Vicinity Transformation

Central to the computation of stresses and strains is the Jacobian matrix which relates the initial and deformed configuration:(1)

$$d{x}_i=\frac{\partial x_i}{\partial X_j}d{X}_j=D_j{x}_{i}d{X}_j=F_{ij}d{X}_j$$

(2)

$$D_j=\frac{\partial }{\partial X_j}$$

The transformation is fully described by the elements of the Jacobian matrix $F$ :(3)

$$F_{ij}\equiv D_j{x}_i$$

So that Equation 1 can be written in matrix notation:(4)

$$dx=FdX$$

The Jacobian, or determinant of the Jacobian matrix, measures the relation between the initial volume $d\Omega$ and the volume in the initial configuration containing the same points:(5)

$$d\Omega =|F|d{\Omega }^0$$

Physically, the value of the Jacobian cannot take the zero value without cancelling the volume of a set of material points. So the Jacobian must not become negative whatever the final configuration. This property insures the existence and uniqueness of the inverse transformation:(6)

$$dX=F^{-1}dx$$

At a regular point whereby definition of the field $u(X)$ is differentiable, the vicinity transformation is defined by:(7)

$$F_{ij}=D_j{x}_i=D_j\left(X_i+u_i(X,t)\right)={\delta }_{ij}+D_j{u}_i$$

or in matrix form:(8)

$$F=1+A$$

So, the Jacobian matrix $F$ can be obtained from the matrix of gradients of displacements:(9)

$$A\equiv D_j{u}_i$$