Abstract
In this chapter we analyze the kinematics of continuous bodies with shape described just by the place they occupy in Euclidean three-dimensional point space. We discuss the large-strain regime and derive small-strain measures and the representation of rigid changes of places as special cases of deformation. We interpret strain measures in terms of spatial and referential metrics and distinguish between contravariant and covariant components of the tensors involved, introducing by stages a language useful to further developments going far beyond the limits of this book. We also discuss motions in actual (Eulerian) and referential (Lagrangian) representations. We include in this chapter several related exercises.
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Notes
- 1.
The closure of a set \(\mathcal{A}\) of a given space (which is itself a set) is the union of \(\mathcal{A}\) with the set of its accumulation points, the elements not belonging to \(\mathcal{A}\) but having elements of \(\mathcal{A}\) in every neighborhood. The notion of open set foresees the preliminary assignment of a topology in the space where \(\mathcal{A}\) is selected, namely a collection of sets (that we call open) containing the empty set, the whole space, and such that the union of every family of open sets is in the class, as is the intersection of every collection of finitely many open sets.
- 2.
Hom(A, B) indicates the space of linear maps from A to B. Here \(T_{x}\mathcal{B}\) indicates the linear space of vectors tangent to all smooth curves on \(\mathcal{B}\) crossing x. It is called the tangent space to \(\mathcal{B}\) at x. Analogously, \(T_{\tilde{y}(x)}\mathcal{B}_{a}\) is the tangent space to \(\mathcal{B}_{a}\) at \(y =\tilde{ y}(x)\).
- 3.
e i is defined by the condition \(e^{i} \cdot e_{j} = e^{i}(e_{j}) =\delta _{ j}^{i}\), with δ j i the Kronecker symbol, which is 1 for i = j and 0 for i ≠ j.
- 4.
Recall that there is a natural isomorphism between \(\mathbb{R}^{3}\) and its dual counterpart, and it is defined by the metric.
- 5.
When we change the basis \(\tilde{e}_{1}\), \(\tilde{e}_{2}\), \(\tilde{e}_{3}\) into another basis, the ith component of F is altered as a vector.
- 6.
When we change the dual basis e 1, e 2, e 3 into another one, the Ath component of F is altered as a covector (the derivative of a function with respect to x, indeed).
- 7.
We can assume that the scalar product is defined with respect to another frame in \(\mathbb{R}^{3}\) that we consider orthogonal.
- 8.
Of course we can consider the dual basis e 1, e 2, e 3 as obtained from e 1, e 2, e 3, by the action of g −1, so that e A = g AB e B .
- 9.
In other words, \(\nabla \tilde{y}(x)\) maps covectors at x in \(\mathcal{B}\) into vectors at \(y =\tilde{ y}(x)\) in the actual shape \(\mathcal{B}_{a}\).
- 10.
We recall that R ∈ SO(3) means that \(R \in \mathrm{ Hom}(\mathbb{R}^{3}, \mathbb{R}^{3})\), \(\det R = +1\), \(R^{-1} = R^{\mathtt{T}}\).
- 11.
R is said to be an orthogonal linear operator from \(\mathbb{R}^{m}\) to itself, and we write R ∈ O(m), when \(R \in \mathrm{ Hom}(\mathbb{R}^{m}, \mathbb{R}^{m})\) and \(\det R = +1\) or − 1. The set of orthogonal linear operators O(m) is a group. Its subset SO(m), characterized by \(\det R = +1\) , is a group as well. When R ∈ SO(m), it describes a rotation in \(\mathbb{R}^{3}\). Otherwise, R represents a reflection.
- 12.
For every n × n real matrix M and real number β, the identity \(\det \beta M =\beta ^{n}\det M\) holds.
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Mariano, P.M., Galano, L. (2015). Bodies, Deformations, and Strain Measures. In: Fundamentals of the Mechanics of Solids. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-3133-0_1
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DOI: https://doi.org/10.1007/978-1-4939-3133-0_1
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4939-3132-3
Online ISBN: 978-1-4939-3133-0
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