Equations of the Infinitesimal Theory of Elasticity

We shall consider only the case of small strains, and infinitesimal velocities and accelerations as compared to some reference values. Thus every particle is always in a small neighborhood of the reference configuration. The reference configuration in which $T_{ij} = 0$ is also called a natural state. Thus, if $x_i$ denotes the position in the natural state of a typical material particle, we assume that $x_i \simeq X_i$, and that the magnitude of the components of the displacement gradient $\frac{\partial u_i}{\partial X_j}$ is much smaller than unity. Since

$$x_i = X_i + u_i,$$

we have

$$\frac{\partial T_{ij}}{\partial X_1} =\ \frac{\partial T_{ij}}{\partial x_k} \frac{\partial x_k}{\partial X_1}\ ,$$

$$=\ \frac{\partial T_{ij}}{\partial x_1} \left(1 + \frac{\partial u_1... ..._1} + \frac{\partial T_{ij}}{\partial x_3} \frac{\partial u_3}{\partial X_1}\ ,$$

$$\simeq\ \frac{\partial T_{ij}}{\partial x_1}\ .$$

Similarly,

$$\frac{\partial T_{ij}}{\partial X_2} \approx \frac{\partial T_{ij... ...{\partial T_{ij}}{\partial X_3} \approx \frac{\partial T_{ij}}{\partial x_3}\ .$$

Therefore, for small deformations,

$$\frac{\partial T_{ij}}{\partial X_k} \approx \frac{\partial T_{ij}}{\partial x_k}\ .$$ (5.3.1)

Since

$$a_i = \frac{\partial^2x_i}{\partial t^2}\bigg\vert _{X_j{\rm -fixed}} = \frac{\partial^2u_i}{\partial t^2}\bigg\vert _{X_j{\rm -fixed}}$$

and

$$\rho = \rho_0(1 - e_{kk})\ ,$$

we have

$$\begin{split}\rho a_i =\ & \rho _0 (1 - e_{kk})\frac{\partial... ...prox\ & \rho_0\frac{\partial^2u_i}{\partial t^2}\ . \end{split}$$ (5.3.2)

Also

$$\rho g_i\simeq \rho_0g_i\ .$$

Thus the balance laws or conservation laws for small deformations of a body take the following form.

$$\ \ \rho = \rho_0(1 - e_{kk})$$ (5.3.3)

$$\ \ \rho_0\frac{\partial^2u_i}{\partial t^2} =\rho_0g_i + \frac{\partial T_{ij}}{\partial X_j}\ .$$ (5.3.4)

For a linear elastic homogeneous and isotropic body, substitution from (5.2.5) and (3.14.3) into (5.3.4) gives

$$\rho_0\frac{\partial^2u_i}{\partial t^2} = \rho_0g_i + (\lambda + \mu ) u_{k,ki} + \mu u_{i,jj}.$$ (5.3.5)

These are three equations for the three unknowns $u_1,u_2$, and $u_3$. After a solution of (5.3.5) has been obtained, one can find the present mass density from eqn. (5.3.3). Note that eqn. (5.3.5) is a system of three coupled partial differential equations. In order to find a solution of (5.3.5) applicable to a given problem, side conditions such as initial conditions and boundary conditions are needed.

In a dynamic problem, one needs the values of $u_i(X_j,0)$ and $\dot u_i (X_j,0)$. That is, the initial displacement and the initial velocity field should be given as smooth functions throughout the body. Note that these initial conditions are not needed in a static problem. However, in both static and dynamic problems one needs boundary conditions which can be one of the following three types. In the boundary condition of traction the stress vector is prescribed at the boundary points of the body. That is, at the points on the boundary

$$T_{ij} n_j = \lambda u_{k,k} n_i + \mu (u_{i,j} + u_{j,i})n_j = f_i (\mathbf{X},t),$$ (5.3.6)

in which $f_i$ is a known function.

In a displacement type boundary condition, displacements are prescribed on the boundary points. For example, a part of the boundary of a body could be glued to a rigid support. In this case, displacements for these boundary points will be zero. The third type of boundary condition is the one in which surface tractions are prescribed on one part and the displacements on the remainder or at a boundary point, tangential components of the stress-vector and the normal component of the displacement vector (or vice-versa) are prescribed. These are known as the mixed type boundary conditions.