Finite Strain Theory - Transformation of A Surface and Volume Element

Transformation of A Surface and Volume Element

To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use Nanson's relation, expressed as

$da~\mathbf{n} = J~dA~\mathbf{F}^{-T}\cdot \mathbf{N} \,\!$

where is an area of a region in the deformed configuration, is the same area in the reference configuration, and is the outward normal to the area element in the current configuration while is the outward normal in the reference configuration, is the deformation gradient, and .

The corresponding formula for the transformation of the volume element is

$dv = J~dV \,\!$

Derivation of Nanson's relation
To see how this formula is derived, we start with the oriented area elements in the reference and current configurations: $d\mathbf{A} = dA~\mathbf{N} ~;~~ d\mathbf{a} = da~\mathbf{n} \,\!$ The reference and current volumes of an element are $dV = d\mathbf{A}^{T}\cdot d\mathbf{L} ~;~~ dv = d\mathbf{a}^{T} \cdot d\mathbf{l} \,\!$ where . Therefore, $d\mathbf{a}^{T} \cdot d\mathbf{l}= dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L} \,\!$ or, $d\mathbf{a}^{T} \cdot \mathbf{F}\cdot d\mathbf{L} = dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L} \,\!$ so, $d\mathbf{a}^{T} \cdot \mathbf{F} = J~d\mathbf{A}^{T} \,\!$ So we get $d\mathbf{a} = J~\mathbf{F}^{-T} \cdot d\mathbf{A} \,\!$ or, $da~\mathbf{n} = J~dA~\mathbf{F}^{-T}\cdot \mathbf{N}\qquad \qquad \square \,\!$

Derivation of Nanson's relation

To see how this formula is derived, we start with the oriented area elements

in the reference and current configurations:

$d\mathbf{A} = dA~\mathbf{N} ~;~~ d\mathbf{a} = da~\mathbf{n} \,\!$

The reference and current volumes of an element are

$dV = d\mathbf{A}^{T}\cdot d\mathbf{L} ~;~~ dv = d\mathbf{a}^{T} \cdot d\mathbf{l} \,\!$

where .

Therefore,

$d\mathbf{a}^{T} \cdot d\mathbf{l}= dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L} \,\!$

or,

$d\mathbf{a}^{T} \cdot \mathbf{F}\cdot d\mathbf{L} = dv = J~dV = J~d\mathbf{A}^{T}\cdot d\mathbf{L} \,\!$

so,

$d\mathbf{a}^{T} \cdot \mathbf{F} = J~d\mathbf{A}^{T} \,\!$

So we get

$d\mathbf{a} = J~\mathbf{F}^{-T} \cdot d\mathbf{A} \,\!$

or,

$da~\mathbf{n} = J~dA~\mathbf{F}^{-T}\cdot \mathbf{N}\qquad \qquad \square \,\!$

Read more about this topic: Finite Strain Theory

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