Governing Equations and Weak Form

In order to study a model PDE, let us consider small displacement elastostatics, which is governed by the equation of equilibrium:

$$\boldsymbol{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{b} = 0 \ \Omega$$ (7)

where

$$\boldsymbol{\sigma} = \mathbf{C}: \boldsymbol{\varepsilon} \, , \ \ \boldsymbol{\varepsilon} = \boldsymbol{\nabla}_s \mathbf{u} \,.$$ (8)

In the above equations, $\Omega \subset {\mathbf{R}\,}^2$ 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

$$\mathbf{u} = \bar{\mathbf{u}} \ \Gamma_u \,, \ \ \mathbf{n} \cdot \boldsymbol{\sigma} = \bar{\mathbf{t}} \ \Gamma_t \,,$$ (9)

where $\Gamma$ is the boundary of $\Omega$, and $\bar{\mathbf{u}}$ and $\bar{\mathbf{t}}$ are prescribed displacements and tractions, respectively.

The weak form (principle of virtual work) is

$$\int_{\Omega} \nabla_s \delta \mathbf{v} : \boldsymbol{\sigma}\, ... ...bf{t}}\, d\Gamma \ \ \forall \ \delta {\mathbf{v}} \in {\cal{H}\,}_E^1 (\Omega)$$ (10)

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:

$${\boldsymbol{K}} {\mathbf{d}} = {\mathbf{f}}^{ext} \,,$$ (11)

where

$${\mathbf{K}}_{IJ} = \int_\Omega {\mathbf{B}}_I^T {\mathbf{D}} {\m... ... \, d \Gamma + \int_\Omega {\boldsymbol{\Phi}}_I^T {\mathbf{b}} \, d \Omega \,.$$ (12)

In the above equation, $\boldsymbol{\Phi}_I$ is the shape function vector and ${\mathbf{B}}_I$ is the matrix of shape function derivatives.