CONTINUOUS AND DISCONTINUOUS QUANTITIES

In a stack of layers certain stress and strain components are continuous across layer boundaries. For a stack of layers perpendicular to the 3-axis (z-axis) the continuous stress components are those acting on the plane normal to the 3-axis, $\sigma_{13},\sigma_{23},\sigma_{33}$. The continuous strain components are those tangential to the 3-axis, $\epsilon_{11},\ \epsilon_{12},\ \epsilon_{22}$. The other stress and strain components are discontinuous from layer to layer. In this paper we rearrange the compressed subscript matrix form of Hooke's law by dividing the stress and strain vectors into sub-vectors, the continuous parts,

$${\sigma}_{N} \equiv \pmatrix{ \sigma_3 \cr \sigma_4 \cr \sig... ...matrix{ \epsilon_{11} \cr \epsilon_{22} \cr \epsilon_{12} \cr }$$ (1)

and the discontinuous parts,

$${\epsilon}_{N} \equiv \pmatrix{ \epsilon_3 \cr \epsilon_4 \c... ... \pmatrix{ \sigma_{11} \cr \sigma_{22} \cr \sigma_{12} \cr }\ .$$ (2)

The relationship between stress and strain for anisotropic media is a generalized version of Hooke's Law. If the medium properties are expressed as stiffnesses it is,

 

$$\pmatrix{ {\sigma}_{T} \cr {\sigma}_{N} } = \pmatrix{ {\bf ... ...f C}_{NN} \cr} \pmatrix{ {\epsilon}_{T} \cr {\epsilon}_{N} \cr}$$ (3)

and similarly for the inverse, compliance, matrix  

$$\pmatrix{ {\epsilon}_{T} \cr {\epsilon}_{N} \cr} = \pmatrix... ...} & {\bf S}_{NN} \cr} \pmatrix{ {\sigma}_{T} \cr {\sigma}_{N} }$$ (4)

Rewriting 3 and 4 we have,  

$$\pmatrix{ {\sigma}_{T} \cr {\epsilon}_{N} \cr} = \pmatrix{ {... ...C}_{NN} }^{-1} \cr} \pmatrix{ {\epsilon}_{T} \cr {\sigma}_{N} }$$ (5)

The constitutive relation using a hybrid (stiffness/compliance) matrix, ${\bf X}_{}$, is  

$$\pmatrix{ {\sigma}_{T} \cr {\epsilon}_{N} \cr} = \pmatrix{ ... ...\bf X}_{NN} \cr} \pmatrix{ {\epsilon}_{T} \cr {\sigma}_{N} \cr}$$ (6)

Note that since ${\bf S} = {\bf C^{-1}}$ it can be shown that ${\bf X}_{NT} = { {\bf C}_{NN} }^{-1} {\bf C}_{NT}$ is the transpose of ${\bf X}_{TN} = { {\bf S}_{TT} }^{-1} {\bf S}_{TN}$. The matrix ${\bf X}_{}$ is therefore symmetric.

This formulation reflects the special geometry of horizontal layers. It relates quantities, ${\sigma}_{N}$ and ${\epsilon}_{T}$,which are continuous to quantities, ${\sigma}_{T}$ and ${\epsilon}_{N}$,which are discontinuous across layer interfaces.