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Notations

Material manifold

Let \(\mathcal{B}\) be manifold. We will refer to \(\mathcal{B}\) as the material manifold.

Reference body

\(\u{\mathcal{B}}_{\rm R}=\left\{\u{\kappa}_{\rm R}(\mathcal{X})\in \mathbb{E}_{\rm R}\,\big|\,\mathcal{X}\in \mathcal{B}\right\} \subset \mathbb{E}_{\rm R}\) (Reference body in \(\mathbb{E}_{\rm R}\), reference Euclidean vector space),

\(\u{\mathcal{B}}_{\rm R}=\u{\kappa}_{\rm R}(\mathcal{B})\) (Reference body in \(\mathbb{E}_{\rm R}\)),

\(\mathcal{B}_{\rm R}=\set{\kappa_{\rm R}(\mathcal{X})\in \mathcal{E}_{\rm R}}{\mathcal{X}\in \mathcal{B}} \subset \mathcal{E}_{\rm R}\) (Reference body in \(\mathcal{E}_{\rm R}\), reference Euclidean point space),

\(\mathcal{B}_{\rm R}=\kappa_{\rm R}(\mathcal{B})\) (Reference body in \(\mathcal{E}_{\rm R}\)),

Remarks:

1. The choice of the reference body is arbitrary.
2. The reference body need not be stress free.

Deformed body

\(\u{\mathcal{B}}_{\tau}=\set{\u{\kappa}_{\tau}(\mathcal{X})\in \m{E}}{\mathcal{X}\in \mathcal{B}} \subset \mathbb{E}\) (Current body or deformed body in \(\mathbb{E}\), Euclidean vector space). We will also be denoting \(\mathcal{B}_{\tau}\).

\(\u{\mathcal{B}}_{\tau}=\u{\kappa}_{\tau}(\mathcal{B})\) (Current body in \(\mathbb{E}\)),

\(\mathcal{B}_{\tau}=\set{\kappa_{\tau}(\mathcal{X})\in \mathcal{E}}{\mathcal{X}\in \mathcal{B}} \subset \mathcal{E}\) (Current body in \(\mathcal{E}_{\rm R}\), Euclidean point space),

\(\mathcal{B}_{\tau}=\kappa_{\tau}(\mathcal{B})\) (Current body in \(\mathcal{E}\)),