Every day we experiment motion and shape change of continuum media, assembly of different space dimension structures, fluids, solids which exhibit numerous evolutions either particular or general: large deformations, incompressibility, cavitation, contact, surface tension, defects, dislocations,... Stretch and rotation are a common feature of these motions and phenomenons. We base a continuum mechanics predictive theory on stretch and rotation quantified by matrices. In this Chapter we identify the informations which are required to predict the motion of a dimension three continuum medium with respect to a frame. The shape and the shape change velocities of the system continuum medium-frame are considered. The classical polar decomposition defines the stretch matrix and the rotation matrix which may be experimented as well as their velocities. It results the angular velocity matrix may be experimented. Virtual stretch velocity matrices and virtual angular velocity matrices are defined: they are velocities we may think of. Considering either a nail firmly hammered in a wall or a paper clip floating on water which are mechanical structures, assumed in this model to be a point and a dimension one line, with velocities, accelerations,... equal to the velocities, accelerations,... volume values in the wall and water. To ensure this physical experimented continuity it appears that the volume velocities, accelerations,... have to be smooth enough, for instance to be differentiable up to the order three. Let us recall a mechanical theory is an nth gradient theory if the powers to change the shape and to change the velocity make use of order n space derivatives of the actual and virtual velocities. Considering the nail-wall system, the paper clip-water system and other engineering structures, for instance in soil mechanics houses lying on soils represented by layers of springs, we are motivated to choose third order theories for dimension three continuum media: actual and virtual stretch and angular matrices velocities, which are order one quantities, have space derivatives up to the order two. We consider a 3D continuum medium either a solid or a fluid as an example. Let us repeat that to speak of motion we need two elements: the continuum medium and a frame. We identify all the informations which are required to predict the motion of the chosen continuum medium with respect to the chosen frame. Following Chaps. 10 and 11 are devoted to the actual prediction of the thermo-mechanical evolution of solids and fluids.

The Motion of a Continuum Medium

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发表日期：

2025年

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Michel Frémond

摘要

Every day we experiment motion and shape change of continuum media, assembly of different space dimension structures, fluids, solids which exhibit numerous evolutions either particular or general: large deformations, incompressibility, cavitation, contact, surface tension, defects, dislocations,... Stretch and rotation are a common feature of these motions and phenomenons. We base a continuum mechanics predictive theory on stretch and rotation quantified by matrices. In this Chapter we identify the informations which are required to predict the motion of a dimension three continuum medium with respect to a frame. The shape and the shape change velocities of the system continuum medium-frame are considered. The classical polar decomposition defines the stretch matrix and the rotation matrix which may be experimented as well as their velocities. It results the angular velocity matrix may be experimented. Virtual stretch velocity matrices and virtual angular velocity matrices are defined: they are velocities we may think of. Considering either a nail firmly hammered in a wall or a paper clip floating on water which are mechanical structures, assumed in this model to be a point and a dimension one line, with velocities, accelerations,... equal to the velocities, accelerations,... volume values in the wall and water. To ensure this physical experimented continuity it appears that the volume velocities, accelerations,... have to be smooth enough, for instance to be differentiable up to the order three. Let us recall a mechanical theory is an nth gradient theory if the powers to change the shape and to change the velocity make use of order n space derivatives of the actual and virtual velocities. Considering the nail-wall system, the paper clip-water system and other engineering structures, for instance in soil mechanics houses lying on soils represented by layers of springs, we are motivated to choose third order theories for dimension three continuum media: actual and virtual stretch and angular matrices velocities, which are order one quantities, have space derivatives up to the order two. We consider a 3D continuum medium either a solid or a fluid as an example. Let us repeat that to speak of motion we need two elements: the continuum medium and a frame. We identify all the informations which are required to predict the motion of the chosen continuum medium with respect to the chosen frame. Following Chaps. 10 and 11 are devoted to the actual prediction of the thermo-mechanical evolution of solids and fluids.