Boundary Conditions

A continuous body is usually deformed in the laboratory either by applying a distributed load on a part of its boundary and/or by specifying displacements on another part of the boundary. For example when a bar is pulled in a universal testing machine, one end of the bar is kept fixed and the other is pulled either at a prescribed rate or with an assigned force. Thus displacemets are prescribed to be zero at the fixed end, surface tractions equal zero on the lateral surface of the bar, and either displacements or tractions are prescribed at the other end. Whereas we usually represent force at the pulled end by a point load, in continuum mechanics (which includes elasticity, plasticity, fluid mechanics etc.) only distributed forces are considered. Let $\Gamma_1$ and $\Gamma_2$ represent complementary parts of the boundary of $\Omega$. The boundary conditions are expressed as
$$\begin{align}&u_i = \bar u_i\ {\rm on}\ \Gamma_1,\ (i = 1,2,3),\tag{7.1}\\ &\sigma_{ji}n_j = f_i\ {\rm on}\ \Gamma_2,\ (i=1,2,3), \tag{7.2} \end{align}$$
where n is a unit outward normal on $\Gamma_2$, and $\bar u_i$ and f_i are given functions of x₁,x₂,x₃ on the boundary of the body. The left-hand sides of (7.1) and (7.2) represent limiting values of the displacement and surface tractions at a point on the boundary as that point is approached from the interior of the body, and the right-hand sides are the prescribed values. f is called surface tractions and has the units of stress or force/area.

The boundary conditions at the inner and the outer surface of a cylindrical pressure vessel subjected to an internal pressure p₁ and external pressure p₂ are expressed as (see Fig. 7.1)

$$\begin{align}&\sigma_{ji}n_j = -p_1n_i,\ (i = 1,2,3)\ \mbox{on the inner surface... ...i}n_j = -p_2n_i,\ (i = 1,2,3)\ \mbox{on the outer surface}.\tag{7.4} \end{align}$$
Since $\mathbf{n} = \displaystyle\left( - \frac{x_1}{a},\ -\frac{x_2}{a}, 0\right)$ on the inner surface, and $\mathbf{n} = \displaystyle\left(\frac{x_1}{b},\ \frac{x_2}{b},\ 0\right)$ on the outer surface, equations (7.3) and (7.4) simplify to

$$\begin{array}{ll} \sigma_{11}x_1 + \sigma_{21} x_2 = -p_1x_1 ... ...3}x_2 = 0 & \mbox{on the inner surface,} \end{array} \tag{7.5}$$ (7.5)

$$\begin{array}{ll} \sigma_{11}x_1 + \sigma_{21} x_2 = -p_2x_1,... ...3}x_2 = 0, & \mbox{on the outer surface.} \end{array}\tag{7.6}$$ (7.6)

For a long prismatic dam (see Fig. 7.2) subjected to water pressure that increases linearly with the depth, the boundary conditions are as follows.

$$\begin{gather}\begin{split} &\sigma_{ji}n_j = -\sigma_{1i} = 0,\ (i = 1,2,3)\ \m... ...\ (i = 1,2,3)\ \mbox{on the surface}\ x_1 = h. \end{split}\tag{7.7} \end{gather}$$
Here $\delta_{ij} = 1$ if i=j and equals 0 when $i\ne j$. That is, $\delta_{ij}$ is the identity matrix. $\rho_\omega$ is the mass density of water and g is the gravitational constant.