Main stretch

In continuum mechanics, the main extensions denote the three main values ​​of the mathematically similar right and left deformation tensor U and v . The main extensions are obtained from the main axis transformation of the deformation tensor by solving the characteristic polynomial . ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}$

In the main axis system of the deformation tensor, the elongations reflect the current length of a line element in relation to its initial length and are therefore related to the elongation : ${\displaystyle \lambda }$ ${\displaystyle l}$ ${\displaystyle l_{0}}$ ${\displaystyle \varepsilon }$

${\displaystyle \lambda ={\frac {l}{l_{0}}}={\frac {l_{0}+\Delta l}{l_{0}}}=1+\varepsilon }$.

With the help of these stretchings, the deformation invariants in solid mechanics ( continuum mechanics of solids ) can also be represented quite easily.

Illustration of the polar decomposition. Unlike in the text, the left stretch tensor is capitalized here.

Because the right and left deformation tensor result from the polar decomposition

${\displaystyle \mathbf {F} =\mathbf {R\cdot U} =\mathbf {v\cdot R} }$

of the deformation gradient F , see picture. Therein, R a is actually orthogonal tensor , which is a rotation and the properties R ^T · R = R · R ^T = 1 and det ( R ) = +1 has ( 1 is the unit tensor ). The deformation gradient transforms line elements in the undeformed body into the line elements of the deformed body: ${\displaystyle \mathrm {d} {\vec {X}}}$ ${\displaystyle \mathrm {d} {\vec {x}}}$

${\displaystyle \mathbf {F} \cdot \mathrm {d} {\vec {X}}=\mathrm {d} {\vec {x}}.}$

The stretching of a line element in the Lagrange's approach is :

${\displaystyle {\frac {|\mathrm {d} {\vec {x}}|}{|\mathrm {d} {\vec {X}}|}}={\frac {|\mathbf {F} \cdot \mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}}={\frac {|\mathbf {R\cdot U} \cdot \mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}}={\frac {|\mathbf {U} \cdot \mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}},}$

because the rotation R leaves the norm untouched. Be eigenvector with eigenvalue λ of positive definite right Stretches sensor U . Then it is calculated ${\displaystyle \mathrm {d} {\vec {X}}}$

${\displaystyle {\frac {|\mathrm {d} {\vec {x}}|}{|\mathrm {d} {\vec {X}}|}}={\frac {|\mathbf {U} \cdot \mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}}={\frac {|\lambda \mathrm {d} {\vec {X}}|}{|\mathrm {d} {\vec {X}}|}}=\lambda ={\frac {l}{l_{0}}}.}$

For the left stretch tensor v , Euler's approach determines :

${\displaystyle \mathbf {F} ^{-1}=\mathbf {R^{\top }\cdot v^{\rm {-1}}} \quad \rightarrow \quad {\frac {|\mathrm {d} {\vec {x}}|}{|\mathrm {d} {\vec {X}}|}}={\frac {|\mathrm {d} {\vec {x}}|}{|\mathbf {F} ^{-1}\cdot \mathrm {d} {\vec {x}}|}}={\frac {|\mathrm {d} {\vec {x}}|}{|\mathbf {R^{\top }\cdot v} ^{-1}\cdot \mathrm {d} {\vec {x}}|}}={\frac {|\mathrm {d} {\vec {x}}|}{|\mathbf {v} ^{-1}\cdot \mathrm {d} {\vec {x}}|}},}$

again because the rotation maintains the norm. Let the eigenvector with eigenvalue λ of the left distance tensor v, which is also positively defined . Then it shows up ${\displaystyle \mathrm {d} {\vec {x}}}$

${\displaystyle \mathbf {v} \cdot \mathrm {d} {\vec {x}}=\lambda \mathrm {d} {\vec {x}}\quad \Leftrightarrow \quad \mathrm {d} {\vec {x}}=\lambda \mathbf {v} ^{-1}\cdot \mathrm {d} {\vec {x}}\quad \Leftrightarrow \quad \mathbf {v} ^{-1}\cdot \mathrm {d} {\vec {x}}={\frac {1}{\lambda }}\mathrm {d} {\vec {x}}}$

and further

${\displaystyle {\frac {|\mathrm {d} {\vec {x}}|}{|\mathrm {d} {\vec {X}}|}}={\frac {|\mathrm {d} {\vec {x}}|}{|\mathbf {v} ^{-1}\cdot \mathrm {d} {\vec {x}}|}}={\frac {|\mathrm {d} {\vec {x}}|}{|{\frac {1}{\lambda }}\mathrm {d} {\vec {x}}|}}=\lambda ={\frac {l}{l_{0}}}.}$

The main elongations in Lagrange's and Euler's approach are the same, but the directions in which the main elongations occur are in accordance with

${\displaystyle \mathbf {U} \cdot {\vec {v}}=\lambda {\vec {v}}\quad \Leftrightarrow \quad \mathbf {R\cdot U} \cdot {\vec {v}}=\mathbf {v\cdot R} \cdot {\vec {v}}=\mathbf {v\cdot (R} \cdot {\vec {v}})=\lambda (\mathbf {R} \cdot {\vec {v}})}$

twisted against each other, as the crosses in the picture suggest.

Footnotes

1. ↑ The variable is case-sensitive. Variables in capital letters relate to the reference state and those in lower case letters to the current state, which can be greatly deformed and twisted compared to the reference state.

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .