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Linear kinematics

Let us consider a deformable body as a collection of points, where position of each point is denoted as $ x$$ \in\Omega$. In a deformed configuration the position of each point is identified by its position vector $ x$$ ^\varphi (x) =$   $ \phi$$ ($$ x$$ )$. The displacement vector is then defined as:

$\displaystyle \mbox{\boldmath$x$}$$\displaystyle ^\varphi ($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle ) =$   $\displaystyle \mbox{\boldmath$x$}$$\displaystyle +$   $\displaystyle \mbox{\boldmath$u$}$$\displaystyle ($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )$ (1.1)

Let us now examine the position in a local neighborhood of a point. The deformed position of such neighbor point with coordinates $ x$$ +d$$ x$ (where d$ x$is infinitisemally small vector) is

$ x$$\displaystyle ^\varphi ($$ x$$\displaystyle +d$$ x$$\displaystyle ) =$   $ x$$\displaystyle +d$$ x$$\displaystyle +$$ u$$\displaystyle ($$ x$$\displaystyle +d$$ x$$\displaystyle ) =$   $ x$$\displaystyle ^\varphi + d$$ x$$\displaystyle ^\varphi $

, where $ d$$ x$$ ^\varphi $ is the mapping of vector $ d$$ x$ onto deformed configuration, see Fig. 1.2.1.
Figure 1.1: Deformed configuration
Image deformedconfiguration
Taking into account the definition of displacement vector 1.1 and using Taylor formula we get

$\displaystyle d$$\displaystyle \mbox{\boldmath$x$}$$\displaystyle ^\varphi =$   $\displaystyle \mbox{\boldmath$x$}$$\displaystyle +d$$\displaystyle \mbox{\boldmath$x$}$$\displaystyle +$$\displaystyle \mbox{\boldmath$u$}$$\displaystyle ($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle +d$$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )-$$\displaystyle \mbox{\boldmath$x$}$$\displaystyle ^\varphi = d$$\displaystyle \mbox{\boldmath$x$}$$\displaystyle -$$\displaystyle \mbox{\boldmath$u$}$$\displaystyle ($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )+$$\displaystyle \mbox{\boldmath$u$}$$\displaystyle ($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle +d$$\displaystyle \mbox{\boldmath$x$}$$\displaystyle ) \approx [$$\displaystyle \mbox{\boldmath$I$}$$\displaystyle +\nabla$   $\displaystyle \mbox{\boldmath$u$}$$\displaystyle ($$\displaystyle \mbox{\boldmath$x$}$$\displaystyle )]d$$\displaystyle \mbox{\boldmath$x$}$ (1.2)

where $ \nabla$   $ u$$ ($$ x$$ )$ is the displacement gradient tensor (in small strain theory we assume $ \vert\vert\nabla$   $ u$$ ($$ x$$ )\vert\vert \ll 1$). The displacement gradient tensor can be decomposed into symetric and antisymmetric parts

$\displaystyle \nabla$   $ u$$\displaystyle =$    $ \varepsilon $$\displaystyle +$$ \omega$$\displaystyle ={1\over 2}(\nabla$   $ u$$\displaystyle +\nabla$   $ u$$\displaystyle ^T)+{1\over 2}(\nabla$   $ u$$\displaystyle -\nabla$   $ u$$\displaystyle ^T) = \nabla ^s$$ u$$\displaystyle +\nabla ^a$$ u$

The antisymmetric part corresponds to infinitesimal rotation. The symmeric part of displacement gradient tensor is therefore the measure of infinitesimal deformation

$\displaystyle d$$ x$$\displaystyle ^\varphi =$    $ \varepsilon $$\displaystyle d$$ x$

next up previous contents
Next: Equlibrium equations Up: Linear elasticity Previous: Linear elasticity   Contents
Borek Patzak 2017-12-30