Principle of Superposition

Let $u^{(1)}_i$ and $u^{(2)}_i$ be two possible displacement fields corresponding to the body force fields $g^{(1)}_i$ and $g^{(2)}_i$ and surface tractions $f^{(1)}_i$ and $f^{(2)}_i$ respectively. Let $T^{(1)}_{ij}$ and $T^{(2)}_{ij}$ be the stress fields corresponding to displacement fields $u^{(1)}_i$ and $u^{(2)}_i$. Then

$$\rho_0\frac{\partial^2u^{(1)}_i}{\partial t^2} = \rho_0g^{(1)}_i + (\lambda + \mu )u^{(1)}_{k,ki} + \mu u^{(1)}_{i,jj},$$ (5.4.1)

$$\rho_0\frac{\partial^2u^{(2)}_i}{\partial t^2} = \rho_0g^{(2)}_i + (\lambda + \mu )u^{(2)}_{k,ki} + \mu u^{(2)}_{i,jj},$$ (5.4.2)

$$\ u^{(1)}_i = h^{(1)}_i (X_j,t),\ \ T^{(1)}_{ij} n_j = f^{(1)}_i,\ ,$$ (5.4.3)

$$\ u^{(2)}_i = h^{(2)}_i (X_j,t),\ \ T^{(2)}_{ij} n_j = f^{(2)}_i,\ .$$ (5.4.4)

Adding (5.4.1) to (5.4.2), and (5.4.3) to (5.4.4) we obtain

$$\rho_0\frac{\partial^2(u^{(1)}_i + u^{(2)}_i)}{\partial t^2} = \r... ...a + \mu ) (u^{(1)}_k + u^{(2)}_k)_{,ki} + \mu (u^{(1)}_i + u^{(2)}_i )_{,jj}\ ,$$

$$\ u^{(1)}_i + u^{(2)}_i = h^{(1)}_i + h^{(2)}_i,\ \ (T^{(1)}_{ij} + T^{(2)}_{ij}) n_j = f^{(1)}_i + f^{(2)}_i\ .$$

Thus $u^{(1)}_i + u^{(2)}_i$ is a possible motion for the same linear elastic body corresponding to the body force $g^{(1)}_i + g^{(2)}_i$ and surface tractions $f^{(1)}_i + f^{(2)}_i$. This is the principle of superposition and is frequently used in the Mechanics of Materials course when solving problems for the combined loads.

One application of this principle in linear elastic problems is that in a given problem, we shall often assume that the body force is absent having in mind that its effect, if not negligible, can always be obtained separately and then superposed onto the solution of the problem with vanishing body force.