PAMM · Proc. Appl. Math. Mech. 11, 377 – 378 (2011) / DOI 10.1002/pamm.201110180

Softening in Nanocrystalline Metals: Modeling of Grain Growth Induced

Creep and Relaxation

Ercan Gürses1,∗ and Tamer El Sayed2

1 Middle East Technical University (METU), Aerospace Engineering Department, Ankara, Turkey2 Computational Solid Mechanics Laboratory (CSML), King Abdullah University of Science and Technology (KAUST),

Thuwal, Kingdom of Saudi Arabia

A variational multiscale model is presented for grain growth in face-centered cubic nanocrystalline (nc) metals. In particular,

grain-growth-induced stress softening and the resulting relaxation and creep phenomena are addressed. The behavior of

the polycrystal is described by a conventional Taylor-type averaging scheme in which the grains are treated as two-phase

composites consisting of a grain interior phase and a grain boundary affected zone. Furthermore, a grain growth law that

captures the experimentally observed characteristics of the grain coarsening phenomena is proposed. The model is shown to

provide a good description of the experimentally observed grain-growth-induced relaxation in nc-copper.

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Description of Grain Growth Model

Following the classical multiplicative decomposition framework, the deformation gradient F = FeF

p is assumed to decom-

pose into an elastic part Fe and a plastic part Fp with J = detF > 0, Jp = detFp > 0. Treating each grain as a composite

material composed of a grain interior (GI) phase and a grain boundary (GB) phase, the free energy function can be expressed

as a simple volume average W = ξWgi + (1− ξ)Wgb, where ξ is the volume fraction of the grain core region, Wgi and Wgb

denote the free energies of the grain interior and boundary phases, respectively. The average first Piola-Kirchhoff stress P

reads similarly P = ξPgi + (1− ξ)Pgb. The volume average stress P, the GI stress Pgi and the GB stress Pgb are computed

from Coleman’s relations by evaluating the partial derivatives of corresponding energies with respect to F. Assuming cubical

grains and a constant thickness dgb of the GB phase, the volume fraction is evaluated as ξ = (d − dgb)3/d3. Following [2],

the flow rule for the GB phase is assumed to be

FpgbF

p−1

gb = ǫpM+ θpN , (1)

where multipliers ǫp and θp are subject to irreversibility constraints. M and N are the directions of the deviatoric and

volumetric plastic deformation rates, respectively. The free energy Wgb is assumed to decompose additively into an elastic

part W egb and a plastic part W p

gb, i.e., Wgb(F,Fpgb, ǫ

p, θp) = W egb(F

egb) +W p

gb(ǫp, θp). The plastic stored energy is assumed

to additively decompose into deviatoric and volumetric parts W pgb(ǫ

p, θp) = W p,devgb (ǫp) + W p,vol

gb (θp), and a conventional

power-law of hardening is employed for the deviatoric part. The volumetric part of the plastic energy is attributed to void

growth. The stored energy for a spherical void in a power-law hardening material was derived by Ortiz and Molinari [3]. The

response of the grain interior region is described by a rate-independent crystallographic multi-surface plasticity model. Thus,

the flow rule has the classical form

FpgiF

p−1

gi =∑

α

γαsα ⊗m

α , (2)

where γα is the rate of crystallographic slip on slip system α; sα and mα are the slip system vectors. The free energy is

assumed to have an additive structure Wgi(F,Fpgi, γ

α) = W egi(F

egi)+W p

gi(γα), where W e

gi(Fegi) and W p

gi(γα) are the elastic

energy density and stored plastic energy, respectively. The plastic stored energy for the crystalline phase of an individual grain

of size d is given by

W pgi(γ

αn+1) = W p

gi,n +∑

α

[

(ταn +ζ(dn − d)

D)∆γα +

1

2∆γα

∑

β

hαβn ∆γβ

]

, (3)

where ταn is the critical resolved shear stress for slip system α at time tn, hαβn is the hardening matrix at time tn, dn is the size

of the grain at tn, ζ is a material parameter and D is the mean grain size, i.e., D = 1

N

∑N

i=1di. We make use of the classical

hardening matrix hαβ(γ) = [q + (1− q)δαβ ]h(γ), where γ =∑

α γα is the accumulated plastic slip on slip systems, q is the

latent hardening coefficient and δαβ is the Kronecker delta. The hardening function h(γ) reads from [2].

∗ Corresponding author: email ercangurses@gmail.com, phone +90 312 210 4257.

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

378 Section 6: Material modeling in solid mechanics

0

200

400

600

800

1000

0 0.02 0.04 0.06 0.08 0.1 0.12

Com

pre

ssiv

e S

tres

s [M

Pa]

Compressive Strain [-]

700

750

800

850

900

950

1000

20 40 60 80 100 120

Com

pre

ssiv

e S

tres

s [M

Pa]

Time [min](a) (b)

Fig. 1 Comparison of the model response against the relaxation test data from Brandstetter et al. [1], where the blue discrete points and

the red solid line correspond to the experimental data and the model predictions, respectively. (a) Comparison of the compressive stress

response at a constant strain rate of ε = 10−4 s−1. (b) Comparison of the stress relaxation behavior for a duration of 90 minutes at a

constant strain of ε = 11%.

We propose a phenomenological grain growth model. Despite its phenomenological nature, the growth law is based on

several experimental observations and captures the main characteristic features of the grain growth phenomena. As a con-

sequence of several experimental works it is well agreed on that the grain coarsening in nc-metals is mainly stress driven.

Therefore, in what follows it is assumed that the grain growth is driven by the stress. Furthermore, it has been observed in

experiments and MD simulations that larger grains exhibit coarsening at the expense of shrinkage and elimination of smaller

grains. Based on these observations, we assume that the rate of grain growth is controlled by the following rule

d =

η(d0 −D0)

D0

(

d0d

)p

exp

(

−||devσ||

scr

)

if ||devσ|| ≥ scr

0 if ||devσ|| < scr

(4)

where σ, d0 and D0 are the Cauchy stress, the initial size of an individual grain in the polycrystalline aggregate and the initial

mean size, respectively. The exponent p and the reference rate η are parameters that control the rate of the grain growth. The

critical stress scr determines when the grain growth initiates. The parameters η and scr typically depend on the impurity level

of the specimen.In other words, compared to a low purity nc-sample, a high purity metal is expected to have a higher scr value

and a lower η value resulting in a larger threshold against grain coarsening and a lower grain boundary mobility, respectively.

Note that as a consequence of the growth rule (4) the grains initially greater than the mean grain size D0 coarsen, whereas the

grains smaller than D0 shrink in accord with experimental observations and MD simulations.

The scale transition from the single grain level to the polycrystalline level is achieved through a conventional Taylor

averaging, i.e., all the grains are subject to the same deformation and, therefore, the compatibility among grains is satisfied

a priori and the macroscopic stress is computed by volume averaging; see [2] for details. Furthermore, we assume that the

initial grain size has a lognormal distribution through the polycrystalline sample.

2 Application to Nanocrystalline Copper

In this section, we utilize the proposed model to simulate the relaxation behavior of nanocrystalline copper and present the

predictive capability of the model through a comparison against the experimental data of Brandstetter et al. [1]. The initial

mean grain size of nc-Cu is D0 = 32 nm as reported in [1]. Next, we provide a brief explanation of the experimental

procedure; see [1] for full details. The experiment consisted of two steps, namely, a compressive loading step and a constant

strain relaxation step. The compressive load was applied to the nc-Cu specimen in a deformation-controlled manner with a

constant strain rate of ε = 10−4 s−1 up to a maximum strain of ε = 11%. Following this loading step, the compressive

strain was kept constant at ε = 11% and the stress relaxation was followed for about 90 min. In Fig. 1(a) the loading step is

depicted where the blue discrete points and the red solid line correspond to the experimental data and the model prediction,

respectively. The stress relaxation that follows the loading step at a constant strain of ε = 11% is evident in Fig. 1(a). The

relaxation behavior for a period of 90 minutes at a strain level of ε = 11% is visualized in Fig. 1(b), where the solid line is

the result of the proposed model. As can be seen from Figs. 1(a) and (b), both the compressive behavior and the relaxation

response, i.e., the amount and the rate of the reduction in stress, are successfully reproduced by the proposed model.

References

[1] S. Brandstetter, K. Zhang, A. Escuadro, J. R. Weertman, and H. Van Swygenhoven. Scr. Mater., 58, 61 (2008).[2] E. Gürses and T. El Sayed, J. Mech. Phys. Solids, 59, 732 (2011).[3] M. Ortiz and A. Molinari. J. Appl. Mech.-T. ASME, 59, 48 (1992).

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com