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Constitutive Models Based on a Quadri-Dimensionnal Formalism For Constitutive models based on a quadri-dimensionnal formalism for numerical simulations of finite transformations Emmanuelle Rouhaud, Benoît Panicaud, Arjen Roos, Najoua Mhenni, Richard Kerner To cite this version: Emmanuelle Rouhaud, Benoît Panicaud, Arjen Roos, Najoua Mhenni, Richard Kerner. Constitutive models based on a quadri-dimensionnal formalism for numerical simulations of finite transformations. 11e colloque national en calcul des structures, CSMA, May 2013, Giens, France. hal-01717080 HAL Id: hal-01717080 https://hal.archives-ouvertes.fr/hal-01717080 Submitted on 25 Feb 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Public Domain CSMA 2013 11e Colloque National en Calcul des Structures 13-17 Mai 2013 Constitutive models based on a quadri-dimensionnal formalism for numerical simulations of finite transformations Emmanuelle ROUHAUD 1, Benoît PANICAUD 2, Arjen ROOS 3, Najoua BEN MHENNI 4, Ri- chard KERNER 5 1 Université de Technologie de Troyes, [email protected] 2 Université de Technologie de Troyes, [email protected] 3 ONERA, [email protected] 4 Université de Technologie de Troyes, [email protected] 5 Université Pierre et Marie Curie, [email protected] Résumé — La description correcte des non-linéarités lors des transformations finies d’un milieu continu est une nécessité pour accrocher un comportement cinématique réaliste [1]. Nous proposons ainsi d’uti- liser les outils mathématiques de la géométrie différentielle dans le cadre d’un formalisme quadridi- mensionnel et curviligne de l’espace-temps pour écrire des modèles de comportement. Cette approche garantie une description dite "covariante" des transformations subies par la matière, c’est-à-dire valable pour tous les systèmes de référence [2]. Cette approche a fait ses preuves en physique, en particulier en relativité générale [3]. Le but est de résoudre les difficultés qui se posent encore avec la notion d’objectivité matérielle, ambigüe en 3D [4]. Le principe de covariance généralisé est donc appliqué à des transformations qui res- tent celles de la mécanique classique des milieux continus (c’est-à-dire sans aucun effet relativiste et en particulier pour des vitesses qui restent toujours bien en deçà de la vitesse de la lumière). Ainsi, appliqué aux modèles de comportement purement macroscopique, il est montré comment les projections de la 4D sur l’espace 3D aboutissent à la construction de relations de comportement, par un formalisme varia- tionnel. Ces relations sont ensuite déclinées pour des comportements particuliers (élasticité linéaire ou non-linéaire, pour différents choix de potentiels). En parallèle, une approche incrémentale est également proposée où le formalisme quadri-dimensionnel apporte des réponses uniques sur le choix de la dérivée "objective", comparé à la multitude des choix possibles en 3D. Les calculs éléments finis sur différentes structures montrent la comparaison entre ces différentes approchent pour différentes sollicitations. Une approche micromécanique pourra être finalement envisagée dans le cadre de l’élasticité pour en montrer à la fois les forces et les faiblesses. Mots clés — transformations finies, géométrie différentielle, objectivité matérielle, élastoplasticité 1 Introduction To say that a constitutive model has to verify the principle of material objectivity to ensure its frame- indifference is presented as common wisdom in the area of continuum mechanics. This problematic has been extensively studied, detailed and discussed. Among the points that have been subjected to debates are the principle itself [5], a (re)definition of the change of observers [5, 6] with inertial/non inertial consideration [7] or its link with the superposition of a rigid body motion [8, 9, 1]. An important issue concerns the definition of material frame-indifferent time derivatives to represent, objectively, the va- riations of a tensor with respect to time. These derivatives appear in incremental forms of constitutive models when the invariance under superposed rigid body motions is indeed observed. But the time deri- vative of an objective tensor is in general not objective. To work out this problem, 3D objective transport operators are defined ; they are also referred to as "objective rates" or "invariant time fluxes". They are applied in particular to the Cauchy stress tensor. The result of each of these transports is proven to be invariant with respect to superposition of rigid body motions [10, 11, 1]. The difficulty resides in the fact that there are "infinitely many possible objective time fluxes that may be used" [1]. In the end, the validity of such an "objective" approach, and the restriction that it imposes on the 1 constitutive models are often questioned and reconsidered ; see for example [12, 13, 14, 6, 15, 7, 9, 16, 5, 17, 18, 19, 20, 21]. Thus, if everyone agrees on the necessity to define frame-indifferent entities and equations, the usage, applications and consequences of the notion of objectivity still constitute an open subject of debate. Differential geometry, also known as Ricci-calculus, [22, 23], offers a convenient framework to mo- del the finite transformations of a material continuum. It proposes a mathematical formalism for the use of tensors expressed in arbitrary coordinate systems within a differentiable manifold. The physical en- tities are then represented with tensor fields of the considered manifold. It is recognized as a formalism of choice to describe the straining motion of material continua within a classical three-dimensional (3D) context. Within such a formalism, tensors are independent of any arbitrary change of coordinate systems and it is possible to define derivative operators that are tensors of the considered space. Within its four- dimensional (4D) context, differential geometry has found a major and essential application in physics with the theory of General Relativity, which has shown its ability to deal properly with space-time trans- formations. The covariance principle guarantees the proper expression of all physical relations in any kind of systems of reference, as developed and reviewed by Landau and Lifshitz [3]. Indeed, as written by Eringen [24] : "Attempts to secure the invariance of the physical relations of motion from the observer have produced one of the great triumphs of twentieth-century physics. (...) Attempts to free the principles of classical mechanics from the motion of an observer were resolved by Einstein in his general theory of relativity...". The only appropriate way to define frame independence is thus to consider four-dimensional quantities, i.e. to construct physics and material mechanics within the scope of 4D physics. The present work thus proposes to use the covariance principle to describe a continuum, in order to guaranty the frame-indifference of tensors, equations and models. A major interest consists in using such a method to construct material constitutive relations. The objective of the present paper is to illustrate the method and use the 4D formalism to construct models for elastic transformations. 2 Material objectivity In three dimensions, a rigid frame of reference is classically associated with an observer and offers the possibility to parametrize the position and instant of an occurring event. The frame of reference is constituted by a 3D Cartesian coordinate system to specify any positions, to which an origin for time and a Galilean chronology are associated. If the same event is now described by two observers with the i i respective coordinates and time (x ;t) and (xe;t), then an orthogonal matrix Q and a vector δ can always be found such that : i i j i xe = Q j(t)x + d (t) (1) where Q and δ correspond respectively to the instantaneous rotation and to the instantaneous translation of one frame with respect to the other. Then a second rank tensor T is said to be frame-indifferent or objective if it verifies the transformation rule : i j i j kl Te = Q k(t)Q l(t)T (2) for any changes of observers. Similar equations may be written for other component types and tensor ranks. Truesdell and Noll [1] define the principle of objectivity as : "it is a fundamental principle of classical physics that material properties are indifferent, i.e., independent of the frame of reference or observer". Nemat Nasser [11] defines it as : "Constitutive relations must remain invariant under any rigid-body ro- tation of the reference coordinate system. This is called objectivity or the material frame indifference." As stated by Liu [25], "the principle of material frame indifference plays an important role in the de- velopment of continuum mechanics, by delivering restrictions on the formulation of the constitutive functions of material bodies. It is embedded in the idea that material properties should be independent of observations made by different observers. Since different observers are related by a time-dependent rigid transformation, known as a Euclidean transformation, material frame-indifference is sometimes in- terpreted as invariance under superposed rigid body motions".
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