Marco Panza

La Qué Podría Haber Sido

Euclides G D Universalidad Para

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O M Sautter

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Euclides Marco Panza

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

tampoco deseo una respuesta que vaya en contra de la filología. Así, comenzaré recolectando evidencia filológica. He revisado las afirmaciones de las 48 proposiciones del primer libro de los Elementos y clasificado los diferentes modos en que ellas, en mi opinión, se refieren a los ítems relevantes.

Dejé de lado las pruebas, tanto por cuestiones de espacio, como también porque la práctica (o técnica) de la ekthesis uniformiza el modo de referencia, de una manera que no ofrece comprensión sino que la requiere, puesto que esconde el universal bajo la forma aparente de un particular.

Algunas proposiciones refieren a los ítems a través del nombre de su tipo, utilizado en el singular y precedido por un artículo, a menudo junto con el calificativo que es usualmente traducido como 'dado', extraído del verbo 'δίδωμι':

• Prop. I.1 Sobre una línea recta dada [επι της δοθεισης ευθειας] construir un triángulo equilátero.1

• Prop. I.2 Poner en un punto dado [τῶ δοθεντι σημειω] (como extremo) una línea recta igual a una línea recta dada [τῆ δοθειση ευθεια].

• Prop. I.9 Bisectar un ángulo rectilíneo dado [τὴν δοθεῖσαν γωνίαν εὐθύγραμμον].

• Prop. I.10 Bisectar una línea recta dada [τὴν δοθεῖσαν εὐθεῖαν πεπερασμένην].

• Prop. I.11 Trazar una línea recta en ángulo recto a una línea recta dada [τῇ δοθείσῃ εὐθείᾳ] desde un punto dado [δοθέντος σημείου] sobre ella.

• Prop. I.12 Para una recta infinita dada [τὴν δοθεῖσαν εὐθεῖαν ἄπειρον], desde un punto dado [τοῦ δοθέντος σημείου] que no está sobre ella, trazar una línea recta perpendicular.

Otros ejemplos son las proposiciones I.23; I.31; I.42; I.44; I.45; I.46.

Algunas proposiciones hacen lo mismo, aunque omitiendo tanto el calificativo 'dado' como el artículo:

• Prop. I.6 Si en [un] triángulo [τριγωνου] dos ángulos son iguales entre sí, los lados que subtienden los ángulos iguales serán también iguales entre sí.

• Prop. I.13 Si [una] línea recta [εὐθεῖα] puesta sobre [una] línea recta [εὐθεῖαν] forma ángulos, formará o bien dos ángulos rectos o bien ángulos iguales a dos ángulos rectos. Finalmente, algunas otras proposiciones utilizan uno de los modos anteriores de exposición para parte de ellas, y otros para otras partes de ellas, tal como ocurre con la recién mencionada Proposición I.14:

• Prop. I.7 Dadas dos líneas rectas [δυο ταις αυταις ευθειαις] constuidas sobre una línea recta [επι της αυτης ευθειας] (desde sus extremos) y que se encuentran en un punto, no es posible constuir sobre la misma línea recta (desde sus extremos), y por el mismo lado de ella, otras dos líneas rectas que se encuentren en otro punto y que sean iguales respectivamente a las dos anteriores, es decir, con los mismos extremos que las líneas rectas dadas.

• Prop. I.27 Si una línea recta [εὐθεῖα] al incidir sobre dos líneas rectas hace [δύο εὐθείας] los ángulos alternos iguales entre sí, las dos rectas serán paralelas entre sí.

• Prop. I.28 Si una línea recta [εὐθεῖα] al incidir sobre dos líneas rectas hace el ángulo externo igual al interno y opuesto del mismo lado, o los dos internos del mismo lado iguales a dos rectos, las líneas rectas [αἱ εὐθεῖαι] serán paralelas entre sí.

• Prop. I.29 (para una parte de) Una línea recta que cae sobre líneas rectas paralelas [ἡ εἰς τὰς παραλλήλους εὐθείας εὐθεῖα ἐμπίπτουσα] hace los ángulos alternos iguales entre sí, el ángulo externo igual al interno y opuesto, y los ángulos internos del mismo lado igual a dos ángulos rectos.

Algunos de estos usos son ciertamente establecidos por restricciones gramaticales. Sin embargo, creo que podrían haber sido uniformizados, si ello hubiese sido requerido. Ello sugiere que cualquier interpretación de la universalidad de las proposiciones de Euclides que esté justificada por algunos de estos usos, lo estaría arbitrariamente, puesto que se podría haber apelado a otros para contradecirla.

Por lo tanto, el único modo de desenmarañar esta cuestión es por medio de una reflexión general sobre la lógica de los argumentos de Euclides (si es que es posible identificar una). Identificar y reconstruir esta lógica es el objeto de una investigación en curso, que estoy llevando a cabo con el colega Alberto Naibo. Aquí solo deseo realizar algunas observaciones sobre el problema de la universalidad.

Mi punto de partida es el que ya he anticipado: deseo evitar toda apelación a cualquier tipo de totalidad fija de ítems geométricos, sean estos:

• Una totalidad de ítems independientes y existentes por sí mismos (presumiblemente ideales o abstractos), simplemente porque, aun si Euclides hubiese estado dispuesto a admitirla, no hubiese tenido manera (o hubiese fallado en proporcionar los medios) de discriminar globalmente entre ellos, o, al menos, de hacer claro lo que distingue a uno de ellos de los otros;

• Una totalidad de ítems construidos, almacenados en algún lugar para su futuro uso o referencia, por razones similares a las anteriores, puesto que el único modo en que Euclides hubiese podido discriminar entre ellos hubiese sido apelando a un acto previo de construcción, planteado como la construcción de un ítem individual, distinto de cualesquiera otros ítems del mismo tipo, y sin ningún tipo de necesidad matemática, sino más bien perjudicando la universalidad de sus construcciones;

• Una totalidad de formas, que se distinguen entre sí por su tamaño y su ubicación en el espacio, puesto que cualquier distinción de esta clase hubiese requerido algún tipo de sistema de referencia en el espacio, que Euclides hubiese podido proporcionar no solo identificando ya un ítem en el espacio, sino también y en general solo localmente apelando meramente a una representación diagramática de este ítem, y haciendo trabajar sus condiciones locales de identidad.

La idea básica para evitar esto es entender a las construcciones de Euclides como regidas por algo así como reglas de introducción para ítems de tipos previamente identificados, cada uno de los cuales puede ser aplicado en todo momento que se requiera tener a disposición uno de estos ítems. Algunas de estas reglas son elementales y vienen con suposiciones liminales de la teoría. Otras son derivadas y se siguen de las elementales.

Utilizando letras romanas mayúsculas, A, B, C, . . . , para puntos, letras romanas minúsculas, a, b, c, . . . , para segmentos, y letras griegas minúsculas, α, β, γ, . . . , para círculos, estas reglas pueden ser formuladas de un modo esquemático como sigue:

A B P.1 a ; a P.2 b c ; A B P.3
α o mejor:

A B P.1 a [A,B] ; a P.2 [a] b [a] c [b] ; A B P.3 α [A,B]
Algunas otras reglas elementales son implícitas. Entre ellas, las más importantes son las siguientes:

• Una regla requerida para comenzar cualquier construcción Com a

• Una regla requerida para extraer puntos a partir de segmentos a

Ex [a] A B [a]
• Una regla (diagramática) requerida para obtener puntos a partir de intersecciones D

In

A [D]
Las reglas derivadas son probadas resolviendo problemas. De acuerdo con la interpretación que sugiero, resolver un problema proporciona, en efecto, una suerte de meta-teorema, que establece que algunos objetos pueden ser construidos si otros son tales, concibiendo a una construcción como el resultado de las reglas de introducción.

El ejemplo más simple y obvio es la Proposición I.1. Tomemos a un triángulo como una configuración apropiada de tres segmentos, y a un triángulo equilátero, como una configuración de este tipo, que satisface las condiciones apropiadas. Supongamos también que las reglas de introducción vienen aparejadas con reglas secuenciales que permiten guardar registro de la introducción sucesiva de ítems, y combinarlas (funcionando, mutatis mutandis, como reglas de inferencia en una deducción), y además con reglas de identificación (que corresponden básicamente a definiciones) que permiten reconocer una cierta configuración, tal como un triángulo equilátero, en el resultado de una construcción (en virtud de una prueba apropiada). Luego, la solución de la Proposición I.1 muestra que un triángulo es construible (puede ser introducido) por la aplicación sucesiva de las reglas: Com; Ex; P.3 (dos veces); In; P.1 (dos veces), que arrojan la siguiente regla derivada:

a I.1
a, b, c Teq Ahora bien, ¿qué tiene todo esto que ver con la universalidad? Mucho, sugiero, puesto que nos proporciona suficientes herramientas conceptuales para delinear una noción de universalidad, que no depende de ninguna apelación a una totalidad fija de ítems geométricos, y es perfectamente compatible con la práctica constructiva y demostrativa de Euclides.

Lo primero que debe observarse es que las reglas previas de introducción son esquemáticas. Pero no en el sentido de que las letras utilizadas para enunciarlas pueden referir a cualquier ítem del tipo apropiado; sino más bien en el sentido de que las letras usadas para enunciarlas pueden ser reemplazadas, en ellas, por cualqueir otra letra del mismo tipo, y en el sentido de que cualquier letra del tipo apropiado puede ser usada para referir a los ítems relevantes, dentro de cualquier argumento, con la única restricción de que la referencia debe ser mantenida fija a lo largo de todo el argumento. Sí, ¿pero cuáles son los ítems relevantes?

Consideremos un ejemplo para comprender mejor el punto que quiero establecer. El más simple es, de nuevo, la proposición I.1. Esta requiere construir un triángulo equilátero con un segmento dado. ¿Qué segmento? Ciertamente no cualquiera, elegido dentro de la totalidad de segmentos. Ciertamente no uno particular, elegido dentro de esta totalidad. Ciertamente no uno genérico que representa cualquier segmento de esta totalidad. Pero, mucho más simple, un segmento que de ningún modo es particular, puesto que no tiene ni una posición particular en el espacio, ni una longitud particular. Y ello, meramente, porque no hay otro ítem en el espacio de la proposición, con respecto al cual su posición pueda ser determinada, mejor, no hay espacio en el cual está inmerso, sino que más bien él mismo proporciona una referencia con respecto a la cual la posición de otros ítems puede ser luego determinada. Y porque no hay ningún otro segmento que tenga una longitud cualquiera con respecto a la cual su propia longitud pueda ser determinada.

Si uno quiere decirlo de la siguiente manera: este es un segmento genérico, pero no porque representa o está en lugar de cualquier otro, sino porque no es de modo alguno particular, no tiene ninguna propiedad específica que lo distinga de los otros. Es con este segmento que la construcción comienza.

Por lo tanto, la Proposición I.1 es universal, puesto que trata del único caso que podía ser considerado: el caso de un solo segmento que es simplemente genérico por no ser particular en modo alguno.

Si ello resulta así, ¿que tipo de resultado universal establece la resolución de este problema? Simplemente, como dije antes, que una nueva regla derivada de introducción puede ser añadida a las elementales: una que autoriza la introducción de triángulos equiláteros. Ello también puede ser dicho de la siguiente manera: los triángulos equiláteros son construibles con los medios de construcción admitidos. Dos objeciones parecerían aquí obvias. Sin embargo, sostengo que cada una de ellas es defectuosa.

La primera es que ahora estoy utilizando un plural para expresar universalidad: digo que los triángulos equiláteros son construibles. En consecuencia, refiero implícitamente a una totalidad. Aquí está mi respuesta: por supuesto, utilizo el plural, e incluso para mí ello resulta esencial, puesto que sin utilizar el plural no sería capaz de expresar universalidad. Y por supuesto, también, al hacerlo refiero implícitamente a una totalidad. Pero esta no es una totalidad de ítems geométricos. No estoy hablando de todos los triángulos equiláteros. Estoy hablando de una totalidad abierta de actos de construcción admitidos dentro de una teoría. Estoy diciendo que la teoría permite introducir (construir) un triángulo equilátero, cada vez que un segmento ha sido introducido (construido o dado).

Ello debería volver manifiesta a la segunda objeción. Se trata de una objeción clásica, pero que no por ello no está fuera de lugar: ¿Cómo podemos estar seguros que una construcción realizada una vez puede ser repetida? Digamos que esta construcción comienza con un segmento que es genérico por no ser particular en modo alguno, por no tener ninguna propiedad específica. Pero ello no es lo que ocurre cuando la construcción puede ser repetida en la solución o prueba de cualquier otra proposición. Por ejemplo, ¿no resulta así, ya en la Proposición I.2, donde un triángulo equilátero es construido sobre un segmento que une los extremos de dos segmentos dados, y entonces es particular tanto por su posición como por su longitud? Por supuesto, ello es así. Pero, ni en la proposición I.2, ni en ninguna otra proposición en la que la Proposición I.1 es o puede ser aplicada, los otros ítems que hacen específico al ítem relevante son relevantes para que la construcción del triángulo equilátero resulte exitosa. Ella simplemente tiene lugar aisladamente de estos otros ítems: ella meramente considera al segmento singular sobre el que se construye un triángulo equilátero, que, como tal, aisladamente, es todavía genérico por no ser particular en modo alguno, por no tener ninguna propiedad específica.

Ahora que la Proposición I.1 ha sido probada, sabemos que un triángulo equilátero puede ser construido cada vez que se lo requiera. Esto es lo que una solución de un problema prueba, y es una afirmación universal. Más aún, en el caso de que la solución sea correcta, la afirmación está correctamente probada por ella, sin ningún salto de lo particular a lo general, sin ninguna suposición de uniformidad, sin ninguna apelación a una cuantificación universal sobre cualquier totalidad de ítems geométricos.

Mutatis mutandis, lo mismo ocurre con cualquier otro problema geométrico de la geometría de Euclides.

Lo que debe ser modificado depende de una simple observación: en la mayoría de los otros problemas, lo que es dado no es un ítem singular, que puede ser tomado como genérico por no ser particular en sentido alguno, sino más bien múltiples ítems que permiten la comparación entre sus respectivos tamaños, y admiten diferentes posiciones relativas; a menudo, lo que es dado es incluso una configuración de ítems, que es particular debido a su estructura interna.

Sin embargo, lo que puede ser diferente en un sentido relevante de un caso posible a otro son, por así decirlo, las características topológicas, que pueden variar en un número finito, usualmente pequeño, de maneras. Por lo tanto, la generalidad solo depende de consideraciones de un número finito de casos. Ciertamente, Euclides generalmente no distingue casos. Pero ello puede ser realizado y, como observó repetidamente Proclo, debe ser realizado para dar una solución a la requerida universalidad. Si Euclides no lo hace, supongo, es meramente porque el caso considerado proporciona suficientes instrucciones para tratar con todos los otros, de modo de obtener el mismo resultado.

Es importante aclarar un punto aquí. Afirmo que lo que puede ser diferente en un sentido relevante de un caso posible a otro, en lo que un problema toma como dado, son las propiedades topológicas. Ello podría ser desafiado. Uno podría observar, por ejemplo, que cuando dos segmentos son dados, su razón puede ser diferente, de un caso a otro caso. Podemos imaginar que nos fueron dados segmentos, uno de los cuales es el doble del otro, o el triple, o inconmensurable con aquel, de acuerdo con cierta secuencia antipherisis, o simplemente más que el doble, el triple, etc. Esto es cierto, pero la corrección de la solución depende justamente de su independencia respecto de estas diferencias, que por cierto son muy a menudo identificables solo en relación a parámetros aritméticos externos. Ello hace que estas diferencias simplemente no sean relevantes.

La solución a cualquier problema en la geometría de Euclides proporciona, de acuerdo con mi interpretación, la prueba de una regla derivada de introducción. Y, al hacerlo, autoriza una extensión de la ontología de esta teoría. Pero no lo hace probando que los nuevos ítems existen; y, menos incluso, añadiendo nuevos ítems a la totalidad de aquellos que ya existen. Y tampoco añadiendo nuevos ítems a la totalidad de aquellos ya construidos. Por el contrario, esta extensión es lograda proporcionando nuevas reglas de introducción, a las que se debe apelar, para construir los ítems relevantes en cualquier caso que esto sea requerido.

¿Pero qué ocurre con la universalidad de los teoremas? ¿Qué prueba el teorema I.5, por ejemplo, si no es acaso que en todos los triángulos isósceles, o en cualquiera de ellos, los ángulos de la base son iguales entre sí? Si no hubiese otra alternativa más que leer estos teoremas de este modo, no podríamos evitar admitir que, a pesar de la dificultad de proporcionar condiciones de identidad globales, Euclides aceptó totalidades fijas de ítems y modos de tratar con ellas. Sin embargo, existe una alternativa, y la comprensión de los problemas que he recién presentado es una indicación de la misma.

En la geometría de Euclides, las pruebas de un teorema comienza con la suposición de que los ítems relevantes, aquellos acerca de los cuales el teorema afirma que cumplen una cierta condición, son dados. Pero su ser dados no es arbitrario. Ello está sujeto a la posibilidad de introducir (construir) estos ítems a través de las cláusulas de introducción aceptadas. No hay prueba alguna, en los Elementos, acerca de un cuadrado igual a un círculo dado, o acerca de la tercera parte de un ángulo.

Esto sugiere que lo que una prueba de un teorema geométrico demuestra es algo acerca del resultado de una construcción. Es decir, que la construcción de cierto ítem se corresponde, o no puede sino corresponderse, con hacer que este ítem satisfaga algunas condiciones, o con la construcción de ítems relacionados que satisfacen ciertas condiciones.

Por ejemplo, lo que la prueba de la Proposición I.5 demuestra es que la construcción de un triángulo isósceles se corresponde con la construcción de dos ángulos iguales. Esto es esencialmente diferente a probar que todos los triángulos isósceles tienen los ángulos de la base iguales entre sí. Simplemente porque en el marco de la geometría de Euclides no hay algo así como todos los triángulos isósceles. Lo que hay son reglas de introducción que, cuando son combinadas entre sí, introducen un triángulo isósceles, que, además de sus propiedades irrelevantes, es genérico, por no ser particular en modo alguno, por no tener ninguna propiedad específica, más allá de ser triangular e isósceles, y, entonces, de tener dos de sus lados iguales.

En un sentido esto es universal, puesto que cada vez que un triángulo isósceles es construido, dos ángulos iguales son así. Pero en otro sentido también es particular, puesto que trata del resultado de una construcción singular, o, posiblemente, de una clase de equivalencia de construcciones, o incluso, para decirlo mejor, solo trata de un número finito de reglas particulares de introducción, y de un modo particular de combinarlas.

Una última observación, para responder preventivamente a una nueva posible objeción.

Uno podría decir que lo que hace que los triángulos isósceles tengan los ángulos de la base iguales no es el modo modo en que están construidos, sino el hecho independiente de que tienen dos lados iguales entre sí.

Sin embargo, esto está simplemente errado. Puesto que, en la geometría de Euclides, lo que decide que dos segmentos sean iguales es que están construidos como los radios de un mismo círculo, o como los radios de círculos cuyos radios son iguales, dado que están construidos para ser así, debido al modo en que la regla P.3 es aplicada. Y, mutatis mutandis, esto es también así para los ángulos, a pesar de que la cuestión aquí es un poco más compleja, en virtud del papel de la proposición I.4

Si estoy acertado, no sólo la generalidad de la geometría de Euclides trata de sus ítems sin ocuparse de cualquier totalidad de ellos, sino también sus reglas de introducción, y meramente dependiendo de sus condiciones locales de identidad, que corresponden a las condiciones de identidad de los diagramas trazados sobre un sustento limitado, o imaginadas para ser trazada sobre tal tipo de sustento. Pero con ello no viene aparejado nada misterioso, puesto que esto no depende de ninguna generalización, o esquema, sino solo del modo en que funcionan las reglas de introducción relevantes.

•

  Prop. I.21 Si sobre uno de los lados de un triángulo [τριγώνου ἐπὶ μιᾶς τῶν πλευρῶν], desde sus extremos, se construyen dos líneas rectas que se encuentren en el interior del triángulo, las líneas rectas construidas serán menores que los lados restantes del triángulo, pero comprenderán un ángulo mayor.• Prop. I.3 Dadas dos líneas rectas desiguales [δυο δοθεισων ευθειων], quitar de la mayor una línea recta igual a la menor.

	Algunas proposiciones siguen los postulados I.1, I.3 y I.4, al utilizar 'πᾶς',
	que resultaría difícil no traducir como 'todo':
	• Post. I.1 Postúlese el trazar una línea recta desde todo punto hasta
	cualquier punto [απο παντος σημειου επι παν σημειον].
	Otros ejemplos son las proposiciones I.41; I.48. • Post. I.3 Y el describir un círculo con todo centro y distancia [παντι κεντρω
	Algunas proposiciones utilizan numerales adecuados, en lugar del artículo και διαστηματι].
	con o sin 'dado': • Post. I.4 Y todos los ángulos rectos [πασας τας ορθας γωνιας] son iguales
	entre sí.
	• I.16 En todo triángulo [παντὸς τριγώνου], si se prolonga uno de sus lados, el
	ángulo externo es mayor que cada uno de los ángulos internos y opuestos.
	• I.17 En todo triángulo [παντὸς τριγώνου] dos ángulos tomados juntos de
	cualquier manera son menores que dos ángulos rectos.
	• I.18 En todo triángulo [παντὸς τριγώνου] el lado mayor subtiende al ángulo
	mayor.
	• I.19 En todo triángulo [παντὸς τριγώνου] al ángulo mayor lo subtiende el
	lado mayor.
	• I.20 En todo triángulo [παντὸς τριγώνου] dos lados tomados juntos de
	cualquier manera son mayores que el restante.
	Otros ejemplos son las proposiciones I.18; I.19; I.20; I.32; I.43.
	Otros ejemplos son las proposiciones I.34; I.35; I.36; I.37; I.38: I.39; 40; I.47.
	La Proposición 14 hace lo mismo con una parte de ella, aunque utilizando,
	en otra parte, el pronombre 'τις', que podría ser mejor traducido como 'algún':
	• Prop. I.14 Si con alguna línea recta [τινι εὐθείᾳ], y en un punto de ella, dos
	líneas rectas [δύο εὐθεῖαι] que no están en el mismo lado (de ella) forman
	los ángulos adyacentes iguales a dos rectos, las dos líneas rectas estarán
	en línea recta entre sí.

• Prop. I.4 Si dos triángulos [δυο τριγωνα] tienen dos lados iguales a dos lados respectivamente, y tienen iguales los ángulos comprendidos por las líneas rectas iguales, tendrán también la base igual a la base, el triángulo será igual al triángulo, y los ángulos restantes serán iguales a los restantes ángulos respectivamente, a saber: los subtendidos por lados iguales.

• I.8 Si dos triángulos [δυο τριγωνα] tienen los dos lados iguales a dos lados respectivamente, y tiene también la base igual a la base, ellos también tendrán iguales los ángulos que son contenidos por las líneas rectas iguales.

• I.15 Si dos líneas rectas [δύο εὐθεῖαι] cortan una a la otra, ellas hacen los ángulos del vértice iguales entre sí.

Otros ejemplos son las proposiciones I.22; I.24; I.25; I.26. Algunas proposiciones directamente usan el nombre del tipo de ítems relevantes en el plural, junto con un artículo:

• Prop. I.5 En los triángulos isósceles [των ισοσκελων τριγωνων] los ángulos de la base son iguales entre sí, y si las líneas rectas iguales son prolongadas, los ángulos bajo la base serán iguales entre sí.

• Prop. I.30 Las líneas rectas [εὐθεῖαι] paralelas a una misma línea recta son también paralelas entre sí.

• Prop. I.33 Las líneas rectas [εὐθεῖαι] que unen líneas rectas iguales y paralelas sobre la misma parte [de ellas] son también ellas mismas iguales y paralelas.

En la medida de lo posible, hemos intentado una traduccion literal al castellano de la traducción al inglés del texto de Euclides por parte del autor de esta nota, con base en la traducción de Heath. El texto griego entre corchetes es introducido por el autor.[N.d.T.]

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Numerical simulation of the reaming process in aluminium 2024-T351 -Analysis of the evolution of the material stress state

Keywords: Reaming, numerical simulation, residual stress, aluminum

In aircraft assemblies, holes for fastening are critical areas from where fatigue damage can be initiated, especially for metal parts. Depending on the operating conditions of the various stages of the machining process of the hole (drilling, reaming, cold expansion, second reaming), the manufacturers observe significantly different fatigue strength of the structures. To optimize the behavior of the aircraft assemblies, the manufacturers want to understand the impact of the whole machining process of the hole on the material characteristics of the part. They investigate particularly the residual stress state in the hole edge zone which can be significantly different depending on the operating conditions. This paper investigates the influence of the reaming process in an aluminum 2024-T351 part with initial material state not clean of mechanical stress. Indeed, before the final reaming, the parts undergo at least a drilling operation, and sometimes a cold expansion operation. For this study, in collaboration with Airbus, the simulated reamed part has been pre-stressed by the cold expansion process. Both processes are simulated using a finite element model with Abaqus. The implemented strategy to simulate the reaming process is based on the progressive deactivation of mesh elements. The obtained results show a low relaxation of the circumferential and radial residual stresses in the part during the reaming simulation.

INTRODUCTION

Fastening holes in mechanical structures are critical areas for the initiation of fatigue damage, especially for metal parts. The experimental study of fatigue strength and failure modes of bolted joints made of aluminum alloy parts is the subject of a lot of experimental works [START_REF] Krishnakumar | Fatigue behaviour of aluminium alloy 7075 bolted joints treated with oily film corrosion compounds[END_REF][START_REF] Croccolo | Failure analysis of bolted joints: Effect of friction coefficients in torque-preloading relationship[END_REF][START_REF] Hoang | Rupture and damage mechanism analysis of a bolted assembly using coupling techniques between A.E. and D.I.C[END_REF][START_REF] Fallahnezhad | Failure mode analysis of aluminium alloy 2024-T3 in double-lap bolted joints with single and double fasteners; A numerical and experimental study[END_REF][START_REF] Samaei | Experimental and numerical study of fatigue crack growth of aluminum alloy 2024-T3 single lap simple bolted and hybrid (adhesive/bolted) joints[END_REF]. Different failure modes are observed. Some failure occurs in the bearing mode, with cracks initiating at the edge of the hole and others occurs in the fretting mode, with cracks initiating away from the hole boundary. Residual stress fields at the vicinity of the hole have an impact on the fatigue performance of holed parts [START_REF] Lai | Fatigue properties of holes with residual stresses[END_REF][START_REF] Withers | Residual stress and its role in failure[END_REF]. Compressive residual stress result in a significant increase in the fatigue lives. Indeed, this kind of stress induces a closure effect on the crack whereas the tensile residual stress results in an opening effect on the crack. Thus, in the aeronautical industry, processes have largely been developed in order to introduce a compressive residual stress field at the vicinity of the holes [START_REF] Sticchi | Review of Residual Stress Modification Techniques for Extending the Fatigue Life of Metallic Aircraft Components[END_REF]. Interference fit fastener [START_REF] Chakherlou | An investigation about interference fit effect on improving fatigue life of a holed single plate in joints[END_REF][START_REF] Lanciotti | The effect of interference-fit fasteners on the fatigue life of central hole specimens[END_REF][START_REF] Taghizadeh | Fatigue behavior of interference fitted Al-alloy 7075-T651 specimens subjected to bolt tightening[END_REF] or cold expansion process [START_REF] Ozdemir | Effect of expansion technique and plate thickness on near-hole residual stresses and fatigue life of cold expanded holes[END_REF][START_REF] Lacarac | Fatigue crack growth from plain and cold expanded holes in aluminium alloys[END_REF][START_REF] Chakherlou | Effect of cold expansion on the fatigue life of Al 2024-T3 in double shear lap joints: Experimental and numerical investigations[END_REF][START_REF] Wang | Investigation on fatigue performance of cold expansion holes of 6061-T6 aluminum alloy[END_REF][START_REF] Emami Geiglou | Investigating the effect of cold expansion process on the fatigue behavior of aluminum alloy 7075-T6 in double-lap shear joints[END_REF], which consist in introducing a conical mandrel with a high level of interference, allow to improve greatly fatigue life of aluminum alloy joints. Compressive residual stress fields introduced by these two processes are well known to be extensive and to reach high values. However, other processes involved in the machining of the fastener hole, such as drilling and reaming, also leave a region of disturbed material along the edges of the hole that can cause a residual stress field. Even if these induced residual stress fields are much lower than those left by a process such as cold expansion, they affect the fatigue behavior of the holed part [START_REF]Assessment of residual stresses and hole quality on the fatigue behavior of aircraft structural joints[END_REF]. Therefore, they need to be better apprehended in order to be accounted for the life-prediction methodology.

Generation of residual stress during machining is a complex phenomenon to simulate numerically. It involves both mechanical and thermal phenomena. On the one hand, simulation of metal cutting, which is associated with large strains, requires Eulerian methods. On the other hand, the prediction of residual stress, which needs knowledge of the loading history of each material particle, requires Lagrangian methods. Thus, several authors choose to develop « decoupled » numerical models, composed of a first model simulating the cutting, allowing them to obtain an equivalent thermomechanical loading, which they apply in a second model directly on the finished part. This is for example the case of Girinon [START_REF] Girinon | 3D numerical simulation of drilling residual stresses[END_REF], who, in his work on numerical simulation of drilling in hard metals, develops a rigid ALE model of material removal, then extracts the thermal loading on the part which is then applied in a second model purely Lagrangian. Others authors, as Ozel [START_REF] Özel | Prediction of machining induced residual stresses in turning of titanium and nickel based alloys with experiments and finite element simulations[END_REF] in his work on the simulation of the turning of hard metals develops a unique Lagrangian model with an updated software in which chip separation from workpiece is achieved with continuous remeshing. Still others, such as Valiorgue [START_REF] Valiorgue | Numerical Modeling of Residual Stresses in Turning of a 27MnCr5 Steel[END_REF] who works also on the turning of hard metals, free themselves from the modeling of material cutting by experimentally identifying the thermomechanical equivalent loading to apply in their Lagrangian model. Finally, other authors adopt even simpler reaming modeling strategies. In order to simulate the reaming process after cold expansion in their finite elements model, Achard [START_REF] Achard | Numerical modelling of the cold expansion process in mechanical stacked assemblies[END_REF] and Huang [START_REF] Huang | An numerical investigation on the effect of the combination of cold expansion and interference fitting on fatigue life improvement of a 7075-T6 aluminum alloy single plate[END_REF], choose to simply deactivate the mesh elements located in the reaming zone of the workpiece and Chakherlou [START_REF] Chakherlou | Numerical simulation of residual stress relaxation around a cold-expanded fastener hole under longitudinal cyclic loading using different kinematic hardening models[END_REF] and Ismonov [START_REF] Ismonov | Three dimensional finite element analysis of a split-sleeve cold expansion process[END_REF] choose to reduce the stiffness of the elements near the hole to almost zero. This paper focuses on the numerical study of the modification of the stress state of an aluminum 2024-T351 part during the reaming process which has been weakly studied. This alloy was selected because of its use in the majority of the current fleet of commercial aircraft. Most authors studying the numerical simulation of machining processes consider a material devoid of mechanical and thermal stress. However, in the aeronautical industry, parts do not present a zero stress state before reaming since they undergo at least a drilling operation, and sometimes a cold expansion operation. Thus, it is interesting to carry out a study on the impact of the reaming process taking into account the history of the loads applied to the part. For this, a numerical model considering a pre-stressed part is developed. The initial stress state of the part is introduced through the simulation of the cold expansion process.

NUMERICAL MODEL

The Finite Elements Model (FEM) representing the cold expansion process developed with Abaqus by Achard [START_REF] Achard | Numerical modelling of the cold expansion process in mechanical stacked assemblies[END_REF] is taken over and adapted to the expansion in an aluminum alloy part in order to generate a stress state in the part before reaming. The model, presented in Figure 1, represents the workpiece and the different parts of the tooling : the conical mandrel, the sleeve and the jaw.

Fig. 1. Finite element model of the cold expansion process

All the parts respect the dimensions imposed by the industrial specifications to obtain a final hole diameter of 6.84 mm and the thickness of the part is 6 mm. The simulated expansion rate is 6% in order to generate important residual stress fields in the part to better observe the influence of the reaming process in the rest of the study. The model is 2D axisymmetric, the part geometries, the loadings and the boundary conditions being compatible with this behavior hypothesis.

MATERIALS

2024-T351 aluminium alloy is widely used in the aircraft industry for its low density, high fatigue performance and high damage tolerance. This alloy is precipitation strengthened, its mechanical properties are determined by a heat treatment (T3) applied to the material after its shaping. This heat treatment induces the controlled precipitation of some material phases and so an increase in mechanical properties by obstructing movement of dislocations. According to the microstructural characterization of 2024-T351 alloy lead by Huda [START_REF] Huda | Characterization of 2024-T3: An aerospace aluminum alloy[END_REF], the high mechanical properties of this alloy are associated to three types of inclusions: Al-Cu, Al-Cu-Fe-Mn and Al-Cu-Fe-Si-Mn.

The T3 heat treatment refers to a three-stage treatment: solution heat treated, quenched and naturally aged. The T-51 treatment indicates that after quenching, the material is stress relieved through controlled stretching (plastic deformation of about 2%).

In the numerical model, the mechanical behavior of the parts is defined according to different constitutive laws, the parts having different materials. The steel mandrel can be considered and modeled as a rigid body in relation to the expanded aluminium part. The material data of the three other parts have been provided by Airbus and are confidential. Therefore, they are not presented in the paper. However, a graph representing the constitutive laws of the parts without units is presented for illustration purposes in Figure 2. The elastic behavior of these three parts is considered as isotropic. The steel jaw which undergoes only weak plastic strain is modelled with a simple elastic constitutive law. The mechanical behavior of the steel sleeve and the aluminum workpiece which undergo severe plastic strain is described by elastoplastic constitutive laws. The plastic behavior of the workpiece is modelled by a linear kinematic hardening and the plastic behavior of the sleeve is modelled by an isotropic hardening.

COLD EXPANSION SIMULATION

In order to simulate the kinematics of the expansion, a displacement of -40 mm along the model axis of revolution (z) is imposed on the mandrel and the lower face of the jaw is fixed. This boundary condition is sufficient to ensure the correct positioning of all the parts between them. Indeed, the axial forces applied on the sleeve and the workpiece during cold expansion simulation are recovered by the fixed jaw through the contact areas. Four contact areas are considered with a "surface-to-surface" discretization method. A contact area between the inner surface of the sleeve and the mandrel is defined with a very low friction coefficient (0.005) in order to apply a reasonable axial load on the sleeve and to consider the split of the sleeve. Indeed, in practice the sleeve is split which limits the axial load applied to the sleeve. A contact area between the outer surface of the sleeve and the workpiece and between the sleeve and the jaw is defined with a friction coefficient of 0.1 in order to ensure the correct positioning of the parts. A friction coefficient of 0.1 is chosen to simulate the contact between the mandrel and the jaw.

The calculation step corresponding to the simulation of the cold expansion process with the axial displacement of the mandrel is followed by a step simulating the withdrawal of the sleeve which is called "recovery". During this step, the contacts between the elements are deactivated and no loading is applied. This allows the part to achieve a new stress state.

REAMING SIMULATION

The residual stress state obtained from the expanded part after the recovery step constitutes the initial stress state for the simulation of the reaming process.

Reaming is simulated by deactivating mesh elements through the "Model change" interaction in Abaqus. This option allows to ramp down to zero the forces and fluxes that the region to be removed is exerting on the remaining part of the model at the nodes on the boundary between them. To simulate reaming in the model, the deactivated elements are located in a band along the height of the part at the hole edge with a width of 0.35 mm. Two strategies for simulating the reaming process are compared. The first one is similar to the one implemented in Achard's work [START_REF] Achard | Numerical modelling of the cold expansion process in mechanical stacked assemblies[END_REF]: all the elements of the reaming area are deactivated in the same calculation step. This modeling strategy allows to obtain the final residual stress state after reaming, it will be called "global reaming" in this study. The second strategy consists in deactivating the elements by several steps along the height of the part in order to simulate the progressive feed of the tool during reaming. 19 calculation steps, each corresponding to the deactivation of around ten elements in the thickness are carried out (Figure 3). This allows obtaining the history of the stress state in the part during reaming. This will be called "progressive reaming".

Fig. 3. Representation of the reaming area and the deactivated elements in the first calculation step for the progressive reaming (dimensions in mm)

Thus, the two modelling strategies do not take into account the thermal effects occurring during reaming. The aim is to study only the impact of material removal on the stress state of the part which cannot be studied separately experimentally.

Both modelling strategies should provide similar results. In the case of progressive reaming, the machining kinetics is exploited to define the calculation steps (height of deactivated elements and step duration). This time effect will fully make sense when another model involving progressive thermal loading will be implemented.

ABAQUS MODEL

The mesh, presented in Figure 4, is refined and composed of biquadratic elements with eight nodes (CAX8) in the areas of high plasticity (sleeve and hole edge area of the part). These elements which imply stress and strain calculations at 8 integration points per element allow providing accurate results that take the plastic phenomena into consideration well. The rest of the workpiece and the jaw are meshed with bilinear elements with four nodes using a reduced integration (CAX4R). These elements which use a reduced integration imply a stress and strain calculation at only one integration point by element. This allows saving computational time without losing accuracy, the areas being slightly plastically deformed.

Fig. 4. Mesh of the numerical model

In this study, the effects of inertia are neglected. Hence, the problem is considered quasi-static and is solved with Abaqus/Standard which uses an implicit time integration scheme. With this iterative method, the calculation of the current quantities in one time step are based on the quantities calculated in the previous time step [START_REF] Krishnakumar | Fatigue behaviour of aluminium alloy 7075 bolted joints treated with oily film corrosion compounds[END_REF]. The advantage of this integration scheme is that even if large time steps are taken, the solution remains stable. However, the disadvantage is that it requires the calculation of the inverse of the non-diagonal stiffness matrix which can be time consuming [START_REF] Sun | Comparison of implicit and explicit finite element methods for dynamic problems[END_REF].

∆𝑢𝑢 (𝑖𝑖+1) = ∆𝑢𝑢 (𝑖𝑖) + 𝐾𝐾 𝑡𝑡 -1 • (𝐹𝐹 (𝑖𝑖) -𝐼𝐼 (𝑖𝑖)
) Δu : increment of displacement Kt : current tangent stiffness matrix F : applied load vector I : internal force vector

RESULTS

During cold expansion simulation, residual stresses are generated in the part along the three directions: axial, radial and circumferential.

Figure 5 shows the stress states in the part for the three directions after cold expansion and global and progressive reaming simulations.

Fig. 5. Radial (a), circumferential (b) and axial (c) stress states obtained in the part at the end of the cold expansion (1 st column), global (2nd column) and progressive (3rd column) reaming simulations

These results show a non-homogeneous distribution of the residual stresses in the part after the cold expansion and reaming processes. The method for deactivating the elements in order to simulate the reaming process does not seem to affect the final stress state of the part.

Axial residual stresses are largely lower than radial and circumferential stresses. Hence, the rest of the study focuses on the latter two.

The low influence of the reaming simulation on the residual stress state of the part can be noted. The radial residual stress field undergoes a progressive offset in the radial direction during the reaming simulation. The two compression points on the entry and exit faces are slightly relaxed and the radial residual stress tends to homogenize in the thickness of the part. This relaxation could be related to the reduction of the axial strain at the hole edge on the entry and exit faces which is generated during the cold expansion process and which is sometimes called "volcano effect". Concerning the circumferential residual stress field, this one is simply truncated in the reaming area. Thus, a higher compressive stress state favoring a longer fatigue life is obtained at the hole edge.

The radial and circumferential residual stresses on the entry face and middle section of the part after cold expansion, global and progressive reaming simulations are shown in Figure 6. Radial stresses are compressive and have a characteristic profile. They cancel at the hole edge and at the free outer edge of the part. Circumferential stresses are of mixed nature with a compressive peak at the hole edge and then a tensile zone far from the hole.

Fig. 6. Evolution of radial (a) and circumferential (b) residual stresses along the entry face (1 st column) and middle section (2 nd column) of the part after cold expansion and global and progressive reaming simulations

A relaxation of the radial residual stress is observed during reaming simulation with a 7 % reduction in the compressive peak on the entry face and 15 % on the middle section. The modification of the circumferential residual stress is significant only on the entry face with a 6 % increase and a small offset of the compressive peak. Even though the level of residual stress changes during the reaming simulation, the heterogeneity of the stress state in the part remains. Indeed, the residual stress curves obtained after the cold expansion and reaming simulations present similar shapes. The changes in residual stress states are located in the hole edge area.

The final results obtained with the two simulation strategies for reaming are identical. This confirms the equivalence of the two methods to obtain the residual stress state of the part at the end of the reaming process. The progressive reaming method seems nevertheless more interesting because it allows accessing to the intermediate stress states during the process.

The evolution of the residual stress fields during progressive reaming is studied in order to better understand the impact of the process on the material characteristics of the part. For this purpose, radial and circumferential stresses are measured on 9 elements distributed according to the height and depth of the part. The location of these elements is described in Figure 7.

Fig. 7. Selected elements locating (center) in mm on the part after cold expansion simulation for constraint results reporting

The residual stress evolution for each element is presented in Figure 8. On these graphs, a dotted line indicates the position of the elements where the stress results are measured in relation to the part depth.

The evolutions of the residual stress during the reaming process present different shapes depending on the elements considered. A stress balancing takes place gradually when the elements bands are deactivated for the elements located at the same level in the part thickness. This balancing is mainly carried out in the radial direction and intensifies by approaching the hole edge. Indeed, the elements located exactly at the hole edge (E1, E4 and E7) show a sudden stress evolution. The elements a little farther away from the hole edge (E2, E5 and E8) present a smoother evolution. Finaly, the elements very far from the hole edge (E3, E6 and E9) show almost linear evolutions.

DISCUSSION

The comparison of the numerical results obtained with experimental results is not yet possible. Indeed, it is impossible to implement experimentally the mechanism of material removal without involving thermal phenomena. Therefore, the results obtained are compared with previous numerical work as a first validation step.

Kang [START_REF] Kang | Three-dimensional finite element analysis of the cold expansion of fastener holes in two aluminum alloys[END_REF] in his work develop a 3D FEM simulating the entire cold expansion process including hole expansion, recovery and finish reaming. A quarter of 2024-T351 aluminum part is modeled with a thickness of 6.35 mm and a final diameter hole of 7.26 mm. The reaming is simulated through the deactivation of mesh elements in the same way as in this study. Kang provides results of circumferential residual stress along the entry face and the middle section. The result curves present a general shape similar to the one obtained in this study with a compressive peak at the hole edge and a tensile zone far from the hole. However, the shape of the compressive peak varies. This can be explained by the number of points considered for the plotting of the result graphs which depends on the mesh size. The residual stress levels obtained by Kang are of the same order of magnitude as those presented in this study. The differences may be due to the different dimensions used.

Thus, the model developed in this study seems relevant and seems more interesting to gain in computing time compared to the 3D model.

In addition to saving computing time, the interest of the developed model with the progressive reaming method compared to the models of the literature performing the deactivation of the mesh elements in a single calculation step [START_REF] Achard | Numerical modelling of the cold expansion process in mechanical stacked assemblies[END_REF][START_REF] Huang | An numerical investigation on the effect of the combination of cold expansion and interference fitting on fatigue life improvement of a 7075-T6 aluminum alloy single plate[END_REF][START_REF] Kang | Three-dimensional finite element analysis of the cold expansion of fastener holes in two aluminum alloys[END_REF] is the access to the intermediate stress states during the reaming process. The numerical study of the evolution of the residual stress in the workpiece due to the material removal during the reaming process shows a gradual stress balancing as the mesh elements are deactivated along the thickness of the part. In practise during the reaming process, material removal is associated with thermal phenomena that some authors try to quantify [START_REF] Han | Heat flux density distribution differences in four machining processes of AlSi7 Block: MQL drilling, tapping, reaming and dry milling[END_REF] and which are not yet taken into account in our model. In order to envisage a more realistic simulation of the reaming process, it would be interesting to consider the intermediate stress states obtained and to gradually apply an equivalent thermal load.

An enhancement in the reaming simulation is proposed compared to Achard's simulation strategy [START_REF] Achard | Numerical modelling of the cold expansion process in mechanical stacked assemblies[END_REF]. This new simulation which consists of progressively deactivating mesh elements allows to access to the history of the residual stress state of the part.

The results obtained show only a small change in the residual stress state of the part during the reaming simulation. It means the material removal has a weak influence on the residual stress state of the part. Thus, if a significant change in the residual stress state occurs during the reaming process, it should mainly be related to the thermo-mechanical phenomena. Therefore, a new strategy of reaming simulation according to the principle of applying a thermo-mechanical load as in Valiorgue's works [START_REF] Valiorgue | Numerical Modeling of Residual Stresses in Turning of a 27MnCr5 Steel[END_REF] is envisaged in order to take into account the mechanical cutting and thermal heating phenomena.

Nevertheless, the interpretation of the results emphasize a certain change of the residual stress state in the part during the reaming simulation especially at the hole edge area. It means to envisage a more realistic reaming simulation by applying an equivalent thermo-mechanical load on the hole edge, this one should not be dissociated from the material removal. Thereby, in the envisaged reaming simulation, the loading and the material removal will be applied synchronously and gradually in order to achieve an accurate final stress state in the part.

A reaming simulation on a pre-stressed part obtained after drilling simulation is also envisaged in order to study the evolution of the residual stress state in the hole edge zone during the whole machining process of the hole.

Finally, following this numerical study, an experimental study is envisaged in collaboration with Airbus in order to correlate the numerical results with the experimental measurements. Reaming tests will be performed with residual stress measurements in the part using the X-ray diffraction method and the incremental hole drilling method.

BIBLIOGRAPHY

Fig. 2 .

 2 Fig. 2. Constitutive laws of the numerical model parts

Fig. 8 .

 8 Fig. 8. Evolution of radial (a,b,c) and circumferential (d,e,f) residual stresses during progressive reaming simulation


domain: math qfin

text:
Introduction

Static game

Dynamic Game

Conclusions and extensions

The one shot model. The static game Two players. Each player endowed with w i and contributes x i ∈ [0, w i ] to a public good. Then w i -x i privately consumed. Preferences: intrinsic utility (absolute level of contribution) positional payoffs(associated with relative contribution) U i (x i , x j ) = u i (x i , X ) + V i (x i -x j ), X = x i + x j .

Global intrinsic utility: u = u i + u j The global utility or social welfare: U = U i + U j .

The intrinsic utility. Properties C1 Individual provision always reduces own welfare:

∂u i ∂x i (x i , X ) < 0, ∀(x i , x j ) ∈ [0, w i ] × [0, w j ].
Nash equilibrium is (0, 0). C2 Individual provision: First unit ↑ u; last unit ↓ u.

C 2a : ∂u ∂x i (0, 0) > 0, C 2b : ∂u ∂x i (w 1 , w 1 + w 2 ) < 0.
Some contribution to the public good and to the private good are socially desirable. C3 Agents contributions are substitutes

C 3 : ∂ 2 u ∂x i ∂X (x i , X ) < 0.

The intrinsic utility + positional concerns

The intrinsic utility is an additively separable function:

u i (x i , X ) = w i -x i + b i (X ), u(X ) = w 1 + w 2 -X + b 1 (X ) + b 2 (X ). b i (X ) = α i X - ε 2 X 2 . α i PG valuation by i, ε satiation Positional payoff V i (x i -x j ) = v P i × (x i -x j ). Positional concern of player i: v P i ≥ 0

Maximization problem

The problem for player i ∈ {1, 2} is:

max 0≤x i ≤w i w i -x i + b i (X ) + v P i (x i -x j ).
The marginal utility of player i now reads as:

∂U i ∂x i = -1 + b ′ i (X )+v P i .
Positionality raises the marginal benefit of private provision. Private provision can become individually rational for player i.

Nash equilibrium

The FOC condition: b ′ i (X ) = 1 -v P i . In equilibrium,

1 v P i < 1 → b ′ i (X ) > 0: PG. 2 v P i > 1 → b ′ i (X ) < 0: PB. Assumption 0 ≤ v P i < 1
Definition (Wished amount)

A i =            0 if v P i < 1 -α i , v P i -(1 -α i ) α i ε otherwise w i + w j if v P i > 1 -α i + α i ε(w 1 + w 2 ). Positive contribution v P i > 1 -α i Nash equilibrium (continued) a) If w 1 + w 2 ≤ min{A 1 , A 2 }, then (x N 1 , x N 2 ) = (w 1 , w 2 ). b) If w 1 + w 2 > min{A 1 , A 2 } and bI) A 1 = A 2 = A > 0, set of Nash equilibria: N = {(x 1 , x 2 )|x 1 + x 2 = A ∧ (x 1 , x 2 ) ∈ [0, min{A, w 1 }] × [0, min{A, w 2 }]} bII) If A i > A j and A i > 0, i, j ∈ {1, 2}, i ̸ = j, (x N i , x N j ) =      (w i , A j -w i ) w i < A j < w i + w j [bII.1], (w i , 0) A j ≤ w i < A i [bII.2], (A i , 0) A i ≤ w i [bII.3]. c) A 1 = A 2 = 0: (x N 1 , x N 2 ) = (0, 0).
Welfare analysis. Position and intrinsic utility bII.1: (w i , A j -w i ), bII.2: (w i , 0), bII.3: (A i , 0) Case of symmetric valuation of PG consumption, α i = α j = α. bII.1: (w i , A j -w i ), bII.2: (w i , 0), bII.3: (A i , 0) Case of symmetric valuation of PG consumption, α i = α j = α.

Introduction Static game

Dynamic Game

Conclusions and extensions

At time t a positional agent gets joy from contributions above the other player at time t -1: +v P i (x it -x jt-1 ).

2

Agents have Inertia from previous actions. Disutility from deviations from the previous action:

-v I i (x it -x it-1 ) 2 /2. Utility at time t: U i (•) = w i -x it + α i x it + x jt - ε 2 (x it + x jt ) 2 +v P i (x it -x jt-1 ) - v I i 2 (x it -x it-1 ) 2 .
At time t a positional agent gets joy from contributions above the other player at time t -1: +v P i (x it -x jt-1 ).

2

Agents have Inertia from previous actions. Disutility from deviations from the previous action:

-v I i (x it -x it-1 ) 2 /2. Utility at time t: U i (•) = w i -x it + α i x it + x jt - ε 2 (x it + x jt ) 2 +v P i (x it -x jt-1 ) - v I i 2 (x it -x it-1 ) 2 .
Alternative dynamics based completely in the present (except for inertia) or completely in the past → no big differences.

The problem of a myopic agent i ∈ {1, 2} is:

max x it U i (x it , x jt , xit , xjt ), s.t.: xit = x it-1 , i ∈ {1, 2}.
The first-order conditions for an interior equilibrium are:

-1 + α i [1 -ε(x it + x jt )] -v I i (x it -xit ) + v P i , = 0 ∀i ∈ {1, 2}.
Dynamic problem: Free evolution with inertia

x it x jt = δt 1 -1 + x i0 x j0 + (1 -λ t 2 ) X * -X 0 α i v I j + α j v I i α j v I i α i v I j ,
where, The effect of a joint increment in the positional concerns of both players is (in bII1 and bII3):

δ = εα i α j (A i -A j ) α i v I j + α j v I i λ 2 = v I i v I j v I i v I j + ε(α i v I j + α j v I i ) X * = α i v I j A i + α j v I i A j α i v I j + α j v I i . X t = X 0 + (1 -λ t 2 )(X * -X 0 ).
dU N dv P i + dU N dv P j ⋛ 0 ⇔ v P i + v P j ⋚ 1.
Simulations: bII.1 (w i , A j -w i ) and bII.2 (w i , 0)

Figure :

 : Figure: Level curves for u(X ): ε = 0.7 (left), ε = 0.4 (right).

Figure :

 : Figure: Level curves for utility: ε = 0.7 (left), ε = 0.4 (right).

  Figure: Individual (left) and total (right) contributions

Figure :

 : Figure: Individual (left) and total (right) contributions

Myopic dynamic problem: X 0 = 0 and A i ≥ A j Asymptotic behavior of the sequence of contributions:

1 Static equilibrium: a fixed point of dynamical system, 2 trajectory of contributions hits the boundary of feasible contributions, where the static equilibrium lies . . .

Conclusions

In the static framework Nash equilibrium for an additive separable public good game with positional concerns. Different equilibria depending on players positional concern (wished amounts) vs endowments. Positional concerns, contributions and social welfare. In the myopic dynamic framework with inertia Convergence to the static case Possibility of overshooting in individual and total contribution.

Extensions

A positional agent versus a conformist (joy from doing like others). Disutility: -v C i (x it -x jt-1 ) 2 /2. Consider another dynamic: the stock of public good. Consider farsighted agents (true dynamic game). More than two players. ...

Proposition

Assume w i = w and α i = α. ε = 2α-1 w α . Then:


Nolwenn Fougeron

Isabelle Rivals

Nathanaël Connesson

Grégory Chagnon

Thierry Alonso

Laurent Pasquinet

Stéphane Auguste

Antoine Perrier

Yohan Payan

Pressure Ulcers and Dressings: A Strain Sensitivity Analysis of the Boundary Conditions of a Finite Element Model

Keywords: finite element analysis, pressure ulcers, dressing, soft tissues, internal strains, sensitivity analysis Fougeron,

Recently, a new bi-layer dressing was proposed by Urgo RID to reduce the healing time of pressure ulcers (PU). This dressing was numerically evaluated in previously published work. In the current work, the influence on the maximal shear strains of modelling parameters such as the dressing local geometry, the pressure applied by the gauze inside the wound, the wound deepness, and the mattress stiffness, was assessed. A sensitivity analysis was performed on these four parameters. Among all experiments, the mean maximal Green-Lagrange shear strain was 0.29. The gauze pressure explained 60% of the model response in terms of the volume of tissues under strains of 0.3, while the wound deepness explained 28%. The mattress had a significant, but low impact, whereas the dressing local geometry had no significant impact. As expected, the wound deepness was one of the most influential parameters. The gauze turned out to be more significant than expected. This may be explained by the large range of values chosen for this study. The results should be extended to more subjects, but still suggest that the gauze is a parameter that might not be neglected. Care should also be taken in clinical practice when using gauze that could have either a positive or negative impact on the soft tissues' strains. This may also depend on the wound deepness.

Introduction

Pressure ulcers (PU) are injuries to the skin and underlying tissues that are common adverse events in healthcare. For example, in intensive care units, PU prevalence reaches almost 27% [START_REF] Labeau | Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: The DecubICUs study[END_REF]. In any healthcare facility, the risk of developing a PU is increased for older patients, patients with spinal cord injuries, or comorbidities [START_REF] Demarré | The cost of prevention and treatment of pressure ulcers: A systematic review[END_REF]. PU have terrible consequences on the quality of life of patients including longer hospitalisation time, social isolation, and pain [START_REF] Graves | Effect of Pressure Ulcers on Length of Hospital Stay[END_REF][START_REF] Theisen | Pressure ulcers in older hospitalised patients and its impact on length of stay: A retrospective observational study[END_REF]. PU are localised wounds that propagate in the soft tissues after a detrimental external loading. They are classified from stage-1, for light wounds, to stage-4, for the most severe wounds. Short time (some minutes), but intense, load application is sufficient to cause a PU, while reduced loads applied for an extended period (2 to 4 h) can also lead to this kind of wound [START_REF] Loerakker | Temporal effects of mechanical loading on deformation-induced damage in skeletal muscle tissue[END_REF]. They have a multifactorial origin, but mechanical loads applied to the tissues are considered to play a significant role in the onset of PU. Pressure or shear loads applied at the skin level may lead to significant internal strains [START_REF] Gefen | Our contemporary understanding of the aetiology of pressure ulcers/pressure injuries[END_REF]. Strains and, more particularly, the Green-Lagrange maximal shear strains, appeared to be a mechanical biomarker for the development of PU [START_REF] Ceelen | Compression-induced damage and internal tissue strains are related[END_REF]. When these strains exceed the cell's ability to deform, in most cases under bony prominences, this eventually leads to cell death and the development of a wound [START_REF] Ceelen | Compression-induced damage and internal tissue strains are related[END_REF][START_REF] Bouten | Compressive deformation and damage of muscle cell subpopulations in a model system[END_REF][START_REF] Oomens | Can loaded interface characteristics influence strain distributions in muscle adjacent to bony prominences?[END_REF]. The sacrum is the most affected area of the human body. In this case, the recommended procedure to treat PU consists of the unloading of the weakened tissues, which can be tedious to do continuously, particularly if several PU are present. Dressings are common medical devices used to improve the healing process of PU and a huge range of products are proposed to clinicians. Yet, the mean healing time of PU is estimated to be 3 weeks and can sometimes exceed 10 weeks [START_REF] Palese | What is the healing time of stage II pressure ulcers? Findings from a secondary analysis[END_REF]. Recently, Urgo RID developed a new concept of dressing to improve the healing of PU by reducing internal strains. This dressing consists of two layers, the first one being the classic Urgo Start Plus Border dressing and the second one consisting of an unloading material. This material is cut into alveoli that can be removed under the wound to relocate the loads outside of the wound region when complete unloading of the PU is not temporally possible. The ability of this dressing to alleviate soft tissues has already been studied previously [START_REF] Fougeron | New pressure ulcers dressings to alleviate human soft tissues: A finite element study[END_REF]; however, question marks remain about the use of the dressing. The impact of the dressing with different wound deepness, quantities of alveoli removed, or mattress stiffnesses still needs to be estimated. Furthermore, the interaction with the gauze that is sometimes applied in the wound to absorb part of the exudate has not been studied yet.

Finite element modelling is a common method applied to compute the soft tissues' internal strains. Ceelen et al. [START_REF] Ceelen | Validation of a numerical model of skeletal muscle compression with MR tagging: A contribution to pressure ulcer research[END_REF] proposed and validated this method on rat models for the estimation of the Green-Lagrange maximal shear strains. Yet, few models of the sacrum region were proposed in the literature [START_REF] Savonnet | Finite element models of the thigh-buttock complex for assessing static sitting discomfort and pressure sore risk: A literature review[END_REF]. These models were mainly proposed to compute internal and external stresses in soft tissues without PU. Some studies also applied the finite element modelling method to evaluate penetrating ulcers in the cardiovascular domain [START_REF] Gefen | The biomechanical efficacy of a dressing with a soft cellulose fluff core in prophylactic use[END_REF], yet few efforts were made for PU in soft tissues such as skin or adipose tissues. To the authors' knowledge, the group of Amit Gefen (Tel Aviv University) was the only one to propose a finite element model of the injured tissues with a stage-4 PU. The authors showed that adding a multilayer dressing allowed the reduction of internal and external stresses around the wound. Several other studies from that group also performed comparative analyses of the finite element model of the sacrum region with various dressings or mattresses. They compared the use of silicone foam dressing with various material parameters and showed that dressings anisotropy helped reduce the internal and external stresses [START_REF] Linder-Ganz | Stress analyses coupled with damage laws to determine biomechanical risk factors for deep tissue injury during sitting[END_REF]. In another study, the authors from this group showed that the increase in mattress stiffness induced an increase in internal stresses [START_REF] Schwartz | The biomechanical protective effects of a treatment dressing on the soft tissues surrounding a nonoffloaded sacral pressure ulcer[END_REF]. They also studied various soft cellulose fluff core dressings with two mattress conditions and two moisture states of the dressings. Few differences could be noted among the dressings, but better performances were obtained with the softest mattress [START_REF] Schwartz | A Computer Modeling Study to Assess the Durability of Prophylactic Dressings Subjected to Moisture in Biomechanical Pressure Injury Prevention[END_REF][START_REF] Macron | Is a simplified Finite Element model of the gluteus region able to capture the mechanical response of the internal soft tissues under compression?[END_REF]. These studies bring interesting insights into how the change of boundary conditions may impact the response of the soft tissue and potentially the onset or propagation of a PU. However, none of the previous studies reported statistical analysis on the relative importance of the studied parameters [START_REF] Call | Enhancing pressure ulcer prevention using wound dressings: What are the modes of action?[END_REF]. Furthermore, the gauze inside the wound has still not been modelled.

The current study aims to estimate the relative impact of the dressing geometry, mattress stiffness, use of gauze, and PU deepness on the soft tissues' maximal shear strains around the wound. A sensitivity analysis was performed on these four parameters using a previously designed parametric model of the sacrum region.

Materials and Methods

Reference Finite Element Model

To reduce the computation time, a parametric approach was adopted. The model consisted of several layers: the skin, adipose tissues, both dressing layers, and a mattress. The skin and adipose tissue thicknesses were set to 1.30 mm and 22.30 mm, respectively [START_REF] Call | Enhancing pressure ulcer prevention using wound dressings: What are the modes of action?[END_REF][START_REF] Plagenhoef | Anatomical Data for Analyzing Human Motion University of Massachusetts -Amherst[END_REF]. To simulate a bony prominence on the median crest of the sacrum, an imprint of the sacrum geometry was approximated by a portion of a sphere with an ellipsoidal volume on top of it. The adipose tissue thickness was thus reduced to 13.30 mm under the bony prominence. The sacrum bone was set as rigid with the pilot node at the centre of the area. A PU from stage-2 to stage-3 was added to the model, with various depths defined after, while the radius was set to 15.00 mm. The dressing was modelled with two layers referred to as dressing layer 1 and dressing layer 2 (Figure 1). Dressing layer 1 is the unloading material cut into alveoli that is in contact with the mattress, and dressing layer 2 is the UrgoStart Plus Border dressing that is in contact with the skin. Both layers were modelled as a cylindrical layer with a radius of 125.00 mm. The thickness of dressing layer 2 was set to 3.50 mm, whereas the thickness of dressing layer 1 was set to 5.20 mm. A mattress with a height of 50.00 mm was added to the model. The diameter of the model was 250.00 mm to avoid boundary effects in the wound area. Symmetry in the sagittal plane was also considered so only half of the model was used for the simulation (Figure 2). All components were meshed with SOLID185 linear hexahedral elements ANSYS APDL (ANSYS 2020 R2 software, ANSYS Inc., Cannonsburg, PA, USA) with a mixed pressure-displacement formulation for the soft tissues. The model was composed, at most, of 6088 elements. The dressing layers were tied together. Tie constraints were also used between the soft tissue layers and between the skin and dressing layer 2. A coefficient of friction of 0.62 was defined between dressing layer 2 and the mattress. This value was computed from friction tests performed at Urgo RID. The dressing, glued on a calibrated weight, was positioned on a rigid plate cover with a clinical sheet. A gradually increasing force was applied to a cable attached to the dressing. The coefficient of friction was the ratio of the force that pulled the dressing and the calibrated weight. Between the skin and the mattress, this coefficient was set to 0.43 [START_REF] Plagenhoef | Anatomical Data for Analyzing Human Motion University of Massachusetts -Amherst[END_REF]. A vertical force of 217 N was applied to the pilot node of the sacrum area, as illustrated in Figure 2a. Considering the symmetry of the model, this corresponded to 47% of the bodyweight of a 94 kg subject [START_REF] Isihara | Statistical theory of rubber-like elasticity. IV. (Two-dimensional stretching)[END_REF]. The bottom nodes of the mattress were fixed in position. Simulations were performed with ANSYS in a quasi-static analysis with an implicit scheme. Soft tissues were modelled with non-linear hyperelastic isotropic constitutive equations. More particularly, the skin was modelled with the law proposed by Isihara et al. [START_REF] Annaidh | Characterization of the anisotropic mechanical properties of excised human skin[END_REF]. The material parameters were optimised using a curve-fitting method from the data of Ni Annaidh et al. [START_REF] Yeoh | Some forms of the strain energy function for rubber[END_REF]. The adipose tissues were modelled with the equation developed by Yeoh [START_REF] Sommer | Multiaxial mechanical properties and constitutive modeling of human adipose tissue: A basis for preoperative simulations in plastic and reconstructive surgery[END_REF] with parameters fitted according to the data of Sommer et al. [START_REF] Mott | The bulk modulus and Poisson's ratio of "incompressible" materials[END_REF]. The soft tissue stiffness was increased close to the wound region, as detailed in Fougeron et al. [START_REF] Fougeron | New pressure ulcers dressings to alleviate human soft tissues: A finite element study[END_REF], to account for the stiffening of the tissues surrounding a PU. The constitutive equation for the different tissues is:

𝑊𝑊 = � 𝐶𝐶 𝑖𝑖0 (𝐼𝐼 1 � -3) 𝑖𝑖 + 𝑛𝑛 𝑖𝑖=1 � 1 𝑑𝑑 𝑘𝑘 (𝐽𝐽 -1) 2𝑘𝑘 𝑛𝑛 𝑘𝑘=1 𝑑𝑑 1 = 𝑑𝑑 2 = 𝑑𝑑 3 = 3(1 -2𝜈𝜈) 2𝐶𝐶 10 (1 + 𝜈𝜈)
where W is the strain energy density function, 𝐶𝐶 𝑖𝑖0 the material parameters, 𝐼𝐼 1 � the first deviatoric invariant of the right Cauchy-Green deformation tensor, J the Jacobian of the deformation gradient, and 𝑑𝑑 𝑘𝑘 the nearly incompressibility parameters expressed with the Poisson's ratio ν by the formula provided by Mott et al. [START_REF] Lee | Finite element modeling for predicting the contact pressure between a foam mattress and the human body in a supine position[END_REF]. The indices i and k are between 1 and 3 for the skin and between 1 and 2 for the adipose tissues. Soft tissue stiffening was considered by multiplying the C10 parameters of the skin and the adipose tissue by a coefficient of 1.0, 1.5, and 2.0 for the soft, medium, and stiff areas, respectively, as detailed in Figure 2.

The value of Poisson's ratio was set to 0.4999 to account for the nearly incompressibility of the soft tissues. Dressing layer 2 was modelled with a linear elastic orthotropic material, whereas layer 1 was defined as a compressible material and modelled with a Blatz-Ko constitutive equation [START_REF] Dickinson | Finite element analysis of the amputated lower limb: A systematic review and recommendations[END_REF].

𝑊𝑊 = 𝜇𝜇 2 � 𝐼𝐼 2 𝐼𝐼 3 + �𝐼𝐼 3 -5�
where 𝐼𝐼 2 and 𝐼𝐼 3 are the second and third invariants of the right Cauchy-Green deformation tensor, respectively, and µ is the initial shear modulus.

The initial shear modulus µ of dressing layer 1 and Young's moduli of dressing layer 2 were computed from compression and tension tests. According to the literature data, the Poisson ratio of dressing layer 2 was set to 0.2560. The mattress was modelled as a linear elastic isotropic material with a Poisson ratio of 0.3000 and a reference Young modulus, E, of 0.23 MPa. The material parameters are detailed in Table 1. Further details about the reference model are provided in Fougeron et al. [START_REF] Fougeron | New pressure ulcers dressings to alleviate human soft tissues: A finite element study[END_REF].

Sensitivity Analysis

Principal stretches λ1, λ2, and λ3 were extracted to compute the Green-Lagrange principal strains (Equation ( 4)). The maximal shear strain, Eshear, was calculated as detailed in Equation [START_REF] Loerakker | Temporal effects of mechanical loading on deformation-induced damage in skeletal muscle tissue[END_REF].

𝐸𝐸 𝑖𝑖 = 1 2 (𝜆𝜆 𝑖𝑖 2 -1), 𝑖𝑖 ∈ [1, 2, 3] 𝐸𝐸 𝑠𝑠ℎ𝑒𝑒𝑒𝑒𝑒𝑒 = 1 2 max (|𝐸𝐸 1 -𝐸𝐸 2 |, |𝐸𝐸 2 -𝐸𝐸 3 |, |𝐸𝐸 3 -𝐸𝐸 1 |)
Green-Lagrange maximal shear strains are recognised as potential mechanical biomarkers to study the onset and development of PU [START_REF] Ceelen | Compression-induced damage and internal tissue strains are related[END_REF]. In the current study, a region of interest (ROI) was defined for the computation of the strains. The ROI included soft tissues under the wound and in the perilesional area within three times the radius of the PU.Experiments performed on rats suggested the possibility to define a threshold of damage that should be subject-specific [START_REF] Ceelen | Compression-induced damage and internal tissue strains are related[END_REF]. Due to the lack of data on human subjects, the threshold was arbitrarily fixed to 0.3 considering that Eshear was below this threshold for healthy tissues.

A sensitivity analysis was performed to assess the relative significance of the model parameters on the volume of healthy tissues. The finite element model was emulated with a polynomial model detailed after, following the method described by Macron et al. [START_REF] Call | Enhancing pressure ulcer prevention using wound dressings: What are the modes of action?[END_REF], to investigate the impact of the following parameters on the volume of healthy tissues: wound deepness, alveoli cutting size, mattress stiffness, and pressure applied by the gauze. The parameters varied between their minimal and maximal values, as detailed in Table 2. After normalisation in [-1, 1], experimental points were chosen according to a three-level full factorial design resulting in 34 combinations (i.e., 81 simulations). Based on the knowledge of expert clinicians, the wound deepness extrema were set to 1.30 mm and 5.00 mm to respectively account for a stage-2 and a stage-3 PU. The recommendations from Urgo about the use of the bi-layer dressing are to remove alveoli around the wound, so this was used as the mean level of the parameter. Then, a layer of alveoli around the wound was added or removed to respectively define the minimal and maximal levels (Figure 3). The mattress stiffness limits were defined according to literature values [START_REF] Schwartz | The biomechanical protective effects of a treatment dressing on the soft tissues surrounding a nonoffloaded sacral pressure ulcer[END_REF][START_REF] Keenan | A review of foot finite element modelling for pressure ulcer prevention in bedrest: Current perspectives and future recommendations[END_REF]. In the clinical routine, the pressure applied by the gauze may significantly vary depending on its saturation in fluid and on the person who inserts the gauze in the wound. As a consequence, it was chosen to model the effect of the gauze by the pressure applied on the wound walls rather than that of the gauze itself. A finite element preliminary study was performed to define the gauze pressure values. To this end, the volume of healthy tissues was analysed with a 5.0 mm deep PU model for multiple values of pressure between 0.00 MPa and 0.08 MPa. A local optimum was found at 0.02 MPa. As a consequence, the minimal and maximal values were set to 0.00 MPa (i.e., no gauze in the wound) and 0.04 MPa.

𝑦𝑦(𝜃𝜃) = 𝜃𝜃 0 + � 𝜃𝜃 𝑖𝑖 𝑥𝑥 𝑖𝑖 𝑚𝑚 𝑖𝑖=1 + � 𝜃𝜃 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑖𝑖 𝑚𝑚 𝑖𝑖=1 2 + � � 𝜃𝜃 𝑖𝑖𝑖𝑖 𝑥𝑥 𝑖𝑖 𝑥𝑥 𝑖𝑖 𝑖𝑖>𝑖𝑖 𝑚𝑚 𝑖𝑖=1
where y is the volume of healthy tissues, m the number of parameters, xi the value of the ith parameter, and θ the vector of the adjustable coefficients, which was estimated with ordinary least squares. The value of two for the degree will be further justified in the results section. The sensitivity of the model to each input (linear term, square, order-two interaction) can be simply defined as the percentage of variance due to this input. Assuming, for simplicity, the m = 4 parameters independent and uniformly distributed in [-1, 1] (i.e., with second-and fourth-order moments of respectively 1/3 and 4/45), it becomes:

i i i i i i ii ii i ii i ii ij ij i j ij i j ij m m m i ii ij i i i j i s x x s x x s x x x x θ θ θ = = = >  = = = ×    = = = ×     = = = ×   = + +    ∑ ∑ ∑∑
The sensitivities to the ith parameter and to its interaction with parameter j are hence given by the percentages: (8)

Results

This section may be divided by subheadings. It should provide a concise and precise 208 description of the experimental results, their interpretation, as well as the experimental 209 conclusions that can be drawn in Table 3.

Table 3. Parameter coefficients and polynomial model sensitivities (>1%) in decreasing order of magnitude.

Parameters

Coefficients θi and θii or θij Sensitivities Si or Sij (%) Gauze pressure -3.9, -10.7 60 Wound deepness -4.3, -3.3 28 Wound deepness * Gauze pressure +4.6 10 Mattress stiffness +1.1, -0.9 1

One may notice that approximately 99% of the model response y was explained by four parameters: the gauze pressure, the wound deepness, the interaction of the wound deepness and the gauze pressure, and the mattress stiffness. More particularly, the gauze pressure explained about 60% of the model response, as illustrated in Figure 4a. Considering dressing layer 1, this layer was shown to reduce the maximal shear strains on one model of a stage-2 PU in a previous study. When close enough to the recommended (i.e., plus or minus one layer of alveoli), the change in the volume of healthy tissues was not significant, as presented in Figure 4. On the contrary, wound deepness was a significant parameter that explained 28% of the response (cf. Figure 4). As expected, the interaction of the wound deepness and the gauze pressure was also important, whereas the mattress stiffness had a significant, but low impact on the volume of healthy tissues (cf. Figure 4). Extreme values of gauze pressure seem to have a negative impact on the volume of healthy tissues (cf. Figure 4), suggesting that an optimal value can be found. Tissues around deep PU tend to have more important strains (cf. Figure 4) and softer mattresses may not be suitable in all cases, since the interquartile range of the volume of healthy tissues is larger than for stiffer mattresses. Worst-case scenarios were defined as the 10% experiments with the highest peak maximal shear strains. Among these nine experiments, the peak maximal shear strains were greater than 0.80 and all were designed with the softest mattress and the maximal gauze pressure with various wound deepness and alveoli cut. To illustrate the results, Green-Lagrange maximal shear strains in the ROI were plotted for some experiments in Figure 5.

Discussion

A new bi-layer dressing has been proposed by Urgo RID to improve the healing of PU. This dressing has previously been studied to evaluate its mechanical impact on the soft tissues in one specific scenario. In this case, the use of the dressing allowed the reduction of internal strains around the wound. Yet, some factors may affect the conclusions: the dressing alveoli cutting, the pressure applied by the gauze inside the wound, the deepness of the wound, and/or the stiffness of the mattress. Thereby, the present study aimed to evaluate the relative importance of these parameters regarding the maximal shear strains around the PU. A sensitivity analysis was performed following a three-level full factorial design.

Among all experiments, the mean maximal shear strain was 0.29 and the peak value reached 0.97. The experiments that reached the highest values of maximal shear strains were all designed with the softest mattress and the maximal gauze pressure. The strain values are in range with the previously published results, but are lower than those obtained by Macron et al., for whom peak values ranged between 1.42 and 4.14. Macron et al. studied the strains under the ischial tuberosities in subjects in a sitting position, which may explain the differences [START_REF] Call | Enhancing pressure ulcer prevention using wound dressings: What are the modes of action?[END_REF]. The computation of the peak maximal shear strain is also local and thus highly sensitive to mesh quality and model non-linearities. Therefore, the volume of healthy tissues was preferred here as a discriminant measure for the sensitivity analysis. The gauze pressure alone explained 60% of the model response, while the wound deepness and the interaction between the gauze pressure and the wound deepness accounted for 28% and 10% of the response, respectively. To the authors' knowledge, this study is the first attempt to assess the impact of these two parameters on the computation of the strains where they both significantly impacted the results. The mattress also had a significant, but low impact. Contrary to the previous studies of Linder-Ganz and Gefen [START_REF] Schwartz | The biomechanical protective effects of a treatment dressing on the soft tissues surrounding a nonoffloaded sacral pressure ulcer[END_REF], the softest mattress did not necessarily reduce the strains in the ROI. This may be due to the use of the bi-layer dressing in this particular study, which adds a cushion layer between the soft tissues and the mattress. Furthermore, the local approach proposed in this study may not be able to capture the impact of the mattress on a large scale, since weight-bearing areas are limited here. It is worth noting that the results could be affected by the levels chosen for the sensitivity analysis. Mattress stiffness is highly dependent on the brand and few data are provided by the manufacturers. The mattress was modelled with linear elastic homogeneous isotropic material properties, which may not be appropriate for all mattress technologies. The use of gauze was modelled as a homogeneous pressure applied inside the PU. Various products are used by clinicians and the filling of the gauze inside the wound is highly dependent on the operator and the exudate of the wound. The use of pressure allows one to model the effect of the gauze without the need to model all types of commercialised products or operators' protocols. The wound deepness is a significant parameter with an important impact, but in the present study, PU 5.3 mm deep at most were designed. Consequently, the conclusion might not be extrapolated to deeper PU. Other parameters could also have been included in the sensitivity analysis. A geometrical description of the PU such as its diameter or the interaction between the PU diameter and the dressing alveoli cutting could modify the strain distribution. Subject-specific parameters were also not studied in this work. As detailed by Macron et al. [START_REF] Call | Enhancing pressure ulcer prevention using wound dressings: What are the modes of action?[END_REF], materials and thicknesses of soft tissues as well as bone geometries may have a significant impact on strain computation [START_REF] Call | Enhancing pressure ulcer prevention using wound dressings: What are the modes of action?[END_REF]. The material parameters of soft tissues were estimated from cadaveric tests of the literature. Therefore, the current study does not account for the variability of the constitutive behaviours that are proposed in the literature [START_REF] Savonnet | Finite element models of the thigh-buttock complex for assessing static sitting discomfort and pressure sore risk: A literature review[END_REF][START_REF] Bonet | Nonlinear Continuum Mechanics for Finite Element Analysis[END_REF][START_REF] Clark | Measurement of soft tissue thickness over the sacrum of elderly hospital patients using B-mode ultrasound[END_REF]. The Poisson ratio was also higher than in most literature studies, but this is in range with the recommendation of Bonet and Wood [START_REF] Bonet | Nonlinear Continuum Mechanics for Finite Element Analysis[END_REF] to be close to incompressibility. The soft tissue thicknesses were fixed in the current study even though values from 4.0 mm to 33.5 mm were reported by Clark et al. [START_REF] Clark | Measurement of soft tissue thickness over the sacrum of elderly hospital patients using B-mode ultrasound[END_REF]. Yet, considering all of the parameters would have entailed too many experiments. As a result, it was decided for this study to focus on one particular case for which the model was previously experimentally evaluated, and to evaluate the parameters relating to the use of the dressing in this particular environment: the alveoli cutting, the gauze pressure, the wound deepness, and the mattress stiffness. The present study was not exhaustive on the studied parameters. Further analyses are necessary to include subject-specific parameters obtained on healthy subjects, but also on subjects with PU. The threshold of the strains used to define healthy tissues could also have an impact on the results. Thus, the same sensitivity analysis was performed with a threshold of 0.65 as prescribed by Ceelen et al. [START_REF] Ceelen | Compression-induced damage and internal tissue strains are related[END_REF]. Small discrepancies, a few percent, were noted in terms of sensitivities, but the relative order of the parameters remained the same.

Finally, the results presented here suggest that care should be taken when filling the wound with gauze. Gauze is important to maintain an optimal environment in the wound, particularly in terms of moisture. However, gauze should not be crammed into the wound or filled with too much fluid at the risk of applying too much pressure inside the wound and thus exacerbating the deformations of already weakened soft tissue. Furthermore, as was expected, the deeper the wound, the more strains. Even though the unloading of soft tissues is always prescribed for PU, special care should be taken when dealing with stage-2 and higher PU. To consolidate the conclusion, future work will include the transfer of the proposed modelling on realistic subject-specific geometries of the sacrum and the heel in several patients. This study is a first attempt to numerically evaluate the effect of new dressing designs and to potentially propose guidelines to industrials and clinicians for the use of these medical devices.

Figure 1 .

 1 Figure 1. The new dressing design developed by Urgo RID.

Figure 2 .

 2 Figure 2. Model geometry and boundary conditions.

Figure 3 .

 3 Figure 3. Minimal, intermediate, and maximal levels of the alveoli cutting (a) and wound deepness (b) parameters.

Figure 4 .

 4 Figure 4. Effect of the four parameters on the volume of healthy tissues (i.e., tissues with strains lower than 0.3), with the other three parameters being set to their intermediary value.

Figure 5 .

 5 Figure 5. Green-Lagrange maximal shear strains in the ROI of some experiments. All parameters were set to the intermediate values except for one that varied according to the defined levels: (a) changes in the gauze pressure, (b) changes in the alveoli cut, (c) changes in the wound deepness, and (d) changes in the mattress stiffness. The ROI appears in grey in (e).

Table 1 .

 1 Material parameters of the model's components.

	Component	C10 (MPa)

C20 (MPa) C30 (MPa) µ (MPa) EX (MPa) EY (MPa) EZ (MPa) d1 (MPa -1 )

										ν
	Adipose tissue	1.3 × 10 -4	0.0	12.2 × 10 -3	-	-	-	-	1.6	0.4999
	Skin	2.7 × 10 -1	1.9	-	-	-	-	-	-	0.4999
	Dressing layer 1	-	-	-	1.0 × 10 -3	-	-	-	-	-
	Dressing layer 2	-	-	-	-	4.4	1.8	2.6 × 10 -2	-	0.2560
	Mattress	-	-	-	-	2.3 × 10 -1	-	-	-	0.3000

Table 2 .

 2 Parameters' minimal, intermediate, and maximal values used as levels for the experimental points of the sensitivity analysis.

	Parameters	Minimal Level	Intermediate Level	Maximal Level
	Wound deep-	1.30 mm	3.20 mm	5.00 mm
	ness
	Alveoli cut	Recommended +1 layer	Recom-mended	Recommended -1 layer
	Mattress	0.03 MPa	0.23 MPa	0.43 MPa
	stiffness
	Gauze pres-	0.00 MPa	0.02 MPa	0.04 MPa
	sure
	Given that the local finite element model is rather a qualitative model, a full polynomial model of degree
	two was considered sufficient to emulate it:

Conflicts of Interest:

This study was financially supported by Urgo RID.


text:
Introduction

Static game

Dynamic Game

Conclusions and extensions

The one shot model. The static game Two players. Each player endowed with w i and contributes x i ∈ [0, w i ] to a public good. Then w i -x i privately consumed. Preferences: intrinsic utility (absolute level of contribution) positional payoffs (associated with relative contrib.) U i (x i , x j ) = u i (x i , X ) + V i (x i -x j ), X = x i + x j conformist payoffs (associated with distance to contrib.)

U j (x i , x j ) = u j (x j , X ) + V j (x i -x j ).
Global intrinsic utility: u = u i + u j The global utility or social welfare:

U = U i + U j .
The intrinsic utility. Properties C1 Individual provision always reduces own welfare:

∂u i ∂x i (x i , X ) < 0, ∀(x i , x j ) ∈ [0, w i ] × [0, w j ].
Nash equilibrium is (0, 0). C2 Individual provision: First unit ↑ u; last unit ↓ u.

C 2a : ∂u ∂x i (0, 0) > 0, C 2b : ∂u ∂x i (w 1 , w 1 + w 2 ) < 0.
Some contribution to the public good and to the private good are socially desirable. C3 Agents contributions are substitutes

C 3 : ∂ 2 u ∂x i ∂X (x i , X ) < 0.

Intrinsic utility, positional + conformist concerns

Functional specifications:

The intrinsic utility is an additively separable function:

u i (x i , X ) = w i -x i + b i (X ) u(X ) = w 1 + w 2 -X + b 1 (X ) + b 2 (X ) b i (X ) = α i X - ε 2 X 2
Public Good valuation by i: α i , satiation ε Positional payoff

V i (x i -x j ) = v P i × (x i -x j ). Positional concern of player i: v P i ≥ 0 Conformist payoff V j (x i -x j ) = - v C j 2 × (x i -x j ) 2 .
Conformist concern of player j: v C j ≥ 0

Maximization problem

The problem for player k ∈ {i, j} is:

max 0≤x i ≤w i w i -x i + b i (X )+v P i (x i -x j ) max 0≤x j ≤w j w j -x j + b j (X )- v C j 2 (x i -x j ) 2
The marginal utility of players i and j now reads as:

∂U i ∂x i = -1 + b i (X )+v P i ∂U j ∂x j = -1 + b j (X )+v C j (x i -x j )
Positionality raises the marginal benefit of private provision. Conformism does also, as long as j contributes less than i.

For the positional player, a similar situation: Best-reply function:

x b i (x j ) =        0 A i ≤ x j A i -x j A i -w i ≤ x j ≤ A i w i x j ≤ A i -w i .
Definition (Wished amount)

A i =            0 if v P i < 1 -α i , v P i -(1 -α i ) α i ε otherwise w i + w j if v P i > 1 -α i + α i ε(w 1 + w 2 ). Positive contribution v P i > 1 -α i Public Good Assumption v P i < 1
Similarities and differences (continued)

For the conformist player, different situations Referring to condition C3:

∂ 2 U j ∂x i ∂x j = -α j ε + v C j 1 v C j < α j ε: contributions are still substitutes 2 v C j > α j ε:
conformism is so strong that contributions from the other agent increases the willingness to contribute: contributions become complements for Agent j. Best-reply function (complementarity case):

x b j (x i ) =                0 x i ≤ B j := 1 -α j v C j -α j ε v C j -α j ε v C j + α j ε (x i -B j ) otherwise w j w j (v C j + α j ε) ≤ (x i -B j )(v C j -α j ε) 10 / 28

Nash equilibrium

Unique Nash equilibrium in all cases. c) If

v P i ≤ 1 -α i , A i = 0: no contribution at all (x N i , x N j ) = (0, 0) a) If v P i > 1 -α i , A i > 0, and v C j ≤ v C j := α j ε + 1 -α j min{A i , w i } then (x N i , x N j ) = (min{A i , w i }, 0) Agent j is not incentivized enough to contribute. This includes the substituability case v C j ≤ α j ε, but also the weak complementarity case v C j ∈ (α j ε, v C j ]. Nash Equilibrium (continued) b) If v P i > 1 -α i , A i > 0, and v C j > v C j (strong complementarity) b int ) unique interior equilibrium x N,PC i int = 1 2 (A i + ∆ x ) x N,PC j int = 1 2 (A i -∆ x ) ∆ x := α i -α j (1 -v P i ) α i v C j b 1 ) (w i , x b j (w i )) b 12 ) (w i , w j ) b 2 ) (A i -w j , w j )

Introduction

Static game Dynamic Game

Conclusions and extensions

Simple ynamic model with inertia (continued)

Utilities at time t:

U i (•) = w i -x it + α i x it + x jt - ε 2 (x it + x jt ) 2 +v P i (x it -x jt-1 )- v I i 2 (x it -x it-1 ) 2 U j (•) = w j -x jt + α j x it + x jt - ε 2 (x it + x jt ) 2 - v C j 2 (x jt -x it-1 ) 2 - v I j 2 (x jt -x jt-1 ) 2
Simple dynamic model with inertia (end)

The problem of a myopic agent k ∈ {i, j} is:

max x kt U k (x kt , x -kt , xkt , x-kt ), s.t.: xkt = x kt-1 .
The first-order conditions for an interior equilibrium are:

εα i + v I i εα i εα j εα j + v I j + v C j x it x jt = v I i 0 v C j v I j x it-1 x jt-1 + -1 + α i + v P i -1 + α j . =⇒ dynamic system x t = Mx t-1 + V 0 .
Illustration 1: no inertia Parameters: ε = 0.4, α 1 = 0.8, α 2 = 0.5, v P 1 = 0.56, v C 2 = 2. Wished amount A 1 = 1.125, A 1 /2 = 0.5625.

A simple dynamic model with inertia

We consider the intrinsic utility function of the static model...

At time t the positional agent gets joy from contributions above the other player at time t -1: +v P i (x it -x jt-1 ).

2

The conformist agent gets joy from contributions close to the other player at time t -1: -v C j (x jt -x it-1 ) 2 /2.

3

Agents have Inertia from previous actions. Disutility from deviations from the previous action:

Dynamic problem: general properties

Properties of the free dynamical system When v I k = 0 at least for one player k ∈ {i, j}, both eigenvalues have modulus < 1 when they are real, they have the same sign the dynamical system x t converges to a unique steady state which is the point

one eigenvalue is equal to -1, the other one is 0 the dynamical system does not converge in general: its trajectories tend asymptotically to a cycle of order 2 which is organized around the point x int

General properties (continued)

Behavior different from the Positional+Positional case Eigenvalues A sufficient condition for both eigenvalues to be complex is:

Spiralling behavior: small inertia of Positional player large inertia or conformism from Conformist player When taking budget constraints into account: Property of the constrained dynamical system If x 0 = (0, 0), then x 1 = (x i1 , 0).

Illustration 2: welfare analysis Welfare analysis: remember the typology of total contribution X :

Conclusions

With respect to the Positional+Positional case, there are common features and qualitative differences.

In the static framework = The standard situation with large endowments is an interior Nash equilibrium

In the myopic dynamic framework with inertia = Convergence to the static equilibrium = Possibility of spiralling or oscillating convergence = Possibility of overshooting = Possibility of transient welfare reduction

Extensions

As in the previous talk: Consider another dynamic: the stock of public good. Consider farsighted agents (true dynamic game). More than two players. ...

Obrigado!