AIMS Workshop on Capacity Building in the Mathematical Sciences 13 – 17 April, 2004

Jean S ALENÇON École polytechnique, Palaiseau, France. Académie des sciences, Paris, France.

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Dimensional analysis The continuum and its coherence Yield design

Dimensional analysis The continuum and its coherence Yield design

Dimensional analysis

Is it possible to increase proportionally the size of a structure? “Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Only referring to RUPTURE

Is it possible to increase proportionally the size of a structure? “Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Only referring to RUPTURE

“Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Only referring to RUPTURE

“Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Volume density of Only referring to RUPTURE Active forces: gravity

“Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Volume density of Only referring to RUPTURE Active forces: gravity Coherence of matter

“Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Volume density of Only referring to RUPTURE Active forces: gravity Resisting forces: Coherence of matter

“Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Volume density of Only referring to RUPTURE Active forces: gravity Resisting forces across sections Surface density of

“Yet I shall say and will affirm that, even if the imperfections did not exist and matter were absolutely perfect, unalterable and free from all accidental variations, still the mere fact that it is matter makes the larger machine, built of the same material and in the same proportion as the smaller, correspond with exactness to the smaller in every respect except that it will not be so strong or so resistant against violent treatment; the larger the machine, the greater its weakness.” Volume density of Only referring to RUPTURE Active forces: gravity Resisting forces across sections Surface density of [scale] 3 [scale] 2 balance

A. Vaschy (1892). Annales télégraphiques, 25-28, DIMENSIONAL ANALYSIS E. Buckingham (1914). Phys. Rev., 4,  Theorem A simple and common wording: Any homogeneous relationship between n quantities, p of which are dimensionally independent from each other, can be substituted by a relationship between (n-p) non dimensional factors  i.

DIMENSIONAL ANALYSIS  Theorem A simple and common wording: Any homogeneous relationship between n quantities, p of which are dimensionally independent from each other, can be substituted by a relationship between (n-p) non dimensional factors  i.

DIMENSIONAL ANALYSIS A common feeling: any physical relationship should be independent of the units the observer chooses for each quantity involved in its writing.

DIMENSIONAL ANALYSIS A more sophisticated approach: A problem is set through a system of field differential equations with boundary conditions depending on the time with given values of quantities or fields.

DIMENSIONAL ANALYSIS A more sophisticated approach: A problem is set through a system of field differential equations with boundary conditions depending on the time with given values of quantities or fields. The system admits an invariance group of unit changes.

DIMENSIONAL ANALYSIS A more sophisticated approach: The solution: a general relationship where the unknown quantities and unknown fields are determined as functions of the data.

DIMENSIONAL ANALYSIS A more sophisticated approach: The solution: a general relationship where the unknown quantities and unknown fields are determined as functions of the data. It must be invariant with respect to the group of unit changes.

DIMENSIONAL ANALYSIS Federman’s Theorem (1911) Assuming the functions to be differentiable and with other restrictive assumptions, proves that the invariance group is a homogeneity group, proves Vaschy’s theorem.

DIMENSIONAL ANALYSIS Martinot-Lagarde (1960) Assuming the functions to be continuous and with less restrictive assumptions, proves that the invariance group is a homogeneity group, thence Vaschy’s theorem.

DIMENSIONAL ANALYSIS The mathematical proof: Is based upon the theory of arc-wise connected subgroup of a vector group (Hayashida, 1949), can be derived from a result by Bourbaki, (1960).

DIMENSIONAL ANALYSIS “… so it would be impossible to build up the bony structures of men, horses, or animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height; for this increase in height can be accomplished only by employing a material which is harder and stronger than usual, or by enlarging the size of the bones, thus changing their shape until the form and appearance of the animals suggest a monstrosity.”

DIMENSIONAL ANALYSIS “… so it would be impossible to build up the bony structures of men, horses, or animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height; for this increase in height can be accomplished only by employing a material which is harder and stronger than usual, or by enlarging the size of the bones, thus changing their shape until the form and appearance of the animals suggest a monstrosity.”

DIMENSIONAL ANALYSIS

“… so it would be impossible to build up the bony structures of men, horses, or animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height; for this increase in height can be accomplished only by employing a material which is harder and stronger than usual, DIMENSIONAL ANALYSIS Reduced scale experiments for structural design