The Polar Formalism
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 devel
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The Polar Formalism
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.
4.1 Introduction: Why the Polar Formalism? In 1979 G. Verchery presented a memory about the invariants of an elasticity-type tensor (Verchery 1982). This short paper marks the birth of the polar formalism or method. We have seen that for anisotropic materials the Cartesian components of a tensor describing a given property all depend upon the direction; moreover, this dependence is rather cumbersome, Sect. 2.2.2, also in the plane case, Eq. (3.27). Hence, when the Cartesian components are used for representing an anisotropic tensor, none of these components are an intrinsic1 quantity: all of them are frame-dependent parameters. In addition, if a privileged direction linked to the anisotropic property exist, it does not appear explicitly. Basically, the polar formalism is an algebraic technique to represent a plane tensor using only tensor invariants and angles (that is why the method is called polar). Hence, the intrinsic quantities describing a given anisotropic property and the direc1 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. © Springer Nature Singapore Pte Ltd. 2018 P. Vannucci, Anisotropic Elasticity, Lecture Notes in Applied and Computational Mechanics 85, DOI 10.1007/978-981-10-5439-6_4
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4 The Polar Formalism
tion directly and explicitly appear in the equations. It is exactly the use of invariants and angles that makes the polar method interesting for analyzing anisotropic phenomena: on one side, the invariants are not linked to the particular choice of the axes, so they give an intrinsic representation of the property. On the other side, the explicit use of angles makes appear directly one of the fundamental aspects of anisotropy: the direction. This is possible because, unlike other tensor representations, the polar method does not use exclusively polynomial invariants. Moreover, we will see that the invariants used in the pol
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