Abstract
This chapter generalizes to three dimensions the one-dimensional viscoelastic constitutive equations derived in earlier chapters. The concepts of homogeneity, isotropy, and anisotropy are introduced and the principle of superposition is used to construct three-dimensional constitutive equations for general anisotropic, orthotropic, and isotropic viscoelastic materials. So-called Poisson’s ratios are introduced, and it is shown that uniaxial tensile and shear relaxation and creep tests suffice to characterize orthotropic viscoelastic solids. A rigorous treatment extends applicability of the Laplace and Fourier transforms to three-dimensional conditions, and constitutive equations in both hereditary integral form and differential form for compressible and incompressible isotropic solids are developed and discussed in detailed.
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Notes
- 1.
For each pair of indices, ij and kl, ranging from 1 to 3, the number of independent components is 3·4/2 = 6.
- 2.
In matrix notation, the superscript T is used to denote the “transpose” of the matrix it is appended to.
- 3.
- 4.
Material principal directions are not to be confused with the principal directions of stress or the principal directions of strain, the latter so-named because along them the stress and, respectively, the strain attain their extreme numerical values.
- 5.
In indicial tensor notation, the summation convention is typically suspended by adding an underscore to the indices excluded from the summation.
- 6.
δ ij  = 1 if i = j, δ ij  = 0 if i ≠ j; δ ii  = 3; A ik ·δ kj  = A i1 δ 1j  + A i2 δ 2j  + A i3 δ 3j  = A ij.
- 7.
According to the summation convention, terms in an expression are summed over the range of their repeated indices, so that, for instance, A ikk  = A i11  + A i22  + A i33 , for i = 1, 2, 3, etc.
References
A.E. Green, W. Zerna, Theoretical Elasticity, 2nd edn. (Dover, NY, 1968), pp. 1–3
L.E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice-Hall, Englewood Cliffs, 1963), pp. 64–80
B.E. Read, G.D. Dean, The Determination of Dynamic Properties of Polymer Composites (Wiley, New York, 1978), pp. 12–17
R.M. Christensen, Mechanics of Composite Materials (Wiley, New York, 1979), pp. 150–160
W.R. Little, Elasticity (Prentice Hall, Englewood Cliffs, 1973), pp. 62–63. 16–17
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Gutierrez-Lemini, D. (2014). Three-Dimensional Constitutive Equations. In: Engineering Viscoelasticity. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8139-3_8
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DOI: https://doi.org/10.1007/978-1-4614-8139-3_8
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