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Continuum Mechanics and Nonlinear Elasticity

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Continuum Modeling in Mechanobiology

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Abstract

This chapter derives the fundamental equations of continuum mechanics and the nonlinear theory of elasticity. Topics include the description of motion of particles and continua; analysis of deformation and stress; balance of mass, momentum, and energy; the entropy inequality; and constitutive relations. To facilitate understanding of the basic concepts, equations are developed in one, two, and three dimensions.

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Notes

  1. 1.

    Since this book focuses primarily on elastic solids, some formulas that may be useful for fluid mechanics are omitted.

  2. 2.

    Some authors use the alternative notation D∕Dt for the material time derivative to emphasize that the derivative follows the motion of a particle in a continuum.

  3. 3.

    For convenience, in the remainder of this book, the time variable is dropped from all equations unless it is needed for clarity or if the analysis is explicitly time-dependent. Consequently, we write x(X) instead of x(X, t). For example, in the problem discussed in Sect. 3.2.1, the motion of the bar is described by x(X, t) = X(1 + αt 2). If we set \(\bar {\alpha }(t)=\alpha t^2\), then \(x(X)=X (1+\bar {\alpha })\) gives a snapshot of the bar at an instant in time, as defined by the value of the new parameter \(\bar {\alpha }\).

  4. 4.

    For x << 1, \(\sin x \cong x\).

  5. 5.

    The left Cauchy-Green deformation tensor, defined as B = F⋅F T, is not used in this book.

  6. 6.

    Some authors define P T as the first Piola-Kirchhoff stress tensor.

  7. 7.

    It is important to remember that the equation of motion involves the summation of forces, not stresses.

  8. 8.

    Similar difficulties are encountered with some theories for remodeling (see Chap. 7).

  9. 9.

    Lagrange multipliers are used in optimization problems to enforce constraint conditions (Belytschko et al. 2000).

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Authors and Affiliations

  1. Department of Biomedical Engineering, Washington University, St. Louis, MO, USA

    Larry A. Taber

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Taber, L.A. (2020). Continuum Mechanics and Nonlinear Elasticity. In: Continuum Modeling in Mechanobiology. Springer, Cham. https://doi.org/10.1007/978-3-030-43209-6_3

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  • DOI: https://doi.org/10.1007/978-3-030-43209-6_3

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  • Print ISBN: 978-3-030-43207-2

  • Online ISBN: 978-3-030-43209-6

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