# Numerical Simulations of the Hall–Petch Relationship in Aluminium Using Gradient-Enhanced Plasticity Model

Authors: Yooseob Song; Jaeheum Yeon; Byoungjoon Na
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

The Hall-Petch relation in aluminium is discussed based on the strain gradient plasticity framework. The thermodynamically consistent gradient-enhanced flow rules for bulk and grain boundaries are developed using the concepts of thermal activation energy and dislocation interaction mechanisms. It is assumed that the thermodynamic microstresses for bulk and grain boundaries have dissipative and energetic contributions, and in turn, both dissipative and energetic material length scale parameters are existent. Accordingly, two-dimensional finite element simulations are performed to analyse characteristics of the Hall–Petch strengthening and the Hall–Petch constants. The proposed flow rules for the grain boundary are validated using the existing experimental data from literatures. An excellent agreement between the numerical results and the experimental measurements is obtained in the Hall–Petch plot. In addition, it is observed that the Hall–Petch constants do not remain unchanged but vary depending on the strain level.


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## Body

## 1. Introduction

Most of the metals and metal alloys have polycrystalline nature. In general, a fine-grained material is stronger and harder than a coarse-grained one. This can be described in the relation between the grain size and yield stress through the Hall–Petch equation as follows [1, 2]:(1)σy=σ0+kD,where σy denotes the yield stress, σ0 denotes the material constant related to the resistance of lattice to dislocation motion, k denotes the Hall–Petch strengthening coefficient, and D denotes the average grain size. A linear relationship between σy and D−1/2 with a slope of k is shown in the Hall–Petch plot.After the pioneering works of Hall [1] for mild steels and Petch [2] for brittle materials, numerous works have been conducted to investigate the Hall–Petch relation through various methods including experiments [3], review/overview [4, 5], theoretical investigations [6], and numerical simulations [7]. In [3], microhardness of nanocrystalline palladium and copper was experimentally investigated according to grain size variation, and significant increases in strength were observed in both materials compared with conventional grain size materials. The predictive capability of the Hall–Petch relation as well as its physical basis was discussed briefly by [4], based on experimental findings. Connections of the Hall–Petch relation to strain rate sensitivities; shear banding; fracture mechanics; fatigue; hardness property; broader stress-strain behaviour of hexagonal close packed, body-centered cubic, and face-centered cubic materials; and ductile-to-brittle transition behaviour of steel and related materials were reported in [5]. Pande and Cooper [6] focused on the inverse Hall–Petch relation, which manifests itself as the softening of nanocrystalline materials of very small mean grain sizes. In [7], the strain gradient crystal plasticity theory and its finite element algorithm were developed to describe the grain size-dependent behaviours of polycrystalline materials.Even in these days, the Hall–Petch relation is of great interest to many researchers. Yu et al. [8] presented a review of the Hall–Petch relationship in magnesium alloys, especially focusing on the Hall–Petch slope (k) and the factors influencing the mechanisms of k. The Hall–Petch relationship in Al-ZnO composites with different matrix grain sizes (D1) relative to interparticle spacing (D2) was studied in [9]. For samples with D1>D2, it was observed that both particle strengthening mechanism and the grain size effects described by the Hall–Petch relationship contribute to the strengthening of the metal matrix composites. In [10], the Hall–Petch breakdown in nanocrystalline ceramics was tested by performing indentation studies on fully dense nanocrystalline ceramics fabricated with grain sizes ranging from 3.6 to 37.5 nm. It was observed that the maximum hardness occurs at a grain size of 18.4 nm, and the inverse (or negative) Hall–Petch relationship reduces the hardness as the grain size is decreased to roughly 5 nm.There are several numerical studies on the Hall–Petch strengthening based on crystal plasticity [11]. However, strain gradient continuum plasticity is rarely used in this area. Voyiadjis and coworkers [12–19] have developed the coupled thermo-mechanical and thermodynamically consistent strain gradient plasticity models to study the characteristics of nano/microscale metallic materials. In this work, strain gradient-enhanced flow rules for bulk and grain boundaries are proposed to investigate the grain size-dependent flow stress of polycrystalline materials.Therefore, the main aim of this work is to show that the proposed strain gradient-enhanced flow rules for bulk and grain boundaries well capture the Hall–Petch relation.


## 2. Gradient-Enhanced Continuum Plasticity

In the current work, the subscriptsi, j, k, l, m, and n are used to denote tensors. The superscripts “dis,” “en,” “ext,” “int,” “GB,” “e,” and “p” stand for specific quantities such as dissipative, energetic, external, internal, grain boundary, elastic state, and plastic state, respectively.


### 2.1. Principle of Virtual Power

The external powerℙext expended by the macrotraction Ti and microtraction m on the external surface ∂Ω0 and the generalized external body force bi acting in Ω0 as follows [15, 18]:(2)ℙext=∫Ω0biu˙idV+∫∂Ω0tiu˙i+mε˙p+aT˙dS,where u˙i denotes the macroscopic velocity. The term εp denotes the accumulated plastic strain, and the terms T and T˙ denote the temperature and its rate, respectively. In addition, for the thermal effect, term a is present in the external power.In the arbitrary regionΩ0, the internal power ℙint is assumed as follows with a combination of macroenergy, microenergy, and thermal energy contributions [15, 18]:(3)ℙint=∫Ω0σijε˙ije+ξε˙p+ℚiε˙,ip+AT˙+BiT˙,idV,where εije is the elastic part of the strain tensor, ξ and ℚi are the thermodynamic microforces conjugate, respectively, to ε˙p, and ε˙,ip, A, and Bi are the micromorphic scalar and vector generalized stresses conjugate to the temperature rate T˙ and the gradient of the temperature rate T˙,i, respectively, and σij is the Cauchy stress tensor.From the relation,ℙext=ℙint and the divergence theorem, the balance equations in volume Ω0 are obtained, respectively, as follows:(4)σij,j+bi=0,σ¯ij=ξ−ℚk,kNij,Bi,i−A=0,where σ¯ij denotes the deviatoric part of σij, σ¯ij=σij−σkkδij/3, where δij denotes the Kronecker delta. Equation (4) represents the macroscopic linear momentum balance equation and the nonlocal microforce balance equation, respectively. The term Nij denotes the direction of plastic flow given by Nij=e˙ijp/ε˙p.The local and nonlocal traction balance equations on∂Ω0 are given as follows:(5)tj=σijni,m=ℚini,a=Bini,where ni denotes the outward unit vector normal to ∂Ω0.Next, a thermodynamically consistent grain boundary flow rule is developed in this work. Consider the two grainsG1 and G2 separated by the grain boundary. The continuous displacement field across the grain boundary is assumed, uiG1=uiG2. It is also assumed on the arbitrary surface SGB over the grain boundary that the grain boundary internal virtual power depends on the grain boundary accumulated plastic strain rates ε˙pGBG1 at SGBG1 and ε˙pGBG2 at SGBG2 [16, 18]:(6)ℙintGB=∫SGBMGBG1ε˙pGBG1+MGBG2ε˙pGBG2dSGB,where the grain boundary micromoment tractions MGBG1 and MGBG2 are assumed to, respectively, expend power over ε˙pGBG1 and ε˙pGBG2. The external power ℙextGB is expended by the macrotractions σijG1−njGB and σijG2njGB and the microtractions ℚkG1−nkGB and ℚiG2nkGB as follows [16, 18]:(7)ℙextGB=∫SGBσijG2njGB−σijG1njGBu˙i+ℚkG2nkGBε˙pGBG2−ℚkG1nkGBε˙pGBG1dSGB,where nGB denotes the unit vector normal to the grain boundary surface. Using ℙintGB=ℙextGB, the grain boundary macroscopic and microscopic force balance equations can be obtained as(8)σijG1−σijG2njGB=0,MGBG1+ℚkG1nkGB=0,MGBG2−ℚkG2nkGB=0.


### 2.2. Thermodynamic Microforces: Energetic and Dissipative Components

The Helmholtz free energyΨ is given through the Legendre transform as Ψ=ℰ−Ts, where ℰ denotes the internal energy and s denotes the entropy [20]. Using this relation, the following Clausius–Duhem inequality is constructed [15]:(9)σijε˙ije+ξε˙p+ℚiε˙,ip+AT˙+BiT˙,i−ρΨ˙−ρsT˙−qiT,iT≥0.In order to take the effect of nonuniform distribution of microdefects into account along with temperature on the homogenized material behaviour, it is assumed that the Helmholtz free energy is a smooth function ofεije, εp, ε,ip, T, and T,i. During the process of deriving the constitutive equations, it is important to make sure that nonnegative dissipation is maintained. It should be mentioned that the Helmholtz free energy is at its minimum in a stable equilibrium state with respect to any isothermal small geometrically admissible virtual displacement field. In this regard, different counterparts of Ψ are locally convex functions of εije, εp, and ε,ip at all points of the body in the considered equilibrium state, and Ψ is a concave function of temperature [15].The thermodynamic microforcesξ, ℚi, and A are assumed to have the energetic and dissipative contributions [12–16, 18]. Thus, ξ=ξen+ξdis, ℚi=ℚien+ℚidis, and A=Aen+Adis. From the Clausius–Duhem inequality and the aforementioned decompositions, the energetic microforces are defined as(10)σij=ρ∂Ψ∂εije,ξen=ρ∂Ψ∂εp,ℚien=ρ∂Ψ∂ε,ip,Aen=ρs+∂Ψ∂T,Bi=ρ∂Ψ∂T,i.The dissipative microstresses are then determined from the dissipation potentialDε˙p,ε˙,ip,T˙,T,i as follows:(11)ξdis=∂D∂ε˙p,ℚidis=∂D∂ε˙,ip,Adis=∂D∂T˙,−qiT=∂D∂T,i.Similarly, the grain boundary energetic and dissipative microforces are defined as(12)MGB,en=ρ∂ΨGB∂εpGB,MGB,dis=∂DGB∂ε˙pGB,where the grain boundary Helmholtz free energy (ΨGB) is a function of εpGB, i.e., ΨGB=ΨGBεpGB and the grain boundary thermodynamic microforce quantitity MGB has the energetic and dissipative contributions, i.e., MGB=MGB,en+MGB,dis. The components MGB,en and MGB,dis are related to the preslip and postslip transfer mechanisms; therefore, the grain boundary accumulated plastic strains for the preslip transfer εpGBpre and the postslip transfer εpGBpost are included, respectively, (εpGB=εpGBpre+εpGBpost). DGB denotes the nonnegative grain boundary dissipation density per unit time DGB=MGB,disε˙pGB≥0. This nonnegative plastic dissipation condition is satisfied in the case that DGB is a convex function of ε˙pGB.


### 2.3. Energetic and Dissipative Constitutive Relations

The Helmholtz free energy function in this work is assumed based on [12, 13, 18] as follows:(13)Ψ=12ρεijeEijklεkle+h0ρr+11−TTynεpr+1+σ∗ρϑ+1ℓen2ε,ipε,ipϑ+1/2−12cεTrT−Tr2−αthρT−Trεijeδij−12ρaT,iT,i,where Eijkl denotes the elastic modulus tensor, αth denotes the thermal expansion coefficient, h0 and r denote the material parameters related to isotropic hardening, Ty and n denote the material parameters related to the thermal effects, σ∗>0 denotes the initial slip resistance scaling parameter, ℓen denotes the energetic material length scale, a denotes the material parameter related to the isotropic heat conduction, ϑ denotes the material parameter related to the nonlinearity of the defect energy, cε denotes the specific heat capacity at the constant stress, and Tr denotes the reference temperature.From equations (10) and (13), the energetic microforces can be obtained as follows:(14)σij=Eijklεkle−αthT−Trδij,ξen=h01−TTynεpr,ℚien=σ∗ℓen2ℓen2ε,kpε,kpϑ−1/2ε,ip,Aen=ρs−cεTrT−Tr−αthT−Trεijeδij−h0εpr+1r+1TTyTTyn−1,Bi=−aT,i.The dissipation potential has the following functional form in this work [12, 13, 18]:(15)D=σ∗ℋ2εp+ℓNGep1−TTynε˙pp˙1m1ε˙p+σ∗1−TTynp˙p˙2m2p˙−ς2T˙2−12kTTT,iT,i,where p˙1>0 and p˙2>0 denote the reference rate parameters, m1>0 and m2>0 denote the rate sensitivity parameters, ς denotes the material parameter related to the energy exchange between electron and phonon, and kT denotes the thermal conductivity coefficient. ℓNG denotes the NG (Nix–Gao) material length scale firstly introduced by Nix and Gao [21]. When ℋεp=1 and ℓNG=0, equation (15) reduces to the one used in [12]. ep is defined as ep=defαij=bρG, where αij denotes the Nye dislocation density tensor, b denotes the magnitude of the Burgers vector, and ρG denotes the density of geometrically necessary dislocations.The scalarp˙ measures the gradient of plastic strain rate, and it is defined as p˙=defℓdisε˙,ip=ℓdisε˙,ipε˙,ip, where ℓdis denotes the dissipative material length scale [12, 13, 18].The strain hardening/softening behaviour is determined through the dimensionless functionℋεp. The following mixed-form hardening function is considered in this work [22]:(16)ℋεp=1+χ−11−exp−ωεp+h0σ∗εp,where ω and χ denote the material constants.Using equations (11) and (15) along with the assumption kT/T=k0=constant, the dissipative microforces can be obtained as follows:(17)ξdis=σ∗ℋ2εp+ℓNGep1−TTynε˙pp˙1m1,ℚidis=σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,ipp˙,Adis=−ςT˙,qiT=k0T,i.The grain boundary free energy per unit surfaceΨGB in this work is assumed to have the general power law form as follows [23]:(18)ΨGBεpGB=12GℓenGBεpGBpre2,where G denotes the shear modulus and ℓenGB denotes the grain boundary energetic length scale. From equations (12) and (18), the grain boundary energetic microforce is determined as(19)MGB,en=GℓenGBεpGBpre.The generalized expression of the grain boundary dissipation potential is put forward in this work as follows [14, 16]:(20)DGB=ℓdisGBmGB+1σ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGBε˙pGBpost≥0,where ℓdisGB denotes the grain boundary dissipative length scale, mGB and p˙GB denote the material parameters related to viscosity, σ∗GB is the stress-dimensioned parameter related to the grain boundary yield stress, h0GB denotes the grain boundary hardening constant, TyGB denotes the grain boundary thermal constant at the onset of yielding, nGB denotes the grain boundary thermal constant. The rate and temperature dependency of the grain boundary energy are presented through the terms ε˙pGBpost/p˙GBmGB and 1−TGB/TyGBnGB, respectively.Substitution of equation (20) into equation (12) gives the grain boundary dissipative microforce MGB,dis as follows:(21)MGB,dis=ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB.Finally, the grain boundary thermodynamic microforceMGB is obtained by combining equations (19) and (21) as follows:(22)MGB=GℓenGBεpGBpre+ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB.From equation (22), it is obvious that the grain boundary acts like free surface when microscopically free boundary condition is imposed (ℓenGB=ℓdisGB=0), while passivated condition at the grain boundary can be described when the microscopically hard boundary condition is imposed (ℓenGB⟶∞ and ℓdisGB⟶∞).


### 2.4. Flow Rules

One can establish the flow rule from the nonlocal microforce balance, equation (4), and the energetic and dissipative microforce quantities. The backstress is considered in this work in the microforce equilibrium, i.e., σ¯ij−−ℚk,kenNij=ξ−ℚk,kdisNij, where Nij=e˙ijp/ε˙p. The following flow rule for the bulk can be obtained [12, 13, 15].(23)σ¯ij−−σ∗ℓen2ℓen2ε,ipε,ipϑ−1/2ε,kkpNij=h01−TTynεpr+σ∗ℋ2εp+ℓNGep1−TTynε˙pp˙1m1−σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,kkpp˙Nij.The substitution of equation (22) into the grain boundary microforce balances, equation (8), gives the grain boundary flow rules as follows [16, 18]:ForSGBG1,(24)σ∗ℓen2ℓen2ε,kpε,kpϑ−1/2ε,ip+σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,ipp˙nkGB+GℓenGBεpGBpre=−ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB.ForSGBG2,(25)σ∗ℓen2ℓen2ε,kpε,kpϑ−1/2ε,ip+σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,ipp˙nkGB−GℓenGBεpGBpre=ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB,where the second terms in left-hand side of equations (24) and (25) indicate the backstress. The grain boundary flow rules, equations (24) and (25), are only applied for the nodes on the grain boundaries.The developed flow rules for the bulk and grain boundaries are numerically implemented through the finite element simulations to address the microstructural material characteristics. In this work, the unknown nodal degrees of freedom are the displacement fieldui and the plastic strain field εp, and they are independently discretized.


## 2.1. Principle of Virtual Power

The external powerℙext expended by the macrotraction Ti and microtraction m on the external surface ∂Ω0 and the generalized external body force bi acting in Ω0 as follows [15, 18]:(2)ℙext=∫Ω0biu˙idV+∫∂Ω0tiu˙i+mε˙p+aT˙dS,where u˙i denotes the macroscopic velocity. The term εp denotes the accumulated plastic strain, and the terms T and T˙ denote the temperature and its rate, respectively. In addition, for the thermal effect, term a is present in the external power.In the arbitrary regionΩ0, the internal power ℙint is assumed as follows with a combination of macroenergy, microenergy, and thermal energy contributions [15, 18]:(3)ℙint=∫Ω0σijε˙ije+ξε˙p+ℚiε˙,ip+AT˙+BiT˙,idV,where εije is the elastic part of the strain tensor, ξ and ℚi are the thermodynamic microforces conjugate, respectively, to ε˙p, and ε˙,ip, A, and Bi are the micromorphic scalar and vector generalized stresses conjugate to the temperature rate T˙ and the gradient of the temperature rate T˙,i, respectively, and σij is the Cauchy stress tensor.From the relation,ℙext=ℙint and the divergence theorem, the balance equations in volume Ω0 are obtained, respectively, as follows:(4)σij,j+bi=0,σ¯ij=ξ−ℚk,kNij,Bi,i−A=0,where σ¯ij denotes the deviatoric part of σij, σ¯ij=σij−σkkδij/3, where δij denotes the Kronecker delta. Equation (4) represents the macroscopic linear momentum balance equation and the nonlocal microforce balance equation, respectively. The term Nij denotes the direction of plastic flow given by Nij=e˙ijp/ε˙p.The local and nonlocal traction balance equations on∂Ω0 are given as follows:(5)tj=σijni,m=ℚini,a=Bini,where ni denotes the outward unit vector normal to ∂Ω0.Next, a thermodynamically consistent grain boundary flow rule is developed in this work. Consider the two grainsG1 and G2 separated by the grain boundary. The continuous displacement field across the grain boundary is assumed, uiG1=uiG2. It is also assumed on the arbitrary surface SGB over the grain boundary that the grain boundary internal virtual power depends on the grain boundary accumulated plastic strain rates ε˙pGBG1 at SGBG1 and ε˙pGBG2 at SGBG2 [16, 18]:(6)ℙintGB=∫SGBMGBG1ε˙pGBG1+MGBG2ε˙pGBG2dSGB,where the grain boundary micromoment tractions MGBG1 and MGBG2 are assumed to, respectively, expend power over ε˙pGBG1 and ε˙pGBG2. The external power ℙextGB is expended by the macrotractions σijG1−njGB and σijG2njGB and the microtractions ℚkG1−nkGB and ℚiG2nkGB as follows [16, 18]:(7)ℙextGB=∫SGBσijG2njGB−σijG1njGBu˙i+ℚkG2nkGBε˙pGBG2−ℚkG1nkGBε˙pGBG1dSGB,where nGB denotes the unit vector normal to the grain boundary surface. Using ℙintGB=ℙextGB, the grain boundary macroscopic and microscopic force balance equations can be obtained as(8)σijG1−σijG2njGB=0,MGBG1+ℚkG1nkGB=0,MGBG2−ℚkG2nkGB=0.


## 2.2. Thermodynamic Microforces: Energetic and Dissipative Components

The Helmholtz free energyΨ is given through the Legendre transform as Ψ=ℰ−Ts, where ℰ denotes the internal energy and s denotes the entropy [20]. Using this relation, the following Clausius–Duhem inequality is constructed [15]:(9)σijε˙ije+ξε˙p+ℚiε˙,ip+AT˙+BiT˙,i−ρΨ˙−ρsT˙−qiT,iT≥0.In order to take the effect of nonuniform distribution of microdefects into account along with temperature on the homogenized material behaviour, it is assumed that the Helmholtz free energy is a smooth function ofεije, εp, ε,ip, T, and T,i. During the process of deriving the constitutive equations, it is important to make sure that nonnegative dissipation is maintained. It should be mentioned that the Helmholtz free energy is at its minimum in a stable equilibrium state with respect to any isothermal small geometrically admissible virtual displacement field. In this regard, different counterparts of Ψ are locally convex functions of εije, εp, and ε,ip at all points of the body in the considered equilibrium state, and Ψ is a concave function of temperature [15].The thermodynamic microforcesξ, ℚi, and A are assumed to have the energetic and dissipative contributions [12–16, 18]. Thus, ξ=ξen+ξdis, ℚi=ℚien+ℚidis, and A=Aen+Adis. From the Clausius–Duhem inequality and the aforementioned decompositions, the energetic microforces are defined as(10)σij=ρ∂Ψ∂εije,ξen=ρ∂Ψ∂εp,ℚien=ρ∂Ψ∂ε,ip,Aen=ρs+∂Ψ∂T,Bi=ρ∂Ψ∂T,i.The dissipative microstresses are then determined from the dissipation potentialDε˙p,ε˙,ip,T˙,T,i as follows:(11)ξdis=∂D∂ε˙p,ℚidis=∂D∂ε˙,ip,Adis=∂D∂T˙,−qiT=∂D∂T,i.Similarly, the grain boundary energetic and dissipative microforces are defined as(12)MGB,en=ρ∂ΨGB∂εpGB,MGB,dis=∂DGB∂ε˙pGB,where the grain boundary Helmholtz free energy (ΨGB) is a function of εpGB, i.e., ΨGB=ΨGBεpGB and the grain boundary thermodynamic microforce quantitity MGB has the energetic and dissipative contributions, i.e., MGB=MGB,en+MGB,dis. The components MGB,en and MGB,dis are related to the preslip and postslip transfer mechanisms; therefore, the grain boundary accumulated plastic strains for the preslip transfer εpGBpre and the postslip transfer εpGBpost are included, respectively, (εpGB=εpGBpre+εpGBpost). DGB denotes the nonnegative grain boundary dissipation density per unit time DGB=MGB,disε˙pGB≥0. This nonnegative plastic dissipation condition is satisfied in the case that DGB is a convex function of ε˙pGB.


## 2.3. Energetic and Dissipative Constitutive Relations

The Helmholtz free energy function in this work is assumed based on [12, 13, 18] as follows:(13)Ψ=12ρεijeEijklεkle+h0ρr+11−TTynεpr+1+σ∗ρϑ+1ℓen2ε,ipε,ipϑ+1/2−12cεTrT−Tr2−αthρT−Trεijeδij−12ρaT,iT,i,where Eijkl denotes the elastic modulus tensor, αth denotes the thermal expansion coefficient, h0 and r denote the material parameters related to isotropic hardening, Ty and n denote the material parameters related to the thermal effects, σ∗>0 denotes the initial slip resistance scaling parameter, ℓen denotes the energetic material length scale, a denotes the material parameter related to the isotropic heat conduction, ϑ denotes the material parameter related to the nonlinearity of the defect energy, cε denotes the specific heat capacity at the constant stress, and Tr denotes the reference temperature.From equations (10) and (13), the energetic microforces can be obtained as follows:(14)σij=Eijklεkle−αthT−Trδij,ξen=h01−TTynεpr,ℚien=σ∗ℓen2ℓen2ε,kpε,kpϑ−1/2ε,ip,Aen=ρs−cεTrT−Tr−αthT−Trεijeδij−h0εpr+1r+1TTyTTyn−1,Bi=−aT,i.The dissipation potential has the following functional form in this work [12, 13, 18]:(15)D=σ∗ℋ2εp+ℓNGep1−TTynε˙pp˙1m1ε˙p+σ∗1−TTynp˙p˙2m2p˙−ς2T˙2−12kTTT,iT,i,where p˙1>0 and p˙2>0 denote the reference rate parameters, m1>0 and m2>0 denote the rate sensitivity parameters, ς denotes the material parameter related to the energy exchange between electron and phonon, and kT denotes the thermal conductivity coefficient. ℓNG denotes the NG (Nix–Gao) material length scale firstly introduced by Nix and Gao [21]. When ℋεp=1 and ℓNG=0, equation (15) reduces to the one used in [12]. ep is defined as ep=defαij=bρG, where αij denotes the Nye dislocation density tensor, b denotes the magnitude of the Burgers vector, and ρG denotes the density of geometrically necessary dislocations.The scalarp˙ measures the gradient of plastic strain rate, and it is defined as p˙=defℓdisε˙,ip=ℓdisε˙,ipε˙,ip, where ℓdis denotes the dissipative material length scale [12, 13, 18].The strain hardening/softening behaviour is determined through the dimensionless functionℋεp. The following mixed-form hardening function is considered in this work [22]:(16)ℋεp=1+χ−11−exp−ωεp+h0σ∗εp,where ω and χ denote the material constants.Using equations (11) and (15) along with the assumption kT/T=k0=constant, the dissipative microforces can be obtained as follows:(17)ξdis=σ∗ℋ2εp+ℓNGep1−TTynε˙pp˙1m1,ℚidis=σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,ipp˙,Adis=−ςT˙,qiT=k0T,i.The grain boundary free energy per unit surfaceΨGB in this work is assumed to have the general power law form as follows [23]:(18)ΨGBεpGB=12GℓenGBεpGBpre2,where G denotes the shear modulus and ℓenGB denotes the grain boundary energetic length scale. From equations (12) and (18), the grain boundary energetic microforce is determined as(19)MGB,en=GℓenGBεpGBpre.The generalized expression of the grain boundary dissipation potential is put forward in this work as follows [14, 16]:(20)DGB=ℓdisGBmGB+1σ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGBε˙pGBpost≥0,where ℓdisGB denotes the grain boundary dissipative length scale, mGB and p˙GB denote the material parameters related to viscosity, σ∗GB is the stress-dimensioned parameter related to the grain boundary yield stress, h0GB denotes the grain boundary hardening constant, TyGB denotes the grain boundary thermal constant at the onset of yielding, nGB denotes the grain boundary thermal constant. The rate and temperature dependency of the grain boundary energy are presented through the terms ε˙pGBpost/p˙GBmGB and 1−TGB/TyGBnGB, respectively.Substitution of equation (20) into equation (12) gives the grain boundary dissipative microforce MGB,dis as follows:(21)MGB,dis=ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB.Finally, the grain boundary thermodynamic microforceMGB is obtained by combining equations (19) and (21) as follows:(22)MGB=GℓenGBεpGBpre+ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB.From equation (22), it is obvious that the grain boundary acts like free surface when microscopically free boundary condition is imposed (ℓenGB=ℓdisGB=0), while passivated condition at the grain boundary can be described when the microscopically hard boundary condition is imposed (ℓenGB⟶∞ and ℓdisGB⟶∞).


## 2.4. Flow Rules

One can establish the flow rule from the nonlocal microforce balance, equation (4), and the energetic and dissipative microforce quantities. The backstress is considered in this work in the microforce equilibrium, i.e., σ¯ij−−ℚk,kenNij=ξ−ℚk,kdisNij, where Nij=e˙ijp/ε˙p. The following flow rule for the bulk can be obtained [12, 13, 15].(23)σ¯ij−−σ∗ℓen2ℓen2ε,ipε,ipϑ−1/2ε,kkpNij=h01−TTynεpr+σ∗ℋ2εp+ℓNGep1−TTynε˙pp˙1m1−σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,kkpp˙Nij.The substitution of equation (22) into the grain boundary microforce balances, equation (8), gives the grain boundary flow rules as follows [16, 18]:ForSGBG1,(24)σ∗ℓen2ℓen2ε,kpε,kpϑ−1/2ε,ip+σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,ipp˙nkGB+GℓenGBεpGBpre=−ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB.ForSGBG2,(25)σ∗ℓen2ℓen2ε,kpε,kpϑ−1/2ε,ip+σ∗ℓdis2m2+11−TTynp˙p˙2m2ε˙,ipp˙nkGB−GℓenGBεpGBpre=ℓdisGBσ∗GB+h0GBεpGBpost1−TGBTyGBnGBε˙pGBpostp˙GBmGB,where the second terms in left-hand side of equations (24) and (25) indicate the backstress. The grain boundary flow rules, equations (24) and (25), are only applied for the nodes on the grain boundaries.The developed flow rules for the bulk and grain boundaries are numerically implemented through the finite element simulations to address the microstructural material characteristics. In this work, the unknown nodal degrees of freedom are the displacement fieldui and the plastic strain field εp, and they are independently discretized.


## 3. Model Validation and Calibration of the Model Parameters

The proposed strain gradient-dependent flow rules will be validated in this section through the comparison against the experimental measurements by [24]. The calibration of some model parameters will also be carried out simultaneously. In [24], the Bauschinger effect in sputter-deposited aluminium (Al) thin film was investigated experimentally. However, the Bauschinger effect will not be studied in this work since it is not of interest; instead, the experimental data from passivated and unpassivated layers will be used for the model validation.The sample preparation method is introduced in [24] in detail. Samples are vacuum-annealed at 300°C for stabilization of the material microstructure. It is observed through transmission electron microscope (TEM) micrographs that the average grain size is 2.1 μm.The material parameters for Al are also calibrated using the experimental measurements. Tables1 and 2 show the general and calibrated material parameters, respectively. As mentioned earlier, the grain boundary can behave like a free surface through the microscopically free boundary condition, under ℓenGB=ℓdisGB=0, whereas the passivated condition on film surfaces can be described with ℓenGB⟶∞ and ℓdisGB⟶∞. Stress-strain responses of passivated and unpassivated films from experiments and simulations are shown in Figure 1. As clearly shown in this figure, the numerical results and experimental measurements correspond with each other closely.Table 1
General material parameters of aluminium for the current numerical simulations [15, 18].
General material parameters
Values
E
GPa
70
ν
0.30
μ
GPa
27
ρ
g
⋅
cm
−
3
2.702
c
ε
J
/
g
⋅
∘
K
0.910
α
th
μ
m
/
m
⋅
∘
K
24.0
p
˙
1
,
p
˙
2
s
−
1
0.04
r
0.6
m
1
0.05
m
2
0.2
T
y
∘
K
933
n
0.3Table 2
Calibrated material parameters of aluminium for the current numerical simulations.
Calibrated material parameters
Values
σ
∗
MPa
100
h
0
MPa
100
ℓ
en
μ
m
1.0
ℓ
dis
μ
m
2.5
ℓ
NG
μ
m
1.0Figure 1
Proposed model validation using the experimental data in [24] on the stress-strain responses in aluminium thin films.


## 4. Numerical Results: Hall–Petch Relation

Materials can be strengthened by decreasing the average grain size. This method is called grain boundary strengthening or Hall–Petch strengthening. Grain boundary impedes the movements of dislocations and how many dislocations are existent in a grain affects on how smoothly they can travel from grain to grain. Hall–Petch strengthening is based on this observation. In this section, Hall–Petch strengthening is investigated using finite element simulations with different grain sizes based on the proposed model.


### 4.1. Problem Description

The schematic illustration of uniaxial strain problem with single-crystal and polycrystalline materials is shown in Figure2. The problem geometry, initial condition, loading condition, macroscopic boundary condition, and finite element mesh are displayed in this figure. The term u†t represents the prescribed displacement. Each grain has an average grain size of D. The whole square is split into several grains by grain boundaries, which are represented by bold lines, as shown in Figure 2. 4096 (64 × 64) elements are used. The general material parameters in Table 1 are used, while the rest of the material parameters are calibrated using another set of experiments for pure aluminium (99.999%) by [25]. Since the experiments were performed for grain sizes from 0.035 to 1.3 mm by [25], six different grain sizes, D = 0.03125 mm (32 × 32 = 1024 grains), D = 0.0625 mm (16 × 16 = 256 grains), D = 0.125 mm (8 × 8 =64 grains), D = 0.25 mm (4 × 4 = 16 grains), D = 0.5 mm (2 × 2 = 4 grains), and D = 1.0 mm (single grain) are considered in the current simulations. In this work, room temperature is assumed as is in [25].Figure 2
Schematic illustration of uniaxial strain problem with single crystal and polycrystalline materials.D is the (average) grain size. Dotted lines and bold lines in single crystal and polycrystalline materials represent finite element mesh and grain boundary, respectively.
(a)
(b)
(c)


### 4.2. Grain Boundary Simulations

To check that the proposed grain boundary flow rules in equations (24) and (25) can properly mimic the two null conditions (free surface and passivated surface), two simulations are performed in single crystal with the microscopically free boundary condition and the microscopically hard boundary condition at the grain boundary. Figure 3 shows the distributions of the accumulated plastic strain εp with the microscopically free boundary condition and the microscopically hard boundary condition. As expected, uniform distribution of εp is observed in case of the microscopically free boundary condition like free surface, while total blockage of dislocation movement at the grain boundary (εp=0) is well described by imposing the microscopically hard boundary condition in the proposed grain boundary flow rules.Figure 3
Distributions of the accumulated plastic strain in single crystal with (a) microscopically free boundary condition and (b) microscopically hard boundary condition [15, 18].
(a)
(b)


### 4.3. Hall–Petch Strengthening

Numerical simulations under the uniaxial tensile loading condition are carried out with six different grain sizes from 0.03125 to 1.0 mm to investigate the Hall–Petch strengthening and to validate the proposed model by comparing with the experimental data of [25]. The Hall–Petch constants such as σ0 and k are also studied in this section. The numerically obtained true stress-true strain curves are shown in Figure 4 with varying grain sizes. As can be seen in this figure, the Hall–Petch strengthening (material hardens with the decreasing grain size) is well observed qualitatively.Figure 4
True stress-true strain responses with varying grain sizes from simulations.The yield stress at a (true) strain of 0.002 and the flow stresses at four selected strains (0.01, 0.05, 0.1, and 0.2) against the reciprocal square root of the grain size are plotted in Figure5. Straight lines are fitted to the points by the least square method. A significant grain size effect is predicted at the onset of yielding, which is in line with experimental finding. Both experimental data [25] and model predictions show a good linear correlation for all the grain sizes. Moreover, model predictions from the simulations are in an excellent agreement with experimental data. Table 3 shows the Hall–Petch constants, σ0 and k, for both experiments and simulations at the different levels of strain obtained from the linear trendlines in Figure 5. Again, simulations show a good agreement with experiments in both parameters.Figure 5
Hall–Petch plot for pure aluminium from experiments [25] and simulations.Table 3
Hall–Petch constants for pure aluminium from experiments [25] and simulations.
True strain
Experiments [25]
Simulations
σ
0
MPa
k
MPa
⋅
mm
1
/
2
σ
0
MPa
k
MPa
⋅
mm
1
/
2
0.002
5.62
1.27
5.39
1.37
0.01
13.83
1.39
13.37
1.49
0.05
28.92
1.58
28.05
1.70
0.1
36.15
1.95
35.07
2.15
0.2
46.32
2.29
44.94
2.60Figure5 and Table 3 imply that the Hall–Petch constants, σ0 and k, increase with increasing strain. It is worth mentioning that the strengthening effect of the grain boundary areas described by k increases with increasing strain.These two parameters are plotted in Figure6 as a function of strain. Besides the presented experimental data at the five strain levels in Figure 5 and Table 3, the other experimental datasets presented in [25] are also plotted in this figure. The good correlations between experiments and simulations are obtained.Figure 6
Hall–Petch constants as a function of strain for pure aluminium from experiments [25] and simulations.This is in line with the findings of [26]. In [26], an empirical relationship of Hall [1] and Petch [2], equation (1), is extended by expressing the Hall–Petch constants, σ0 and k, to depend on the strain level, such that σy=σ0ε+kεD−n, where the exponent n typically ranges from 0.3 to 1.0 (the most reported value is 0.5).Due to the formation of dislocation pile-ups at grain boundaries, flow stress can be enhanced and yielding occurs when flow stress is large enough to cause the slip to propagate from one grain to the adjacent grain. In order to underpin this behaviour, physically based strain gradient plasticity models are in demand. The current model is phenomenological, and physically based models have not been attained despite its importance. Furthermore, the pioneering measurements made by [27] were reported without assessment in terms of the Hall–Petch constants, σ∗ and k, but nevertheless show on examination decreasing values of both parameters with increase in temperature, more so in σ∗. A weak strain rate dependence of the flow stress was also measured by [27]. Effects of temperature and strain rate on the Hall–Petch constants will be investigated in the future. In addition, strain gradient crystal plasticity is another interest of the authors so that its pros and cons compared with the strain gradient continuum plasticity covered in this work will be explored in the future work.


## 4.1. Problem Description

The schematic illustration of uniaxial strain problem with single-crystal and polycrystalline materials is shown in Figure2. The problem geometry, initial condition, loading condition, macroscopic boundary condition, and finite element mesh are displayed in this figure. The term u†t represents the prescribed displacement. Each grain has an average grain size of D. The whole square is split into several grains by grain boundaries, which are represented by bold lines, as shown in Figure 2. 4096 (64 × 64) elements are used. The general material parameters in Table 1 are used, while the rest of the material parameters are calibrated using another set of experiments for pure aluminium (99.999%) by [25]. Since the experiments were performed for grain sizes from 0.035 to 1.3 mm by [25], six different grain sizes, D = 0.03125 mm (32 × 32 = 1024 grains), D = 0.0625 mm (16 × 16 = 256 grains), D = 0.125 mm (8 × 8 =64 grains), D = 0.25 mm (4 × 4 = 16 grains), D = 0.5 mm (2 × 2 = 4 grains), and D = 1.0 mm (single grain) are considered in the current simulations. In this work, room temperature is assumed as is in [25].Figure 2
Schematic illustration of uniaxial strain problem with single crystal and polycrystalline materials.D is the (average) grain size. Dotted lines and bold lines in single crystal and polycrystalline materials represent finite element mesh and grain boundary, respectively.
(a)
(b)
(c)


## 4.2. Grain Boundary Simulations

To check that the proposed grain boundary flow rules in equations (24) and (25) can properly mimic the two null conditions (free surface and passivated surface), two simulations are performed in single crystal with the microscopically free boundary condition and the microscopically hard boundary condition at the grain boundary. Figure 3 shows the distributions of the accumulated plastic strain εp with the microscopically free boundary condition and the microscopically hard boundary condition. As expected, uniform distribution of εp is observed in case of the microscopically free boundary condition like free surface, while total blockage of dislocation movement at the grain boundary (εp=0) is well described by imposing the microscopically hard boundary condition in the proposed grain boundary flow rules.Figure 3
Distributions of the accumulated plastic strain in single crystal with (a) microscopically free boundary condition and (b) microscopically hard boundary condition [15, 18].
(a)
(b)


## 4.3. Hall–Petch Strengthening

Numerical simulations under the uniaxial tensile loading condition are carried out with six different grain sizes from 0.03125 to 1.0 mm to investigate the Hall–Petch strengthening and to validate the proposed model by comparing with the experimental data of [25]. The Hall–Petch constants such as σ0 and k are also studied in this section. The numerically obtained true stress-true strain curves are shown in Figure 4 with varying grain sizes. As can be seen in this figure, the Hall–Petch strengthening (material hardens with the decreasing grain size) is well observed qualitatively.Figure 4
True stress-true strain responses with varying grain sizes from simulations.The yield stress at a (true) strain of 0.002 and the flow stresses at four selected strains (0.01, 0.05, 0.1, and 0.2) against the reciprocal square root of the grain size are plotted in Figure5. Straight lines are fitted to the points by the least square method. A significant grain size effect is predicted at the onset of yielding, which is in line with experimental finding. Both experimental data [25] and model predictions show a good linear correlation for all the grain sizes. Moreover, model predictions from the simulations are in an excellent agreement with experimental data. Table 3 shows the Hall–Petch constants, σ0 and k, for both experiments and simulations at the different levels of strain obtained from the linear trendlines in Figure 5. Again, simulations show a good agreement with experiments in both parameters.Figure 5
Hall–Petch plot for pure aluminium from experiments [25] and simulations.Table 3
Hall–Petch constants for pure aluminium from experiments [25] and simulations.
True strain
Experiments [25]
Simulations
σ
0
MPa
k
MPa
⋅
mm
1
/
2
σ
0
MPa
k
MPa
⋅
mm
1
/
2
0.002
5.62
1.27
5.39
1.37
0.01
13.83
1.39
13.37
1.49
0.05
28.92
1.58
28.05
1.70
0.1
36.15
1.95
35.07
2.15
0.2
46.32
2.29
44.94
2.60Figure5 and Table 3 imply that the Hall–Petch constants, σ0 and k, increase with increasing strain. It is worth mentioning that the strengthening effect of the grain boundary areas described by k increases with increasing strain.These two parameters are plotted in Figure6 as a function of strain. Besides the presented experimental data at the five strain levels in Figure 5 and Table 3, the other experimental datasets presented in [25] are also plotted in this figure. The good correlations between experiments and simulations are obtained.Figure 6
Hall–Petch constants as a function of strain for pure aluminium from experiments [25] and simulations.This is in line with the findings of [26]. In [26], an empirical relationship of Hall [1] and Petch [2], equation (1), is extended by expressing the Hall–Petch constants, σ0 and k, to depend on the strain level, such that σy=σ0ε+kεD−n, where the exponent n typically ranges from 0.3 to 1.0 (the most reported value is 0.5).Due to the formation of dislocation pile-ups at grain boundaries, flow stress can be enhanced and yielding occurs when flow stress is large enough to cause the slip to propagate from one grain to the adjacent grain. In order to underpin this behaviour, physically based strain gradient plasticity models are in demand. The current model is phenomenological, and physically based models have not been attained despite its importance. Furthermore, the pioneering measurements made by [27] were reported without assessment in terms of the Hall–Petch constants, σ∗ and k, but nevertheless show on examination decreasing values of both parameters with increase in temperature, more so in σ∗. A weak strain rate dependence of the flow stress was also measured by [27]. Effects of temperature and strain rate on the Hall–Petch constants will be investigated in the future. In addition, strain gradient crystal plasticity is another interest of the authors so that its pros and cons compared with the strain gradient continuum plasticity covered in this work will be explored in the future work.


## 5. Conclusions

The Hall–Petch relation and its strengthening effects on the flow stress of deformed metals were investigated based on the strain gradient plasticity model. The thermodynamically consistent strain gradient dependent plasticity flow rules for bulk and grain boundaries were developed. The proposed theory was implemented through the finite element simulations. Through the comparison with the existing experimental data in the literature, the model proposed was validated.To study the Hall–Petch relation and characteristics of the grain boundary, the uniaxial strain problem was solved. The two null boundary conditions at the grain boundary, the microscopically free and hard boundary conditions, were well captured through the proposed grain boundary flow rule. The six different grain sizes were considered in this work, and the Hall–Petch equation was well described. The good correlations between the simulated results and the experimental measurements were presented at the five selected strain levels. Lastly, it was clearly observed that the Hall–Petch constants change according to the variation of strain level.



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*Source: 7356581-2019-12-05.xml*