Relation between area elements in finite deformation theory (continuum mechanics)

Question

There are relations for the line and volume elements in continuum mechanics. For example:
begin{align}
             frac{V}{V_0}&={rm det}(F)tag{1}
lambda^2&=(F^TFe_1cdot e_1)tag{2}
end{align}

with $F$ being the deformation gradient,

$$lambda=delta x/delta X tag{3}$$

is the stretch and $e_1$ is the unit vector in direction where stretch is to be found.

Is there a similar relation between infinitesimal areas (for ratio of deformed and undeformed areas)?

rdt2 · Answer

Your first two equations can be written in other forms, so how about this?   In 1-D:

$$lambda=frac{L}{^0L}$$

(I use a pre-superscript for initial values and post-subscripts for spatial directions).   In 2-D:

$$lambda_1lambda_2=frac{L_1}{^0L_1}frac{L_2}{^0L_2}=frac{A}{^0A}$$

In 3-D:

$$lambda_1lambda_2lambda_3=frac{L_1}{^0L_1}frac{L_2}{^0L_2}frac{L_3}{^0L_3}=frac{V}{^0V}$$

Can't recall ever using the 2-D case but it might arise in a plane stress or plane strain calculation.

aghostinthefigures · Answer

There is, and it is usually referred to as Nanson's formula/equation.

Consider an area element with initial size $da$ and normal orientation vector $vec{n}$ that is deformed into an area element of size $dA$ with normal orientation vector $vec{N}$.

The relationship describing these oriented area elements before/after deformation is:

$$vec{n} da = (det{pmb{F}})(pmb{F}^{-T}cdot vec{N})dA$$

where $pmb{F}$ is the deformation gradient tensor. See A.J.M. Spencer's book on continuum mechanics or this page for a derivation of it.

Relation between area elements in finite deformation theory (continuum mechanics)

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