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## Continuity of field components

Depending on the constitutive relation, some components of the fields and stays continuous everywhere in a laminate while the other component are only piece-wise continuous.

Example

For example, consider conducting materials. The normal component of the current and the tangent component(s) of the electrical field are continuous, the remaining components are discontinuous.

1. Generally, the vector field is subject to some differential constraints (the vectors may be divergencefree, curlfree, etc.) These constraints imply that some linear combinations of its components are continuous on the boundary between layers,
 (3)

Here is a matrix projector on the subspace of continuous components.

The components of are constant in the laminate and the the supplementary orthogonal components of jump on the boundary between layers.

 (4)

Projections and are mutually orthogonal:

2. The dual vector has the properties similar to the vector , but its discontinuous components are supplementary to those of the prime vector ; in other words, the continuous and discontinuous components of and switch their places:
 (5)

The next Table shows discontinuous components of the fields in the equations of interest. These component are completely defined by the normal to the laminate. The matrix projects vector onto as follows:

Table 1: Space of discontinuous components of the fields in various equilibria
 Process Properties tensor Field vector Subspace of discontinuous components Conductivity Conductivity tensor Curlfree field normal Conductivity Resistivity tensor Divergencefree current tangent(s) 2D Elasticity Stiffness tensor Strain the tensorial component 2D Elasticity Compliance tensor Stress two tensorial components 3D Elasticity Stiffness tensor Strain Three tensorial components 3D Elasticity Compliance tensor Stress Three tensorial components Maxwell equations .

Next: Effective properties and local Up: Simple laminates Previous: Equilibrium
Andre Cherkaev
2001-07-31