Nonadditive strengthening functions for cold-worked cubic metals: Experiments and constitutive modeling

International Journal of Plasticity xxx (xxxx) xxx

Contents lists available at ScienceDirect

International Journal of Plasticity journal homepage: http://www.elsevier.com/locate/ijplas

Nonadditive strengthening functions for cold-worked cubic metals: Experiments and constitutive modeling Fulin Jiang a, b, *, Setsuo Takaki b, Takuro Masumura b, Ryuji Uemori b, c, Hui Zhang a, Toshihiro Tsuchiyama b, c, ** a

College of Materials Science and Engineering, Hunan University, Changsha, 410082, China Research Center for Steel, Kyushu University, 744 Motooka Nishi-ku, Fukuoka, 819-0395, Japan c Department of Materials Science and Engineering, Kyushu University, 744 Motooka Nishi-ku, Fukuoka, 819-0395, Japan b

A R T I C L E I N F O

A B S T R A C T

Keywords: Mechanical properties Dislocations Nonadditive strengthening Steels Aluminum alloys Constitutive modeling

Strong metals are greatly desired for lightweight and energy-efficient industrial design. The strengthening of metals is traditionally accomplished by the additive contributions of various obstacle families (e.g., solid solutions, particles, and grain boundaries) and dislocation selfinteractions that impede dislocation motion. In the present work, unlike a traditional additive understanding, a distinctive nonadditive strengthening mixture rule for obstacles and dislocations were validated based on experimental and modeling analyses in numerous cold-worked steels and aluminum alloys. Concretely, numerous well-annealed body-centered cubic steels and facecentered cubic aluminum alloys were prepared, in which the hierarchical strength levels of solid solutions, grain boundaries, and/or particles were estimated. The above specimens were then cold rolled to various strain levels. Dislocation densities were quantified by utilizing X-ray diffraction line-profile-analysis, and the dislocation density was found to increase faster with an increasing strain level when a high strength was presented in well-annealed specimens. When plotting the yield stresses as a function of the square roots of the dislocation densities in massive distorted samples based on the Taylor hardening law, it is interesting to note that an approximate single linearity was obtained. Individual dislocation strengthening was found to respond to the total strength in the deformed specimens, which indicated that the full nonadditive strengthening mixture rule was employed between the obstacle families and the dislocation contributions. The mechanisms of the observed nonadditive strengthening were also discussed by implementing additional experiments and transmission electron microscopy observations. Then, the modified constitutive models based on both one and two internal parameters’ Knocks-Mecking models were developed respectively, which excellently captured the effects of the various obstacle families on dislocation storage processes. The developed models also rationalized the observed nonadditive strengthening mixture rule.

* Corresponding author. College of Materials Science and Engineering, Hunan University, Changsha, 410082, China. ** Coresponding author. Research Center for Steel, Kyushu University, 744 Motooka Nishi-ku, Fukuoka, 819-0395, Japan. E-mail addresses: [email protected], [email protected] (F. Jiang), [email protected] (T. Tsuchiyama). https://doi.org/10.1016/j.ijplas.2020.102700 Received 16 July 2019; Received in revised form 20 January 2020; Accepted 3 February 2020 Available online 11 February 2020 0749-6419/© 2020 Elsevier Ltd. All rights reserved.

Please cite this article as: Fulin Jiang, International Journal of Plasticity, https://doi.org/10.1016/j.ijplas.2020.102700

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

1. Introduction Metallic structure materials have been gaining widespread industrial applications owing to their excellent properties. Strong metals are substantially desired in lightweight and energy-efficient industrial designs such as the extensive applications of high-strength steels and aluminum (Al) alloys in automobiles, trains, and planes (Bhadeshia and Honeycombe, 2017; Lumley, 2018). In most industrial alloy production and modern alloy design strategies, multiple obstacle families (for instance solid solutions, particles, and grain boundaries) and dislocations are employed to increase strength (Lu et al., 2009; Deschamps and Brechet, 1998; Khan and Liu, 2016; He et al., 2017). For example, He et al. (2017) utilized a processing mechanism to create a “forest” of line defects in manganese steel. This deformed and partitioned steel was produced by cold-rolling and low-temperature annealing, which resulted in a dislocation network that improved both strength and ductility. Ma et al. (2016) prepared ultrafine grained Al–Zn–Mg–Cu alloys with an extremely high ultimate tensile strength of approximately 878 MPa and uniform elongation of 4.1% by coupling the additional functions of dislo­ cations and precipitates. Other outstanding implementations in designing high strength alloys included ultrahigh strength steel by Xu et al. (2019), Jiang et al. (2017) and Galindo-Nava et al. (2016), hierarchically structured materials by Devaraj et al. (2016) and Ming et al. (2019), architectural materials by Azizi et al. (2018), high-entropy alloys by Li et al. (2016) and Fang et al. (2019), and so on. Classical concepts of strengthening metallic materials depend on an intrinsic function of obstructing a line defect (dislocation) slip by utilizing various obstacle families and dislocation self-interactions (junctions) (Queyreau et al., 2010; Soare and Curtin, 2008; Devincre et al., 2008). For instance, solid solutions hinder the movement of dislocations through a matrix lattice due to a misfit volume and elastic mismatch (Soare and Curtin, 2008). The strengthening contributions of particles (precipitates or dispersions) are intro­ duced irrespective of whether dislocations cut through or loop around the particles (Queyreau et al., 2010; Tsuchiyama et al., 2016). A dislocation pile-up is formed when its motion is hindered by incoherent grain boundaries (Hall, 1951; Hu et al., 2017; Takaki et al., 2014; Farrokh and Khan, 2009). Furthermore, dislocation self-interactions are introduced by accumulated junctions during plastic deformation, which yields a classical Taylor hardening law (Taylor, 1934; Devincre et al., 2008; Zecevic and Knezevic, 2015; Li et al., 2017). The interactions of a dislocation with single precipitates, grain boundaries, forest dislocations, irradiation defects and solid solutes have been widely investigated (Olmsted et al., 2006; Khater et al., 2014). However, the strengthening mixture rules associated with simultaneous operations of multiple obstacle types and forest dislocations are more complicated and less clear (Queyreau et al., 2010; Dong et al., 2010; Borovikov et al., 2017). Traditional ideas generally employ the additive (linear or nonlinear) mixtures rule of the above strengthening contributions (Bhadeshia and Honeycombe, 2017; Deschamps and Brechet, 1998; Galindo-Nava et al., 2016; Steinmetz et al., 2013). In such cases, the Taylor hardening law (Taylor, 1934) accounting for dislocation strengthening is expressed as: (1)

σy ¼ σ0 þ MT α μ b ρ1/2

where σy is the yield stress (strength) and σ 0 is believed to be the additivity of friction stress (σfriction), solid solution strengthening (σss), particle strengthening (σp), and/or grain boundary (GB) strengthening (σ GB). For instance, the linearly additive rule gives, either σ 0 ¼ σfriction þ σ ss þ σp þ σ GB or σ 0 ¼ σ friction þ σ ss þ σp. MT α μ b ρ1/2 represents the total dislocation strengthening (σ d), where MT is the Taylor factor, α is a factor that relies on dislocation type and distribution, μ is the shear modulus, b is the magnitude of the Burgers vector and ρ is the dislocation density. In recent simulation attempts (Queyreau et al., 2010; Monnet and Devincre, 2006) and crystal plasticity or constitutive theory (Soare and Curtin, 2008; Steinmetz et al., 2013; Dong et al., 2010), the complex cases of strengthening superposition have attracted increasing attention. Consequently, an integrated mixture rule (nonlinear) of various strengthening ef­ fects has been proposed as shown in the following: h X i1=k σy ¼ σkj (2) where j represents the number of strengthening functions involved and k is a complicated constant depending on the strengthening superposition level. The value of k is usually supposed to be between 1 and 2. Nevertheless, the frequently applied additive strengthening mechanisms encounter certain challenges or mismatches in experi­ mental, modeling and simulation works (Kocks and Mecking, 2003; Jiang et al., 2017; Steinmetz et al., 2013; Takaki et al., 2004; He et al., 2017; Fan et al., 2018; Vinogradov and Estrin, 2018). The validations of various strengthening mixture rules are difficult to obtain, although many attempts have been made (Reppich et al., 1986; Büttner et al., 1987; Deschamps and Brechet, 1998; Dong et al., 2010; Queyreau et al., 2010). Key difficulties involve a precise measurement of each of the separate strengthening contributions and predominant roles of the phenomenological parameters in various models. For example, dislocation dynamics simulation analyses of Monnet and Devincre (2006) and Queyreau et al. (2010) indicated the complex superposition between solid solutions/Orowan strengthening and forest hardening. Soare and Curtin (2008) developed a constitutive model to account for dynamic strain aging in face-centered cubic (FCC) metals, which revealed the origin of the nonadditivity of solute and forest strengthening. In addition, by introducing the kinematical barriers concept, Steinmetz et al. (2013) established a physics-based strain rate- and temperature-sensitive dislocation density-based model for the strain-hardening behavior in twinning-induced plasticity steels. Although the introduced ε-Cu particles in ferritic steels brought remarkable strengthening contributions (Tsuchiyama et al., 2016), precipitation strengthening was found experimentally to be suppressed in Fe–C–Mn–Cu martensitic steels by Takaki et al. (2004) in which massive and fine ε-Cu particles were observed by atom probe tomography (APT). In the present work, to tackle the problem of forest hardening in conjunction with the strengthening of various obstacle families, a wide range of well-annealed (WA) body-centered cubic (BCC) steels and FCC Al alloys are prepared, in which hierarchical strength 2

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

levels of σ ss, σGB, and/or σ p are observed and quantitatively estimated. Then, the WA specimens are plastically deformed to various strain levels by cold rolling. In cases of both the steels and Al alloys, the quantified dislocation density is found to accumulate faster with increased plastic deformation level when the strengthening contributions of solid solutions, particles, and GBs are high in WA specimens. When plotting the yield stresses as a function of the square roots of the quantified dislocation densities in the deformed samples based on the Taylor hardening law (Taylor, 1934), an approximate single linearity is obtained, which indicates that the nonadditive strengthening mixture rule is employed completely between the obstacle families and the dislocation junction contri­ butions that are present in the distorted cubic metals. To explore the mechanisms of the observed nonadditive strengthening, a modified constitutive model based on the Knocks-Mecking approach is developed to capture the experimental results. The current nonadditive understanding is of essential importance in modern alloy design strategies, as well as modeling and simulation explo­ rations for material strengthening and work hardening. 2. Experimental procedure and method 2.1. Determination of the Hall-Petch relation in ferritic steels Various steel ingots (thickness: 80 mm), with their main chemical compositions given in Table A1 (Appendix A), were solution treated at 1473 K for 3.6 � 103 s, and then hot-rolled (1223 K–1473 K) to 12 mm thick plates. After grinding the surface (descaling), the plates were cold rolled (CR) at ambient temperature to 1 mm sheets. To obtain particular ferritic grain sizes, the CR sheets were annealed in a temperature range of 973 K–1173 K for various times. The recrystallized specimens were further subjected to a “standard” treatment of 873 K/1.8 � 103 s followed by water quenching (Akama et al., 2014; Takeda et al., 2008). The 1% Cu steel samples were not subjected to a “standard” treatment to avoid ε-Cu particle effects (Tsuchiyama et al., 2016). In the present steels with additions of various substitutional alloy elements, a small amount of Ti (Table A1) was also added to fix a few carbon and nitrogen as Ti (C, N) (Akama et al., 2014; Takeda et al., 2008). Tensile tests were carried out on standard specimens according to JIS13B with a strain rate of 1.0 � 10 3 s 1. The yield stress was evaluated at 0.2% proof stress. The average grain sizes were quantitatively evaluated based on massive optical microstructures, which were further confirmed by electron backscatter diffraction (EBSD). The Hall-Petch relations in IF steel, 0.0056% C steel, and 3% Ni steel have been reported in our recent works (Takaki et al., 2014; Akama et al., 2014; Takeda et al., 2008). 2.2. Preparation of cold-rolled ferritic steels By combining hot rolling (1223 K–1473 K) and unidirectional cold rolling (70% reduction), a series of ferritic steel (Table A1) sheets with thicknesses ranging from 1 mm to 10 mm were prepared first. Then, the sheets were fully recrystallized under particular conditions (973 K–1173 K) followed by water quenching to obtain the WA samples. The average grain sizes are summarized in Table A2 (Appendix A). WA 1% Cu and 3% Cu steels were also tempered at 773 K and 873 K for 3.6 � 104 s to introduce ε-Cu particle strengthening (Tsuchiyama et al., 2016). Unidirectional CR was applied to these WA/tempered steels (different thicknesses) to obtain 1 mm sheets for mechanical and microstructural examinations, which achieved CR reductions from 5% to 90% (Table A2). 2.3. Preparation of martensitic steels Industrial Fe–2Mn-0.5Si-xC (x ¼ 0.1, 0.2 and 0.3, Table A1) alloy plates (thickness: 15 mm) were annealed at 1273 K for 1.8 � 103 s, quenched in water and immediately held in liquid nitrogen for 1.8 � 103 s to obtain full martensitic microstructures. A group of asquenched (AQ) martensitic steels (MS) were also CR (10% reduction) to reduce the fraction of mobile dislocations (Akama et al., 2016; He et al., 2017; Azizi et al., 2018). Before mechanical and microstructural examinations, sufficient surface layers were removed to eliminate decarburization effects. 2.4. Preparation of cold-rolled pure Al and Al alloys Commercial pure Al plates (Table A1) with a thickness of 5 mm were processed to various thicknesses of 1 mm–2.5 mm by combining hot rolling (553 K–773 K) and common CR (50% reduction). Full recrystallization treatments at 698 K for 1.2 � 103 s were applied to the CR pure Al sheets to obtain WA samples (grain size: ~57 μm). For industrial 5383 Al alloy plates (thickness: 10 mm), hot rolling (553 K–773 K) and common CR (70% reduction) were adopted to prepare sheets with thicknesses ranging from 1 mm to 2.5 mm. The sheets were then annealed at 683 K for 1.2 � 103 s to achieve an average grain size of approximately 6 μm. Industrial 7150 Al alloy plates (thickness: 10 mm) were hot-rolled (553 K–773 K) and identical CR (50% reduction) to 1–2.5 mm sheets for further annealing at 743 K/900 s (grain size: ~51 μm). The AQ 7150 Al alloy samples were immediately tempered (at 443 K and 523 K for 105 s respectively) or CR to minimize natural aging effects. Tempering was adopted to introduce precipitate strengthening in the 7150 Al alloy (Deschamps and Brechet, 1998; Jiang et al., 2016). CR (5%–60% reductions) was carried out on the prepared pure Al, 5383 Al alloy, and 7150 Al alloy specimens to introduce dislocations (Table A2). The finished sheets had an identical thickness of approxi­ mately 1 mm for mechanical and microstructural characterizations. In addition, a special group of deformed 7150 Al alloy specimens (0%–60% reductions) were further tempered at 393 K for 8.6 � 104 s to examine the precipitate strengthening functions in deformed specimens. The grain sizes in pure Al and Al alloys were checked by EBSD. The precipitates and dislocations in steels and Al alloys were examined by transmission electron microscopy (TEM) on an FEI Tecnai G2 F20 instrument operating at 200 kV. Twin-jet 3

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

electropolishing was carried out on the prepared thin films for TEM observations. 2.5. X-ray diffraction experiments for examining dislocation density in cold rolled specimens The prepared specimens (CR steels, pure Al and Al alloys, and martensitic steels) were cut to dimensions of 15 mml � 12 mmw � 1 mmt for X-ray diffraction experiments. The specimens were ground to remove approximately 150 μm depths from the surface (normal direction, ND) using various sand papers. To fully remove the mechanically affected layers, approximately 50 μm thicknesses from the topmost surface were further removed by combining mechanical polishing and finishing with electrolytic polishing (Ung� ar and Borb� ely, 1996; Jiang et al., 2018). X-ray diffraction measurements were performed on an X-ray diffractometer (RINT2100, Rigaku Co. Ltd.) equipped with a Cu-Kα radiation source (40 kV, 40 mA) and using a step size of 0.02� . In BCC steels, six diffraction reflections were measured, i.e., 110, 200, 211, 220, 310, and 222. In FCC pure Al and Al alloys, nine diffraction reflections were checked, i.e., 111, 200, 220, 311, 222, 400, 331, 420, and 422. For line-profile-analysis (LPA), only Kα1 diffraction patterns were adopted based on a refinement with PDXL software (Rigaku Co. Ltd.) (Akama et al., 2016; Jiang et al., 2019). Instrumental effects were eliminated by utilizing the diffraction pattern of the NIST SRM 660c LaB6 standard powder (Jiang et al., 2019). The present results of yield stress and diffraction analysis were the average of three tested samples, and the standard deviations were given as error bars. 2.6. Quantitative dislocation density evaluation based on X-ray diffraction line-profile-analysis In the present work, the X-ray diffraction line profile broadening of distorted specimens can be primarily assumed to be the dominating source of crystallite size and distortions (mainly dislocations) (Williamson and Smallman, 1956; Wilkens, 1970; Ung� ar and �r and Borb�ely, Borb� ely, 1996). Then, the Fourier transform (A(L)) of the line profiles can be given as (Warren and Averbach, 1950; Unga 1996): LnAðLÞ ffi LnAsL þ LnADL ¼ LnAsL

(3)

2π2 g2 L2 < ε2g;L >

where AsL is the size Fourier coefficient, ADL is the distortion Fourier coefficient, L is the Fourier parameter, and g is the diffraction �r and Borb�ely, 1996), the mean square strain < ε2g;L > in dislocated crystals vector. According to Wilkens’s model (Wilkens, 1970; Unga can be expressed as the following function: � �2 b < ε2g;L >¼ ρCf ðηÞ (4) 2π where ρ, b and C represent the density, Burgers vector magnitude and average contrast factor of dislocations, respectively; f(η) is the L dependence of the mean square strain for dislocations; and η ¼ RLe , in which Re is the effective outer cut-off radius of the dislocations. To pffiffi better characterize the dislocation arrangement, a dimensionless parameter (M) is introduced: M¼ Re ρ (Wilkens, 1970). Accordingly, effective LPA methods have been developed to evaluate dislocation density and characteristics, which are typically �r and Borb� �rik et al., 2004; Scardi and Leoni, 2005). The analytical methods (Unga ely, 1996) and modeling-based methods (Riba �this et al., 2011; Dirras et al., 2012; Szabo � et al., 2015; Shi et al., 2015; preferred advantages of LPA have been gradually confirmed (Ma Khajouei-Nezhad et al., 2017), especially for highly dislocated crystals, because the TEM method normally provides reliable results around a dislocation density magnitude of 1014 m 2 (Dingley and McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Lan et al., 1992). In the present WA specimens, the line profile broadening was too small to be analyzed by LPA methods. Therefore, in this case, the dislocation densities in the WA specimens were assumed to be 2 � 1012 m 2 (Williamson and Smallman, 1956). In this work, crystallite size and dislocations are the main contributors to the line profile breadth. It is reasonable to fit experimental profiles using Voigt functions, which follow the summarized properties of Cauchy (Lorentzian) and Gaussian functions in analytical LPA methods (Langford, 1978; Sato et al., 2013). Then, the effective full width at half maximum (FHWM, β) and Fourier transform of the entire profile can be conveniently implemented by removing instrumental broadening. By introducing a dislocation contrast factor (C), the strain anisotropy in the classical Williamson-Hall (WH) method can be effectively corrected, which is called the modified Williamson-Hall (mWH) method (Warren and Averbach, 1950; Ung� ar and Borb�ely, 1996): ! ! ! � � 1=2 πAb2 πA’b2 1=2 2 ΔK ¼ α þ K C ρ1=2 KC1=2 þ (5) Q 2 2 kc θ where ΔK ¼ β cosλ θ and K ¼ 2 sin λ ; θ and λ are the diffraction angle and wavelength of X-rays, respectively; α ¼ d , where kc is a constant 0 (~0.9) and d is the crystallite size. A and A are constants determined by the effective outer cut-off radius of the dislocations; and the Q parameter is related to the two-particle correlations in the dislocation ensemble. In actual applications, the average dislocation contrast factor is generally applied based on the estimated Ch00 value (Borb� ely et al., 2003) and the following equation (Ung� ar and Borb� ely, 1996):

Chkl ¼ Ch00 ð1

(6)

qΓÞ

where q is the constant related to the elastic constants and character of dislocations, which can be estimated experimentally; Γ ¼ 4

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al. h2 k2 þh2 l2 þk2 l2 ðh2 þk2 þl2 Þ2

is the orientation parameter; and h, k and l are the Miller indices. The obtained orientation-dependent average contrast

factors can be further used to correct the anisotropy in the modified Warren-Averbach (mWA) method, which is called the modified �r and Borb�ely, 1996). Quantified dislocation density and Williamson-Hall and modified Warren-Averbach (mWH/mWA) method (Unga dislocation arrangement parameter can be primarily estimated, which are adopted to determine more precise input parameters in the following convolutional multiple whole profile (CMWP) method (Rib� arik et al., 2004; M� athis et al., 2011; Shi et al., 2015). In the current work, the dislocation density and arrangement parameter, which are quantitatively estimated from the modern �rik et al., 2004; Ma �this et al., CMWP method (modeling-based LPA method), are finally employed, as shown in the main text (Riba 2011; Shi et al., 2015). In the CMWP method, the experimental patterns are physically modeled and matched by the �rik et al., 2004): Marquard-Levenberg nonlinear least squares method and integrated modeling (Riba ! X I PM 2θ ¼ I Shkl I Dhkl I Inst (7) hkl þ IBG hkl

where IPM ð2θÞ represents the modeled patterns, IShkl is the size-related profile function assuming a spherical size distribution, ID hkl is the distortion-related profile function on the basis of Wilkens’s model, IInst hkl is the experimental instrumental profile function and IBG is the background of the pattern. The initial input parameters are estimated based on an analysis in the mWH/mWA method. The output results are also approximately verified by comparing with the primary results from the mWH/mWA method. 3. Experimental results and discussions 3.1. Hall-Petch relations in well-annealed ferritic steels Fig. 1 presents the Hall-Petch relations for ferritic steels (FS) which include little particle strengthening effect (Hall, 1951). Different solid solution strengthening abilities (intercepts) of various substitutional elements in steels are observed first. The intercept in interstitial-free (IF) steel is close to the friction stress of pure iron (~50 MPa). The 3% Si steel is found to yield the maximum σss value

Fig. 1. (a) EBSD images showing typical grain sizes in 1% Mn steel for determining the Hall-Petch relation. (b) Hall-Petch plots of 3% Si, 4% Al, 1% Cu, 1% Mn, and IF steels. The solid lines represent the fitted Hall-Petch relations from experimental data. The values of the Hall-Petch coefficients (slopes) are summarized in Table A2. 5

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

of approximately 237 MPa (Table A2). The Hall-Petch coefficient values (slopes of the plots in Fig. 1 (b)) in 3% Si steel, 4% Al steel, and 1% Mn steel are also larger than that of IF steel, which may have been caused by GB segregations of the added alloy elements (Hu et al., 2017; Takaki et al., 2014; Akama et al., 2014; Borovikov et al., 2017). Hence, the individual strengthening contribution of σ ss and σGB in WA specimens can be generally estimated under a particular grain size based on the obtained Hall-Petch relations (Table A2) (Zhu et al., 2014). 3.2. Evolutions of yield stress and quantified dislocation density in deformed specimens Fig. 2 (a) and (d) show selected engineering stress-strain curves of WA steels, pure Al and Al alloys, which indicate hierarchical strengths before cold rolling. The primary strengthening contributions in ferritic steels are σss and σGB, as summarized in Table A2. Meanwhile, additional strengthening functions from ε-Cu particles are also induced by tempering 1% Cu and 3% Cu steels (Fig. 2 (a) and A1 (a)) (Tsuchiyama et al., 2016). In the WA 5383 Al alloy, σ ss is dominated by a Mg addition and σGB is dominated by fine grains (Table A2) (Soare and Curtin, 2008; Furukawa et al., 1996). For the WA 7150 Al alloy, the primary σss from Zn, Mg, and Cu atoms are introduced when compared to pure Al. Furthermore, 7150 Al alloy specimens are tempered at 443 K and 523 K, respectively, to simultaneously introduce σp from η0 and η precipitates and relieve certain σss effects (Deschamps and Brechet, 1998; Jiang et al., 2016). To introduce dislocation strengthening, cold rolling is applied to the above WA samples. The CR reductions are summarized in

Fig. 2. (a) Engineering stress-strain curves of WA 3% Cu, 3% Si, 4% Al, 1% Cu, 1% Mn and IF steels for cold rolling. WA 3% Cu steel was tempered at 773 K (T773) for 3.6 � 104 s to introduce ε-Cu particles. The evolutions of yield stress (b) and dislocation density (c) with CR reductions. (d) Engineering stress-strain curves of WA pure Al, 5383 Al alloy, 7150 Al alloy and further tempered 7150 Al alloys (T443 and T523). The evolutions of yield stress (e) and dislocation density (f) with CR reductions are also given after deforming the alloys in (d). 6

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Table A2. The yield stresses of the CR specimens, as shown in Fig. 2 (b), (e) and A1 (b), increase more rapidly when high strength levels are included in the corresponding WA specimens. In addition, as shown in Fig. A2, the breadths of the diffraction peaks and the slopes in the mWH plots are proportional to the strength magnitudes of the primary obstacles in the WA samples under the same cold rolling �rik reduction, which qualitatively supports the magnitudes of dislocation density. Based on the analysis of the CMWP method (Riba et al., 2004; M� athis et al., 2011; Shi et al., 2015), the quantitatively estimated dislocation densities in deformed metals are also found to increase faster when the corresponding WA specimens display higher strengths (Fig. 2 (c), (f) and A1 (c)). The selected experimental and CMWP method fitted patterns of the distorted steels and Al alloys are given in Fig. A3. Very good matches are indicated by the slight differences that are presented in the CMWP method analysis. The estimated values of the dislocation arrangement parameter (M) from the CMWP method in all deformed specimens are summarized in Fig. 3, which are in a range of 0–1. In general, a large M value (M > 1) indicates a random dislocation distribution, wherein the dipole character and the screening of the displacement field for dislocations are relatively weak. When the M value is smaller than the unity (M < 1), cell structures or dislocation dipoles are formed which leads to strong dislocation interactions (Wilkens, �r and Borb� 1970; Unga ely, 1996). Therefore, the overall decreases of M values with increased rolling reduction (Fig. 3), which indicated enhanced dislocation interactions, also agree well with published observations of dislocations from TEM (Dingley and McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Lan et al., 1992; Segall and Partridge, 1959) and results from LPA (M� athis et al., 2011; � et al., 2015; Khajouei-Nezhad et al., 2017) in deformed steels and Al alloys. In Fig. 4, TEM micrographs Dirras et al., 2012; Szabo showing the dislocation evolutions of the present IF steel and pure Al under various deformation levels are also presented. At lower cold rolling reductions, both IF steel and pure Al exhibit dispersed dislocations. With increased reductions, cell boundaries are gradually formed and are accompanied be high dislocation density and strong interactions. The observations coincide very well with the estimated results of LPA in Figs. 2 and 3. 3.3. Summarized plots based on the Taylor hardening law Fig. 5 (a) summarizes the plot of yield stresses and square roots of the dislocation densities in deformed steels (given in Fig. 2) based

Fig. 3. Output values of the dislocation arrangement parameter (M) from the CMWP method: (a) CR ferritic steels and (b) CR pure Al and Al alloy specimens. Small M values represent strong dislocation interactions. The average M values (from all samples) are 0.43 for steels and 0.29 for pure Al and Al alloys. 7

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Fig. 4. TEM microstructures of IF steel at cold rolling reductions of (a) 3%, (b) 10%, (c) 20% and (d) 40% and pure Al at cold rolling reductions of (e) 5% and (f) 30%, which show typical dislocation characteristics under cold rolling.

on the Taylor hardening law (Taylor, 1934). Interestingly, the plots show an almost single linearity, which maintains a sole σ0 value approximating the friction stress of pure iron (σ 0 ¼ σ friction � 50 MPa) (Cracknell and Petch, 1955; Takaki et al., 2014). In other words, individual dislocation strengthening can respond to total strength. The strengthening contributions of various obstacles (Figs. 1 and 2, Table A2) almost vanish completely in the deformed steels due to the predominant strengthening of dislocation junctions (forest hardening). However, classical (both linear and nonlinear) additive understanding adds various obstacle strengthening contributions (σss, σp and/or σ GB) to the σ 0 value (Bhadeshia and Honeycombe, 2017; Deschamps and Brechet, 1998; Galindo-Nava et al., 2016; Steinmetz et al., 2013), which does not fit the present plots. To correlate the data of WA samples with that of deformed steels, particular dashed curves are presented for 3% Si steel and 3% Cu steel (T773), which tend to coincide gradually with a least squares fitting line. Therefore, the present results indicate a full nonadditivity of dislocation strengthening and strengthening from various obstacle families in the deformed steels, but they do not conform to the classical additive understanding. Fig. 5 (b) shows the extended plots up to higher (double) strength level and dislocation density by utilizing martensitic steels (MS) containing 0.1 to 0.3 wt% carbon. In AQ-MS, weak dislocation interactions are indicated by M values between 0.85 and 1, owing to the presence of massive mobile dislocations in the thermally formed MS (Akama et al., 2016; He et al., 2017; Azizi et al., 2018; Shi et al., 2015). After a 10% CR reduction, dislocations are rearranged to enhance the dislocation self-interactions, form dislocation cells, and result in M values (0.28–0.29) identical to those of deformed FS (Fig. 3 (a)) (Akama et al., 2016; He et al., 2017). It is also observed that CR-MS correlates to the previous fitting line in Fig. 5 (a) fairly well, while AQ-MS deviates slightly. Images of random dislocations with weak interactions are also frequently identified through TEM in FS when the deformation level is low (Fig. 4) (Dingley and McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Lan et al., 1992). The slightly decreased M values with increased CR reductions (Fig. 3 (a)) are a response to the enhanced dislocation interactions. Hence, the correlated dashed curves (Fig. 5) should be attributed to the gradual vanishing of mobile dislocations and enhancement of dislocation interactions (Fig. 4). The present plots are further verified by summarizing the results in various references, which quantitatively estimated dislocation density by utilizing TEM (Dingley and 8

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Fig. 5. Plots of yield stresses and square roots of dislocation densities in various steels. (a) Ferritic steels (FS). The selected dashed curves of 3% Si and 3% Cu (T773) steels were deduced for correlating the data of WA samples with that of deformed steels. (b) An extended figure of (a) to a high strength by coupling the results of AQ and CR martensitic steels (MS) with a carbon (C) content of 0.1–0.3 (wt. %). The high portion of mobile dislocation in AQ-MS is revealed by M values of 1, 0.98, and 0.85 in the 0.1, 0.2 and 0.3% C samples, respectively, leading to certain deviations from the fitted line. CR-MS shows small M values of 0.29, 0.29, and 0.28 in the 0.1, 0.2 and 0.3% C MS specimens, respectively and is in better agreement with the fitted plot in (a). The reported results from various references are also compared, which estimated dislocation density by utilizing TEM (Dingley and McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Lan et al., 1992) or LPA (M� athis et al., 2011; Dirras et al., 2012; Shi et al., 2015). Details of the calculated solution strengthening level in MS are given in Table A2.

�this et al., 2011; Dirras et al., 2012; Shi et al., McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Lan et al., 1992) or LPA (Ma 2015). It is observed that most of the results in published works commendably fit the present plots (Fig. 5 (b)). Fig. 6 presents the global strengthening superposition phenomenon in deformed pure Al and Al alloys. The different strengthening contributions in WA specimens (Fig. 2 (b)) mainly result from either fine grains in the 5383 Al alloy (Fig. 6 (a)) (Soare and Curtin, 2008; Furukawa et al., 1996) or precipitates in the 7150 Al alloy (Fig. 5 (b)) (Deschamps and Brechet, 1998; Jiang et al., 2016). In Fig. 6 (c), a similar trend to that of steels (Fig. 5) is exhibited with pure Al and Al alloys, showing the fully nonadditive strengthening functions when the obtained yield stresses (Fig. 2 (e)) are plotted against the square roots of dislocation densities (Fig. 2 (f)). A σ0 value, which is identical to the friction stress of high-purity Al, is shared (σ0 ¼ σ friction � 5 MPa) (Hansen, 1977). Such results clearly reveal the full nonadditivity of primary obstacles strengthening and dislocation strengthening in the deformed specimens. The results from various publications (Segall and Partridge, 1959; Dirras et al., 2012; Khajouei-Nezhad et al., 2017) are also compared and are observed to match well with the obtained results currently (Fig. 6 (c)). 3.4. Discussions on the strengthening mixture rule For a better understanding of the present results, Fig. 7 shows the comparison between the classical linear or nonlinear additive mixture rules and the present nonadditive strengthening functions. In the classical concepts, σ 0 in Eq. (1) is believed to include the contributions of σss, σ p and/or σ GB in the WA alloys (Bhadeshia and Honeycombe, 2017; Deschamps and Brechet, 1998; Galindo-Nava et al., 2016; Steinmetz et al., 2013). Alternatively, the strengthening contributions of different obstacles and dislocations can be in­ tegrated by Eq. (2) with k values between 1 and 2 (Soare and Curtin, 2008; Steinmetz et al., 2013; Dong et al., 2010). However, the experimental results shown here present a complete nonadditivity of strengthening contributions between various obstacle families and immobile (forest) dislocations. As shown in Figs. 5 and 6, the primary obstacle strengthening contributions in the WA specimens are not included in the total strength of the cold-worked steels and Al alloys. However, a strengthening overlap is observed, as illustrated in Fig. 7 (b). The original attributions are that during cold working, the obstacles remarkably accelerate the accumulation rate of dislocations under the same deformation conditions (Fig. 2). The influences of various obstacles on the accumulation of dislocations during cold working have been regularly reported. In ferritic steels, a 3% Si addition was found to intensively enhance dislocation accumulation as observed by TEM when comparing with low-carbon steels during cold working (Griffiths and Riley, 1966; Dingley and McLean, 1967; Evans and Rawlings, 1969). Waldron (1965) examined the dislocation development of Al–Mg alloys with different Mg contents by TEM during tensile testing and found that high Mg contents responded with a fast increase in dislocation density. Rib� arik et al. (2004) estimated the dislocation density in ball-milled Al–Mg alloys by XRD LPA methods and indicated that an increased Mg addition notably increased dislocation densities and decreased crystallite size. For instance, after ball milling for 3 h, the dislocation densities quantified by the CMWP method in pure Al, Al-3 wt.% Mg and Al-6 wt.% Mg alloys were 12 � 1014 m 2, 44 � 1014 m 2 and 100 � 1014 m 2, respectively. For the effects of grain size, Evans and Rawlings (1969) and Bailon et al. (1971) observed that the decreased grain size accelerated the accumulation of 9

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Fig. 6. Microstructures and mechanical responses in hierarchically strengthened pure Al and Al alloys. (a) EBSD microstructures of the WA 7150 Al alloy, 5383 Al alloy, and pure Al. (b) TEM images indicating the main η0 and η precipitates in the WA 7150 Al alloy specimens after additional tempering at 443 K (T443) and 523 K (T523) for 105 s, respectively. (c) Plots of yield stresses and square roots of dislocation densities of the pure Al and Al alloy specimens. Published results (Segall and Partridge, 1959; Dirras et al., 2012; Khajouei-Nezhad et al., 2017) based on TEM or LPA are also summarized and compared.

dislocations in ferritic steels. Recently, Tanaka et al. (2018) also confirmed that dislocation density increased in proportion to the inverse of the grain size of ferrite. When plotting the yield stresses in the cold worked specimens with the dislocation densities, the total strengths in the cold worked steels only depended on the strengthening contributions of dislocation density, regardless of initial grain size. In the TEM observations of a precipitate hardening Al alloy by Khadyko et al. (2016), a lower dislocation density was presented in the precipitate-free zones than in the regions with massive precipitates. In Fig. 8 (a), the introduced precipitates in 1% Cu steel are observed to strongly pin dislocations at cold rolling reductions of 5%. In the AQ 7150 Al alloy, a very high dislocation density is observed along the thick cell boundaries in Fig. 8 (b). In the tempered 7150 Al alloy (T443), as shown in Fig. 8 (c), the simultaneous functions of solid solutions and precipitates alter the uniformity of the dislocation distribution. In the regions with massive dislocations, numerous precipitates are also indicated and marked in Fig. 8 (c). In the TEM observations, it should also be noted that the dislocations for which bg ¼ 0 have not been indicated (or are invisible). Hence, the enhanced accumulation of dislocations by the introduced obstacles of solid solution strengthening, particle strengthening and GB in Fig. 2 and A1 agrees well with the published works (Griffiths and Riley, 1966; Dingley and McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Khadyko et al., 2016; Tanaka et al., 2018) and the present TEM observations. However, the underlying mixture rules and physically-based mechanisms of the strengthening functions between dislocations and various obstacle families are still in dispute and have been less investigated. Additive mixture rules have been better accepted and employed in massive published works to interpret the strength and work hardening behaviors found in metallic materials (Bhadeshia and Honeycombe, 2017; Deschamps and Brechet, 1998; Galindo-Nava et al., 2016; Steinmetz et al., 2013; He et al., 2017; Keller and Hug, 2017; Harjo et al., 2017; Vaucorbeil et al., 2013). Among massive attempts in exploring the complicated mixture rule of strengthening contributions and the physical mechanisms, two main strategies have been adopted in experimental, modeling and simulation works. In the first strategy, a particular mixture rule is assumed in advance based on Eq. (2) and a subsequent estimation of individual strengthening contributions (Deschamps and Brechet, 1998; Galindo-Nava et al., 2016; Liddicoat et al., 2010; Holmen et al., 2017). For instance, Deschamps and Brechet (1998) developed an integrated precipitation and strengthening model to fit the aging hardening responses in a predeformed Al–Zn–Mg alloy 10

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Fig. 7. Comparison between (a) the classical linear or nonlinear additive mixture rules and (b) the present nonadditive strengthening mechanisms. The WA “strengthened metal” sample indicates that the WA pure metal is strengthened by introducing obstacle families (for instance solid solutions, particles, and grain boundaries) for their strengthening contributions, such as the steels and Al alloys in Fig. 2.

Fig. 8. TEM bright field (BF) images showing the dislocation characteristics in (a) the 1% Cu steel (T873) with a cold rolling reduction of 5%; (b) the AQ 7150 Al alloy with a cold rolling reduction of 15% and (c) the tempered 7150 Al alloy (T443) with a cold rolling reduction of 5%.

11

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

based on a hypothesis of the linear additive mixture rule. Similarly, Galindo-Nava et al. (2016) added individual strengthening contributions directly to predict the strength of maraging steels. The reported theoretical and experimental approaches have also proposed empirical fits to select the best value for k in Eq. (2) (Queyreau et al., 2010; Dong et al., 2010; Vaucorbeil et al., 2013). Owing to considerable errors in the estimation of individual strengthening and phenomenological parameter evaluation for various strengthening models, alternative values of k (k ¼ 1 or k ¼ 2) can be adopted to fit a wide range of data. A definitive mixture rule and physical mechanism for multiple strengthening functions are still in dispute. In the studies of dynamic strain aging, the nonadditivity concept was noted and interpreted by the strong interaction between the solute hardening and strain hardening (Kocks et al., 1985; Soare and Curtin, 2008; Olmsted et al., 2006; Khater et al., 2014; Jobba et al., 2015). In the constitutive model developed by Soare and Curtin (2008), the origin of the nonadditivity of solute and forest strengthening was revealed by a mechanism of cross-core diffusion within a dislocation core that controlled the aging of both the solute and forest strengthening. The recent ideas of kinematical barriers were applied by Steinmetz et al. (2013) in their new constitutive model to reveal the strain-hardening behavior of twinning-induced plasticity steels. In the second strategy, the plot of yield stresses and square roots of the quantified dislocation densities is provided to account for the mixture rule of multiple strengthening functions based on the Taylor hardening law (Taylor, 1934; Dingley and McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Monnet and Devincre, 2006; Devincre et al., 2008; Mughrabi, 2016; Harjo et al., 2017). The primary argument is the assessment of the intercepts in the plot, i.e., σ 0 in Eq. (1), which has a distinct influence on the estimated value of the key parameter α. Though σ GB is suggested to be removed during the fitting process (Segall and Partridge, 1959; Dingley and McLean, 1967; Evans and Rawlings, 1969; Bailon et al., 1971; Lan et al., 1992), experimental yield stress in the corresponding WA specimens is frequently employed (Mughrabi, 2016; Harjo et al., 2017). Accordingly, the obtained values of α are generally explained based on dislocation characteristics and arrangements. As reviewed by Mughrabi (2016), effective values of the α factor were conventionally in a range of 0.1–0.4. In Mughrabi’s composite model, α was proportional to the square root of the cell wall volume fraction. Recently, Harjo et al. (2017) found that the values of α in a MS increased rapidly at the beginning of plastic deformation and then gradually tended to be constant with the progression of tensile deformation by utilizing in situ neutron diffraction experiments. Moreover, the M values gradually decreased to a constant level with increasing strain level, which indicated enhanced dislocation interactions. These results agree with the present plots of MS in Fig. 5 (b) very well. Monnet and Devincre (2006) also reported gradually decreasing α values at the initial deformation stage which tended to a constant value at large strain levels by utilizing

Fig. 9. (a) Effect of additional tempering (at 393 K for 8.6 � 104 s) for deformed 7150 Al alloys (0%–60% CR reductions) on mechanical behaviors. The precipitate microstructures of the tempered 7150 Al alloy (T393, 0% CR reduction) are also shown, which indicate the main η0 precipitate (Deschamps and Brechet, 1998; Jiang et al., 2016). TEM BF and SAED images of the tempered 7150 Al alloys with previous cold rolling reductions of (b) 5% and (c) 60%. 12

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

dislocation dynamics simulations and assuming a constant saturated dislocation density. They attributed this phenomenon to the progressively eliminated contribution of solid solution friction to strain hardening when dislocation strengthening was enhanced during plastic deformation. The remaining shortcomings of this strategy in the above experimental works arise from the limited data of examined alloys and notable errors in the quantified dislocation density. The present work employs a similar approach to that in the second strategy. At the same time, unfixed σ0 values and a wide range of hierarchical-strengthened alloys are adopted to minimize the effects from preconceived assumptions and experimental errors. Based on the least square fitting lines in Figs. 5 and 6, the values of the key factor α in the Taylor hardening law are estimated to be 0.36 for the steels and 0.65 for pure Al and Al alloys by hypothesizing the constant parameters (such as MT ¼ 2.9, b ¼ 0.248 nm, and μ ¼ 80 GPa for steels (Bhadeshia and Honeycombe, 2017; He et al., 2017) and MT ¼ 3.06, b ¼ 0.286 nm, and μ ¼ 26 GPa for pure Al and Al alloys (Furukawa et al., 1996; Deschamps, and Brechet, 1998)). The larger α value for the pure Al and Al alloys is attributed to the strong dislocation interactions indicated by the average M values (i.e., 0.43 for the steels and 0.29 for the pure Al and Al alloys in Fig. 3). Values of the α factor estimated in this work are also slightly larger than those in published works since (Mughrabi, 2016; Kocks and Mecking, 2003; Bailon et al., 1971; Lan et al., 1992; Harjo et al., 2017): (i) larger σ 0 values, which include the strengthening con­ tributions of various obstacles, are usually adopted to determine the α value in published works; (ii) higher forest dislocation density is quantified in many works but not the present total dislocation density; (iii) the initial dislocation accumulation stages at low strain levels (dashed curves in Figs. 5 and 6 and TEM in Fig. 4) are not examined in this work. To examine the observed nonadditive mixture rule, an additional experimental testing was implemented. Additional tempering treatments (at 393 K for 8.6 � 104 s) were carried out on the cold rolled AQ 7150 Al alloys. As shown in Fig. 9 (a), before tempering, the strength levels are found to be proportional functions of the initial CR reductions. When tempering the deformed alloys with various cold rolling reductions, it is observed that the strengthening magnitudes of the precipitates decreased with increased reduction. In the nondeformed alloy (0% reduction), the strength increases remarkably after tempering. However, at reduction of 60%, the strengths are almost identical in the CR specimens with and without additional tempering treatment. The reasons should be due to the heteroge­ neous precipitates in the distorted alloys and their competition with the dislocation junctions (Deschamps and Brechet, 1998; Liddicoat et al., 2010; Antolovich and Armstrong, 2014; Kuzmina et al., 2015). According to the TEM BF and selected area electron diffraction

Fig. 10. Schematic diagram for the new nonadditive understanding of various elementary strengthening contributions in distorted cubic metals. (a) Mobile dislocations pile-up owing to obstruction of various obstacle families (e.g., solid solutions, particles and grain boundaries) in non-distorted specimens. (b) Typical dislocation developments during plastic straining as observed under TEM. (c) Overview of the nonadditive strengthening functions between various obstacle families and immobile dislocations due to enhanced dislocations junctions and their preferred concentrations around obstacles. σ obs represents the strengthening contributions from various obstacle families. (d) and (e) schematic illustrations for the potential accounts of the nonadditive strengthening mechanisms. 13

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

(SAED) images in Fig. 9, dominant η0 precipitates are identified in all specimens. In the tempered 7150 Al alloy without predeformation (0% reduction), uniform precipitates are observed. When tempering the alloy with a previous 5% cold rolling reduction, numerous precipitates are observed in dislocation-free areas. Fine precipitates are difficult to observe around dark fields that are accompanied by massive dislocations. It has been confirmed that precipitation processes were accelerated by plastic deformation owing to a preferred nucleation and pipe diffusion effect at the dislocations (Deschamps and Brechet, 1998; Liddicoat et al., 2010). In Fig. 9 (c), the pre­ cipitates in the tempered alloy with a previous 60% cold rolling reduction are identified by a Moir�e pattern in the BF images and the patterns in the SAED. Therefore, at low CR reductions, the massive precipitates that form at dislocation-free zones should be effective in impeding dislocation motions, which retaining the strengthening of the tangled dislocations strengthening leads to a high strength (Fig. 9). In the severely deformed Al alloy (60% reduction), most precipitates coincide with the primary dislocation junctions, thus losing their strengthening abilities. Hence, the strength increase decreases when the tempering is applied to severely deformed samples. In summary, the above results verified the completely nonadditive strengthening contributions between various obstacles families and immobile dislocations in distorted cubic metals. The general terms of the new nonadditive understanding are interpreted in the schematic diagram of Fig. 10. Although the detailed physical mechanisms on the strengthening overlaps of dislocations and various obstacles are still unclear and additional experimental, modeling and simulation analyses are needed to demonstrate, three potential accounts are discussed here. First, the converted role (stabilizers) of primary obstacles for blocking immobile dislocation cross-slip in distorted metals may be employed, which results from the preferred dislocation concentration and interaction around such obstacles (Fig. 10(c) and (d)). As per the existing models (Deschamps and Brechet, 1998; Soare and Curtin (2008); Lu et al., 2009; Queyreau et al., 2010), various obstacle families anchored mobile dislocations in nondeformed samples. Consequently, the multiplication and entanglement of dislocations are enhanced during plastic deformation. Furthermore, the interactions or junctions of tangled dislo­ cations are concentrated more intensively around such obstacles because of strain localization (Fig. 8) (Antolovich and Armstrong, 2014; Kocks and Mecking, 2003; Khadyko et al., 2016); dislocation strengthening contributions then begin to gradually dominate. The surrounding obstacles lose their ability to impede fresh mobile dislocations (generated by the Frank-Read source under loading) due to faded internal interactions (Zhu et al., 2014; Kassner et al., 2013; Monnet and Devincre, 2006; Queyreau et al., 2010) or “isolation effects” (Antolovich and Armstrong, 2014; Kocks and Mecking, 2003) as illustrated in Fig. 10 (d). These obstacles solely stabilize immobile dislocations and retard the recovery in distorted specimens. Only the contributions of some residual obstacles (AS-MS in Fig. 5 (b)) or newly introduced obstacles (the tempered 7150 Al alloy specimens in Fig. 9), which are away from the dislocation junctions or not trapped by forest dislocations, are added to the total strength. The initial red dashed curves in Figs. 5 and 6 (stage II in Fig. 10 (c)) represent the integrated outcome of the retained obstacle strengthening and effective strengthening from immobile dis­ locations. Second, the primary obstacles may dominate the functions of specific amounts of immobile dislocations, as shown in Fig. 10 (e). During plastic deformation, the existence of obstacle families might limit the zipping processes of dislocation junctions (Monnet and Devincre, 2006; Queyreau et al., 2010; Vaucorbeil et al., 2013). The overlaps of initial obstacles with particular dislocation junctions can also be introduced due to the Gibbs system’s energy and the internal stress stabilization (Kassner et al., 2013; Kuzmina et al., 2015). Thus, the primary obstacle contributions may subsequently cut off certain dislocation strengthening functions. Finally, as proposed in modern crystal plasticity or constitutive models (Steinmetz et al., 2013; Soare and Curtin (2008); Vaucorbeil et al., 2013), and simulations (Monnet and Devincre, 2006; Queyreau et al., 2010), primary obstacles can produce kinematical barriers to dislo­ cation motion during plastic deformation. Consequently, the improved integrated model for such interactions also provide promising accounts for the presence of present nonadditive strengthening (Dong et al., 2010; Vaucorbeil et al., 2013). 4. Constitutive modeling 4.1. One-internal-variable models From a microscopic point of view, plastic deformation in metallic materials reflects the collective behavior of a vast number of dislocations. The different storage processes of statistical dislocations (Fig. 2) during cold rolling, which should be affected by various obstacle families, are the essential source for the nonadditive strengthening mixture rule in Figs. 5 and 6. According to the KocksMecking model, the evolution of dislocation density with the plastic deformation strain (ε) results from the competition of the pro­ duction and annihilation (rearrangement) of dislocations (Kocks et al., 1975; Kocks and Mecking, 2003; Estrin and Mecking, 1984; Bouaziz and Guelton, 2001; Keller and Hug, 2017; Vinogradov and Estrin, 2018), which is also known as the one-internal-variable model and can be expressed as follows: � � dρ 1 1 � k2 ρ ¼ MT (8) b Λ dε where ddρε indicates the storage rate of dislocations, Λ denotes the mean-free path of gliding dislocations and k2 is the dynamic recovery coefficient, which generally depends on the strain rate, temperature, and stacking fault energy in pure metals. In polycrystalline alloys, Λ could be governed by obstacles of various kinds at which gliding dislocations become immobilized. For instance, when precipitate particles and solute atoms are present, such barriers impede the mobile dislocations and affect the athermal storage rate. Hence, for the present steels and Al alloys with cold rolling reductions above 5% (Table A2), Λ1 can be shown as the following (Estrin and Mecking, 1984; Estrin, 1996; Bouaziz and Guelton, 2001; Vinogradov and Estrin, 2018):

14

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

1 X1 ¼ Λ Λi i P1 i

Λi

(9)

includes the effective contributions of the dislocation interactions and additional obstacles (e.g., grain boundaries, particles,

and solid solutions) in the mean free paths for collecting dislocations. The contribution of dislocation interactions is associated with the athermal storage of moving dislocations that become immobilized after having travelled a distance proportional to the average spacing between the dislocations. In addition to the frictional effects of the metallic matrix, the dislocation storage, owing to their interactions, can also be enhanced by supersaturated solid solutes (Cheng et al., 2003; Anjabin et al., 2013; Moghanaki and Kazeminezhad, 2016). Hence, Λ1f , which results from effective dislocation interactions, is the function of dislocation density and can be expressed by the following (Estrin and Mecking, 1984): 1 pffiffi ¼ kf ρ Λf

(10)

where kf is a constant related to the rates of the self-trapping of dislocation, which can be affected by the frictional function of the metal matrix, the characteristics (arrangement and edge/screw constituent) of dislocations and the introduced supersaturated alloy solutes. Grain boundaries can influence strain hardening by an additional contribution to the storage rate of dislocations. The mean free path (Λ1GB ) is generally prescribed by the grain size (d), as shown in the following equation (Estrin and Mecking, 1984; Bouaziz and

Guelton, 2001; Wang et al., 2014; Keller and Hug, 2017; Vinogradov and Estrin, 2018). 1 kGB ¼ ΛGB d

(11)

where kGB is a constant and is generally set as 1. In addition, the introduced particles/precipitates were also reported to work on the storage of dislocations during plastic defor­ mation (Kocks et al., 1985; Estrin, 1996; Roters et al., 2000; Queyreau et al., 2010; Vaucorbeil et al., 2013; Khadyko et al., 2016; Moghanaki and Kazeminezhad, 2016). As discussed by Estrin (1996), only nonshearable particles could account for the accumulation of dislocations, which worked on Λ by the average spacing. In Fe–Cu alloys, ε-Cu particles were softer than the ferrite matrix, which were found to be plastically deformed in step with the deformation of the ferrite matrix during cold working (Imanami et al., 2009; Tsuchiyama et al., 2016). In the present Fe–Cu alloys, as shown in Fig. A1 (c), deformable ε-Cu particles were also observed to remarkably enhance dislocation storage (Fig. 8 (a)). Especially at low strain levels (<20% reduction), both dislocation density and flow stress (Fig. A1) increase rapidly in tempered Fe–Cu alloys. When tempered Fe–Cu alloys were severely deformed, a low work hardening rate and slow increase in dislocation density were indicated because of the simultaneous deformation of soft ε-Cu particles (Imanami et al., 2009; Tsuchiyama et al., 2016). In contrast, with the tempered 7150 Al alloys (Fig. 2 (f)), the introduction of η0 and η precipitates could impede the movement of dislocations by nonshearable mechanisms (Deschamps and Brechet, 1998; Cheng et al., 2003). To capture the effects of particles/precipitates on dislocation storage, several models have been proposed (Estrin, 1996; Roters et al., 2000; Cheng et al., 2003; Queyreau et al., 2010; Moghanaki and Kazeminezhad, 2016; Anjabin et al., 2013). For instance, Queyreau et al. (2010) employed an Orowan loop storage mechanism in their dislocation dynamics simulation analysis. Following the concept of Roters et al. (2000), the dislocation slip length could be determined by particles that the mobile dislocations encounter on their way through the crystal. Hence, the slip length decreases with increasing particle density (small particle spacing). By taking into account the effective influence of the particle/precipitate ensemble, the effective mean free paths (Λ1P ) can be formulated as follows (Roters et al., 2000):

1 kP ¼ ΛP LP

(12)

where kP is a constant that depends on shearable or nonshearable particles and LP is the particle spacing estimated by the following (Deschamps and Brechet, 1998; Cheng et al., 2003): sffiffiffiffiffiffi 2π LP ¼ 1:15 r (13) 3f v where fv represents the volume fraction of the particles and r is the average size of the particles. In addition, the dynamic recovery rate in Eq. (8) is also dependent on the introduced dislocation, solute content and precipitates (Cheng et al., 2003; Anjabin et al., 2013; Moghanaki and Kazeminezhad, 2016). According to an irreversible thermodynamics approach (Vinogradov and Estrin, 2018), the dynamic recovery coefficient can account for the activation energy (ΔG) associated with the dislocation climb and average dislocation glide velocity (< v >) at given temperature (T). k2 ¼

2 þ 2α bν0 e 1 þ 2α < v >

(14)

ΔG kB T

where ν0 is the Debye frequency. Therefore, to capture the integrated influences of the above functions on the collected dislocations during cold rolling, a comprehensive constitutive model is obtained as follows: 15

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

� � � dρ 1 pffiffi kGB kP kf ρ þ ¼ MT þ b dε d LP

� (15)

fk2 ρ

where f represents a modifying factor due to the effects of solutes and particles on the dynamic recovery. Fig. 11 (a), (c) and (e) present the comparison of experimental and modeled dislocation evolutions during cold rolling. The established model (Eq. (15)) is found to rationalize the present experimental results of the deformed steels and Al alloys nicely. The Kocks-Mecking model with one structure parameter relating to the dislocation density has been demonstrated to provide very satis­ factory descriptions of plastic deformations whose characteristic strains exceed the transient strains (Kocks et al., 1975; Kocks and Mecking, 2003). Due to neglecting further state parameters (for instance mobile dislocations) whose fast relaxation may be responsible for transient behavior, the model fails to account for the short transients present in the beginning of plastic deformation (Estrin and Mecking, 1984; Estrin, 1996; Bouaziz and Guelton, 2001; Vinogradov and Estrin, 2018). In the present experimental work, the cold rolling reductions are larger than 5%. Hence, it is reasonable to ignore the presence of mobile dislocations and assume the dominance of immobile (forest) dislocations. As a result, the sole dislocation densities of deformed specimens above cold rolling reductions of 5% are shown in Fig. 11 (a), (c) and (e). The modeling results confirm the experimental observations that the introduced solid solutions,

Fig. 11. Comparisons of the experimental and modeling results of (a), (c) and (e) dislocation and (b), (d) and (f) yield stress in the cold rolled steels and Al alloys. 16

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

particles, and grain boundaries enhance the accumulation rate of dislocations in the steels and Al alloys. As summarized in Table A3, the value of parameter kf in IF steel is larger than that of pure Al owing to the higher friction stress of IF steel (Cracknell and Petch, 1955; Takaki et al., 2014; Hansen, 1977). Due to a higher stacking fault energy, a larger k2 value for the dynamic recovery of pure Al is obtained compared to that of the IF steel. By adding alloy elements, both kf and f values increase, which depend on the alloy content and alloy element. Therefore, the introduction of supersaturated alloy solutes affects both the dislocation generation and dynamic recovery. When tempering the Fe–Cu alloys or 7150 Al alloy, the introduced particles (ε-Cu particles or η0 and η precipitates) contribute to a rapid accumulation of dislocations, especially at low strain levels. Meanwhile, the kf value is decreases due to the decomposition of the supersaturated solid solution during tempering. The values of the kP parameter in the tempered Fe–Cu alloys are smaller than those in the tempered 7150 Al alloy. The reason for the above is due to the soft ε-Cu particles which can be plastically deformed under high levels of strain (Imanami et al., 2009; Tsuchiyama et al., 2016). In addition, the dynamic recovery is also found to be enhanced with a large f value in the tempered alloys. By employing the modeling dislocation density results, σy values are also calculated based on the Taylor hardening law (Eq. (1)), and these values are summarized in Fig. 11 (b), (d) and (f). The values of σ 0, which only include the friction stress of the IF steel or pure Al, are adopted for calculation, as observed in the present experiments (Figs. 5 and 6). The calculated σ y values in the deformed samples agree fairly well with the experimental results in Fig. 11 (b), (d) and (f). Therefore, the modeling results also reveal that individual dislocation strengthening responds to the evolution of yield strength in the deformed steels and Al alloys. By summarizing all the modeling results of yield stress as a function of the square roots of the dislocation densities, Fig. 12 (a) shows that the modeling results coincide fairly well with the experimental observations. The approximate single linearity is also identified by the established model. When the dislocation density is fixed, steels can obtain a higher strength than Al alloys. By plotting σ y with the group of MT μ b ρ1/2, a larger value of parameter α for the Al alloys compared to that for the steels is clearly indicated in Fig. 12 (b), which should be attributed to the different dislocation gliding and accumulation characteristics (Mughrabi, 2016), as shown in Figs. 4 and 8. 4.2. Two-internal-variable models As discussed above, the models employing single internal parameter of dislocation density is not sufficient for the histories involving lower strain levels, the changes of deformation path or the AQ martensitic steels containing numerous mobile dislocation densities (Estrin and Mecking, 1984; Estrin, 1996; Bouaziz and Guelton, 2001; Harjo et al., 2017; He et al., 2017; Vinogradov and Estrin, 2018). To devise a more flexible model for the purpose of distinguishing dislocation types whose densities evolved with different rates, the two-internal-variable models have been proposed by considering the mobile dislocation density (ρm) and the relatively immobile or forest dislocation density (ρf). The coupled differential equations for the evolution of mobile and forest dis­ locations read (Estrin and Kubin, 1986; Estrin, 1996; Vinogradov and Estrin, 2018): � � � � dρm C 1 ρf C3 pffiffiffiffi C 2 ρm ¼ MT 2 ρf dε b b ρm (16) � dρf C3 pffiffiffiffi ¼ MT ½C2 ρm þ ρf C 4 ρf dε b where C1 specifies the generation of mobile dislocations by the forest sources; C2 takes into account the decrease of mobile dislocation density by interactions between mobile dislocations; C3 represents the immobilization of mobile dislocations with a mean free path proportional to p1ffiρffiffi by assuming a spatially organized forest structure; C4 is associated with dynamic recovery by the rearrangement and f

annihilation of forest dislocations.

Fig. 12. Comparisons of the plots between (a) the yield stresses and square roots of the dislocation densities and (b) the yield stresses and the aggregate of MT μ b ρ1/2 based on the Taylor hardening law from the modeling results of the cold rolled steels and Al alloys. 17

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

In the practical metallic materials, the contributions of various obstacles to the dislocation accumulation were included during plastic deformation as verified in section 3 and discussed in section 4.1. Therefore, Eq. (16) could be modified by the similar ap­ proaches in section 4.1 and then assumes the following form: � � � � dρm C 1 ρf C3 pffiffiffiffi kGB kP ¼ MT 2 ρf C2 ρm dε b bd bLP b ρm (17) � dρf C3 pffiffiffiffi kGB kP 0 ¼ MT ½C2 ρm þ ρf þ C 4 ρf þ dε b bd bLP The modified parameter of C’4 in Eq. (17) takes the effects of solutes and particles on dynamic recovery into account. It is known that the slip of dislocation for strengthening metals could be impeded by various obstacle families and dislocation junctions. The additional parameters for the effects of various obstacles in Eq. (17) were estimated in the one-internal-variable models. After distinguishing dislocation types, the strengthening model based on Taylor hardening law (Eq. (1)) could be improved as below: � � � ρf pffiffiffiffi (18) σy ¼ σfricton þ σ ss þ σp þ σGB 1 þ MT αμb ρf ρm þ ρf where the full nonadditive strengthening mixture rule between the obstacle families and dislocations contributions was accounted by the fraction of forest dislocations. σ ss, σp and σ GB could be estimated from experimental results or classical strengthening models (Hall, 1951; Bhadeshia and Honeycombe, 2017; Deschamps and Brechet, 1998; Dong et al., 2010; Galindo-Nava et al., 2016; Steinmetz et al., 2013). Eq. (18) indicates that the introduced forest dislocation strengthening gradually dominates the strengthening contributions from various obstacle families, as illustrated in Fig. 10. Fig. 13 (a) and (c) show the comparisons of measured dislocation densities and modeled dislocation densities (ρm, ρf and total dislocation density, ρtotal ¼ ρm þρf) in selected steels and Al alloys. The established two-internal-variable model (Eq. (17)) is found to fit the experimental results fair well. In addition, the dislocation developments at low strain levels are also captured. The introduced solid solutions, particles, and grain boundaries are found to accelerate the accumulation rate of ρf and ρtotal. However, for mobile dislocation density (ρm), little effects from the introduced obstacles are indicated. At high strain levels, the magnitude of ρm is also observed to be much lower than that of ρf. In Fig. 13 (b) and (d), the modeled yield stresses by utilizing Eqs. (17) and (18) are found to commendably capture the experimental result. By collecting the modeling results of yield stress as a function of the square roots of the forest dislocation densities, Fig. 13 (e) presents that the modeling results coincide nicely with the experimental observations in both steels and Al alloys. The evolutions at low strains (stage II in Fig. 10 (c)) are rationalized in the developed two-internal-variable models as well. During initial straining stage, the introduced strengthening effects of forest dislocation gradually dominate the strengthening functions of primary obstacles due to the decreased fraction of mobile dislocations (Fig. 10 (c), Fig. 13 (a) and (c)). At high strain levels, forest dislocations are predominant and dominate the strengthening functions in metals. Therefore, the complete nonadditive strengthening mixture rule is suggested based on the present experimental works and constitutive models; interestingly, the above rule differs remarkably from the classical linear or nonlinear additive mixture rules (Bhadeshia and Honeycombe, 2017; Deschamps and Brechet, 1998; Galindo-Nava et al., 2016; Steinmetz et al., 2013). Further works are still needed to examine the strengthening functions under low strain levels (<5%) and interpret the physically-based mechanisms. 5. Conclusions In this work, crucial experimental outcomes and modeling were shown to demonstrate the distinctive nonadditive strengthening of elementary strengthening contributions in distorted BCC steels and FCC Al alloys. The main conclusions are summarized as follows: (a) By combining the experimental and referenced Hall-Petch relations of steels and Al alloy, hierarchical strength levels of solid solutions, GBs, and/or particles were estimated in the prepared WA specimens. In the cold worked steels and Al alloy samples, the accumulated dislocation densities were found to increase faster with an increasing plastic deformation level when the primary strengthening contributions of the solid solutions, GBs, and/or particles were high in the corresponding WA specimens. (b) By summarizing the experimental plots of the yield stresses and square roots of the dislocation densities based on the Taylor hardening law, an approximate single linearity was obtained, which also agreed with the published results very well. Individual dislocation strengthening was found to respond to the total strength in the deformed specimens from these plots. The quan­ titatively estimated strength levels of σss, σGB, and/or σ p in the WA specimens were not included in the total strength of the deformed alloys. As a result, the full nonadditive strengthening mixture rule between the contributions of various obstacle families and dislocation junctions was observed. (c) An incomplete strengthening overlap was further confirmed by utilizing martensitic steels and tempering predeformed 7150 Al alloys. The AQ-MS deviated slightly from the observed single linear relation because of the presence of mobile dislocations. In contrast, the CR-MS corresponded to the previous fitting line fairly well. When tempering the AQ 7150 Al alloys with various predeformation levels, a decrease in the strengthening magnitudes of the precipitates were observed when the cold rolling reduction was increased. (d) The modified constitutive models based on both one and two internal parameters’ Knocks-Mecking models were developed respectively, which excellently captured the effects of the various obstacle families on the accumulation rates of dislocations. 18

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Fig. 13. Comparisons of the experimental and modeling results of (a) & (c) dislocation densities and (b) & (d) yield stresses in selected steels and Al alloys. The modeled mobile dislocation density (ρm), the relatively immobile or forest dislocation density (ρf) and total dislocation density (ρtotal ¼ ρm þρf) are presented. (e) Comparisons of the plots between the yield stresses and square roots of the forest dislocation densities based on twointernal-variable models.

The developed models also rationalized the obtained nonadditive strengthening mixture rule in the deformed steels and Al alloys. In addition, the two-internal-variable models could reconstruct the competitive strengthening processes from low to high strain levels. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Fulin Jiang: Conceptualization, Data curation, Formal analysis, Investigation, Validation, Writing - original draft, Writing - review & editing. Setsuo Takaki: Conceptualization, Funding acquisition, Supervision, Writing - review & editing. Takuro Masumura: Data curation, Investigation, Software. Ryuji Uemori: Investigation, Methodology, Validation. Hui Zhang: Conceptualization, Investiga­ tion, Funding acquisition, Writing - review & editing. Toshihiro Tsuchiyama: Conceptualization, Funding acquisition, Supervision, Writing - review & editing. Acknowledgments This work was supported by the National Natural Science Foundation of China (51674111 & 51874127), the Fundamental Research Funds for the Central Universities (531118010353) and the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant (JP15H05768). Appendix B. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijplas.2020.102700. Appendix A Table A1 Chemical compositions (wt. %) of steels, pure Al and Al alloys in this work. “-”: Not examined. Steels (continued on next page)

19

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Table A1 (continued ) Steels Specimens

Si

Al

Cu

Mn

Ni

C

N

SþP

Ti

Specimens

Si

Al

Cu

Mn

Ni

C

N

SþP

Ti

3% Si steel 4% Al steel 1% Cu steel 3% Cu steel 1% Mn steel 3% Ni steel 0.0056% C steel IF steel MS-0.1C MS-0.2C MS-0.3C

3.14 0.022 0.004 0.004 0.016 0.01 <0.003 <0.001 0.50 0.50 0.50

0.066 3.97 0.035 0.05 0.053 – 0.004 0.005 – – –

0.005 0.002 1.13 3.08 0.003 – – – – – –

0.002 0.0049 0.002 0.003 1.07 0.0005 <0.003 <0.003 1.99 1.98 1.98

0.003 – – 0.01 0.003 3.02 – – – – –

0.001 0.0008 0.007 0.009 0.001 0.0007 0.0056 <0.001 0.10 0.20 0.31

0.0005 0.0006 0.0008 0.0005 0.0009 0.0034 0.0011 <0.0001 – – –

<0.0005 <0.0005 <0.0005 <0.0005 <0.005 <0.007 <0.003 <0.003 <0.004 <0.004 <0.004

0.028 0.019 0.0235 0.026 0.0278 0.076 – 0.014 – – –

Specimens

Zn

Mg

Cu

Mn

Cr

Ti

Si

Fe

–

7150 Al alloy 5383 Al alloy Pure Al

6.38 0.1 –

2.32 4.9 –

2.11 – 0.01

– 0.85 –

– 0.13 –

0.09 0.09 0.01

0.06 – 0.08

– – 0.31

– – –

Pure Al and Al alloys

Table A2 Summary of average grain size (d), Hall-Petch coefficient (ky), friction stress (σ friction), GB strengthening (σGB), solid solution strengthening (σ ss), particle strengthening (σP) and experimental yield stress (σ y) of annealed ferritic steels, martensitic steels, pure Al, and Al alloy specimens. The values of ky in interstitial-free (IF) steel, 0.0056% C steel and 3% Ni steel were given in our recent works (Takaki et al., 2014; Akama et al., 2014; Takeda et al., 2008). ky values of pure Al and Al alloys are summarized from references (Hansen, 1977; Furukawa et al., 1996). Solid solution strengthening ability of carbon in BCC steels is assumed to be 4500 MPa/1 wt% (Cracknell and Petch, 1955). σGB and σss values are estimated by Hall-Petch relation (Fig. 1b). The values of σP are estimated from the residuals of experimental σy (after subtracting other contributions). The detailed CR reductions for dislocation estimations are also summarized. “-”: Not estimated. Specimens

d (μm)

ky (MPa.μm 2)

σfriction (MPa)

σGB (MPa)

σss, (MPa)

σP (MPa)

σy (MPa)

CR reductions (%)

47 39 33 36 49 59 59 49 49 16 42 117 – – –

150 409 244 350 516 166 166 166 166 330 330 330 – – –

50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

22 65 42 58 74 22 22 24 24 83 51 31 – – –

– 237 136 44 11 57 162 288 555 14 14 14 1441 946 446

– – – – – –

76 347 230 163 124 134 234 352 629 134 107 86 – – –

5, 10, 20, 90 5, 10, 20, 70, 90 10, 20, 40, 60, 90 20, 80 5, 10, 20, 40, 60, 80 5, 10, 20, 50 5, 10, 20, 50 5, 10, 20, 50 5, 10, 20, 50 10, 20, 50, 90 10, 20, 50, 80 10, 20, 50, 80 0, 10 0, 10 0, 10

57 51 51 51 6

105 149 149 149 149

5 5 5 5 5

5 7 7 7 20

15 190 345 111 137

25 202 357 123 162

15, 30, 45, 60 5, 15, 35, 60 5, 15, 35, 60 5, 15, 35, 60 15, 35, 60

Steels IF steel 3% Si steel 4% Al steel 3% Ni steel 0.0056% C steel 1% Cu steel 1% Cu steel (T873) 3% Cu steel (T873) 3% Cu steel (T773) 1% Mn steel (16 μm) 1% Mn steel (42 μm) 1% Mn steel (117 μm) MS-0.1C MS-0.2C MS-0.3C

– – – – – –

Pure Al and Al alloys Pure Al 7150 Al 7150 Al 7150 Al 5383 Al

alloy alloy (T443) alloy (T523) alloy

– –

–

Table A3 Summary of parameters used in the one-internal-variable model (Eq. (15)). “-”: Not included. Specimens

k2

f

kf

kP (107)

2.36 2.36 2.36 2.36 2.36 2.36 2.36

1 1.12 1.07 1.25 1.23 1.26 1.16

0.011 0.013 0.014 0.013 0.012 0.017 0.023

– – – – – – –

Steels IF steel 0.0056% C steel 1% Mn steel (16 μm) 1% Mn steel (42 μm) 1% Mn steel (117 μm) 3% Ni steel 4% Al steel

(continued on next page)

20

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Table A3 (continued ) Specimens

k2

f

kf

kP (107)

3% 1% 1% 3% 3%

2.36 2.36 2.36 2.36 2.36

1.34 2.34 2.01 3.07 2.19

0.035 0.023 0.017 0.032 0.025

– – 2.1 2.6 1.9

3.02 3.02 3.02 3.02 3.02

1 1.12 2.19 1.53 1.91

0.029 0.017 0.051 0.019 0.022

– – – 2.9 3.3

Si steel Cu steel Cu steel (T873) Cu steel (T773) Cu steel (T873)

Pure Al and Al alloys Pure Al 5383 Al 7150 Al 7150 Al 7150 Al

alloy alloy alloy (T523) alloy (T443)

Table A4 Summary of parameters used in the two-internal-variable model (Eq. (117)). Specimens

C1

C2

C3

C4

IF steel 3% Cu steel (T873) 3% Si steel Pure Al 5383 Al alloy 7150 Al alloy (T443)

0.2 � 10 5 1 � 10 5 1.3 � 10 5 5 � 10 6 21 � 10 6 5.7 � 10 5

2 1 1 12 18 173

0.005 0.00265 0.0043 0.00095 0.00053 0.00011

1.2 6 4 5.6 3.2 13.9

0

Fig. A1. Effects of tempering treatments on mechanical properties and dislocation density in 1% Cu and 3% Cu steels. (a) Engineering stress-strain curves of WA 1% Cu steels, tempered 1% Cu and 3% Cu steels (at 773 K and 873 K for 3.6 � 104 s). The enhanced strength is clearly indicated because of the introduced ε-Cu particle strengthening, as reported in previous publications (Tsuchiyama et al., 2016). Evolutions of yield stress (b) and dislocation density (c) with CR reductions are indicated as well.

21

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Fig. A2. Qualitative evaluations of dislocation magnitudes by line profile broadening and mWH plots in various steels, pure Al, and Al alloys. (a) Peak broadening of reflection 222 in selected CR steels (common 20% reduction). Full width at half maximum (FHWM) is also indicated, which is proportional to the strength magnitude (Fig. 2 (b)). Instrumental broadening has been removed (Akama et al., 2016; Jiang et al., 2019). (b) mWH plots in selected 20% CR steels. The slopes generally represent the magnitudes of dislocation density. (c) Peak broadening of reflection 331 and (d) mWH plots in selected 15% CR pure Al and Al alloys.

22

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Fig. A3. Experimental and CMWP method fitted patterns coupled with their differences in selected steels and Al alloys: (a) IF steel with 10% CR reduction, (b) 90% CR 3% Si steel, (c) AQ-MS containing 0.3% C and (d) 15% CR 7150 Al alloy. The output dislocation density (ρ) and arrangement parameter (M) from the CMWP method are also indicated.

References Akama, D., Nakada, N., Tsuchiyama, T., Takaki, S., Hironaka, A., 2014. Discontinuous yielding induced by the addition of nickel to interstitial-free steel. Scripta Mater. 82, 13–16. https://doi.org/10.1016/j.scriptamat.2014.03.012. Akama, D., Tsuchiyama, T., Takaki, S., 2016. Change in dislocation characteristics with cold working in ultralow–carbon martensitic steel. ISIJ Int. 56, 1675–1680. https://doi.org/10.2355/isijinternational.ISIJINT-2016-140. Anjabin, N., Taheri, A.K., Kim, H.S., 2013. Constitutive modeling of hot deformation behavior of the AA6063 alloy with different precipitates. Metall. Mater. Trans. 44, 5853–5860. https://doi.org/10.1007/s11661-013-1916-3. Antolovich, S.D., Armstrong, R.W., 2014. Plastic strain localization in metals: origins and consequences. Prog. Mater. Sci. 59, 1–160. https://doi.org/10.1016/j. pmatsci.2013.06.001. Azizi, H., Zurob, H.S., Embury, J.D., Wang, X., Wang, K., Bose, B., 2018. Using architectured materials to control localized shear fracture. Acta Mater. 143, 298–305. https://doi.org/10.1016/j.actamat.2017.10.027. Bailon, J.P., Loyer, A., Dorlot, J.M., 1971. The relationships between stress, strain, grain size and dislocation density in Armco iron at room temperature. Mater. Sci. Eng. 8, 288–298. https://doi.org/10.1016/0025-5416(71)90095-4. Bhadeshia, H.K.D.H., Honeycombe, R.W.K., 2017. Steels: Microstructure and Properties, fourth ed. Butterworth-Heinemann, Oxford. Borb�ely, A., Dragomir-Cernatescu, J., Rib� arik, G., Ung� ar, T., 2003. Computer program ANIZC for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals. J. Appl. Crystallogr. 36, 160–162. https://doi.org/10.1107/S0021889802021581. Borovikov, V., Mendelev, M.I., King, A.H., 2017. Effects of solutes on dislocation nucleation from grain boundaries. Int. J. Plast. 90, 146–155. https://doi.org/ 10.1016/j.ijplas.2016.12.009. Bouaziz, O., Guelton, N., 2001. Modelling of TWIP effect on work-hardening. Mater. Sci. Eng. A 319, 246–249. https://doi.org/10.1016/S0921-5093(00)02019-0. Büttner, N., Fusenig, K.D., Nembach, E., 1987. On the additivity of precipitation and solid solution hardening in under-and over-aged single crystals of (CuAu)-Co. Acta Metall. 35, 845–854. https://doi.org/10.1016/0001-6160(87)90161-1. Cheng, L.M., Poole, W.J., Embury, J.D., Lloyd, D.J., 2003. The influence of precipitation on the work-hardening behavior of the aluminum alloys AA6111 and AA7030. Metall. Mater. Trans. 34, 2473–2481. https://doi.org/10.1007/s11661-003-0007-2. Cracknell, A., Petch, N.J., 1955. Frictional forces on dislocation arrays at the lower yield point in iron. Acta Metall. 3, 186–189. https://doi.org/10.1016/0001-6160 (55)90090-0. Deschamps, A., Brechet, Y., 1998. Influence of predeformation and ageing of an Al-Zn-Mg alloy—II. Modeling of precipitation kinetics and yield stress. Acta Mater. 47, 293–305. https://doi.org/10.1016/S1359-6454(98)00296-1. Devaraj, A., Joshi, V.V., Srivastava, A., Manandhar, S., Moxson, V., Duz, V.A., Lavender, C., 2016. A low-cost hierarchical nanostructured beta-titanium alloy with high strength. Nat. Commun. 7 https://doi.org/10.1038/ncomms11176, 11176-1-8. De Vaucorbeil, A., Poole, W.J., Sinclair, C.W., 2013. The superposition of strengthening contributions in engineering alloys. Mater. Sci. Eng. A 582, 147–154. Devincre, B., Hoc, T., Kubin, L., 2008. Dislocation mean free paths and strain hardening of crystals. Science 320, 1745–1748. https://doi.org/10.1126/ science.1156101. Dingley, D.J., McLean, D., 1967. Components of the flow stress of iron. Acta Metall. 15, 885–901. https://doi.org/10.1016/0001-6160(67)90371-9. Dirras, G., Chauveau, T., Abdul-Latif, A., Gubicza, J., Ramtani, S., Bui, Q., Heged} us, Z., Bacroix, B., 2012. Ultrafine-grained aluminum processed by a combination of hot isostatic pressing and dynamic plastic deformation: microstructure and mechanical properties. Metall. Mater. Trans. 43, 1312–1322. https://doi.org/ 10.1007/s11661-011-1028-x. Dong, Y., Nogaret, T., Curtin, W.A., 2010. Scaling of dislocation strengthening by multiple obstacle types. Metall. Mater. Trans. 41, 1954–1960. https://doi.org/ 10.1007/s11661-010-0229-z. Estrin, Y., 1996. In: Krausz (Ed.), Unified Constitutuve Laws of Plastic Deformation. Academic Press, Cambridge chap. 2. Estrin, Y., Kubin, L.E., 1986. Local strain hardening and nonuniformity of plastic deformation. Acta Metall. 34, 2455–2466. https://doi.org/10.1016/0001-6160(86) 90148-3. Estrin, Y., Mecking, H., 1984. A unified phenomenological description of work hardening and creep based on one-parameter models. Acta Metall. 32, 57–70. https:// doi.org/10.1016/0001-6160(84)90202-5. Evans, J.T., Rawlings, R., 1969. Work hardening in low-carbon steel deformed at room temperature. Mater. Sci. Eng. 4, 297–303. https://doi.org/10.1016/0025-5416 (69)90006-8. Fan, H., Ngan, A.H.W., Gan, K., El-Awady, J.A., 2018. Origin of double-peak precipitation hardening in metallic alloys. Int. J. Plast. 111, 152–167. https://doi.org/ 10.1016/j.ijplas.2018.07.016. Fang, Q., Chen, Y., Li, J., Jiang, C., Liu, B., Liu, Y., Liaw, P.K., 2019. Probing the phase transformation and dislocation evolution in dual-phase high-entropy alloys. Int. J. Plast. 114, 161–173. https://doi.org/10.1016/j.ijplas.2018.10.014. Farrokh, B., Khan, A.S., 2009. Grain size, strain rate, and temperature dependence of flow stress in ultra-fine grained and nanocrystalline Cu and Al: synthesis, experiment, and constitutive modeling. Int. J. Plast. 25, 715–732. https://doi.org/10.1016/j.ijplas.2008.08.001. Furukawa, M., Horita, Z., Nemoto, M., Valiev, R.Z., Langdon, T.G., 1996. Microhardness measurements and the Hall-Petch relationship in an AlMg alloy with submicrometer grain size. Acta Mater. 44, 4619–4629. https://doi.org/10.1016/1359-6454(96)00105-X. Galindo-Nava, E.I., Rainforth, W.M., Rivera-Díaz-del-Castillo, P.E.J., 2016. Predicting microstructure and strength of maraging steels: elemental optimization. Acta Mater. 117, 270–285. https://doi.org/10.1016/j.actamat.2016.07.020. Griffiths, D., Riley, J.N., 1966. Dislocation arrangements in deformed polycrystalline 3% silicon-iron. Acta Metall. 14, 755–773. https://doi.org/10.1016/0001-6160 (66)90123-4. Hall, E.O., 1951. The deformation and ageing of mild steel: III discussion of results. Proc. Phys. Soc., London B 64, 747–753. Hansen, N., 1977. The effect of grain size and strain on the tensile flow stress of aluminium at room temperature. Acta Metall. 25, 863–869. https://doi.org/10.1016/ 0001-6160(77)90171-7. Harjo, S., Kawasaki, T., Tomota, Y., Gong, W., Aizawa, K., Tichy, G., Shi, Z., Ung� ar, T., 2017. Work hardening, dislocation structure, and load partitioning in lath martensite determined by in situ neutron diffraction line profile Analysis. Metall. Mater. Trans. 48, 4080–4092. https://doi.org/10.1007/s11661-017-4172-0. He, B.B., Hu, B., Yen, H.W., Cheng, G.J., Wang, Z.K., Luo, H.W., Huang, M.X., 2017. High dislocation density-induced large ductility in deformed and partitioned steels. Science 357, 1029–1032. https://doi.org/10.1126/science.aan0177. Holmen, J.K., Frodal, B.H., Hopperstad, O.S., Børvik, T., 2017. Strength differential effect in age hardened aluminum alloys. Int. J. Plast. 99, 144–161. https://doi. org/10.1016/j.ijplas.2017.09.004. Hu, J., Shi, Y.N., Sauvage, X., Sha, G., Lu, K., 2017. Grain boundary stability governs hardening and softening in extremely fine nanograined metals. Science 355, 1292–1296. https://doi.org/10.1126/science.aal5166. Imanami, Y., Murakami, M., Nakada, N., Tsuchiyama, T., Takaki, S., 2009. Contribution of soft copper particle on work hardening behavior in ferritic iron. ISIJ Int. 49, 1225–1228. https://doi.org/10.2355/isijinternational.49.1225.

23

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Jiang, F., Zurob, H.S., Purdy, G.R., Zhang, H., 2016. Characterizing precipitate evolution of an Al-Zn-Mg-Cu-based commercial alloy during artificial aging and nonisothermal heat treatments by in situ electrical resistivity monitoring. Mater. Char. 117, 47–56. Jiang, F., Masumura, T., Hirata, K., Tsuchiyama, T., Takaki, S., 2019. A new diffraction line profile breadth analysis approach for evaluating plastic lattice strain anisotropy in cold-worked nickel under various strain paths. Int. J. Plast. 112, 89–107. https://doi.org/10.1016/j.ijplas.2018.08.006. Jiang, S., Wang, H., Wu, Y., Liu, X., Chen, H., Yao, M., Gault, B., Ponge, D., Raabe, D., Hirata, A., Chen, M., 2017. Ultrastrong steel via minimal lattice misfit and highdensity nanoprecipitation. Nature 544, 460–464. https://doi.org/10.1038/nature22032. Jiang, F., Hirata, K., Masumura, T., Tsuchiyama, T., Takaki, S., 2018. Effect of the surface layer strained by mechanical grinding on X-ray diffraction analysis. ISIJ Int. 58, 376–378. https://doi.org/10.2355/isijinternational.ISIJINT-2017-578. Jobba, M., Mishra, R.K., Niewczas, M., 2015. Flow stress and work-hardening behaviour of Al–Mg binary alloys. Int. J. Plast. 65, 43–60. https://doi.org/10.1016/j. ijplas.2014.08.006. Kassner, M.E., Geantil, P., Levine, L.E., 2013. Long range internal stresses in single-phase crystalline materials. Int. J. Plast. 45, 44–60. https://doi.org/10.1016/j. ijplas.2012.10.003. Keller, C., Hug, E., 2017. Kocks-Mecking analysis of the size effects on the mechanical behavior of nickel polycrystals. Int. J. Plast. 98, 106–122. https://doi.org/ 10.1016/j.ijplas.2017.07.003. Khadyko, M., Marioara, C.D., Ringdalen, I.G., Dumoulin, S., Hopperstad, O.S., 2016. Deformation and strain localization in polycrystals with plastically heterogeneous grains. Int. J. Plast. 86, 128–150. https://doi.org/10.1016/j.ijplas.2016.08.005. Khajouei-Nezhad, M., Paydar, M.H., Ebrahimi, R., Jenei, P., Nagy, P., Gubicza, J., 2017. Microstructure and mechanical properties of ultrafine-grained aluminum consolidated by high-pressure torsion. Mater. Sci. Eng. A 682, 501–508. https://doi.org/10.1016/j.msea.2016.11.076. Khan, A.S., Liu, J., 2016. A deformation mechanism based crystal plasticity model of ultrafine-grained/nanocrystalline FCC polycrystals. Int. J. Plast. 86, 56–69. https://doi.org/10.1016/j.ijplas.2016.08.001. Khater, H.A., Monnet, G., Terentyev, D., Serra, A., 2014. Dislocation glide in Fe–carbon solid solution: from atomistic to continuum level description. Int. J. Plast. 62, 34–49. https://doi.org/10.1016/j.ijplas.2014.06.006. Kocks, U.F., Mecking, H., 2003. Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48, 171–273. https://doi.org/10.1016/S0079-6425 (02)00003-8. Kocks, U.F., Argon, A.S., Ashby, M.F., 1975. Thermodynamics and kinetics of slip. Prog. Mater. Sci. 19, 1–291. https://doi.org/10.1016/0079-6425(75)90005-5, 1975. Kocks, U.F., Cook, R.E., Mulford, R.A., 1985. Strain aging and strain hardening in Ni C alloys. Acta Metall. 33, 623–638. Kuzmina, M., Herbig, M., Ponge, D., Sandl€ obes, S., Raabe, D., 2015. Linear complexions: confined chemical and structural states at dislocations. Science 349, 1080–1083. https://doi.org/10.1126/science.aab2633. Lan, Y., Klaar, H.J., Dahl, W., 1992. Evolution of dislocation structures and deformation behavior of iron at different temperatures: Part II. dislocation density and theoretical analysis. Metall. Trans. A 23, 545–549. https://doi.org/10.1007/BF02801172. Langford, J.I., 1978. A rapid method for analysing the breadths of diffraction and spectral lines using the Voigt function. J. Appl. Crystallogr. 11, 10–14. https://doi. org/10.1107/S0021889878012601. Li, J., Weng, G.J., Chen, S., Wu, X., 2017. On strain hardening mechanism in gradient nanostructures. Int. J. Plast. 88, 89–107. https://doi.org/10.1016/j. ijplas.2016.10.003. Li, Z., Pradeep, K.G., Deng, Y., Raabe, D., Tasan, C.C., 2016. Metastable high-entropy dual-phase alloys overcome the strength-ductility trade-off. Nature 534, 227–230. https://doi.org/10.1038/nature17981. Liddicoat, P.V., Liao, X.Z., Zhao, Y., Zhu, Y., Murashkin, M.Y., Lavernia, E.J., Valiev, R.Z., Ringer, S.P., 2010. Nanostructural hierarchy increases the strength of aluminium alloys. Nat. Commun. 1 https://doi.org/10.1038/ncomms1062, 63-1-7. Lu, K., Lu, L., Suresh, S., 2009. Strengthening materials by engineering coherent internal boundaries at the nanoscale. Science 324, 349–352. https://doi.org/ 10.1126/science.1159610. Lumley, R., 2018. Fundamentals of Aluminium Metallurgy: Recent Advances. Woodhead Publishing, Cambridge. Ma, K., Hu, T., Yang, H., Topping, T., Yousefiani, A., Lavernia, E.J., Schoenung, J.M., 2016. Coupling of dislocations and precipitates: impact on the mechanical behavior of ultrafine grained Al–Zn–Mg alloys. Acta Mater. 103, 153–164. https://doi.org/10.1016/j.actamat.2015.09.017. M� athis, K., Kraj� n� ak, T., Ku�zel, R., Gubicza, J., 2011. Structure and mechanical behaviour of interstitial-free steel processed by equal-channel angular pressing. J. Alloys Compd. 509, 3522–3525. https://doi.org/10.1016/j.jallcom.2010.12.142. Ming, K., Bi, X., Wang, J., 2019. Strength and ductility of CrFeCoNiMo alloy with hierarchical microstructures. Int. J. Plast. 113, 255–268. https://doi.org/10.1016/j. ijplas.2018.10.005. Moghanaki, S.K., Kazeminezhad, M., 2016. Modeling of the mutual effect of dynamic precipitation and dislocation density in age hardenable aluminum alloys. J. Alloys Compd. 683, 527–532. https://doi.org/10.1016/j.jallcom.2016.05.133. Monnet, G., Devincre, B., 2006. Solute friction and forest interaction. Philos. Mag. A 86, 1555–1565. https://doi.org/10.1080/14786430500398425. Mughrabi, H., 2016. The α-factor in the Taylor flow-stress law in monotonic, cyclic and quasi-stationary deformations: dependence on slip mode, dislocation arrangement and density. Curr. Opin. Solid State Mater. Sci. 20, 411–420. Olmsted, D.L., Hector Jr., L.G., Curtin, W.A., 2006. Molecular dynamics study of solute strengthening in Al/Mg alloys. J. Mech. Phys. Solid. 54, 1763–1788. https:// doi.org/10.1016/j.jmps.2005.12.008. Queyreau, S., Monnet, G., Devincre, B., 2010. Orowan strengthening and forest hardening superposition examined by dislocation dynamics simulations. Acta Mater. 58, 5586–5595. https://doi.org/10.1016/j.actamat.2010.06.028. Reppich, B., Kühlein, W., Meyer, G., Puppel, D., Schulz, M., Schumann, G., 1986. Duplex γ0 particle hardening of the superalloy Nimonic PE 16. Mater. Sci. Eng. 83, 45–63. https://doi.org/10.1016/0025-5416(86)90173-4. Rib� arik, G., Gubicza, J., Ung� ar, T., 2004. Correlation between strength and microstructure of ball-milled Al-Mg alloys determined by X-ray diffraction. Mater. Sci. Eng. A 387, 343–347. https://doi.org/10.1016/j.msea.2004.01.089. Roters, F., Raabe, D., Gottstein, G., 2000. Work hardening in heterogeneous alloys—a microstructural approach based on three internal state variables. Acta Mater. 48, 4181–4189. https://doi.org/10.1016/S1359-6454(00)00289-5. Sato, S., Wagatsuma, K., Suzuki, S., Kumagai, M., Imafuku, M., Tashiro, H., Kajiwara, K., Shobu, T., 2013. Relationship between dislocations and residual stresses in cold-drawn pearlitic steel analyzed by energy-dispersive X-ray diffraction. Mater. Char. 83, 152–160. https://doi.org/10.1016/j.matchar.2013.06.017. Scardi, P., Leoni, M., 2005. Diffraction whole-pattern modelling study of anti-phase domains in Cu3Au. Acta Mater. 53, 5229–5239. https://doi.org/10.1016/j. actamat.2005.08.002. Segall, R.L., Partridge, P.G., 1959. Dislocation arrangements in aluminium deformed in tension or by fatigue. Philos. Mag. 4, 912–919. https://doi.org/10.1080/ 14786435908238266. Shi, Z.M., Gong, W., Tomota, Y., Harjo, S., Li, J., Chi, B., Pu, J., 2015. Study of tempering behavior of lath martensite using in situ neutron diffraction. Mater. Char. 107, 29–32. https://doi.org/10.1016/j.matchar.2015.06.040. Soare, M.A., Curtin, W.A., 2008. Solute strengthening of both mobile and forest dislocations: the origin of dynamic strain aging in fcc metals. Acta Mater. 56, 4046–4061. https://doi.org/10.1016/j.actamat.2008.04.027. Steinmetz, D.R., J€ apel, T., Wietbrock, B., Eisenlohr, P., Gutierrez-Urrutia, I., Saeed–Akbari, A., Hickel, T., Roters, F., Raabe, D., 2013. Revealing the strain-hardening behavior of twinning-induced plasticity steels: theory, simulations, experiments. Acta Mater. 61, 494–510. https://doi.org/10.1016/j.actamat.2012.09.064. Szab� o, P.J., Field, D.P., J� oni, B., Horky, J., Ung� ar, T., 2015. Bimodal grain size distribution enhances strength and ductility simultaneously in a low-carbon low-alloy steel. Metall. Mater. Trans. 46, 1948–1957. https://doi.org/10.1007/s11661-015-2783-x. Takaki, S., Fujioka, M., Aihara, S., Nagataki, Y., Yamashita, T., Sano, N., Adachi, Y., Nomura, M., Yaguchi, H., 2004. Effect of copper on tensile properties and grainrefinement of steel and its relation to precipitation behavior. Mater. Trans. 45, 2239–2244. https://doi.org/10.2320/matertrans.45.2239.

24

International Journal of Plasticity xxx (xxxx) xxx

F. Jiang et al.

Takaki, S., Akama, D., Nakada, N., Tsuchiyama, T., 2014. Effect of grain boundary segregation of interstitial elements on Hall-Petch coefficient in steels. Mater. Trans. 55, 28–34. https://doi.org/10.2320/matertrans.MA201314. Takeda, K., Nakada, N., Tsuchiyama, T., Takaki, S., 2008. Effect of interstitial elements on Hall-Petch coefficient of ferritic iron. ISIJ Int. 48, 1122–1125. https://doi. org/10.2355/isijinternational.48.1122. Tanaka, Y., Takaki, S., Tsuchiyama, T., Uemori, R., 2018. Effect of grain size on the yield stress of cold worked iron. ISIJ Int. 58, 1927–1933. https://doi.org/10.2355/ isijinternational.ISIJINT-2018-371. Taylor, G.I., 1934. The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. R. Soc. Lond. - Ser. A Contain. Pap. a Math. Phys. Character 145, 362–387. https://doi.org/10.1098/rspa.1934.0106. Tsuchiyama, T., Yamamoto, S., Hata, S., Murayama, M., Morooka, S., Akama, D., Takaki, S., 2016. Plastic deformation and dissolution of ε-Cu particles by cold rolling in an over-aged particle dispersion strengthening Fe-2mass% Cu alloy. Acta Mater. 113, 48–55. https://doi.org/10.1016/j.actamat.2016.03.018. Ung� ar, T., Borb� ely, A., 1996. The effect of dislocation contrast on x-ray line broadening: a new approach to line profile analysis. Appl. Phys. Lett. 69, 3173–3175. https://doi.org/10.1063/1.117951. Vinogradov, A., Estrin, Y., 2018. Analytical and numerical approaches to modelling severe plastic deformation. Prog. Mater. Sci. 95, 172–242. https://doi.org/ 10.1016/j.pmatsci.2018.02.001. Waldron, G.W.J., 1965. A study by transmission electron microscopy of the tensile and fatigue deformation of aluminum-magnesium alloys. Acta Metall. 13, 897–906. https://doi.org/10.1016/0001-6160(65)90081-7. Wang, J., Beyerlein, I.J., Tom� e, C.N., 2014. Reactions of lattice dislocations with grain boundaries in Mg: implications on the micro scale from atomic-scale calculations. Int. J. Plast. 56, 156–172. https://doi.org/10.1016/j.ijplas.2013.11.009. Warren, B.E., Averbach, B.L., 1950. The effect of cold-work distortion on X-ray patterns. J. Appl. Phys. 21, 595–599. https://doi.org/10.1063/1.1699713. Wilkens, M., 1970. The determination of density and distribution of dislocations in deformed single crystals from broadened X-ray diffraction profiles. Phys. Status Solidi 2, 359–370. https://doi.org/10.1002/pssa.19700020224. Williamson, G.K., Smallman, R.E., 1956. III. Dislocation densities in some annealed and cold-worked metals from measurements on the X-ray debye-scherrer spectrum. Philos. Mag. 1, 34–46. https://doi.org/10.1080/14786435608238074. Xu, S.S., Zhao, Y., Chen, D., Sun, L.W., Chen, L., Tong, X., Liu, C.T., Zhang, Z.W., 2019. Nanoscale precipitation and its influence on strengthening mechanisms in an ultra-high strength low-carbon steel. Int. J. Plast. 113, 99–110. https://doi.org/10.1016/j.ijplas.2018.09.009. Zecevic, M., Knezevic, M., 2015. A dislocation density based elasto-plastic self-consistent model for the prediction of cyclic deformation: application to AA6022-T4. Int. J. Plast. 72, 200–217. https://doi.org/10.1016/j.ijplas.2015.05.018. Zhu, Y., Wang, H., Zhu, X., Xiang, Y., 2014. A continuum model for dislocation dynamics incorporating Frank–Read sources and Hall–Petch relation in two dimensions. Int. J. Plast. 60, 19–39. https://doi.org/10.1016/j.ijplas.2014.04.013.

25