In this study we use the theory of eigenstrains to investigate how diﬀerent sources of anisotropy aﬀected the results of shot peen forming experiments reported in Part 1. The specimens consisted of 4 .9 mm thick 2024–T3 alu- minum sheets uniformly shot peened on one face that were either free to deform or held onto a prestressing jig during peening. Potential sources of anisotropy included the plastic anisotropy of rolled aluminum, anisotropic initial stresses that redistribute when their equilibrium is disturbed by peen- ning, the geometry of the specimens, and externally applied prestress. For the alloy and peening conditions considered, plastic anisotropy had no discern- able inﬂuence on the resulting shape of the peen formed specimens. Initial residual stresses, on the other hand, caused slightly larger bending loads in the rolling direction of the alloy. Although the magnitude of these loads was approximately 30 times smaller than peening-induced loads, it was suf- ﬁcient to overcome the geometric preference for rectangular sheets to bend along their long side and cause all unconstrained specimens to bend along the rolling direction instead. Once the sheets started to deform, larger plas- tic strains developed in the bending direction. We show that this eﬀect is equivalent to that used in the variant of the process called stress peen form- ing where parts are elastically prestressed during peening to obtain larger plastic strains in directions in which the material is stretched. ∗Corresponding author Email address: frederick.gosselin@polymtl.ca (Fr´ ed´ erick P. Gosselin) URL: http://www.polymtl.ca/lm2 (Fr´ ed´ erick P. Gosselin) Preprint submitted to engrXiv April 23, 2021 Keywords: Shot peening, peen forming, residual stress, initial stress, 2024–T3 aluminum, eigenstrains 1. Introduction Lower wing skins of most commercial aircraft are shaped using shot peen forming. The process consists in bombarding thin metal parts with hard shot, usually steel spheres less than one millimeter in diameter, propelled at velocities ranging from 20 to 100 m s−1to plastically deform the material over a few tenths of a millimeter. In the plastically deformed layer, the material is compressed in the direction normal to the surface and stretched in the direction parallel to the surface, which causes the part to bend and elongate, as shown schematically in Figure 1a–b. Although peen forming has been used since the 1950s, the literature review conducted in Part 1 Faucheux et al. (2021) found very few experimental studies of the process. It also highlighted two major limitations of these studies. First, the vast majority of publicly available data were acquired on 76×19 mm strips for peening conditions such that the deﬂection of the strips remained small when compared to their thickness. Consequently, these experiments did not capture nonlinear geometric eﬀects that become signiﬁcant when forming large panels (see Part 1 Faucheux et al. (2021)). Secondly, residual stresses were seldom measured in more than one direction even though several phenomena could induce anisotropic residual stresses. These include plastic anisotropy of the target material and prestress, a term used to designate external loads deliberately applied to a part being peened to increase plastic ﬂow in directions in which the material is in tension. The manufacturing process of heat treatable aluminum plates in which most peen formed parts are machined is also known to induce anisotropic initial stresses that could aﬀect the deformed shape of the parts and should, therefore, be characterized. In Part 1 Faucheux et al. (2021), we reported the results of a peen form- ing experimental campaign designed to address these issues. The specimens consisted in 508 and 1016 mm long rectangular 2024–T3 aluminum sheets of diﬀerent aspect ratios, sourced from the same lot, and peened with the same treatment. Some sheets were free to deform during peening while others were prestressed in a four-point bending jig. A noteworthy result was that 2 Figure 1. Peen forming of a metal plate. (a) Small and hard shot propelled at a metal surface at high velocity plastically deform a thin layer of material. As the material plastically ﬂows in the direction parallel to the surface, it stretches the part and induces distortions. If controlled, this eﬀect can be used to shape thin components. (b) At the scale of a shot, the distribution of plastic strains is highly irregular due to the stochastic nature of the process. (c) To visualize how plastic strains cause residual stresses and distortions, it is convenient to imagine cutting the plate into inﬁnitesimally thin layers. In this (virtual) conﬁguration, each layer is homogeneous, unconstrained, and therefore stress-free. Gluing the layers back together requires stretching some layers and compressing others, which induces internal stresses and distortions. (d) Typical in-plane plastic strain proﬁles after averaging in the plane of the plate. The proﬁles have a maximum at or near the surface and smoothly decrease to zero. Also shown is a typical in-plane residual stress proﬁle. 3 all sheets that were free to deform bent in the rolling direction of the alloy, irrespective of their aspect ratio or of the peening trajectory. A similar phe- nomenon was reported in Kulkarni et al. (1981) for 2024 and 7075 aluminum although, in this case, 2024 aluminum specimens bent in the transverse di- rection of the alloy. In a study about thin bilayers, Alben et al. (2011) showed that rectangular plates of uniform thickness made of an isotropically expanding active layer on top of a passive layer spontaneously bend along their long direction. This eﬀect is purely geometrical and comes from regions of double curvatures that develop near the free edges, thus reducing the elastic energy of the system and making long side bending energetically favorable. That all our specimens bent along the rolling direction instead of the long direction therefore suggests that, for the alloy and peening conditions considered, material anisotropy was suﬃciently strong to overcome this eﬀect. In this article, we aim to clarify the contribution of each source of anisotropy to explain why all plates in Part 1 Faucheux et al. (2021) bent along the rolling direction. The paper is structured as follows. After a brief summary of the experiments conducted in Part 1 Faucheux et al. (2021), Section 2 introduces the concept of eigenstrains on which the rest of the analysis is based. Section 3 presents equations needed to relate eigenstrains to residual stresses and strains in uniformly shot peened plates. Section 4 presents the procedures used to characterize eigenstrains present in our specimens as well as numerical models used to interpret the result. We then successively investigate the inﬂuence of prestress, plastic anisotropy, initial eigenstrains, and geometric eﬀects on the deformed shape of the specimens in Sections 5 and 6. 2. Background 2.1. Summary of peen forming experiments conducted in Part 1 Faucheux et al. (2021) All peen forming experiments reported in Part 1 Faucheux et al. (2021) were conducted on rectangular specimens cut from 4 .9 mm thick 2024–T3 aluminum sheets from the same lot. Uniaxial tensile tests at 0, 45, and 90°from the rolling direction revealed isotropic elastic properties but a mild plastic anisotropy: the elastic modulus was 71 500 MPa and the yield stress at 0.2 % was 381 MPa along the rolling (L) direction, 349 MPa at 45 °of the 4 Figure 2. Residual stresses measured with the slitting method in 51 ×51 mm coupons removed from an as-rolled sheet. rolling direction, and 339 MPa along the transverse (T) direction. Residual stresses in as-received sheets were measured with the slitting method. These initial stresses were anisotropic with a magnitude of less than 20 MPa both in tension and in compression, as shown in Figure 2. Both conventional and stress peen forming experiments were conducted. For the former, specimens were free to deform during peening. For the latter, the specimens were elastically prestressed on the four-point bending jig shown in Figure 3a, shot peened, then released. Sheets of 1:4, 1:2, 1:1, 2:1, and 4:1 aspect ratio, here deﬁned as the ratio of the dimension in the L direction to the dimension in the T direction, 1016 mm along their long side were used for conventional peen forming experiments. Smaller 508 ×127 mm strips were used for stress peen forming experiments. Prestress curvatures ranged from 0 (the strips were held ﬂat) to 10 .54×10−4mm−1, which corresponds to a radius of curvature of 949 mm. All sheets were peened to full coverage in an automated compressed air peen- ing cabinet with SCCW28 steel shot, 0 .71 mm in diameter. The average ve- locity of shot exiting the nozzle was 41 m s−1, which corresponds to an Almen intensity of 12 .0 A(0.30 mmA). The treatment was applied progressively over the course of 12 peening cycles, each cycle delivering a small increment of coverage to the whole surface. For each geometry, one specimen was peened with strokes parallel to the L direction and another identical specimen was peened with strokes parallel to the T direction. Figure 4 shows the ﬁnal deformed shape of some of the sheets used for con- ventional peen forming experiments. All sheets, including those not shown, 5 Figure 3. (a) Four-point bending jig used to hold 508 ×127 mm strips in a bent shape during stress peen forming experiments. The height and the spacing of the support beams could be varied to obtain diﬀerent prestress curvatures. After peening, the strips were unclamped, their curvatures were measured, and a 254 ×127 mm coupon was removed from the central region for residual stress measurements. (b) Cross-sectional view of the jig. See supplementary material of Part 1 (Faucheux et al., 2021) for dimensions. 6 Figure 4. 3D scans of 4 .9 mm thick and 1016 mm long 2024-T3 aluminum sheets of 4:1, 2:1, 1:1, 1:2, and 1:4 aspect ratio shot peened to full coverage with SCCW28 shot at an Almen intensity of 12 .0 A(0.30 mmA). The specimens were free to deform during peening. Deﬂection magniﬁed by a factor of 4. had their largest curvature along the L direction, irrespective of their as- pect ratio or of the peening trajectory. While sheets of 1:1 and 2:1 aspect ratios deformed into cylindrical shapes with almost zero curvature in the T direction, other sheets assumed a compound curvature. Because of size constraints, residual stresses could not be measured directly in as-peened specimens. Instead, smaller coupons were removed from the center of the specimens listed in Table 1 with a jigsaw and residual stresses were measured at the center of these coupons with the hole drilling method. Residual stresses were recorded at 51 µm intervals down to a depth of 1 mm and the data were ﬁt with a sixth order polynomial to obtain continuous residual stress proﬁles. The coupons, the dimensions of which were chosen as large as possible to minimize the eﬀect of cutting induced plastic deformations and heat- ing on measurement spots, were 254 ×254 mm for free-to-deform sheets and 254×127 mm for prestressed strips. Upon removal, stress rebalancing caused the coupons to deform. For example, sheet 1:1–L assumed a cylin- drical shape with 3 .46×10−4mm−1curvature in the L direction and almost zero curvature in the T direction while the coupon removed from it assumed an almost spherical shape with 2 .93×10−4mm−1curvature in the L direc- tion and 1.81×10−4mm−1curvature in the T direction. As a result, residual stresses measured in the coupons diﬀered from those that would have been measured in as-peened specimens. Their analysis constitutes the bulk of the following sections. 7 Table 1 List of specimens peen formed in Part 1 (Faucheux et al., 2021) for which residual stress measurements are available. Specimen IDDimensions Prestress curvatureAlong L Along T (mm) (mm) ( ×10−4mm) Sheet 4:1–L 1016 254 Free to deformSheet 2:1–T 1016 508 Sheet 1:1–L 1016 1016 Sheet 1:2–L 508 1016 Sheet 1:4–L 254 1016 Strip K0–L–A 508 127 0 Strip K1–L–A 508 127 2.82 Strip K2–L–A 508 127 5.11 Strip K3–L–A 508 127 8.28 Strip K4–L–A 508 127 10.54 2.2. Eigenstrains Recall that our aim is to clarify the contribution of diﬀerent sources of anisotropy on the ﬁnal deformed shape of the aluminum sheets peen formed in Part 1 (Faucheux et al., 2021). This analysis must be carried out from the only experimental data available, namely residual stresses and curva- tures measured on coupons removed from the specimens. Because of stress rebalancing, these stresses and curvatures diﬀer from those in the original specimens. On the other hand, if the coupon were carefully removed, and provided that no reverse yielding occurred during springback, then plastic strains were not aﬀected by this removal. Consequently, characterizing plas- tic strains in the coupons provides a direct means to study the eﬀect of the peening treatment on the original specimens. At the scale of a shot, the distribution of plastic strains induced by shot peening treatments is highly irregular as it depends on the precise sequence of impacts and on the distribution of impact velocities. On average, however, symmetry and translational invariance require that the plastic strain ﬁeld "plonly depends on the coordinate normal to the surface and that shear components be zero. In the rectangular coordinate system of Figure 1a, this 8 implies that "pl(z) = εpl xx(z) 0 0 0εpl yy(z) 0 0 0 εpl zz(z) . (1) Furthermore, the fact that plastic deformations take place at constant volume requires that εpl zz=−(εpl xx+εpl yy). Since peening stretches the material in the direction parallel to the surface and compresses it in the direction normal to the surface, εxxandεyyare positive while εzzis negative. To visualize what residual stresses and distortions a peening treatment does induce, it is convenient to imagine cutting a shot peened part into inﬁnitesi- mally thin layers parallel to its surface and letting the system relax, as shown schematically in Figure 1c. In this virtual conﬁguration, the layers are not constrained by surrounding material; the strains they undergo are equal to the plastic strains and residual stresses are zero everywhere. Since the strains from one layer to the next are incompatible, reassembling the structure re- quires stretching some layers and compressing others, which induces residual stresses and distortions. Figure 1d shows typical plastic strain and residual stress proﬁles in uniformly shot peened plates. Plastic strains are the largest at, or near, the peened surface then gradually decrease to zero. Residual stresses vary linearly in the bulk owing to the bending and stretching of the plate. They reach a maximum near the transition with the peening-aﬀected layer, located at z=d, then decrease until they become compressive near the surface. There exist a general framework to study the relationship between incompat- ible strains, residual stresses, and distortions called the theory of eigenstrains (Korsunsky, 2017). The term eigenstrains refers to all permanent nonelastic strains present in a structure. Thus, for small strains "="el+"∗, (2) where"are the total strains, "elare the elastic strains, and "∗are the eigenstrains. Similarly, the eigenstrains can themselves be additively decom- posed into thermal strains, plastic strains, and any other source of nonelastic strains. Many analytical solutions have been reported that relate speciﬁc dis- tributions of eigenstrains to the residual stresses and distortions they induce, some of which are reviewed in Mura (1987). Conversely, several authors ad- dressed the inverse problem of identifying eigenstrains present in a structure 9 from limited experimental measurements; see for example in Jun and Kor- sunsky (2010). Since plastic deformation is the only source of eigenstrains in shot peened parts, these inverse identiﬁcation procedures can be used with- out modiﬁcations to compute peening-induced plastic strains. Therefore, here we consider "∗="pl. In general, identifying eigenstrains is an iterative process that involves ﬁnite element simulations and optimization algorithms. For simple geometries, however, closed-form solutions that relate eigenstrains to residual stresses and total strains are available. 3. A model to relate eigenstrains, stresses, and strains in uniformly shot peened plates Consider a plate of uniform thickness h, made of an isotropic material with elastic modulus Eand Poisson ratio ν, and subjected to a distribution of eigenstrains of the form of Equation (1). Using the coordinate system of Figure 1a, let αi=/integraldisplayh 0ε∗ ii(z)dzandβi=/integraldisplayh 0ε∗ ii(z)zdz, i ={x,y} (3) be the resulting eigenstrains and the ﬁrst eigenstrains moment with respect to the peened surface. Within the hypotheses of the theory of linear elasticity, (residual) stresses are related to the strains through εxx=axz+bx, ε yy=ayz+by, (4) εel xx=εxx−ε∗ xx, εel yy=εyy−ε∗ yy, (5) σxx=E 1−ν2/bracketleftBigg εel xx+νεel yy/bracketrightBigg , σ yy=E 1−ν2/bracketleftBigg εel yy+νεel xx/bracketrightBigg , (6) where ai= (12βi−6αih)/h3, b i= (4αih−6βi)/h2. (7) This solution can be derived by adopting the kinematic assumption of Equa- tion (4) – material lines perpendicular to the undeformed midsurface remain straight during deformation – and by using the stress-strain relation of Equa- tion (6) to enforce that resulting axial loads and bending moments acting on any cut through the plate must be zero, as was done in Korsunsky (2005) 10 for equibiaxial eigenstrains. Another derivation without a priori kinematic assumptions is presented in the supplementary material. In this simple case, the above equations can be inverted to yield the eigen- strains in terms of residual stresses and total strains. Solving Equation (6) for the strains yields εel xx= (σxx−νσyy)/E, εel yy= (σyy−νσxx)/E. (8) Finally, combining Equations (4), (5), and (8), and noticing that ai=−κi, (9) whereκiis the curvature of the plate in the idirection, yields ε∗ xx= (εxx|z=0−κxz)−(σxx−νσyy)/E, ε∗ yy=/parenleftBig εyy|z=0−κyz/parenrightBig −(σyy−νσxx)/E,(10) where"|z=0are total strains measured on the upper face of the plate. 4. Methods 4.1. Near-surface eigenstrains The near-surface eigenstrains present in our specimens were computed from residual stress and curvature measurements using Equation (10). Here and in what follows, we used E= 71500 MPa and ν= 0.33. Because the in-plane stretch of the coupons by the peening treatment has not been characterized, the constant terms εii|z=0in Equation (10) could not be determined directly. To enable comparing eigenstrain proﬁles, the missing constant was set such that the eigenstrains were equal to zero at the end of the peening-aﬀected layer, that is, ε∗ ii(d) = 0, as shown in Figure 5a. Since residual stress mea- surements showed that dvaried little across specimens, we used d= 0.5 mm in all cases. (Recall from Section 2.2 that the depth of the peening-aﬀected layer can be estimated from the location of the tensile residual stress peak; also see Korsunsky (2005).) Although it is sensitive to experimental errors and oscillations in the proﬁles, this approach is suﬃcient for the purpose of this analysis. 11 Figure 5. Typical eigenstrain proﬁles. (a) Near-surface eigenstrains computed from curvature and residual stress measurements using Equation (10). (b) Peening-induced eigenstrains obtained by truncating the proﬁles in (a) at the end of the peening aﬀected layer. (c) Through-thickness proﬁles obtained by extending the proﬁles in (a) with piece- wise constant functions. The magnitude of the piecewise constant parts is chosen such that if these proﬁles were input in a model of a small coupon, they would induce the same axial stretch and the same curvatures as those that would be measured. 12 4.2. Accounting for initial eigenstrains Equation (10) only returns eigenstrain proﬁles up to the depth at which resid- ual stresses have been measured—in this case zmax= 1 mm. However, the shape of residual stress proﬁles measured in as-rolled sheets (Figure 2) indi- cates that anisotropic incompatible eigenstrains are initially present through the entire thickness of the specimens. To estimate how eigenstrains deeper than zmaxaﬀected the ﬁnal deformed shape of free-to-deform specimens, we ﬁrst input the previously identiﬁed peening-induced eigenstrains, i.e., only the z <d part or the proﬁles, into a ﬁnite element model of the specimens and solved for equilibrium. Figure 5b shows a typical proﬁle used for these simulations. This provided an estimate of the shape that the specimens would have assumed if no initial eigenstrains were present. These results were then compared to a second set of simulations that included the contribution of initial eigenstrains. A simple way to perform these simulations consists in constructing idealized eigenstrain proﬁles ˆ ε∗ ii(z) that, when input into a model of the coupons, induce the same in-plane stretch and the same curvatures as those that were experimentally measured, then to input these proﬁles into a model of the sheets. This process may be interpreted as reassembling a sheet by joining together several coupons, as illustrated in Figure 6; it is the opposite of coupons removal. Since all quantities are explicitly integrated over the thickness in conventional thin plate theories, any eigenstrain proﬁles having the same resulting values and ﬁrst moments will yield the same deformed shape. Therefore, the only conditions that idealized proﬁles must satisfy to induce the desired axial stretch and curvatures is /integraldisplayh 0ˆε∗ ii(z)dz=αi,/integraldisplayh 0ˆε∗ ii(z)zdz=βi, i ={x,y}. (11) In what follows, we use ˆε∗ ii(z) =  ˇε∗ ii(z) ifz≤zmax, θtop i ifzmax<z≤(zmax+h)/2, θbtm i if (zmax+h)/2<z≤h,(12) where ˇε∗ iiare the eigenstrains identiﬁed in Section 4.1 and θtop iandθbtm iare 13 constants evaluated to satisfy Equation (11). Figure 5c shows a typical proﬁle obtained in this way. The reason for choosing such proﬁles instead of simpler ones is that, by construction, they not only induce the correct deformations but also the correct residual stresses in the z≤zmaxregion. Consequently, they enable estimation residual stresses present in the z≤zmaxregion of the original sheets before the coupons were removed—in other words, to correct for the stress relaxation that occured during removal. An explicit expression for the constants can be obtained by substituting Equation (12) into (11), integrating, and solving for the constants. This yields θtop i= [(αi−ˇαi)(3h+zmax)−4(βi−ˇβi)]/(h−zmax)2, θbtm i=−[(αi−ˇαi)(3zmax+h)−4(βi−ˇβi)]/(h−zmax)2,(13) where ˇαi=/integraldisplayzmax 0ˇε∗ ii(z)dz and ˇβi=/integraldisplayzmax 0ˇε∗ ii(z)zdz. (14) Finally, using the inverse of Equation (7), αi= (aih2+ 2bih)/2, β i= (2aih3+ 3bih2)/6, (15) and recalling that ai=−˜κiandbi= ˜εii|z=0, where a ‘tilde’ is used to indicate that a quantity is measured on a coupon, the constants can be expressed in terms of quantities that can be determined experimentally. Again, because the in-plane stretch of the coupons had not been charac- terized, the value of ˜ εii|z=0was unknown. However, for moderately curved uniformly shot peened plates, this term has very little inﬂuence on com- puted curvatures and residual stresses. Indeed, it has the same eﬀect as an homogenous in-plane expansion which, by analogy with homogenous thermal expansion, only causes the structure to expand without inducing stresses. In fact, curvatures are independent of bi(= ˜εii|z=0) in the linear domain, that is, for very small curvatures, as shown by Equation (9). In what follows, we arbitrarily used ˜ εii|z=0=κih/2, which corresponds to a scenario where the midplane of the plates does not stretch. Running the simulations for any ˜εii|z=0in the±5×10−3mm mm−1range, where 5×10−3mm mm−1is the typical magnitude of peening induced eigenstrains (see later sections), yielded virtually identical results. 14 Figure 6. Thought experiment where a large peen formed plate is cut into many small coupons and the coupons are then joined back together. If neither cutting nor springback induce plastic deformation, then the process is reversible. 4.3. Finite element simulations All ﬁnite element simulations were done with Abaqus 6.14. Taking advantage of symmetries, only one quarter of the geometries were meshed, as shown in Figure 7a. Eigenstrain proﬁles were input as pseudo-thermal strains with the UEXPAN subroutine as described in Jun and Korsunsky (2010). The distribution of integration points through the thickness was as shown in Fig- ure 7c, with 99 integration points over 0 ≤z≤zmax, 3 integration points over zmax≤z≤(zmax+h)/2, and 3 integration points over ( zmax+h)/2≤z≤h. Geometric nonlinearities were included in all analyses. Gravity was neglected as additional simulations showed that it caused variations in curvatures of less than 5 %. Similar simulations were used to compute residual stresses in prestressed strips while the strips were still held onto the prestressing jig. Figure 7b shows the 1/4th symmetry model used for these simulations. The contact between the strips, clamps, and support beams was modeled by prescribing z-displacements along the lines where the strips touched support beams. 4.4. Deﬁnitions To facilitate the discussion, we ﬁrst introduce relevant dimensionless param- eters. As discussed in Audoly and Pomeau (2010), it is usually possible 15 Figure 7. Finite element models used to simulate (a) conventional peen forming and (b) stress peen forming experiments. For the latter model, contact between prestressed strips, clamps, and support beams was modeled by prescribing z-displacements along the lines where the strips touched support beams (see Figure 3). The mesh used for simulations was three times ﬁner than that shown, i. e., had 9 times more elements. (c) Distribution of integration points through the thickness of the shell elements. 16 to characterize the response of thin structures with very few dimensionless parameters with the same parameters being valid for both small and large deformations. Here, we propose to use ¯B= 12(1+ν)BavgL2/h4, χ = (Blong−Bavg)/Bavg,¯κ=κL2/h, ν, (16) whereLis a characteristic length (here taken as the dimension along the longest side), his the thickness, κis any curvature measured on the plate, Bavg= (Blong+Bshort)/2, (17) andBlongandBshort are the ﬁrst eigenstrain moments with respect to the midplane of the plate in the long and short directions. Using the axis conven- tion of Figure 1a, the ﬁrst eigenstrain moment with respect to the midplane of the plate along the idirection is given by Bi=/integraldisplayh 0ε∗ ii(z)(h/2−z)dz. (18) The parameter ¯Bcorresponds to the ratio of bending loads over bending rigidity rescaled by ( L/h)2,χcharacterizes the anisotropy of eigenstrains, and ¯κis a dimensionless curvature. Simple idealized eigenstrain proﬁles of the form ε∗ ii(z) =  θiifz≤h/2, −θiifz >h/ 2,(19) where θi=¯Bh3 3(1 +ν)L2(1 +χ) (20) in the long direction and θi=¯Bh3 3(1 +ν)L2(1−χ) (21) in the short direction were used to obtain the desired dimensionless loads andχ. 17 5. Results The left panels of Figures 8 and 9 show near-surface eigenstrains identiﬁed in the coupons removed from free-to-deform and prestressed strips, respectively. The shape of the proﬁles is typical of that induced by peening treatments: eigenstrains peak at or near the peened surface then decrease gradually until they reach a plateau, approximately 0 .5 mm below the surface, which corre- sponds to the end of the peening-aﬀected layer. In all cases, eigenstrains are larger in the directions in which the specimens bent the most, that is, along the L direction for free-to-deform sheets, and along the prestress direction for prestressed strips. Inputting these eigenstrains (including the piecewise constant extension not shown in the ﬁgures) into a model of the coupons yields the residual stresses shown as solid lines in the middle panels. In all cases, computed stresses match experimental data, which conﬁrms that eigenstrains present in the coupons were correctly identiﬁed. Estimates of residual stresses in the original specimens, prior to coupons removal, are shown in the right panels. For free-to-deform specimens, the only apparent diﬀerence between these proﬁles and those in the coupons is that residual stresses in the L and T directions in sheets 1:1–L and 2:1–T become almost superimposed over the ﬁrst 0 .2 mm below the peened surface owing to the bending of the specimens. Similarly, residual stresses in the L and T direction of prestressed strips are closer to one another than residual stresses in the coupons and they exhibit almost the same minimum in both directions. Finally, Table 2 lists experimentally measured and simulated curvatures for free-to-deform sheets. The relative error RE is deﬁned as: RE =κisim−κiexp 1 2(κxexp+κyexp)×100%, (22) wherei={x,y}and represents the L and T directions, κisimandκiexpare simulated and experimental curvatures in L and T directions, respectively. These simulations qualitatively reproduce the response of the sheets. The rel- ative error between measured and simulated curvatures ranges between −20 and 15 % in the L direction and between −25 and 4 % in the T direction. On the other hand, simulations that neglected the contribution of initial eigen- 18 Figure 8. Eigenstrains and residual stresses in specimens used for conventional peen forming experiments. Left: Peening-induced eigenstrains. Middle: Residual stresses mea- sured with the hole drilling method at the center of the coupons. Also shown are residual stresses obtained when the complete eigenstrain proﬁles (i.e., after they were extended to include the contribution of initial eigenstrains; see Section 2.2) are input into a ﬁnite element model of the coupons. Right: Residual stresses in the sheets prior to removal of the coupons—estimated by inputting the same eigenstrain proﬁles into ﬁnite element models of the sheets. 19 Figure 9. Eigenstrains and stresses in specimens used for stress peen forming experi- ments. Left: Peening-induced eigenstrains. Middle: Residual stresses measured with the hole drilling method at the center of the coupons and residual stresses obtained when the complete eigenstrain proﬁles are input into a ﬁnite element model of the coupons. Right: Stresses in the sheets prior to removal of the coupons while the sheets are still held on the prestressing jig—estimated by inputting the same eigenstrain proﬁles into ﬁnite element models of the strips. 20 Table 2 Experimental and simulated curvatures of sheets used for conventional peen forming ex- periments. SpecimenExperimental curvaturesκiexp (×10−4mm−1)Simulated curvatures κisim(×10−4mm−1) Relative error RE (%) Including initial eigenstrainsNeglecting initial eigenstrainsIncluding initial eigenstrainsNeglecting initial eigenstrains Along L Along T Along L Along T Along L Along T Along L Along T Along L Along T Sheet 4:1–L 2 .50 2.19 2.54 1.72 2.18 2.27 2 -20 -14 3 Sheet 2:1–T 3 .06 0.11 3.30 0.03 2.30 0.28 15 -5 -48 11 Sheet 1:1–L 3 .46−0.01 3.25−0.07 2.64−0.08 -12 -3 -48 -4 Sheet 1:2–L 3 .56 0.30 3.18 0.38 0.40 2.18 -20 4 -164 97 Sheet 1:4–L 3 .00 1.81 3.07 1.24 2.26 2.65 3 -25 -31 33 strains display larger errors in most of the cases. Errors range from −164 to −14 % in the L direction and from −4 to 97 % in the T direction. The curva- tures of prestressed strips (not shown) were identical to prestress curvatures as jig constraints prevented the strips to deform. The fact that simulations that included the contribution of initial eigenstrains do not perfectly match experiments suggests that some plastic yielding occurred during coupons removal. Indeed, experimental and simulated deformed shapes should be identical if eigenstrains in the coupons and in the sheets were the same. 6. Discussion 6.1. Inﬂuence of prestress That larger eigenstrains developed in the direction in which specimens bent the most conﬁrms that prestress, whether it results from externally applied loads or from the progressive deformation of an unconstrained specimen, does aﬀect the distribution of peening-induced plastic strains. This is further evi- denced by Figure 10 which shows the same eigenstrain proﬁles as in Figure 9 but superimposed. Whereas eigenstrains along the transverse direction are almost superimposed, the magnitude of eigenstrains along the prestress di- rection increases with prestress curvature. Similar results were reported by Hu et al. (2015) for laser peening simulation of 2024–T3 aluminum sheets. Similarly, sheets 2:1–T and 1:1–L, which deformed into cylindrical shapes, display larger eigenstrains along the bending (L) direction whereas other un- constrained sheets, which deformed into elliptical or nearly spherical shapes, display similar eigenstrains in both directions (see Figure 8). 21 Figure 10. Peening-induced eigenstrains in stress peen formed strips. 22 Although it can induce anisotropic eigenstrains, prestress alone, however, does not explain why all unconstrained specimens bent in the same direction. Consider for example the 1:1 aspect ratio sheet shown in Figure 4. This sheet was peened with strokes parallel to the L direction and bent in L direction. If prestress alone did determine the bending direction, we would expect an identical sheet peened in the same conditions with strokes parallel to the T direction to bend along T. Experiments conducted in Part 1 (Faucheux et al., 2021) showed that this was not the case; all specimens bend along L, irrespective of the peening trajectory. 6.2. Inﬂuence of plastic anisotropy One phenomenon that could explain why unconstrained specimens preferen- tially bent in the L direction is the known plastic anisotropy of the 2024–T3 aluminum alloy (Bron and Besson, 2004; Seidt and Gilat, 2013). Mild plas- tic anisotropy was observed during uniaxial tensile tests conducted in Part 1 Faucheux et al. (2021). Clear evidence of how indenting a mildly plastically anisotropic material can result in a highly anisotropic response was reported in Prime (2017) where the authors measured residual stresses 40 % larger in the T direction than in the L direction at the center of a 10 mm thick 2024– T3 aluminum disk after it had been indented with two opposing cylindrical punches. This result was attributed to preferential plastic ﬂow along the T direction as compression tests showed that stress-strain curves in the T direction were about 40 MPa below the curves in the L direction. Uniaxial compression tests reported in Seidt and Gilat (2013) for plates of the same alloy and thickness also identiﬁed the T direction as the weaker one, as did the tensile tests of Part 1 Faucheux et al. (2021). These observations, how- ever, should not be used be extrapolate the behavior of heavily shot peened specimens since laboratory tests do not capture the eﬀect of the complex loading paths experienced by material points close to the peened surface. Whether plastic ﬂow occurred preferentially in the L or T direction can be assessed from eigenstrain proﬁles in the strip that was held ﬂat during peen- ing (Figure 9a). For this strip, curvatures, hence the eﬀect of prestress, were the same in all directions. Consequently, any observed anisotropy in near- surface eigenstrains would result from plastic anisotropy. Similarly, markedly anisotropic eigenstrains in free-to-deform 1:4 and 4:1 aspect ratio specimens (Figure 8a, e) would betray plastic anisotropy since the geometry of these specimens constrains them to deform into quasi-spherical shape, thus induc- 23 Depth (mm)Figure 11. Eigenstrains in an as-received sheet computed from the residual stress proﬁles in Figure 2 using Equation 10. When a plate in which such eigenstrains are initially present is peened on the face located at z= 0, initial eigenstrains in the peening aﬀected layer (region 1) are replaced by positive peening-induced eigenstrains about one order of magnitude larger than initial eigenstrains which causes the structure to bend. Eigenstrains present deeper into the material are left unaﬀected by the treatment. ing approximately the same prestress in all directions. The proﬁles, however, show no clear sign of plastic anisotropy that would favor L side bending: eigenstrains are almost the same in both directions in free-to-deform spec- imens and they are slightly larger in the T direction for the specimen held ﬂat during peening, which does favor T side bending. 6.3. Inﬂuence of initial eigenstrains Another source of material anisotropy that could explain why free-to-deform specimens bent along the L direction is initial eigenstrains. Figure 11 shows eigenstrains in an as-rolled sheet computed from the ini- tial stress proﬁles in Figure 2 using Equation (10). If a specimen in which these eigenstrains are initially present is peened on the surface located at z= 0, large positive eigenstrains will develop in the peening aﬀected layer and induce positive curvatures. Initial eigenstrains present deeper into the material, however, are left unaﬀected by the treatment. Let dbe the depth the peening-aﬀected layer. Because they are symmetric with respect to the midplane of the plate, initial eigenstrains in the d≤z≤h−dregion (re- gion 2) induce no bending loads hence have no eﬀect on curvatures. On the other hand, eigenstrains in the z≥h−dregion (region 3) decrease curvatures if they are positive and increase curvatures if they are negative. In this case, eigenstrains in region 3 are slightly larger in the T direction than 24 in the L direction, thus favoring L side bending. However, their magnitude, hence the bending moment they induce, is one to two order of magnitude smaller than peening-induced eigenstrains. Although this suggests that ini- tial eigenstrains have very little inﬂuence on the ﬁnal deformed shape of the specimens, ﬁnite element simulations summarized in Table 2 showed that this was not the case as neglecting initial eigenstrains caused simulated cur- vatures to vary by several tens of percent for all specimens and caused an inversion in the bending direction of the 1:2 aspect ratio sheet. Considering the low magnitude of initial eigenstrains, these results show that the response of uniformly shot peened plates, and in particular that of 1:2 aspect ratio plates, is highly sensitive to small variations in eigenstrains. 6.4. Competition between geometric eﬀects and eigenstrains anisotropy This sensitivity is surprising since, for specimens of aspect ratio smaller than one, L side bending can only occur if the eigenstrains are suﬃciently anisotropic to compete against the geometric preference of rectangular sheets to bend along their long direction. This eﬀect, which was described in Al- ben et al. (2011), comes from regions of double curvatures that develop near the free edges of plates loaded with biaxial eigenstrains and that lowers the elastic energy of the system, thereby making long side bending energetically favorable. In this section, we investigate the level of eigenstrain anisotropy required to overcome this eﬀect and trigger short side bending. When eigenstrains are equibiaxial, these parameters coincide with those used in Faucheux et al. (2018) which, closely resemble those used by Freund (2000) to characterize the response of thin ﬁlm bilayers. In what follows, ¯Band ¯κ are referred to as the dimensionless load and the dimensionless curvature, respectively. Figure 12 shows the evolution of dimensionless curvatures as a function of the dimensionless load for rectangular sheets of 1:1, 2:1, and 4:1 aspect ratio for χ=−6 %, 0.1 %, and 6 %. These curves were obtained from ﬁnite element simulations using the same quarter plate model as described in Section 4. As in Section 4, the eigenstrains were applied uniformly. For χ= 0.1 %, which corresponds to almost equibiaxial eigenstrains, simulations predict that all sheets bend preferentially along the long side, which is consistent with the analysis in Alben et al. (2011). For small dimensionless load (i.e., for small eigenstrain moments and/or thick plates), all sheets deform into spherical shapes. For larger dimensionless loads, out-of-plane deﬂections are no longer 25 Figure 12. Master curves showing the relationship between dimensionless curvatures ¯ κ and dimensionless loads ¯Bfor diﬀerent levels of eigenstrain anisotropy χfor plates of 1:1, 2:1, and 4:1 aspect ratio. The curves were obtained by ﬁnite element simulations. For nearly equibiaxial eigenstrains (middle), square plates assume spherical shapes for dimen- sionless loads smaller than 25 then transition suddenly to an elliptical then cylindrical shapes due to an elastic instability. For all other aspect ratios and values of χ, the tran- sition is gradual, and even more so as the strip is elongated or χdeparts from 0. In the range of parameters considered here, plates always bend along the long side for χ>0 and they always bend along the short side for χ <−5%. For intermediate values of χ, the direction of largest curvature depends on the aspect ratio and on the dimensionless load. small compared to the thickness and geometric nonlinearities become signif- icant: the stress stiﬀening that results causes the curves to bend downward. Perfect 1:1 aspect ratio sheets maintain a spherical shape far into the nonlin- ear domain, up to a dimensionless load of about 25, then suddenly transition to cylindrical conﬁgurations due to an elastic instability. All other geometries display a smooth transition between spherical and cylindrical shapes with curvatures in the long and short directions gradually separating. Increas- ingχonly causes the curves to separate faster and the elastic instability for sheets of 1:1 aspect ratio to be replaced by a smooth transition. Decreasing χ, however, causes a sudden inversion of the bending direction for sheets of 1:1 and 2:1 aspect ratio. The transition occurs at χ= 0 for the former, and at aboutχ=−5 % for the latter. In other words, for sheets of 2:1 aspect ratio, the ﬁrst eigenstrain moment along the short direction must be about 10 % larger than that along the long direction for eigenstrain anisotropy to overcome the geometric preference of the sheets to bend along the long side. The value of the dimensionless parameters for our specimens can be deter- 26 Table 3 Dimensionless load ¯Band eigenstrains anisotropy χinduced by all eigenstrains present in the specimens and by peening-induced eigenstrains only. The former were estimated from the curvature of the coupons while the latter where obtained by integrating previously identiﬁed eigenstrain proﬁles. Diﬀerences between the two sets of parameters are due to initial eigenstrains present below the peening-aﬀected layer. Specimen IDLong side aligned withAll eigenstrainsPeening-induced eigenstrains only ¯B χ ¯B χ (-) (%) (-) (%) Sheet 4:1–L L 67 11.0 70.8 -4.4 Sheet 2:1–T L 65 34.5 59.1 9.7 Sheet 1:1–L - 66 23.7 61.5 7.6 Sheet 1:2–L T 72 -18.6 67.0 -1.1 Sheet 1:4–L T 66 -34.5 80.8 1.5 mined experimentally by noticing that, for a coupon small enough such that its response is not aﬀected by geometric nonlinear eﬀects, Bi= ˜κih3/12, (23) where ˜κiis the curvature of the coupon in the idirection (see Appendix A for details). The third and fourth columns of Table 3 list the values of ¯Bandχthus computed from the curvatures of coupons used for residual stress measurements. In this case, ¯Branges from 65 to 72, χis positive for specimens with their long side aligned with the L direction, and it is less than−18 % for specimens with their short side aligned with the T direction. To put these values into perspective, the last two columns of Table 3 list the ¯Bandχinduced by peening-induced eigenstrains only—i.e., neglecting the contribution of initial eigenstrain present below the peening-aﬀected layer. These values were obtained by integrating the previously identiﬁed near- surface eigenstrain proﬁles with Equation (18) from z= 0 toz=d= 0.5 mm. Whereas ¯Bdoes not markedly change, χdrops by several percent, even changing signs for sheets 4:1–L and 1:4–L. This shows that χis quite sensitive to small variations in eigenstrains and that only slightly anisotropic eigenstrains are needed to overcome the geometric preference for the sheets to bend along the long side. 27 The competition between geometry and eigenstrain anisotropy is further il- lustrated by the map in Figure 13a which shows the direction of largest curvature as a function of the aspect ratio and χat¯B= 70. For perfect 1:1 aspect ratio plates, the boundary between the two regions is located at χ= 0 and any small perturbation causes bending in either direction. The deformed shape on either side of the boundary are identical, but with the bending direction inverted, as illustrated in Figure 13b. For small aspect ratios, the boundary asymptotically approaches the χ= 0 line. This is be- cause the bending along the long side and the short side of a narrow plate become decoupled. In this domain, the curvature in one direction is simply proportional to the ﬁrst eigenstrain moment in this direction and there is no sharp transition when crossing the boundary: curvature in both directions are almost identical when χ= 0 and gradually separate when χdeparts from zero. This is illustrated in Figure 13d which shows the deformed shape of 1:8 aspect ratio plates for χ=−2 % (left) and χ= 2 % (right). For inter- mediate aspect ratios, the boundary between the two regions moves towards negativeχsince larger eigenstrains along the short direction are required to induce short side bending. The point the furthest away from the χ= 0 line is reached for an aspect ratio of (approximately) 1:2.4, which suggests that the edge eﬀects that favor long side bending described in Alben et al. (2011) are most pronounced for this speciﬁc geometry at ¯B= 70. 7. Conclusion The purpose of this paper was to clarify how diﬀerent sources of anisotropy contributed to the ﬁnal deformed shape of 2024–T3 aluminum rectangular sheets shot peened to full coverage with steel shot 0 .71 mm in diameter and propelled at 41 m s−1. In particular, we tried to explain why sheets that were free to deform during peening always bent in the rolling direction of the alloy, no matter what their aspect ratio or their alignment with respect to the rolling direction was. We analysed two sources of anisotropy likely to explain the observed deformed shapes: the plastic anisotropy of 2024–T3 aluminum, and the anisotropy of initial stresses inherited from the sheet manufacturing process. These resid- ual stress initially present in the ﬂat sheets redistribute when their equilib- rium is disturbed by peening. The analysis of eigenstrain proﬁles identiﬁed from residual stress and curvature measurements showed no clear sign of ei- 28 Figure 13. (a) Map showing the direction of largest curvature of rectangular plates of uniform thickness for ¯B= 70 andν= 0.33. The map is made of 3400 pixels, each pixel corresponding to one ﬁnite element simulation. The eigenstrains are larger along the long direction when χ>0 and vice versa. Plates of 1:1 aspect ratio and very elongated strips have their largest curvature in the direction along which the eigenstrains are the largest. For plates of intermediate ratio, there exist a domain in which the plates bend along their long side even when eigenstrains are largest along the short side. (b) Deformed shapes of plates of 1:1 aspect ratio for χ=±1 %, (c) of plates of 1:2 aspect ratio for χ=−6 % and χ=−4 %, and (d) of plates of 1:8 aspect ratio for χ=±2 %. Deﬂection magniﬁed by a factor of 4. 29 ther mechanism prompting L side bending: peening-induced plastic strains were almost equibiaxial or slightly larger along the T direction—which would have favored T side bending—and bending moments induced by the rebal- ancing of initial stresses were one to two orders of magnitude smaller than those induced by the peening treatment. Further investigation revealed that the response of uniformly shot peened rectangular plates of uniform thick- ness is highly sensitive to small variations in internal loads and identiﬁed initial stresses as the most likely cause of L side bending. This sensitivity, however, is a feature of the simple geometries of the sheets and we believe that complex production parts such as pocketed or integrally stiﬀened panels should behave in a more controlled manner. Once the sheets started to deform, signiﬁcantly larger plastic strains devel- oped in the bending direction. This eﬀect is similar to that used in the variant of the process called stress peen forming where parts are held onto a curved jig during peening to obtain larger curvatures along the prestress di- rection. Whereas prestressed parts are ﬁrmly held into place during peening, the shape of unconstrained parts is continuously evolving; it depends on the precise peening sequence and on the geometry of the part. Capturing this ef- fect in a numerical model of the process will likely require computer-intensive simulations. Alternatively, one could imagine implementing a feedback loop, possibly coupled to a learning system, to monitor the shape of a part as it is being peened and adjust the treatment accordingly. This kind of ap- proach is well established in the industry (see for example Kittel et al. (1999); Lundquist et al. (2015)) but remains largely unexplored in the literature. 8. Acknowledgements The authors gratefully acknowledge ﬁnancial support from Airbus, from the Rio Tinto group through a graduate scholarship, from the Canada Research Chairs program, and from the Natural Sciences and Engineering Research Council of Canada (NSERC; funding reference number 175791953). The prestressing jig used for stress peen forming experiments was courtesy of A´ erosph` ere and Centre Technologique en A´ erospatiale (CTA). The authors would also like to thank Dr. P. Faucheux for some of the original ideas behind this work and for the numerical results of section 6.4. 30 Appendix A. Estimating ﬁrst eigenstrain moments from the cur- vature of small coupons Consider a uniformly shot peened plate with a distribution of eigenstrains of the form in Equation (1). The ﬁrst eigenstrain moments with respect to themidplate of the plate introduced in Equation (18), Bi, is related to the resulting eigenstrains, αi, and the ﬁrst eigenstrain moments with respect to thesurface of the plate, βi, through Bi=αih/2−βi. (A.1) Now, assume that a coupon is removed from the shot peened plate and that removal does not alter the distribution of eigenstrains. For example, imagine that a 254×254×4.9 mm coupon is removed from a 1016 ×1016×4.9 mm plate peen formed to a dimensionless load ¯B= 70. For ¯B= 70, the response of the plate is well into the nonlinear regime, as shown in Figure 12. However, since ¯Bscales with L2, as in Equation (16), ¯Bdrops to approximately 4 .4 for the coupon, which is very nearly in the linear domain. In other words, removing a small coupon from a larger plate is a way to do away with geometric nonlinearities. In the linear domain, the analytical solution presented in Section 3 applies. In particular, it can be used to express the αiandβiin terms of measur- able quantities. For example, substituting Equations (15) and (9) in Equa- tion (A.1) yields Bi= ˜κih3/12, (A.2) where a tilde is used to indicate that the curvatures are measured on the small coupon—and not directly on the plate.