*Corresponding author. Tel.: +49615176941, E-mail address: pouriayevali@mfm.tu-darmstadt.de

A Study of Hardening Behavior Based on a Finite-Deformation

Gradient Crystal-Plasticity Model

Habib Pouriayevali*, Bai-Xiang Xu

Mechanics of Functional Materials Division, Institute of Material Science, Technische

Universität Darmstadt, Jovanka-Bontschits-Strasse 2, D-64287 Germany

Keywords: Gradient Crystal Plasticity; Hardening; Finite Deformation; Gradient Strengthening.

Abstract

A systematic study on the different roles of the governing components of a well-defined finite-

deformation gradient crystal-plasticity model proposed by Gurtin (Int. J. Plasticity 24, pp. 702-

725, 2008) is carried out, in order to visualize the capability of the model in the prediction of a

wide range of hardening behaviors as well as rate-dependent, scale-variation and Bauschinger-

like responses in a single crystal. A function of accumulation rates of dislocations is employed

and viewed as a measure of formation of short-range interactions which impede dislocation

movements within a crystal. The model is first represented in the reference configuration for the

purpose of numerical implementation, and then implemented in the FEM software ABAQUS via

a user-defined subroutine (UEL). Moreover, plastic flows in predefined slip systems and

expansion of accumulation of GNDs are distinctly observed in varying scales and under different

loading conditions.

July 2016

2

1. Introduction

Crystal lattices are disrupted by point, line and surface defects such as dislocations,

disclinations, voids and interstitial atoms (Clayton, 2011; Hull and Bacon, 2011). Experimental

results have shown that accumulation of defects give rise to an intrinsic size-dependent response

of crystalline materials along with inhomogeneous plastic flows on the micro-scale level (Abu

Al-Rub and Voyiadjis, 2004; Fleck et al., 1994; Hall, 1951; Hutchinson, 2000; Maaß et al., 2009;

Petch, 1953; Raabe et al., 2001; Yefimov and van der Giessen, 2005). Prediction of such a size-

dependent response requires incorporation of atomistic slip systems, gradient description and

length-scale parameters into classical plasticity models.

Pertinent works on the development of classical crystal-plasticity theory have been

presented in e.g., Asaro and Rice (1977), Rice (1971) and Teodosiu and Sidoroff (1976). The

general approach is an incorporation of the Schmid law into inelastic constitute models.

Concerning the size-dependence modelling, it is well accepted that extension of conventional

plasticity theories to strain-gradient plasticity is a promising approach. Comprehensive reviews

of strain-gradient theories have been presented in Aifantis (2003), Fleck and Hutchinson (1997),

Gudmundson (2004) and Gurtin and Anand (2009). Studies on the non-uniform straining in

crystalline materials show that flow and accumulation of dislocations play an important role in

the accommodation of deformation in a crystal lattice (Ashby, 1970; Nye, 1953). Furthermore, it

is well accepted that statistically stored dislocation (SSD) and geometrically necessary

dislocation (GND) densities could be simply described by plastic flow and its gradient,

respectively (Gao et al., 1999; Han et al., 2005; Hutchinson, 2000; Nix and Gao, 1998).

Geometrical foundations of dislocation density tensors associated with the torsion of crystal

connection have been detailed in e.g., Clayton (2011) and Steinmann (2015). Studies on

relationship between dislocation densities and material shear-strength can be found in Gao et al.

(1999), Nix and Gao (1998) and Taylor (1934). A simplest form of small-deformation higher-

order strain-gradient plasticity theory was proposed by Aifantis (1984), and further developed in

Aifantis (1987) and Zbib and Aifantis (1991). Aifantis introduced the Laplacian of equivalent

plastic strain into a constitutive expression. Later, a class of higher-order strain-gradient theories

using a couple stress (Toupin, 1962) work-conjugate to strain gradient was proposed (Fleck and

Hutchinson, 1993; Fleck et al., 1994) and further generalized in Fleck and Hutchinson (1997,

2001) and Gao et al. (1999). In addition to the density of line defects, a measure of density of

3

point defects was also proposed by Naghdi and Srinivasa (1994) in a finite-deformation gradient

crystal plasticity framework. It is worth mentioning that Acharya and Bassani developed a

conventional plasticity theory with an incorporation of a strain-gradient term as an internal

variable in hardening modulus. This theory has standard boundary conditions and no higher-

order stresses (Acharya and Bassani, 1996; Bassani, 2001).

In addition, introducing a definition of GNDs tensor into strain-gradient continuum

mechanics is of great interest (Menzel and Steinmann, 2000; Svendsen, 2002), and it reveals the

physical interpretation behind the concept of strain-gradient crystal-plasticity theories. Cermelli

and Gurtin studied a variety of GND tensors (Cermelli and Gurtin, 2001, 2002). Moreover,

Gurtin (2000) introduced a framework in development of gradient crystal-plasticity model with

incorporation of a system of microscopic stresses, which is consistent with microforce balance

for each independent kinematic processes (Fleck and Hutchinson, 1997; Fried and Gurtin, 1993;

Fried and Gurtin, 1994; Gurtin et al., 2010). This framework was greatly followed later (Cermelli

and Gurtin, 2002; Gurtin, 2003, 2006, 2008, 2010; Gurtin and Anand, 2005a, 2005b; Gurtin and

Anand, 2007; Gurtin and Anand, 2009; Gurtin et al., 2007; Lele and Anand, 2008; Lele and

Anand, 2009). Cermelli and Gurtin (2002) proposed a small-deformation gradient crystal-

plasticity model which employs the microforce framework and a defect energy based on the

GND tensor. In this study, microscopic force is simply decomposed into distributed Peach–

Koehler forces for the edge and screw dislocations (Arsenlis and Parks, 1999; Hirth and Lothe,

1982; Teodosiu, 1982). Gurtin and Anand modified the strain-gradient theory detailed above for

an irrotational-plasticity isotropic material in the framework of small and finite deformations

(Gurtin and Anand, 2005a, 2005b). A detailed study on the comparison of stain-gradient theory

in small and finite deformations has been presented by Gurtin (2006). In the gradient plasticity

frameworks, length-scale parameters are employed to keep the dimensional consistency (Fleck

and Hutchinson, 1993, 1997). Voyiadjis and co-workers investigated the identification of

material length-scale parameters from micro- and nano-scale experiments (Abu Al-Rub and

Voyiadjis, 2004; Voyiadjis and Al-Rub, 2005; Voyiadjis and Faghihi, 2012), the resulting model

is a physically based model of length-scale parameters for gradient isotropic hardening plasticity.

As a review of recent studies on the large-deformation gradient crystal-plasticity, Clayton et

al. (2006) employed a three-term multiplicative decomposition of the deformation gradient and

studied the distortion of crystal lattice via a framework including the concept of dislocations and

4

disclinations. Levkovitch and Svendsen proposed a reformulation of a non-local single-crystal

model which is accompanied with an additional kinematic-like hardening due to GNDs

(Levkovitch and Svendsen, 2006). Furthermore, a crystal-plasticity model incorporating

additional differential equations for the evolution of GND densities was introduced in Kuroda

(2011) and Kuroda and Tvergaard (2008). A physically based strain-gradient crystal-plasticity

theory was proposed via concept of micro-stress framework (Ertürk et al., 2009). Svendsen and

Bargmann studied a possible extension of selected small-deformation extended crystal-plasticity

models to the large-deformation ones via variational approach (Svendsen and Bargmann, 2010).

Finally, Gurtin (2008) introduced a finite-deformation gradient crystal-plasticity model for a

single crystal. This thermodynamically consistent model is based on the system of microforce

balances and derived from the principle of virtual power. The free energy comprises two parts: a

hyperelastic description for large-deformation compressible material and a recoverable defect-

energy in a frame-invariance form based on a net dislocation density (Gurtin et al., 2007; Ohno

and Okumura, 2007). Moreover, a non-local rate-dependence flow rule is introduced, which

incorporates an energetic back-stress, a dissipative gradient-strengthening as well as dissipative

self- and latent-hardening. Energetic and dissipative length-scale parameters and a function of

accumulation rates of SSDs and GNDs are also taken into account.

As for numerical solution of large-deformation strain-gradient theories, a study on isotropic

material was presented by Lele and Anand (2009), and an implementation method which

introduces GND density as an additional nodal degree of freedom has been widely employed

(Bargmann et al., 2011; Ekh et al., 2007; El-Naaman et al., 2016; Evers et al., 2004; Klusemann

et al., 2013; Kuroda, 2011; Kuroda and Tvergaard, 2008; Wulfinghoff and Böhlke, 2015). Based

on this method, Bargmann et al. (Bargmann and Reddy, 2011; Bargmann et al., 2014) studied the

effect of dissipative gradient-hardening in the extension of two small-deformation yield-

functions proposed in Ekh et al. (2007) and Gurtin and Reddy (2014). Kuroda also employed

similar implementation method in a study of a large-strain gradient crystal-plasticity model

developed via an additional higher-order back-stress description (Kuroda, 2011; Kuroda and

Tvergaard, 2008).

It can be concluded from the literature review that an understanding of large-deformation

gradient crystal-plasticity is of great interest. To the best of authors’ knowledge, there is a lack of

systematic study on the governing components of a well-defined large-deformation gradient

5

crystal-plasticity model, even though there are great efforts on the development of theoretical

models. The model proposed by Gurtin (2008) was claimed to cover a number of hardening

behaviors. But there exists no numerical implementation of this model, due to its complexity. In

the current work, the authors represent this model in the reference configuration for the

convenience of numerical implementation. A function of accumulation rates of SSDs and GNDs

is employed and viewed as a measure of formation of short-range interactions which impede

dislocation movements within a crystal. The reference version of the model is then implemented

in the FEM software ABAQUS via a user-defined subroutine (UEL). The aim of the present

work is to reveal the capability of the model in the prediction of a wide range of hardening

behaviors as well as rate-dependent, scale-variation and Bauschinger-like responses in a single

crystal. Furthermore, plastic flows in predefined slip systems and accumulation of GNDs are

distinctly observed in the single crystal under different loading conditions. Effects of scale

variation in expansion of directional flows and accumulation of GNDs within the single crystal

are also investigated and simply linked to the phenomenon ‘smaller is stronger’.

The present work is organized as follows. In Section 2, the constitutive model with respect to

the reference configuration is detailed, Section 3 explains the implementation approach; a plane-

strain quadratic element in which displacement components and dislocation densities are treated

as nodal degrees of freedom, is defined. The flow rule is applied to the integration points and

solved to obtain the plastic flow in each slip system via the Newton-Raphson scheme. In Section

4, numerical results are presented, compared and discussed.

6

2. Formulation

In this section, a finite-deformation gradient crystal-plasticity model which has been

proposed by Gurtin (2008) in the deformed configuration is represented with respect to the

reference configuration, in order to facilitate the numerical implementation and development of a

user-defined subroutine (UEL) in the finite-element software ABAQUS.

2.1. Kinematics in a single crystal

As a starting point, the Kröner-Lee decomposition (Kroner, 1960; Lee, 1969) is employed

and yields

𝐅 = 𝐅e𝐅p, det 𝐅p = 1, 𝐂e = 𝐅eT𝐅e, 𝐄e =1

2(𝐂e − 𝐈) and 𝐋 = 𝐋e + 𝐅e𝐋p𝐅e−1, (1)

where 𝐅 represents the deformation gradient tensor, 𝐅e denotes the elastic distortion and

characterizes vector mapping from the intermediate space (Gurtin and Reddy, 2016) to the

deformed configuration. 𝐅p represents an inelastic deformation and characterizes vector mapping

from the reference configuration to the intermediate space. det stands for the determinant of a

tensor. 𝐂e is the right Cauchy-Green elastic tensor, 𝐄e is the Green-Lagrangian elastic-strain

tensor, and 𝐈 is the identity tensor. 𝐋e and 𝐋p represent the elastic and plastic distortion-rate

tensors, and 𝐋p is expressed as

where 𝛼 is a slip-system number, 𝜐𝛼 is a scalar defining the flow rate, and 𝕊𝛼 is the Schmid

tensor which is defined by

Here, 𝐬𝛼 and 𝐦𝛼 are the slip direction and the normal vector of slip plane which are constant

vectors in the intermediate space and vary in the reference and deformed configurations (Gurtin,

2006). 𝐬𝛼, 𝐦𝛼 and 𝐥𝛼 are given in the reference configuration by

1 Glide direction of screw dislocations (Section 2.4)

𝐋p = ∑ 𝜐𝛼 𝕊𝛼𝛼 , (2)

𝕊𝛼 = 𝐬𝛼⨂ 𝐦𝛼, |𝐬𝛼| = |𝐦𝛼| = 1, trace 𝕊𝛼 = 0 and 1𝐥𝛼 = 𝐦𝛼 × 𝐬𝛼. (3)

7

𝐬𝑟𝛼 = 𝐅p−1𝐬𝛼, 𝐦𝑟

𝛼 = 𝐅pT𝐦𝛼 and 𝐥𝑟𝛼 = 𝐅p−1 𝐥𝛼.

2 (4)

As an explanation of this vector mapping, Gurtin et al. stated that “Some readers may disagree

with this: In the literature one sometimes finds the assertion, either verbally or implicitly via a

figure, that the undistorted lattice resides in the reference space (and is hence material). We

believe this to be a misconception. Physically, because a flow of dislocations involves a flow of

material relative to the undistorted lattice, lattice vectors cannot be material” (Section 91.2 in

Gurtin et al. (2010)).

2.2. Internal and external power

Consider an arbitrary subregion 𝑃 of the reference body, internal and external power

expenditures 𝒲int and 𝒲ext are expressed in the following forms3

𝒲int(𝑃) = ∫ 𝐏e ∶ ��e 𝑑𝑉 + ∑ ∫ ( π𝛼𝜐𝛼 + 𝛏𝛼 ∙ ∇𝜐𝛼)𝑃

𝑑𝑉 𝛼𝑃, (5)

𝒲ext(𝑃) = ∫ 𝐛 ∙ 𝒗𝑃

𝑑𝑉 + ∫ 𝐩(𝐧)𝜕𝑃

∙ 𝒗 𝑑𝐴 + ∑ ∫ 𝛏 (𝐧)𝛼𝜐𝛼𝜕𝑃

𝑑𝐴 𝛼 , (6)

where 𝐏e represents the first Piola elastic-stress tensors, π𝛼 is a scalar microforce and 𝛏𝛼 is a

microstress vector. 𝐛 is a macroscopic body force per unit reference volume, 𝐩(𝐧) is a

macroscopic surface traction, 𝒗 is a velocity vector, 𝐧 is an outward unit vector on the boundary

and 𝛏 (𝐧)𝛼 is a scalar microtraction.

2.3. Macroscopic and microscopic force balances

Guided by the principle of virtual power (Lele and Anand, 2009), macroscopic and

microscopic force balances as well as macroscopic and microscopic traction conditions are

respectively defined in

Div 𝐏 + 𝐛 = 𝟎, 𝜏𝛼 − π𝛼 + Div 𝛏𝛼 = 0, (7)

2 𝑟 stands for the reference configuration in this study.

3 In consistence with Gurtin’s derivation, ∇, Div, and Curl denote the gradient, divergence, and curl operations in the

reference configuration, and grad, div and curl are employed in the deformed configuration. ∇⋕ denotes gradient

operation in the intermediate (lattice) space.

8

𝐩(𝐧) = 𝐏𝐧 and 𝛏 (𝐧)𝛼 = 𝛏𝛼 ∙ 𝐧. (8)

Here, 𝐏 = 𝐏e𝐅p−T represents the first Piola-Kirchhoff stress and 𝜏𝛼 = 𝐅eT𝐏e: 𝕊𝛼 = 𝐂e𝐒e: 𝕊𝛼 is

the resolved shear stress, where 𝐒e represents the second Piola elastic-stress.

2.4. Burgers tensor and scalar measures of dislocation densities

In the finite-deformation crystal plasticity description, the Burgers tensor 𝐆 is defined in the

intermediate space in the form (Cermelli and Gurtin, 2001; Gurtin, 2006, 2008; Gurtin and

Anand, 2005a)

Plastically convected rate of 𝐆 in a frame-invariance form4 is given by

By considering ∇⋕𝜐𝛼 = 𝐅p−T∇𝜐𝛼 and Eq. (4), plastically convected rate of 𝐆 is derived with

respect to the reference configuration in

where ��⊢𝛼 = −𝐬𝑟

𝛼 ∙ ∇𝜐𝛼 = −𝐬𝛼 ∙ ∇⋕𝜐𝛼 and ��⊙𝛼 = 𝐥𝑟

𝛼 ∙ ∇𝜐𝛼 = 𝐥𝛼 ∙ ∇⋕𝜐𝛼 are directional derivatives

of the slip rate in the glide directions −𝐬𝛼 and 𝐥𝛼, 𝜌⊢𝛼 and 𝜌⊙

𝛼 denote the edge and screw

dislocation densities5.

2.5. Free-energy imbalance

Guided by the second law of thermodynamics and the balanced power (Lele and Anand,

2009), the local dissipation inequality may be rewritten as

Ψ − 𝐒e ∶ ��e − ∑ (πα𝜐α + 𝛏α ∙ ∇𝜐α)𝛼 ≤ 0, (12)

4 Oldroyd rate-derivation, Section 5.3 in Holzapfel (2000), Eq. (107.23) in Gurtin et al. (2010) and Eq. (158) in

Svendsen (2002). 5 𝜌⊢

𝛼 and 𝜌⊙𝛼 are in units of length

−1, and are signed. To emphasize a difference in notation, each continuum-

mechanical density 𝜌𝛼 can be converted to a materials-science density 𝜚𝛼 via the transformation 𝜌𝛼 = b 𝜚𝛼 , b is

the magnitude of material Burgers vector (Gurtin, 2010).

𝐆 = 𝐅pCurl 𝐅p. (9)

𝐆 = 𝐅p (𝐅p−1𝐆𝐅p−T

) 𝐅pT = �� − 𝐋p𝐆 − 𝐆𝐋pT. (10)

𝐆𝑟 = 𝐅p−1𝐆𝐅p

−T= ∑ (��⊙

𝛼 𝐬𝑟𝛼⨂𝐬𝑟

𝛼 + ��⊢𝛼𝐥𝑟𝛼⨂𝐬𝑟

𝛼)𝛼 , (11)

9

where Ψ = Ψe +Ψ𝜌 denotes a stored free-energy per unit reference volume and comprises two

frame-invariance terms – an elastic strain-energy Ψe and a defect energy Ψ𝜌. Ψe is given in a

form of compressible Neo-Hookean energy density by

where I1, I3 are the first and third invariants of 𝐂e, μ the shear modulus and λ the Lame

parameter (Bonet and Wood, 2008). Motivated by an idea that a defect energy which is a

function of dislocation densities could simply contribute the gradient of flow rate to the free

energy, a recoverable uncoupled defect-energy6 Ψ𝜌 is introduced in a frame-invariance form

where 𝑆0 is a slip resistance and 𝐿1 is an energetic length-scale parameter. Defect-energy rate Ψ𝜌

takes the form

2.6. Constitutive model

By considering the free energy Ψ defined in the last section, the local dissipation inequality

Eq. (12) is rewritten in

(𝜕Ψe

𝜕𝐄e − 𝐒e) : ��e + 𝑆0𝐿1

2 ∑ (−𝜌⊢𝛼𝐬𝑟

𝛼 + 𝜌⊙𝛼 𝐥𝑟𝛼) ∙ ∇𝜐𝛼𝛼 − ∑ (πα𝜐α + 𝛏α ∙ ∇𝜐α)𝛼 ≤ 0. (16)

The energetic and dissipative microstresses are defined via 𝛏α = 𝛏engα + 𝛏dis

α , thus a reduced

inequality is derived in

Eq. (17) shows that the energetic microstress 𝛏engα is an in-slip-plane vector. Consequently, 𝛏dis

α

is assumed to be tangential to the slip plane and the reduced inequality in Eq. (17) is redefined in

6 A coupled defect energy associated with coupled Peach-Koehler forces is detailed in Section 105.5 in Gurtin et al.

(2010), an introduction of recoverable and irrecoverable defect energies has been presented in Gurtin and Ohno

(2011).

Ψe =μ

2(I1 − 3 − ln I3) +

λ

2(ln I3

1/2)2, (13)

Ψ𝜌 = Ψ𝜌(𝜌⊢𝛼 , 𝜌⊙

𝛼 ) =1

2𝑆0𝐿1

2 ∑ (|𝜌⊢𝛼|2 + |𝜌⊙

𝛼 |2)𝛼 , (14)

Ψ𝜌 =∑ (𝜕Ψ𝜌

𝜕𝜌⊢𝛼 ��⊢

𝛼 +𝜕Ψ𝜌

𝜕𝜌⊙𝛼 ��⊙

𝛼 )𝛼

= 𝑆0𝐿12∑ (−𝜌⊢

𝛼𝐬𝑟𝛼 + 𝜌⊙

𝛼 𝐥𝑟𝛼)

𝛼∙ ∇𝜐𝛼.

(15)

𝜕Ψe

𝜕𝐄e= 𝐒e , 𝛏eng

α = 𝑆0𝐿12(−𝜌⊢

𝛼𝐬𝑟𝛼 + 𝜌⊙

𝛼 𝐥𝑟𝛼) , ∑ (πα𝜐α + 𝛏dis

α ∙ ∇𝜐α)𝛼 ≥ 0. (17)

10

where ∇α𝜐α = ℙ ∇𝜐α = ∇𝜐α − (��𝑟𝛼 ∙ ∇𝜐α)��𝑟

𝛼, ℙ = 𝐈 − ��𝑟𝛼⨂��𝑟

𝛼 and ��𝑟𝛼 =

𝐦𝑟𝛼

|𝐦𝑟𝛼|

.

2.6.1. Constitutive relations for 𝛑𝛂 and 𝛏𝐝𝐢𝐬𝛂

A strong form of the reduced inequality in Eq. (18) is considered by

The conventional power-law model which is widely used in the viscoplasticity is employed to

define πα and 𝛏disα 7 in

where 𝑞 and 𝑚 are rate-sensitivity parameters, 𝜐0 and 𝑑0 are constant positive-valued reference

flow-rates, 𝐿2 is a dissipative length-scale parameter, 𝑆𝛼 is a positive-valued slip resistance, 𝜙𝛽

is a measure of accumulation rate of dislocations, 𝑆𝑦 is an initial coarse-grain shear resistance,

𝕂𝛼𝛽 is a stiffness matrix, ℎ(𝑆𝛽) is a self-hardening function and 𝜘𝛼𝛽 is an interaction constant.

𝜘𝛼𝛽 = 1 if 𝐦𝛼 = ±𝐦𝛽 and 𝜘𝛼𝛽 = 𝜅 otherwise, 𝜅 represents a ratio of latent-hardening rate to

self-hardening rate8. Self–hardening describes the hardening interaction between a slip system 𝛼

defined in the slip plane Π𝛼 and other slip systems sharing Π𝛼. Latent-hardening describes the

hardening on the slip system 𝛼 due to other slip systems having non-tangential slip plane to

Π𝛼. 𝜐𝑎𝑐𝑐𝛼 and ��𝑎𝑐𝑐

𝛼 represent the accumulation rate of SSDs and GNDs, respectively (Anand et al.,

2015; Gurtin, 2010; Gurtin and Reddy, 2014; Ohno et al., 2008),

7 Here, an alternative form of power law in πα and 𝛏dis

α is employed, it is similar to the formulation in Lele and

Anand (2009) and in the spirit of formulation in Gurtin (2008). 8 A discussion on the self-hardening function as well as alternative forms of 𝜅 = |𝒔𝛼 ∙ 𝒔𝛽||𝒎𝛼 ×𝒎𝛽| and 𝜅 =

0.5(|𝒎𝛼 ∙ 𝒔𝛽| + |𝒔𝛼 ∙𝒎𝛽|)|𝒎𝛼 ×𝒎𝛽| have been respectively discussed in Kalidindi et al. (1992), Gurtin and

Reddy (2014) and Gurtin and Ohno (2011).

∑ (πα𝜐α + 𝛏disα ∙ ∇α𝜐α)𝛼 ≥ 0, (18)

πα𝜐α ≥ 0 and 𝛏disα ∙ ∇α𝜐α ≥ 0. (19)

πα = 𝑆𝛼 (𝜐𝛼

𝜐0)𝑚 𝜐𝛼

|𝜐𝛼| , πα𝜐α = 𝑆𝛼 (

𝜐𝛼

𝜐0)𝑚

|𝜐𝛼| ≥ 0,

𝛏disα = 𝐿2

2𝑆𝛼 (𝑑𝛼

𝑑0)𝑞 ∇α𝜐𝛼

𝑑𝛼 , 𝛏dis

α . ∇α𝜐α = 𝐿22𝑆𝛼 (

𝑑𝛼

𝑑0)𝑞

|∇α𝜐𝛼|2

𝑑𝛼 ≥ 0,

(20)

𝑑𝛼 = 𝐿2|∇α𝜐𝛼|, ��𝛼 = ∑ 𝕂𝛼𝛽 𝜙𝛽𝛽 , 𝑆𝛼(𝑥, 0) = 𝑆𝑦,

𝕂𝛼𝛽 = [𝜘𝛼𝛽 ℎ(𝑆𝛽)], 𝜙𝛽 = √𝜐𝑎𝑐𝑐

𝛽 2+ 𝐿2

2 ��𝑎𝑐𝑐𝛽 2

,

11

where ℙ = 𝐈 −𝐦𝛼⨂𝐦𝛼.

2.6.2. Flow rule

The microscopic force balance Eq. (7)2 takes the following form via 𝛏α = 𝛏engα + 𝛏dis

α ,

𝜏𝛼 + Div𝛏engα = πα − Div𝛏dis

α . (22)

Substitution of Eqs. (17)2, (20) into Eq. (22) results in

𝜏𝛼 + 𝑆0𝐿12Div(−𝜌⊢

𝛼𝐬𝑟𝛼 + 𝜌⊙

𝛼 𝐥𝑟𝛼)⏟

(I)

= 𝑆𝛼 (𝜐𝛼

𝜐0)

𝑚 𝜐𝛼

|𝜐𝛼|⏟ (II)

− 𝐿22𝑆𝛼 (

𝑑𝛼

𝑑0)

𝑞 Div∇α𝜐𝛼

𝑑𝛼⏟ (III)

. (23)

The flow rule in Eq. (23) is governed by three different components. The term (I) represents

an energetic hardening defined via the recoverable defect-energy and is independent of 𝜐𝛼 sign.

Therefore, term (I) is expected to show back-stress effect and the Bauschinger-like

phenomenon9. The term (II) represents a rate-dependent dissipative hardening, and term (III)

defines a dissipative gradient-strengthening. 𝑆𝛼 incorporates the initial coarse-grain shear-

resistance 𝑆𝑦 as well as self- and latent-hardening which are evolved via ∑ 𝕂𝛼𝛽 𝜙𝛽𝛽 defined in

Eq. (20). In this study, 𝜙𝛽 is defined as an accumulation rate of dislocations and represents a

measure of formation of short-range interactions between all dislocations, i.e. both SSDs and

GNDs. These interactions impede the dislocation movements and render a dissipative hardening

(Evers et al., 2004). Similar to many other gradient theories, the length-scale parameters 𝐿1 and

𝐿2 are ad hoc parameters (Gurtin et al., 2010) and keep the dimensional consistency (Fleck and

Hutchinson, 1993, 1997). However, a built-in gradient length-scale μb

�� in a small-deformation

theory based on a GND self-energy near the grain boundaries was proposed by Ohno and

Okumura, (2007) and Ohno et al. (2008), where μ is the shear modulus, b is the magnitude of

material Burgers vector and 𝑆 is the initial slip resistance (Section 8 in Gurtin, (2010)). This

small-deformation theory showed good agreements with a series of experiments from submicron

to several microns range of grain sizes (Ohno and Okumura, 2007) and yielded an observation of

9 Div𝛏eng

α in Eq. (22) plays a role of energetic back-stress; Div𝛏engα = − 𝜎𝑏𝑎𝑐𝑘𝑠𝑡𝑟𝑒𝑠𝑠 .

𝜐𝑎𝑐𝑐𝛽= |𝜐𝛽| and ��𝑎𝑐𝑐

𝛽= √|��⊢

𝛽|2

+ |��⊙𝛽|2

= |ℙ𝐅p−T∇α𝜐α|, (21)

12

size dependency of the initial slip resistance which reveals the physical interpretation behind the

concept of length-scale parameters.

2.6.3. Microscopic boundary conditions

Considering a higher-order strain-gradient theory requires concomitantly a microscopic

boundary condition and results in an expenditure of microscopic power ∫ 𝛏 (𝐧)𝛼𝜐𝛼𝜕𝑃

𝑑𝐴 defined

in Eq. (6). In this study, a microscopically noninteractive boundary with a null power-

expenditure is defined by

𝛏 (𝐧)𝛼𝜐𝛼 = (𝛏𝛼 ∙ 𝐧) 𝜐𝛼 = 0, (24)

and it is simply satisfied via considering two idealized boundary conditions – microscopically

soft boundary through 𝛏𝛼 ∙ 𝐧 = 0, and microscopically hard boundary via 𝜐𝛼 = 0.

2.6.4. Two-dimensional version of the model

A two-dimensional version of the model is summarized in this section. Here, 𝐦𝛼 and 𝐬𝛼 are

in-plane unit vectors, and 𝐞 = 𝐦𝛼 × 𝐬𝛼 is an out-of-plane unit vector. The rates of dislocation

densities are given by

��⊢𝛼 = −𝐬𝑟

𝛼 ∙ ∇𝜐𝛼 and ��⊙𝛼 = 𝐞 ∙ ∇𝜐𝛼 = 0, (25)

where 𝜌⊢𝛼(𝑥, 0) = 𝜌⊙

𝛼 (𝑥, 0) = 0. Recalling Eqs. (20, 21, 23, 25) represents the flow rule in the

form

where ∇α𝜐𝛼 = (𝐬𝑟𝛼 ∙ ∇𝜐𝛼) 𝐬𝑟

𝛼/|𝐬𝑟𝛼|2, 𝜘𝛼𝛽 = 1 if 𝐦𝑟

𝛼 = ±𝐦𝑟𝛽

, and 𝜘𝛼𝛽 = 𝜅 otherwise.

𝜏𝛼 + 𝑆0𝐿12Div(−𝜌⊢

𝛼𝐬𝑟𝛼)⏟

(I)

= 𝑆𝛼 (𝜐𝛼

𝜐0)𝑚 𝜐𝛼

|𝜐𝛼|⏟ (II)

− 𝐿22𝑆𝛼 (

𝑑𝛼

𝑑0)𝑞 Div ∇α𝜐𝛼

𝑑𝛼⏟ (III)

,

𝑑𝛼 = 𝐿2|∇α𝜐𝛼|, ��𝛼 = ∑ 𝕂𝛼𝛽 𝜙𝛽𝛽 , 𝑆𝛼(𝑥, 0) = 𝑆𝑦,

𝕂𝛼𝛽 = [𝜘𝛼𝛽 ℎ(𝑆𝛽)], 𝜙𝛽 = √|𝜐𝛽|2 + 𝐿2

2 |ℙ𝐅p−T∇α𝜐α|2,

(26)

13

3. Numerical implementation

In the current work, the two-dimensional version of the model is implemented via a user-

defined subroutine (UEL) in the finite-element software ABAQUS. An implementation method

introducing the rate of dislocation density ��⊢𝛼 = −𝐬𝑟

𝛼 ∙ ∇𝜐𝛼 as an additional nodal degree of

freedom, is employed. A plane-strain quadratic-element (8-node element with 9 integration

points) is defined and the flow rule Eq. (26) is solved at the integration points to obtain the

plastic flow in each slip system via the Newton-Raphson scheme. This implementation method

has been widely used by others (Bargmann et al., 2014; Ekh et al., 2007; Ertürk et al., 2009;

Evers et al., 2004; Klusemann et al., 2013; Kuroda, 2011; Kuroda and Tvergaard, 2008).

3.1. Calculation within an element

The nodal degrees of freedom: the displacement components 𝑢1, 𝑢2 and ��⊢𝛼, are given by the

finite-element software. At a time step ∆𝑡 = 𝑡𝑛+1 − 𝑡𝑛, 𝜌⊢𝑛+1𝛼 = ∆𝑡 ��⊢

𝛼+𝜌⊢𝑛𝛼 is updated, and the

deformation gradient 𝐅𝑛+1,

are computed at the integration points. The quantity 𝐅𝑛p which has been already saved in the last

time-step, is employed, and trial quantities

are calculated. The flow rule Eq. (26) is solved at the integration points and the flow rate 𝜐𝑛+1𝛼 is

obtained via the Newton-Raphson scheme, 𝐅𝑛+1p

may be updated via a semi-implicit Euler

method in the form

10

Effect of Div 𝐬𝑟𝛼 is discussed in the Appendix.

Div ∇α𝜐𝛼 = Div(−��⊢𝛼𝐬𝑟

𝛼/|𝐬𝑟𝛼|2) = −∇��⊢

𝛼 ∙ 𝐬𝑟𝛼/|𝐬𝑟

𝛼|2 − ��⊢𝛼 Div(𝐬𝑟

𝛼/|𝐬𝑟𝛼|2) 10

Div(−𝜌⊢𝑛+1𝛼 𝐬𝑟

𝛼) = −∇𝜌⊢𝑛+1𝛼 ∙ 𝐬𝑟

𝛼 − 𝜌⊢𝑛+1𝛼 Div 𝐬𝑟

𝛼 and (27)

𝜏𝑛+1𝛼 𝑡𝑟𝑖𝑎𝑙

= 𝐂𝑛+1e 𝑡𝑟𝑖𝑎𝑙

𝐒𝑛+1e 𝑡𝑟𝑖𝑎𝑙

: 𝕊𝛼

𝐅𝑛+1e 𝑡𝑟𝑖𝑎𝑙

= 𝐅𝑛+1𝐅𝑛p−1

,

𝐂𝑛+1e 𝑡𝑟𝑖𝑎𝑙

= 𝐅𝑛+1e T 𝑡𝑟𝑖𝑎𝑙 𝐅𝑛+1

e 𝑡𝑟𝑖𝑎𝑙 and

(28)

𝐅𝑛+1p

= 𝚪(∆𝑡 ∑ 𝜐𝑛+1𝛼 𝕊𝛼𝛼 + 𝐈)𝐅𝑛

p , (29)

14

where 𝚪(𝐀) = (det𝐀)−𝟏

𝟑 𝐀 enforces det 𝐅p = 1. Finally, 𝐅𝑛+1p

is saved for a new calculation in

the next time-step, and 𝐅𝑛+1e and 𝐂𝑛+1

e are calculated and employed in the derivation of elemental

stiffness matrix and elemental residual which are detailed in the following section.

3.2. Weak forms and elemental residual

Here, weak forms of the macroscopic force balance and the rate of dislocation density ��⊢𝛼 are

detailed. The macroscopic force balance via a vectorial set of virtual velocity descriptor 𝝋 is

redefined in

∫ (𝐏 ∶ ∇𝝋) 𝑑𝑉 𝑃

− ∫ 𝐛 ∙ 𝝋𝑃

𝑑𝑉 − ∫ 𝐩(𝐧)𝜕𝑃

∙ 𝝋 𝑑𝐴 = 0. (30)

Following the variational approach leading to the finite-element discretization technique

(Section 8 in Holzapfel (2000)), a virtual displacement field 𝝋 = 𝛿𝒖 is introduced. In the

absence of the body force and considering a null external-work due to the boundary conditions,

the variational equation is given by

𝛿Π = 𝛿𝑊𝑖𝑛𝑡 − 𝛿𝑊𝑒𝑥𝑡 = ∑ ∑ 𝛿𝒖𝑖𝑒T ∫ (𝐁𝑖

T: 𝐒 ) 𝑑𝑉 Ω𝑗

𝑛𝑜𝑑𝑒𝑠𝑖

𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠𝑗 , (31)

where 𝑗 and 𝑖 stand for the element and node numbers, 𝛿𝒖𝑖𝑒 represents a virtual displacement

vector. 𝐁𝑖 = 𝓓𝑟𝐍𝑖, where 𝓓𝑟 is a nonlinear differential operating matrix with respect to the

reference configuration, 𝐍𝑖 is an elemental shape function (Section 5 in Zienkiewicz and Taylor

(2005)), and 𝐒 is the second Piola-Kirchhoff stress. Therefore, the elemental residual is defined

in the form

𝑅𝑑𝒖𝑒 = ∑ 𝑅𝑑𝒖

𝑒 𝑖𝑛𝑜𝑑𝑒𝑠𝑖 = ∑ ∫ (𝐁𝑖

T: 𝐒 ) 𝑑𝑉 𝑛𝑜𝑑𝑒𝑠𝑖 . (32)

Furthermore, the rate of dislocation density ��⊢𝛼 = −𝐬𝑟

𝛼 ∙ ∇𝜐𝛼 is formulated in a weak form by

∫ ��⊢𝛼𝛿��⊢

𝛼 𝑑𝑉 𝑃

− ∫ ∇ ∙ (𝛿��⊢𝛼𝐬𝑟

𝛼)𝜐𝛼 𝑑𝑉 𝑃

+ ∫ 𝜐𝛼(𝛿��⊢𝛼𝐬𝑟𝛼) ∙ 𝐧𝜕𝑃 𝑑𝐴 𝜕𝑃

= 0. (33)

By considering the boundary conditions introduced in Section 2.6.3, Eq. (33) may be rewritten in

∫ ��⊢𝛼𝛿��⊢

𝛼 𝑑𝑉 𝑃

− ∫ (𝐬𝑟𝛼 ∙ ∇𝛿��⊢

𝛼)𝜐𝛼 𝑑𝑉 − ∫ (𝛿��⊢𝛼 Div𝐬𝑟

𝛼 )𝜐𝛼 𝑑𝑉 𝑃𝑃

= 0. (34)

15

Thus the elemental residual associated with the rate of dislocation density is expressed as

𝑅𝑑𝜌⊢𝛼𝑒 = ∑ 𝑅𝑑𝜌⊢𝛼

𝑒 𝑖𝑛𝑜𝑑𝑒𝑠𝑖 = ∑ ∫ (𝐍𝑙��⊢

𝛼𝑙𝐍𝑖 − 𝜐𝛼(𝐬𝑟

𝛼 ∙ 𝐁𝑖 + Div𝐬𝑟𝛼𝐍𝑖)) 𝑑𝑉

𝑛𝑜𝑑𝑒𝑠𝑖 . (35)

The elemental stiffness matrix 𝐊𝑒 = [𝐾𝑖𝑗𝑒 =

𝜕𝑅𝑑𝑖𝑒

𝜕𝑑𝑗] and the elemental residual are assembled over

all elements and expressed in

[𝐾𝑢 𝑢 𝐾𝑢 𝜌⊢𝛼𝐾𝜌⊢𝛼 𝑢 𝐾𝜌⊢𝛼 𝜌⊢𝛼

] [𝑑𝑢𝑑𝜌⊢𝛼

] = − [𝑅𝑑𝑢𝑅𝑑𝜌⊢𝛼

]. (36)

16

4. Results

In this section, the capability of the two-dimensional version of the model in the prediction of

a wide range of hardening behaviors as well as rate-dependent and scale-variation responses in a

single crystal, is investigated. Furthermore, directional plastic-flows in predefined slip systems

and accumulation of GNDs are distinctly observed in the single crystal under three different

loading conditions. Effects of scale variation in expansion of directional flows and accumulation

of GNDs are also studied and simply linked to the phenomenon ‘smaller is stronger’. A two-

dimensional square single-crystal in a size of D = 1 𝜇𝑚 is employed. In order to show the

strength of the implementation method for a distinct observation of dislocation glides in different

directions, four arbitrary slip systems in a variety of slip directions (Table 1) are incorporated

into the crystal. Here, the grain boundary is idealized considering the concept of hard-boundary

introduced in Section 2.6.3. A surrounding boundary-layer with a thickness of 0.01 𝜇𝑚 enforces

the microscopically hard boundary condition via 𝜐𝛼 = 0. A common set of material coefficients

and modelling parameters are given in Table 2. The parameters μ and λ are taken from a

common material with an elastic modulus E = 200 GPa and a poison ratio 𝜈 = 0.3, other

parameters are employed in an acceptable range. In order to gain insight into the spirit of the

model, different loading conditions and case studies which are detailed in Tables 3, 4, are carried

out. The modelling parameters in the flow rule Eq. (26) may vary or vanish in different case

studies.

4.1. Different displacement-loading conditions

Loading scenarios (1, 2, 3) are detailed in Table 3. Here, it is aimed to distinctly observe

directional plastic-flows associated with the predefined slip systems as well as accumulation of

GNDs in the single crystal. Three different displacement-loading conditions depicted in Fig. 1

are considered. The single crystal is discretized by 5180 elements. A common set of material

parameters and modelling coefficients given in Table 2, is employed. A high magnitude of the

energetic resistance 𝑆0 = 100𝑒7 is considered, which results in a distinct observation of plastic

flows as expected. Numerical results based on the loading scenarios (1-3) are illustrated in Figs.

2-4, respectively. Plastic flow ∫ 𝜐𝛼𝑑𝑡 for each predefined slip direction, a combination of all

plastic flows (∑ (∫ 𝜐𝛼𝑑𝑡)2𝛼 )0.5 and accumulation of edge dislocations (∑ (∫ ��⊢𝛼 𝑑𝑡)2𝛼 )0.5 are

simply observed. Figures 2-4 show that the plastic flows in an area close to the hard-boundary

17

are quite low and the edge dislocations are highly accumulated towards the boundary due to the

hard-boundary condition. In addition, directional plastic-flows and accumulation of GNDs under

a cyclic loading condition are investigated here. Figure 5 shows the stress-strain response of

single crystal under a cyclic version of loading scenario (2), four different red points: A, B, C

and D which correspond respectively to strains ɛ = 2, −0.67, −2 and 0.16% are marked on the

stress-strain curve. Figure 6 illustrates the directional plastic-flow, a combination of plastic flows

and accumulation of GNDs at different strains. Similar responses and behaviors are observed at

points A and C. However, the directional plastic-flows at A and C are respectively captured in

positive and negative magnitudes as expected. Point B is located at a zero plastic-flow position

(Fig. 5), no plastic flows and no accumulation of GNDs are consequently represented at this

point. Point D defines the single crystal at a positive strain location, although the directional

plastic-flow is observed in a negative magnitude. A small value of combination of plastic flows

and accumulation of GNDs are captured at point B.

4.2. Different case studies

Case studies (1-6) are detailed in Table 4. In the case studies (1-5), the capability of the two-

dimensional version of the model Eq. (26) in the prediction of a wide range of rate-dependent

hardening behaviors via simulation of the single crystal under a cyclic simple-shear loading, is

studied. Figure 1(b) depicts the simple-shear loading. In a similar vein with effect of cold work

(Gurtin and Ohno, 2011), the case study (4) investigates the dissipative gradient-strengthening as

a source of isotropic-hardening behavior. The case study (6) aims to validate the phenomenon

‘smaller is stronger’ in the single crystal under a large-strain simple-shear loading. In the case

studies (1-6) discretization of single crystal by 96 elements results in an acceptable range of

mesh-size insensitivity response along with an efficient solution time11

.

Case study (1): 𝐿2 = 0, 𝑆𝛼 = 𝑆𝑦 and ��𝛼 = 0. The effect of energetic gradient-

strengthening (term (I) in Eq. (26)) is studied here. The dissipative gradient-strengthening

(term(III)) is not involved and only the initial coarse-grain yield-threshold 𝑆𝑦 is considered.

Different 𝐿1/D ratios are also employed. Figure 7(a) shows the effect of term (I) in the stress-

strain responses, it can be concluded that the energetic gradient-strengthening represents the

kinematic hardening which is accompanied with a Bauschinger-like phenomenon. It is worth 11

Increasing the number of element from 96 to 320, improves the stress-strain results by less than 8%, but solution

time increases almost three times.

18

mentioning that GND densities which are evolved via the directional gradient of slip-rates

(Section 2.4) are independent of sign of 𝜐𝛼. Therefore, term (I) plays a role of energetic back-

stress and gives rise to the kinematic-hardening response. Figure 7(b) shows that changing in the

energetic length-scale 𝐿1 has no effect on the onset of plastic flow. However, a higher strain-

hardening response is observed with a larger value of 𝐿1.

Case study (2): 𝑆𝛼 = 𝑆𝑦, ��𝛼 = 0, and 𝐿1/D and 𝐿2/D vary. Effect of the dissipative

gradient-strengthening (term(III)) is investigated here. Figure 8 shows the stress-strain results. It

can be seen that the dissipative gradient-strengthening represents the yield strengthening. In

order to gain insight into the effect of dissipative length-scale 𝐿2, a varying 𝐿2/D ratio along

with 𝐿1 = D are considered here. Figure 9 shows that an increase in 𝐿2 yields a higher yield

strength in the onset of plastic flow which is in contrast to the response observed from the

energetic length-scale in Fig. 7(b). Similar behaviors were already observed in the small-

deformation studies (Gurtin et al., 2007; Lele and Anand, 2009; Niordson and Legarth, 2010).

Bear in mind that ��𝛼 = 0 is considered here, thus the post-yield strain-hardening is not induced.

The effect of ��𝛼 ≠ 0 in term (III) is studied in the case study (4).

Case study (3): ��𝛼 ≠ 0. Evolution of self- and latent-hardening is activated here. First,

only self-hardening without latent one is considered (��𝛼 ≠ 0, 𝜅 = 0). Figure 10(a) shows the

effect of dissipative self-hardening in the stress-strain results, it can be seen that self-hardening is

a source of isotropic-hardening behavior and yields a post-yield strain-hardening response that

consistently increases by plastic work. Second, the latent-hardening is taken into account

(��𝛼 ≠ 0, 𝜅 ≠ 0). Figure 10(b) indicates a comparison of stress-strain results obtained by

considering self- and latent-hardening with the results obtained without latent-hardening (Fig.

10(a)). An additional isotropic-hardening response is easily observed via employing the latent-

hardening. It is worth mentioning that self- and latent-hardening are evolved via 𝜙𝛽 (Eq. (20))

which is a function of accumulation rates of SSDs and GNDs (Eq. (21)). Evers (Evers et al.,

2004) emphasized that evolution of SSDs during crystallographic slips increases formation of

short-range interactions between all dislocations, i.e. both SSDs and GNDs. These interactions

turn to impede the dislocation movements and yield the isotropic-hardening responses which are

observed in this case study.

19

Case study (4), Gurtin and Ohno (2011) proposed a new small-deformation gradient

crystal-plasticity model which incorporates an irrecoverable stored-energy of cold work and

represents a new form of irrecoverable microstress and irrecoverable gradient-strengthening (Eq.

(13.5-6) in Gurtin and Ohno (2011)). Gurtin emphasized that the irrecoverable stored-energy

may lead to isotropic hardening which is induced via the new form of irrecoverable gradient-

strengthening. Moreover, the dissipative microstress which is introduced in the current large-

deformation theory plays the same role as the new form proposed in the new small-deformation

model. Gurtin highlighted that there is no way of differentiating between the new model and a

theory in which the irrecoverable stored-energy vanishes and a dissipative term is considered.

Motivated by this statement, a source of isotropic-hardening behavior is also investigated in the

dissipative gradient-strengthening (term (III)). A complete form of Eq. (26) accompanied with

��𝛼 = 0 in term (II) and ��𝛼 ≠ 0 in term (III) is considered. Figure 11 compares the stress-strain

responses obtained here with the results depicted in Fig. 7(a) (case study (1): 𝐿2 = 0, ��𝛼 = 0 in

terms (II) and (III)). Based on the results presented in Fig. 11, it can be concluded that the

dissipative gradient-strengthening (term (III)) results in a yield-strengthening response which has

been already observed in the case study (2), as well as an isotropic-hardening behavior due to

formation of short-range interactions and entanglements which are taken into account by the

function of accumulation rates of dislocations and induced via ��𝛼 ≠ 0 in term (III).

Case study (5), two rate-sensitivity parameters 𝑚, 𝑞 are employed in Eq. (26). Effect of

𝑚 which is widely used in the conventional viscoplasticity and supposed to be quite small in

simulation of metals under quasi-static loading12

, is studied here. Note that term (III) in Eq. (26)

produces an additional hardening in region of inhomogeneous deformations (Lele and Anand,

2009). Therefore, 𝑞 behaves differently from 𝑚, and requires a high enough value compared to

𝑚 to produce a measurable effect of term (III). Figure 12 compares the stress-strain responses

obtained based on a complete form of Eq. (26) and the coefficients and parameters detailed in

Table 2 along with varying value of 𝑚. It can be concluded that considering 𝑚 = 0.08 in this

study, results in a nearly rate-independent response and has no effect in the investigation of

hardening responses discussed in the previous case studies. Figure 12 shows that an increase in

12

It is generally accepted that metals show a nearly rate-independent response under a quasi-static loading at room-

temperature condition.

20

𝑚 value in a range of greater than 0.12 results in a significant change in the onset of plastic flow

and the post-yield response.

Case study (6), employing all three governing components (terms (I − III)) in Eq. (26)

with the common set of material coefficients and modelling parameters detailed in Table 2,

yields a stress-strain response which shows a combination of isotropic and kinematic-like

hardening behaviors as well as yield-strengthening. In this case study, a phenomenon which is

so-called ‘smaller is stronger’ and refers to an increase in strengthening response with decrease

in the crystal size, is investigated. A complete form of the flow rule Eq. (26) is employed to

investigate size-dependent behavior of single crystals which are simulated with different crystal

sizes D under a large-strain simple-shear loading. Figure 13 shows the stress-strain responses and

indicate that the onset of plastic flow and the post-yield behavior are highly size-dependent.

However, in a range of crystal size close to 8 𝜇𝑚, where 𝐿1/𝐷 = 𝐿2/𝐷 = 0.125, a nearly size-

independent response is observed, and it can be concluded that the effect of gradient-

strengthenings (terms(I, III)) is insignificant in this range.

4.3. Effect of scale variation in directional plastic-flows and accumulation of

GNDs

In order to observe the effect of scale variation in directional plastic-flows and accumulation of

GNDs, loading scenario (2 ( is recalled and applied to two more single crystals with different

crystal sizes D = 0.5 𝜇𝑚 and 2 𝜇𝑚. Figure 14 compares the numerical results obtained here with

the results presented in Fig. 3 for a single crystal with D = 1 𝜇𝑚. A high magnitude of the

energetic resistance 𝑆0 = 100𝑒7 is considered here. Based on the results depicted in Fig. 14-

rows (a, b), it can be concluded that an increase in the crystal size results in a larger magnitude of

plastic flows as well as a more expansion of plastic flows towards the boundary. Smaller crystals

show a higher aggregation of plastic flows towards the center of crystals. Results presented in

Fig. 14-row (c) indicate the greatest magnitude of GND density (dislocations pile-up) in an area

close to the hard-boundary. It can also be concluded that a decrease in the crystal size yields an

increase in the accumulation of GNDs towards the boundary. A smaller crystal shows a more

expansive accumulation of GNDs and is subjected to a larger value of gradient strengthening,

thus the phenomenon ‘smaller is stronger’ is validated.

21

5. Summary and Conclusion

In the current work, the well-defined model by Gurtin (2008) was numerically shown to be

capable of describing a wide range of hardening behaviors as well as rate-dependent and scale-

variation responses. The model was represented first with respect to the reference configuration

which facilitates the numerical implementation. The numerical results reveal that the energetic

and dissipative gradient-strengthenings yield respectively the kinematic-hardening behavior and

yield-strengthening response. It is also concluded that the dissipative and energetic length-scales

act differently; the former affects the onset of plastic flow, and the latter results in a change of

the post-yield strain-hardening response. Moreover, evolution of self- and latent-hardening is

induced via a function of accumulation rates of SSDs and GNDs which is viewed as a measure

of formation of short-range interactions between SSDs and GNDs. These interactions turn to

impede the dislocation movements and yield isotropic-hardening responses. The simulation

results reveal that the dissipative gradient-strengthening is also identified as a source of

isotropic-hardening behavior, which represents the effect of cold work introduced by Gurtin and

Ohno (2011). In addition, the rate-dependent stress-strain response of single crystal is simply

observed here, and directional plastic-flows and accumulation of GNDs towards the hard-

boundary are distinctly observed in a single crystal under three different loading conditions.

Furthermore, the phenomenon ‘smaller is stronger’ is easily validated in different crystal sizes,

smaller crystals show a more expansive accumulation of GNDs along with a larger value of

gradient strengthening, and larger crystals present a larger magnitude of plastic flows as well as a

more expansion of plastic flows towards the boundary.

Finally, the current model can predict a wide variety of behaviors and responses of single

crystals which have been already observed in experiments and it is envisaged that current work

could be further developed to study interactions between grain and grain-boundaries in multi

crystals.

Acknowledgment

The support rendered by the German Research Foundation (DFG) for research grant SFB 666 is

cordially acknowledged.

22

Appendix

It is worth noting that evaluation of Div 𝐬𝑟𝛼 represented in Eqs. (27) and (35) requires an

intricate analysis at the integration points, which is time consuming and cannot be simply

accomplished via a user-defined subroutine (UEL). In this study, Div 𝐬𝑟𝛼 is evaluated via a

common block function which is linked to UEL subroutine along with an averaging method on

the nodes.

23

References

Abu Al-Rub, R.K., Voyiadjis, G.Z., 2004. Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. Int J Plasticity 20, 1139-1182. Acharya, A., Bassani, J.L., 1996. On Non-Local Flow Theories that Preserve the Classical Structure of Incremental Boundary Value Problems, in: Pineau, A., Zaoui, A. (Eds.), IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials. Springer Netherlands, pp. 3-9. Aifantis, E.C., 1984. On the Microstructural Origin of Certain Inelastic Models. J Eng Mater-T Asme 106, 326-330. Aifantis, E.C., 1987. The physics of plastic deformation. Int J Plasticity 3, 211-247. Aifantis, E.C., 2003. Update on a class of gradient theories. Mechanics of Materials 35, 259-280. Anand, L., Gurtin, M.E., Reddy, B.D., 2015. The stored energy of cold work, thermal annealing, and other thermodynamic issues in single crystal plasticity at small length scales. Int J Plasticity 64, 1-25. Arsenlis, A., Parks, D.M., 1999. Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater 47, 1597-1611. Asaro, R.J., Rice, J.R., 1977. Strain localization in ductile single crystals. J Mech Phys Solids 25, 309-338. Ashby, M.F., 1970. Deformation of Plastically Non-Homogeneous Materials. Philos Mag 21, 399-&. Bargmann, S., Reddy, B.D., 2011. Modeling of polycrystals using a gradient crystal plasticity theory that includes dissipative micro-stresses. European Journal of Mechanics - A/Solids 30, 719-730. Bargmann, S., Reddy, B.D., Klusemann, B., 2014. A computational study of a model of single-crystal strain-gradient viscoplasticity with an interactive hardening relation. International Journal of Solids and Structures 51, 2754-2764. Bargmann, S., Svendsen, B., Ekh, M., 2011. An extended crystal plasticity model for latent hardening in polycrystals. Comput Mech 48, 631-645. Bassani, J.L., 2001. Incompatibility and a simple gradient theory of plasticity. J Mech Phys Solids 49, 1983-1996. Bonet, J., Wood, R.D., 2008. Nonlinear continuum mechanics for finite element analysis, 2nd ed. Cambridge University Press, Cambridge, UK ; New York. Cermelli, P., Gurtin, M.E., 2001. On the characterization of geometrically necessary dislocations in finite plasticity. J Mech Phys Solids 49, 1539-1568. Cermelli, P., Gurtin, M.E., 2002. Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations. International Journal of Solids and Structures 39, 6281-6309. Clayton, J.D., 2011. Nonlinear mechanics of crystals, Solid mechanics and its applications v 177. Springer,, Dordrecht, pp. xiii, 700 p. Clayton, J.D., McDowell, D.L., Bammann, D.J., 2006. Modeling dislocations and disclinations with finite micropolar elastoplasticity. Int J Plasticity 22, 210-256. Ekh, M., Grymer, M., Runesson, K., Svedberg, T., 2007. Gradient crystal plasticity as part of the computational modelling of polycrystals. Int J Numer Meth Eng 72, 197-220. El-Naaman, S.A., Nielsen, K.L., Niordson, C.F., 2016. On modeling micro-structural evolution using a higher order strain gradient continuum theory. Int J Plasticity 76, 285-298. Ertürk, İ., van Dommelen, J.A.W., Geers, M.G.D., 2009. Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories. J Mech Phys Solids 57, 1801-1814. Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D., 2004. Non-local crystal plasticity model with intrinsic SSD and GND effects. J Mech Phys Solids 52, 2379-2401. Fleck, N.A., Hutchinson, J.W., 1993. A Phenomenological Theory for Strain Gradient Effects in Plasticity. J Mech Phys Solids 41, 1825-1857.

24

Fleck, N.A., Hutchinson, J.W., 1997. Strain Gradient Plasticity, in: John, W.H., Theodore, Y.W. (Eds.), Advances in Applied Mechanics. Elsevier, pp. 295-361. Fleck, N.A., Hutchinson, J.W., 2001. A reformulation of strain gradient plasticity. J Mech Phys Solids 49, 2245-2271. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., 1994. Strain gradient plasticity: Theory and experiment. Acta Metall Mater 42, 475-487. Fried, E., Gurtin, M.E., 1993. Continuum theory of thermally induced phase transitions based on an order parameter. Physica D: Nonlinear Phenomena 68, 326-343. Fried, E., Gurtin, M.E., 1994. Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D: Nonlinear Phenomena 72, 287-308. Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W., 1999. Mechanism-based strain gradient plasticity— I. Theory. J Mech Phys Solids 47, 1239-1263. Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. J Mech Phys Solids 52, 1379-1406. Gurtin, M.E., 2000. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J Mech Phys Solids 48, 989-1036. Gurtin, M.E., 2003. On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients. Int J Plasticity 19, 47-90. Gurtin, M.E., 2006. The Burgers vector and the flow of screw and edge dislocations in finite-deformation single-crystal plasticity. J Mech Phys Solids 54, 1882-1898. Gurtin, M.E., 2008. A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations. Int J Plasticity 24, 702-725. Gurtin, M.E., 2010. A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations. Int J Plasticity 26, 1073-1096. Gurtin, M.E., Anand, L., 2005a. A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part II: Finite deformations. Int J Plasticity 21, 2297-2318. Gurtin, M.E., Anand, L., 2005b. A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J Mech Phys Solids 53, 1624-1649. Gurtin, M.E., Anand, L., 2007. A gradient theory for single-crystal plasticity. Model Simul Mater Sc 15, S263-S270. Gurtin, M.E., Anand, L., 2009. Thermodynamics applied to gradient theories involving the accumulated plastic strain: The theories of Aifantis and Fleck and Hutchinson and their generalization. J Mech Phys Solids 57, 405-421. Gurtin, M.E., Anand, L., Lele, S.P., 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J Mech Phys Solids 55, 1853-1878. Gurtin, M.E., Fried, E., Anand, L., 2010. The mechanics and thermodynamics of continua. Cambridge University Press, New York. Gurtin, M.E., Ohno, N., 2011. A gradient theory of small-deformation, single-crystal plasticity that accounts for GND-induced interactions between slip systems. J Mech Phys Solids 59, 320-343. Gurtin, M.E., Reddy, B.D., 2014. Gradient single-crystal plasticity within a Mises–Hill framework based on a new formulation of self- and latent-hardening. J Mech Phys Solids 68, 134-160. Gurtin, M.E., Reddy, B.D., 2016. Some issues associated with the intermediate space in single-crystal plasticity. J Mech Phys Solids 95, 230-238. Hall, E.O., 1951. The Deformation and Ageing of Mild Steel .3. Discussion of Results. P Phys Soc Lond B 64, 747-753. Han, C.-S., Gao, H., Huang, Y., Nix, W.D., 2005. Mechanism-based strain gradient crystal plasticity—I. Theory. J Mech Phys Solids 53, 1188-1203. Hirth, J.P., Lothe, J., 1982. Theory of dislocations, 2nd ed. Wiley, New York.

25

Holzapfel, G.A., 2000. Nonlinear solid mechanics : a continuum approach for engineering. Wiley, Chichester ; New York. Hull, D., Bacon, D.J., 2011. Introduction to dislocations, 5th ed. Butterworth-Heinemann, Oxford ; Burlington, MA. Hutchinson, J.W., 2000. Plasticity at the micron scale. International Journal of Solids and Structures 37, 225-238. Kalidindi, S.R., Bronkhorst, C.A., Anand, L., 1992. Crystallographic texture evolution in bulk deformation processing of FCC metals. J Mech Phys Solids 40, 537-569. Klusemann, B., Svendsen, B., Bargmann, S., 2013. Analysis and comparison of two finite element algorithms for dislocation density based crystal plasticity. GAMM-Mitteilungen 36, 219-238. Kroner, E., 1960. Allgemeine Kontinuumstheorie Der Versetzungen Und Eigenspannungen. Arch. Rational Mech. Anal. 4, 273-334. Kuroda, M., 2011. On large-strain finite element solutions of higher-order gradient crystal plasticity. International Journal of Solids and Structures 48, 3382-3394. Kuroda, M., Tvergaard, V., 2008. A finite deformation theory of higher-order gradient crystal plasticity. J Mech Phys Solids 56, 2573-2584. Lee, E.H., 1969. Elastic-Plastic Deformation at Finite Strains. J Appl Mech 36, 1-&. Lele, S.P., Anand, L., 2008. A small-deformation strain-gradient theory for isotropic viscoplastic materials. Philos Mag 88. Lele, S.P., Anand, L., 2009. A large-deformation strain-gradient theory for isotropic viscoplastic materials. Int J Plasticity 25, 420-453. Levkovitch, V., Svendsen, B., 2006. On the large-deformation- and continuum-based formulation of models for extended crystal plasticity. International Journal of Solids and Structures 43, 7246-7267. Maaß, R., Van Petegem, S., Ma, D., Zimmermann, J., Grolimund, D., Roters, F., Van Swygenhoven, H., Raabe, D., 2009. Smaller is stronger: The effect of strain hardening. Acta Mater 57, 5996-6005. McBride, A., Bargmann, S., Reddy, B.D., 2015. A computational investigation of a model of single-crystal gradient thermoplasticity that accounts for the stored energy of cold work and thermal annealing. Comput Mech 55, 755-769. Menzel, A., Steinmann, P., 2000. On the continuum formulation of higher gradient plasticity for single and polycrystals. J Mech Phys Solids 48, 1777-1796. Naghdi, P.M., Srinivasa, A.R., 1994. Some General Results in the Theory of Crystallographic Slip. Z Angew Math Phys 45, 687-732. Niordson, C.F., Legarth, B.N., 2010. Strain gradient effects on cyclic plasticity. J Mech Phys Solids 58, 542-557. Nix, W.D., Gao, H., 1998. Indentation size effects in crystalline materials: A law for strain gradient plasticity. J Mech Phys Solids 46, 411-425. Nye, J.F., 1953. Some geometrical relations in dislocated crystals. Acta Metallurgica 1, 153-162. Ohno, N., Okumura, D., 2007. Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations. J Mech Phys Solids 55, 1879-1898. Ohno, N., Okumura, D., Shibata, T., 2008. Grain-Size Dependent Yield Behavior under Loading, Unloading and Reverse Loading. International Journal of Modern Physics B 22, 5937-5942. Petch, N.J., 1953. The Cleavage Strength of Polycrystals. J Iron Steel I 174, 25-28. Raabe, D., Sachtleber, M., Zhao, Z., Roters, F., Zaefferer, S., 2001. Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation. Acta Mater 49, 3433-3441. Rice, J.R., 1971. Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity. J Mech Phys Solids 19, 433-455.

26

Steinmann, P., 2015. Geometrical foundations of continuum mechanics : an application to first- and second-order elasticity and elasto-plasticity, Lecture notes in applied mathematics and mechanics volume 2. Springer,, Heidelberg, pp. 1 online resource (xxiv, 517 pages). Svendsen, B., 2002. Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. J Mech Phys Solids 50, 1297-1329. Svendsen, B., Bargmann, S., 2010. On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. J Mech Phys Solids 58, 1253-1271. Taylor, G.I., 1934. The Mechanism of Plastic Deformation of Crystals. Part I. Theoretical. Teodosiu, C., 1982. Elastic models of crystal defects. Springer-Verlag, Berlin ; New York. Teodosiu, C., Sidoroff, F., 1976. A theory of finite elastoviscoplasticity of single crystals. International Journal of Engineering Science 14, 165-176. Toupin, R.A., 1962. Elastic materials with couple-stresses. Arch. Rational Mech. Anal. 11, 385-414. Voyiadjis, G.Z., Al-Rub, R.K.A., 2005. Gradient plasticity theory with a variable length scale parameter. International Journal of Solids and Structures 42, 3998-4029. Voyiadjis, G.Z., Faghihi, D., 2012. Thermo-mechanical strain gradient plasticity with energetic and dissipative length scales. Int J Plasticity 30–31, 218-247. Wulfinghoff, S., Böhlke, T., 2015. Gradient crystal plasticity including dislocation-based work-hardening and dislocation transport. Int J Plasticity 69, 152-169. Yefimov, S., van der Giessen, E., 2005. Size effects in single crystal thin films: nonlocal crystal plasticity simulations. European Journal of Mechanics - A/Solids 24, 183-193. Zbib, H.M., Aifantis, E.C., 1991. On the Structure and Width of Shear Bands in Finite Elastoplastic Deformations, in: Boehler, J.-P., Khan, A. (Eds.), Anisotropy and Localization of Plastic Deformation. Springer Netherlands, pp. 99-102. Zienkiewicz, O.C., Taylor, R.L., 2005. The finite element method for solid and structural mechanics, 6th ed. Elsevier Butterworth-Heinemann, Amsterdam ; Boston.