )˜ q (cid:105) , d ˜ χ d γ = K χ τ ˜ q ( τ, ˜ ρ ) µ ˜ q (cid:20) − ˜ χ ˜ χ (cid:21) , d ˜ ρ d γ = K ρ τµ ˜ ν ( θ, ˜ ρ, ˜ q ) ˜ q ( τ, ˜ ρ )˜ q (cid:20) − ˜ ρ ˜ ρ ss ( χ ) (cid:21) . (22)The corresponding equations of the small strain theory ared β d γ = q ( τ, ρ ) q , d τ d γ = µ (cid:104) cos 2 ϕ − ˜ q ( τ, ˜ ρ )˜ q (cid:105) , d ˜ χ d γ = K χ τ ˜ q ( τ, ˜ ρ ) µ ˜ q (cid:20) − ˜ χ ˜ χ (cid:21) , d ˜ ρ d γ = K ρ τµ ˜ ν ( θ, ˜ ρ, ˜ q ) ˜ q ( τ, ˜ ρ )˜ q (cid:20) − ˜ ρ ˜ ρ ss ( χ ) (cid:21) . (23)
5. Numerical simulations
In order to simulate the theoretical stress-strain curves, we need values forfive system-specific parameters and two initial conditions from each sample.The five basic parameters are the following: the activation temperature T P ,the stress ratio r , the steady-state scaled effective temperature ˜ χ , and thetwo dimensionless conversion factors K χ and K ρ . We also need initial valuesof the scaled dislocation density ˜ ρ i and the effective disorder temperature ˜ χ i ;all of which are determined by the sample preparation. The basic parametersfor copper are chosen as follows [27] T P = 40800 K , r = 0 . , ˜ χ = 0 . , K χ = 350 , K ρ = 96 . . We choose also the initial conditions τ (0) = 0 , ˜ ρ (0) = 10 − , ˜ χ (0) = 0 . , β (0) = 0 . .2 0.4 0.6 0.8 1.0100200300400 τ (MPa) γ Figure 3: Resolved shear stress τ (in MPa) versus shear strain γ for copper at the roomtemperature 298 K, at ϕ = 30 ◦ , and at three shear rates 0 . / s (red), 10 / s (blue), and1000 / s (green): (i) finite strain (bold line), (ii) small strain theory (dashed line). We take the shear modulus for copper to be µ = 50000 MPa. The plotsof the resolved shear stress τ ( γ ) as function of γ for copper found by thenumerical integration of (22) (bold lines) and (23) (dashed lines) for threedifferent resolved shear rates 0 . / s (red), 10 / s (blue), and 1000 / s (green), atroom temperature 298 K, and at ϕ = 30 ◦ are shown in Fig. 3. It can be seenthat τ ( γ ) is rate-sensitive. Besides, the steady-state stresses τ ss calculatedby the finite deformation TDT are somewhat smaller than those computedby the small strain theory. τ (MPa) γ Figure 4: Resolved shear stress τ (in MPa) versus shear strain γ for copper at the shearrate 10 / s, at ϕ = 30 ◦ , and at three temperatures 273 K (red), 473 K (blue), and 673 K(green): (i) finite strain (bold line), (ii) small strain theory (dashed line). τ ( γ ) as function of γ for copper foundby the numerical integration of (22) (bold lines) and (23) (dashed lines) atthe shear rate 10 / s, at ϕ = 30 ◦ , and at three temperatures 273 K (red), 473 K(blue), and 673 K (green). We see that the higher the temperature, the loweris the hardening rate and the corresponding steady-state stresses. τ (MPa) γ Figure 5: Resolved shear stress τ (in MPa) versus shear strain γ for copper at roomtemperature 298 K, at the shear rate 10 / s, and at three angles of slip ϕ = 0 ◦ (red), ϕ = 15 ◦ (blue), and ϕ = 30 ◦ (green): (i) finite strain (bold line), (ii) small strain theory(dashed line). Fig. 5 plots the resolved shear stress τ ( γ ) as function of γ for copper foundby the numerical integration of (22) (bold lines) and (23) (dashed lines) atroom temperature 298 K, at the shear rate 10 / s, and at three angles of slip ϕ = 0 ◦ (red), ϕ = 15 ◦ (blue), and ϕ = 30 ◦ (green). Note that at ϕ = 0 ◦ the results of small strain and large strain theories coincide. As the angle ϕ increases and approaches 45 ◦ , the resolved shear stress diminishes to zero.The latter becomes negative if the angle ϕ is larger than 45 ◦ .Although the resolved shear stress decreases with the increasing slip angle(for ϕ < ◦ ), the true shear stress σ increases with the increasing slip angle.To see this, let us first compute the true Cauchy stress σ = J − F e · µ ( I − C e − ) · F eT = µ ( F e · F eT − I ) . With (20) we find that σ = 12 µ [(2 + β ) γ − β (2 + βγ ) cos 2 ϕ + β ( β − γ ) sin 2 ϕ ] . σ = µ ( γ − β cos 2 ϕ ) . Since σ = µβ sin 2 ϕ, σ = − µβ sin 2 ϕ, according to the small strain theory, it is easy to check that the resolvedshear stress τ = s i σ ij m j = µ ( γ cos 2 ϕ − β ) as expected. Fig. 6 show the plotsof the shear stress σ ( γ ) for copper as function of γ computed according to(22) (bold lines) and (23) (dashed lines) at room temperature 298 K, at theshear rate 10 / s, and at three angles of slip ϕ = 0 ◦ (red), ϕ = 15 ◦ (blue),and ϕ = 30 ◦ (green). We see clearly that, as ϕ increases the shear stress σ increases too. However, this is valid only for ϕ < ◦ . γσ Figure 6: Shear stress σ (in MPa) versus shear strain γ for copper at room temperature298 K, at the shear rate 10 / s, and at three angles of slip ϕ = 0 ◦ (red), ϕ = 15 ◦ (blue), and ϕ = 30 ◦ (green): (i) finite strain (bold line), (ii) small strain theory (dashed line).
6. Conclusions and discussions
In this paper, we have developed the extension of the LBL-theory to non-uniform finite plastic deformations for single crystals deforming in single slip.For single crystals deforming in multiple slips, the set of state variables mustbe enlarged to include multiple densities of dislocation populations, each de-scribing different types of dislocations with different orientations, and wewill need separate equations of motion for each of these populations. But19ven in this case, we can assume that the dislocation populations will re-tain their identity. Consequently, the generalization to a multi-slip versionof the present theory for single crystals should not pose fundamentally newproblems. The stress-strain curves of finite strain constrained shear defor-mation show sensitivity to temperature, strain rate and orientation of theslip system. However, this case study serves only as an illustration of thetheory. To compare with real experiments, tension/compression and torsiontests should be considered instead. This and another problem of finite de-formation with excess dislocations exhibiting kinematic work hardening andBauschinger and size effects will be discussed in our forthcoming papers.