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8126 results in Fluid dynamics and solid mechanics

Page 198 of 407

  • PART XIII - THEORY OF ISOTROPIC PLASTIC SOLIDS UNDERGOING SMALL DEFORMATIONS

  • 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 415-416

  • Summary

    The theory of elasticity furnishes a simple and elegant vehicle for illustrating the basic ideas of continuum mechanics.Among itsmany uses, elasticity theory is widely utilized for modeling the response of rubber-like, elastomeric materials at large strains. However, elasticity theory can be applied to the description of metals only for extremely small strains, typically not exceeding 10−3. Larger deformations in metals lead to flow, permanent set, hysteresis, and other interesting and important phenomena that fall naturally within the purview of plasticity.

  • 7 - Stretch, Strain, and Rotation

  • 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 69-74

  • Summary

    Stretch and Rotation Tensors. Strain

    Consider the (pointwise) polar decomposition

    of the deformation gradient F into a rotation R and positive-definite symmetric tensors U and V; using terminology motivated in §7.3, we refer to U as the right stretch tensor and to V as the left stretch tensor. The tensors U and V, which have the explicit representations

    are useful in theoretical discussions but are often problematic to apply because of the square root. For that reason, we introduce the right and left Cauchy–Green (deformation) tensors C and B defined by

    Then, by (7.1),

    For future reference,we list the properties of the stretch and Cauchy–Green tensors:

    A tensor useful in applications is the Green–St. Venant strain tensor

    Note that E vanishes when F is a rotation, for then F⊤F = 1. This property of E is often adopted as one necessary for a tensor to qualify as a meaningful measure of strain.

    Next, by (M1) and (M2) on page 65 and (7.3) and (7.6),

    (M3) U, C, andEmap material vectors to material vectors;

    (M4) VandBmap spatial vectors to spatial vectors;

    (M5) Rmaps material vectors to spatial vectors.

    To verify (M3), consider F⊤F: (M1) implies that F maps a material vector fR to a spatial vector FfR and, by (M2), this spatial vector is mapped back to a material vector. Thus C = F⊤F maps material vectors to material vectors and since UU = C, with U symmetric, U must also map material vectors to material vectors.

Page 198 of 407