Vicinity Transformation

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

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

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

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

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

So that 式 1 can be written in matrix notation:

$$dx=FdX$$

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:

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

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:

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

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

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

or in matrix form:

$$F=1+A$$

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

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