Restrictions on Continuous Deformation of a Deformable Body.

For $t = 1$, eqns. (3.3.2) give

$$x_1 =\ 2(X_1 + X_2),$$

$$x_2 =\ 2(X_1 + X_2),$$

$$x_3 =\ \frac{3}{2} X_3.$$

Thus points $(a,\ -a,\ 0)$ in the reference configuration move to $(0,0,0)$ at time $t = 1$. This implies that distinct particles which occupy different places in the reference configuration are deformed into the same place in the present configuration (at time $t = 1$). This amounts to the collision of various material particles. Even though in particle mechanics collisions are allowed, in continuum mechanics, such a possibility is assumed to be ruled out to start with. Thus in continuum mechanics it is assumed that different material particles always occupy distinct places. This is equivalent to the requirement that

$$x_i = x_i (X_1,\ X_2,\ X_3,\ t)$$

be a one-to-one mapping of $R$ onto the present configuration. Since in Continuum Mechanics we will need to differentiate functions with respect to $x_i$ or $X_i$, we will henceforth assume that the mapping

$$x_i = x_i (X_1,\ X_2,\ X_3,\ t)$$

is continuously differentiable and has a continuously differentiable inverse given by

$$X_i = X_i (x_1,\ x_2,\ x_3,\ t).$$

This is so if and only if the Jacobian $J$ defined by

$$J = {\rm det}\left[\begin{array}{ccc}\displaystyle\frac{\partia... ... \displaystyle\frac{\partial x_3}{\partial X_3}\end{array}\right]$$ (3.4.1)

is non-zero for all points in $R$ and for every value of $t$. Since

$$J(X_1,\ X_2,\ X_3,\ 0) = 1,$$

and $J$ is a continuous function of $t$, therefore, $J$ must be positive for every $t$. Using the index notation, we can write $J$ as

$$J = {\rm det}\left[\frac{\partial x_i}{\partial X_j}\right] = {\rm det}\left[\delta_{ij} + \frac{\partial u_i}{\partial X_j}\right].$$ (3.4.2)

In summary, we can conclude from the preceding discussion that a necessary and sufficient condition for a continuous deformation to be physically admissible is that the Jacobian $J$ be greater than zero.

A displacement field satisfying the condition $J > 0$ is said to be proper and admissible, or simply admissible.

Thus, for an admissible deformation of a medium the displacement components $(u_1,\ u_2,\ u_3)$ must satisfy $J > 0$. For example, a piece of rubber can not be subjected to displacement components $u_1 = -2X_1,\ u_2 = 0,\ u_3 = 0$ since then $J = -1$. This displacement is called a reflection about the $(X_2,\ X_3)$ plane, since the point $(x_1,\ x_2,\ x_3)$ is the image of the point $(X_1,\ X_2,\ X_3)$ in a mirror that lies in the plane $x_1 = 0$.

Exercise: Determine whether or not

$$u_1 = k(X_2-X_1),\ u_2 = k(X_1-X_2),\ u_3 = kX_1X_3,$$

where $k$ is a constant, are possible displacement components for a continuous medium.