Abstract
The polar formalism, a mathematical technique used to represent plane tensors by invariants and angles, is introduced in this chapter. The theory is fully developed in detail, starting from the pioneer, founding cworks of Verchery to the more recent developments. The algebra of the method is completely given and different topics are developed: the decomposition of the strain energy and the bounds on the polar invariants, a full analysis of all the possible elastic symmetries in plane elasticity, the cases of special plane anisotropic materials, the theory of polar projectors, some cases of interaction between geometry and anisotropy, plane piezoelectricity, anisotropy induced by damage, the polar invariant formulation of strength criteria for anisotropic layers. The chapter ends with different examples of plane anisotropic materials.
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Notes
- 1.
We will use the word intrinsic as synonymous of invariant. While invariant has a clear and precise mathematical meaning, a tensorial quantity whose value is preserved under frame changes, the word intrinsic has a more physical signification: it indicates a quantity that characterizes intrinsically a physical property, that belongs, in some sense, to it.
- 2.
Any plane symmetry in 3D corresponds to a symmetry with respect to a straight line in 2D; for the sake of simplicity, and for recalling that we are not in 3D, we will call mirror symmetry any symmetry with respect to a straight line.
- 3.
For the sake of simplicity, we continue to use the standard notation of tensors also when these are considered as matrices, like in Sect. 3.10. As we use the tensorial components also for matrices, i.e. we do not use the Kelvin’s notation, there is no risk of mistakes.
- 4.
To make a comparison, the transformation normally used, cf. Green and Zerna, is defined by the equations \(X^1=z,\ X^2=\overline{z}\). Following the same procedure used here for the Verchery’s transformation, it is easy to check that in this case all the listed properties are no more valid.
- 5.
The order in which the components of a tensor appear in the column is not arbitrary, but obeys to the following rule: the first component is that whose indexes are all 1 and the successive components increase the indexes starting from the right: 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222 and so on.
- 6.
We continue to indicate a fourth-rank tensor, even in the case of an elastic tensor, by the letter \(\mathbb {T}\) to maintain a wide generality, because the polar representation is valid for any tensor, not only for the stiffness elasticity tensor.
- 7.
In \(\mathbb {R}^2\), a frame is fixed by the choice of a unique parameter: an angle measured from a direction chosen conventionally. Hence, if a tensor has n distinct components, it can have at most \(n-1\) independent invariants.
- 8.
A syzygy is a relation between two or more tensor invariants. The search for syzygies is a crucial point in determining which are the dependent invariants; unfortunately, no general method exists for finding the syzygies.
- 9.
The definition of \(\kappa \) is the same for isotropic or anisotropic materials, and Eq. (4.83) is valid in any case; nevertheless, the mechanical idea in the definition of \(\kappa \) is to measure the mechanical hydrostatic pressure to be done on the material to obtain a unitary change of volume, tr\(\varvec{\varepsilon }=1\); it is somewhat understood, in doing this, that all the mechanical stress produce uniquely a change of volume, not of shape, i.e. that the deformation itself is spherical. This fact is always true not only for isotropic materials, but also for a square symmetric material, Sect. 2.5, hence when \(r_1=0\); in fact, in such a case \(S_{1111}(\theta )=S_{2222}(\theta )\) and \(S_{1112}(\theta )+S_{1222}(\theta )=0\ \forall \theta \). That is why, though Eq. (4.83) has a general validity, the same notion of \(\kappa \) is usually restricted to the use with isotropic materials.
- 10.
The reader should remark that \(V_s\ne V_{sph}\) and \(V_d\ne V_{dev}\), cf. Sect. 2.5.
- 11.
Remembering Eq. (4.54)\(_{3,4}\), we remark that \(\mathsf {T}^{cont}\in \mathbb {R}\) for a material having an axis of mirror symmetry tilted of \(\pi /4\) on the axis of \(x_1\).
- 12.
It is important to preserve, in the set of the independent invariants, the invariant of the highest degree, that is why we keep \(C_1\) in the list.
- 13.
- 14.
“Apparently” because if one makes experimental tests on the components of \(\mathbb {S}\) or traces the directional diagrams of its components, they look like those of an ordinarily orthotropic material with \(k=0\), the difference is in the special value get by \(r_0\), Eq. (4.158).
- 15.
We remember, see Sect. 2.1, that we call tensor or index symmetry any equivalence of the positions of an index for two or more components of the elastic tensor.
- 16.
The Kelvin decomposition (Kelvin 1856, 1878), basically consists in the diagonalization of the elasticity matrix as defined in Eq. (2.24); some simple algebraic passages show that the elasticity tensor \(\mathbb {E}\) can be represented as
$$\begin{aligned} \mathbb {E}=\lambda ^i\mathbb {P}^i,\ \ \ i=1,...,6, \end{aligned}$$where the Kelvin projectors \(\mathbb {P}^i\) are fourth-rank dimensionless tensors defined as (no summation over i)
$$\begin{aligned} \mathbb {P}^i=\mathbf {E}^i\otimes \mathbf {E}^i. \end{aligned}$$The scalars \(\lambda ^i\) and the second-rank tensors \(\mathbf {E}^i\) are the couples eigenvalue-eigenvector of the equation (no summation over i)
$$\begin{aligned} \mathbb {E}\mathbf {E}^i=\lambda ^i\mathbf {E}^i. \end{aligned}$$The \(\lambda ^i\) are homogeneous to a modulus, and are called the Kelvin moduli, while the \(\mathbf {E}^i\) are homogeneous to a strain and are called the Kelvin modes.
Calling Kelvin strains and Kelvin stresses respectively each one of the tensors
$$\begin{aligned} \varvec{\varepsilon }^i=\mathbb {P}^i\varvec{\varepsilon }, \ \ \varvec{\sigma }^i=\mathbb {P}^i\varvec{\sigma }, \end{aligned}$$by their same construction, Kelvin strains and stresses are mutually orthogonal:
$$\begin{aligned} \varvec{\varepsilon }^i\cdot \varvec{\varepsilon }^j=0,\ \ \varvec{\sigma }^i\cdot \varvec{\sigma }^j=0. \end{aligned}$$Let us consider now the strain energy V stored in an elastic body:
$$\begin{aligned} V=\frac{1}{2}\varvec{\varepsilon }\cdot \mathbb {E}\varvec{\varepsilon }=\frac{1}{2}\lambda ^i\varvec{\varepsilon }^i\cdot \mathbb {P}^i\varvec{\varepsilon }=\frac{1}{2}\lambda ^i\varvec{\varepsilon }^i\cdot \varvec{\varepsilon }^i. \end{aligned}$$V can hence be decomposed into three terms \(V^i\),
$$\begin{aligned} V^i=\frac{1}{2}\lambda ^{\underline{i}}\varvec{\varepsilon }^{\underline{i}}\cdot \varvec{\varepsilon }^{\underline{i}}\ \ \ \forall i\in \{\mathrm {I, II, III}\}, \end{aligned}$$each one of these three terms being associated to the corresponding Kelvin mode. For this reason, we will denote them as Kelvin modal energies.
For more details about the Kelvin decomposition, see Rychlewski (1984), Mehrabadi and Cowin (1990), François (1995, 2012), Desmorat and Marull (2011), de Saxcé and Vallée (2013), Desmorat and Vannucci (2014).
- 17.
In this section, some concepts introduced extensively in Chap. 5 are used; the reader is hence addressed to this chapter to have details, concepts and nomenclature on laminates.
- 18.
The compliance of a structure is defined as the overall work of the external forces acting upon the structure; from the Clapeyron’s Theorem, we have that
$$\begin{aligned} J=\int _\varOmega \mathbf {f}\cdot \mathbf {u}\ d\omega =\int _\varOmega \varvec{\sigma }\cdot \varvec{\varepsilon }\ d\omega . \end{aligned}$$It is evident that, for a given set of applied forces f, the less the compliance, the less the displacements u, i.e. the highest the stiffness. That is why the minimization of the compliance J is often used, in structural optimization, as a standard formulation for the problems of stiffness maximization. This has also some mathematical nice properties: unlike the minimization of the displacement of some specific points of the structure, that are hold by local functionals, J is a global functional, which has important, positive consequences in variational calculus, see for instance (Banichuk 1983).
- 19.
For a proof of this statement, see Vannucci and Desmorat (2015), Sect. 4 and Appendix.
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Vannucci, P. (2018). The Polar Formalism. In: Anisotropic Elasticity . Lecture Notes in Applied and Computational Mechanics, vol 85. Springer, Singapore. https://doi.org/10.1007/978-981-10-5439-6_4
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