Physics::Transformations Namespace Reference

Namespaces
	Contravariant
	Covariant
	Piola
	Rotations

Detailed Description

A collection of operations to assist in the transformation of tensor quantities from the reference to spatial configuration, and vice versa. These types of transformation are typically used to re-express quantities measured or computed in one configuration in terms of a second configuration.

Notation

We will use the same notation for the coordinates \(\mathbf{X}, \mathbf{x}\), transformations \(\varphi\), differential operator \(\nabla_{0}\) and deformation gradient \(\mathbf{F}\) as discussed for namespace Physics::Elasticity.

As a further point on notation, we will follow Holzapfel (2007) and denote the push forward transformation as \(\chi\left(\bullet\right)\) and the pull back transformation as \(\chi^{-1}\left(\bullet\right)\). We will also use the annotation \(\left(\bullet\right)^{\sharp}\) to indicate that a tensor \(\left(\bullet\right)\) is a contravariant tensor, and \(\left(\bullet\right)^{\flat}\) that it is covariant. In other words, these indices do not actually change the tensor, they just indicate the kind of object a particular tensor is.

Note

For these transformations, unless otherwise stated, we will strictly assume that all indices of the transformed tensors derive from one coordinate system; that is to say that they are not multi-point tensors (such as the Piola stress in elasticity).

Author

Jean-Paul Pelteret, Andrew McBride, 2016

Function Documentation

nansons_formula()

template<int dim, typename Number >

Tensor<1, dim, Number> Physics::Transformations::nansons_formula	(	const Tensor< 1, dim, Number > &	N,
		const Tensor< 2, dim, Number > &	F
	)

Return the result of applying Nanson's formula for the transformation of the material surface area element \(d\mathbf{A}\) to the current surfaces area element \(d\mathbf{a}\) under the nonlinear transformation map \(\mathbf{x} = \boldsymbol{\varphi} \left( \mathbf{X} \right)\).

The returned result is the spatial normal scaled by the ratio of areas between the reference and spatial surface elements, i.e.

\[ \mathbf{n} \frac{da}{dA} \dealcoloneq \textrm{det} \mathbf{F} \, \mathbf{F}^{-T} \cdot \mathbf{N} = \textrm{cof} \mathbf{F} \cdot \mathbf{N} \, . \]

Parameters

[in]	N	The referential normal unit vector \(\mathbf{N}\)
[in]	F	The deformation gradient tensor \(\mathbf{F} \left( \mathbf{X} \right)\)

Returns

The scaled spatial normal vector \(\mathbf{n} \frac{da}{dA}\)

Note

For a discussion of the background of this function, see G. A. Holzapfel: "Nonlinear solid mechanics. A Continuum Approach for Engineering" (2007), and in particular formula (2.55) on p. 75 (or thereabouts).

For a discussion of the background of this function, see P. Wriggers: "Nonlinear finite element methods" (2008), and in particular formula (3.11) on p. 23 (or thereabouts).