Post on 19-Jun-2018
*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
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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.
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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
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