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In order to study a model PDE, let us consider
small displacement elastostatics, which is
governed by the equation of equilibrium:
$\displaystyle \boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{b} = 0 \ $ in $\displaystyle \Omega$
(7)
where
$\displaystyle \boldsymbol{\sigma} = \mathbf{C}: \boldsymbol{\varepsilon} ,円 , \ \ \boldsymbol{\varepsilon} = \boldsymbol{\nabla}_s \mathbf{u} ,円.$
(8)
In the above equations,
[
画像:$ \Omega \subset {\mathbf{R},円}^2$] is the
domain of the body,
$ \boldsymbol{\sigma}$ is the Cauchy
stress tensor,
$ \boldsymbol{\varepsilon}$ is the small strain
tensor,
$ \mathbf{b}$ is the body force per unit volume,
$ \mathbf{C}$ is the material moduli tensor,
$ \mathbf{u}$
is the displacement,
$ \boldsymbol{\nabla}$ is the gradient operator, and
$ \boldsymbol{\nabla}_s$ is the symmetric gradient operator.
The essential and natural boundary conditions are
where $ \Gamma$ is the boundary of $ \Omega,ドル and
$ \bar{\mathbf{u}}$ and
$ \bar{\mathbf{t}}$ are prescribed displacements and
tractions, respectively.
The weak form (principle of virtual work) is
On substituting the trial and test
functions in the above equation and using the arbitrariness
of nodal variations, the following
discrete system of equations is obtained:
where
In the above equation,
$ \boldsymbol{\Phi}_I$ is the shape function
vector and
$ {\mathbf{B}}_I$ is the matrix of shape function derivatives.
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N. Sukumar
