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2019 | OriginalPaper | Chapter

6. Constitutive Equations

Author : Zdeněk Martinec

Published in: Principles of Continuum Mechanics

Publisher: Springer International Publishing

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Abstract

The equations listed in the preceding chapters apply to any material body that deforms under the action of external forces. Mathematically, they do not, by themselves, have a unique solution because the deformations are not related to internal contact forces.

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previous chapter Moving Reference Frames

next chapter Entropy Principles

This is a significant advantage of the Lagrangian description because material properties are always associated with a material particle ${\mathcal {X}}$ with the position $\vec X$, while in the Eulerian description, various material particles may pass through a given present position $\vec x$.

Note that if higher-order gradients in (6.9) are retained, we obtain non-simple materials of various classes. For example, we obtain a polar material of couple stress by including the second-order gradients into argument of ${\mathcal {F}}$.

A group is a set of abstract elements A, B, C, ⋯, with a defined operation A ⋅B (as, for instance, ‘multiplication’) such that: (1) the product A ⋅B is defined for all elements A and B of the set, (2) this product A ⋅B is itself an element of the set for all A and B (the set is closed under the operation), (3) the set contains an identity element I such that I ⋅A = A = A ⋅I for all A, (4) every element has an inverse A ⁻¹ in the set such that A ⋅A ⁻¹ = A ⁻¹ ⋅A = I.

The temperature, as any other variable, has both an Eulerian and a Lagrangian representation. For example, the Lagrangian temperature is defined as $\Theta (\vec X,t):=\theta (\vec x(\vec X,t),t)$, where $\theta (\vec x,t)$ is the Eulerian temperature. We make, however, an exception in the notation and use $\theta (\vec X,t)$ instead of $\Theta (\vec X,t)$ for the Lagrangian representation of temperature.

Note that this is spatial isotropy, not material isotropy as discussed in the preceding sections.

The functions in the argument of constitutive functionals may not be so smooth as to admit truncated Taylor series expansion. Nevertheless, the constitutive functionals may be such as to smooth out past discontinuities in these argument functions and/or their derivatives. The principle of bounded memory, in this context also called the principle of fading memory, is then a requirement on the smoothness of constitutive functionals.

The principle of fading memory, mathematically formulated by Coleman and Noll (1960), starts with the assumption that the so-called Frechét derivatives of constitutive functionals up to an order n exist and are continuous in the neighbourhood of histories at time t in the Hilbert space normed by an influence function of order greater than $n+{1\over 2}$. Then, the constitutive functionals can be approximated by linear functionals for which explicit mathematical representations are known. The most important result of the fading memory theory is the possibility to approximate asymptotically sufficiently slow strain and temperature histories by a Taylor series expansion.

Every second-order tensor A can be written as the sum of a spherical tensor a I (i.e., a scalar multiple of the identity tensor) and a deviator A ^D:

$$\displaystyle \begin{aligned} {\boldsymbol{A}}=a{\boldsymbol{I}}+{\boldsymbol{A}}^D. \end{aligned}$$

Under the choice of a := tr A∕3, the trace of the deviator vanishes, i.e., tr A ^D = 0.

The stress tensor t, as any other tensor, can be decomposed into the spherical tensor σ I and the deviatoric stress tensor t ^D:

$$\displaystyle \begin{aligned} {\boldsymbol{t}}=\sigma{\boldsymbol{I}}+{\boldsymbol{t}}^D, \end{aligned}$$

where σ = tr t∕3 and tr t ^D = 0. The scalar σ is thus the mean of the normal-stress components and is called the mechanical pressure. A characteristic feature of all fluids at rest is that they cannot support shear stresses (see Sects. 7.3 and 7.7). Consequently, the deviatoric stress identically vanishes. Choosing p := −σ, we obtain t = −p I. As shown in Sect. 6.14, the stress in a fluid at rest, called hydrostatic pressure, depends on the density alone.

Title: Constitutive Equations
Author: Zdeněk Martinec
Publisher: Springer International Publishing
Book: Principles of Continuum Mechanics
Print ISBN: 978-3-030-05389-5

Electronic ISBN: 978-3-030-05390-1

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-05390-1_6

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