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36965 results in Cambridge Textbooks

Page 1628 of 1849

  • 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.

  • PART V - BASIC THERMODYNAMICAL 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 181-182

  • Summary

    In introducing basic versions of the first two laws of thermodynamics appropriate to continua, we emphasize that

    • like force, we view energy, entropy, heat flow, and entropy flow as primitive objects;

    • a priori notions of “equilibrium” and “state” are not employed.

    We find it most convenient to use a spatial formulation–that is, a formulation in terms of quantities measured per unit volume and area in the observed space.

  • PART XI - THERMOELASTICITY

  • 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 331-332

  • Summary

    It is well known that heating or cooling an unconfined solid specimen generally leads to dimensional changes of the specimen. For confined specimens, the deformation produced by heating may generate complex stress distributions, and the peak magnitudes of such thermally induced stresses are often substantial. Conversely, temperature changes and distributions generated in the mechanical loading of metals may also be important. We now present a framework for the coupled thermal and mechanical response of solids, restricting our attention to situations in which the deformation is elastic. The general framework — known as the theory of thermoelasticity — is broad enough to describe both metals and rubber-like elastomeric materials, including the anomalous contraction of a stressed rubber-like material on heating, known as a Gough–Joule effect.

  • 34 - General Considerations

  • from PART VII - INTERLUDE: BASIC HYPOTHESES FOR DEVELOPING PHYSICALLY MEANINGFUL CONSTITUTIVE THEORIES
  • 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 223-223

  • Summary

    The balance laws for mass, momentum, and energy and the imbalance law for entropy represent fundamental principles of continuum thermomechanics and as such are presumed to hold for all bodies, whether they be solid, liquid, or gas. In contrast, the constitution of a class of bodies composed of a particular material is specified by constitutive equations. Such equations limit the class of “processes” that bodies comprised of a given material may undergo.

    In the words of Truesdell & Noll (1965, §1). “The general physical laws in themselves do not suffice to determine the deformation or motion of a body subject to given loading. Before a determinate problem can be formulated, it is usually necessary to specify the material of which the body is made. In the program of continuum mechanics, such specification is stated by constitutive equations, which relate the stress tensor … to the motion. For example, the classical theory of elasticity rests upon the assumption that the stress tensor at a point depends linearly on the changes in length and mutual angle suffered by elements at that point …, while the classical theory of viscosity is based on the assumption that the stress tensor depends linearly on the instantaneous rates of change of length and mutual angle. These statements … are definitions of ideal materials. The former expresses in words the constitutive equation that defines a linearly and infinitesimally elastic material; the latter a linearly viscous fluid. […]

Page 1628 of 1849