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Research Papers

Pertinence of the Grain Size on the Mechanical Strength of Polycrystalline Metals OPEN ACCESS

[+] Author and Article Information
N. A. Zontsika

Laboratoire des Sciences des Procédés et
des Matériaux (LSPM–UPR CNRS 3407),
Université Paris 13,
99 Avenue Jean-Baptiste Clément,
Villetaneuse 93430, France

A. Abdul-Latif

Laboratoire Quartz,
Supméca, 3,
rue Fernand Hainaut,
St Ouen Cedex 93407, France;
IUT de Tremblay,
Université Paris 8,
Tremblay-en-France 93290, France
e-mail: aabdul@iu2t.univ-paris8.fr

S. Ramtani

Laboratoire des Sciences des Procédés et des
Matériaux (LSPM–UPR CNRS 3407),
Université Paris 13,
99 Avenue Jean-Baptiste Clément,
Villetaneuse 93430, France

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 10, 2016; final manuscript received September 9, 2016; published online February 9, 2017. Assoc. Editor: Taehyo Park.

J. Eng. Mater. Technol 139(2), 021017 (Feb 09, 2017) (10 pages) Paper No: MATS-16-1176; doi: 10.1115/1.4035489 History: Received June 10, 2016; Revised September 09, 2016

Motivated by the already developed micromechanical approach (Abdul-Latif et al., 2002, “Elasto-Inelastic Self-Consistent Model for Polycrystals,” ASME J. Appl. Mech., 69(3), pp. 309–316.), a new extension is proposed for describing the mechanical strength of ultrafine-grained (ufg) materials whose grain sizes, d, lie in the approximate range of 100 nm < d < 1000 nm as well as for the nanocrystalline (nc) materials characterized by d100nm. In fact, the dislocation kinematics approach is considered for characterizing these materials where grain boundary is taken into account by a thermal diffusion concept. The used model deals with a soft nonincremental inclusion/matrix interaction law. The overall kinematic hardening effect is described naturally by the interaction law. Within the framework of small deformations hypothesis, the elastic part, assumed to be uniform and isotropic, is evaluated at the granular level. The heterogeneous inelastic part of deformation is locally determined. In addition, the intragranular isotropic hardening is modeled based on the interaction between the activated slip systems within the same grain. Affected by the grain size, the mechanical behavior of the ufg as well as the nc materials is fairly well described. This development is validated through several uniaxial stress–strain experimental results of copper and nickel.

FIGURES IN THIS ARTICLE
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Since the early work of Gleiter [1], the ufg and nc materials gained a reputation due to their varied range of advanced and growing properties. Despite the classification of these materials into two groups with respect to their grain sizes, as stated in certain works, one might find such a distinction not so rigorous. In some studies, the ufg materials may denote an average grain size of 100–1000 nm, e.g., Ref. [2], whereas in other studies it may refer to grain sizes confined to 250–1000 nm, e.g., Ref. [3]. There are also publications that argue that the lower bound of ufg size could extend up to 10 nm [4,5].

Historically, the impact of grain size on the mechanical properties has originally been reported by Hall [6] and later on by Petch [7], where the yield strength of polycrystalline materials increases linearly with the inverse square root of the grain size. Among potential applications of this class of materials, the possibility for high strain rate superplasticity at medium and elevated temperatures can be mentioned [3,8]. Because of the relatively higher volume fraction of grain boundaries in nc materials, research interests have been developed in investigating the deformation mechanisms at the grain boundaries, such as sliding. Such grain boundaries play a significant role induced by complex dislocation mechanisms [9].

The responses of ufg and nc structures are also affected by several intra and intergranular mechanisms, which explain different deformation mechanisms.

On the other hand, it is generally observed that the enhancement in yield strength is accompanied by lack of ductility [4]. The ufg materials are further characterized by low work hardening rate and high fatigue resistance (e.g., Ref. [3]). Moreover, molecular dynamics simulations show that partial dislocation movement within grain boundaries becomes a dominant mechanism in the grain sizes range of 10–50 nm [9,10]. A reverse Hall–Petch law (i.e., a negative slope) is observed for grain sizes below 10 nm. Such a behavior has been interpreted either by the Coble creep of grain-boundary diffusion [11,12]. The concept of critical grain size has been proposed considering the equilibrium position of dislocation pile-up under external stress. In ufg copper, for example, predictions indicated a critical grain size ∼19.3 nm (e.g., Refs. [11] and [1315]). In another similar study and by using the theoretical shear strength and Peierls strength, Wang et al. [16] concluded that the computed critical grain size for copper is 8.2 nm. In this range of grain sizes, further features are also reported such as more sensitivity due to porosity, tension/compression asymmetry, and shear bands localization [17], increase in strain rate sensitivity especially for face-centered cubic (fcc) metals [18,19], high dislocation density, and high volume fraction of grain boundaries. Such impressive mechanical features have turned the subject into an appealing one with intensive research developments.

From the modeling point of view, the nonlocal dislocation mechanics approaches based on the concept of geometrically necessary dislocations (GNDs) have been developed to successfully predict the dependence of strain-hardening rate on the grain size (see Refs. [20] and [21], and many others). Several other works (e.g., Refs. [15] and [22]) focused on the grain size distribution effect as an internal parameter of the heterogeneous microstructure. Micro–macro models consider the grain size effect that has been developed (e.g., Refs. [2325]) whereby a Hall–Petch-type equation with a single-valued grain size is considered at the microscale. The mixture model of two phases (grain and grain boundary) has been developed, e.g., Refs. [24], [2632], and [3336] among others. Likewise, other modeling approaches have been recently developed for describing size effects in crystalline metals. These are the discrete dislocation dynamics (see Refs. [3739] and others) and molecular dynamic simulations (e.g., Refs. [4043]). In the current modeling, another modeling approach that is used assumes that the grain is deformed plastically via dislocation glide; whereas the plasticity of grain boundary takes place through a boundary diffusion mechanism [4446]. The latter describes the strain hardening behavior with sensitivity to the grain size and strain rate.

In this work, an extension of an already developed micromechanical approach [47,48] is proposed to predict the grain size effect on the mechanical strength of metals. As a new development, dislocation kinetics equations look interesting for characterizing the ufg and nc materials where the grain boundary is taken into account using the thermal diffusion concept. In the used self-consistent model, the overall kinematic hardening effect is naturally described by the interaction law. The elastic part is considered at the granular level assumed to be uniform, isotropic, and compressible. The heterogeneous inelastic deformation is locally determined by the activated slip systems. The intragranular isotropic hardening is taken into account. The model describes fairly well the effect of wide range of grain sizes on the strain–stress behavior of polycrystalline metals. The model is validated by copper and nickel experimental data.

Length Scales and Internal Variables.

Based on the elasto-inelastic self-consistent model developed in Refs. [47] and [48], three different length scales are recognized: (i) macroscale (grains aggregate), (ii) mesoscale (each grain), and (iii) microscale crystallographic slip systems (css).

Assumed to be initially isotropic, the polycrystal is usually viewed as an aggregate of numerous grains (single-phase or polyphase) having different orientations with respect to the loading axis. For polycrystalline metals, the configuration of random crystal distribution is used. Emphasis is placed on single-phase polycrystals where the mechanical properties of each grain are considered to be identical with respect to the crystallographic reference frame. At the mesoscale, the heterogeneity comes from the differences in the orientation of the grains. The morphology and spatial distribution of the grains are not taken into account.

The mechanical state for each length scale can be mathematically described with the aid of the following internal state variables:

  • At the macroscale: there are no state variables at this level. The second-order overall stress tensor Σ¯¯ (overall strain tensor E¯¯) is the simple average of the second-order granular stress tensor σ¯¯ (granular strain tensor ε¯¯).

  • At the mesoscale: the second-order granular elastic tensor ε¯¯e is introduced as internal state variable and its dual thermodynamic associated force variable is σ¯¯.

  • At the microscale: a couple of internal state variables (qs, Rs) is used as descriptors of the intragranular isotropic hardening for each css; with the isotropic hardening variable qs, the dual scalar variable Rs is the thermodynamic associated force. It is assumed that this hardening is affected by interactions among the slip systems within the same single crystal.

For several metals, two sources of kinematic hardening are experimentally observed at two length scales. The first one comes from the plastic strain incompatibility at the grain boundary level (intergranular). This leads to nonuniform distribution of stresses and strains at the mesoscale. The second source (intragranular) represents the long-range interactions at the css. However, in the present modeling, the global effects of these hardening sources can be implicitly described by the interaction law via the inelasticity-hardening coupling parameter (Θ).

With the octahedral slips, the inelastic deformation of single crystal is defined.

Grains within the aggregate are considered to have the same elastic coefficients. The overall elastic strain tensor E¯¯e together with the overall inelastic strain tensor E¯¯in is determined through the micro–macro methodology.

Throughout this paper, the index s {1,2,….,n} is associated to the system rank, with n being the maximum number of octahedral systems in the grain (n = 12 for fcc and n = 24 for bcc). Similarly, the index g {1, 2,…., Ng} describes the grain rank, with Ng being the maximum number of grains contained in the grains aggregate.

Single Crystal Behavior Modeling.

Since the granular elastic part is assumed to be uniform, isotropic, and compressible, its free energy per unit volume is written as a classical isotropic quadratic function of the granular elastic strain tensor ε¯¯e. Thermodynamically, its associated variable is the granular stress tensor σ¯¯. Thus, the elastic granular specific free energy ψeg is expressed as Display Formula

(1)ρψeg(ε¯¯e)=12λ(trε¯¯e)2+μtr(ε¯¯e)2

where ρ is the granular material density, and λ and μ are the classical Lame's constants of the grain.

The associated stress variable σ¯¯ is given by Display Formula

(2)σ¯¯=ρψegε¯¯e=2με¯¯e+λ(trε¯¯e)I¯¯

where I¯¯ is the second-order identity tensor.

Under isothermal condition, the granular coefficients λ and μ remain always constants. With the time derivative of Eq. (2), the granular elastic strain rate is therefore defined by Display Formula

(3)ε˙¯¯e=σ˙¯¯2μλ2μ(2μ+3λ)tr(σ˙¯¯)I¯¯

The granular inelastic part of the state potential ρψing, which is assumed to be exclusively dependent on the intragranular isotropic hardening, is expressed as quadratic function of the internal variable qs by [49] Display Formula

(4)ρψing=12r=1ns=1nHrsQqrqs

where Q is the intragranular isotropic hardening modulus considered to be the same for all slip systems. The hardening interaction matrix (Hrs) is supposed to describe dislocation–dislocation interaction allowing the introduction of the cross effect of the slip of the system s on the hardening of the system r. Using the state law, the dual variable Rs is derived from Eq. (4) given by Display Formula

(5)Rs=ρψingqs=Qr=1nHrsqr

This hardening tends to saturate on each slip system. It depends on the amount of the slip on the same system (self-hardening) as well as on the other systems of the same grain (cross hardening).

Neglecting the grain size effect, the yield surface Schmid type of microplasticity for each css is defined in terms of the resolved shear stress τs together with the hardening expressed by Display Formula

(6)fs=|τs|Rsτos

where τos is the initial value of the critical resolved shear stress (friction stress).

The local inelastic flow determination is made when the absolute value of the resolved shear stress τs is greater than the new actual flow surface radius, i.e., |τs| ≥  Rs+τos. The slip rate can be determined as long as the stress and the hardening variables are known. Each resolved shear stress τs is computed by projecting the granular stress σ¯¯ on each slip system using Display Formula

(7)τs=σ¯¯:m¯¯s
where m¯¯s is the orientation tensor and “:” is the twice-contracted tensorial product.

The rates of change of the inelastic strain and intragranular isotropic hardening are obtained by introducing the yield function fs together with dissipation potential Fs adopting the nonassociated plasticity concept. Fs is thus expressed by [49] Display Formula

(8)Fs=fs+bsqsRs

where bs is a material constant characterizing the nonlinearity of the intragranular isotropic hardening assumed to be the same for all slip systems.

The evolution equations of the granular inelastic strain and the isotropic hardening variable are deduced from the generalized normality rule Display Formula

(9)ε˙¯¯in=s=1nλ˙sFsσ¯¯=s=1nλ˙ssign(τs)m¯¯s=s=1nγ˙sm¯¯s

with Display Formula

(10)γ˙s=λ˙ssign(τs)

where γ˙s is the slip rate on the css.

The orientation tensor m¯¯s is defined as Display Formula

(11)m¯¯s=12(lsns+nsls)

where ls and ns are the slip direction and slip plane normal unit vectors, respectively.

After determining the associated force Rs (Eq. (5)), the rate equation of q˙s is then derived as follows: Display Formula

(12)q˙s=r=1nλ˙rFrRs=λ˙s(1bsqs)

In the framework of viscoplasticity, the value of pseudomultiplier λ˙s occurring along closed-packed crystallographic slip planes and directions is a power function of the distance to the yield point defined by the local yield criterion fs (without grain size effect) Display Formula

(13)λ˙s=fsKz=|τs|RsτosKz

where K and z are material constants describing the local viscous effect of the material.

Since the rate-independent models do not possess the uniqueness in the numerical applications, the rate-dependent slip is thus adopted to resolve such numerical difficulties used previously in several research programs (e.g., Refs. [4850]). Although, the developed model is a rate-dependent type, the rate-independent case can be practically obtained by choosing a high value of viscous exponent z and a low value of the coefficient K, leading to low and constant viscous stress σvs described by σvs=K|γ˙s|1/z.

Interaction Law.

A generalization of grain/matrix interaction law for fully anisotropic and compressible elastoviscoplastic behavior with small strain hypothesis has been developed in Ref. [51] assuming that the granular and macroscopic behaviors follow Maxwell elastic–viscoplastic relation. They proposed an approximate solution keeping the same structure of the incompressible interaction law developed in Ref. [52], i.e., with elastic and viscoplastic parts. The generalized elastic–viscoplastic interaction law is expressed by Display Formula

(14)(J¯¯¯¯s-1+C¯¯¯¯):-1(σ˙¯¯Σ˙¯¯)+(R¯¯¯¯s-1+A¯¯¯¯):-1(s¯¯S¯¯)=(ε˙¯¯E˙¯¯)
where J¯¯¯¯s and R¯¯¯¯s are the fourth-order symmetrical Eshelby's tensor and inelastic interaction tensor, respectively. Their definition is limited to spherical inclusions. C¯¯¯¯ is the fourth-order global stiffness tensor. A¯¯¯¯ is the fourth-order macroscopic tangent modulus controlled by the overall inelastic strain rate E˙¯¯in. s¯¯ and S¯¯ are the granular and macroscopic deviatoric Cauchy stress tensors, respectively. ε˙¯¯ and E˙¯¯ are, respectively, the total granular and overall strain rates. J¯¯¯¯s and R¯¯¯¯s can be computed using A¯¯¯¯ and C¯¯¯¯ using Green function and integral method, respectively.

In the case where the elastic response dominates, the viscoplastic term (R¯¯¯¯s-1+A¯¯¯¯)-1 becomes negligible; therefore, the interaction law (Eq. (14)) can be written as [47] Display Formula

(15)(J¯¯¯¯s-1+C¯¯¯¯):-1(σ˙¯¯Σ˙¯¯)=(ε˙¯¯eE˙¯¯e)

Based on the solution of the general interaction law of elasticity for heterogeneous media using Green function method, it is demonstrated that the solution for a spherical inclusion embedded in an infinite homogeneous matrix having elastic properties defined by λ and μ, the fourth-order interaction tensor is determined in Ref. [53]. Therefore, it can be defined by executing the following steps:

First, we have Display Formula

(16)(J¯¯¯¯s-1+C¯¯¯¯)=-1N¯¯¯¯1

For a spherical inclusion embedded in a finite homogenous isotropic matrix, J¯¯¯¯s is defined as Display Formula

(17)Jsijkl=115μ(3λ+6μ)[(3λ+3μ)δijδkl3(3λ+8μ)Iijkl]

The isotropic macroscopic stiffness tensor is classically expressed as Display Formula

(18)Cijkl=2μIijkl+λδijδkl

Therefore, one can obtain the following relation: Display Formula

(19)Nijkl1=2AIijkl+Bδijδkl

where the constants A and B are defined by Lame's coefficients μ and λ as follows: Display Formula

(20)A=(8μ+3λ)2μ(14μ+9λ)
Display Formula
(21)B=(6μ+λ)(3λ+8μ)μ(448μ2+456μλ+108λ2)

As a result, the elastic part of the interaction law can be written as Display Formula

(22)2A(σ˙¯¯Σ˙¯¯)+Btr(σ˙¯¯Σ˙¯¯)1¯=(ε˙¯¯eE˙¯¯e)

For a fully viscoplastic behavior dominating at stationary state (in the long range response), the term (σ˙¯¯Σ˙¯¯) is practically vanished. Therefore, Eq. (14) can be approximately written as [47] Display Formula

(23)(R¯¯¯¯s-1+A¯¯¯¯):-1(s¯¯S¯¯)=(ε˙¯¯inE˙¯¯in)

This equation represents a self-consistent approach developed in Ref. [54] describing the viscoplastic behavior of polycrystals under large deformations condition.

In the case of viscoplastic behavior for spherical inclusion where the matrix is assumed isotropic and incompressible, the tangent modulus is approximated by Display Formula

(24)Aijkl=2ηoIijkl

and the interaction tensor R¯¯¯¯s is given by Display Formula

(25)Rsijkl=15ηoIijkl

ηo is the scalar macroscopic viscous tangent modulus dependent on the deformation history.

The viscoplastic part of the general interaction law (Eq. (14)) takes therefore the following form: Display Formula

(26)13ηo(s¯¯S¯¯)=(ε˙¯¯inE˙¯¯in)

Consequently, the modified interaction law of the elastoviscoplastic behavior of polycrystal (Eq. (14)) in the case of spherical inclusion is equal to the sum of the two approximated parts (elastic: Eq. (22) and viscoplastic: Eq. (26)). Thus, one obtains Display Formula

(27)2A(σ˙¯¯gΣ˙¯¯)+Btr(σ˙¯¯Σ˙¯¯)1¯13ηo(s¯¯S¯¯)=(ε˙¯¯E˙¯¯)

The same problem of ηο adjustment is encountered in order to satisfy the self-consistency conditions especially under multiaxial loading paths [47]. Hence, the term (1/3ηo) is replaced with a positive phenomenological parameter (Θ > 0) responsible for inelasticity-hardening coupling and the modified interaction law can be written as Display Formula

(28)2A(σ˙¯¯Σ˙¯¯)+Btr(σ˙¯¯Σ˙¯¯)1¯Θ(s¯¯gS¯¯)=(ε˙¯¯E˙¯¯)

Θ can implicitly and naturally describe the global kinematic hardening effects.

Grain Size Modeling.

As a basic concept to start this modeling, the relationship between dislocation density and crystal plasticity of ufg materials is adopted. Then, a uniform distribution of dislocations is considered [44,55,56]. The evolution of average dislocation density ρ versus slip γ is given by Display Formula

(29)dρdγ+(ka+kb)ρ=kfρ

where kf is a constant accounting for dislocation storage; ka is a coefficient of dislocation annihilation by slip mechanisms determining the dynamic restoration of the intensity in a deformed metal; kb=B/d2 is the dislocation annihilation coefficient occurring at the grain boundaries, d being grain size and B is a constant related to the thermal activation mechanism expressed by Display Formula

(30)B=4ηDgbmε˙

where η is the thermal activation coefficient, and Dgb denotes the diffusion coefficient at the grain boundaries. Its value at room temperature varies between 10−19 and 10−22 m2/s. Note that for fcc metals, B values are close to each other.

The thermal activation coefficient η is defined by Display Formula

(31)η=μb3kBθ
where b is the Burgers vector, b3 is the activation volume, kB is the Boltzmann constant, θ is the temperature, and μ is the shear modulus.

These coefficients can be directly obtained from experimental data describing the behavior of metals. The sum of ka and kb defines the global loss of dislocations density due to dislocation annihilation.

To solve the nonlinear differential equation (29), it is considered as a Bernoulli type. So, its solution for any r varying between 0 and 1 is expressed by u=ρ1r=ρ112=ρ12 and dρ/dγ=2u(du/dγ). This gives a first-order differential equation for which the exact solution is stated as [44] Display Formula

(32)u=ρ=βobdexp(ka+Bd22γ)+kfka+Bd2[1exp(ka+Bd22γ)]

In addition, some quantities require to be defined Display Formula

(33)τ=αμbρ
Display Formula
(34)σ=mτ
Display Formula
(35)u=ταμb

where α is a dimensionless parameter. Its values range from 0.05 to 0.26 for different materials [57].

The most condensed form can be written as follows: Display Formula

(36)σy=σo+Ddexp(A+Bd22mεy)+CA+Bd2[1exp(A+Bd22mεy)]

where d is the average grain size, σo is the flow stress, and m is the Taylor factor, and εy is a macroscopic strain offset of 0.002.

It is worth emphasizing that Eq. (36) presented at the macroscale is derived from Eq. (29) based on average values (average dislocation density, slip, etc.) across the grain. Several mechanisms such as accumulation and annihilation of dislocations and thermal diffusion, which depend on grain size, lead to build local barriers across the grain. By analogy, it is assumed for the sake of simplicity that the same concept in Eq. (36) can be reformulated at the microscale. Note that a new reference granular shear strain (γog) is proposed as material parameter replacing (y) in Eq. (36). Therefore, the grain size effect presented at the css level is expressed by Display Formula

(37)τts=τos+Ddexp(A+Bd22γog)+CA+Bd2[1exp(A+Bd22γog)]

where τts is the initial flow stress and τos represents the flow stress for an infinite grain size. The above equation is a kind of generalized Hall–Petch law because under certain conditions, the classical Hall–Petch law can be simply retrieved as given below.

The second and the third terms (noted henceforth χGB) in the right-hand side of Eq. (37) describe the grain size effect. These two terms are based on the concept that smaller the grain size corresponds to grater the importance of the grain boundary phase where the deformation is not dominated by the dislocation glide mechanism within the grain Display Formula

(38)χGB=Ddexp(A+Bd22γog)+CA+Bd2[1exp(A+Bd22γog)]

The constants A, C, and D have the respective expressions Display Formula

(39)A=ka
Display Formula
(40)C=αμbmkf
Display Formula
(41)D=αμmβob

The local yield surface (Eq. (6)) is now affected by the grain size defined by Display Formula

(42)τts=τos+χGB

It is worth emphasizing that in the case of coarse grains this leads to Display Formula

(43)exp(A+Bd22γog)1

One can obtain Display Formula

(44)χGBDd

Therefore, the form of classical Hall–Petch relation τts=τos+D/d is retrieved.

Therefore, the new local yield surface affected by the grain size is written as Display Formula

(45)fs=|τs|Rsτts

Polycrystal Constitutive Relations.

It is well known that the overall properties are a function of grain properties. Thus, each grain is assumed to be embedded in a homogenous equivalent medium exhibiting a response equal to the average response of all grains. In this context, the overall elastic and inelastic strain tensors are obtainable through the micro–macro methodology, which means that there are no state variables at this level. Hence, the macroscopic Cauchy stress tensor Σ¯¯ is also deduced from the same concept. In fact, after determining the granular inelastic strain rate as the sum of the contributions from all activated slip systems, the transition from the single to polycrystal response is performed by following the well-known averaging procedures. For homogenous elastic media, the overall stress Σ¯¯ (the overall strain E˙¯¯e) is simply the volume average of the granular stresses σ¯¯ (granular strains ε¯¯e) [58,59]. For the overall inelastic strain, the averaging procedure is usually not straightforward since it involves localization tensors [60]. In the special case of a single-phase polycrystal with isotropic elasticity, as in the actual case of elasto-inelastic behavior with homogeneous elasticity, the overall elastic and inelastic strain rates (E˙¯¯e and E˙¯¯in) are equal to the average of granular rates (i.e., ε˙¯¯e and ε˙¯¯in) [60]. The rates of change of overall elastic and inelastic strains are therefore computed, respectively, as follows: Display Formula

(46)E˙¯¯e=g=1Ngvgε˙¯¯e
Display Formula
(47)E˙¯¯in=g=1Ngvgε˙¯¯in

where vg denotes the volume fraction of co-oriented grains. ε˙¯¯e and ε˙¯¯in are determined by Eqs. (3) and (9), respectively.

Since the finite strains are involved, the total overall strain rate (E˙¯¯) can be decomposed as follows: Display Formula

(48)E˙¯¯=E˙¯¯e+E˙¯¯in

In this paragraph, the abilities of the model in describing the grains size effect on the elasto-inelastic behavior of polycrystals for spherical grains are demonstrated. The identification procedure for determining different material coefficients is carried out by relying on the prepared experimental databases. Two distinct metals, namely nickel (experimental results provided via ANR-mimic, 2008–2012) and copper [61] are studied.

Choice of the Microstructure.

The impact of the grain size on different length scales behavior is predicted under uniaxial quasi-static load of strain rate of 10−4/s using a random single-phase fcc crystal distribution of 400 grains. This grains aggregate (number and orientation of grains) represents an acceptable compromise between minimizing the calculation time (CPU) and describing correctly the polycrystalline microstructure [48]. With a single-valued grain size, the orientation of each grain is defined by the Euler angles shown in the inverse-pole figure (Fig. 1). Besides, the distribution is suitably covered almost all space directions.

As experimental data, three average grain diameters of 0.049, 0.11, and 20 μm for copper [61] are employed here. It is worth noting that Sanders et al. [61] have been observing the impact of the grain size on Young's modulus. However, the independence of the model elastic part on the grain size is adopted. Its standard value is therefore used (Table 1). For nickel, two different average grain size diameters of 0.22 and 3.79 μm are utilized. Their overall mechanical behavior has been studied by means of uniaxial compression tests conducted at room temperature and strain rate of 10−4/s.

As a forthcoming work, other loading configurations for different ufg and nc metallic materials will be predicted showing the model robustness.

Determination of the Model Constants.

Using these experimental data, the model parameters are calibrated for the two metals. The model constitutive equations are programmed into a computer code using the well-known second-order Runge–Kutta algorithm with adaptive step size. It is a single-step method directly derived from the Euler method which has the advantage of being easy to implement and stable for those functions used in physics. With a nonstiff problem, this algorithm consists of improving the accuracy of the solution y(t). Its principle basis is the same as for the Euler method, whereas the difference is based on the introduction of additional correction terms to the iterative relation for reducing rounding errors. This approach is largely used for solving ordinary differential equations systems.

The model constitutive equations are presented as: (1) intragranular isotropic hardening variables qs; (2) viscoplastic pseudomultiplier λs for each css; (3) granular stress tensor σ¯¯ obtained via the interaction law; (4) granular elastic strain tensor ε¯¯e ; (5) granular inelastic strain tensor ε¯¯in ; and (6) overall total strain tensor E¯¯.

Y˙¯ is considered as a vector defined by

Y˙¯=E˙¯¯,σ˙¯¯,ε˙¯¯e,ε˙¯¯in,λ˙s,q˙s

Since an fcc polycrystalline structure is considered in this work, the number of scalar equations to be solved is Neq = 13 + 42 × Ng. The first number corresponds to 6 × 2 + 1 = 13 (components of elastic E¯¯e, inelastic E¯¯in tensors and overall accumulated inelastic strain), Ng × 6 × 3 (σ˙¯¯, ε˙¯¯eandε˙¯¯in granular variables), as well as Ng × 12 × 2 (λ˙sandq˙s variables). The initial conditions are effectively assumed to be totally known (Y¯tn=t0=Y¯0).

It is considered therefore that at the instant tn+1 = tn + Δt, an increment of total overall strain ΔE is applied to the grains aggregate. It is assumed that Y˙¯tn=E˙¯¯,σ˙¯¯,ε˙¯¯e,ε˙¯¯in,λ˙s,q˙s is known at the instant tn. The following variables and measurements ΔY¯=ΔE¯¯,Δσ¯¯,Δε¯¯e,Δε¯¯in,Δλs,Δqs have to be defined in accurate space increments. From the algorithm, the numerical solution Y¯tn+1 for each time increment is expressed by

Y¯tn+1=Y¯tn+ΔY¯

Figure 2 shows the flowchart for the model programming.

Young's modulus (E) and Poisson's ratio (ν) are directly determined via the experimental data of the used metals. For the sake of simplicity, these overall elasticity constants are equal to these of granular ones due to the isotropic elasticity. Note that the elasticity is assumed to be independent of grain size. Thus, the Lame's coefficients (λ and μ) are classically computed based on the overall Young's modulus and the Poisson's ratio using the classical elasticity relations λ = νΕ / (1 + ν)(1 − 2ν) and μ = Ε / 2(1 + ν). In addition, the model constants ka, βo, kf, μ, b, m, and kB are defined using physical standard values summarized in Table 1. The strain rate (ε˙) is equal to10−4/s and the value of reference granular shear strain γog  = 0.006 for the two metals.

As discussed above, the plastic behavior can be described by choosing a relatively high value of viscous exponent z and low value of the coefficient K, i.e., minimizing the viscosity effect. The parameter Θ, the initial yield stress of the slip system τos, and the intragranular isotropic hardening coefficients (Q and bs) should also be calibrated. In addition, six coefficients of the Hrs matrix (h1, h2,…., h6) also require identification. In fact, the diagonal term (h1) is always equal to one. For simplification, it is considered that h2 = h3 = h4 = h5 = h1 = 1.0, whereas, h6 requires calibration. To begin the identification procedure, the initial values of the model parameters are chosen. In order to achieve this operation satisfactorily, the following steps are applied:

  1. (1)Identification of the elastoplastic behavior coefficients without considering the grain size effect can be made using therefore the highest grain size for each employed metal. In fact, this step is achieved based on the experimental data with d = 20 μm and 3.79 μm for copper and nickel, respectively. It is important to note that C is postulated as a constant. Hence, the coefficients τos, Q, bs, h6, Θ, z, K, h6, Dgb, α, and C should be numerically identified.
  2. (2)For the grain refinement coefficients (Dgb and α) calibration, this procedure is made by fixing all the coefficients identified in the previous step. After several numerical iterations, such parameters are then determined. The optimized material coefficients are finally calibrated by having the best fit between prediction and experimental data. The identified parameters for the two materials are summed up in Table 2Table 2

    Identified parameters of the micromechanical model for the two employed metals

    ΘτOS (MPa)zKQbsh6DgbαCCopper1.58 × 10−61.4310.27955,22249001.041.45 × 10−200.05120.02Nickel5 × 10−815.628.555.396.60.013.043.87 × 10−210.2520.02.

Parametric Study.

The new model parameters (Dgb and α) are numerically studied demonstrating their impact on the flow stress evolution for 500μmd3nm. As a result, four values of Dgb (10−22, 10−21, 10−20 and 10−19 m2/s) and four ones of α (0.05, 0.12, 0.2, and 0.26) are tested. The selected values are based on their ranges given in Refs. [3] and [28] for Dgb and in Ref. [57] for α.Other already calibrated parameters (Tables 1 and 2) of copper are employed.

The effect of Dgb is first studied by fixing α = 0.12 (Fig. 3). Then, α is varied for Dgb = 10−20 (Fig. 4). Interactions between these two parameters are observed leading consequently to a significant flow stress evolution due to grain size. Note that the flow stress is defined using a macroscopic strain offset of 0.2%. Figure 3 reveals that the greater the value of Dgb, the lower becomes the overall flow stress and the greater is the variation of the Hall–Petch plot. The reverse Hall–Petch does not capture at Dgb = 10−22, whereas it is clearly observed at Dgb = 10−20 and 10−19, for d5nm and d20nm, respectively. It is recognized that the critical value of grain size is influenced by Dgb (Fig. 3). Besides, when d50nm, Hall–Petch slope is not affected by Dgb. In the case of copper, Dgb should be 10−20 with α = 0.12 to describe suitably the Hall–Petch plot.

The effect of αon the flow stress is numerically tested. So, Fig. 4 illustrates the yield stress evolution for different values of α using Dgb = 10−20. It is concluded that the greater the value of α, the greater is the Hall–Petch slopes (positive and negative) and the greater is the variation of the yield strength. In addition, whatever the value of α, the same critical grain size (d = 5 nm) is recorded.

As a conclusion, these two coefficients (Dgb and α) and their interactions can describe their impact on the flow stress evolution.

Quantitative Study.

After parameters calibration, comparisons between the predictions and the experimental data at the macroscale are shown in Figs. 5 and 6 for the copper and nickel, respectively. The principal features of grain size effect on the elasto-inelastic behaviors for these metals are predicted showing a good agreement with the experimental results.

The impact of the copper grain size for d = 0.049, 0.11, and 20 μm on the mechanical strength is investigated at the different length scales. Hence, the granular distributions of axial elastic and inelastic strains and axial stress are recorded at the end of loading. An examination of Fig. 7 highlights the fact that the granular behavior has a heterogeneous nature whatever the grain size. This reflects the aptitude of this approach in describing the heterogeneity. The parameter Θ (=1.58 × 10−6) leads to an instantaneous elastic effect and viscoplastic relaxation at the steady-state. If these two conditions have been respected, the overall kinematic and isotropic hardenings will be successfully reproduced [47]. Hence, the term Θ(s¯¯gS¯¯) gives certain equilibrium between the first two terms in the left-hand side of the interaction law (28) leading therefore to relatively smooth interactions between grains and their matrix for a given grain size. The heterogeneity of the granular elastic strain (Fig. 7(a)) dictated by the interaction law increases with the grain size decrease. Figure 7(b) shows also the role of grain size on the granular inelastic strain heterogeneity. In fact, this heterogeneity is inversely proportional with respect to the grain size. Figure 7(c) reveals a significant heterogeneity of granular axial stress distribution and its dependence on the grain size. In fact, the greater the value of the grain size, the lower is the granular stress heterogeneity.

To describe the predicted the heterogeneities at the microscale and their dependence on the grain size, the slip (λs), the intragranular isotropic hardening (Rs), and shear stress (τs) evolutions are studied. A given slip system (No. 11 defining by (11¯1) × [011]) is beforehand selected for the grain no. 80, which is favorably oriented with respect to the external loading. Figure 8(a) shows that for d = 20 μm, the slip began to be active first, and its rate of change is almost linear; whereas for other grain sizes, the slip evolves in nonlinear manner. This nonlinearity is inversely proportional to the grain size.

Concerning Rs, its evolution versus slip is nonlinear whatever the grain size (Fig. 8(b)). When the grain size decreases, the evolution demonstrates a plateau especially at the beginning of straining. This plateau decreases with the grain size increasing and it vanishes when d = 20 μm. One can conclude that the greater the grain size, the greater becomes the isotropic hardening. This is due to the cross effect of the slip on the hardening of other slips.

τs evolves differently during deformation for these three grain sizes (Fig. 8(c)). Whatever the grain size, its negative evolution is related to the selected slip system particularly its orientation with respect to external loading. As in Rs case, a plateau is clearly observed especially for d = 110 and 49 nm. The nonlinearity becomes evident and its value is totally conditioned by the grain size, i.e., the lower the grain size, the greater is the shear stress (τs). This is mainly induced by the yield surface evolution (Eq. (45)).

A wide range of grain sizes (500μmd3nm) is numerically tested showing the model ability in describing its effect on the mechanical strength. The prediction of copper grain size effect on the flow stress is demonstrated in Fig. 9. In fact, the yield stress (Σy) increases proportionally with decreasing the inverse square root of grain size for conventional, ufg and nc materials following the Hall–Petch plot. However, as the grain size further decreases below 10 nm, the yield stress starts to decrease leading to the reverse Hall–Petch plot. The plastic behavior in the reverse Hall–Petch represents up to now a debated question due to a softening mechanism related to processes taking place in the grain boundaries of nc materials (e.g., Refs. [11,44,45], and [6163]). Moreover, the experimental yield stresses for the three grain sizes [61] are integrated in this figure as a comparison with the plotted prediction. To demonstrate the role of χGB (Eq. (38)), its evolution has the same trend as for the yield stress but with different slopes regarding the positive and the negative Hall–Petch relation (Fig. 9).

A new extension of the micromechanical approach already developed [47] is proposed to predict the grains size effect on the enhancement of the mechanical strength of metallic polycrystals. In fact, the dislocation kinematics concept is adopted for characterizing the ufg and the nc materials, where grain boundary is taken into account by a thermal diffusion concept. Thus, this model describes the inelastic behavior of ufg and nc metals whatever the grain size. Within the framework of small strain hypothesis, the impact of the grains size on the overall mechanical behavior of polycrystals is faithfully described by the model. The model highlights the fact that the meso and microscale heterogeneous behaviors are suitably described. Predictions point out the fundamental influence of the grain size on the response of the model length scales.

It is recognized that the two coefficients related to grain size (Dgb and α) and their interactions show a significant impact on the yield stress evolution for a given temperature and strain rate.

Comparisons between copper and nickel experimental results and the model predictions reveal the model capability to appropriately reproduce the behavior of these metals and their sensitivity to the grain size.

As important issues, the effects of strain rate and temperature on Hall–Petch relationship will be considered in a future work.

The authors are grateful to The French National Research Agency, ANR (ANR 09-BLAN-0010-01) for supporting this work.

Gleiter, H. , 1982, “ On the Microstructure of Grain Boundaries in Metals,” Mater. Sci. Eng., 52(2), pp. 91–131. [CrossRef]
Kumar, K. S. , Van Swygenhon, H. , and Suresh, S. , 2003, “ Mechanical Behavior of Nanocrystalline Metals and Alloys,” Acta Mater., 51(19), pp. 5743–5774. [CrossRef]
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Hall, E. O. , 1951, “ Macroscopic Aspect of Lüders Band Deformation in Mild Steel,” Proc. R. Soc. London B, 64(1), p. 474.
Petch, N. J. , 1954, “ Fracture of Metals,” Prog. Met. Phys., 5(1), pp. 1–52. [CrossRef]
Tjong, S. C. , and Chen, H. , 2004, “ Nanocrystalline Materials and Coatings,” Mater. Sci. Eng.: R, 45(1–2), pp. 1–88. [CrossRef]
Swygenhoven, H. , Derlet, P. M. , and Hasnaoui, A. , 2002, “ Atomic Mechanism for Dislocation Emission From Nano-Sized Grain Boundaries,” Phys. Rev. B, 66(2), p. 024101. [CrossRef]
Swygenhoven, H. , Spaczer, M. , Caro, A. , and Farkas, D. , 1999, “ Competing Plastic Deformation Mechanisms in Nanophase Metals,” Phys. Rev. B, 60(1), p. 22. [CrossRef]
Chokshi, A. H. , Rosen, A. , Karch, J. , and Gleiter, H. , 1989, “ On the Validity of the Hall-Petch Relationship in Nanocrystalline Materials,” Scr. Metall., 23(10), pp. 1679–1683. [CrossRef]
Lu, K. , Wei, W. D. , and Wang, J. T. , 1990, “ Microhardness and Fracture Properties of Nanocrystalline Ni-P Alloy,” Scr. Metall. Mater., 24(12), pp. 2319–2323. [CrossRef]
Hahn, H. , and Padmanabhan, K. A. , 1997, “ A Model for the Deformation of Nanocrystalline Materials,” Philos. Mag. B, 76(4), pp. 559–571. [CrossRef]
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Champion, Y. , 2013, “ Computing Regimes of Rate Dependent Plastic Flow in Ultrafine Grained Metals,” Mater. Sci. Eng., 560, pp. 315–320. [CrossRef]
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Aoyagi, Y. , and Shizawa, K. , 2007, “ Multiscale Crystal Plasticity Modeling Based on Geometrically Necessary Crystal Defects and Simulation on Fine-Graining for Polycrystal,” Int. J. Plast., 23(6), pp. 1022–1040. [CrossRef]
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References

Gleiter, H. , 1982, “ On the Microstructure of Grain Boundaries in Metals,” Mater. Sci. Eng., 52(2), pp. 91–131. [CrossRef]
Kumar, K. S. , Van Swygenhon, H. , and Suresh, S. , 2003, “ Mechanical Behavior of Nanocrystalline Metals and Alloys,” Acta Mater., 51(19), pp. 5743–5774. [CrossRef]
Meyers, M. A. , Mishra, A. , and Benson, D. J. , 2006, “ Mechanical Properties of Nanocrystalline Materials,” Prog. Mater. Sci., 51(4), pp. 427–556. [CrossRef]
Cheng, S. , Spencer, J. A. , and Milligan, W. W. , 2003, “ Strength and Tension/Compression Asymmetry in Nanostructured and Ultrafine-Grain Metals,” Acta Mater., 51(15), pp. 4505–4518. [CrossRef]
Conrad, H. , 2003, “ Grain Size Dependence of Plastic Deformation Kinetics in Copper,” Mater. Sci. Eng., 341(1–2), pp. 216–228. [CrossRef]
Hall, E. O. , 1951, “ Macroscopic Aspect of Lüders Band Deformation in Mild Steel,” Proc. R. Soc. London B, 64(1), p. 474.
Petch, N. J. , 1954, “ Fracture of Metals,” Prog. Met. Phys., 5(1), pp. 1–52. [CrossRef]
Tjong, S. C. , and Chen, H. , 2004, “ Nanocrystalline Materials and Coatings,” Mater. Sci. Eng.: R, 45(1–2), pp. 1–88. [CrossRef]
Swygenhoven, H. , Derlet, P. M. , and Hasnaoui, A. , 2002, “ Atomic Mechanism for Dislocation Emission From Nano-Sized Grain Boundaries,” Phys. Rev. B, 66(2), p. 024101. [CrossRef]
Swygenhoven, H. , Spaczer, M. , Caro, A. , and Farkas, D. , 1999, “ Competing Plastic Deformation Mechanisms in Nanophase Metals,” Phys. Rev. B, 60(1), p. 22. [CrossRef]
Chokshi, A. H. , Rosen, A. , Karch, J. , and Gleiter, H. , 1989, “ On the Validity of the Hall-Petch Relationship in Nanocrystalline Materials,” Scr. Metall., 23(10), pp. 1679–1683. [CrossRef]
Lu, K. , Wei, W. D. , and Wang, J. T. , 1990, “ Microhardness and Fracture Properties of Nanocrystalline Ni-P Alloy,” Scr. Metall. Mater., 24(12), pp. 2319–2323. [CrossRef]
Hahn, H. , and Padmanabhan, K. A. , 1997, “ A Model for the Deformation of Nanocrystalline Materials,” Philos. Mag. B, 76(4), pp. 559–571. [CrossRef]
Nieh, T. G. , and Wadsworth, J. , 1991, “ Hall-Petch Relation in Nanocrystalline Solids,” Scr. Metall. Mater., 25(4), pp. 955–958. [CrossRef]
Lian, J. , Baudelet, B. , and Nazarov, A. A. , 1993, “ Model for the Prediction of the Mechanical Behavior of Nanocrystalline Materials,” Mater. Sci. Eng.: A, 172(1–2), pp. 23–29. [CrossRef]
Wang, N. , Wang, Z. , Aust, K. T. , and Erb, U. , 1995, “ Effect of Grain Size on the Mechanical Properties on Nanocrystalline Materials,” Acta Metall. Mater., 43(2), pp. 519–528. [CrossRef]
Cao, W. Q. , Dirras, G. , Benyoucef, M. , and Bacroix, B. , 2007, “ Room Temperature Deformation Mechanisms in Ultrafine-Grained Materials Processed by Hot Isostatic Pressing,” Mater. Sci. Eng., 462(1–2), pp. 100–105. [CrossRef]
Wang, Y. M. , Hamza, A. V. , and Ma, E. , 2006, “ Temperature-Dependent Strain Rate Sensitivity and Activation Volume of Nanocrystalline Ni,” Acta Mater., 54(10), pp. 2715–2726. [CrossRef]
Champion, Y. , 2013, “ Computing Regimes of Rate Dependent Plastic Flow in Ultrafine Grained Metals,” Mater. Sci. Eng., 560, pp. 315–320. [CrossRef]
Acharya, A. , and Beaudoin, A. J. , 2000, “ Grain-Size Effect in Viscoplastic Polycrystals at Moderate Strains,” J. Mech. Phys. Solids, 48(10), pp. 2213–2230. [CrossRef]
Aoyagi, Y. , and Shizawa, K. , 2007, “ Multiscale Crystal Plasticity Modeling Based on Geometrically Necessary Crystal Defects and Simulation on Fine-Graining for Polycrystal,” Int. J. Plast., 23(6), pp. 1022–1040. [CrossRef]
Zhu, B. , Asaro, R. , and Krysl, P. , 2006, “ Effects of Grain Size Distribution on the Mechanical Response of Nanocrystalline Metals: Part II,” Acta Mater., 54(12), pp. 3307–3320. [CrossRef]
Weng, G. J. , 1983, “ A Micromechanical Theory of Grain-Size Dependence in Metal Plasticity,” J. Mech. Phys. Solids, 31(3), pp. 193–203. [CrossRef]
Jiang, B. , and Weng, G. J. , 2004, “ A Generalized Self Consistent Polycrystal Model for the Yield Strength of Nanocrystalline Materials,” J. Mech. Phys. Solids, 52(5), pp. 1125–1149. [CrossRef]
Abdul-Latif, A. , Dirras, G. F. , Ramtani, S. , and Hocini, A. , 2009, “ A New Concept for Producing Ultrafine Grained Metallic Structures Via an Intermediate Strain Rate: Experiments and Modeling,” Int. J. Mech. Sci., 51(11–12), pp. 797–806. [CrossRef]
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Figures

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Fig. 1

Inverse-pole figure for the used 400 grains aggregate showing the initial grains distribution

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Fig. 2

Programming flowchart of the model

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Fig. 3

Influence of the diffusion coefficient (Dgb) on the copper flow stress for 500 μm≥d≥3 nm

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Fig. 4

Influence of the model parameter (α) on the copper flow stress for 500 μm≥d≥3 nm

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Fig. 5

Comparison between the overall predicted response and the experiment result for the copper showing the grain size impact on the tensile stress–strain behavior

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Fig. 6

Comparison between the experimental results and the model predictions of the compressive stress–strain for nickel

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Fig. 7

Axial granular responses at the end of loading: (a) elastic, (b) inelastic, and (c) stress showing the impact of the grain size on their evolutions

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Fig. 8

Predicted evolution of the: (a) slip, (b) intragranular isotropic hardening, and (c) resolved shear stress, for a given activated slip system using the three different grain sizes

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Fig. 9

Dependence of the yield stress on the copper grain size

Tables

Table Grahic Jump Location
Table 1 Standard physical constants of the micromechanical model for the two employed metals

Errata

Discussions

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