# Size effects on the plastic behavior of polycrystalline materials: Grain size, precipitation state and free-surface effects

Damien Texier, Julien Genée, Vincent Velay, Antonio Castro Moreno, Daniel Monceau, Eric Andrieu

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Damien Texier, Julien Genée, Vincent Velay, Antonio Castro Moreno, Daniel Monceau, et al.. Size effects on the plastic behavior of polycrystalline materials: Grain size, precipitation state and free-surface effects. International Journal of Plasticity, 2025, 188, pp.104284. 10.1016/j.ijplas.2025.104284. hal-04980104

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Submitted on 6 Mar 2025

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Size effects on the plastic behavior of polycrystalline materials: grain size,
precipitation state and free-surface effects

Damien Texier<sup>a,b,*</sup>, Julien Genée<sup>a</sup>, Vincent Velay<sup>a</sup>, Antonio Castro Moreno<sup>c</sup>, Daniel Monceau<sup>b</sup>, Eric Andrieu<sup>b</sup>

<sup>a</sup>Institut Clement Ader (ICA) - UMR CNRS 5312, Université de Toulouse, CNRS, INSA, UPS, Mines Albi, ISAE-SUPAERO, Campus Jarlard, 81013 Albi Cedex 09, France

<sup>b</sup>CIRIMAT-UMR CNRS 5085, Université de Toulouse, ENSIACET, 4 Allée Emile Monso, BP 44362, 31030, Toulouse Cedex 4, France

<sup>c</sup>IRT Saint-Exupery, 3, Rue Tarfaya, 31400 Toulouse, France

**Abstract**

Surface effects were investigated using ultrathin specimens with thicknesses in the order of the grain size of the material. The candidate material was a polycrystalline Ni-based superalloy (Alloy 718) purposely heat treated to document both the effects of the grain size and the metallurgical state, i.e., solid solution and pre-cipitation hardened state, on the polycrystalline-to-multicrystalline behavior. Ultrathin tensile specimens were prepared with a dedicated technique to obtain specimens with thicknesses ranging between 20 and 550 µm, then tensile tested at room temperature. The polycrystalline-to-multicrystalline transition (PMT) was found to depend on the material grain size relative to the specimen thickness and to impair severely the tensile strength of the material. The yield strength, ultimate tensile strength (maximal stress on the stress-strain curve) and strain-to-failure severely dropped for specimens thinner than approximately two times the grain size of the material regardless of the metallurgical state. Such a decrease in tensile properties is mainly attributed to free-surface effects acting as an escape sink of dislocations, thus leading to a significant decrease of the primary dislocations density within the surface grains in comparison with the core grains. Interestingly, difference in work-hardening behavior with size reduction was found between both precipitation states, the solid solution state being more sensitive with the size reduction. The decrease in tensile properties was not found as expected from the commonly reported "thickness/grain size (t/D)" ratio. Therefore, a numerical approach using a modified Berveiller-Zaoui self-consistent model based on a continuum crystal plasticity approach was conducted in the present paper to distinguish microstructural features acting as strengthening (dislocation accumulation) and softening (dislocation escape at the free-surface) features. 3D numerical materials were produced using Voronoi tessellation methods to represent the fraction of "core grains" versus "surface grains". These fractions were then used as microstructural parameters for the identification of a crystal plasticity model using mean-field homogenization with different populations of grains, i.e., core versus surface features. The present work aimed at distinguishing the mechanical behavior of surface grains from core grains in Alloy 718 Ni-based superalloys using various thicknesses of specimens and different microstructure and metallurgical state variants.

**Keywords:** Polycrystal, Micromechanics, Grain boundary (GB), free-surface, Size effect

# 1. INTRODUCTION

The impetus for size reduction and even miniaturization of industrial components was motivated by trends to manufacture smaller, lighter, cheaper, material-saving, and power-saving products. Micro-sized components were particularly encountered in microelectronics, integrated optoelectronics, and micro-electromechanical systems (MEMS) applications [1]. Size reduction also concern structural components in various transportation applications for weight reduction purposes. Those applications require reliable and high performance components, whose efficiency strongly depends on their mechanical properties, and generally in

<footnote>*Corresponding author</footnote>

<footnote>Email address: damien.texier@mines-albi.fr (Damien Texier)</footnote>
complex environments. Size reduction amplifies the impact of surface-dependent behavior since the "surface/volume" ratio increases with the thickness reduction [2–6]. "Surface grains" in polycrystalline metallic materials are thus in front line position regarding surface evolution due to deformation, i.e., lower strain-hardening and dislocation densification due to dislocation escape [7–10], roughness evolution assisted by strain localization and dislocation escape (step height due to slip activity, persistent slip bands (PSB) and extrusion/intrusion under cyclic loading [11–14]), but also surface reactivity, i.e., oxidation [5, 14–19]. For instance, oxidation can promote depletion of strengthening precipitates due to selective consumption of elements to form the oxide scale, as encountered in Ni-based superalloys [20, 21]. "Surface grains" subjected to mechanical stresses and/or severe environments are thus particularly prone to early damage. Therefore, assessing and dissociating the mechanical behavior of "surface grains" versus "core grains" is particularly important for structural applications, especially when cyclic and mechanochemical coupling with the environment is of concern. The present paper intends to quantitatively assess the tensile behavior of "surface grains" and "core grains" using ultrathin tensile specimen with different surface/volume ratios to identify microstructural strengthening characteristics of a polycrystalline Ni-based superalloys (Alloy 718) at room temperature and low plastic deformation.

Making products smaller and smaller might raise material issues since component sizes can be as small as some microstructural features, i.e., grains, phases. These microstructural features participate in the mechanical strength of the materials at the grain/phase level and subsequently on the macroscopic mechanical behavior of the material. Size reduction in metallic materials is generally associated with a loss of mechanical strength, especially for few-grain-thick components due to dislocation escape at free-surface. Such size effect was widely investigated since the 1960's on various high purity single- or polycrystalline metals such as iron, nickel, copper, and aluminum [7–10, 22–26]. Below a given thickness, polycrystalline specimens show a behavior modification, exhibiting a lower mechanical strength. Thompson defined this critical thickness as the transition from a polycrystalline-to-multicrystalline behavior [23]. This transition in mechanical behavior was attributed to the increase of the "free-surface/volume" ratio of thin components and to differences in dislocation structures and/or size in the near-surface region compared to the core of specimens. An important metric to describe such size effects and the polycrystalline-to-multicrystalline transition is the thickness over grain diameter ratio, $t/D$ ratio. It approximately corresponds to the number of grains comprised in the specimen gage. This ratio has been reported in several studies and generally ranged between 3 and 32, depending on the studied material [7–10, 22–26]. In the near-surface region, the continuous loss of dislocation gliding out of crystals during straining contributes to a delay in dislocation clustering then tangling in dislocation cells compared to the core region, independent of the macroscopic work-hardening stage. Transmission electron microscopy observations of deformed single crystals [7, 8] and polycrystals [9, 24, 27] evidenced that the dislocation density increased significantly from the surface towards the inner region of the deformed components, resulting in a soft-surface effect. This soft-surface effect was reported to be particularly strong for grains oriented for single glide compared to multiple glide oriented grains [28].

The polycrystalline-to-multicrystalline transition (PMT) behavior was generally investigated for high-purity metals since they can be manufactured as rolled and thin products with high tolerances on the thickness. Performing additional heat treatments for grain growth and/or testing materials with different thicknesses makes such an investigation possible on high-purity materials. However, size-effect investigation of "bulky" materials with process-inherited microstructure and/or texture requires the development of machining processes for mesoscale specimens. Developments in microelectromechanical systems (MEMS) applications enabled both specimen micromachining and mechanical testing of microsize specimens [29]. Femtolaser machining [30], photochemical machining [31], micro-EDM carving [32] and electrodischarge machining (EDM) followed by tripod/precision jig polishing [5, 33, 34] would allow machining of millimeter-size specimens compatible with the PMT investigation. Here, laser stencil techniques combined with precision jig polishing was used in the present study to investigated size effects on centimeter long specimens suitable for representative elemental volume testing.

While soft-surface effects were intensively investigated from a metallurgical and experimental standpoint, mechanical and numerical approaches demonstrate a growing interest in modeling these size-effect and free-surface effects [10, 35–37]. Either semi-analytical approach considering a crystallographic inclusion in a semi-infinite and homogeneous medium (self-consistent model) or full-field finite element calculations of polycrystalline aggregate using either strain gradient crystal plasticity models or classical crystal plasticity models are capable to numerically reproduce the loss of deformation confinement in the presence of a free-surface. Geers et al. [35] investigated first-order, i.e., conventional continuum gradient
plasticity models, and second-order, i.e., strain gradient dependent crystal plasticity models, weakening effects in miniaturized polycrystalline components due to grain size dependence and dislocation density gradients at the sub-grain level. Both the models were necessary to computationally translate sample geometry-dominated and grain-dominated size effects due to influence of the grain size, orientation and grain boundaries. In addition, Keller et al. [36] performed numerical simulations of tensile tests for pure Ni specimens with different thicknesses and grain sizes using a strain gradient crystalline plasticity model. They successfully reproduced size effect softening associated with surface effects in relation to grain orientations, dislocation mean free path and long-range back-stress. Guilhem et al. [37] also implemented a computational term to characterize extrusion/intrusion steps for further documentation on favorable surface and sub-surface initiation sites under cyclic loading. However, despite the advance and complexity of these models, none of them considers dislocation annihilation at the free-surface, thus impacting the strengthening and work-hardening behavior of "surface grains". Therefore, the present study intends to develop a simple self-consistent model with different mechanical responses for "surface grains" and "core grains" due to difference in boundary conditions and capability of dislocation escape at the free-surface.

In the present study, the tensile response of a polycrystalline Ni-based superalloy (Alloy 718) has been experimentally investigated both in an annealed solid solution state and in a precipitation-hardened state. Different grain size microstructures (grain size ranging from 18 to 137 µm) and specimen thicknesses ranging from 550 down to 20 µm were studied to evaluate the polycrystalline-to-multicrystalline transition. Numerical microstructures representative of tested specimens were generated using 3D Voronoi tessellation to quantify "surface grain" versus "core grains" fractions. This fraction was taken as a simulation input in a modified Berveiller-Zaoui self-consistent model [38] based on a continuum crystal plasticity approach. Computational simulations aimed to dissociate the tensile response of "surface grains" and "core grains" to account for the experimentally reported size effects for the different specimen thickness/grain size ratios. The present model reproduced the present experiments on multicrystalline and polycrystalline specimens at low deformation levels (up to 1.0 % plastic strain), deformation domain particularly important for structural applications.

## 2. EXPERIMENTAL SECTION

### 2.1. *Experimental methods*

#### 2.1.1. *As received material*

A 635 µm-thick rolled sheet of Alloy 718 Ni-based superalloy was provided by Huntington Alloys® in a solution annealed condition with the nominal composition as follows: Ni-18.5Cr-18.3Fe-5.0(Nb+Ta)-3.0Mo-1.0Ti-0.6Al-0.03C (wt. %). The "as-received" microstructure consists of a polycrystalline γ-Ni solid solution with a grain size of 18 µm (excluding twins). Flat dogbone-shape tensile specimens were machined from the sheet by a laser-cut stencil technique with the longitudinal direction of the specimen, i.e., the further loading direction (LD), parallel to the rolling direction (RD). The geometry of the dogbone-shape tensile specimens consists in a 2 mm-wide and 20 mm-long gage area and 8 mm-wide and 10 mm-long gripping heads (2 mm radius between heads and gage area). Edges were gritted using SiC paper from P320 down to P4000 grades in order to remove the thermally affected region due to the laser-cut stencil technique and residual stresses.

#### 2.1.2. *Heat treatments for controlled microstructures and metallurgical states*

Heat treatments were performed on the 635 µm-thick tensile specimens in order to produce and investigate six different microstructural and metallurgical states: three different grain size microstructures and two intragranular precipitation states for each grain size microstructure (solid solution (SS) and precipitation hardened (PH) states). The fine grain (FG: $D_{FG} = 18$ µm – ASTM 8.5 (excluding twins)) microstructure corresponds to the as-received 18 µm-grain size material. The two coarser grain-size microstructures, i.e., medium grains (MG: $D_{MG} = 76$ µm – ASTM 4.5 (excluding twins)) and coarse grains (CG: $D_{CG} = 137$ µm – ASTM 3 (excluding twins)), were obtained via dwells at 1050 °C during 30 and 210 minutes, respectively. Thermal heat treatments were conducted under vacuum ($P_{tot} = 1 \times 10^{-5}$ mbar) on the flat dogbone-shape tensile specimens beforehand wrapped into a FeCrAlY foil for protection from severe oxidation. The temperature was controlled with a S-type thermocouple spot welded on a factitious Alloy 718 specimen inside the FeCrAlY envelop. Specimens were finally quenched in liquid-nitrogen to keep the high temperature SS microstructure at room temperature. Half of the quenched specimens were then aged with a two-step standard aging heat treatment in order to confer the precipitation hardened (PH) state for the three different grain
size microstructures: 720 °C-8 h-controlled cooling at 0.83 °C.min⁻¹ + 620 °C-8 h-furnace cooling. This standard aging (S.A.) heat treatment does not affect the grain size but ensures intragranular precipitation of around 4 % of ordered L₁₂-γ'-Ni₃(Ti,Al) and 15 % of coherent ordered DO₂₂-γ''-Ni₃Nb phases [39]. The different heat treatments, the associated microstructures and metallurgical states are reported in Table 1. Grain size will be accounted for equiaxed grains excluding twins by EBSD post-treatment (pseudo symmetry correction). This consideration is by the fact that (i) slip transmission at Σ3 twin boundaries is much easier than at general grain boundaries, leading to reduced work-hardening [40, 41], and (ii) intense strain localization near Σ3 twin boundaries generally occur in a fully transgranular manner in Ni-based superalloy [19], and especially in Alloy 718 [42].

### 2.1.3. Preparation of the thin tensile specimens

Thin dogbone-shape tensile specimens were prepared in pairs with a LOGITECH CL50® lapping machine combined with a PP5GT precision Jig to obtain various specimen thicknesses ranging from 20 to 550 µm. SiC papers down to a P4000 grad paper were used as an abrasive. Specimens were finally polished with diamond pastes down to 1 µm. The mounting of the tensile specimens as well as their disposition on the lapped glass support are central operations to limit vibration of the precision Jig during polishing. Specimens were polished two-by-two to limit surface to be polished and subsequently improve the material removal rate (removal rate between 30 to 120 µm.h⁻¹ depending on the grit paper), so that the misalignment of the precision jig is reduced between two rectifications. Having at least two specimens or one specimen with spare material is necessary to limit inclination and edge blunting on the specimens. Furthermore, mounting of the specimen was only done on specimens thicker than 100 µm prior to polishing to limit undulation of the high-compliant specimens because of difficulties in thermal wax escape. This gentle thinning down technique aimed to prepare centimeter long specimens with a relatively low variation of thickness (± 1.0 µm for specimens thicker than 100 µm and ± 0.5 µm for specimens thinner than 100 µm). This preparation technique is fully detailed in refs. [34, 43]. Based on the investigated grain size in this present work, specimens with *t/D* ratio between 30.6 and 0.2 will be tensile tested.

### 2.1.4. Tensile testing

Tensile experiments were conducted at room temperature with a MTS Synergie 1000 frame equipped with a 5 kN load-cell and self-tightening grips. Due to the wide range of specimen thicknesses, the load-cell was calibrated at five calibration steps in the low load domain (load <50 N). Tensile tests were performed at a constant crosshead-displacement rate to apply a strain rate of approximately 5 × 10⁻⁴ s⁻¹. Specimen elongation was measured with a non-contact extensometer (KEYENCE LS7030M) by following the relative displacement between two fiducial markers glued on the specimen gage. Two specimens were tested for each condition, one up to fracture and one up to 1 % plastic strain.

### 2.1.5. Characterization techniques

Grain-microstructure observations and fractographic analyses were performed using a LEO 435VP scanning electron microscope (SEM) operating at an accelerating tension of 15 kV in secondary and backscattered electron modes.

Large electron backscattering diffraction (EBSD) scans were conducted using a conventional JEOL 6100 SEM equipped with an EDAX acquisition setup operating with the EDAX OIM Analysis™. Large scans sampled a 280 × 500 µm², 2000 × 1800 µm², and 4000 × 2500 µm² areas with 0.25 µm-, 1 µm-, and 2 µm-step-size squared patterns for the FG-, MG- and CG-microstructures, respectively.

<table><caption>Table 1: Microstructural characteristics and associated heat treatment for the different metallurgical states</caption><thead><tr><td>Reference</td><td>Heat treatment</td><td>Grain size (µm)</td><td>Metallurgical state</td></tr></thead><tbody><tr><td>FG-SS</td><td>As received (A.R.)</td><td>18 ± 7</td><td>Solid solution</td></tr><tr><td>FG-PH</td><td>A.R. + S.A.</td><td>18 ± 7</td><td>Precipitate hardened</td></tr><tr><td>MG-SS</td><td>30 min – 1050 °C</td><td>76 ± 36</td><td>Solid solution</td></tr><tr><td>MG-PH</td><td>30 min – 1050 °C + S.A.</td><td>76 ± 36</td><td>Precipitate hardened</td></tr><tr><td>GG-SS</td><td>3 h30 – 1050 °C</td><td>137 ± 75</td><td>Solid solution</td></tr><tr><td>GG-PH</td><td>3 h30 – 1050 °C + S.A.</td><td>137 ± 75</td><td>Precipitate hardened</td></tr></tbody></table>
Transmission electron microscopy (TEM) observations states were performed using a JEOL JEM 2100F TEM apparatus operating at 200 kV in bright-field and dark-field observation modes in order to ascertain the precipitation state of the different metallurgical states investigated in the present study. Prior to TEM observations, 100 µm-thick samples were dimpled on both sides then ion-milled with a low angle (0 - 10°) GATAN precision ion-beam polishing system (PIPS).

## 2.2. *Experimental results*

<figure><img src="image_2.png" /><figcaption>Figure 1: Microstructure of the three grain-size microstructures represented as inverse pole figures (IFP) maps in reference to the further loading direction (LD) corresponding to the rolling direction (RD): (a) Fine grain microstructure (FG), (b) Medium grain microstructure (MG), and (c) Coarse grain microstructure (CG). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: {100}, {101} and {111} pole figures for the three grain size microstructure showing similar and weak crystallographic textures. This texture will be necessary for the numerical section. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article)</figcaption></figure>
### 2.2.1. Microstructure and metallurgical state characterizations

The three grain size microstructures are depicted in Figure 1 via inverse pole figure (IPF) representations in reference to the longitudinal direction, i.e., the further loading direction during tensile tests. All material variants exhibited equiaxed grains containing transverse annealing twins in a similar proportion, and non-preferentially oriented grains (multiple of uniform density (MUD) less than 3.7, as shown in Figure 2). It is worth noting that the heat treatment for grain growth did not alter neither the texture nor the annealed twin proportion. Grain size were computed from EBSD measurements, and the mean grain diameters ± 2 × standard deviation are reported in Table 1 for each microstructure.

<figure><img src="image_4.png" /><figcaption>Figure 3: TEM observations of the different metallurgical states. Low magnification bright field observation of (a) the FG-SS microstructure, (b) the MG-SS microstructure, (c) the FG-PH microstructure and (e) the MG-PH microstructure, and their associated diffraction patterns. The superimposed diffraction structures resulting from the $\gamma'$ and $\gamma''$ precipitates are visible for PH metallurgical states (d and e) and not for SS metallurgical states (a and b). High-resolution observation of the $\gamma$, $\gamma'$ and $\gamma''$ structure for (c) SS microstructure and (f) PH microstructure.</figcaption></figure>

TEM observations were performed to ascertain the intragranular precipitation states of all the metallurgical variants. As illustrated in Figure 3, no intra- and intergranular precipitation were observed for the solid solution metallurgical state (SS). Diffraction patterns are representative of the face-centered-cubic (FCC) austenitic structure of the $\gamma$-Ni solid solution (Figure 3(a) to (c)). Materials subjected to the standard aging (S.A.) heat treatment exhibited a homogeneously dispersed precipitation of $\gamma$'-Ni$_3$(Ti,Al) and $\gamma$''-Ni$_3$Nb phases (Figure 3(d) and (e)). High resolution TEM observations clearly showed the difference in atomic arrangement between the $\gamma$''-Ni$_3$Nb precipitates and the $\gamma$-Ni matrix (Figure 3(f)). Isolated disc-shaped $\gamma''$ precipitates as well as $\gamma''$-$\gamma'$ co-precipitates ($\gamma''$-single sided $\gamma'$ precipitates) were found in the PH-metallurgical state. The occurrence of this intragranular precipitation state is also visible via the superimposed diffraction structures evidenced on the diffraction patterns (Figure 3(d and e)). The heat treatments performed in the study successfully aimed to generate all the materials and metallurgical variants previously described and necessary for the present investigation.
### 2.2.2. Reference tensile behavior of the different metallurgical states

Tensile experiments were performed on thick specimens (thickness >500 µm) to assess the macroscopic polycrystalline behavior of each material (Figure 4(a)). These tensile tests and resulting tensile properties are considered as reference tests for each microstructural variants since the probed volume is closer to a representative elementary volume. As expected, the solid-solution materials (red plots in Figure 4(a)) exhibit a significantly lower mechanical strength and a higher ductility than the precipitation-hardened materials (blue plots in Figure 4(a)).

<figure><img src="image_5.png" /><figcaption>Figure 4: (a) Reference tensile behavior for the different metallurgical states performed on the thickest tensile specimens, (b and c) Stress-strain behavior of the MG-PH microstructure for various thicknesses ranging from 550 to 20 µm, and (d and e) Stress-strain behavior of the MG-SS microstructure for various thicknesses ranging from 550 to 20 µm. The color code was adapted to the specimen thickness for the MG-PH and MG-SS specimens. Black/gray and reddish curves correspond to specimen with a *t/D* ratio higher and lower than 2, respectively. Interestingly, a strong difference in stress-strain behavior was found for tensile specimens having less than two grains in the thickness regardless of the precipitation state. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article)</figcaption></figure>

In addition, the mechanical strength of the materials decreases for coarser grain microstructures for a given precipitation state, accordingly to Hall-Petch's expectations [44, 45]. The 0.2 % offset yield strength (Y.S.) decreases from 1247 MPa to 1003 MPa and from 433 MPa to 288 MPa with an increase of the grain-size microstructure from 18 to 137 µm for precipitation hardened (PH) materials and solid solution (SS) materials, respectively. In the following sections of the paper, all the properties measured on the very-thin tensile specimens will be compared to the ones obtained on reference specimens.

### 2.2.3. Thickness effect on the tensile behavior

A total of 120 microtensile tests were performed in the present investigation in order to evaluate the impact on mechanical properties debit for all the microstructure variants when decreasing the specimen thickness down to less than one complete grain in the thickness. In the present section, attention will be paid on the tensile behavior of the medium-grain size microstructure for both the precipitation state variants ((Figure 4(b) to (e)). Grayish curves and reddish curves correspond to specimen with a *t/D* ratio higher and lower than 2, respectively. Lighter graylevel curves correspond to thinner specimen but with more than two grains in the thickness. Darker reddish curves correspond to thinner specimen but with less than two grains in the thickness. It is worth noting that the strength properties, i.e., both the yield strength and the
ultimate tensile strength, and the strain-to-failure of the microtensile specimens decreased with the thickness reduction, especially for specimens having less than two grains in the thickness (red curves in Figure 4(b) to (e)). The "two grains in the thickness" $-t/D$ ratio of 2 – threshold is visually used owing to the existing literature review on size effects on the polycrystalline-to-multicrystalline transition behavior. The decrease in ultimate tensile strength with decrease in specimen thickness is intimately related to the loss in ductility and change in the work-hardening behavior of the Alloy 718 alloy, both in SS- and PH-states. Failure was found quite abrupt after the ultimate tensile strength for all the specimens, the solid solution metallurgical state having a slightly smoother failure.

### 2.2.4. Fractography

Fracture surfaces were investigated using scanning electron microscopy and all exhibited typical ductile features, such as dimples and area reduction due to necking (Figure 5). As aforementioned, the failure was found quite abrupt on the macroscopic stress-strain curve. Necking was yet evidenced on the fracture area on which the ductile fracture surface (region with several dimples) is surrounded by surfaces with slip marks on it, corresponding to the lateral faces of the specimen. This observation demonstrates that area reduction related to necking is indeed found. The thinner the specimen, the thinner the final fracture zone with dimples was. In the case of medium grain-size microstructure in a PH-state, a thin region with dimples is still observed in the core of specimens having at least half a grain in the thickness (Figure 5(c)). Thinner specimens exhibited a complete necking (Figure 5(d)). Furthermore, grains morphology was clearly observed on the fracture surface, highlighting deformation confinement within grains, especially for relatively thin specimens (Figure 5(b) to (d)).

<figure><img src="image_6.png" /><figcaption>Figure 5: Fracture surface of MG-PH tensile specimens having different $t/D$ ratio: $t/D = (a) 3.5, (b) 2.0, (c) 0.5, and (d) 0.2$. The final rupture surface was shown to decrease with the thickness, leading to a complete necking for $t/D$ ratio of 0.2.</figcaption></figure>

### 2.2.5. Tensile behavior in the early plastic deformation domain

In the present investigation, a particular focus was given to the evolution of the tensile behavior in the first five percents of strain as a function of the thickness reduction for all the metallurgical and microstruc-
<figure><img src="image_7.png" /><figcaption>Figure 6: Evolution of the 0.2 % offset Y.S. as a function of the specimen thickness for the three grain-size microstructure variants and for (a) the precipitation hardened and (b) the solid solution (SS) metallurgical states. Interestingly, a progressive transition in yielding behavior is noticed for specimen as thin as approximately two-to-four times the grain size (excluding twins) for both the PH and SS metallurgical states.</figcaption></figure>

tural variants (Figure 4(c) and (e)). As an example, the 0.2 % offset Y.S. was measured for all the specimens and are plotted as a function of the thickness of the specimens (Figure 6). Trends are added as a colored domain for each grain-size and metallurgical state variant to better visualize the evolution of the yield strength as a function of the thickness of the specimens. In addition, the "two-times the grain diameter" is shown in Figure 6 for a geometrical indication. As aforementioned, the yielding behavior of the specimens evolves similarly for all the variants, i.e., a stabilized mechanical strength over a given thickness (the polycrystalline behavior) and a progressive decrease of the mechanical strength below a threshold thickness (the multicrystalline behavior), depending on the microstructural and metallurgical variants. Interestingly, the transition from polycrystalline-to-multicrystalline behavior slightly differed between solid solution metallurgical state and precipitation hardened metallurgical state. Based on the slight discrepancy in yield strength results for such thin specimens, the polycrystalline-to-multicrystalline transition (PMT) is not trivial to determine. The transition has been defined as the intersection of the stress plateau related to the polycrystalline behavior and the slope related to the multicrystalline behavior. The PMT is highlighted for each microstructure variant in Figure 6. With this definition, the polycrystalline-to-multicrystalline transition was found of approximately 90 µm, 140 µm, and 180 µm for the FG-PH, MG-PH, and GG-PH metallurgical states, respectively. For the solid solution metallurgical state, the polycrystalline-to-multicrystalline transition was found of approximately 50 µm, 130 µm, and 200 µm for the FG-SS, MG-SS, and GG-SS conditions, respectively. The previously introduced *t/D* ratio of 2 does not perfectly match with the yield strength transition. However, there is an effect of the *t/D* ratio on the polycrystalline-to-multicrystalline transition. In addition, the slope in the multicrystalline behavior domain was found equivalent to 1.2 MPa. µm⁻¹ for all the PH-states, while the slopes decrease with the increase in grain size from 1.3 to 0.12 MPa. µm⁻¹ for fine grain to coarse grain microstructure in SS-states, respectively. The transition thickness from polycrystalline-to-multicrystalline behavior was also found to decrease with the grain-size of the specimens. The transition between the polycrystalline behavior and the multicrystalline behavior is progressive. The exact transition thickness is not straightforward to assess, especially for the coarse grain variants. Vertical black dashed lines were added on the graphs to represent specimens having two grains in the thickness, as aforementioned. This transition was found to particularly match with the reference of two-to-four grains in the specimen thickness for both the solid solution and the precipitation hardened metallurgical states. Such evolutions of the polycrystalline-to-multicrystalline behavior with the specimen thickness were observed
for different plastic strain levels.

To better illustrate the dependency of the polycrystalline-to-multicrystalline transition with both the thickness and the grain size, the "specimen thickness" over "grain diameter" ratio (t/D) was considered. As mentioned in the *materialsandmethods* section, thicker specimens were considered as reference specimens since they have a probed volume the closest to the representative elementary volume. The 0.2 pct. offset yield strength identified for all the tensile tests was normalized by the 0.2 pct. offset yield strength of the thicker specimen for a given metallurgical state. The normalized yield strength was plotted as a function of the "t/D" ratio, as shown in Fig. 7. All the curves gathered using this representation, demonstrating the importance of the t/D ratio to describe the polycrystalline-to-multicrystalline transition.

<figure><img src="image_8.png" /><figcaption>Figure 7: Evolution of the Normalized 0.2 % offset Yield strength as a function of the t/D ratio.</figcaption></figure>

As aforementioned, size effects tend to shift down the mechanical strength of the material for a given deformation level, as clearly depicted in Figures 4 and 6. In addition to this offset in yielding response with the reduction of the specimen thickness, the evolution of the work-hardening was also investigated for all the microstructure and metallurgical variants. Figure 4(c) and (e) illustrates the stress-strain response of the medium-grain size microstructure for the PH- and SS-states in the first five percents of deformation, respectively. PH-states and SS-states were found to slightly differ in work-hardening behavior with the reduction of the specimen thickness. The slope of the stress-strain curve at given plastic deformation levels were thus plotted as a function of the specimen thickness (Figure 8). The work-hardening properties of the PH-states were similar in the investigated range of thickness (Figure 8(a)). The work-hardening properties of the SS-states were found lower of approximately 10 % for specimens with less than two grains in the thickness (Figure 8(b)). The transition is not clear to affirm a progressive or stepped as a function of the specimen thickness.

### 3. NUMERICAL SECTION

Based on the experimental results, the present section intends to focus on (i) the development of a numerical model integrating microstructural parameters to describe size effects in polycrystalline materials related to the fraction of surface grains and core grains, and (ii) the identification of the material properties at the grain level for surface grains and core grains using mean field homogenization techniques. Despite the interior of the grains should have a similar mechanical behavior, surface grains and core grains does not have the same boundary conditions, leading to softening and less advanced dislocation networks, as
reported in the literature [7–10, 22–26]. Such free-surface effects, i.e., the surface softening due to dislocation escape and evolution of the dislocation landscape, are not necessarily considered in the literature for polycrystalline aggregate even in full-field numerical models. Therefore, the present model will (i) consider two populations of grains, i.e., surface grains and core grains, (ii) consider similar crystal plasticity law, and (iii) attribute different model parameters for each grain categories to consider early softening but also difference in work-hardening due to either dislocation annihilation at the free-surface or dislocation accumulation at grain boundaries using a modified Berveiller-Zaoui self-consistent model [38]. For information, if the same crystal plasticity parameters are used for both the surface grains and core grains, the present model evidenced no influence of the thickness of the specimen, i.e., fraction of surface/core grains, on the macroscopic tensile response.

### 3.1. Voronoï-tessellated microstructures for the calculation of surface/core grains fractions

As aforementioned, one of the input parameters of the modified Berveiller-Zaoui self-consistent model is the fraction of surface grains and core grains in order to dissociate their mechanical behavior. To have access to this microstructural parameter as a function of the thickness, numerical polycrystalline microstructures were generated using Voronoï tessellation techniques [46] in three dimensions (3D). Different 3D aggregates were generated using Neper open-source software [47] to have access of surface grains and core grains for different *t/D* ratios. Dimensions of the 3D aggregates were of 2 mm wide ×5 mm long ×thickness ranging between 550 and 25 µm. Seeds were randomly distributed in 3D following a uniform distribution compatible with the three-materials grain size. Finite domain of space was adapted to the thickness of the microtensile specimens. Examples of numerical microstructure for three different specimen thicknesses and three grain sizes are depicted in Figure 9. Semi-transparent reddish regions correspond to grain boundaries in the specimen volume while gray color regions correspond to the interior of the grains. The *t/D* ratio was reported for all the thickness/grain size variants. Peripheral grains, i.e., grain having a free-surface, and core grains, i.e., grains fully included in the volume, were labeled and the fraction of core grains was measured for each thickness/grain size condition. The evolution of the fraction of core grains as a function of the specimen thickness for the three different grain-size materials is depicted in Fig. 10(a). The fraction of the surface grains corresponds to the complementary response of the fraction of the core grains. The fraction of core grains was nearly stabilized for thick samples and progressively decreased with the thickness decrease. The severe decrease in fraction of core grains yet appears for thicker specimens (about 10 times the grain size). It is worth noting that the coarse-grain microstructure does not converge to 100 % of core grains

<figure><img src="image_9.png" /><figcaption>Figure 8: Evolution of the work-hardening at different plastic strain level as a function of the specimen thickness for the medium grain microstructure for (a) the precipitation hardened, (b) the solid solution (SS) metallurgical state.</figcaption></figure>
based on the restricted volume of the tensile specimen, showing that the mechanical response of the thickest specimens for the coarse-grain microstructure is significantly affected by the free-surface.

The normalized yield strength was plotted as a function of the fraction of core grains in Figure 10(b). Three different trends were identified for each grain-size microstructure; these trends better match with a power-law function. The variation in tensile properties (15 %, 12 %, and 8 % in normalized yield strength) compared to the fraction of core grains (100 %, 62 %, and 42 % in fraction of core grains) makes the modified Berveiller-Zaoui self-consistent homogenization techniques feasible. Interestingly, the fraction of core grains deviates from the 100 % value for the reference specimens (thickest specimens) for all the grain-size microstructure microstructures, the deviation being stronger for coarser-grain microstructure. The fraction of core grains is equal to 97 %, 69 %, and 43 % for fine-grain, medium-grain, and coarse-grain reference specimens, respectively. In other words, fine-grain, medium-grain, and coarse-grain reference specimens are affected by free-surface effects by 3 %, 31 %, and 57 %, respectively, and are thus more or less affected by the multicrystalline response.

## 3.2. Numerical model

In the present approach, a modified version of the model proposed by Berveiller and Zaoui [38] and derived from the self-consistent homogenization is used to describe the macroscopic response of polycrystalline microstructures using a crystal plasticity model. It is worth mentioning that the homogenization scheme is valid for spherical inclusion – here grains – embedded in an infinite continuous medium and constitutes limitation of the approach. The crystal plasticity model was considered to account for the initial crystallographic texture of the material and better describe intragranular slip activity. Based on the motivation of the present investigation related to structural application, the present model only considers the onset

<figure><img src="image_10.png" /><figcaption>Figure 9: Schematic illustration of the grain boundary network as a function of the specimen thickness and the material grain size. These numerical microstructures using three-dimension Voronoï tessellation techniques are used to evaluate the ratio of internal grain boundaries surface area over sample surface area</figcaption></figure>
of plasticity (up to 1.0 % plastic strain). This model limitation allows some simplifications, especially the interaction matrix between the different slip systems, and is not applicable for greater deformation levels. Only single slip activity without interaction with other systems, leading to self-hardening, was assumed. Under the assumption of a macroscopically isotropic elasto-plastic behavior of the polycrystal, the scale transition between the macroscale and the scale of the grain is expressed through the following localization rule:

$$ \underline{\sigma}^g = \underline{\Sigma} + 2\mu\alpha(1 - \beta)(\underline{E}^p - \underline{\varepsilon}_p^g) \quad (1) $$

Here, the uniform stress $\underline{\sigma}^g$ in each grain $g$ is function of the macroscopic applied stress $\underline{\Sigma}$ and the difference between the macroscopic average plastic stain $\underline{E}^p$ and $\underline{\varepsilon}_p^g$ the plastic strain in the same grain. $\mu$ is the shear modulus and $\beta = \frac{2(4-5\nu)}{15(1-\nu)}$ is close to $\frac{1}{2}$ for Poisson ratio $\nu \approx 0.3$. $\alpha$ is usually referred to as a factor of accommodation between the homogeneous equivalent medium and the considered grain. In the present case, a simplified form is used for its expression:

$$ \alpha \approx \frac{1}{1 + \mu \frac{3\underline{E}^p}{2\underline{\Sigma}}} \quad (2) $$

Conventional crystal plasticity equations are considered for the intragranular constitutive behavior. As slip is the predominant plastic deformation mode, the plastic strain in each grain $g$ is obtained from the sum of the shear strains acting on each slip system $s$, as illustrated by Eq. 3.

$$ \underline{\varepsilon}_p^g = \sum_{s=1}^{N_s} \dot{\gamma}^s \frac{1}{2} (\underline{l}^s \otimes \underline{n}^s + \underline{n}^s \otimes \underline{l}^s) \quad (3) $$

$\underline{n}^s$ is the unit vector normal to the slip plane, $\underline{l}^s$ the slip direction and $N_s$ the number of activated slip systems. The expression of the shear strain rate on each slip system follows the general framework of viscoplasticity:

$$ \dot{\gamma}^s = \left\langle \frac{|\tau^s| - r^s - \tau_c^s}{K} \right\rangle^n \text{sign}(\tau^s) = \dot{\bar{v}}^s \text{sign}(\tau^s) \quad (4) $$

In the present investigation, time-independent behavior is considered as a limit case by fixing a high value for $n$ and a low value for $K$. $\tau^s$ is the resolved shear stress, i.e., the projection of the stress tensor on each slip system $s$ (see Eq. 5).

<figure><img src="image_11.png" /><figcaption>Figure 10: (a) Evolution of the fraction of core grains as a function of the specimen thickness for the three different grain-size materials from two-dimension Voronoï tessellation analyses. (b) Evolution of the normalized yield strength as a function of the fraction of core grains.</figcaption></figure>
$$ \tau^s = \underline{\underline{\sigma}}^g : \frac{1}{2} (\underline{l}^s \otimes \underline{n}^s + \underline{n}^s \otimes \underline{l}^s) \qquad (5) $$

$\tau_c^s$ corresponds to the critical resolved shear stress required to activate the slip system s. $r^s$ is the isotropic hardening variable used to translate the interactions between dislocations. Its evolution follows an exponential function of the accumulated plastic strain $v^s$, as shown in Eq. 6. Such an evolution of the isotropic hardening is correct for the low deformation levels investigated in the present study.

$$ r^s = H \sum_{k=1}^{N_s} h_{sk} v^s \qquad (6) $$

$H$ is a material parameter related to the hardening behavior. Assuming that only octahedral slip is present, $h_{sk}$ is a 12 × 12 interaction matrix that represents the strength of the interaction between the slip systems s and k among the 12 face-centered cubic (FCC) slip systems. Here, only self-hardening is considered (i.e., $h_{sk} = \delta_{sk}$), based on the observations and interest at low deformation levels (less than 1.0 % plastic strain). Indeed, for Alloy 718, either in a solid solution or a precipitation hardened state, single slip is the most representative deformation mechanism encountered up to 1 % plastic strain [42, 48]. Only some grains experience multiple slip. It is worth noting that modeling the stress-strain response at greater deformation levels would require a more complex interaction matrix, considering different scenarios, i.e., co-planar slip, collinear slip but on different slip planes, slip systems with perpendicular Burgers vector, and other deformation mechanisms [49]. In the present model, no kinematic hardening variable is considered as this study focuses on monotonous loadings and Berveiller-Zaoui model is only suited for such loadings.

In this approach, two sets of parameters ($\tau_c^s$, $H$) are determined to allow the differentiation of hardening behavior between “surface grains” (grain category $g_1$ associated to volume fraction $f_{g1}$) and “core grains” (grain category $g_2$ associated to volume fraction $f_{g2}$). In the mean-field framework, grains are characterized only by their orientation. Here, the same population of 60 grains of random orientation is considered for each category, weighted by volume fractions $f_{g1}$ and $f_{g2}$ informed from previous section. The macroscopic plastic strain $E^p$ and the macroscopic stress $\underline{\underline{\Sigma}}$ are provided by the summations over all grains in the aggregate, as illustrated by Eq. 7 and Eq. 8.

$$ \underline{\underline{E}}^p = \langle \underline{\underline{\varepsilon}}_p^g \rangle_V = \sum_{g \in g_1 \cup g_2} \frac{f_g}{N_g} \underline{\underline{\varepsilon}}_p^g \qquad (7) $$

$$ \underline{\underline{\Sigma}} = \langle \underline{\underline{\sigma}}_1^g \rangle_V = \sum_{g \in g_1 \cup g_2} \frac{f_g}{N_g} \underline{\underline{\sigma}}_1^g \qquad (8) $$

with

$$ \sum_{g \in g_1 \cup g_2} f_g = 1 $$

### 3.3. Numerical simulations and model identification

Numerical simulations were performed using the modified Berveiller-Zaoui self-consistent model with volume fraction of the "surface grains" and "core grains" and an optimization procedure using a Levenberg-Marquardt algorithm was used for the parameter identification. The elastic anisotropy of the Alloy 718 was considered using the following parameters of the stiffness tensor $C$ [50]:

$C_{11} = 202$ GPa, $C_{12} = 124$ GPa, and $C_{11} = 106$ GPa, i.e., $E_{<001>} = 108$ GPa, $v_{<001>} = 0.38$, and $A = 2.7$

With $E_{<001>}$ and $v_{<001>}$, the directional Young's modulus and the Poisson's ratio along the $<001>$ crystallographic direction, respectively. A is the Zener ratio.

To consider the effect of grain size, an additional constraint was added in order to verify the Hall-Petch relationship for each metallurgical state (see Eq. 10).

$$ \tau_c^s = \tau_0^s + \frac{K^{HP}}{\sqrt{D}} \qquad (9) $$

The critical resolved shear stress $\tau_c^s$ is expressed as a function of the initial friction stress $\tau_0^s$ and the grain size $D$. $\tau_0^s$ and $K^{HP}$, the Hall-Petch constant, are considered as additional parameters of the model. Based
on the slight discrepancy in the experimental stress-strain response, several tensile tests were considered for the parameter identification of the model. The set of identified parameters is summarized in Table 2. Similarly, the monotonic tendency of the work-hardening parameter H with the grain size allows for the further affine equation:

$$H = H_0 + KH \times D \qquad (10)$$

As already mentioned, *n* and *K* parameters from the viscous contribution are fixed to respectively high and low values (*n*=11 and *K*=1). This set of parameters was chosen to limit strain rate effects during the mechanical loading but conserve the continuity of deformation gradient in the elasto-plastic transition. Based on the limited experimental results on the metallurgical states in the present investigation (one solid solution state and one precipitation hardened state), the cumulative strengthening effects related to solid solution and precipitation hardening will not be considered here. Indeed, no pertinent internal lengths between core grains and surface grains are capable to be dissociated for both these strengthening contributions. Parameter *H* will thus be identified for a given metallurgical state but will only depend on the grain size.

<table><caption>Table 2: Parameters of the numerical mean-field model to consider size effect in polycrystalline materials</caption><thead><tr><th>Ref.</th><th>τ<sup>s</sup><sub>0-Core</sub><br>(MPa)</th><th>K<sup>HP</sup><sub>Core</sub><br>(MPa.mm<sup>1/2</sup>)</th><th>H<sub>0-Core</sub><br>(b)</th><th>KH<sub>Core</sub><br>(Q)</th><th>τ<sup>s</sup><sub>0-Surf</sub><br>(MPa)</th><th>K<sup>HP</sup><sub>Surf</sub><br>(MPa.mm<sup>1/2</sup>)</th><th>H<sub>0-Surf</sub><br>(b)</th><th>KH<sub>Surf</sub><br>(Q)</th></tr></thead><tbody><tr><td>PH</td><td>373.9</td><td>22.7</td><td>-5460.3</td><td>32.4</td><td>341.1</td><td>15.1</td><td>-2454.2</td><td>7.7</td></tr><tr><td>SS</td><td>99.1</td><td>7.8</td><td>989.4</td><td>4.4</td><td>76.5</td><td>7.4</td><td>1096.1</td><td>3.3</td></tr></tbody></table>

Experimental (markers) and simulated (solid curves) stress-strain curves for the MG-PH and MG-SS are compared for different specimen thicknesses in Figure 11(a) and (b), respectively. Good agreement was found between simulations and experiments for different specimen thicknesses, i.e., different fractions of surface/core grains. In other words, the identified tensile behavior of both the surface-grain and core-grain populations satisfyingly represents the effect of the specimen size on the macroscopic tensile response for different fractions of the populations. The identified stress-strain curves of the surface-grain and core-grain populations were plotted in Figure 11 in light blue/orange and black, respectively. For the thinnest specimens in the medium-grain condition, the specimen gage was fully constituted of surface grains, as shown in Fig. 10. Therefore, the simulated tensile curve of the "surface-grain" population matches relatively well with the experimental data of the thinnest specimens, i.e., the fully "surface-grain" specimen. For the thickest specimens, 70 % of the gage section was constituted of core grains, the remaining fraction being attributed to surface grains. Therefore, the macroscopic tensile stress-strain curve for a "fictive" specimen fully constituted of core grains is slightly stronger than the one of the thickest specimens, i.e., the reference specimens. The simulated tensile curve of the "core grain" experiment corresponds to the tensile response of the polycrystalline material exempt of free-surface. Similar stress-strain identifications were done for all the microstructure variants. The greater the grain size, the greater the deviation in tensile response between the "core-grain" population and the "reference specimens" due to geometrical constraints imposed by the specimen geometry. Differences were noticed for thick FG-SS specimens due to their cold-work state compared to the restored states after either precipitation or grain growth treatments. To better illustrate the validity of the numerical simulation, experimental and numerical stresses for a given plastic strain are reported in Figure 12. The studied plastic strain levels are 0.2 %, 0.5 %, and 1.0 %. Trends obtained with numerical simulations are well describing the experimental evolutions.

# 4. DISCUSSION

Size effects were investigated in the present paper both in an experimental and simulation manner. These two aspects are discussed in the following sections.

## 4.1. *Specimen preparation: Long but thin specimens to document size effects*

In the present study, the experimental approach was to investigate size effects on a given material, the Alloy 718, by exploring different configurations of grain size and the precipitation state. To limit material artifacts, the same Alloy 718 plate was used to prepare all the specimens, leading to the same chemistry. In
addition, the texture was found insensitive to the grain growth and precipitation treatments. Therefore, sim-
ilar chemical composition, precipitation states, grain sizes, crystallographic textures were obtained for the
materials. The main difficulty in the present investigation was the specimen preparation in order to provide
acceptable dimensional tolerances and repeatable surface finish for reproducibility in mechanical response.
The present specimen preparation technique was tested in previous studies to investigate the mechanical
behavior of tens-of-micrometer-thick specimens from room temperature up to 1100 °C [5, 34, 51–54]. Me-
chanical tests were found coherent for specimens as thin as 30 µm but can evolve with the thickness based
on the polycrystal-to-multicrystal transition. Therefore, preliminary studies ascertain the potential repro-
ducibility and low discrepancy in mechanical response in the present investigation. The high-precision and
automatic polishing finish with an identical procedure for the surface finishing lead to a repeatable surface
state in terms of roughness and low residual stresses (the surface diffracts backscattered electrons using
EBSD). For the geometrical tolerances, repeated rectification of the polishing setup aimed to limit thickness
variation, as described in Ref. [34]. Despite particular attentions during ultrathin specimens preparation, the
thickness variation in the specimen gage was of ±0.5 to 1 µm, i.e., the resolution of the micrometer used
to measure the specimen thickness. The thickness variation along the specimen gage decreased with the
decrease of the specimen thickness due to stabilization of the glass support with the lubricant. Despite these
geometrical tolerances are little, they can yet correspond to an error of ±10 to 20 % on specimens thinner
than 20 µm. Additional specimens were thus prepared, tested but not considered for this study because of
thickness variations along the specimen gage being too important relative to the specimen thickness (thickness
variation greater than 5 %). In addition, having a thickness variation in the specimen gage can lead to
a local stress concentration and subsequent mesoscopic strain localization, especially for materials having
low work-hardening behavior. Local thickness variation can play a role on the thickness evaluation and
stress increase in minimal section locations (error on stress evaluation) but also intense macroscopic strain
localization due to section area inconsistency leading to error in strain evaluation (strain undervaluation).
Strain undervaluation might also lead to work hardening overvaluation and ductility undervaluation. Therefore,
work-hardening decrease with the decrease specimen thickness is a physical response of the size effect
opposite to the one expected from defect in geometrical tolerances. As aforementioned, several studies have
already investigated size effects on various materials with different approaches [7–10, 22–26]. From a technical
point of view, size effects were investigated using sheet products with thicknesses comparable to the
grain size and thus prone to the investigation of the polycrystal to multicrystal transition. Rolled sheet (or
wires) were often used with different thicknesses (resp. diameter). These approaches have the advantage of
highly controlled-thickness sheet/plate, leading to a more homogeneous thickness in the specimen gage and
less mesoscopic strain localization. Despite accurate geometrical tolerances, reducing the specimen size of
a specimen from 10 to 2.5 grains in the thickness lead to a two time reduction of the strain-to-failure for
the thinner specimen [27]. Interestingly, such a strain-to-failure reduction is similar to the one experienced
in the present investigation 4(b) and (d), testifying the application validity of our specimen preparation

<figure><img src="image_12.png" /><figcaption>Figure 11: Experimental and simulated stress-strain curves for three different specimen thicknesses of the medium grain microstructure: (a) Precipitate hardened state, (b) Solid solution state.</figcaption></figure>
technique. When dealing with rolled sheets/plates, the chemical composition of the sheet/plate and the texture, the grain size morphology, etc., can depend on the material batch and thickness reduction. This latter point might affect the recrystallization process and ultimately the texture and strengthening mechanisms of the material. In addition, such an approach is limited to rolled, extruded, or thin-walled materials, i.e., thin-manufactured components, and cannot be applied to any materials. Therefore, our present specimen preparation technique can be applied to any flat material. New developments in micromachining make it possible to machine microtensile specimens with dimensions as small as tens of micrometer for millimeter size specimens [55]. However, finishing operations or further surface treatments have to be carefully carried out to limit the machining-affected layer (recrystallized grains, work-hardening, etc.). Micromilling,

<figure><img src="image_13.png" /><figcaption>Figure 12: Evolution of the true stress for different macroscopic plastic strain level as a function of the specimen thickness: Stress at 0.2 pct plastic strain for (a) PH state, (b) SS-state, Stress at 0.5 pct plastic strain for (c) PH state, (d) SS-state, Stress at 1.0 pct plastic strain for (e) PH state, (f) SS-state.</figcaption></figure>
microturning, microelectrodischarge machining are very promising tools but require additional operations to provide "free-surface" conditions. At an even smaller scales, ion based techniques are also used to machine and prepare tens-of-micrometer size specimens. The stencil mask methodology paired with deep-reactive ion etching [56] aims to produce millimeter long specimens on the same specimen preparation batch. This technique is very interesting for thin-sheet components and have the advantage to prepare hundreds of specimens at the time. However, it has the drawback of potential and consequent material redeposition, thus affecting the intrinsic mechanical properties of the material to be tested. For non-flat specimens, femtolaser machining and photochemical machining are prominent solutions to manufacture miniaturized specimens but the machining affected layer remains an open question.

## 4.2. Size and surface effects in experimental studies

Working with specimens with reduced number of grains in the surface can exacerbate surface effects since "surface grains" do not experience similar boundary conditions than "core grains", due to the presence of the free-surface. In other words, "core grains" are more subject to grain incompatibility with surrounding grains and dislocation storage due to difficulties in plastic transmission at grain boundaries. Therefore, microstructural criterion such as the *t/D* ratio (the specimen thickness over the grain size diameter ratio), corresponding roughly to the number of grains confined in the specimen thickness, is generally used to describe the polycrystalline-to-multicrystalline transition. In the literature, similar evolution of yield strength are reported with specimen thickness reduction due to softening effects related to the presence of a free-surface [7–10, 22–26]. However, no consensus is found for the critical value of *t/D* ratio for the PMT transition in the different studies, this parameter being material dependent. Miyazaki et al. [24] performed a systematic analysis on various high purity sheet of metals (Al, Cu, Al-Cu alloy, Fe) and found critical values of *t/D* ratio ranging between 5 and 32 and varies with the grain size, larger grain size microstructures for a given material leading to smaller critical values of *t/D* ratio. Keller et al. [9] performed similar investigations on high purity Ni sheets and found a critical value of *t/D* ratio of 4 regardless of the grain size. Below this critical value, the authors attribute the change in yield strength and strain-hardening to a decrease in the contribution of the grain boundaries. In the present investigation, the critical value of *t/D* ratio was found between 2 and 4 (Figure 6).

In the plastic domain, the solid-solution metallurgical state demonstrated a decrease of the strain hardening with the specimen size reduction. This strain-hardening behavior is very comparable to the high-purity materials for which, grain boundaries plays a significant contribution in the hardening process, following Mecking's formulation of free-surface contribution to the strain hardening behavior of polycrystals [57]. For precipitation-hardened precipitation states, the strain-hardening behavior is less affected by the thickness reduction, and subsequently the critical value of *t/D* ratio. Homogeneous, fine precipitation brings another strengthening length that interfere with the role of grain boundaries in the hardening behavior. For interest, few recent papers are considering cumulative strengthening effects of solid solution and precipitation hardening due to weak/strong pair-coupling or Orowan process [58, 59]. Despite the high relevance of these papers, these models are developed for bulk materials without contribution of dislocation escape with a free-surface. Size effects are still present at the grain scale due to incompatibility of deformation from grains to grains. Such findings were not reported in the literature due to the solid-solution metallurgical state of high-purity metals. Therefore, the present paper demonstrates important and original capabilities in the size effect investigation of all type of materials.

As far as the mechanical testing is concerned, ultrathin and small size specimens limit the use of contact extensometers. Optical extensometers and digital image correlation techniques are well established to inform either on contactless macroscopic extensometry or local information using full-field measurements. This point will be further investigated to consider the mesoscopic strain localization, leading to a significant reduction of the strain-to-failure on tens-of-micrometer thick specimens. This point is important since very-thin specimens experienced a nearly infinite necking, as observed in Figure 5. Early failure for thin specimen can also be related to the greater statistical occurrence of a location along the specimen gage with few adjacent grains having a weaker orientations. Therefore, strain heterogeneity and mesoscopic strain localization might be found within the specimen gage for thin specimens. Furthermore, the increased variability in experimental strain-to-failure strain and fracture behavior with thinner specimens could also become more important with the Schmid-like behavior of the surface grains reported by Guilhem et al. [37]. Therefore, full-field kinematics assessment using digital image correlation during the tensile experiments
would better document this heterogeneous deformation and better inform on the true stress-true strain behavior of the materials with small dimensions. Another advantage of the full-field method is to consider intragranular stress/strain heterogeneity related to neighboring effects with adjacent grains that the present self-consistent model cannot depict.

### 4.3. Size and surface effects in numerical simulations

The evaluation of surface grains deformation is of high interest since it is the scene that can be (i) easily captured during the deformation process using optical or electronic observation means, (ii) compared to phenomenological models [60], (iii) a critical site of crack initiation under low cycle fatigue loading, and (iv) a population of grains to better consider for miniaturized and thin-walled components. Assessment of surface kinematics fields is very important for identification of crystal plasticity parameters in an integrated manner [61, 62] or in experimental-simulation direct comparison [48, 63]. However, the plasticity-free-surface has to be well considered in simulation works since the macroscopic stress-strain response as well as the heterogeneity of the sub-granular deformation is affected with the aggregate thickness for a given set of crystal plasticity parameter, as shown in Refs. [35, 36, 64]. Therefore, knowing the mechanical behavior of "surface grains" due to special surface boundary conditions – even for millimeter-size specimens – is central since crack initiation sites at the specimen surface and sub-surface are generally encountered under low cycle fatigue conditions. Developing simple models accounting for such distinctions in mechanical response between surface grains and core grains can be a step forward for the prediction of the fatigue life of structural components. The modified Berveiller-Zaoui homogenization coupled with crystal plasticity criterion is a simple and low-cost computational model that can be implemented in fatigue model after appropriate identification of potential isotropic and kinematics hardening/softening. The present experimental data offers an opportunity for a full description of the plastic behavior using some extra considerations of the interaction matrix $h_{sk}$ and potentially on the identified parameters of the linear hardening behavior. However, the present numerical model was limited to low deformation levels (1 % plastic strain) not to consider dislocation interaction since the evolution of the dislocation structures in surface grains and core grains with the macroscopic deformation was not characterized in the present study.

Size effects and free-surface effects are considered in different models: discrete dislocation dynamics [65–67], image decomposition method coupled with Leknitskii-Eshelby-Stroh (LES) formalism [68], Kocks-Mecking model [10], crystal plasticity with mean-field homogenization scheme [69–72], crystal plasticity with full-field simulations with or without internal deformation lengths [35, 37, 64, 73–77]. Full-field simulations with internal deformation lengths are also capable to capture microstructural size effects such as Hall-Petch effects. Such approaches were recently applied onto the Alloy 718 to account for precipitation and size effect on the plastic flow response of the material under tensile and fatigue stresses using dislocation density based crystal plasticity model [78]. This latter behavior is partly associated with deformation incompatibilities from grains to grains leading to local strengthening, as recently reported in several studies [79–82]. However, most of or nearly all these advanced crystal plasticity models does not account for a physical consideration of the free-surface with possibility of dislocation annihilation at the free-surface and limiting work-hardening of surface grains compared to core grains due to dislocation network maturity. Even full-field crystal plasticity simulations see evolution of "free-surface" effects as a result of air being a softer and more accommodating medium to deformation compared to an adjacent grain, leading to less stress concentration and thus lower dislocation density when GND are integrated in crystal plasticity equations. In the present work, the mean-field homogenization scheme was used to consider crystallographic texture, grain size, and specimen size effects to ascertain the mechanical response of thin components. The present modified Berveiller-Zaoui model intends to consider change in elastoviscoplastic behavior between surface grains and core grains to consider the dislocation annihilation capability of surface grains. This model considers different fraction of surface grains and core grains depending on the specimen thickness. Grains were considered fully embedded in a continuum medium, without particular surface conditions, except that crystal plasticity parameters were found distinctly for surface grains and core grains. In other terms, the surface grains and core grains consideration was done using a two-phase composite approach or microstructure-informed homogenization scheme for elasto-viscoplastic heterogeneous materials, as for Refs. [72, 83]. In addition, the Berveiller-Zaoui model initially considers spherical grains that grain truncation at the free-surface does not ensure. The present model approximation is considered as little in comparison with the change in free-surface behavior of surface grains. Full-field crystal plasticity simulations and modification of the present homogenization model considering non-spherical grains at the free-surface, specific mechanical conditions and a finite medium, will be considered for a further study. The present model
considers only the average grain size as an input parameter, and not the grain size distribution. Increase in
grain size scattering was found to affect the effective Hall-Petch tendency and should also be a survey to
be continued, as reported by Berbenni et al. [84, 85]. The identification of the parameters of the crystal-
plasticity law aimed at assessing a distinct mechanical behavior for the surface grains and core grains, and
subsequently the difference in local stresses/deformation in surface grains despite bulky specimens and de-
viation from the polycrystalline behavior. Further full-field crystal plasticity simulations will better help in
the inverse identification of the mechanical responses based on particular mechanical conditions imposed
by a free-surface compared to a fully-embedded medium. As far as surface effects are considered in mean-
field homogenization techniques, Pilvin [69] and Sauzay et al. [70, 71] proposed semi-analytical approaches
with good estimation of the average stress/strain microplasticity compared to finite element analyses solu-
tions. The microplasticity in the surface grains is first calculated using the Eshelby analytical solution of
a grain embedded in an infinite matrix. Surface effects are then considered by multiplying the analytical
solution by a grain-orientation and loading-condition factor. The mean-field homogenization technique is
of high interest to assess both the macroscopic behavior and the average stress-strain response of individ-
ual grains for low computational costs. However, neighboring effects due to deformation incompatibility
from grains to grains and local, sub-grain stress-strain heterogeneity are not simulated. To do so, full-field
numerical techniques are needed, such as finite element [35–37] or Fast Fourier transform [64] methods
considering either conventional or strain gradient crystalline plasticity model. Up to now, these advanced
models depicts free-surface effect as a less constraint surface than a grain boundary but does not account for
dislocation escape at the free-surface. Geers et al. [35] used two crystal plasticity models to investigate first-
order weakening effects due to grain size dependency, specimen thickness/grain diameter effects involving
strain gradients and slip gradients and second-order size strengthening such as lattice curvature effects and
boundary effects due to dislocations pile-up at grain boundaries: (i) the conventional continuum gradient
plasticity model, and (ii) the strain gradient dependent crystal plasticity model. Conventional continuum
gradient plasticity model was not capable to depict grain size dependency related to the first-order weaken-
ing effects due to lack in internal scale in the model. The strain gradient dependent crystal plasticity model
evidenced both the first-order weakening effects due to either grain size increase or specimen size reduction
and second-order strengthening due to deformation incompatibility from grains to grains leading to local
strain hardening. Therefore, the strain gradient dependent crystal plasticity model is particularly well suited
to consider all the aforementioned size effects. Keller et al. [36] confirmed similar trends for the simulation
of a high-purity Ni polycrystal. Larger aggregates were recently probed using FFT simulation with both
conventional continuum gradient plasticity and strain gradient dependent crystal plasticity models [64]. The
authors found similar insensitivity of the conventional continuum gradient plasticity to either grain size and
specimen size effect; the strain gradient dependent crystal plasticity model was capable to reproduce the
size effects and the polycrystal to multicrystal transition behavior. Despite conventional continuum gradi-
ent plasticity model is not adequate to capture such size effects, Guilhem et al. [37] evidenced interesting
response of surface grains experiencing a plastic behavior closer to the one of single crystal due to the
presence of a free-surface. Their plastic behavior was found to respond particularly well to the Schmid
factor criterion while core grains microplasticity was found more affected by deformation incompatibility
between adjacent grains, leading to particular stress/strain localization at grain boundaries. Similarly, Bing
et al. used full-field crystal plasticity modeling and simulation to investigate the role of a free-surface on
the slip activity evolution in the vicinity of the surface and in the specimen core [86]. Different average
cumulative shear were observed for slip systems for a hexagonal Close-Packed (HCP) and Body-Centered
Cubic (BCC) material, the slip activity depending on both the macroscopic deformation level and possibility
of slip activation. Feng et al. [87] also investigated the scatter in mechanical response in surface and core
grains for different specimen thickness/grain diameter ratio. Low ratio lead to more intense discrepancy in
stress-strain results compared to the normal stress-strain distribution. Depending on the parameters of the
crystal plasticity model, they were capable to identify polycrystal to multicrystal transition for specimen
thickness/grain diameter ratio ranging between 3 to 11. Similar numerical results were previously reported
by Zhang et al. [88, 89]. Further works should be continued on such full-field simulation to better account
for sub-grain strain heterogeneity and neighboring effects. Comparison with surface kinematics measure-
ments should also experimentally inform on the sub-grain strain heterogeneity in a quantitative manner to
better identify crystal plasticity parameters. Another important element to consider in the present study con-
cerns the consideration of internal lengths due to free-surface effect softening but also strengthening lengths
related to solid solution or precipitation strengthening [58, 59]. Despite important development in modeling
solid solution and precipitation hardening, their interaction with free-surface and grain-size effects was not
considered in the literature, to the best of the authors' knowledge.

## 5. CONCLUSIONS

Size effects due to the presence of a free-surface were experimentally and numerically investigated for the combination of three different grain size microstructures and two precipitation states (6 metallurgical variants). Thin specimens of Alloy 718 Ni-based superalloy with thicknesses ranging from 20 and 550 µm were tensile tested. This corresponds to specimens having a t/D ratio of 30 down to less than one grain in the thickness. The polycrystalline-to-multicrystalline transition (PMT) was gradual and appeared for specimens having 2 to 4 grains in the thickness. For thicknesses smaller than the PMT, specimens experienced a progressive decrease in yield strength, ultimate tensile strength and strain-to-failure. Interestingly, the precipitation state had an influence on the lower-order hardening and subsequently on the strain-hardening for thin specimens: strain-hardening of solid solution states being more sensitive to size effects compared to precipitation states. The two-phase mean-field model considering the crystalline texture and crystal plasticity formulation was capable to simulate both size and Hall-Petch effects taking into the fraction of surface grains and core grains as a function of the specimen thickness.

## 6. Acknowledgements

This work was supported by the European Research Council [project HT-S₄DefOx - Grant number 948007]. D. Texier is particularly grateful to D. Hourcade for technical assistance in specimen preparation.

## 7. License

This research was funded, in part, by The European Research Council, project HT-S₄DefOx - Grant number 948007. A [CC-BY public copyright license] has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions. The authors acknowledge S. Berbenni for fruitful and interesting discussions.

## 8. Data Availability

The raw/processed data required to reproduce these findings cannot be shared at this time, as the data also forms part of an ongoing study.

## 9. Declaration of competing interests

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.

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