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38480 results in Cambridge Textbooks

Page 1703 of 1924

  • 9 - Material and Spatial Descriptions of Fields

  • from PART III - KINEMATICS
  • Morton E. Gurtin, Carnegie Mellon University, Pennsylvania, Eliot Fried, McGill University, Montréal, Lallit Anand, Massachusetts Institute of Technology
  • Book:
    The Mechanics and Thermodynamics of Continua
    Published online:
    05 June 2012
    Print publication:
    19 April 2010, pp 80-85

  • Summary

    Consider a motion χ. Since the mapping x = χ(X, t) is invertible in X for fixed t, it has an inverse X = χ−1(x, t), the reference map defined in (5.5) and (5.6). χ−1 associates with each time t and spatial point x in Bt, a material point X = χ−1(x, t) in B.

    Using the reference map, we can describe the velocity as a function v(x, t) of the spatial point x and t:

    The field v represents the spatial description of the velocity; v(x, t) is the velocity of the material point that at time t occupies the spatial point x.

    More generally, let ϕ denote a scalar, vector, or tensor field defined on the body for all time. We generally consider ϕ to be a function ϕ(X, t) of the material point X and the time t; this is called the material description of ϕ. But, as with the velocity, we may also consider ϕ to be a function φ(x, t) of the spatial point x and t; this is called the spatial description and is related to the material description through

    Similarly, a field φ(x, t) described spatially may be considered as a function ϕ(X, t) of the material point X and t; this is called the material description and is given by

    • When there is no danger of confusion we use the same symbol for both the material and spatial descriptions.

  • 19 - Forces and Moments. Balance Laws for Linear and Angular Momentum

  • from PART IV - BASIC MECHANICAL PRINCIPLES
  • Morton E. Gurtin, Carnegie Mellon University, Pennsylvania, Eliot Fried, McGill University, Montréal, Lallit Anand, Massachusetts Institute of Technology
  • Book:
    The Mechanics and Thermodynamics of Continua
    Published online:
    05 June 2012
    Print publication:
    19 April 2010, pp 131-145

  • Summary

    The current entrenched, facile conception of force in terms of “pushes” and “pulls” has fostered a view of force as a “real quantity” rather than a mathematical concept. In the words of Pierce (1934, p. 262): [Force is] “the great conception which, developed in the early part of the seventeenth century from the rude idea of a cause, and constantly improved upon since, has shown us how to explain all the changes of motion which bodies experience, and how to think about physical phenomena; which has given birth to modern science; and which … has played a principal part in directing the course of modern thought … It is, therefore, worth some pains to comprehend it.”

    Those who believe the notion of force is obvious should read the scientific literature of the period following Newton. Truesdell (1966) notes that “D'Alembert spoke of Newtonian forces as ‘obscure and metaphysical beings, capable of nothing but spreading darkness over a science clear by itself,’” while Jammer (1957, pp. 209, 215) paraphrases a remark of Maupertis, “we speak of forces only to conceal our ignorance,” and one of Carnot, “an obscure metaphysical notion, that of force.”

    Within the framework of continuum mechanics, the basic balance laws for linear and angular momentum assert that, given any spatial region Pt convecting with the body,

    1. (i) the net force on Pt is balanced by temporal changes in the linear momentum of Pt;

    2. (ii) the net moment on Pt is balanced by temporal changes in the angular momentum of Pt.

Page 1703 of 1924