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A four-dimensional formalism encompassing Eulerian and Lagrangian approaches to constitutive Models

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URI: http://hdl.handle.net/2042/56997
Title: A four-dimensional formalism encompassing Eulerian and Lagrangian approaches to constitutive Models
Author: Rouhaud, Emmanuelle; Panicaud, Benoit; Kerner, Richard
Abstract: The covariance principle of General Relativity ensures the validity of any equation and physical models through any changes of coordinates and any changes of frame of reference, because of the definition of the 4D space-time, the use of 4D tensors, 4D operations and 4D operators. Frame indifference being a critical concept for the construction of material constitutive models within the area of three dimensional classical continuum mechanics, we propose here to write this constitutive model using the covariance principle and 4D tensor formalism. A general method is then proposed to build a constitutive material model with such a four-dimensional point of view. Within this 4D formalism, the choice of the 4D coordinate system defines the frame of reference and change of reference frames corresponds to change of 4D coordinates. It is then possible to define a curvilinear coordinate system deforming like the material and corresponding to convective coordinates. The 4D formalism enables to change coordinates from inertial to convective hence allowing to consider the Lagrangean formulation as a specific set of components of an Eulerian description. Then, a choice of four-dimensional variables leads to a model for a general thermo-elastic isotropic material that encompasses the 3D Eulerian and Lagrangean thermo-elastic isotropic or anisotropic models for finite deformations.
Subject: Lagrangean formulation; covariance principle; constitutive models
Publisher: AFM, Association Française de Mécanique
Date: 2015

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