Eduard Laurentiu Nitu
email: eduard.nitu@upit.ro

Daniela Monica Iordache

Claudiu Badulescu

Orcid Id

Numerical investigation of the radial cold rolling process of the grooves

Keywords: Radial cold rolling, groove profile, rolling device, finite element modeling, ABAQUS

The radial cold rolling process is widely used in industry due to the advantages of chip removing processing. In this study, circular complex profiles were formed by cold-rolling with radial feed, using a patented device with two roller-tools. By achieving several numerical simulations of the radial rolling process, the study aimed to establish the influence that the maximum set force and the diameter of the workpiece have on the productivity of the process and the quality of the formed trapezoidal and metric grooves. The numerical simulations were performed with ABAQUS/Explicit software and by using a previously validated finite element model. The penetration curve of the roller-tools in the material was introduced in the simulation as an analytical function and was determined based on experimental researches. To express the dependency of the penetration curve coefficients on other rolling conditions, a multivariable analysis using a design of experiments technique was performed. The parameters that result from numerical simulations and were analyzed in this study are the profile forming time, the maximum radial force, the distribution of the equivalent strains in the axial section of the profiles, and the profile dimensions. The conclusions that were drawn from the analysis of the results regarding the influence of the set maximum force and the workpiece diameter on the analyzed parameters of the numerical simulations, together with the definition of efficient process criteria, allowed also the identification of the optimal conditions of the rolling process.

Introduction

Cold forming presents major advantages compared to the cutting processes [START_REF] Neagu | Cold volumetric pressing of parts with thread and teeth (in romanian)[END_REF][START_REF] Grob | Cold forming solid materials[END_REF][START_REF] Escofier | Roulage de Filetages -Moletages[END_REF][START_REF]Thread rolling heads[END_REF][START_REF] Machinery | Thread form rolling applications[END_REF][START_REF] Lindsay | Rolled vs cut thread bolts[END_REF] : (i) more productive, (ii) no production of chips, and (iii) high quality of the generated surface. Moreover, because the deformation occurs at ambient temperature, the initial properties of the material are irreversibly changed, forming a continuous fiber in the material of the part, which ensures better mechanical properties, [START_REF] Iacomi | Comparative study concerning the generation of some circular grooves by Figure 10. Distribution of the equivalent plastic strains (PEEQ) with respect to the workpiece diameter, d 0 . cold rolling with wedge tools and cutting[END_REF][START_REF] Nitu | Comparative study concerning the generation of some circular grooves by radial cold rolling with roller tools and cutting[END_REF][START_REF] Peng | Geometry optimization of the rolling tool for gear roll-forming process[END_REF] such as: roughness, micro-hardness, and fatigue resistance.

Radial cold rolling with two roller-tools is increasingly used in industry to obtain parts with complex profiles (threads, grooves, etc.) on revolution pieces, generally for series production, [START_REF] Grob | Cold forming solid materials[END_REF][START_REF] Escofier | Roulage de Filetages -Moletages[END_REF][START_REF]Thread rolling heads[END_REF][START_REF] Machinery | Thread form rolling applications[END_REF][START_REF] Lindsay | Rolled vs cut thread bolts[END_REF] and are in continuous development and competition concerning the quality and cost of products. The trend is oriented toward developing and implementing numerical modeling which represents a very useful and efficient tool for successful the experimental researches.

The finite element modeling of the cold rolling process started in the 1990s by Rowe et al., [START_REF] Rowe | Finite-element plasticity and metalforming analysis[END_REF] but the high volume of calculations and the limited capacity of computers to handle a large amount of data, within a reasonable time, restricted these studies to the understanding of the deformation process. Martin 11,[START_REF] Martin | A finite element evaluation of residual stress in a thread form generated by a cold-rolling process[END_REF] used finite element analysis based on MARC finite element code and, using a 2D plane strain finite element model, has established an adequate mesh to investigate the residual stress for a thread profile obtained by radial cold rolling. This process was also investigated by Domblesky and Feng through several numerical models and methods implemented in the finite element code DEFORM. First, by using a 2D plane strain finite element model, Domblesky and Feng [START_REF] Domblesky | Two-dimensional and threedimensional finite element models of external thread rolling[END_REF] analyzed the effect of workpiece diameter, friction on thread formation, flow stress, and thread form on material flow and thread profile in external thread rolling of large diameter workpieces. Furthermore, they have developed a three-dimensional numerical model for fat-die thread rolling to incorporate the effects of both groove orientation and workpiece movement in the simulation. [START_REF] Domblesky | A parametric study of process parameters in external thread rolling[END_REF] Kamouneh et al. [START_REF] Kamouneh | Investigation of work hardening of flat-rolled helical-involute gears through grain-flow analysis, FE-modeling and strain signature[END_REF] investigated the work hardening effect of flat rolling on the critical regions of gear through the use of hardness maps, grain-flow photographs, and finite element models developed in ABAQUS and DEFORM. Also, using numerical simulations, they have optimized the shape and the size of the workpiece, [START_REF] Kamouneh | Diagnosis of involumetric issues in flat rolling of external helical gears through the use of finite-elements models[END_REF] to reduce the negative effects such as the ''rabbit ear'' phenomenon. Song et al. [START_REF] Song | Analysis and experiments on the thread rolling process for micro-sized screws, Part I: process parameter analysis by finite-element simulation[END_REF] used numerical simulations to study the influence of process parameters on small threads obtained by rolling. Kowalik and Trzepiecin´ski 18 estimated the deformation force and the influence of friction on the material flow in the rolling process of the longitudinal channels using numerical simulation and MARC software. Kukie1ka et al. [START_REF] Kukie1ka | 3D numerical analysis the state of elastic/viscoplastic strain in the external round thread rolled on cold[END_REF] have developed an application of numerical analysis in the ANSYS program to both determine and analyze the effects of friction coefficient, initial yield stress, and plastic hardening modulus that are used in the numerical simulation of the cold rolling of the external round thread. Cui et al. [START_REF] Cui | Deformation mechanism and performance improvement of spline shaft with 42CrMo steel by axial-infeed incremental rolling process[END_REF] have investigated the deformation mechanism and have improved the spline shaft performance obtained by axial-infeed incremental rolling by using 3D DEFORM software. Ma et al. [START_REF] Ma | On the pitch error in the initial stage of gear roll-forming with axial-infeed[END_REF][START_REF] Ma | Geometric design of the rolling tool for gear roll-forming process with axialinfeed[END_REF] have investigated the gear roll-forming process with axial-infeed using Finite Element Method (FEM) simulations (3D DEFORM software), have developed an analytical model for predicting the pitch error of gear, and also have optimized the geometric design of rolling tool to reduce the deflection and root stress of rolling tool's teeth, to both eliminate the scratches on tooth flank of the formed gear and to reduce the height of rabbit ear. Kramer and Groche [START_REF] Kramer | Defect detection in thread rolling processes -experimental study and numerical investigation of driving parameters[END_REF] have modeled the thread rolling process using Simufact Forming v13 and have made a systematic study of the influence of the process configuration and the specimen preparation toward the occurrence of defects in rolling processes. Results show that the tribological system influences the rolling process. Yuan et al. [START_REF] Yuan | The influence of roller motion on profile and hardness of T2 worm[END_REF] have investigated the influence of roller motion on profile and hardness of T2 worm using numerical modeling (software Forgeä) and experiments. Zhang et al. [START_REF] Zhang | Deformation characteristics of self-infeed rolling processfor thread shaft[END_REF] have investigated the deformation mechanism in the self-infeed rolling process of thread shaft using DEFORM 3D software, showing the behavior of material flow and material hardening. They also investigated [START_REF] Zhang | The formation mechanism of protrusion of long thread by axial self-infeed rolling process[END_REF] the protrusion generation in the forming process of long threads by axial self-infeed rolling process, establishing that to reduce the protrusion height, the pre-rolling angle should be 2 0 .

In concluding this survey, it may be stated that numerical analysis is a powerful and reliable tool for manufacturing process investigation. The accumulated knowledge has enabled the forming industry to improve process competitiveness, product performance, and service life.

Through numerical simulation in the ABAQUS environment, this paper examines the influence that the maximum set force and diameter of the workpiece have on both the process productivity and the quality of circular profiles obtained by radial cold rolling with a patented device. The elements that are analyzed in the study are the profile forming time, the maximum radial force, the profile dimensions, and the equivalent strains distribution in the axial section of the profiles. Three criteria are defined to evaluate the process efficiency: (i) process productivity, (ii) material hardening, and (iii) profile dimensions.

Experimental work

Processed material and profiles

The material used in this investigation was AISI 1015. Its chemical composition and mechanical characteristics are given in Table 1. The workpieces were obtained by turning and grinding from a hot extruded bar.

The profiles formed by radial cold rolling using two roller-tools and in-feed method were a concentric channels surface as in Figure 1(a), with five similar grooves in axial section for metric thread M20x2, respectively, for trapezoidal thread Tr20x2. Geometric characteristics of the two types of profiles are shown in Figure 1(b) and (c).

Experimental system

The radial cold rolling process was performed by applying the kinematic scheme in Figure 2(a) on a lathe using an experimental system represented schematically in Figure 2(b). The patented device, Ungureanu et al., [START_REF] Ungureanu | Roller tool device and system for machining profiled circular grooves on cylindrical pieces by cold plastic deformation[END_REF] is shown in Figure 2(c) and is described in detail by Iordache et al. [START_REF] Iordache | Device and experimental stand to process profiles by radial cold rolling with roller tools[END_REF] This device is fixed on a lathe carriage and the working cycle (performed by the two roller-tools) is semiautomated: penetration, calibration, withdrawal. The roller-tools (RT) are entrained in rotation by the workpiece at the speed (n r ) (rpm) as a result of the contact between the rolls and the workpiece. The radial feed of the roller-tools (s r ), the same for the two rolls, but in the opposite direction, is generated by two hydraulic motors Figure 2(d). The maximum force developed by each hydraulic engine, denoted as maximum set force (F m ), can be adjusted by setting an equivalent pressure (p h ) in the two hydraulic engines. The rolling speed (v) can be adjusted by the lathe speed setup (n w ), while the radial feed of the roller-tools (s r ) cannot be directly adjusted, except through the pressure p in the hydraulic motors. The penetration depth of the roller-tools (h) can be limited using buffers. When this depth is reached (h max ), the calibration of the profile can be performed by maintaining in rotation the roller-tools in the final position to give the piece a high geometrical and surface quality.

The experimental system allows monitoring the roller-tools displacement, the radial force, and the calibration time. Measurement of the roller-tools displacement is carried out with a displacement transducer WAL20, Figure 2(e), and with a resolution of 1 mm. The radial force (force evolution in the displacement direction of the roller-tools) (F r ) is measured using a force transducer KMR 200 WAL20, Figure 2(e), whose measurement error is equal to 1%. The calibration phase is controlled by a proximity transducer PY 152, allowing the adjustment of the roller-tools to be held in time in this position.

Penetration curve of the roller-tools

In the finite element model, it is necessary for the numerical simulation of the rolling process, to define the penetration curve of the roller-tools in the material. Our rolling process is performed with variable radial feed (s r ) because the time required to reach the set maximum pressure in the experimental system is comparable to the time required to obtain the profile. The experimental shapes of the penetration curves of the rollertools into the material, Figure 3, are similar to the two processed profiles, three areas are being highlighted:

-first area (I) with progressive acceleration, at the beginning of the process, is associated with the elastic deformation of the system; -second area (II) with an almost constant evolution speed and associated with the plastic deformations of the workpiece;

-third area (III) with a deceleration to a possible saturation and which appears when the profile of the piece has been made or the tool-holder sledge reached the buffer.

The penetration depth of the roller-tools in the material (h) depends on several factors and the most important are the rolling speed (v), the maximum set force (F m ), and the maximum penetration depth of the roller-tools (h max ). The form of the law that best describes the evolution of the roller-tools penetration in the workpiece was established after several attempts as follows [START_REF] Nitu | Analytical modelling of the process of cold rolling by in-feed method[END_REF] :

h t, v, F m ð Þ= h max (1 À exp( À b(v, F m ) Á t a v, F m ð Þ )) ð1Þ
where: t is the process time and the coefficients a(v, F m ) and b v, F m ð Þ depends on the rolling conditions. The values of the coefficients a(v, F m ) and b v, F m ð Þ were determined by regression analysis using the experimental curves of the penetration of the roller-tools in the workpiece. The values of the coefficient of determination R 2 were over 0.98, thereby confirming the accuracy of the proposed model.

To express the dependency of the coefficients a and b on other rolling conditions (v, F m ), we performed a multivariable analysis using a design of experiments technique, [START_REF] Goupy | Methods for experimental design: principles and applications for physicists and chemists[END_REF][START_REF] Goupy | Les plans d'experiences: Optimissation du choix des essais et de l'interpretation des resultats[END_REF] aiming to determine the parameters of functions as:

a(v, F m ) = C a Á v X a F m Y a ð2Þ b(v, F m ) = C b Á v X b F m Y b ð3Þ
For each type of profile, 12 experiments were performed consisting of the variation of the rolling speed (v) and the set maximum force (F m ). First, the a and b coefficients for each experimental case were determined by regression analysis. Thus, based on the values of the coefficients a and b previously determined, the six coefficients C a , X a , Y a , C b , X b , Y b were identified (the confidence bound was set to 95%), Table 2.

Thus, the curve of penetration of the roller-tools used in the numerical model can be obtained through

Numerical procedure

Numerical model

The numerical simulations were performed with Abaqus/Explicit software. The numerical model used in this study was presented in detail by Nitu et al. [START_REF] Nitu | FE-modeling of cold rolling by in-feed method of circular grooves[END_REF] A synthetic overview is presented hereafter.

The workpiece used in the simulation has the same (cylindrical) shape and dimensions (diameter and length) as the one used in the experiments. The best compromise between computing resources and results accuracy consists of a different meshing in the axial and radial direction, depending on the profile deformation degree. In the deformed areas C3D8R (8-nodes solid hexahedral elements with reduced integration) are used while in the middle of the workpiece (undeformed area) C3D4R (4-nodes solid tetrahedral elements with reduced integration) are used for the meshing. The different dimensions of the elements in these areas were established based on the profile step p.

The roller-tools have a profile associated with the profile to be obtained and are modeled by analytical rigid surfaces (Figure 4). The workpiece is free and in contact with the piece-support, and the roller-tools has two degrees of freedom: a rotation around its axis (given by the rolling speed v), and an in-feed translation defined by the curve of penetration h(t), equation (1).

The type of contact between the surfaces is a ''surface-to-surface'' one, the friction coefficient between the rolls and the workpiece was set at 0.3, and 0.01 between the workpiece and the base. These values were chosen based on the literature review [START_REF] Mcconnell | Friction in cold rolling of a low carbon steel with lubricants[END_REF][START_REF] Fares | Fatigue design of bolted assembly using multiaxial fatigue criteria[END_REF] and preliminary simulations.

The behavior of the material is defined as follows:

-the elastic behavior of the workpiece is modeled by assuming isotropic elasticity, with the values of Young's modulus, E = 210 GPa and Poisson's ratio, n = 0.3. -the plastic behavior is described using the von Mises criterion with the assumption of isotropic hardening. Accordingly, the yield function is given by:

f = ffiffiffiffiffiffiffiffiffiffiffi 3 2 s ij s ij r À s ð4Þ
where s ij are the deviatoric stress components, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 ð Þs ij s ij p is the von Mises equivalent stress, and s is the current yield stress.

The current yield stress s can be calculated using the constitutive equation at the deformation of the material. A five-parameter constitutive equation that combines Hollomon and Voce's laws was used:

s = Ke n + S(1 À A exp ( À Be)) ½ ð 5Þ
where K, n, S, A, and B are material parameters, and e is the effective strain. The identification of the five parameters, equation ( 5), was performed by fitting the compression test data with a gradient method implemented in a FORTRAN program. [START_REF] Nitu | FE-modeling of cold rolling by in-feed method of circular grooves[END_REF] It was obtained: K = 542.5 MPa, n = 0.135, S = 217.6 MPa, A = 0.99, and B = 9.91. The numerical model validation was performed by comparing the simulation results with the experimental ones, under the rolling conditions presented in Table 3, where d 0 is the diameter of the workpiece.

The compared output parameters were: profile dimensions (d e and d i ), depth of deformation (h d ), profile height (H), maximum radial force (F r max ), and the HV 0,3 microhardness level, which is measured in the axial section in the zone of maximum hardening (the transition region of the flanks with the base of the profile).

h d = d 0 À d i ð Þ=2 ð6Þ
H = d e À d i ð Þ=2 ð7Þ
A Toolmaster 310-P profile projector, with an accuracy of 60.5 mm, was used to measure the geometric parameters of the profile (d 0 , d e , and d i ).

Micro-hardness measurements were made by the Vickers method (300 g load) in the axial section of the tooth, using a Zwick Roell ZHV equipment. These values were expressed in (MPa) by multiplying them with the gravitational acceleration value (9.81 m/s 2 )). The HV 0,3 (micro-hardness measurements) and e(equivalent strains -PEEQ) have been converted based on some assumptions, Tabor, [START_REF] Tabor | The hardness and strength of materials[END_REF] in the quantities directly comparable, in stress respectively, by expressing them in the same unit (MPa).

A summary of the comparisons between simulated results versus experimental ones is shown in Table 4 and Figure 5. Thus, based on these comparisons it is concluded that all calculated outputs were in very good agreement with their experimental values.

Simulations plan

The developed numerical model in the ABAQUS environment was used to analyze the influence of the maximum set force (F m ) and the diameter of the workpiece (d 0 ) on the productiveness and the quality of circular profiles. The penetration curves of the roller-tools used in this model are defined by the equation ( 1), the values of the coefficients a(v, F m ) and b(v, F m ) being calculated with the formulas presented in Table 2.

For the maximum set force (F m ), we selected three values in geometric progression 42, 63, and 94.5 kN that are higher than those used in the experiments. For the workpiece diameter (d 0 ), we selected two values that are symmetrically arranged around the one used in the experiments (determined by the law of volume constancy 1 ): 18.75 and 18.85 mm for the metric profile, respectively, 18.82 and 18.92 mm for the trapezoidal profile. The different numerical simulations are detailed in Table 5.

Results and discussions

The simulations results that were analyzed in this study are:

-the profile forming time (t d ), the maximum radial force (F r max ), and the distribution of equivalent deformations e(PEEQ) in the axial section of the profiles, for the simulations in which the maximum set force (F m ) was varied (simulated cases 3-6, see Table 5); -the profile dimensions (internal diameter d i and outer diameter d e ), the depth of deformation (h d ), and the equivalent strain distribution e(PEEQ) in the axial section of the profiles, for the simulations in which the workpiece diameter (d 0 ) was varied (simulated cases 1-3, see Table 5).

Influence of the maximum set force

The profile forming time (t d ) was estimated by imposing in equation ( 1) the condition of forming a depth as close as possible to the theoretical maximum (h max ).

For a forming degree of the profile of 99% (h = 0.99Áh max ), this condition leads to the following relation:

t d = À ln 0:01 b 1 a ð8Þ
The dependence of the profile forming time (t d ) on the maximum set force (F m ) is obtained by inputting in the relation (8) the values of the coefficients a and A that are expressed by the relations ( 2) and (3):

t d = À ln 0:01 C Á v X b F m Y b 1 Ca Áv Xa Fm Ya = C 1 F m Y b C 2 Fm Ya ð9Þ
where C 1 and C 2 are invariable in relation to the maximum set force F m :

C 1 = À ln 0:01 C Á v X b and C 2 = 1 C a Á v X a ð10Þ
Considering the values expressed in Table 2 for the six coefficients C a , X a , Y a , C b , X b , Y b it results that the dependence of t d on F m follows a hyperbolic shape, which is represented graphically in Figure 6.

The analysis of these dependencies allows drawing the following conclusions:

-For a given value of the maximum set force (F m ) the trapezoidal profile forming time is lower than the metric one. Because deformation during thread rolling closely resembles the indentation of a semiinfinite body, [START_REF] Domblesky | Two-dimensional and threedimensional finite element models of external thread rolling[END_REF][START_REF] Kim | Metal flow in wedge indentation of V-and W-shaped tools[END_REF] the favorable geometry of the trapezoidal profile (lower flank angle) explains this evolution, leading to the rapid penetration of the roller-tools in the workpiece. -There is a slight flattening trend of the two curves, which is found at the values of the maximum set force F m greater than 60 kN for the metric profile, respectively, 90 kN for the trapezoidal profile. This finding is related to the hyperbolic shape of the dependence between t d and F m (see equation ( 10)). These evolutions of the profile forming time are due to the behavior of the material, namely its hardening: in the plastic deformation domain, according to the constitutive equation of the material, [START_REF] Iordache | Determination of constitutive equation at the deformation of materials using the compression test[END_REF] the increase in tension becomes smaller and smaller as the deformation increases, thus slowing down the intensity of the deformation process and, consequently, the increase of the profile forming duration.

The dependency of the maximum radial force (F r max ) on the maximum set force F m is shown graphically in Figure 7. It is found that for the same value of the maximum set force (F m ), the maximum radial force (F r max ) is higher for the trapezoidal profile than for the metric one. This is because the flat base of the trapezoidal indenter (roller-tools) leads to deeper penetration and hardening of the workpiece. [START_REF] Domblesky | A parametric study of process parameters in external thread rolling[END_REF] There is a trend of flattening for the two curves which appears when the maximum set force (F m ) exceeds 42 kN. The levels of these flattenings are 2 kN for the metric profile and 32 kN for the trapezoidal profile. This flattening is because the maximal radial force depends on the penetration depth of the tool into the workpiece (h) and this maximum penetration depth is limited to a required value (h max ) which depends on d 0 and d i (see equation ( 6)).

Figure 8 shows the distribution of the equivalent plastic strains e(PEEQ) in the axial section of the profiles and the dependence between the maximum values PEEQ and the maximum set force value. Values and distribution of the PEEQ in the formed profile give information about the degree of deformation of the profile and its hardness level, and its analysis allows us to draw the following conclusions:

-The largest strains always appear at the bottom of the profile (the maximum value being in the root of the flank) and are propagated radially toward the interior of the material. Although these strains within the tooth are lower, the tooth is hardened ( e ø 0.5 ) HV ø 2130 MPa). -For the metric profile, the material hardening is significantly influenced by the maximum set force (F m ). For example, the most hardened area ( e ø 2 ) HV ø 2440 MPa), (i) differs in terms of size and position: it is greater and more present in the bottom of the profile and at the root of the flank, when processing with a lower maximum set force, and (ii) it becomes smaller and smaller and localizes at the root of the flanks with the increase of the maximum set force.

-For the trapezoidal profile, the material hardening is less dependent on the maximum set force, the area strongly hardened ( e ø 2) varying less. -The maximum material hardening is obtained in the cold rolling process with lower maximum set forces, meaning that the penetration of the roller-tools is performed at lower feeds. The material hardening is greater in the cold rolling process for the trapezoidal profile than the metric one and this can be explained by the greater forming depth of the profile. [START_REF] Domblesky | Two-dimensional and threedimensional finite element models of external thread rolling[END_REF][START_REF] Domblesky | A parametric study of process parameters in external thread rolling[END_REF] Influence of the workpiece diameter

The profile's dimensions obtained by simulations when the workpiece diameter was varied are presented in Figure 9. The increase in the workpiece diameter (d 0 ) Regarding the tolerance margin of both the inner and the outer diameters of the profiles, it can be seen that:

-For the metric profile, the resulted dimensions are adequate if using a workpiece with a smaller or equal diameter to the one determined by the law of volume constancy (d 0 418.80 mm); -For the trapezoidal profile, the obtained dimensions are appropriate if using a workpiece with a diameter smaller than the one determined by the law of volume constancy (d 0 \ 18.87 mm).

The distribution of the equivalent plastic strains e(PEEQ) in the axial section of the profiles and the dependence of the maximum values PEEQ with respect to the workpiece diameter are shown in Figure 10. A slight increase can be noticed in the maximum level of the equivalent plastic strains as well as in the depth of the zone strongly deformed when increasing the workpiece diameter. This fact is due to the increase in the deformation depth (h d ) (see equation ( 6)), which leads to an increase in the degree of deformation of the profile. [START_REF] Herold | Berechnung des Umformgrades beim Gewindewalzen[END_REF]

Summary and conclusions

The research presented in the paper consisted of running several numerical simulations of the cold-rolling process with two roller-tools, of radial grooves with metric and trapezoidal profile, using a numerical model previously validated. The penetration curve of the roller-tools in the material was introduced in the simulation as an analytical function and was determined based on experimental researches. The dependence of the curve penetration coefficients on the rolling conditions (rolling speed, v, and maximum set force, F m ) was established by a multivariable analysis using the design of the experiment technique.

The varied parameters in the simulations were the maximum set force (F m ) and the diameter of the workpiece (d 0 ). The analyzed results were the profile forming time (t d ), the maximum radial force (F m ), the distribution of the equivalent plastic strains in the axial section of the profiles (PEEQ), and the profile dimensions (d e and d i ).

The main conclusions drawn from this work can be summarized as follows:

-rolling with higher maximum set forces leads to (i) decreasing the profile forming time (t d ) up to a limit level, (ii) increasing the radial maximum force (F r max ) up to a limit level, and (iii) decreasing the area strongly hardened and the maximum values of equivalent plastic strains (PEEQ) in the axial section of the rolling profiles. -the use of workpieces with a smaller diameter than the one determined by the law of volume constancy leads to profiles with conforming dimensions (within the specified tolerance field), while the inner diameter of the profile is directly influenced by the adjusting dimension and the stiffness of the technological system.

To define the optimal process conditions, the following criteria are considered:

-Productivity, which can be appreciated with the profile forming time (t d ); -Material hardening, which can be appreciated based on the distribution and the level of the equivalent plastic strains (PEEQ); -Profile dimensions, described by the inside diameter d i and the outer diameter d e .

Based on the results analyzed in section 4 and on the optimality criteria defined above, we can establish the optimum process conditions:

-to obtain a good compromise between high productivity (lower profile forming time) and a higher and more uniform material hardening (PEEQ maximum), the value of the maximum set force (F m ) should not be too high. In this study, the following values are considered as suitable: F m between 40 and 50 kN for the rolling of the metric profile, and F m between 50 and 60 kN for the rolling of the trapezoidal profile, respectively. -to obtain dimensions of the profile within the specified tolerance field, the diameter d 0 of the workpiece must be between 18.75 and 18.80 mm for the rolling of the metric profile, and between 18.79 and 18.84 mm for the rolling of the trapezoidal profile, respectively.

Further research can be done by considering another parameter of the process, for example, the radius of the tip of the tool, or by extending the investigations to other profiles or materials used in this field.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 1 .

 1 Figure 1. Form of the cold rolling profiles: (a) 3D drawing, (b) X detail, and (c) dimensions and deviations of the two types of profiles.

Figure 2 .

 2 Figure 2. Experimental system for the radial cold-rolling: (a) kinematic scheme, (b) general scheme, (c) general photography, (d) CAD model of the rolling device, and (e) positioning of transducers.

Figure 3 .

 3 Figure 3. The penetration curves of the roller-tools for rolling conditions n = 1600 rpm, p = 36 bar: (a) metric profile and (b) trapezoidal profile.

Table 2 .

 2 The functions of the coefficients a and A of penetration curves of the roller-tools. Profile a b M 20 a = 0.316601Áv 0.1981 ÁF m 0.6257 b = 4.92E 2 08Áv 0.8455 ÁF m 5.0172 Tr 20 a = 0.738448Áv 0.3142 ÁF m 0.3334 b = 5.99E 2 16Áv 2.1117 ÁF m 9.6635

Figure 4 .

 4 Figure 4. Overview of ABAQUS model for numerical simulation: (a) the roller-tools and (b) kinematic of the process.

Figure 5 .

 5 Figure 5. The part obtained by radial cold-rolling: (a) experimental shape and (b) FE simulated shape.

Figure 6 .

 6 Figure 6. The dependence between the profile forming time, t d , and the maximum set force, F m .

Figure 7 .

 7 Figure 7. The dependency between the maximum radial force F r max and the maximum set force, F m .

Figure 8 .

 8 Figure 8. Distribution of the equivalent plastic strains (PEEQ) with respect to the maximum set force F m .

Figure 9 .

 9 Figure 9. Dependence of the profiles diameters obtained by radial cold rolling with respect to the workpiece diameter: (a) metric profile and (b) trapezoidal profile.

Table 1 .

 1 Chemical composition and initial mechanical characteristics of the steel AISI 1015.

	Chemical composition (wt %)				Mechanical characteristics
	C	M n	C r	S i	N i	M o	p, S	Rp 0,2 (N/mm 2 )	Rm (N/mm 2 )	A5 (%)	HV 0,3 (MPa)
	0.15	0.65	0.11	0.27	0.08	0.01	\ 0.035	298	475	15	139

Table 3 .

 3 The rolling conditions to validate the numerical models.

	Profile	d 0 (mm)	v (m/min)	F m (kN)
	M 20	18.80	94.50	28
	Tr 20	18.87	94.85

Table 4 .

 4 Synthesis of the comparisons: simulated results versus experimental results.

	Profile	d e (mm)		d i (mm)		h d (mm)		H (mm)		F r max (kN)	HV (MPa)
		exp.	sim.	exp.	sim.	exp.	sim.	exp.	sim.	exp.	sim.	exp.	sim.
	M 20	19.858	19.934	17.818	17.831	0.491	0.484	1.02	1.051	22.94	22.13	. 2370	. 2440
	Tr 20	20.232	19.838	17.481	17.495	0.694	0.682	1.375	1.171	30.14	27.91	. 2470	. 2440

Table 5 .

 5 Variation of the process parameters in the numerical simulations.

	Simulated case nb. Profile M20	Profile Tr20
	F 1 28	18.75	28	18.82
	2		18.85		18.92
	3*		18.80		18.87
	4	4 2		4 2
	5	6 3		6 3
	6	94.5		94.5

m (kN) d 0 (mm) F m (kN) d 0 (mm) * Experimental case.

Proc IMechE Part B: J Engineering Manufacture 00(0)

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix

List of notations


Mélissa C Allé
email: melissa.alle@univ-lille.fr

Dorthe

Dorthe Berntsen

Self-isolation, Psychotic Symptoms and Cognitive Problems During the COVID-19 Worldwide Outbreak

Keywords: COVID-19, self-isolation, confinement, psychotic symptoms, cognitive problems

Self-isolation, psychotic symptoms and cognitive problems during the COVID-19 worldwide outbreak

Self-isolation, Psychotic Symptoms and Cognitive Problems

During the COVID-19 Worldwide Outbreak

Introduction

During the spring 2020, the worldwide outbreak of COVID-19 forced millions of people into social isolation, or even confinement, in their own homes with little social and physical contact with people outside their household. Such widespread social isolation was unprecedented in modern history, and in marked conflict with the highly social nature of human beings [START_REF] Tomasello | The ultra-social animal[END_REF]. Due to its exceptional nature, we are lacking knowledge about its short-and long-term consequences.

Previous studies have shown negative psychological effects of quarantine during health crisis, including post-traumatic stress symptoms, confusion and anger (for review, see [START_REF] Brooks | The psychological impact of quarantine and how to reduce it: rapid review of the evidence[END_REF]. Since the beginning of the COVID-19 pandemic, many studies sought to better understand the nature of this impact and have documented an increase of depressive symptoms, stress and sleep disorders during stay-at-home restrictions, in both individuals with and without psychiatric history [START_REF] Fiorillo | The consequences of the COVID-19 pandemic on mental health and implications for clinical practice[END_REF][START_REF] Huang | Generalized anxiety disorder, depressive symptoms and sleep quality during COVID-19 outbreak in China: a web-based cross-sectional survey[END_REF][START_REF] Wang | Immediate Psychological Responses and Associated Factors during the Initial Stage of the 2019 Coronavirus Disease (COVID-19) Epidemic among the General Population in China[END_REF].

Surprisingly, this research has largely ignored psychotic symptoms and cognitive problems. Psychotic symptoms are present, not only in patients with schizophrenia, but also in other clinical populations and, in an attenuated form, among some individuals in non-clinical populations [START_REF] Mcgrath | Psychotic Experiences in the General Population: A Cross-National Analysis Based on 31,261 Respondents From 18 Countries[END_REF], reflecting a "psychosis continuum" [START_REF] Van Os | A systematic review and meta-analysis of the psychosis continuum: evidence for a psychosis proneness-persistence-impairment model of psychotic disorder[END_REF].

Past research has documented the development of these symptoms in contexts of solitary confinement, social isolation or sensory deprivation even in healthy individuals. Specifically, it has been demonstrated that situations of severe solitary or group confinement (in the context of polar, submarine or insular expeditions) are associated with an increase of psychotic symptoms, paranoia and hallucination experiences in healthy populations, who were especially selected and trained to survive in extreme conditions [START_REF] Cochrane | Working in Arctic and Sub-Arctic Conditions: Mental Health Issues*: Can[END_REF][START_REF] Gunderson | Adaptation of small groups to extreme environments[END_REF][START_REF] Strange | Emotional and social adjustment of recent US winter-over parties in isolated Antarctic stations**The opinions expressed herein are those of the authors and cannot be construed as reflecting the views of the Navy Department or of the Naval Service at large[END_REF]. Similarly, some patients isolated in intensive care units also develop a psychotic syndrome including visual hallucinations and paranoid delusion, unrelated to their neurological condition [START_REF] Granberg-Axèll | Clinical signs of ICU syndrome/delirium: an observational study[END_REF].

A series of studies have shown similar psychological effects of social isolation in prisoners kept in supermaximum security units in the USA (Arrigo & Bullock, 2008). It has been well documented that such confinement can lead to severe changes in the sense of reality, problems with memory and concentration, problems with impulse control, selfmutilation, perceptual distortions, hallucination and derealization (Arrigo & Bullock, 2008;Grassian, 1983). It is noteworthy that these symptoms were observed in individuals with no history of psychiatric disorders and that several factors influenced their occurrence: the sensory deprivation severity, the isolation duration, the individual's personality traits, the reasons justifying the confinement, and the individual's psycho-cognitive-social functioning [START_REF] Grassian | Effects of sensory deprivation in psychiatric seclusion and solitary confinement[END_REF].

Overall, the relationship between social isolation (or loneliness) and psychotic symptoms appears robust, as it has been described in various conditions: borderline personality disorder [START_REF] Slotema | Hallucinations in patients with borderline personality disorder: characteristics, severity, and relationship with schizotypy and loneliness[END_REF], Alzheimer disease [START_REF] El Haj | Hallucinations, loneliness, and social isolation in Alzheimer's disease[END_REF], psychotic continuum [START_REF] Chau | Loneliness and the psychosis continuum: a metaanalysis on positive psychotic experiences and a meta-analysis on negative psychotic experiences[END_REF] and the general population [START_REF] Narita | Loneliness and psychotic experiences in a general population sample[END_REF].

To explain this relationship, [START_REF] Hoffman | A social deafferentation hypothesis for induction of active schizophrenia[END_REF] hypothesized that social withdrawal critically reduces sensory input and could therefore mirror the effects of sensory deafferentation on psychotic symptoms [START_REF] Menon | Complex visual hallucinations in the visually impaired: the Charles Bonnet Syndrome[END_REF]. Indeed, it is well described that sensory loss or impairment (i.e. sensory deafferantation) is responsible for the emergence of hallucinations in the impaired sensory modality (for example the Charles Bonnet syndrom, the phamtom limb, or auditory hallucination in case of hearing loss, [START_REF] Linszen | Auditory hallucinations in adults with hearing impairment: a large prevalence study[END_REF]. Another potential explanation is that reduced social contacts may reduce social reality testing, which can make the distinction between reality and imagination difficult, leading to perception distortions, derealization, and hallucination [START_REF] Haney | Reforming Punishment: Psychological Limits to the Pains of Imprisonment[END_REF][START_REF] White | The social context of insight in schizophrenia[END_REF].

Although the social isolation associated with the COVID-19 crisis is considerably milder than the one enforced upon incarcerates in correction facilities or extreme explorers, it shares important similarities. Like these extreme forms, it may involve loneliness, little (or markedly reduced) face-to-face contact, little (or markedly reduced) physical contact, and little (or markedly reduced) possibilities of physical activities, as well as uncertainty about the duration of the isolation. Thus, there are reasons to believe that lockdown-related social isolation may have important psychological consequences in terms of an increase of psychotic symptoms and cognitive problems.

Supporting this hypothesis, two recent articles reported case series of First Episode Psychosis (FEP) admitted to hospitals in Italy in the second month of national lockdown [START_REF] D'agostino | Brief Psychotic Disorder during the national lockdown in Italy: an emerging clinical phenomenon of the coronavirus pandemic[END_REF][START_REF] Finatti | Psychosis in Patients in COVID-19-Related Quarantine: A Case Series[END_REF]. All patients, with no psychiatric history, presented religious/spiritual delusions and hallucinatory contents. Some of them also presented a somatic delusion of having contracted the SARS-CoV-2 infection. The authors suggest that intense psycho-social stress associated with a novel, potentially fatal disease and national lockdown restrictions might be a trigger for FEP. [START_REF] Escolà-Gascón | Pseudoscientific beliefs and psychopathological risks increase after COVID-19 social quarantine[END_REF] showed that perceptual disturbances (mainly derealization and depersonalization), subclinical psychotic symptoms and beliefs in the pseudoscience have increased after the social quarantine in Spain [START_REF] Escolà-Gascón | Pseudoscientific beliefs and psychopathological risks increase after COVID-19 social quarantine[END_REF]. However, this study did not assess other indicators that could explain this increase.

An additional two studies explored whether the first lockdown might have worsened pre-existing psychotic symptoms or caused the emergence of such symptoms in the general population. Using an experimental design, Lopes and collaborators (2020) showed that paranoia were more frequent in students and employed people than unemployed or selfemployed people during the pandemic. The presence of hallucinations and paranoia was observed only in people with negative affect (fear of COVID-19) and low political trust. The relationship was mediated by COVID-19 news exposure. In addition, Bortolon and collegues (2021) explored how paranoia and hallucinations were related to different stress-related factors during the lockdown in the UK. They showed that both hallucination and paranoia were predicted by repetitive thoughts and anxiety. Hallucinations were also related to sleep quality whereas paranoia was predicted by loneliness. However, none of the two studies observed a relationship between psychotic symptoms and the length of isolation during the lockdown.

The present study examined the relationship between psychotic symptoms and a wide range of potential explanatory factors, with a particular focus on living conditions and social isolation in a US-bases population. We hypothesized that some characteristics of living conditions during the pandemic (such as living space per person, the duration of the selfisolation, the frequency and quality of social interactions) would be related to the frequency of psychotic symptoms and cognitive problems experienced in a general population. More precisely, based on the existing literature, we hypothesized that psychotic symptoms and cognitive problems would be higher in individuals with less social interactions, more prolonged self-isolation and smaller living space.

Methods

Participants

Two hundred eleven participants were recruited through Amazon's Mechanical Turk.

Participants under 18 years old and non-native English speakers, were excluded. The validity of the collected data was ensured using several safeguards. Several attentional checks were included in the online survey to ensure participants answered the questions attentively and thoroughly, and did not rush through the survey. Those who failed to correctly answer one of the control questions and those who did not fully complete the survey were excluded from the analyses. The dataset was carefully checked for stereotyped responses and short completion time. Dubious participants were removed to avoid poor data quality (n = 16).

The final sample consisted of 80 women and 131 men, all living in the USA.

Participants' mean age was 38.33 years, and age ranged from 18 to 74 years. There was a roughly even number of men and women, and 85.3% of them were currently employed (n =180). Hence 14,7% of our sample was unemployed at the time of the questionnaire completion, which largely matches USA unemployment rate (= 13.3%) in May 2020 (data from U.S. Bureau of labor statistics). Also, the years of education showed a substantial variability (from 9 to 20). For this study, each Mechanical Turk worker received USD $ 2.

Procedure and Materials

The study was conducted online using Qualtrics Survey ® Software. The questions related to the coronavirus-crisis were presented by the end of a larger survey on psychotic symptoms and autobiographical memory. Data were collected during the period of April 27 and May 20, 2020 which overlapped with the time for stay-at-home restrictions in many US states. Participants were asked to answer a series of questions on the current COVID-19 pandemic (see Table 1 for details). The questions were about participants' current life situation during the coronavirus-crisis (i.e., economic situation, social isolation, changes in life situation, time since the last conversations, time since they last had left home, the frequency of social interactions and the current worries about COVID-19).

The frequency of psychotic symptoms (i.e. reality or perception distortions, hallucinations, alienation, delusion) and cognitive problems (i.e. memory or concentration difficulties, impulse control problems, thought intrusion) were assessed on 5-point Likert scales (see Table 1), adapted from the Community Assessment of Psychic Experiences questionnaire (CAPE ; [START_REF] Stefanis | Evidence that three dimensions of psychosis have a distribution in the general population[END_REF]. At the end of these questions, an additional question assessed whether the psychotic symptoms and the cognitive problems reported were currently more frequent, as frequent or less frequent than before the coronavirus-crisis.

By the end of the survey, the lockdown conditions were also assessed. First, participants were asked whether they were presently self-isolating. Following this yes/no question, participants who answered 'yes' were asked to report the duration of their selfisolation, the number of persons in their household, the number of square feet of the place in which they were self-isolating, and whether or not they had outdoor access.

Finally, we measured state anxiety (STAI, state subscale only; [START_REF] Spielberger | State-trait anxiety inventory: a comprehensive bibliography[END_REF], as this is one of the main outcome assessed in previous studies on mental health and confinement [START_REF] Brooks | The psychological impact of quarantine and how to reduce it: rapid review of the evidence[END_REF].

Data analysis

Statistical analyses were performed using SPSS software (version 26). We first conducted Pearson correlations between the frequency of psychotic symptoms, cognitive problems and anxiety, and second between measures of individuals' current life situations and lockdown conditions.

We next transformed open-ended measures of conditions of isolation into similarly sized groups. Between-group comparisons of psychotic symptoms and cognitive controls were carried out based on these groupings and using separate one-way analyses of variance (ANOVAs). A first series of group comparisons considered psychotic symptoms and cognitive problems as a function of different conditions of self-isolation. A second series of ANOVAs compared individuals having reported an increase versus a decrease of psychotic and cognitive symptoms.

Results

Descriptive statistics

Among the 211 participants, 192 participants (91%) reported living in a state where lockdown was endorsed and 129 participants (61,1%) reported being currently self-isolating.

Only 6 had lost their job because of the coronavirus outbreak (2.8%), and 180 (85,3%) reported being currently employed. See all descriptive statistics in supplemental material (Tables S1, S2 and S4).

Mental health measures

Correlation analyses

Scores of questions assessing psychotic symptoms correlated highly with each other's (rs > 0.60; ps < .001; see supplemental material Table S1) and were then analyzed aggregated together under the label "Psychotic symptoms", Cronbach Alpha = 0.95. Similarly, scores of questions on cognitive problems also correlated with each other (rs > 0.23; ps < .01; see supplemental material Table S2) and were analyzed aggregated together under the label "Cognitive problems", Cronbach Alpha = 0.82. Psychotic symptoms correlated with both cognitive problems and state anxiety (rs > 0.14; ps < .04). Cognitive problems and state anxiety, however, did not correlate with each other (r = 0.7; p = .30). Psychotic symptoms and cognitive problems were not correlated with age (rs < -0.07; ps > .33) or education (rs < 0.11; ps > .11). State anxiety correlated negatively with education (r = -0.15; p = .03) but not age (r = 0.06; p = .39). Some of the variables assessing the current life situation and the lockdown conditions also correlated with each other (see supplemental material Table S3).

We found a surprising negative correlation between the self-reported duration of selfisolation and frequency of psychotic symptoms (r = -0.25; p =.004). Because the self-reported duration of self-isolation was unrelated to the official lockdown duration endorsed by the participants' state of residence (r = 0.07; p = .44), and because this discrepancy may reflect attitudes concerning preventive activities, personal resources, or political inclination (CNN, n.d.; BBC news, 2020), we calculated the ratio between self-isolation duration and official lockdown duration. This ratio was thought to be a good indicator of the propensity of individuals to self-isolate themselves compared with the official rules endorsed by each state in the USA. Scores above 1.0 reflect a tendency to self-isolate more than required by the official rules, whereas scores under 1.0 reflect the opposite tendency. Negative correlations were observed between this ratio and psychotic symptoms, suggesting that people who selfisolated more than they were asked to do had less frequent psychotic symptoms (r = -0.27; p =.003). The same pattern of correlations was observed for cognitive problems (r = -0.25; p =.004), in particular for impulse control and memory problems (rs < -0.21; ps <.005).

Groups differences in psychotic symptoms and cognitive problems

Groups based on measures of isolation conditions.

To examine associations between key measures of life conditions during the COVID-19 pandemic and psychotic symptoms as well as cognitive problems, we transformed the open-ended key measures of isolation conditions (i.e., days since last conversation, days since they had last left home and square feet per person) into approximately equally sized groups, based on an inspection of the frequency distributions of these variables. We did so, because for such open-ended frequency measures, raw correlations are notoriously noisy and thus less sensitive. This is in part because the differences between the raw numbers likely do not map directly onto psychological importance. For example, if a person had no face-to-face conversation the last 20 days, this may not suggest more severe social isolation than not having had one for 25 days, whereas a difference between 0 and 5 days of having no face-toface conversation would seem more psychologically significant, although in raw numbers, the difference is the same. Hence, this relationship is likely not linear and thus not properly described with correlation analyses (nonetheless, all group effects to be reported in the following were replicated by correlational analyses based on the raw frequency measures, see supplemental material, Tables S4).

On the basis of these transformed variables, we conducted group comparisons of the frequency of psychotic symptoms, and cognitive problems, by days since last face-to-face conversation (four groups), days since they last left home (three groups) and by square feet per person (eight groups) (see Figure 1, the first three upper plots). We observed significant group effects of these three factors : ( 1 Importantly, these groups did not differ regarding economic impact of COVID-19

(except for groups based on days since they last left home, p = .01), level of education, or frequency of having had a psychiatric diagnosis in the past (all ps > .053). Thus, these potential confounding variables did not drive the results. Days since last conversation, F(2,208) = 4.00; p =.008; ŋ 2 p = .06, and Days since last outing, F(2,208) = 8.83; p <.0001; ŋ 2 p = .08, were associated with level of COVID-19 worries, but living space per person was not, F(7,121) = 0.90; p = .51; ŋ 2 p = .05.

Groups based on self-reported increase or decrease of mental problems.

A substantial subclass (n = 31) indicated that the reported symptoms and problems had become worse during the time of lockdown, while another substantial subclass (n = 27) reported a reduction in symptoms and problems. Most participants (n = 153) detected no change. Using one-way ANOVA, we compared the profile of these three groups of participants on psychotic symptoms, cognitive problems and anxiety related to (see all statistics in supplemental material Table S5).

In summary, several significant group effects were observed on the frequencies of both psychotic symptoms, F(2,208) = 11,41; p <.001; ŋ 2 p = .10, and cognitive problems, F(2,208) = 20,00; p <.001; ŋ 2 p = .16, on the worries on COVID-19, F(2,208) = 3,19; p = .04; ŋ 2 p = .03. Posthoc analyses showed that the group of participants reporting an increase of the frequency of their psychological symptoms also presented a higher level of psychotic symptoms, p = .003, and cognitive problems, p <.001 (see Figure 1, the lowest plots), and higher worries on COVID-19, p = .02, in comparison to the "as frequent" group.

The group of participants reporting less frequent psychological symptoms during the coronavirus crisis did not report lower frequencies of psychotic symptoms or cognitive problems, ps >.16. Instead, their scores were comparable with the "as frequent" group (see Figure 1, the lowest plots).

The group reporting more frequent symptoms did not differ from the two others groups on age, education, income, or economic impact of the COVID-19 crisis, Fs(2,208) < 1; ps >.46. However, we observed significant differences between the "more frequent" group and the other groups on square feet per person, F(2,126) = 3,96; p =.02; ŋ 2 p = .06, and days since last conversation, F(2,208) = 13,40; p <.001; ŋ 2 p = .11. More precisely, post hoc analyses showed that the "more frequent" group reported fewer square feet per person compared to the "as frequent" group, p = .02, and more days since last conversation compared to the "less frequent" group, p <.001.

Discussion

The present study examined the relationship between social isolation, psychotic symptoms and cognitive problems in the context of the worldwide COVID-19 pandemic.

Overall, we observed relationships between the frequency of both psychotic symptoms and cognitive problems and life conditions (living space and days since the last outing and the last conversation) during a period of widespread stay-at-home restrictions in a general population.

A substantial number of the participants reporting an increase in the frequency of their psychotic symptoms and cognitive problems during this period also displayed higher levels of these symptoms in comparison to other participants.

First, group comparisons showed that the severity of both psychotic symptoms and cognitive problems differed as a function of the time since participants' last conversation, since they last left home and as a function of the living space. We observed that the frequency of both psychotic symptoms and cognitive problems was higher when participants lived in smaller homes or had less social interactions or outings (reflected by the number of days since last conversation or last outing). This relationship was stronger during the first days of absence of interactions or outing, suggesting the importance of the lack of interactions from the very first days of isolation. Similarly, the impact of living space on psychotic symptoms and cognitive problems was higher for smaller living space.

Hence, individuals experiencing psychotic symptoms and cognitive disorders more frequently were overall more isolated during the lockdown. These results also suggest that the first days without social interactions or outings are particularly important when considering psychotic symptoms or cognitive problems. The findings are in line with the literature showing strong relationships between loneliness and psychotic symptoms [START_REF] Bortolon | Persecutory ideation and anomalous perceptual experiences in the context of the COVID-19 outbreak in France: what's left one month later?[END_REF][START_REF] Chau | Loneliness and the psychosis continuum: a metaanalysis on positive psychotic experiences and a meta-analysis on negative psychotic experiences[END_REF][START_REF] Narita | Loneliness and psychotic experiences in a general population sample[END_REF]. They also underscore the relevance of scientific, medical and societal attention to such symptoms in situations of forced or recommended social isolation, in particular during the first days of isolation and for people living in small spaces.

Importantly, education, economic impact of COVID-19 and former psychiatric diagnosis were not related to the number of days since the last conversation or days since last outing, or to square feet per person. This finding rules out some of the potential confounding factors that could provide alterative explanations for the observed effects.

Along all the analyses, we found that the measures of psychotic symptoms and cognitive problems were similarly associated with the parameters of life situations or conditions of self-isolation, consistently with earlier findings showing that cognitive deficits are highly frequent in psychotic disorders and play an important role in patients' social functioning [START_REF] Green | Social cognition in schizophrenia[END_REF][START_REF] Jabben | Cognitive alterations in groups at risk for psychosis: neutral markers of genetic risk or indicators of social disability?[END_REF].

Second, a substantial number of participants, (15%, n = 31) reported an increase in the frequency of their psychotic symptoms and cognitive problems during the pandemic and stayat-home restrictions, in comparison with the period before the lockdown. These results are in agreement with the increase of perceptual disturbances (mainly derealization and depersonalization), subclinical psychotic symptoms and beliefs in the pseudoscience observed after the lockdown in Spain [START_REF] Escolà-Gascón | Pseudoscientific beliefs and psychopathological risks increase after COVID-19 social quarantine[END_REF]. In the present study, those participants also reported a higher frequency of psychotic symptoms, cognitive problems and higher worries about the COVID-19, in comparison to other participants. However, they did not differ from other participants on age, education, income or economic impact of the COVID-19 crisis, ruling out lower education or social class as potential explanations for this increase of symptoms. The only detected differences were a smaller living space and a longer time since last conversation compared with the other two groups. Although the present study was not longitudinal, we consider these findings of prime importance as the subjective experience of an increase reflects either a worsening of earlier psychotic symptoms and cognitive problems or the emergence of new psychotic symptoms and cognitive problems during self-isolation.

Conversely, 13% of the participants (n = 27) reported experiencing a decrease in psychotic symptoms and cognitive problems during this period in comparison to the time before the coronavirus crisis. These participants did not report lower levels of psychotic symptoms or cognitive problems. Not knowing the initial frequency of both psychotic symptoms and cognitive problems in this group prevents us from drawing any conclusion on the hypothetical beneficial effects of social isolation on these participants' mental health.

The main limitation of the present study is its cross-sectional design. This prevents us from establishing whether the results reflect specific relationships emerging during a worldwide outbreak or whether they reflect a broader reality as to how psychotic symptoms, quite common in the general population [START_REF] Van Os | A systematic review and meta-analysis of the psychosis continuum: evidence for a psychosis proneness-persistence-impairment model of psychotic disorder[END_REF], relate to individuals' social functioning. To better understand the relationship between the development of psychotic symptoms and cognitive problems during a lockdown period, longitudinal studies are needed.

Longitudinal studies examining depressive and anxiety symptoms showed no worsening of mental disorders during the first COVID-19 lockdown [START_REF] Wang | Immediate Psychological Responses and Associated Factors during the Initial Stage of the 2019 Coronavirus Disease (COVID-19) Epidemic among the General Population in China[END_REF]Fancourt et al., 2020). However, these studies did not assess psychotic symptoms. Nonetheless, while not clarifying the direction of causality, the present findings adds to the literature by showing a clear relation between social isolation and psychotic symptoms as well as cognitive problems, and by showing that a substantial number of participants experienced these problems as having increased during the lockdown. This suggests that psychotic symptoms and cognitive problems are relatively common dysfunctional reactions that lawmakers should take into consideration when issuing societal lockdown. Speculatively, such reactions may help to fuel conspiracy theories and irrational fears in the general public during a health crisis (see also [START_REF] Lopes | Paranoia, hallucinations and compulsive buying during the early phase of the COVID-19 outbreak in the United Kingdom: A preliminary experimental study[END_REF].

The present study did not use standardized questionnaires to measure conditions of isolation, cognitive impairments and psychotic symptoms. This is because we wanted to word the questions specifically to address the extraordinary event we were facing in terms of COVID-19 lockdowns. For this reason, we adapted some of the standard CAPE questions on psychotic symptoms to the current situation, and we selected items for cognitive problems that previously had been associated with social isolation. Both sets of items showed satisfactory internal reliability. For the sake of brevity, not all variables usually described as related to psychotic symptoms (e.g. quality of sleep) were included in the present study.

Future studies should expand the range of questions to include such variables.

To conclude, the present findings identified important societal and psychological factors to take into account in periods of lockdown and stay-at-home restrictions. The findings suggest that some individuals might be more at risk for experiencing psychotic symptoms and cognitive problems than others, namely, those who have few face-to-face interactions, rarely leave home and have small living space per person. Importantly, the present study identified psychotic symptoms as an important, but so far neglected, negative effect of self-isolation and confinement during pandemics.

  ) days since last face-to-face conversation on psychotic symptoms, F(3,207) = 19.89; p <.0001; ŋ 2 p = .22, and cognitive problems, F(3,207) = 4.14; p =.007; ŋ 2 p = .06, (2) days since last left home on psychotic symptoms, F(2,208) = 5.45; p =.005; ŋ 2 p = .05, and cognitive problems, F(2,208) = 3.77; p =.02; ŋ 2 p = .03, and (3) square feet per person on psychotic symptoms, F(7,121) = 8.58; p <.0001; ŋ 2 p = .33, and cognitive problems, F(7,121) = 2.52; p =.02; ŋ 2 p = .13.

Figure 1 .

 1 Figure1. Mean ratings of psychotic symptoms (left plots) and cognitive problems (right plots) as a function of the number of days since the last conversation (first row), since participants last left home (second row), of living space (third row), and of the development of these symptoms during the coronavirus crisis (fourth row). Error bars represent standard errors.

Acknowledgements

This work was supported by a grant from the Danish National Research Foundation (DNRF89).